# How Much Computational Power Does It Take to Match the Human Brain?

Author: Joseph Carlsmith
Published September 11, 2020

Open Philanthropy is interested in when AI systems will be able to perform various tasks that humans can perform (“AI timelines”). To inform our thinking, I investigated what evidence the human brain provides about the computational power sufficient to match its capabilities. This is the full report on what I learned. A medium-depth summary is available here. The executive summary below gives a shorter overview.

## Introduction

### Executive summary

Let’s grant that in principle, sufficiently powerful computers can perform any cognitive task that the human brain can. How powerful is sufficiently powerful? I investigated what we can learn from the brain about this. I consulted with more than 30 experts, and considered four methods of generating estimates, focusing on floating point operations per second (FLOP/s) as a metric of computational power.

These methods were:

1. Estimate the FLOP/s required to model the brain’s mechanisms at a level of detail adequate to replicate task-performance (the “mechanistic method”).1
2. Identify a portion of the brain whose function we can already approximate with artificial systems, and then scale up to a FLOP/s estimate for the whole brain (the “functional method”).
3. Use the brain’s energy budget, together with physical limits set by Landauer’s principle, to upper-bound required FLOP/s (the “limit method”).
4. Use the communication bandwidth in the brain as evidence about its computational capacity (the “communication method”). I discuss this method only briefly.

None of these methods are direct guides to the minimum possible FLOP/s budget, as the most efficient ways of performing tasks need not resemble the brain’s ways, or those of current artificial systems. But if sound, these methods would provide evidence that certain budgets are, at least, big enough (if you had the right software, which may be very hard to create – see discussion in section 1.3).2

Here are some of the numbers these methods produce, plotted alongside the FLOP/s capacity of some current computers.

These numbers should be held lightly. They are back-of-the-envelope calculations, offered alongside initial discussion of complications and objections. The science here is very far from settled.

For those open to speculation, though, here’s a summary of what I’m taking away from the investigation:

• Mechanistic method estimates suggesting that 1013-1017 FLOP/s is enough to match the human brain’s task-performance seem plausible to me. This is partly because various experts are sympathetic to these estimates (others are more skeptical), and partly because of the direct arguments in their support. Some considerations from this method point to higher numbers; and some, to lower numbers. Of these, the latter seem to me stronger.3
• I give less weight to functional method estimates, primarily due to uncertainties about (a) the FLOP/s required to fully replicate the functions in question, (b) what the relevant portion of the brain is doing (in the case of the visual cortex), and (c) differences between that portion and the rest of the brain (in the case of the retina). However, I take estimates based on the visual cortex as some weak evidence that the mechanistic method range above (1013-1017 FLOP/s) isn’t much too low. Some estimates based on recent deep neural network models of retinal neurons point to higher numbers, but I take these as even weaker evidence.
• I think it unlikely that the required number of FLOP/s exceeds the bounds suggested by the limit method. However, I don’t think the method itself airtight. Rather, I find some arguments in the vicinity persuasive (though not all of them rely directly on Landauer’s principle); various experts I spoke to (though not all) were quite confident in these arguments; and other methods seem to point to lower numbers.
• Communication method estimates may well prove informative, but I haven’t vetted them. I discuss this method mostly in the hopes of prompting further work.

Overall, I think it more likely than not that 1015 FLOP/s is enough to perform tasks as well as the human brain (given the right software, which may be very hard to create). And I think it unlikely (<10%) that more than 1021 FLOP/s is required.4 But I’m not a neuroscientist, and there’s no consensus in neuroscience (or elsewhere).

I offer a few more specific probabilities, keyed to one specific type of brain model, in the appendix.5 My current best-guess median for the FLOP/s required to run that particular type of model is around 1015 (note that this is not an estimate of the FLOP/s uniquely “equivalent” to the brain – see discussion in section 1.6).

As can be seen from the figure above, the FLOP/s capacities of current computers (e.g., a V100 at ~1014 FLOP/s for ~$10,000, the Fugaku supercomputer at ~4×1017 FLOP/s for ~$1 billion) cover the estimates I find most plausible.6 However:

• Computers capable of matching the human brain’s task performance would also need to meet further constraints (for example, constraints related to memory and memory bandwidth).
• Matching the human brain’s task-performance requires actually creating sufficiently capable and computationally efficient AI systems, and I do not discuss how hard this might be (though note that training an AI system to do X, in machine learning, is much more resource-intensive than using it to do X once trained).7

So even if my best-guesses are right, this does not imply that we’ll see AI systems as capable as the human brain anytime soon.

### Caveats

(This section discusses some caveats about the report’s epistemic status, and some notes on presentation. Those eager for the main content, however uncertain, can skip to section 1.3.)

Some caveats:

• Little if any of the evidence surveyed in this report is particularly conclusive. My aim is not to settle the question, but to inform analysis and decision-making that must proceed in the absence of conclusive evidence, and to lay groundwork for future work.
• I am not an expert in neuroscience, computer science, or physics (my academic background is in philosophy).
• I sought out a variety of expert perspectives, but I did not make a rigorous attempt to ensure that the experts I spoke to were a representative sample of opinion in the field. Various selection effects influencing who I interviewed plausibly correlate with sympathy towards lower FLOP/s requirements.8
• For various reasons, the research approach used here differs from what might be expected in other contexts. Key differences include:
• I give weight to intuitions and speculations offered by experts, as well as to factual claims by experts that I have not independently verified (these are generally documented in conversation notes approved by the experts themselves).
• I report provisional impressions from initial research.
• My literature reviews on relevant sub-topics are not comprehensive.
• I discuss unpublished papers where they appear credible.
• My conclusions emerge from my own subjective synthesis of the evidence I engaged with.
• There are ongoing questions about the baseline reliability of various kinds of published research in neuroscience and cognitive science.9 I don’t engage with this issue explicitly, but it is an additional source of uncertainty.

A few other notes on presentation:

• I have tried to keep the report accessible to readers with a variety of backgrounds.
• The endnotes are frequent and sometimes lengthy, and they contain more quotes and descriptions of my research process than is academically standard. This is out of an effort to make the report’s reasoning transparent to readers. However, the endnotes are not essential to the main content, and I suggest only reading them if you’re interested in more details about a particular point.
• I draw heavily on non-verbatim notes from my conversations with experts, made public with their approval and cited/linked in endnotes. These notes are also available here.
• I occasionally use the word “compute” as a shorthand for “computational power.”
• Throughout the rest of the report, I use a form of scientific notation, in which “XeY” means “X×10Y.” Thus, 1e6 means 1,000,000 (a million); 4e8 means 400,000,000 (four hundred million); and so on. I also round aggressively.

### Context

(This section briefly describes what prompts Open Philanthropy’s interest in the topic of this report. Those primarily interested in the main content can skip to Section 1.4.)

This report is part of a broader effort at Open Philanthropy to investigate when advanced AI systems might be developed (“AI timelines”) – a question that we think decision-relevant for our grant-making related to potential risks from advanced AI.10 But why would an interest in AI timelines prompt an interest in the topic of this report in particular?

Some classic analyses of AI timelines (notably, by Hans Moravec and Ray Kurzweil) emphasize forecasts about when available computer hardware will be “equivalent,” in some sense (see section 1.6 for discussion), to the human brain.11

A basic objection to predicting AI timelines on this basis alone is that you need more than hardware to do what the brain does.12 In particular, you need software to run on your hardware, and creating the right software might be very hard (Moravec and Kurzweil both recognize this, and appeal to further arguments).13

In the context of machine learning, we can offer a more specific version of this objection: the hardware required to run an AI system isn’t enough; you also need the hardware required to train it (along with other resources, like data).14 And training a system requires running it a lot. DeepMind’s AlphaGo Zero, for example, trained on ~5 million games of Go.15

Note, though, that depending on what sorts of task-performance will result from what sorts of training, a framework for thinking about AI timelines that incorporated training requirements would start, at least, to incorporate and quantify the difficulty of creating the right software more broadly.16 This is because training turns computation, data, and other resources into software you wouldn’t otherwise know how to make.

What’s more, the hardware required to train a system is related to the hardware required to run it.17 This relationship is central to Open Philanthropy’s interest in the topic of this report, and to an investigation my colleague Ajeya Cotra has been conducting, which draws on my analysis. That investigation focuses on what brain-related FLOP/s estimates, along with other estimates and assumptions, might tell us about when it will be feasible to train different types of AI systems. I don’t discuss this question here, but it’s an important part of the context. And in that context, brain-related hardware estimates play a different role than they do in forecasts like Moravec’s and Kurzweil’s.

### FLOP/s basics

(This section discusses what FLOP/s are, and why I chose to focus on them. Readers familiar with FLOP/s and happy with this choice can skip to Section 1.5.)

Computational power is multidimensional – encompassing, for example, the number and type of operations performed per second, the amount of memory stored at different levels of accessibility, and the speed with which information can be accessed and sent to different locations.18

This report focuses on operations per second, and in particular, on “floating point operations.”19 These are arithmetic operations (addition, subtraction, multiplication, division) performed on a pair of floating point numbers – that is, numbers represented as a set of significant digits multiplied by some other number raised to some exponent (like scientific notation). I’ll use “FLOPs” to indicate floating point operations (plural), and “FLOP/s” to indicate floating point operations per second.

My central reason for focusing on FLOP/s is that various brain-related FLOP/s estimates are key inputs to the framework for thinking about training requirements, mentioned above, that my colleague Ajeya Cotra has been investigating, and they were the focus of Open Philanthropy’s initial exploration of this topic, out of which this report emerged. Focusing on FLOP/s in particular also limits the scope of what is already a fairly broad investigation; and the availability of FLOP/s is one key contributor to recent progress in AI.20

Still, the focus on FLOP/s is a key limitation of this analysis, as other computational resources are just as crucial to task-performance: if you can’t store the information you need, or get it where it needs to be fast enough, then the units in your system that perform FLOPs will be some combination of useless and inefficiently idle.21 Indeed, my understanding is that FLOP/s are often not the relevant bottleneck in various contexts related to AI and brain modeling.22 And further dimensions an AI system’s implementation, like hardware architecture, can introduce significant overheads, both in FLOP/s and other resources.23

Ultimately, though, once other computational resources are in place, and other overheads have mostly been eliminated or accounted for, you need to actually perform the FLOP/s that a given time-limited computation requires. In order to isolate this quantity, I proceed on the idealizing assumption that non-FLOP resources are available in amounts adequate to make full use of all of the FLOP/s in question (but not in unrealistically extreme abundance), without significant overheads.24 All talk of the “FLOP/s sufficient to X” assumes this caveat.

This means you can’t draw conclusions about which concrete computers can replicate human-level task performance directly from the FLOP/s estimates in this report, even if you think those estimates credible. Such computers will need to meet further constraints.25

Note, as well, that these estimates do not depend on the assumption that the brain performs operations analogous to FLOPs, or on any other similarities between brain architectures and computer architectures.26 The report assumes that the tasks the brain performs can also be performed using a sufficient number of FLOP/s, but the causal structure in the brain that gives rise to task-performance could in principle take a wide variety of unfamiliar forms.

### Neuroscience basics

(This section reviews some of the neural mechanisms I’ll be discussing, in an effort to make the report’s content accessible to readers without a background in neuroscience.27 Those familiar with signaling mechanisms in the brain – neurons, neuromodulators, gap junctions – can skip to Section 1.5.1).

The human brain contains around 100 billion neurons, and roughly the same number of non-neuronal cells.28 Neurons are cells specialized for sending and receiving various types of electrical and chemical signals, and other non-neuronal cells send and receive signals as well.29 These signals allow the brain, together with the rest of the nervous system, to receive and encode sensory information from the environment, to process and store this information, and to output the complex, structured motor behavior constitutive of task performance.30

We can divide a typical neuron into three main parts: the soma, the dendrites, and the axon.31 The soma is the main body of the cell. The dendrites are extensions of the cell that branch off from the soma, and which typically receive signals from other neurons. The axon is a long, tail-like projection from the soma, which carries electrical impulses away from the cell body. The end of the axon splits into branches, the ends of which are known as axon terminals, which reach out to connect with other cells at locations called synapses. A typical synapse forms between the axon terminal of one neuron (the presynaptic neuron) and the dendrite of another (the postsynaptic neuron), with a thin zone of separation between them known as the synaptic cleft.32

The cell as a whole is enclosed in a membrane that has various pumps that regulate the concentration of certain ions – such as sodium (Na+), potassium (K+) and chloride (Cl-) – inside it.33 This regulation creates different concentrations of these ions inside and outside the cell, resulting in a difference in the electrical potential across the membrane (the membrane potential).34 The membrane also contains proteins known as ion channels, which, when open, allow certain types of ions to flow into and out of the cell.35

If the membrane potential in a neuron reaches a certain threshold, then a particular set of voltage-gated ion channels open to allow ions to flow into the cell, creating a temporary spike in the membrane potential (an action potential).36 This spike travels down the axon to the axon terminals, where it causes further voltage-gated ion channels to open, allowing an influx of calcium ions into the pre-synaptic axon terminal. This calcium can trigger the release of molecules known as neurotransmitters, which are stored in sacs called vesicles in the axon terminal.37

These vesicles merge with the cell membrane at the synapse, allowing the neurotransmitter they contain to diffuse across the synaptic cleft and bind to receptors on the post-synaptic neuron. These receptors can cause (directly or indirectly, depending on the type of receptor) ion channels on the post-synaptic neuron to open, thereby altering the membrane potential in that area of that cell.38

The expected size of the impact (excitatory or inhibitory) that a spike through a synapse will have on the post-synaptic membrane potential is often summarized via a parameter known as a synaptic weight.40 This weight changes on various timescales, depending on the history of activity in the pre-synaptic and post-synaptic neuron, together with other factors. These changes, along with others that take place within synapses, are grouped under the term synaptic plasticity.41 Other changes also occur in neurons on various timescales, affecting the manner in which neurons respond to synaptic inputs (some of these changes are grouped under the term intrinsic plasticity).42 New synapses, dendritic spines, and neurons also grow over time, and old ones die.43

There are also a variety of other signaling mechanisms in the brain that this basic story does not include. For example:

• Other chemical signals: Neurons can also send and receive other types of chemical signals – for example, molecules known as neuropeptides, and gases like nitric oxide – that can diffuse more broadly through the space in between cells, across cell membranes, or via the blood.44 The chemicals neurons release that influence the activity of groups of neurons (or other cells) are known as neuromodulators.45
• Glial cells: Non-neuronal cells in the brain known as glia have traditionally been thought to mostly perform functions to do with maintenance of brain function, but they may be involved in task-performance as well.46
• Electrical synapses: In addition to the chemical synapses discussed above, there are also electrical synapses that allow direct, fast, and bi-directional exchange of electrical signals between neurons (and between other cells). The channels mediating this type of connection are known as gap junctions.
• Ephaptic effects: Electrical activity in neurons creates electric fields that may impact the electrical properties of neighboring neurons.47
• Other forms of axon signaling: The process of firing an action potential has traditionally been thought of as a binary decision.48 However, some recent evidence indicates that processes within a neuron other than “to fire or not to fire” can matter for synaptic communication.49
• Blood flow: Blood flow in the brain correlates with neural activity, which has led some to suggest that it might be playing a role in information-processing.50

This is not a complete list of all the possible signaling mechanisms that could in principle be operative in the brain.51 But these are some of the most prominent.

#### Uncertainty in neuroscience

I want to emphasize one other meta-point about neuroscience: namely, that our current understanding of how the brain processes information is extremely limited.52 This was a consistent theme in my conversations with experts, and one of my clearest take-aways from the investigation as a whole.53

One problem is that we need better tools. For example:

• Despite advances, we can only record the spiking activity of a limited number of neurons at the same time (techniques like fMRI and EEG are much lower resolution).54
• We can’t record from all of a neuron’s synapses or dendrites simultaneously, making it hard to know what patterns of overall synaptic input and dendritic activity actually occur in vivo.55
• We also can’t stimulate all of a neuron’s synapses and/or dendrites simultaneously, making it hard to know how the cell responds to different inputs (and hence, which models can capture these responses).56
• Techniques for measuring many lower-level biophysical mechanisms and processes, such as possible forms of ion channel plasticity, remain very limited.57
• Results in model animals may not generalize to e.g. humans.58
• Results obtained in vitro (that is, in a petri dish) may not generalize in vivo (that is, in a live functioning brain).59
• The tasks we can give model animals like rats to perform are generally very simple, and so provide limited evidence about more complex behavior.60

Tools also constrain concepts. If we can’t see or manipulate something, it’s unlikely to feature in our theories.61 And certain models of e.g. neurons may receive scant attention simply because they are too computation-intensive to work with, or too difficult to constrain with available data.62

But tools aren’t the only problem. For example, when Jonas and Kording (2017) examined a simulated 6502 microprocessor – a system whose processing they could observe and manipulate to arbitrary degrees – using analogues of standard neuroscientific approaches, they found that “the approaches reveal interesting structure in the data but do not meaningfully describe the hierarchy of information processing in the microprocessor” (p. 1).63 And artificial neural networks that perform complex tasks are difficult (though not necessarily impossible) to interpret, despite similarly ideal experimental access.64

We also don’t know what high-level task most neural circuits are performing, especially outside of peripheral sensory/motor systems. This makes it very hard to say what models of such circuits are adequate.65

It would help if we had full functional models of the nervous systems of some simple animals. But we don’t.66 For example, the nematode worm Caenorhabditis elegans (C. elegans) has only 302 neurons, and a map of the connections between these neurons (the connnectome) has been available since 1986.67 But we have yet to build a simulated C. elegans that behaves like the real worm across a wide range of contexts.68

All this counsels pessimism about the robustness of FLOP/s estimates based on our current neuroscientific understanding. And it increases the relevance of where we place the burden of proof. If we start with a strong default view about the complexity of the brain’s task-performance, and then demand proof to the contrary, our standards are unlikely to be met.

Indeed, my impression is that various “defaults” in this respect play a central role in how experts approach this topic. Some take simple models that have had some success as a default, and then ask whether we have strong reason to think additional complexity necessary;69 others take the brain’s biophysical complexity as a default, and then ask if we have strong reason to think that a given type of simplification captures everything that matters.70

Note the distinction, though, between how we should do neuroscience, and how we should bet now about where such science will ultimately lead, assuming we had to bet. The former question is most relevant to neuroscientists; but the latter is what matters here.

### Clarifying the question

Consider the set of cognitive tasks that the human brain can perform, where task performance is understood as the implementation of a specified type of relationship between a set of inputs and a set of outputs.71 Examples of such tasks might include:

• Reading an English-language description of a complex software problem, and, within an hour, outputting code that solves that problem.72
• Reading a randomly selected paper submitted to the journal Nature, and, within a week, outputting a review of the paper of quality comparable to an average peer-reviewer.73
• Reading newly-generated Putnam Math competition problems, and, within six hours, outputting answers that would receive a perfect score by standard judging criteria.74

Defining tasks precisely can be arduous. I’ll assume such precision is attainable, but I won’t try to attain it, since little in what follows depends on the details of the tasks in question. I’ll also drop the adjective “cognitive” in what follows.

I will also assume that sufficiently powerful computers can in principle perform these tasks (I focus solely on non-quantum computers – see endnote for discussion of quantum brain hypotheses).75 This assumption is widely shared both within the scientific community and beyond it. Some dispute it, but I won’t defend it here.76

The aim of the report is to evaluate the extent to which the brain provides evidence, for some number of FLOP/s F, that for any task T that the human brain can perform, T can be performed with F.77 As a proxy for FLOP/s numbers with this property, I will sometimes talk about the FLOP/s sufficient to run a “task-functional model,” by which I mean a computational model that replicates a generic human brain’s task-performance. Of course, some brains can do things others can’t, but I’ll assume that at the level of precision relevant to this report, human brains are roughly similar, and hence that if F FLOP/s is enough to replicate the task performance of a generic human brain, roughly F is enough to replicate any task T the human brain can perform.78

The project here is related to, but distinct from, directly estimating the minimum FLOP/s sufficient to perform any task the brain can perform. Here’s an analogy. Suppose you want to build a bridge across the local river, and you’re wondering if you have enough bricks. You know of only one such bridge (the “old bridge”), so it’s natural to look there for evidence. If the old bridge is made of bricks, you could count them. If it’s made of something else, like steel, you could try to figure out how many bricks you need to do what a given amount of steel does. If successful, you’ll end up confident that e.g. 100,000 bricks is enough to build such a bridge, and hence that the minimum is less than this. But how much less is still unclear. You studied an example bridge, but you didn’t derive theoretical limits on the efficiency of bridge-building.

That said, Dr. Paul Christiano expected there to be at least some tasks such (a) the brain’s methods of performing them are close to maximally efficient, and (b) these methods use most of the brain’s resources (see endnote).79 I don’t investigate this claim here, but if true, it would make data about the brain more directly relevant to the minimum adequate FLOP/s budget.

The project here is also distinct from estimating the FLOP/s “equivalent” to the human brain. As I discuss in the report’s appendix, I think the notion of “the FLOP/s equivalent to the brain” requires clarification: there are a variety of importantly different concepts in the vicinity.

To get a flavor of this, consider the bridge analogy again, but assume that the old bridge is made of steel. What number of bricks would be “equivalent” to the old bridge? The question seems ill-posed. It’s not that bridges can’t be built from bricks. But we need to say more about what we want to know.

I group the salient possible concepts of the “FLOP/s equivalent to the human brain” into four categories:

1. FLOP/s required for task-performance, with no further constraints on how the tasks need to be performed.80
2. FLOP/s required for task-performance + brain-like-ness constraints – that is, constraints on the similarity between how the AI system does it, and how the brain does it.
3. FLOP/s required for task-performance + findability constraints – that is, constraints on what sorts of training processes and engineering efforts would be able to create the AI system in question.
4. Other analogies with human-engineered computers.

All these categories have their own problems (see section A.5 for a summary chart). The first is closest to the report’s focus, but as just noted, it’s hard (at least absent further assumptions) to estimate directly using example systems. The second faces the problem of identifying a non-arbitrary brain-like-ness constraint that picks out a unique number of FLOP/s, without becoming too much like the first. The third brings in a lot of additional questions about what sorts of systems are what sorts of findable. And the fourth, I suggest, either collapses into the first or second, or raises its own questions.

In the hopes of avoiding some of these problems, I have kept the report’s framework broad. The brain-based FLOP/s budgets I’m interested in don’t need to be uniquely “equivalent” to the brain, or as small as theoretically possible, or accommodating of any constraints on brain-like-ness or findability. They just need to be big enough, in principle, to perform the tasks in question.

A few other clarifications:

• Properties construed as consisting in something other than the implementation of a certain type of input-output relationship (for example, properties like phenomenal consciousness, moral patienthood, or continuity with a particular biological human’s personal identity – to the extent they are so construed) are not included in the definition of the type of task-performance I have in mind. Systems that replicate this type of task-performance may or may not also possess such properties, but what matters here are inputs and outputs.81
• Many tasks require more than a brain. For example, they may require something like a body, or rely partly on information-processing taking place outside the brain.82 In those cases, I’m interested in the FLOP/s sufficient to replicate the brain’s role.

### Existing literature

(This section reviews existing literature.83 Those interested primarily in the report’s substantive content can skip to Section 2.)

A lot of existing research is relevant to estimating the FLOP/s sufficient to run a task-functional model. But efforts in the mainstream academic literature to address this question directly are comparatively rare (a fact that this report does not alter). Many existing estimates are informal, and they often do not attempt much justification of their methods or background assumptions. The specific question they consider also varies, and their credibility varies widely.84

#### Mechanistic method estimates

The most common approach assigns a unit of computation (such as a calculation, a number of bits, or a possibly brain-specific operation) to a spike through a synapse, and then estimates the rate of spikes through synapses by multiplying an estimate of the average firing rate by an estimate of the number of synapses.85 Thus, Merkle (1989),86 Mead (1990),87 Freitas (1996),88 Sarpeshkar (1997),89 Bostrom (1998),90 Kurzweil (1999)),91 Dix (2005),92 Malickas (2007),93 and Tegmark (2017)94 are all variations on this theme.95 Their estimates range from ~1e12 to ~1e17 (though using basic different units of computation),96 but the variation results mainly from differences in estimated synapse count and average firing rate, rather than differences in substantive assumptions about how to make estimates of this kind.97 In this sense, the helpfulness of these estimates is strongly correlated: if the basic approach is wrong, none of them are a good guide.

Other estimates use a similar approach, but include more complexity. Sarpeshkar (2010) includes synaptic conductances (see discussion in section 2.1.1.2.2), learning, and firing decisions in a lower bound estimate (6e16 FLOP/s);98 Martins et al. (2012) estimate the information-processing rate of different types of neurons in different regions, for a total of ~5e16 bits/sec in the whole brain;99 and Kurzweil (2005) offers an upper bound estimate for a personality-level simulation of 1e19 calculations per second – an estimate that budgets 1e3 calculations per spike through synapse to capture nonlinear interactions in dendrites.100 Still others attempt estimates based on protein interactions (Thagard (2002), 1e21 calculations/second);101 microtubules (Tuszynski (2006), 1e21 FLOP/s),102 individual neurons (von Neumann (1958), 1e11 bits/second);103 and possible computations performed by dendrites and other neural mechanisms (Dettmers (2015), 1e21 FLOP/s).104

A related set of estimates comes from the literature on brain simulations. Ananthanarayanan et al. (2009) estimates >1e18 FLOP/s to run a real-time human brain simulation;105 Waldrop (2012) cites Henry Markram as estimating 1e18 FLOP/s to run a very detailed simulation;106 Markram, in a 2018 video (18:28), estimates that you’d need ~4e29 FLOP/s to run a “real-time molecular simulation of the human brain”;107 and Eugene Izhikevich estimates that a real-time brain simulation would require ~1e6 processors running at 384 GHz.108

Sandberg and Bostrom (2008) also estimate the FLOP/s requirements for brain emulations at different levels of detail. Their estimates range from 1e15 FLOP/s for an “analog network population model,” to 1e43 FLOP/s for emulating the “stochastic behavior of single molecules.”109 They report that in an informal poll of attendees at a workshop on whole brain emulation, the consensus appeared to be that the required level of resolution would fall between “Spiking neural network” (1e18 FLOP/s), and “Metabolome” (1e25 FLOP/s).110

Despite their differences, I group all of these estimates under the broad heading of the “mechanistic method,” as all of them attempt to identify task-relevant causal structure in the brain’s biological mechanisms, and quantify it in some kind of computational unit.

#### Functional method estimates

A different class of estimates focus on the FLOP/s sufficient to replicate the function of some portion of the brain, and then attempt to scale up to an estimate for the brain as a whole (the “functional method”). Moravec (1988), for example, estimates the computation required to do what the retina does (1e9 calculations/second) and then scales up (1e14 calc/s).111 Merkle (1989) performs a similar retina-based calculation and gets 1e12-1e14 ops/sec.112

Kurzweil (2005) offers a functional method estimate (1e14 calcs/s) based on work by Lloyd Watts on sound localization,113 another (1e15 calcs/s) based on an cerebellar simulation at the University of Texas;114 and a third (1e14 calcs/s), in his 2012 book, based on the FLOP/s he estimates is required to emulate what he calls a “pattern recognizer” in the neocortex.115 Drexler (2019) uses the FLOP/s required for various deep learning systems (specifically: Google’s Inception architecture, Deep Speech 2, and Google’s neural machine translation model) to generate various estimates he takes to suggest that 1e15 FLOP/s is sufficient to match the brain’s functional capacity.116

#### Limit method estimates

Sandberg (2016) uses Landauer’s principle to generate an upper bound of ~2e22 irreversible operations per second in the brain – a methodology I consider in more detail in Section 4.117 De Castro (2013) estimates a similar limit, also from Landauer’s principle, on perceptual operations performed by the parts of the brain involved in rapid, automatic inference (1e23 operations per second).118 I have yet to encounter other attempts to bound the brain’s overall computation via Landauer’s principle,119 though many papers discuss related issues in the brain and in biological systems more broadly.120

#### Communication method estimates

AI Impacts estimates the communication capacity of the brain (measured as “traversed edges per second” or TEPS), then combines this with an observed ratio of TEPS to FLOP/s in some human-engineered computers, to arrive an estimate of brain FLOP/s (~1e16-3e17 FLOP/s).121 I discuss methods in this broad category – what I call, the “communication method” – in Section 5.

Let’s turn now to evaluating the methods themselves. Rather than looking at all possible ways of applying them, my discussion will focus on what seem to me like the most plausible approaches I’m aware of, and the most important arguments/objections.

## The mechanistic method

The first method I’ll be discussing – the “mechanistic method” – attempts to estimate the computation required to model the brain’s biological mechanisms at a level of detail adequate to replicate task performance.

Simulating the brain in extreme detail would require enormous amounts of computational power.122 Which details would need to be included in a computational model, and which, if any, could be left out or summarized?

The approach I’ll pursue focuses on signaling between cells. Here, the idea is that for a process occurring in a cell to matter to task-performance, it needs to affect the type of signals (e.g. neurotransmitters, neuromodulators, electrical signals at gap junctions, etc.) that cell sends to other cells.123 Hence, a model of that cell that replicates its signaling behavior (that is, the process of receiving signals, “deciding” what signals to send out, and sending them) would replicate the cell’s role in task-performance, even if it leaves out or summarizes many other processes occuring in the cell. Do that for all the cells in the brain involved in task-performance, and you’ve got a task-functional model.

I’ll divide the signaling processes that might need to be modeled into three categories:

1. Standard neuron signaling.124 I’ll divide this into two parts:
• Synaptic transmission. The signaling process that occurs at a chemical synapse as a result of a spike.
• Firing decisions. The processes that cause a neuron to spike or not spike, depending on input from chemical synapses and other variables.
2. Learning. Processes involved in learning and memory formation (e.g., synaptic plasticity, intrinsic plasticity, and growth/death of cells and synapses), where not covered by (1).
3. Other signaling mechanisms. Any other signaling mechanisms (neuromodulation, electrical synapses, ephaptic effects, glial signaling, etc.) not covered by (1) or (2).

As a first-pass framework, we can think of synaptic transmission as a function from spiking inputs at synapses to some sort of output impact on the post-synaptic neuron; and of firing decisions as (possibly quite complex) functions that take these impacts as inputs, and then produce spiking outputs – outputs which themselves serve as inputs to downstream synaptic transmission. Learning changes these functions over time (though it can involve other changes as well, like growing new neurons and synapses). Other signaling mechanisms do other things, and/or complicate this basic picture.

This isn’t an ideal carving, but hopefully it’s helpful regardless.125 Here’s the mechanistic method formula that results:

Total FLOP/s = FLOP/s for standard neuron signaling +
FLOP/s for learning +
FLOP/s for other signaling mechanisms

I’m particularly interested in the following argument:

1. You can capture standard neuron signaling and learning with somewhere between ~1e13-1e17 FLOP/s overall.
2. This is the bulk of the FLOP/s burden (other processes may be important to task-performance, but they won’t require comparable FLOP/s to capture).

I’ll discuss why one might find (I) and (II) plausible in what follows. I don’t think it at all clear that these claims are true, but they seem plausible to me, partly on the merits of various arguments I’ll discuss, and partly because some of the experts I engaged with were sympathetic (others were less so). I also discuss some ways this range could be too high, and too low.

### Standard neuron signaling

Here is the sub-formula for standard neuron signaling:

FLOP/s for standard neuron signaling = FLOP/s for synaptic transmission + FLOP/s for firing decisions

I’ll budget for each in turn.

#### Synaptic transmission

Let’s start with synaptic transmission. This occurs as a result of spikes through synapses, so I’ll base this budget on spikes through synapses per second × FLOPs per spike through synapse (I discuss some assumptions this involves below).

##### Spikes through synapses per second

How many spikes through synapses happen per second?

As noted above, the human brain has roughly 100 billion neurons.126 Synapse count appears to be more uncertain,127 but most estimates I’ve seen fall in the range of an average of 1,000-10,000 synapses per neuron, and between 1e14 and 1e15 overall.128

How many spikes arrive at a given synapse per second, on average?

• Maximum neuron firing rates can exceed 100 Hz,129 but in vivo recordings suggest that neurons usually fire at lower rates – between 0.01 and 10 Hz.130
• Experts I engaged with tended to use average firing rates of 1-10 Hz.131
• Energy costs limit spiking. Lennie (2003), for example, uses energy costs to estimate a 0.16 Hz average in the cortex, and 0.94 Hz “using parameters that all tend to underestimate the cost of spikes.”132 He also estimates that “to sustain an average rate of 1.8 spikes/s/neuron would use more energy than is normally consumed by the whole brain” (13 Hz would require more than the whole body).133
• Existing recording methods may bias towards active cells.134 Shoham et al. (2005), for example, suggests that recordings may overlook large numbers of “silent” neurons that fire infrequently (on one estimate for the cat primary visual cortex, >90% of neurons may qualify as “silent”).135

Synthesizing evidence from a number of sources, AI Impacts offers a best guess average of 0.1-2 Hz. This sounds reasonable to me (I give most weight to the metabolic estimates). I’ll use 0.1-1 Hz, partly because Lennie (2003) treats 0.94 Hz as an overestimate, and partly because I’m mostly sticking with order-of-magnitude level precision. This suggests an overall range of ~1e13-1e15 spikes through synapses per second (1e14-1e15 synapses × 0.1-1 spikes per second).136

Note that many of the mechanistic method estimates reviewed in 1.6.1 assume a higher average spiking rate, often in the range of 100 Hz.137 For the reasons listed above, I think 100 Hz too high. ~10 Hz seems more possible (though it requires Lennie (2003) to be off by 1-2 orders of magnitude, and my best guess is lower): in that case, we’d add an orders of magnitude to the high-end estimates below.

##### FLOPs per spike through synapse

How many FLOPs do we need to capture what matters about the signaling that occurs when a spike arrives at a synapse?

###### A simple model

A simple answer is: one FLOP. Why might one think this?

One argument is that in the context of standard neuron signaling (setting aside learning), what matters about a spike through a synapse is that it increases or decreases the post-synaptic membrane potential by a certain amount, corresponding to the synaptic weight. This could be modeled as a single addition operation (e.g., add the synaptic weight to the post-synaptic membrane potential). That is, one FLOP (of some precision, see below).138

We can add several complications without changing this picture much:139

• Some estimates treat a spike through a synapse as multiplication by a synaptic weight. But spikes are binary, so in a framework based on individual spikes, you’re really only “multiplying” the synaptic weight by 0 or 1 (e.g., if the neuron spikes, then multiply the weight by 1, and add it to the post-synaptic membrane potential; otherwise, multiply it by 0, and add the result – 0 – to the post-synaptic membrane potential).
• In artificial neural networks, input neuron activations are sometimes analogized to non-binary spike rates (e.g., average numbers of spikes over some time interval), which are multiplied by synaptic weights and then summed.140 This would be two FLOPs (or one Multiply-Accumulate). But since such rates take multiple spikes to encode, this analogy plausibly suggests less than two FLOPs per spike through synapse.

How precise do these FLOPs need to be?141 That depends on the number of distinguishable synaptic weights/membrane potentials. Here are some relevant estimates:

A standard FLOP is 32 bits, and half-precision is 16 – well in excess of these estimates. Some hardware uses even lower-precision operations, which may come closer. I’d guess that 8 bits would be adequate.

If we assume 1 (8-bit) FLOP per spike through synapse, we get an overall estimate of 1e13-1e15 (8-bit) FLOP/s for synaptic transmission. I won’t continue to specify the precision I have in mind in what follows.

###### Possible complications

Here are a few complications this simple model leaves out.

Stochasticity

Real chemical synaptic transmission is stochastic. Each vesicle of neurotransmitter has a certain probability of release, conditional on a spike arriving at the synapse, resulting in variation in synaptic efficacy across trials.148 This isn’t necessarily a design defect. Noise in the brain may have benefits,149 and we know that the brain can make synapses reliable.150

Would capturing the contribution of this stochasticity to task performance require many extra FLOP/s, relative to a deterministic model? My guess is no.

• The relevant probability distribution (a binomial distribution, according to Siegelbaum et al. (2013c), (p. 270)), appears to be fairly simple, and Dr. Paul Christiano, one of our technical advisors, thought that sampling from an approximation of such a distribution would be cheap.151
• My background impression is that in designing systems for processing information, adding noise is easy; limiting noise is hard (though this doesn’t translate directly into a FLOPs number).
• Despite the possible benefits of noise, my guess is that the brain’s widespread use of stochastic synapses has a lot to do with resource constraints (more reliable synapses require more neurotransmitter release sites).152
• Many neural network models don’t include this stochasticity.153

That said, one expert I spoke with (Prof. Erik De Schutter) thought it an open question whether the brain manipulates synaptic stochasticity in computationally complex ways.154

Synaptic conductances

The ease with which ions can flow into the post-synaptic cell at a given synapse (also known as the synaptic conductance) changes over time as the ion channels activated by synaptic transmission open and close.155 The simple “addition” model above doesn’t include this – rather, it summarizes the impact of a spike through synapse as a single, instantaneous increase or decrease to post-synaptic membrane potential.

Sarpeshkar (2010), however, appears to treat the temporal dynamics of synaptic conductances as central to the computational function of synapses.156 He assumes, as a lower bound, that “the 20 ms second-order filter response due to each synapse is 40 FLOPs,” and that such operations occur on every spike.157

I’m not sure exactly what Sarpeshkar (2010) has in mind here, but it seems plausible to me that the temporal dynamics of a neuron’s synaptic conductances can influence membrane potential, and hence spike timing, in task-relevant ways.158 One expert also emphasized the complications to neuron behavior introduced by the conductance created by a particular type of post-synaptic receptor called an NMDA-receptor – conductances that Beniaguev et al. (2020) suggest may substantially increase the complexity of a neuron’s I/O (see discussion in Section 2.1.1.2).159 That said, two experts thought it likely that synaptic conductances could either be summarized fairly easily or left out entirely.160

Sparse FLOPs and time-steps per synapse

Estimates based on spikes through synapses assume that you don’t need to budget any FLOPs for when a synapse doesn’t receive a spike, but could have. Call this the “sparse FLOPs assumption.”161 In current neural network implementations, the analogous situation (e.g., artificial neuron activations of 0) creates inefficiencies, which some new hardware designs aim to avoid.162 But this seems more like an engineering challenge than a fundamental feature of the brain’s task-performance.

Note, though, that for some types of brain simulation, budgets would be based on time-steps per synapse instead, regardless of what is actually happening at synapse over that time. Thus, for a simulation of a 1e14-1e15 synapses run at 1 ms resolution (1000 timesteps per second), you’d get 1e17-1e18 timesteps per synapse – a number that would then be multiplied by your FLOPs budget per time-step at each synapse; and smaller time-steps would yield higher numbers. Not all brain simulations do this (see, e.g., Ananthanarayanan et al. (2009), who simulate time-steps at neurons, but events at synapse),163 but various experts use it as a default methodology.164

Going forward, I’ll assume that on simple models of synaptic transmission where the synaptic weight is not changing during time-steps without spikes, we don’t need to budget any FLOPs for those time-steps (the budgets for different forms of synaptic plasticity are different story, and will be covered in the learning section). If this is wrong, though, it could increase budgets by a few orders of magnitude (see Section 2.4.1).

Others

There are likely many other candidate complications that the simple model discussed above does not include. There is intricate molecular machinery located at synapses, much of which is still not well-understood. Some of this may play a role in synaptic plasticity (see Section 2.2 below), or just in maintaining a single synaptic weight (itself a substantive task), but some may be relevant to standard neuron signaling as well.165

Higher-end estimate

I’ll use 100 FLOPs per spike through synapse as a higher-end FLOP/s budget for synaptic transmission. This would at least cover Sarpeshkar’s 40 FLOP estimate, and provide some cushion for other things I might be missing, including some more complex manipulations of synaptic stochasticity.

With 1 FLOP per spike through synapse as a low-end, and 100 FLOPs as a high end, we get 1e13-1e17 FLOP/s overall. Firing rate models might suggest lower numbers; other complexities and unknowns, along with estimates based on time-steps rather than spikes, higher numbers.

#### Firing decisions

The other component of standard neuron signaling is firing decisions, understood as mappings from synaptic inputs to spiking outputs.

One might initially think these likely irrelevant: there are 3-4 orders of magnitude more synapses than neurons, so one might expect events at synapses to dominate the FLOP/s burden.166 But as just noted, we’re counting FLOPs at synapses based on spikes, not time-steps. Depending on the temporal-resolution we use (this varies across models), the number of time-steps per second (often ≥1000) plausibly exceeds the average firing rate (~0.1-1 Hz) by 3-4 orders of magnitude as well. Thus, if we need to compute firing decisions every time-step, or just generally more frequently than the average firing rate, this could make up for the difference between neuron and synapse count (I discuss this more in Section 2.1.2.5). And firing decisions could be more complex than synaptic transmission for other reasons as well.

Neuroscientists implement firing decisions using neuron models that can vary enormously in their complexity and biological realism. Herz et al. (2006) group these models into five rough categories:167

1. Detailed compartmental models. These attempt detailed reconstruction of a neuron’s physical structure and the electrical properties of its dendritic tree. This tree is modeled using many different “compartments” that can each have different membrane potentials.
2. Reduced compartmental models. These include fewer distinct compartments, but still more than one.
3. Single compartment models. These ignore the spatial structure of the neuron entirely and focus on the impact of input currents on the membrane potential in a single compartment.
1. The Hodgkin-Huxley model, a classic model in neuroscience, is a paradigm example of a single compartment model. It models different ionic conductances in the neuron using a series of differential equations. According to Izhikevich (2004), it requires ~120 FLOPs per 0.1 ms of simulation – ~1e6 FLOP/s overall.168
2. My understanding is that “integrate-and-fire”-type models – another classic neuron model, but much more simplified – would also fall into this category. Izhikevich (2004) suggests that these require ~5-13 FLOPs per ms per cell, 5000-13,000 FLOP/s overall.169
4. Cascade models. These models abstract away from ionic conductances, and instead attempt to model a neuron’s input-output mapping using a series of higher-level linear and non-linear mathematical operations, together with sources of noise. The “neurons” used in contemporary deep learning can be seen as variants of models in this category.170 These cascade models can also incorporate operations meant to capture transformations of synaptic inputs that occur in dendrites.171
5. Black box models. These neglect biological mechanisms altogether.

Prof. Erik De Schutter also mentioned that greater computing power has made even more biophysically realistic models available.172 And models can in principle be arbitrarily detailed.

Which of these models (if any) would be adequate to capture what matters about firing decisions? I’ll consider four categories of evidence: the predictive success of different neuron models; some specific arguments about the computational power of dendrites; a collection of other considerations; and expert opinion/practice.

##### Predicting neuron behavior

Let’s first look at the success different models have had in predicting neuron spike patterns.

###### Standards of accuracy

How accurate do these predictions need to be? The question is still open.

In particular, debate in neuroscience continues about whether and when to focus on spike rates (e.g., the average number of spikes over a given period), vs. the timings of individual spikes.173

• Many results in neuroscience focus on rates,174 as do certain neural prostheses.175
• In some contexts, it’s fairly clear that spike timings can be temporally precise.176
• One common argument for rates appeals to variability in a neuron’s response to repeated exposure to the same stimulus.177 My impression is that this argument is not straightforward to make rigorous, but it seems generally plausible to me that if rates are less variable than timings, they are also better suited to information-processing.178
• A related argument is that in networks of artificial spiking neurons, adding a single spike results in very different overall behavior.179 This plausibly speaks against very precisely-timed spiking in the brain, since the brain is robust to forms of noise that can shift spike timings180 as well as to our adding spikes to biological networks.181

My current guess is that in many contexts, but not all, spike rates are sufficient.

Even if we settled this debate, though, we’d still need to know how accurately the relevant rates/timings would need to be predicted.182 Here, a basic problem is that in many cases, we don’t know what tasks a neuron is involved in performing, or what role it’s playing. So we can’t validate a model by showing that it suffices to reproduce a given neuron’s role in task-performance - the test we actually care about.183

In the absence of such validation, one approach is to try to limit the model’s prediction error to within the trial-by-trial variability exhibited by the biological neuron.184 But if you can’t identify and control all task-relevant inputs to the cell, it’s not always clear what variability is or is not task-relevant.185

Nor is it clear how much progress a given degree of predictive success represents.186 Consider an analogy with human speech. I might be able to predict many aspects of human conversation using high-level statistics about common sounds, volume variations, turn-taking, and so forth, without actually being able to replicate or generate meaningful sentences. Neuron models with some predictive success might be similarly off the mark (and similar meanings could also presumably be encoded in different ways: e.g., “hello,” “good day,” “greetings,” etc.).187

###### Existing results

With these uncertainties in mind, let’s look at some existing efforts to predict neuron spiking behavior with computational models (these are only samples from a very large literature, which I do not attempt to survey).188

Many of these come with important additional caveats:

• Many model in vitro neuron behavior, which may differ from in vivo behavior in important ways.189
• Some use simpler models to predict the behavior of more detailed models. But we don’t really know how good the detailed models are, either.190
• We are very limited in our ability to collect in vivo data about the spatio-temporal input patterns at dendrites. This makes it hard to tell how models respond to realistic input patterns.191 And we know that certain behaviors (for example, dendritic non-linearities) are only triggered by specific input patterns.192
• We can’t stimulate neurons with arbitrary input patterns. This makes it hard to test their full range of behavior.193
• Models that predict spiking based on current injection into the soma skip whatever complexity might be involved in capturing processing that occurs in dendrites.194

A number of the results I looked at come from the retina, a thin layer of neural tissue in the eye, responsible for the first stage of visual processing. This processing is largely (though not entirely) feedforward:195 the retina receives light signals via a layer of ~100 million photoreceptor cells (rods and cones),196 processes them in two further cell layers, and sends the results to the rest of the brain via spike patterns in the optic nerve – a bundle of roughly a million axons of neurons called retinal ganglion cells.197

I focused on the retina in particular partly because it’s the subject of a prominent functional method estimate in the literature (see Section 3.1.1), and partly because it offers advantages most other neural circuits don’t: we know, broadly, what task it’s performing (initial visual processing); we know what the relevant inputs (light signals) and outputs (optic nerve spike trains) are; and we can measure/manipulate these inputs/outputs with comparative ease.199 That said, as I discuss in Section 3.1.2, it may also be an imperfect guide to the brain as a whole.

Here’s a table with various modeling results that purport to have achieved some degree of success. Most of these I haven’t investigated in detail, and don’t have a clear sense of the significance of the quoted results. And as I discuss in later sections, some of the deep neural network models (e.g., Beniaguev et al. (2020), Maheswaranathan et al. (2019), Batty et al. (2017)) are very FLOP/s intensive (~1e7-1e10 FLOP/s per cell).200 A more exhaustive investigation could estimate the FLOP/s costs of all the listed models, but I won’t do that here.

Source Model type Thing predicted Stimuli Results
Beniaguev et al. (2020) Temporally convolutional network with 7 layers and 128 channels per layer Spike timing and membrane potential of a detailed model of a Layer 5 cortical pyramidal cell Random synaptic inputs “accurately, and very efficiently, capture[s] the I/O of this neuron at the millisecond resolution … For binary spike prediction (Fig. 2D), the AUC is 0.9911. For somatic voltage prediction (Fig. 2E), the RMSE is 0.71mV and 94.6% of the variance is explained by this model”
Maheswaranathan et al. (2019) Three-layer convolutional neural network Retinal ganglion cell (RGC) spiking in isolated salamander retina Naturalistic images >0.7 correlation coefficient (retinal reliability is 0.8)
Ujfalussy et al. (2018) Hierarchical cascade of linear-nonlinear subunits Membrane potential of in-vivo validated biophysical model of L2/3 pyramidal cell In vivo-like input patterns “Linear input integration with a single global dendritic nonlinearity achieved above 90% prediction accuracy.”
Batty et al. (2017) Shared two-layer recurrent network RGC spiking in isolated primate retina Natural images 80% of explainable variance.
2016 talk (39:05) by Markus Meister Linear-non-linear RGC spiking (not sure of experimental details) Naturalistic movie 80% correlation with real response (cross-trial correlation of real responses was around 85-90%).
Naud et al. (2014) Two compartments, each modeled with a pair of non-linear differential equations and a small number of parameters that approximate the Hodgkin-Huxley equations In vitro spike timings of layer 5 pyramidal cell Noisy current injection into the soma and apical dendrite “The predicted spike trains achieved an averaged coincidence rate of 50%. The scaled coincidence rate obtained by dividing by the intrinsic reliability (Jolivet et al. (2008a); Naud and Gerstner (2012b)) was 72%, which is comparable to the state-of-the performance for purely somatic current injection which reaches up to 76% (Naud et al. (2009)).”
Bomash et al. (2013) Linear-non-linear RGC spiking in isolated mouse retina Naturalistic and artificial “the model cells carry the same amount of information,” “the quality of the information is the same.”
Nirenberg and Pandarinath (2012) Linear-non-linear RGC spiking in isolated mouse retina Natural scenes movie “The firing patterns … closely match those of the normal retina,”; brain would map the artificial spike trains to the same images “90% of the time.”
Naud and Gerstner (2012a) Review of a number of simplified neuron models, including Adaptive Exponential Integrate and Fire (AdEx) and Spike Response Model (SRM) In vitro spike timings of various neuron types Simulating realistic conditions in vitro by injecting a fluctuating current into the soma “Performances are very close to optimal,” considering variation in real neuron responses. “For models like the AdEx or the SRM, [the percentage of predictable spikes predicted] ranged from 60% to 82% for pyramidal neurons, and from 60% to 100% for fast-spiking interneurons.”
Gerstner and Naud (2009) Threshold model In vivo spiking activity of neuron in the lateral geniculate nucleus (LGN) Visual stimulation of the retina Predicted 90.5% of spiking activity
Gerstner and Naud (2009) Integrate-and-fire model with moving threshold In vitro spike timings of (a) a pyramidal cell, and (b) an interneuron Random current injection 59.6% of pyramidal cell spikes, 81.6% of interneuron spikes.
Song et al. (2007) Multi-input multi-output model Spike trains in the CA3 region of the rat hippocampus while it was performing a memory task Input spike trains recorded from rat hippocampus “The model predicts CA3 output on a msec-to-msec basis according to the past history (temporal pattern) of dentate input, and it does so for essentially all known physiological dentate inputs and with approximately 95% accuracy.”
Pillow et al. (2005) Leaky integrate and fire model RGC spiking in in vitro macaque retina Artificial (“pseudo-random stimulus”) “The fitted model predicts the detailed time structure of responses to novel stimuli, accurately capturing the interaction between the spiking history and sensory stimulus selectivity.”
Brette and Gerstner (2005) Adaptive Exponential Integrate-and-fire Model Spike timings for detailed, conductance-based neuron model Injection of noisy synaptic conductances “Our simple model predicts correctly the timing of 96% of the spikes (+/- 2 ms)…”
Rauch et al. (2003) Integrate-and-fire model with spike-frequency-dependent adaptation/facilitation In vitro firing of rat neocortical pyramidal cells In vivo-like noisy current injection into the soma. “the integrate-and-fire model with spike-frequency- dependent adaptation /facilitation is an adequate model reduction of cortical cells when the mean spike frequency response to in vivo–like currents with stationary statistics is considered.”
Poirazi et al. (2003) Two-layer neural network Detailed biophysical model of a pyramidal neuron “An extremely varied, spatially heterogeneous set of synaptic activation patterns” 94% of variance explained (a single-layer network explained 82%)
Keat et al. (2001) Linear-non-linear RGC spiking in salamander and rabbit isolated retinas, and retina/LGN spiking in anesthetized cat Artificial (“random flicker stimulus’) “The simulated spike trains are about as close to the real spike trains as the real spike trains are across trials.”

Figure 7: List of some efforts to predict neuron behavior that appear to have had some amount of success.

What should we take away from these results? Without much of an understanding of the details here, my current high-level take-away is that it seems like some models do pretty well in some conditions, but in many cases, these conditions aren’t clearly informative about in vivo behavior across the brain, and absent better functional understanding and experimental access, it’s hard to say exactly what level of predictive accuracy is required, in response to what types of inputs. There are also incentives to present research in an optimistic light, and contexts in which our models do much worse won’t have ended up on the list (though note, as well, that additional predictive accuracy need not require additional FLOP/s – it may be that we just haven’t found the right models yet).

Let’s look at some other considerations.

##### Dendritic computation

Some neuron models don’t include dendrites. Rather, they treat dendrites as directly relaying synaptic inputs to the soma.

A common objection to such models is that dendrites can do more than this.201 For example:

• The passive membrane properties of dendrites (e.g. resistance, capacitance, and geometry) can create nonlinear interactions between synaptic inputs.202
• Active, voltage-dependent channels can create action potentials within dendrites, some of which can backpropagate through the dendritic tree.203

Effects like these are sometimes called “dendritic computation.”204

My impression is that the importance of dendritic computation to task-performance remains somewhat unclear: many results are in vitro, and some may require specific patterns of synaptic input.205 That said, one set of in vivo measurements found very active dendrites: specifically, dendritic spike rates 5-10x larger than somatic spike rates,206 which the authors take to suggest that dendritic spiking might dominate the brain’s energy consumption.207 Energy is scarce, so if true, this would suggest that dendritic spikes are important for something. And dendritic dynamics appear to be task-relevant in a number of neural circuits.208

How many extra FLOP/s do you need to capture dendritic computation, relative to “point neuron models” that don’t include dendrites? Some considerations suggest fairly small increases:

• A number of experts thought that models incorporating a small number of additional dendritic sub-units or compartments would likely be adequate.209
• It may be possible to capture what matters about dendritic computation using a “point neuron” model.210
• Some active dendritic mechanisms may function to “linearize” the impact at the soma of synaptic inputs that would otherwise decay, creating an overall result that looks more like direct current injection.211
• Successful efforts to predict neuron responses to task-relevant inputs (e.g., retinal responses to natural movies) would cover dendritic computation automatically (though at least some prominent forms of dendritic computation don’t happen in the retina).212

Tree structure

One of Open Philanthropy’s technical advisors (Dr. Dario Amodei) also suggests a more general constraint. Many forms of dendritic computation, he suggests, essentially amount to non-linear operations performed on sums of subsets of a neuron’s synaptic inputs.213 Because dendrites are structured as a branching tree, the number of such non-linearities cannot exceed the number of inputs,214 and thus the FLOP/s costs they can impose is limited.215 Feedbacks created by active dendritic spiking could complicate this picture, but the tree structure will still limit communication between branches. Various experts I spoke with were sympathetic to this kind of argument,216 though one was skeptical.217

Here’s a toy illustration of this idea.218 Consider a point neuron model that adds up 1000 synaptic inputs, and then passes them through a non-linearity. To capture the role of dendrites, you might modify this model by adding, say, 10 dendritic subunits, each performing a non-linearity on the sum of 100 synaptic inputs, the outputs of which are summed at the soma and then passed through a final non-linearity (multi-layer approaches in this broad vicinity are fairly common).219

If we budget 1 FLOP per addition operation, and 10 per non-linearity (this is substantial overkill for certain non-linearities, like a ReLU),220 we get the following budgets:

Point neuron model:
Soma: 1000 FLOPs (additions) + 10 FLOPs (non-linearity)
Total: 1010 FLOPs
Sub-unit model:
Dendrites: 10 (subunits) × (100 FLOPs (additions) + 10 FLOPs (non-linearity))
Soma: 10 FLOPs (additions) + 10 FLOPs (non-linearity)
Total: 1120 FLOPs

The totals aren’t that different (in general, the sub-unit model requires 11 additional FLOPs per sub-unit), even if the sub-unit model can do more interesting things. And if the tree-structure caps the number of non-linearities (and hence, sub-units) at the number of inputs, then the maximum increase is a factor of ~11×.221 This story would alter if, for example, subunits could be fully connected, with each receiving all synaptic inputs, or all the outputs from subunits in a previous layer. But this fits poorly with a tree structured physiology.

Note, though, that the main upshot of this argument is that dendritic non-linearities won’t add that much computation relative to a model that budgets 1 FLOP per input connection per time-step. Our budget for synaptic transmission above, however, was based on spikes through synapses per second, not time-steps per synapse per second. In that context, if we assume that dendritic non-linearities need to be computed every time-step, then adding e.g. 100 or 1000 extra dendritic non-linearities per neuron could easily increase our FLOP/s budget by 100 or 1000x (see endnote for an example).222 That said, my impression is that many actual ANN models of dendritic computation use fewer sub-units, and it may be possible to avoid computing firing decisions/dendritic non-linearities every time-step as well – see brief discussion in section 2.1.2.5.

Cortical neurons as deep neural networks

What about evidence for larger FLOP/s costs from dendritic computation? One interesting example is Beniaguev et al. (2020), who found that they needed a very large deep neural network (7 layers, 128 channels per layer) to accurately predict the outputs of a detailed biophysical model of a cortical neuron, once they added conductances from a particular type of receptor (NMDA receptors).223 Without these conductances, they could do it with a much smaller network (a fully connected DNN with 128 hidden units and only one hidden layer), suggesting that it’s the dynamics introduced by NMDA-conductances in particular, as opposed to the behavior of the detailed biophysical model more broadly, that make the task hard.224

This 7-layer network requires a lot of FLOPs: roughly 2e10 FLOP/s per cell.225 Scaled up by 1e11 neurons, this would be ~2e21 FLOP/s overall. And these numbers could yet be too small: perhaps you need greater temporal/spatial resolution, greater prediction accuracy, a more complex biophysical model, etc., not to mention learning and other signaling mechanisms, in order to capture what matters.

I think that this is an interesting example of positive evidence for very high FLOP/s estimates. But I don’t treat it as strong evidence on its own. This is partly out of general caution about updating on single studies (or even a few studies) I haven’t examined in depth, especially in a field as uncertain as neuroscience. But there are also a few more specific ways these numbers could be too high:

• It may be possible to use a smaller network, given a more thorough search. Indeed, the authors suggest that this is likely, and have made data available to facilitate further efforts.226
• They focus on predicting both membrane potential and individual spikes very precisely.
• This is new (and thus far unpublished) work, and I’m not aware of other results of this kind.

The authors also suggest an interestingly concrete way to validate their hypothesis: namely, teach a cortical L5 pyramidal neuron to implement a function that this kind of 7-layer network can implement, such as classifying handwritten digits.227 If biological neurons can perform useful computational tasks thought to require very large neural networks to perform, this would indeed be very strong evidence for capacities exceeding what simple models countenance.228 That said, “X is needed to predict the behavior of Y” does not imply that “Y can do anything X can do” (consider, for example, a supercomputer and a hurricane).

Overall, I think that dendritic computation is probably the largest source of uncertainty about the FLOP/s costs of firing decisions. I find the Beniaguev et al. (2020) results suggestive of possible lurking complexity; but I’m also moved somewhat by the relative simplicity of some common abstract models of dendritic computation, by the tree-structure argument above, and by experts who thought dendrites unlikely to imply a substantial increase in FLOP/s.

##### Crabs, locusts, and other considerations

Here are some other considerations relevant to the FLOP/s costs of firing decisions.

Other experimentally accessible circuits

The retina is not the only circuit where we have (a) some sense of what task it’s performing, and (b) relatively good experimental access. Here are two others I looked at that seem amenable to simplified modeling.

• A collection of ~30 neurons in the decapod crustacean stomach create rhythmic firing patterns that control muscle movements. Plausibly, maintaining these rhythms is the circuit’s high-level task.229 Such rhythms can be modeled well using single-compartment, Hodgkin-Huxley-type neuron models.230 And naively, it seems to me like they could be re-implemented directly without using neuron models at all.231 What’s more, very different biophysical parameters (for example, synapse strengths and intrinsic neuron properties) result in very similar overall network behavior, suggesting that replicating task-performance does not require replicating a single set of such parameters precisely.232 That said, Prof. Eve Marder, an expert on this circuit, noted that the circuit’s biophysical mechanisms function in part to ensure smooth transitions between modes of operation – transitions that most computational models cannot capture.233
• In a circuit involved in locust collision avoidance, low-level biophysical dynamics in the dendrites and cell body of a task-relevant neuron are thought to implement high-level mathematical operations (logarithm, multiplication, addition) that a computational model could replicate directly.234

I expect that further examination of the literature would reveal other examples in this vein.235

Selection effects

Neuroscientific success stories might be subject to selection effects.236 For example, the inference “A, B, and C can be captured with simple models, therefore probably X, Y, and Z can too” is bad if the reason X, Y, and Z haven’t yet been so captured is that they can’t be.

However, other explanations may also be available. For example, it seems plausible to me we’ve had more success in peripheral sensory/motor systems than deeper in the cortex because of differences in the ease with which task-relevant inputs and outputs can be identified, measured, and manipulated, rather than differences in the computation required to run adequate models of neurons in those areas.237 And FLOP/s requirements do not seem to be the major barrier to e.g. C. elegans simulation.238

Evolutionary history

Two experts (one physicist, one neuroscientist) mentioned the evolutionary history of neurons as a reason to think that they don’t implement extremely complex computations. The basic thought here seemed to be something like: (a) neurons early in evolutionary history seem likely to have been doing something very simple (e.g., basic stimulus-response behavior), (b) we should expect evolution to tweak and recombine these relatively simple components, rather than to add a lot of complex computation internal to the cells, and (c) indeed, neurons in the human brain don’t seem that different from neurons in very simple organisms.239 I haven’t looked into this, but it seems like an interesting angle.240

Communication bottlenecks

A number of experts mentioned limitations on the bits that a neuron receives as input and sends as output (limitations imposed by e.g. firing precision, the number of distinguishable synaptic states, etc.) as suggestive of a relatively simple input-output mapping.241

I’m not sure exactly how this argument works (though I discuss one possibility in the communication method section). In theory, very large amounts of computation can be required to map a relatively small number of possible inputs (e.g., the product of two primes, a boolean formula) to a small a number of possible outputs (e.g., the prime factors, a bit indicating whether the formula is satisfiable).242 For example, RSA-240 is ~800 bits (if we assume 1000-10,000 input synapses, each receiving 1 spike/s in 1 of 1000 bins, a neuron would be receiving ~10-100k bits/s),243 but it took ~900 core years on a 2.1 Ghz CPU to factor.244 And the bits that the human brain as a whole receives and outputs may also be quite limited relative to the complexity of its information-processing (Prof. Markus Meister suggested ~10-40 bits per second for various motor outputs).245

Of course, naively, neurons (indeed, brains) don’t seem to be factorizing integers. Indeed, in general, I think this may well be a good argument, and I welcome attempts to make it more explicit and quantified. Suppose, for example, that a neuron receives ~10-100k bits/s and outputs ~10 bits/s. What would this suggest about the FLOP/s required to reproduce the mapping, and why?

Ability to replicate known types of neuron behavior

According to Izhikevich (2004), some neuron models, such as simple integrate-and-fire models, can’t replicate known types of neuron behaviors, some of which (like adaptations in spike frequency over time, and spike delays that depend on the strength of the inputs)246 seem to me plausibly important to task-performance:247

Note, though, that Izhikevich suggests that his own model can capture these behaviors, for 13 FLOPs per ms.

Simplifying the Hodgkin-Huxley model

Some experts argue that the Hodgkin-Huxley model can be simplified:

• Prof. Dong Song noted that the functional impacts of its ion channel dynamics are highly redundant, suggesting that you can replicate the same behavior with fewer equations.248
• Izhikevich (2003) claims that “[His simplified neuron model] consists of only two equations and has only one nonlinear term, i.e., v2. Yet … the difference between it and a whole class of biophysically detailed and accurate Hodgkin–Huxley-type models, including those consisting of enormous number of equations and taking into account all possible information about ionic currents, is just a matter of coordinate change.”249

ANNs and interchangeable non-linearities

Artificial neural networks (ANNs) have led to breakthroughs in AI, and we know they can perform very complex tasks.250 Yet the individual neuron-like units are very simple: they sum weighted inputs, and their “firing decisions” are simple non-linear operations, like a ReLU.251

The success of ANNs is quite compatible with the biological neurons doing something very different. And comparisons between brains and exciting computational paradigms can be over-eager.252 Still, knowing that ANN-like units are useful computational building-blocks makes salient the possibility that biological neurons are useful for similar reasons. Alternative models, including ones that incorporate biophysical complications that ANNs ignore, cannot boast similar practical success.

What’s more, the non-linear operations used in artificial neurons are, at least to some extent, interchangeable.253 That is, instead of a ReLU, you can use e.g., a sigmoid (though different operations have different pros and cons). If we pursue the analogy with firing decisions, this interchangeability might suggest that the detailed dynamics that give rise to spiking are less important than the basic function of passing synaptic inputs through some non-linearity or other.

On a recent podcast, Dr. Matthew Botvinick also mentions a chain of results going back to the 1980s showing that the activity in the units of task-trained deep learning systems bears strong resemblance to the activity of neurons deep in the brain. I discuss a few recent visual cortex results in this vein in Section 3.2, and note a few other recent results in Section 3.3.254 Insofar as a much broader set of results in this vein is available, that seems like relevant evidence as well.

Intuitive usefulness

One of our technical advisors, Dr. Paul Christiano, noted that from a computer science perspective, the Hodgkin-Huxley model just doesn’t look very useful. That is, it’s difficult to describe any function for which (a) this model is a useful computational building block, and (b) its usefulness arises from some property it has that simpler computational building blocks don’t.255 Perhaps something similar could be said of even more detailed biophysical models.

Note, though, advocates of large compute burdens need not argue that actual biophysical models themselves are strictly necessary; rather, they need only argue for the overall complexity of a neuron’s input-output transformation.

Noise bounds

Various experts suggest that noise in the brain may provide an upper bound on the compute required to do what it does.256 However, I’m not sure how to identify this bound, and haven’t tried.

##### Expert opinion and practice

There is no consensus in neuroscience about what models suffice to capture task-relevant neuron behavior.257

A number of experts indicated that in practice, the field’s emphasis is currently on comparatively simple models, rather than on detailed modeling.258 But this evidence is indirect. After all, the central question a neuroscientist needs to ask is not (a) “what model is sufficient, in principle, to replicate task-relevant behavior?”, but rather (b) “what model will best serve the type of neuroscientific understanding I am aiming to advance, given my constraints?”.

Indeed, much discussion of model complexity is practical: it is often said that biophysical models are difficult to compute, fit to data, and understand; that simpler models, while better on these fronts, come at the cost of biological realism; and that the model you need depends on the problem at hand.259 Thus, answers to (a) and (b) can come apart: you can think that ultimately, we’ll need complex models, but that simpler ones are more useful given present constraints; or to that ultimately, simplifications are possible, but detailed modeling is required to identify them.260

Still, some experts answer (a) explicitly. In particular:

• A number of experts I spoke to expected comparatively simple models (e.g., simpler than Hodgkin-Huxley) to be adequate.261 I expect many computational neuroscientists who have formed opinions on the topic (as opposed to remaining agnostic) to share this view.262
• Various experts suggest that some more detailed biophysical models are adequate.263
• In an informal poll of participants at a 2007 workshop on Whole Brain Emulation, the consensus appeared to be that a level of detail somewhere between a “spiking neural network” and the “metabolome” would be adequate (strong selection effects likely influenced who was present).264

A number of other experts I spoke with expressed more uncertainty, agnosticism, and sympathy towards higher end estimates.265 And many (regardless of specific opinion) suggested that views about this topic (including, sometimes, their own) can emerge in part from gut feeling, a desire for one’s own research to be important/tractable, and/or from the tradition and assumptions one was trained in.266

##### Overall FLOP/s for firing decisions

Where does this leave us in terms of overall FLOP/s for firing decisions? Here’s a chart with some examples of possible levels of complexity, scaled up to the brain as a whole:

Figure 10: FLOP/s budgets for different models of neuron firing decisions
Anchor FLOPs Size of timestep FLOP/s for 1e11 neurons
ReLU 1 FLOP per operation267 10 ms268 1e13
Izhikevich spiking neuron model 13 FLOPs per ms269 1 ms270 ~1e15
Single compartment Hodgkin-Huxley model 120 FLOPs per .1 ms271 .1 ms272 ~1e17
Beniaguev et al. (2020) DNN 1e7 FLOPs per ms273 1 ms ~1e21
Hay et al. (2011) detailed L5PC model 1e10 FLOPs per ms?274 ? 1e24?

Even the lower-end numbers here are competitive with the budgets for synaptic transmission above (1e13-1e17 FLOP/s). This might seem surprising, given the difference in synapse and neuron count. But as I noted at the beginning of the section, the budgets for synaptic transmission were based on average firing rates; whereas I’m here assuming that firing decisions must be computed once per time-step (for some given size of time-step).275

This assumption may be mistaken. Dr. Paul Christiano, for example, suggested that it would be possible to accumulate inputs over some set of time-steps, then calculate what the output spike pattern would have been over that period.276 And Sarpeshkar (2010) appears to assume that the FLOP/s he budgets for firing decisions (enough for 1 ms of Hodgkin-Huxley model) need only be used every time the neuron spikes.277 If something like this is true, the numbers would be lower.

Other caveats:

• I’m leaning heavily on the FLOPs estimates in Izhikevich (2004), which I haven’t verified.
• Actual computation burdens for running e.g. a Hodgkin-Huxley model depend on implementation details like platform, programming language, integration method, etc.278
• In at least some conditions, simulations of integrate-and-fire neurons can require very fine grained temporal resolution (e.g., 0.001 ms) to capture various properties of network behavior.279 Temporal resolutions like this would increase the numbers above considerably. However, various other simulations using simplified spiking neuron models, such as the leaky-integrate-and-fire simulations run by Prof. Chris Eliasmith (which actually perform tasks like recognizing numbers and predicting sequences of them), use lower resolutions.280
• The estimate above for Hay et al. (2011) is especially rough.281
• The high end of this chart is not an upper bound on modeling complexity. Biophysical modeling can in principle be arbitrarily detailed.

Overall, my best guess is that the computation required to run single-compartment Hodgkin-Huxley models of every neuron in the brain (1e17 FLOP/S, on the estimate above) is overkill for capturing the task-relevant dimensions of firing decisions. This is centrally because:

• Efforts to predict neuron behavior using simpler models (including simplified models of dendritic computation) appear to have had a decent amount of success (though these results also have many limitations, and I’m not in a great position to evaluate them).
• With the exception of Beniaguev et al. (2020), I don’t see much positive evidence that dendritic computation alters this picture dramatically.
• I find some of the considerations canvassed in Section 2.1.2.3 (other simple circuits; the success of ANNs with simple, interchangeable non-linearities) suggestive; and I think that others I don’t understand very well (e.g., communication bottlenecks, mathematical results showing that the Hodgkin-Huxley equations can be simplified) may well be quite persuasive on further investigation.
• My impression is that a substantial fraction (maybe a majority?) of computational neuroscientists who have formed positive opinions about the topic (as opposed to remaining agnostic) would also think that single-compartment Hodgkin-Huxley is overkill for capturing task-performance (though it may be helpful for other forms of neuroscientific understanding).

Thus, I’ll use 1e17 FLOP/s as a high-end estimate for firing decisions.

The Izhikevich spiking neuron model estimate (1e15 FLOP/s) seems to me like a decent default estimate, as it can capture more behaviors than a simple integrate-and-fire model, for roughly comparable FLOP/s (indeed, Izhikevich seems to argue that it can do anything a Hodgkin-Huxley model can). And if simpler operations (e.g., a ReLU) and/or lower time resolutions are adequate, we’d drop to something like 1e13 FLOP/s, possibly lower. I’ll use 1e13 FLOP/s as a low end, leaving us with an overall range similar to the range for synaptic transmission: 1e13 to 1e17 FLOP/s.

### Learning

Thus far, we have been treating the synaptic weights and firing decision mappings as static over time. In reality, though, experience shapes neural signaling in a manner that improves task performance and stores task-relevant information. I’ll call these changes “learning.”

Some of these may proceed via standard neuron signaling (for example, perhaps firing patterns in networks with static weights could store short-term memories).282 But the budgets thus far already cover this. Here I’ll focus on processes that we haven’t yet covered, but which are thought to be involved in learning. These include:

• Synaptic weights change over time (“synaptic plasticity”). These changes are often divided into categories:
• Short-term plasticity (e.g., changes lasting from hundreds of milliseconds to a few seconds).
• Long-term plasticity (changes lasting longer).283
• The type of synaptic plasticity neurons exhibit can itself change (“meta-plasticity”).
• The electric properties of the neurons (for example, ion channel expression, spike threshold, resting membrane potential) also change (“intrinsic plasticity”).284
• New neurons, synapses, and dendritic spines grow over time, and old ones die.285

Such changes can be influenced by many factors, including pre-synaptic and post-synaptic spiking,286 receptor activity in the post-synaptic dendrite,287 the presence or absence of various neuromodulators,288 interactions with glial cells,289 chemical signals from the post-synaptic neuron to the pre-synaptic neuron,290 and gene expression.291 There is a lot of intricate molecular machinery plausibly involved,292 which we don’t understand well and which can be hard to access experimentally293 (though some recent learning models attempt to incorporate it).294 And other changes in the brain could be relevant as well.295

Of course, many tasks (say, tying your shoes) don’t require much learning, once you know how to do them. And many tasks are over before some of the mechanisms above have had time to have effects, suggesting that such mechanisms can be left out of FLOP/s budgets for those tasks.296

But learning to perform new tasks, sometimes over long timescales, is itself a task that the brain can perform. So a FLOP/s estimate for any task that the brain can perform needs to budget FLOP/s for all forms of learning.

How many FLOP/s? Here are a few considerations.

#### Timescales

Some of the changes involved in learning occur less frequently than spike through synapses. Growing new neurons, synapses, and dendritic spines is an extreme example. At a glance, the number of new neurons per day in adult humans appears to be on the order of hundreds or less;297 and Zuo et al. (2005) report that over two weeks, only 3%-5% of dendritic spines in adult mice were eliminated and formed (though Prof. Erik De Schutter noted that networks of neurons can rewire themselves over tens of minutes).298 Because these events are so comparatively rare, I expect modeling their role in task-performance to be quite cheap relative to e.g. 1e14 spikes through synapses/sec.299 This holds even if the number of FLOPs required per event is very large, which I don’t see strong reason to expect.

Something similar may apply to some other types of changes to e.g. synaptic weights and intrinsic neuron properties:

• Some long-term changes require building new biochemical machinery (receptors, ion channels, etc.), which seems resource-intensive relative to e.g. synaptic transmission (though I don’t have numbers here).300 This suggests limitations on frequency.
• If a given type of change lasts a long time in vivo (and hence, is not “reset” very frequently) or is triggered primarily by relatively rare events (e.g., sustained periods of high-frequency pre-synaptic spiking), this could also suggest such limitations.301
• It seems plausible that some amount of stability is required for long-term information storage.302

More generally, some biochemical mechanisms involved in learning are relatively slow-moving. The signaling cascades triggered by some neuromodulators, for example, are limited by the speed of chemical diffusion, which Koch (1999) suggests extends their timescales to seconds or longer;303 Bhalla (2014) characterizes various types of chemical computation within synapses as occurring on timescales of seconds;304 and Yap and Greenberg (2018) characterize gene transcription taking place over minutes as “rapid.”305 This too might suggest limits on required FLOP/s.

I discuss arguments that appeal to timescales in more detail in Section 2.3. As I note there, I don’t think these arguments are conceptually airtight, but I find them suggestive nonetheless, and I expect them to apply to many processes involved in learning.

That said, the frequency with which a given change occurs does not necessarily limit the frequency with which biophysical variables involved in the process need to be updated, or decisions made about what changes to implement as a result.306 What’s more, some forms of synaptic plasticity occur on short timescales, reflecting rapid changes in e.g. calcium or neurotransmitter in a synapse;307 and Bhalla (2014) notes that spike-timing dependent plasticity “requires sharp temporal discrimination of the order of a few milliseconds” (p. 32).

#### Existing models

There is no consensus model for how the brain learns,308 and the training required to create state of the art AI systems seems in various ways comparatively inefficient.309 There is debate over comparisons with learning algorithms like backpropagation310 (along with meta-debate about whether this debate is meaningful or worthwhile).311

Still, different models can at least serve as examples of possible FLOP/s costs. Here are a few that came up in my research.

Figure 11: Some example learning models
Learning model Description FLOP/s costs Expert opinion
Hebbian rules Classic set of models. A synapse strengthens or weakens as a function of pre-synaptic spiking and post-synaptic spiking, possibly together with some sort of external modulation/reward.312 3-5 FLOPs per synaptic update?313 Prof. Anthony Zador expected the general outlines to be correct.314 Prof. Chris Eliasmith uses a variant in his models.315
Benna and Fusi (2016) Models synapses as a dynamical system of variables interacting on multiple timescales. May help resolve the “stability-plasticity dilemma,” on which overly plastic synapses are too easily overwritten, but overly rigid synapses are unable to learn. May also help with online learning. ~2-30x the FLOPs to run a model with one parameter per synapse? (very uncertain)316 Some experts argue that shifting to synaptic models of this kind, involving dynamical interactions, is both theoretically necessary and biologically plausible.317
First order gradient descent methods Use slope of the loss function to minimize the loss.318 Widespread use in machine learning. Contentious debate about biological plausibility. ~2× a static network. The learning step is basically a backwards pass through the network, and going forward and backward come at roughly the same cost.319 Prof. Konrad Kording, Prof. Barak Pearlmutter, and Prof. Blake Richards favored estimates based on this anchor/in this range of FLOP/s costs.320
Second order gradient descent methods Take into account not just the slope of the loss function, but also the curvature. Arguably better than gradient descent methods, but require more compute, so used more rarely.321 Large. Compute per learning step scales as a polynomial with the number of neurons and synapses in a network.322 Dr. Paul Christiano thought it very implausible that the brain implements a rule of this kind.323 Dr. Adam Marblestone had not seen any proposals in this vein.324
Node-perturbation algorithms Involves keeping/consolidating random changes to the network that result in reward, and getting rid of changes that result in punishment. As the size of a network grows, these take longer to converge than first-order gradient methods.325 <2× a static network (e.g., less than first-order gradient descent methods).326 Prof. Blake Richards thought that humans learn with less data than this kind of algorithm would require.327

Caveats:

• This is far from an exhaustive list.328
• The brain may be learning in a manner quite dissimilar from any known learning models. After all, it succeeds in learning in ways we can’t replicate with artificial systems.
• I haven’t investigated these models much: the text and estimates above are based primarily on comments from experts (see endnotes for citations). With more time and expertise, it seems fairly straightforward to generate better FLOP/s estimates.
• Synaptic weights are often treated as the core learned parameters in the brain,329 but alternative views are available. For example, Prof. Konrad Kording suggested that the brain could be optimizing ion channels as well (there are considerably more ion channels than synapses).330 Thus, the factor increase for learning need not be relative to a static model based on synapses.
• As noted above, some of what we think of as learning and memory may be implemented via standard neuron signaling, rather than via modifications to e.g. synaptic weights/firing decisions.

With that said, a number of these examples seem to suggest relatively small factor increases for learning, relative to some static baseline (though what that baseline should be is a further question). Second-order gradient methods would be more than this, but I have yet to hear anyone argue that the brain uses these, or propose a biological implementation. And node perturbation would be less (though this may require more data than humans use).

#### Energy costs

If we think that FLOP/s costs correlate with energy expenditure in the brain, we might be able to estimate the FLOP/s costs for learning via the energy spent on it. For example, Lennie (2003) estimates that >50% of the total energy in the neocortex goes to processes involved in standard neuron signaling – namely, maintaining resting potentials in neurons (28%), reversing Na+ and K+ fluxes from spikes (13%), and spiking itself (13%).331 That would leave <50% for (a) other learning process beyond this and (b) everything else (maintaining glial resting potentials is another 10%). Very naively, this might suggest less than a 2× factor for learning, relative to standard neuron signaling.

Should we expect FLOP/s costs to correlate with energy expenditure? Generally speaking, larger amounts of information-processing take more energy, so the thought seems at least suggestive (e.g., it’s somewhat surprising if the part of your computer doing 99% of the information-processing is using less than half the energy).332 In the context of biophysical modeling, though, it’s less obvious, as depending on the level of detail in question, modeling systems that use very little energy can be very FLOP/s intensive.

#### Expert opinion

A number of experts were sympathetic to FLOP/s budgets for learning in the range of 1-100 FLOPs per spike through synapse.333 Some of this sympathy was based on using (a) Hebbian models, or (b) first-order gradient descent models as an anchor.

Sarpeshkar (2010) budgets at least 10 FLOPs per spike through synapse for synaptic learning.334 Other experts expressed agnosticism and/or openness to much higher numbers;335 and one (Prof. Konrad Kording) argued for estimates based on ion-channel plasticity, rather than synaptic plasticity.336

#### Overall FLOP/s for learning

Of the many uncertainties afflicting the mechanistic method, the FLOP/s required to capture learning seems to me like one of the largest. Still, based on the timescales, algorithmic anchors, energy costs, and expert opinions just discussed, my best guess is that learning does not push us outside the range already budgeted for synaptic transmission: e.g., 1-100 FLOPs per spike through synapse.

• Learning might well be in the noise relative to synaptic transmission, due to the timescales involved.
• 1-10 FLOPs per spike through synapse would cover various estimates for short-term synaptic plasticity and Hebbian plasticity; along with factors of 2× or so (à la first order gradient descent anchors, or the run-time slow-down in Kaplanis et al. (2018)) on top of lower-end synaptic transmission estimates.
• 100 FLOPs per spike through synapse would cover the higher-end Benna-Fusi estimate above (though this was very loose), as well as some cushion for other complexities.

To me, the most salient route to higher numbers uses something other than spikes through synapses as a baseline. For example, if we used timesteps per second at synapses instead, and 1 ms timesteps, then X FLOPs per timestep per synapse for learning would imply X × 1e17-1e18 FLOP/s (assuming 1e14-15 synapses). Treating learning costs as scaling with ion channel dynamics (à la Prof. Konrad Kording’s suggestion), or as a multiplier on higher-end standard neuron signaling estimates, would also yield higher numbers.

I could also imagine being persuaded by arguments of roughly the form: “A, B, and C simple models of learning lead to X theoretical problems (e.g., catastrophic forgetting), which D more complex model solves in a biologically plausible way.” Such an argument motivates the model in Benna and Fusi (2016), which boasts some actual usefulness to task-performance to boot (e.g. Kaplanis et al. (2018)). There may be other models with similar credentials, but higher FLOP/s costs.

I don’t, though, see our ignorance about how the brain learns as a strong positive reason, just on their own, to think larger budgets are required. It’s true that we don’t know enough to rule out such requirements. But “we can’t rule out X” does not imply “X should be our best guess.”

### Other signaling mechanisms

Let’s turn to other signaling mechanisms in the brain. There are a variety. They tend to receive less attention than standard neuron signaling, but some clearly play a role in task-performance, and others might.

Our question, though, is not whether these mechanisms matter. Our question is whether they meaningfully increase a FLOP/s budget that already covers standard neuron signaling and learning.337

As a preview: my best guess is that they don’t. This is mostly because:

1. My impression is that most experts who have formed opinions on the topic (as opposed to remaining agnostic) do not expect these mechanisms to account for the bulk of the brain’s information-processing, even if they play an important role.338
2. Relative to standard neuron signaling, each of the mechanisms I consider is some combination of (a) slower, (b) less spatially-precise, (c) less common in the brain (or, not substantially more common), or (d) less clearly relevant to task-performance.

But of course, familiar caveats apply: there’s a lot we don’t know, experts might be wrong (and/or may not have given this issue much attention), and the arguments aren’t conclusive.

Arguments related to (a)-(d) will come up a few times in this section, so it’s worth a few general comments about them up front.

Speed

If a signaling mechanism X involves slower-moving elements, or processes that take longer to have effects, than another mechanism Y, does this suggest a lower FLOP/s budget for X, relative to Y? Heuristically, and other things equal: yes, at least to my mind. That is, naively, it seems harder to perform lots of complex, useful information-processing per second using slower elements/processes (computers using such elements, for example, are less powerful). And various experts seemed to take considerations in this vein quite seriously.339

That said, other things may not be equal. X signals might be sent more frequently, as a result of more complex decision-making, with more complex effects, etc.340 What’s more, the details of actually measuring and modeling different timescales in the brain may complicate arguments that appeal to them. For example, Prof. Eve Marder noted that traditional views about timescales separations in neuroscience emerge in part from experimental and computational constraints: in reality, slow processes and fast processes interact.341

It’s also generally worth distinguishing between different lengths of time that can be relevant to a given signaling process, including:

• How long it takes to trigger the sending of a signal X.
• How long it takes for a signal X to reach its target Y.
• How long it takes for X’s reaching Y to have effect Z.
• How frequently signals X are sent.
• How long effect Z can last.
• How long effect Z does in fact last in vivo.

These can feed into different arguments in different ways. I’ll generally focus on the first three.

Spatial precision

If a signaling mechanism X is less spatially precise than another mechanism Y (e.g., signals arise from the combined activities of many cells, and/or affect groups of cells, rather than being targeted at individual cells), does this suggest lower FLOP/s budgets for X, relative to Y? Again: heuristically, and other things equal, I think it does. That is, naively, units that can send and receive individualized messages seem to me better equipped to implement more complex information-processing per unit volume. And various experts took spatial precision as an important indicator of FLOP/s burdens as well.342 Again, though, there is no conceptual necessity here: X might nevertheless be very complex, widespread, etc. relative to Y.

Number/frequency

If X is less common than Y, or happens less frequently, this seems to me a fairly straightforward pro tanto reason to budget fewer FLOP/s for it. I’ll treat it as such, even though clearly, it’s no guarantee.

The central role of standard neuron signaling in task-performance is well established. For many of these alternative signaling mechanisms, though, the case is weaker. Showing that you can make something can happen in a petri dish, for example, is different from showing that it happens in vivo and matters to task-performance (let alone that it implies a larger FLOP/s budget than standard neuron signaling). Of course, in some cases, if something did happen in vivo and matter to task-performance, we couldn’t easily tell. But I won’t, on these grounds, assume that every candidate for such a role plays it.

Let’s look at the mechanisms themselves.

#### Other chemical signals

The brain employs many chemical signals other than the neurotransmitters involved in standard neuron signaling. For example:

• Neurons release larger molecules known as neuropeptides, which diffuse through the space between cells.343
• Neurons produce gases like nitric oxide and carbon monoxide, as well as lipids known as endocannabinoids, both of which can pass directly through the cell membrane.344

Chemicals that neurons release that regulate the activity of groups of neurons (or other cells) are known as neuromodulators.345

Chemical signals other than classical neurotransmitters are very common in the brain,346 and very clearly involved in task performance.347 For example, they can alter the input-output function of individual neurons and neural circuits.348

However, some considerations suggest limited FLOP/s budgets, relative to standard neuron signaling:

• Speed: Signals that travel through the extracellular space are limited by the speed of chemical diffusion, and some travel distances much longer than a 20 nm synaptic cleft.349 What’s more, nearly all neuropeptides act via metabotropic receptors, which take longer to have effects on a cell than the ionotropic receptors involved in standard neuron signaling.350
• Spatial precision: Some (maybe most?) of these chemical signals act on groups of cells. As Leng and Ludwig (2008) put it: “peptides are public announcements … they are messages not from one cell to another, but from one population of neurones to another.”351
• Frequency: Neuropeptides are released less frequently than classical neurotransmitters. For example, Leng and Ludwig (2008) suggest that the release of a vesicle containing neuropeptide requires “several hundred spikes,” and that oxytocin is released at a rate of “1 vesicle per cell every few seconds.”352 This may be partly due to resource constraints (neuropeptides, unlike classic neurotransmitters, are not recycled).353
• Because neuromodulators play a key role in plasticity, some of their contributions may already fall under the budget for learning.

This is a coarse-grained picture of a very diverse set of chemical signals, some of which may not be so e.g. slow, imprecise, or infrequent. Still, a number of experts treat these properties as reasons to think that the FLOP/s for chemical signaling beyond standard neuron signaling would not add much to the budget.354

#### Glia

Neurons are not the only brain cells. Non-neuron cells known as glia have traditionally been thought to mostly act to support brain function, but there is evidence that they can play a role in information-processing as well.355

This evidence appears to be strongest with respect to astrocytes, a star-shaped type of glial cell that extend thin arms (“processes”) to enfold blood vessels and synapses.

• Mu et al. (2019) suggest that zebra fish astrocytes “perform a computation critical for behavior: they accumulate evidence that current actions are ineffective and consequently drive changes in behavioral states.”356
• Astrocytes exhibit a variety of receptors, activation of which leads to increases in the concentration of calcium within the cell and consequently the release of transmitters.357
• Changes in calcium concentrations can propagate across networks of astrocytes (a calcium “wave”) enabling a form of signaling over longer-distances.358 Sodium dynamics appear to play a signaling role as well.359
• Astrocytes can also signal to neurons by influencing concentrations of ions or neurotransmitters in space between cells.360 They can regulate neuron activity, a variety of mechanisms exist via which they can influence short-term plasticity, and they are involved in both long-term plasticity and in the development of new synapses.361
• Human astrocytes also appear to be larger, and to exhibit more processes, than those of rodents, which has led to speculation that they play a role in explaining the human brain’s processing power.362

Other glia may engage in signaling as well. For example:

• NG2 protein-expressing oligodendrocyte progenitor cells can receive synaptic input from neurons, form action potentials, and regulate synaptic transmission between neurons.363
• Glial cells involved in the creation of myelin (the insulated sheath that surrounds axons) can detect and respond to axonal activity.364

Would FLOP/s for the role of glia in task-performance meaningfully increase our budget? Here are some considerations:

• Speed: Astrocytes can respond to neuronal events within hundreds of milliseconds,365 and they can detect individual synaptic events.366 However, the timescales of other astrocyte calcium dynamics are thought to be slower (on the order of seconds or more), and some effects require sustained stimulation.367
• Spatial resolution: Previous work assumed that astrocyte calcium signaling could not be spatially localized to e.g. a specific cellular compartment, but this appears to be incorrect.368
• Number: The best counting methods available suggest that the ratio of glia to neurons in the brain is roughly 1:1 (it was previously thought to be 10:1, but this appears to be incorrect).369 This ratio varies across regions of the brain (in the cerebral cortex, it’s about 3:1).370 Astrocytes appear to be about 20-40% of glia (though these numbers may be questionable);371 and NG2 protein-expressing oligodendrocyte progenitor cells discussed above are only 2-8% of the total cells in the cortex.372 If the average FLOP/s cost per glial cell were the same as the average per neuron, this would likely less than double our budget.373 That said, astrocytes may have more connections to other cells, on average, than neurons.374
• Energy costs: Neurons consume the majority of the brain’s energy. Zhu et al. (2012) estimate that “a non-neuronal cell only utilizes approximately 3% of that [energy] used by a neuron in the human brain” – a ratio which they take to suggest that neurons account for 96% of the energy expenditure in human cortical grey matter, and 68% in white matter.375 Attwell and Laughlin (2001) also predict a highly lopsided distribution of signaling-related energy consumption between neurons and glia in grey matter – a distribution supported by the observed distribution of mitochondria they suggest is found in Wong-Riley (1989) (see figure below). If glial cells were doing more information-processing than neurons, they would have to be doing it using much less energy – a situation in which, naively, it would appear metabolically optimal to have more glial cells than neurons. To me, the fact that neurons receive so much more of a precious resource suggests that they are the more central signaling element.376

Overall, while some experts are skeptical of the importance of glia to information-processing, the evidence that they play at least some role seems to me fairly strong.378 How central of a role, though, is a further question, and the total number of glial cells, together with their limited energy consumption relative to neurons, does not, to me, initially suggest that capturing this role would require substantially more FLOP/s than capturing standard neuron signaling and learning.

#### Electrical synapses

In addition to the chemical synapses involved in standard neuron signaling, neurons (and other cells) also form electrical synapses – that is, connections that allow ions and other molecules to flow directly from one cell into another. The channels mediating these connections are known as gap junctions.

These have different properties than chemical synapses. In particular:

• Electrical synapses are faster, passing signals in a fraction of a millisecond.379
• Electrical synapses can be bi-directional, allowing each cell to influence the other.380
• Electrical synapses allow graded transmission of sub-threshold electrical signals.381

My impression is that electrical synapses receive much less attention in neuroscience than chemical synapses. This may be because they are thought to be some combination of:

• Much less common.382
• More limited in the behavior they can produce (chemical synapses, for example, can amplify pre-synaptic signals).383
• Involved in synchronization between neurons, or global oscillation, that does not imply complex information-processing.384
• Amenable to very simple modeling.385

Still, electrical synapses can play a role in task-performance,386 and one expert suggested that they could create computationally expensive non-linear dynamics.387 What’s more, if they are sufficiently fast, or require sufficiently frequent updates, this could compensate for their low numbers. For example, one expert suggested that you can model gap junctions as synapses that update every timestep.388 But if chemical synapses only receive spikes, and hence update, ~once per second, and we use 1 ms timesteps, you’d need to have ~1000x fewer gap junctions in order for their updates not to dominate.

Overall, my best guess is that incorporating electrical synapses would not substantially increase our FLOP/s budget, but this is centrally based on a sense that experts treat their role in information-processing as relatively minor.

#### Ephaptic effects

Neuron activity creates local electric fields that can have effects on other neurons. These are known as ephaptic effects. We know that these effects can occur in vitro (see especially Chiang et al. (2019))389 and entrain action potential firing,390 and Chiang et al. (2019) suggest that they may explain slow oscillations of neural activity observed in vivo.391

A recent paper, though, suggests that the question of whether they have any functional relevance in vivo remains quite open,392 and one expert thought them unlikely to be important to task-performance.393

One reason for doubt is that the effects on neuron membrane potential appear to be fairly small (e.g., <0.5 mV, compared with the ~15 mV gap between resting membrane potential and the threshold for firing),394 and may be drowned out by noise artificially lacking in vitro.395

Even if they were task-relevant, though, they would be spatially imprecise – arising from, and exerting effects on, the activity of groups of neurons, rather than on individual cells. Two experts took this as reason to think their role in task-performance would not be computationally expensive to capture.396 That said, actually modeling electric fields seems plausibly quite FLOP/s-intensive.397

#### Other forms of axon signaling

Action potentials are traditionally thought of as binary choices – a neuron fires, or it doesn’t – induced by changes to somatic membrane potential, and synaptic transmission as a product of this binary choice.398 But in some contexts, this is too simple. For example:

• The waveform of an action potential (that is, its amplitude and duration) can vary in a way that affects neurotransmitter release.399
• Variations in the membrane potential that occur below the threshold of firing (“subthreshold” variations) can also influence synaptic transmission.400
• Certain neurons – for example, neurons in early sensory systems,401 and neurons in invertebrates402 – also release neurotransmitter continually, in amounts that depend on non-spike changes to pre-synaptic membrane potential.403
• Some in vitro evidence suggests that action potentials can arise in axons without input from the soma or dendrites.404

Do these imply substantial increases to FLOP/s budgets? Most of the studies I looked at seemed to be more in the vein of “here is an effect that can be created in vitro” than “here is a widespread effect relevant to in vivo task-performance,” but I only looked into this very briefly, the possible mechanisms/complexities are diverse, and evidence of the latter type is rare regardless.

Some effects (though not all)405 also required sustained stimulation (e.g., “hundreds of spikes over several minutes,”406 or “100 ms to several seconds of somatic depolarization”407); and the range of neurons that can support axon signaling via sub-threshold membrane potential fluctuations also appears somewhat unclear, as the impact of such fluctuations is limited by the voltage decay along the axon.408

Overall, though, I don’t feel very informed or clear about this one. As with electrical synapses, I think the central consideration for me is that the field doesn’t seem to treat it as central.

#### Blood flow

Blood flow in the brain correlates with neural activity (this is why fMRI works). This is often explained via the blood’s role in maintaining brain function (e.g., supplying energy, removing waste, regulating temperature).409 Moore and Cao (2008)), though, suggest that blood flow could play an information-processing role as well – for example, by delivering diffusible messengers like nitric oxide, altering the shape of neuron membranes, modulating synaptic transmission by changing brain temperatures, and interacting with neurons indirectly via astrocytes.410 The timescales of activity-dependent changes in blood flow are on the order of hundreds of milliseconds (the effects of such changes often persist after a stimulus has ended, but Moore and Cao believe this is consistent with their hypothesis).411

My impression, though, is that most experts don’t think that blood flow plays a very direct or central role in information-processing.412 And the spatial resolution appears fairly coarse regardless: Moore and Cao (2008) suggest resolution at the level of a cortical column (a group of neurons413), or an olfactory glomerulus (a cluster of connections between cells).414

#### Overall FLOP/s for other signaling mechanisms

Here is a chart summarizing some of the considerations just canvassed (see the actual sections for citations).

Mechanism Description Speed Spatial precision Number/frequency Evidence for task-relevance
Other chemical signals Chemical signals other than classical neurotransmitters. Includes neuropeptides, gases like nitrous oxide, endocannabinoids, and others. Limited by the speed of chemical diffusion, and by the timescales of metabotropic receptors. Imprecise. Affect groups of cells by diffusing through the extracellular space and/or through cell membranes, rather than via synapses. Very common. However, some signal broadcasts are fairly rare, and may take ~400 spikes to trigger. Strong. Can alter circuit dynamics and neuron input-output functions, role in synaptic plasticity.
Glia Non-neuron cells traditionally thought to play a supporting role in the brain, but some of which may be more directly involved in task-performance. Some local calcium responses within ~100 ms; other calcium signaling on timescales of seconds or longer. Can respond locally to individual synaptic events. ~1:1 ratio with neurons (not 100:1). Astrocytes (the most clearly task-relevant type of glial cell) are only 20-40% of glia. Moderate. Role in zebrafish behavior. Plausible role in plasticity, synaptic transmission, and elsewhere. However, glia have a much smaller energy budget than neurons.
Electrical synapses Connections between cells that allow ions and other molecules to flow directly from one to the other. Very fast. Can pass signals in a fraction of a millisecond. Precise. Signals are passed between two specific cells. But may function to synchronize groups of neurons. Thought to be less common than chemical synapses (but may be passing signals more continuously, and/or require more frequent updates?). Can play a role, but thought to be less important than chemical synapses? More limited range of signaling behaviors.
Ephaptic effects Local electrical fields that can impact neighboring neurons. ? Some oscillations that ephaptic effects could explain are slow-moving. Unsure about speed of lower-level effects. Imprecise. Arises from activity of many cells, effects not targeted to specific cells. ? Weak? Small effects on membrane potential possibly swamped by noise in vivo.
Other forms of axon signaling Processes in a neuron other than a binary firing decision that impact synaptic transmission. ? Some effects required sustained stimulation (minutes of spiking, 100 ms to seconds of depolarization). Others arose more quickly (15-50 ms of hyperpolarization). Precise, proceeds via axons/individual synapses. Unclear what range of neurons can support some of the effects (e.g., sub-threshold influences on synaptic transmission). Some effects relevant in at least some species/contexts. Other evidence mostly from in vitro studies?
Blood flow Some hypothesize that blood flow in the brain is involved in information-processing. Responses within hundreds of ms, which persist after stimulus has ended. Imprecise. At the level of a cortical column, or a cluster of connections between cells. ? Weak. Widely thought to be epiphenomenal.

Figure 13: Factors relevant to FLOP/s budgets for other signaling mechanisms in the brain.

Obviously, my investigations were cursory, and there is a lot of room for uncertainty in each case. What’s more, the list is far from exhaustive,415 and other mechanisms may await discovery.416

Still, as mentioned earlier, my best guess is that capturing the role of other signaling mechanisms (known and unknown) in task-performance does not require substantially more FLOP/s than capturing standard neuron signaling and learning. This guess is primarily grounded in a sense that computational neuroscientists generally treat standard neuron signaling (and the plasticity thereof) as the primary vehicle of information-processing in the brain, and other mechanisms as secondary.417 An initial look at the speed, spatial precision, prevalence, and task-relevance of the most salient of these mechanisms seems compatible with such a stance, so I’m inclined to defer to it, despite the possibility that it emerges primarily from outdated assumptions and/or experimental limitations, rather than good evidence.

### Overall mechanistic method FLOP/s

Here are the main numbers we’ve discussed thus far:

Standard neuron signaling: ~1e13-1e17 FLOP/s
Synaptic transmission: 1e13-1e17 FLOP/s
Spikes through synapse per second: 1e13-1e15
FLOPs per spike through synapse:
Low: 1 (one addition and/or multiply operation, reflecting impact on post-synaptic membrane     potential)
High: 100 (covers 40 FLOPs for synaptic conductances, plus cushion for other complexities)
Firing decisions: 1e13-1e17 FLOP/s
Number of neurons: 1e11
FLOP/s per neuron:
Low: 100 (ReLU, 10 ms timesteps)
Middle: 10,000 (Izhikevich model, 1 ms timesteps)
High: 1,000,000 (single compartment Hodgkin-Huxley model, 0.1 ms timesteps)
Learning: <1e13 - 1e17 FLOP/s
Spikes through synapse per second: 1e13-1e15
FLOPs per spike through synapse:
Low: <1 (possibly due to slow timescales)
Middle: 1-10 (covers various learning models – Hebbian plasticity, first-order gradient     methods, possibly Benna and Fusi (2016) – and expert estimates, relative to low end baselines)
High: 100 (covers those models with more cushion/relative to higher baselines).
Other signaling mechanisms: do not meaningfully increase the estimates above.

Overall range: ~1e13-1e17 FLOP/s418

To be clear: the choices of “low” and “high” here are neither principled nor fully independent, and I’ve rounded aggressively.419 Indeed, another, possibly more accurate way to summarize the estimate might be:

“There are roughly 1e14-1e15 synapses in the brain, receiving spikes about 0.1-1 times a second. A simple estimate budgets 1 FLOP per spike through synapse, and two extra orders of magnitude would cover some complexities related to synaptic transmission, as well as some models of learning. This suggests something like 1e13-1e17 FLOP/s. You’d also need to cover firing decisions, but various simple neuron models, scaled up by 1e11 neurons, fall into this range as well, and the high end (1e17 FLOP/s) would cover a level of modeling detail that I expect many computational neuroscientists to think unnecessary (single compartment Hodgkin-Huxley). Accounting for the role of other signaling mechanisms probably doesn’t make much of a difference to these numbers.”

That is, this is meant to be a plausible ballpark, covering various types of models that seem plausibly adequate to me.

#### Too low?

Here are some ways it could be too low:

• The choice to budget FLOP/s for synaptic transmission and learning based on spikes through synapses, rather than timesteps at synapses, is doing a lot of work. If we instead budgeted based on timesteps, and used something like 1 ms resolution, we’d start with 1e17-1e18 FLOP/s as a baseline (1 FLOP per timestep per synapse). Finer temporal resolutions, and larger numbers of FLOPs per time-step, would drive these numbers higher.
• Some neural processes are extremely temporally precise. For example, neurons in the owl auditory system can detect auditory stimulus timing at a precision of less than ten microseconds.420 These cases may be sufficiently rare, or require combining a sufficient number of less-precise inputs, that they wouldn’t make much of a difference to the overall budget. However, if they are indicative of a need for much finer temporal precision across the board, they could imply large increases.
• Dendritic computation might imply much larger FLOP/s budgets than single-compartment Hodgkin-Huxley models.421 Results like Beniaguev et al. (2020) (~1e10 FLOP/s per neuron), discussed above, seem like some initial evidence for this.
• Some CNN/RNN models used to predict the activity of retinal neurons are very FLOP/s intensive as well. I discuss this in Section 3.1.
• Complex molecular machinery at synapses or inside neurons might implement learning algorithms that would require more than 100 FLOPs per spike through synapse to replicate.422 And I am intrigued by theoretical results showing that various models of synaptic plasticity lead to problems like catastrophic forgetting, and that introducing larger numbers of dynamical variables at synapses might help with online learning.423
• One or more of the other signaling mechanisms in the brain might introduce substantially additional FLOP/s burdens (neuromodulation and glia seem like prominent candidates, though I feel most uncertainty about the specific arguments re: gap junctions and alternative forms of axon signaling).
• Processes in the brain that take place over longer timescales involve interactions between many biophysical variables in the brain that are not normally included in e.g. simple models of spiking. The length of these timescales might limit the compute burdens such interactions imply, but if not, updating all relevant variables at a frequency similar to the most frequently updated variables could imply much larger compute burdens.424
• Some of the basic parameters I’ve used could be too low. The average spike rate might be more like 10 Hz than 0.1-1 Hz (I really doubt 100 Hz); synapse count might be >1e15; Hodgkin-Huxley models might require more FLOP/s than Izhikevich (2004) budgets, etc. Indeed, I’ve been surprised at how uncertain many very basic facts about the brain appear to be, and how wrong previous widely-cited numbers have been (for example, a 10:1 ratio between glia and neurons was widely accepted until it was corrected to roughly 1:1).425

There are also broader considerations that could incline us towards higher numbers by default, and/or skepticism of arguments in favor of the adequacy of simple models:

• We might expect evolution to take advantage of every possible mechanism and opportunity available for increasing the speed, efficiency, and sophistication of its information-processing.426 Some forms of computation in biological systems, for example, appear to be extremely energy efficient.427 Indeed, I think that further examination of the sophistication of biological computation in other contexts could well shift my default expectations about the brain’s sophistication substantially (though I have tried to incorporate hazy forecasts in this respect into my current overall view).428
• It seems possible that the task-relevant causal-structure of the brain’s biology is just intrinsically ill-suited to replication using digital computer hardware, even once you allow for whatever computational simplifications are available (though neuromorphic hardware might do better). For example, the brain may draw on analog physical primitives,429 continuous (or very fine-grained) temporal dynamics,430 and/or complex biochemical interactions that are cheap for the brain, but very expensive to simulate.431
• Limitations on tools and available data plausibly do much to explain the concepts and assumptions most prominent in neuroscience. As these limitations loosen, we may identify much more complex forms of information-processing than the field currently focuses on.432 Indeed, it might be possible to extrapolate from trends in this vein, either in neuroscience or across biology more broadly.433
• Various experts mentioned track-records of over-optimism about the ease of progress in biology, including via computational modeling;434 overly-aggressive claims in favor of particular neuroscientific research programs;435 and over-eagerness to think of the brain via in terms of the currently-most-trendy computational/technological paradigms.436 To the extent such track records exist, they could inform skepticism about arguments and expert opinions in a similar reference class (though on their own, they seem like only very indirect support for very large FLOP/s requirements, as many other explanations of such track records are available).

And of course, more basic paradigm mistakes are possible as well.437

This is a long list of routes to higher numbers; perhaps, then, we might expect at least one of them to track the truth. However:

• Some particular routes are correlated: for example, worlds in which the brain can implement very sophisticated, un-simplifiable computation at synapses seem more likely to be ones in which it can implement such computation within dendrites as well.438
• My vague impression is that experts tend to be inclined towards simplification vs. complexity across the board, rather than in specific patterns that differ widely. If this is true, then the reliability of the assumptions and methods these experts employ might be a source of broader correlations.
• Some of these routes are counterbalanced by corresponding routes to lower numbers (e.g., basic parameters could be too high as well as too low; relevant timescales could be more coarse-grained rather than more fine-grained; etc). And there are more general routes to lower numbers as well, which would apply even if some of the considerations surveyed above are sound (see next section).

#### Too high?

Here are a number of ways 1e13-1e17 FLOP/s might be overkill (I’ll focus, here, on ways that are actively suggested by examination of the brain’s mechanisms, rather than on the generic consideration that for any given way of performing a task, there may be a more efficient way).

##### Neuron populations and manifolds

The framework above focuses on individual neurons and synapses. But this could be too fine-grained. For example, various popular models in neuroscience involve averaging over groups of neurons, and/or treating them as redundant representations of high-level variables.439

Indeed, in vivo recording shows that the dimensionality of the activity of a network of neurons is much smaller than the number of neurons themselves (Wärnberg and Kumar (2017) suggest a subspace spanned by ~10 variables, for local networks consisting of thousands of neurons).440 This kind of low-dimensional subspace is known as a “neural manifold.”441

Some of this redundancy may be about noise: neurons are unreliable elements, so representing high-level variables using groups of them may be more robust.442 Digital computers, though, are noise-free.

In general, the possibility of averaging over or summarizing groups of neurons suggests smaller budgets than the estimates above – possibly much smaller. If I had more time for this project, this would be on the top of my list for further investigation.

##### Transistors and emulation costs

If we imagine applying the mechanistic method to a digital computer we don’t understand, we plausibly end up estimating the FLOP/s required to model the activity of very low-level components: e.g. transistors, logic gates, etc (or worse, to simulate low-level physical processes within transistors). This is much more than the FLOP/s the computer can actually perform.

For example: a V100 has about 2e10 transistors, and a clock speed of ~1e9 Hz.443 A naive mechanistic method estimate for a V100 then, might budget 1 FLOP per clock-tick per transistor: 2e19 FLOP/s. But the chip’s actual computational capacity is ~1e14 FLOP/s – a factor of 2e5 less.

The costs of emulating different computer systems at different levels of detail may also be instructive here. For example, one attempt to simulate a 6502 microprocessor (original clock speed of ~1 Mhz) at the transistor level managed to run the simulated chip at 1 Khz using a computer running at ~1 GHz, suggesting a factor of ~1e6 slow-down.444

Of course, there is no easy mapping between computer components and brain components; and there are components in the brain at lower-levels than neurons (e.g., ion channels, proteins, etc). Still, applying the mechanistic method to digital computers suggests that when we don’t know how the system works, there is no guarantee that we land on right level of abstraction, and hence that estimates based on counting synapses, spikes, etc. could easily be overkill relative to the FLOP/s requirements of the tasks the brain can actually perform (I discuss this issue more in the appendix).

How much overkill is harder to say, at least using the mechanistic method alone: absent knowledge of how a V100 processes information, it’s not clear to me how to modify the mechanistic method to arrive at 1e14 FLOP/s rather than 2e19. Other methods might do better.

Note, though, that applying the mechanistic method without a clear understanding of whether models at the relevant level of abstraction could replicate task-performance at all could easily be “underkill” as well.

##### Do we need the whole brain?

Do we need the whole brain? For some tasks, no. People with parts of their brains missing/removed can still do various things.

A dramatic example is the cerebellum, which contains ~69 billion neurons – ~80% of the neurons in the brain as a whole.445 Some people (a very small number) don’t have cerebellums. Yet there are reports that in some cases, their intelligence is affected only mildly, if at all (though motor control can also be damaged, and some cognitive impairment can be severe).446

Does this mean we can reduce our FLOP/s budget by 80%? I’m skeptical. For one thing, while the cerebellum accounts for a large percentage of the brain’s neurons, it appears to account for a much smaller percentage of other things, including volume (~10%),447 mass (~10%),448 energy consumption (<10%),449 and maybe synapses (and synaptic activity dominates many versions of the estimates above).450

More importantly, though, we’re looking for FLOP/s estimates that apply to the full range of tasks that the brain can perform, and it seems very plausible to me that some of these tasks (neurosurgery? calligraphy?) will rely crucially on the cerebellum. Indeed, the various impairments generally suffered by patients without cerebellums seem suggestive of this.

This last consideration applies across the board, including to other cases in which various types of cognitive function persist in the face of missing parts of the brain,451 neuron/synapse loss,452 etc. That is, while I expect it to be true of many tasks (perhaps even tasks important to AI developers, like natural language processing, scientific reasoning, social modeling, etc.) that you don’t need the whole brain to do them, I also expect us to be able to construct tasks that do require most of the brain. It also seems very surprising, from an evolutionary perspective, if large, resource-intensive chunks of the brain are strictly unnecessary. And the reductions at stake seem unlikely to make an order-of-magnitude difference anyway.

##### Constraints faced by evolution

In designing the brain, evolution faced many constraints less applicable to human designers.453 For example, constraints on:

• The brain’s volume.
• The brain’s energy consumption.
• The growth and maintenance it has to perform.454
• The size of the genome it has to be encoded in.455
• The comparatively slow and unreliable elements it has to work with.456
• Ability to redesign the system from scratch.457

It may be that these constraints explain the brain’s functional organization at sufficiently high-levels that if we understood the overarching principles at work, we would see that much of what the brain does (even internally) is comparatively easy to do with human computers, which can be faster, bigger, more reliable, more energy-intensive, re-designed from scratch, and built using external machines on the basis of designs stored using much larger amounts memory.458 This, too, suggests smaller budgets.

#### Beyond the mechanistic method

Overall, I find the considerations pointing to the adequacy of smaller budgets more compelling than the considerations pointing to the necessity of larger ones (though it also seems, in general, easier to show that X is enough, than that X is strictly required – an asymmetry present throughout the report). But the uncertainties in either direction rightly prompt dissatisfaction with the mechanistic method’s robustness. Is there a better approach?

## The functional method

Let’s turn to the functional method, which attempts to identify a portion of the brain whose function we can already approximate with artificial systems, together with the computational costs of doing so, and then to scale up to an estimate for the brain as a whole.

Various attempts at this method have been made. To limit the scope of the section, I’m going to focus on two categories: estimates based on the retina, and estimates based on the visual cortex. But I expect many problems to generalize.

As a preview of my conclusion: I give less weight to these estimates than to the mechanistic method, primarily due to uncertainties about (a) what the relevant portion of the brain is doing (in the case of the visual cortex), (b) differences between that portion and the rest of the brain (in the case of the retina), and (c) the FLOP/s required to fully replicate the functions in question. However, I take visual cortex estimates as some weak evidence that the mechanistic method range above (1e13-1e17 FLOP/s) isn’t much too low. Some estimates based on recent deep neural network models of retinal neurons point to higher numbers. I take these on their own as even weaker evidence, but I think they’re worth understanding better.

### The retina

As I discussed in Section 2.1.2.1.2, the retina is one of the best-understood neural circuits.459 Could it serve as a basis for a functional method estimate?

#### Retina FLOP/s

We don’t yet have very good artificial retinas (though development efforts are ongoing).460 However, this has a lot to do with engineering challenges – e.g., building devices that interface with the optic nerve in the right way.461 Even absent fully functional artificial retinas, we may be able to estimate the FLOP/s required to replicate retinal computation.

Moravec (1988, 1998, and 2008) offers some estimates in this vein.462 He treats the retina as performing two types of operations – a “center surround” operation, akin to detecting an edge, and a “motion detection” operation – and reports that in his experience with robot vision, such operations take around 100 calculations to perform.463 He then divides the visual field into patches, processing of which gets sent to a corresponding fiber of the optic nerve, and budgets ten edge/motion detection operations per patch per second (ten frames per second is roughly the frequency at which individual images become indistinguishable for humans).464 This yields an overall estimate of:

1e6 ganglion cells × 100 calculations per edge/motion detection × 10 edge/motion detections per second = 1e9 calculations/sec for the whole retina

Is this right? At the least, it’s incomplete: neuroscientists have catalogued a wide variety of computations that occur in the retina, other than edge and motion detection (I’m not sure how many were known at the time). For example: the retina can anticipate motion,465 it can signal that a predicted stimulus is absent,466 it can adapt to different lighting conditions,467 and it can suppress vision during saccades.468 And further computations may await discovery.469

But since Moravec’s estimates, we’ve also made progress in modeling retinal computation. Can recent models provide better estimates?

Some of these models were included in Figure 7. Of these, it seems best to focus on models trained on naturalistic stimuli, retinal responses to which have proven more difficult to capture than responses to more artificial stimuli.470 RNN/CNN neural network models appear to have more success at this than some other variants,471 so I’ll focus on two of these:

1. Maheswaranathan et al. (2019), who train a three-layer CNN to predict the outputs of ganglion cells in response to naturalistic stimuli, and achieve a correlation coefficient greater than 0.7 (retinal reliability is 0.8).
2. Batty et al. (2017), use a shared, two-layer RNN on a similar task, and capture around ~80% of explainable variance across experiments and cell types.

These models are not full replications of human retinal computation. Gaps include:

• Their accuracy can still be improved, and what’s missing might matter.472
• The models have only been trained on a very narrow class of stimuli.473
• Inputs are small (50 × 50 pixels or less) and black-and-white (though I think they only need to be as large as the relevant ganglion cell’s receptive field).
• These models don’t include adaptation, either (though one expert did not expect adaptation to make much of a difference to overall computational costs).474
• We probably need to capture correlations across cells, in addition to individual cell responses.475
• Maheswaranathan et al. (2019) use salamander retinal ganglion cells, results from which may not generalize well to humans (Batty et al. (2017) use primate cells, which seem better).476
• There are a number of other possible gaps (see endnote).477

What sort of FLOP/s budgets would the above models imply, if they were adequate?

• The CNN in Maheswaranathan et al. (2019) requires about 2e10 FLOPs to predict the output of one ganglion cell over one second.478 However, adding more ganglion cells only increases the costs in the last layer of the network. A typical experiment involves 5-15 cells, suggesting ~2e9 FLOP/s per cell, and one of the co-authors on the paper (Prof. Baccus) could easily imagine scaling up to 676 cells (the size of the last layer), which would cost ~20.4 billion FLOP/s (3e7 per cell); or 2500 cells (the size of the input), which would cost 22.4 billion FLOP/s (~1e7 per cell).479 I’ll use this last number, which suggests ~1e7 FLOP/s per retinal ganglion cell. However, I don’t feel that I have a clear grip on how to pick an appropriate number of cells.
• I estimate that the RNN in Batty et al. (2017) requires around 1e5 FLOP for one 0.83 ms bin.480 I’m less clear on how this scales per ganglion cell, so I’ll assume one cell for the whole network: e.g., ~1e8 FLOP/s per retinal ganglion cell.

These are much higher than Moravec’s estimate of 1000 calculations/s per ganglion cell, and they result in much higher estimates for the whole retina: 1e13 FLOP/s and 1e14 FLOP/s, respectively (assuming 1e6 ganglion cells).481 But it’s also a somewhat different task: that is, predicting retinal spike trains, as opposed to motion/edge detection more broadly.

Note, also, that in both cases, the FLOPs costs are dominated by the first layer of the network, which processes the input, so costs would scale with the size of the input (though the input size relevant to an individual ganglion cell will presumably be limited by the spatial extent of its receptive field).482 And in general, the scale-up to the whole retina here is very uncertain, as I feel very uninformed about what it would actually look like to run versions of these models on such a scale (how much of the network could be reused for different cells, what size of receptive field each cell would need, etc).

#### From retina to brain

What does it look like to scale up from these estimates to the brain as a whole? Here a few ways of doing so, and the results:

Basis for scaling Rough scaling factor Applied to: Moravec estimate (1e9 calcs/s) Applied to: Maheswaranathan et al. (2019) estimate (1e13 FLOP/s) Applied to: Batty et al. (2019) estimate (1e14 FLOP/s)
Mass 4e3-1e5483 4e12-1e14 4e16-1e18 4e17-1e19
Volume 4e3-1e5484 4e12-1e14 4e16-1e18 4e17-1e19
Neurons 1e3-1e4485 1e12-1e13 1e16-1e17 1e17-1e18
Synapses 1e5-1e6486 1e14-1e15 1e18-1e19 1e19-1e20
Energy use 4e3487 4e12 4e16 4e17
Overall range 1e3-1e6 1e12-1e15 1e16-1e19 1e17-1e20
Figure 14. Estimates of the FLOP/s to replicate retinal computation, scaled up to the whole brain based on various factors.

The full range here runs from 1e12 calc/s (low-end Moravec) to 1e20 FLOP/s (high-end Batty et al. (2017)). Moravec argues for scaling based on a combination of mass and volume, rather than neuron count, on the grounds that the retina’s neurons are unusually small and closely packed, and that the brain can shrink neurons while keeping overall costs in energy and materials constant.488 Anders Sandberg objects to volume, due to differences in “tissue structure and constraints.”489 He prefers neuron count.490

Regardless of how we scale, though, the retina remains different from the rest of the brain in many ways. Here are a few:

• The retina is probably less plastic.491
• The retina is highly specialized for performing one particular set of tasks.492
• The retina is subject to unique physical constraints.493
• Retinal circuitry has lower connectivity, and exhibits less recurrence.494
• We are further from having catalogued all the cell types in the brain than in the retina.495
• Some of the possible complications discussed in the mechanistic method section (for example, some forms of dendritic computation, and some alternative signaling mechanisms like ephaptic effects) may not be present in the retina in the same way.496

Not all of these, though, seem to clearly imply higher FLOP/s burdens per unit something (cell, synapse, volume, etc.) in the brain than in the retina (they just suggest possible differences). Indeed, Moravec argues that given the importance of vision, the retina may be “evolutionarily more perfected, i.e. computationally dense, than the average neural structure.”497 And various retina experts were fairly sympathetic to scaling up from the retina to the whole brain.498

Where does this leave us? Overall, I think that the estimates based on the RNN/CNN models discussed above (1e16-1e20 FLOP/s) are some weak evidence for FLOP/s requirements higher than the mechanistic method range discussed above (1e13-1e17 FLOP/s). And these could yet be under-estimates, either because more FLOP/s are required to replicate retinal ganglion cell outputs with adequate accuracy across all stimuli; or because neural computation in the brain is more complicated, per relevant unit (volume, neuron, watt, etc.) than in the retina (the low plasticity of the retina seems to me like an especially salient difference).

Why only weak evidence? Partly because I’m very uncertain about how it actually looks like to scale these models up to the retina as a whole. And as I discussed in Section 2.1.2.2, I’m wary of updating too much based on a few studies I haven’t investigated in depth. What’s more, it seems plausible to me that these models, while better than current simpler models at fitting retinal spike trains, use more FLOP/s (possibly much more) than are required to do what the retina does. Reasons include:

• The FLOP/s budgets for these RNN/CNN retina models depend on specific implementation choices (for example, input size and architecture) that don’t seem to reflect model complexity that has yet been found necessary. Bigger models will generally allow better predictions, but our efforts to predict retinal spikes using deep neural networks seem to be in early stages, and it doesn’t seem like we yet have enough data to ground strong claims about the network size required for a given level of accuracy (and we don’t know what level of accuracy is necessary, either).
• I’m struck by how much smaller Moravec’s estimate is. It’s true that this estimate is incomplete in its coverage of retinal computation – but it surprises me somewhat if (a) his estimates for edge and motion detection are correct (Prof. Barak Pearlmutter expected Moravec’s robotic vision estimates to be accurate),499 but (b) the other functions he leaves out result in an increase of 4-5 orders of magnitude. Part of the difference here might come from his focus on high-level tasks, rather than replicating spike trains.
• The CNN in Maheswaranathan et al. (2019) would require ~2e10 FLOP/s to predict the outputs of 2500 cells in response to a 50 × 50 input. But various vision models discussed in the next section take in larger inputs (224 × 224 × 3),500 and run on comparable FLOP/s (~1e10 FLOP/s for an EfficientNet-B2 run at 10 Hz). It seems plausible to me these vision models cover some non-trivial fraction of what the retina does (e.g., edge detection), along with much that it doesn’t do.

That said, these CNN/RNN results, together with the Beniaguev et al. (2020) results discussed in Section 2.1.2.2, suggest a possible larger pattern: recent DNN models used to predict the outputs of neurons and detailed neuron models appear to be quite FLOP/s intensive. It’s possible these DNNs are overkill. But they could also indicate complexity that simpler models don’t capture. Further experiments in this vein (especially ones emphasizing model efficiency) would shed helpful light.

### Visual cortex

Let’s turn to a different application of the functional method, which treats deep neural networks (DNNs) trained on vision tasks as automating some portion of the visual cortex.501

Such networks can classify full-color images into 1000 different categories502 with something like human-level accuracy.503 They can also localize/assign pixels to multiple identified objects, identify points of interest in an image, and generate captions, but I’ll focus here on image classification (I’m less confident about the comparisons with humans in the other cases).504

What’s more, the representations learned by deep neural networks trained on vision tasks turn out to be state-of-the-art predictors of neural activity in the visual cortex (though the state of the art is not obviously impressive in an absolute sense505).506 Example results include:

• Cadena et al. (2019): a model based on representations learned by a DNN trained on image classification can explain 51.6% of explainable variance of spiking activity in monkey primary visual cortex (V1, an area involved in early visual processing) in response to natural images. A three-layer DNN trained to predict neural data explains 49.8%. The authors report that these models both outperform the previous state of the art.507
• Yamins et al. (2014) show that layers of a DNN trained on object categorization can be used to achieve what was then state of the art prediction of spiking activity in the monkey Inferior Temporal cortex (IT, an area thought to be involved in a late stage of hierarchical visual processing) – ~50% of explainable variance explained (though I think the best models can now do better).508 Similar models can also be used to predict spiking activity in area V4 (another area involved in later-stage visual processing),509 as well as fMRI activity in IT.510 The accuracy of the predictions appears to correlate with the network’s performance on image classification (though the correlation weakens for some of the models best at the task).511

We can also look more directly at the features that units in an image classifier detect. Here, too, we see interesting neuroscientific parallels. For example:

• Neurons in V1 are sensitive to various low-level features of visual input, such as lines and edges oriented in different ways. Some units in the early layers of image classifiers appear to detect similar features. For example, Gabor filters, often used to model V1, are found in such early layers.512
• V4 has traditionally been thought to detect features like colors and curves.513 These, too, are detected by units in image classifiers.514 What’s more, such networks can be used to create images that can predictably drive firing rates of V4 neurons beyond naturally occurring levels.515

Exactly what to take away from these results isn’t clear to me. One hypothesis, offered by Yamins and DiCarlo (2016), is that hierarchically organized neural networks (a class that includes both the human visual system, and these DNNs) converge on a relatively small set of efficiently-learnable solutions to object categorization tasks.516 But other, more trivial explanations may be available as well,517 and superficial comparisons between human and machine perception can be misleading.518

Still, it seems plausible that at the very least, there are interesting similarities between information-processing occurring in (a) the visual cortex and (b) DNNs trained on vision tasks. Can we turn this into a functional method estimate?

Here are a few of the uncertainties involved.

#### What’s happening in the visual cortex?

One central problem is that there’s clearly a lot happening in the visual cortex other than image classification of the kind these models perform.

In general, functional method estimates fit best with a traditional view in systems neuroscience, according to which chunks of the brain are highly specialized for particular tasks. But a number of experts I spoke to thought this view inaccurate.519 In reality, cortical regions are highly interconnected, and different types of signals show up all over the place. Motor behavior in mice, for example, predicts activity in V1 (indeed, such behaviors are represented using the same neurons that represent visual stimuli);520 and V1 responses to identical visual stimuli alter based on a mouse’s estimate of its position in a virtual-reality maze.521 Indeed, Cadena et al. (2019) recorded from 307 monkey V1 neurons, and found that only in about half of them could more than 15% of the variance in their spiking be explained by the visual stimulus (the average, in those neurons, was ~28%).522

Various forms of prediction are also reflected in the visual system, even in very early layers. For example, V1 can fill in missing representations in a gappy motion stimulus.523 Simple image classifiers don’t do this. Neurons in the visual cortex also learn over time, whereas the weights in a typical image classifier are static.524 And there are various other differences besides.525

More generally, as elsewhere in the brain, there’s a lot we don’t know about what the visual cortex is doing.526 And “vision” as a whole, while hard to define clearly, intuitively involves much more than classifying images into categories (for example, visual representations seem closely tied to behavioral affordances, 3D models of a spatial environment, predictions, high-level meanings and associations, etc.).527

#### What’s human level?

Even if we could estimate what percentage of the visual cortex is devoted to image recognition of the type these models perform, it’s also unclear how much such models match human-level performance on that task. For example:

• DNNs are notoriously vulnerable to adversarial examples,528 some of which are naturally occurring.529 The extent to which humans are analogously vulnerable remains an open question.530
• DNN image classifiers can generalize poorly to data sets they weren’t trained on. Barbu et al. (2019), for example, report a 40-45% drop in performance on the ObjectNet test set, constructed from real-world examples (though Kolesnikov et al. (2020) recently improved the ObjectNet state of the art by 25%, reaching 80% top-five accuracy).531 See figure below, and endnote, for some other examples.532

• The common ILSVRC benchmark involves classifying images from 1000 categories. But humans can plausibly classify objects from more (much more?) than 10,000 categories, including very particular categories like “that one mug” or “the chair from the living room.”533 Indeed, it’s unclear to me, conceptually, how to draw the line between classifying an object (“house,” “dog,” “child”) and thinking/feeling/predicting (“house I’d like to live in,” “dog that I love,” “child in danger”).534 That said, it’s possible that all of these categories draw on similar low-level visual features detected in early stages of processing.
• The resolution of the human visual system may be finer than the resolution of typical ImageNet images. The optic nerve has roughly 1 million retinal ganglion cells that carry input from the retina, and the retina as a whole has about 100 million photoreceptor cells.535 A typical input to an image classifier is 224 × 224 × 3: ~150,000 input values (though some inputs are larger).536

That said, DNNs may also be superior to the human visual system in ways. For example, Geirhos et al. (2018) compared DNN and human performance at identifying objects presented for 200 ms, and found that DNNs outperformed humans by >5% classification accuracy on images from the training distribution (humans generally did better overall when the images were altered).537 And human vision is subject to its own illusions, blind spots, shortcuts, etc.538 And I certainly don’t know that many species of dog. Overall, though, the human advantages here seem more impressive to me.

Note, also, that the question here is not whether DNNs are processing visual information exactly like humans do. For example, in order to qualify as human-level, the models don’t need to make the same sorts of mistakes humans do. What matters is high-level task performance.

#### Making up some numbers

Suppose we forge ahead with a very loose functional method estimate, despite these uncertainties. What results?

An EfficientNet-B2, capable of a roughly human-level 95% top-five accuracy on ImageNet classification, takes 1e9 FLOPs for a forward pass – though note that if we assume sparse FLOPs (e.g., no costs for multiplying by or adding 0), as we did for the mechanistic method, this number would be lower;539 and it might be possible to prune/compress the model further (though EfficientNet-B2 is already optimized to minimize FLOPs).540

Humans can recognize ~ten images per second (though the actual process of assigning labels to ImageNet images takes much longer).541 If we ran EfficientNet-B2 ten times per second, this would require ~1e10 FLOP/s.

On one estimate from 1995, V1 in humans has about 3e8 neurons.542 However, based on more recent estimates in chimpanzees, I think this estimate might be low, possibly by an order to magnitude (see endnote for explanation).543 I’ll use 3e8-3e9 – e.g., ~0.3%-3% of the brain’s neurons.

On an initial search, I haven’t been able to find good sources for neuron count in the visual cortex as a whole, which includes areas V2-V5.544 I’ll use 1e9-1e10 neurons – e.g., ~1-10% of the brain’s neurons as a whole – but this is just a ballpark.545

If we focused on percentage of volume, weight, energy consumption, and synapses, the relevant percentages might be larger (since the cortex accounts for a larger percentage of these than of the brain’s neurons).546

We can distill the other uncertainties from 3.2.1 and 3.2.2 into two numbers:

1. The percentage of its information-processing capacity that the visual cortex devotes to tasks analogous to image classification, when it performs them.
2. The factor increase in FLOP/s required to reach human-level performance on this task (if any), relative to the FLOP/s costs of an EfficientNet-B2 run 10 times per second.

Absent a specific chunk of the visual cortex devoted exclusively to this task, the percentage in (1) does not have an obvious physiological interpretation in terms of e.g. volume or number of neurons.547 Still, something like percentage of spikes or of signaling-based energy consumption driven by performing the task might be a loose guide.548

Of course, the resources that a brain uses in performing a task are not always indicative of the FLOP/s the task requires. Multiplying two 32-bit numbers in your head, for example, uses lots of spikes, energy, etc., but requires only one FLOP. And naively, it seems unlikely that the neural resources used in playing e.g. Tic-Tac-Toe, Checkers, Chess, and Go will be a simple function of the FLOP/s that have thus far been found necessary to match human-level performance. However, the brain was not optimized to multiply large numbers or play board games. Identifying visual objects (e.g. predators, food) seems like a better test of its computational potential.549

Can we say anything about (1)? Obviously, it’s difficult. The variance in the activity in the visual cortex explained by DNN image classifiers might provide some quantitative anchor (this appears to be at least 7% in V1, and possibly much higher in other regions), but I haven’t explored this much.550 Still, to the extent (1) makes sense at all, it should be macroscopic enough to explain the results discussed at the beginning of this section (e.g., it should make interesting parallels between the feature detection in DNNs and the visual cortex noticeable using tools like fMRI and spike recordings), along with other modeling successes in visual neuroscience I haven’t explored.551 I’ll use 1% of V1 as a low end,552 and 10% of the visual cortex as a whole as a high end, with 1% of the visual cortex as a rough middle.

My biggest hesitation about these numbers comes from the conceptual ambiguities involved in estimating this type of parameter at all. Consider: “what fraction of a horse’s legs does a wheelbarrow automate?”553 It’s not clear that “of course it’s hard to say precisely, but surely at least a millionth, right?” is a sensible answer – and the problem isn’t that the true answer is a billionth instead. It seems possible that comparisons between DNNs and the visual cortex are similar.

We also need to scale up the size of the DNN in question by (2), to reflect the FLOPs costs of fully human-level image classification. What is (2)? I haven’t looked into it much, and I feel very uncertain. Some of the differences discussed in 3.2.2 – for example, differences in input size, or in number of categories (assuming we can settle on a meaningful estimate for the number of categories humans can recognize) – might be relatively easy to adjust for.554 But others, such as the FLOPs required to run models that are only as vulnerable to adversarial examples as humans are, or that can generalize as well as humans can, might involve much more involved and difficult extrapolations.

I’m not going to explore these adjustments in detail here. Here are a few possible factors:

• 10x (150k input values vs. ~1 million retinal ganglion cells)
• 100x (~factor increase in EfficientNet-B2 FLOPs required to run a BiT-L model, which exhibits better, though still imperfect, generalization to real-world datasets like ObjectNet).555
• 1000x (10x on top of a Bit-L model, for additional improvements. I basically just pulled this number out of thin air, and it’s by no means an upper bound).

Putting these estimates for (1) and (2) together:

Estimate type Assumed percentage of visual cortex information-processing capacity used for tasks analogous to image classification, when performed Implied percentage of the whole brain’s capacity (based on neuron count) Assumed factor increase in 10 Hz EfficientNet-B2 FLOP/s (1e10) required to reach fully human-level image classification Whole brain FLOP/s estimate resulting from these assumptions
Low-end 10% 0.1%-1% 10x 1e13-1e14
Middle 1% 0.01%-0.1% 100x 1e15-1e16
High-end 0.3% (1% of V1) 0.003%-0.03% 1000x 3e16-3e17
Figure 16: Functional method estimates based on the visual cortex.

Obviously, the numbers for (1) and (2) here are very made-up. The question of how high (2) could go, for example, seems very salient. And the conceptual ambiguities involved in comparing what the human visual system is doing when it classifies an image, vs. what a DNN is doing, caution against relying on what might appear to be conservative bounds.

What’s more, glancing at different models, image classification (that is, assigning labels to whole images) appears to require fewer FLOPs than other vision tasks in deep learning, such as object detection (that is, identifying and localizing multiple objects in an image). For example: an EfficientDet-D7, a close to state of the art object-detection model optimized for efficiency, uses 3e11 FLOPs per forward pass - 300x more than an EfficientNet-B2.556 So using this sort of model as a baseline instead could add a few orders of magnitude. And such a choice would raise its own questions about what human-level performance on the relevant task looks like.

Overall, I hold functional method estimates based on current DNN vision models very lightly – even more lightly, for example, than the mechanistic method estimates above. Still, I don’t think them entirely uninformative. For example, it is at least interesting to me that you need to treat an EfficientNet-B2 as running on e.g. ~0.1% of the FLOPs of a model that would cover ~1% of V1, in order to get whole brain estimates substantially above 1e17 FLOP/s – the top end of the mechanistic method range I discussed above. This weakly suggests to me that such a range is not way too low.

### Other functional method estimates

There are various other functional method estimates in the literature. Here are three:557

Source Task Artificial system Costs of human-level performance Estimated portion of brain Resulting estimate for whole brain
Drexler (2019)558 Speech recognition DeepSpeech2 1e9 FLOP/s >0.1% 1e12 FLOP/s
Drexler (2019)559 Translation Google Neural Machine Translation 1e11 FLOP/s (1 sentence per second) 1% 1e13 FLOP/s
Kurzweil (2005)560 Sound localization Work by Lloyd Watts 1e11 calculations/s 0.1% 1e14 calculations/s
Figure 17: Other functional method estimates in the literature.

I haven’t attempted to vet these estimates. And we can imagine others. Possibly instructive recent work includes:

That said, I expect other functional method estimates to encounter difficulties analogous to those discussed in section 3.2: e.g., difficulties identifying (a) the percentage of the brain’s capacity devoted to a given task, (b) what human-level performance looks like, and (c) the FLOP/s sufficient to match this level.

## The limit method

Let’s turn to a third method, which attempts to upper bound required FLOP/s by appealing to physical limits.

Some such bounds are too high to be helpful. Lloyd (2000), for example, calculates that a 1 kg, 1 liter laptop (the brain is roughly 1.5 kg and 1.5 liters) can perform a maximum of 5e50 operations per second, and store a maximum of 1e31 bits. Its memory, though, “looks like a thermonuclear explosion.”564 For present purposes, such idealizations aren’t informative.

Other physical limits, though, might be more so. I’ll focus on “Landauer’s principle,” which specifies the minimum energy costs of erasing bits (more description below). Standard FLOPs (that is, the FLOPs performed by human-engineered computers) erase bits, which means that an idealized computer running on the brain’s energy budget (~20W) can only perform so many standard FLOP/s: specifically, ~7e21 (~1e21 if we assume 8-bit FLOPs, and ~1e19 if we assume current digital multiplier implementations).565

Does this upper bound the FLOP/s required to match the brain’s task-performance? In principle, no. The brain need not be performing operations that resemble standard FLOPs, and more generally, bit-erasures are not a universal currency of computational complexity.566 In theory, for example, factorizing a semiprime requires no bit-erasures, since the mapping from inputs to outputs is 1-1.567 But we’d need many FLOPs to do it. Indeed, in principle, it appears possible to perform arbitrarily complicated computations with very few bit erasures, with manageable algorithmic overheads (though there is at least some ongoing controversy about this).568

Absent a simple upper bound, then, the question is what we can say about the following quantity:

FLOP/s required to match the brain’s task performance ÷ bit-erasures/s in the brain

Various experts I spoke to about the limit method (though not all569) thought it likely that this quantity is less than 1 – indeed, multiple orders of magnitude less.570 They gave various arguments, which I’ll roughly group into (a) algorithmic arguments (Section 4.2.1), and (b) hardware arguments (Section 4.2.2). Of these, the hardware arguments seem to me stronger, but they also don’t seem to me to rely very directly on Landauer’s principle in particular.

Whether the bound in question emerges primarily from Landauer’s principle or not, though, I’m inclined to defer to the judgment of these experts overall.571 And even if their arguments to do not treat the brain entirely as a black box, a number of the considerations these arguments appeal to seem to apply in scenarios where more specific assumptions employed by other methods are incorrect. This makes them an independent source of evidence.

Note, as well, that e.g. 1e21 FLOP/s isn’t too far from some of the numbers that have come up in previous sections. And some experts either take numbers in this range or higher seriously, or are agnostic about them.572 In this sense, the bound in question, if sound, would provide an informative constraint.

### Bit-erasures in the brain

#### Landauer’s principle

Landauer’s principle says that implementing a computation that erases information requires transferring energy to the environment – in particular, k × T × ln2 per bit erased, where k is Boltzmann’s constant, and T is the absolute temperature of the environment.573

I’ll define a computation, here, as a mapping from input logical states to probability distributions over output logical states, where logical states are sets of physical microstates treated as equivalent for computational purposes;574 and I’ll use “operation” to refer to a comparatively basic computation implemented as part of implementing another computation. Landauer’s principle emerges from the close relationship between changes in logical entropy (understood as the Shannon entropy of the probability distribution over logical states) and thermodynamic entropy (understood as the natural logarithm of the number of possible microstates, multiplied by Boltzmann’s constant).575

In particular, if (given an initial probability distribution over inputs) a computation involves decreasing logical entropy (call a one bit decrease a “logical bit-erasure”),576 then implementing this computation repeatedly using a finite physical system (e.g., a computer) eventually requires increasing the thermodynamic entropy of the computer’s environment – otherwise, the total thermodynamic entropy of the computer and the environment in combination will decrease, in violation of the second law of thermodynamics.577

Landauer’s principle quantifies the energy costs of this increase.578 These costs arise from the relationship between the energy and the thermodynamic entropy of a system: broadly, if a system’s energy increases, it can be in more microstates, and hence its entropy increases.579 Temperature, fundamentally, is defined by this exchange rate.580

There has been some controversy over Landauer’s principle,581 and some of the relevant physics has been worked out more rigorously since Landauer’s original paper.582 But the basic thrust emerges from very fundamental physics, and my understanding is that it’s widely accepted by experts.583 A number of recent results also purport to have validated Landauer’s principle empirically.584

#### Overall bit-erasures

Let’s assume that Landauer’s principle caps the bit-erasures the brain can implement. What bit-erasure budget does this imply?

Most estimates I’ve seen of the brain’s energy budget vary between ~10-20W (Joules/second).585 But not all of this energy goes to computation:

• Loose estimates suggest that 40% of energy use in the brain,586 and 25% in cortical gray matter,587 goes towards non-signaling tasks.588
• Some signaling energy is plausibly used for moving information from one place to another, rather than computing with it. Harris and Attwell (2012), for example, estimate that action potentials use 17% of the energy in grey matter (though much less in white matter).589

That said, these don’t initially appear to be order-of-magnitude level adjustments. I’ll use 20W as a high end.

The brain operates at roughly 310 Kelvin, as does the body.590 Even if the air surrounding the body is colder, Dr. Jess Riedel suggested that it’s the temperature of the skull and blood that’s relevant, as the brain has to push entropy into the environment via these conduits.591

At 310 K, k × T × ln2 Joules results in a minimum energy emission of 3e-21 Joules per bit erasure.592 With a 20W budget, this allows no more than 7e21 bit erasures per second in the brain overall.593 This simple estimate passes over some complexities (see endnote), but I’ll use it as a first pass.594

### From bit-erasures to FLOP/s

Can we get from this to a bound on required FLOP/s?

If the brain were performing standard FLOPs, it would be easy. A standard FLOP takes two n-bit numbers, and produces another n-bit number. So absent active steps to save the inputs, you’ve erased at least n bits.595 7e21 bit-erasures/s, then, would imply a maximum of e.g. ~2e21 4-bit FLOP/s, 9e20 8-bit FLOP/s, and so forth, for a computer running on 20W at 310 Kelvin.

And the intermediate steps involved in transforming inputs into outputs erase bits as well. For example, Hänninen et al. (2011) suggest that on current digital multiplier implementations, the most efficient form of n-bit multiplication requires 8 × n2 bit-erasures – e.g., 128 for a 4-bit multiplication, and 512 for an 8-bit multiplication.596 This would suggest a maximum of ~5e19 4-bit digital multiplications, and ~1e19 8-bit multiplications (though analog implementations may be much more efficient).597

And FLOPs in actual digital computers appear to erase even more bits than this – ~1 bit-erasure per transistor switch involved in the operation.598 Sarpeshkar (1998) suggests 3000 transistors for an 8-bit digital multiply (though only 4-8 in for analog implementations);599 Asadi and Navi (2007) suggest >20,000 for a 32-bit multiply.600

Perhaps for some, comfortable assuming that the brain’s operations are relevantly like standard FLOPs, this is enough. But a robust upper bound should not assume this. The brain implements some causal structure that allows it to perform tasks, which can in principle be replicated using FLOP/s, but which itself could in principle take a wide variety of unfamiliar forms. Landauer’s principle tells us that this causal structure, represented as a set of (possibly stochastic) transitions between logical states, cannot involve erasing more than 7e21 bits/second.601 It doesn’t tell us anything, directly, about the FLOP/s required to replicate the relevant transitions, and/or perform the relevant tasks.602

Here’s an analogy. Suppose that you’re wondering how many bricks you need to build a bridge across the local river, and you know that a single brick always requires a pound of mortar. You learn that the “old bridge” across the river was built using no more than 100,000 pounds of mortar. If the old bridge is made of bricks, then you can infer that 100,000 bricks is enough. If the old bridge is made of steel, though, you can’t: even assuming that a brick can do anything y units of steel can do, y units of steel might require less (maybe much less) than a pound of mortar, so the old bridge could still be built with more than 100,000×y units of steel.

Obviously, the connection between FLOPs, bit-erasures, and the brain’s operations may be tighter than that between bricks, mortar, and steel. But conceptually, the point stands: unless we assume that the brain performs standard FLOPs, moving from bit-erasures to FLOPs requires further arguments. I’ll consider two types.

#### Algorithmic arguments

We might think that any algorithm useful for information-processing, whether implemented using standard FLOPs or no, will require erasing lots of logical bits.

In theory, this appears to be false (though there is at least some ongoing controversy, related to the bit-erasures implied by repeatedly reading/writing inputs and outputs).603 Any computation can be performed using logically reversible operations (that is, operations that allow you to reconstruct the input on the basis of the output), which do not erase bits.604 For example, in theory, you can make multiplication reversible just by saving one of the inputs.605 And my understanding is that the algorithmic overheads involved in using logically reversible operations, instead of logically irreversible ones – e.g., additional memory to save intermediate results, additional processing time to “rewind” computations606 – are fairly manageable, something like a small multiplicative factor in running time and circuit size.607

In practice, however, two experts I spoke with expected the brain’s information-processing to involve lots of logical bit-erasures. Reasons included:

• When humans write software to perform tasks, it erases lots of bits.608
• Dr. Jess Riedel suggested that processing sensory data requires extracting answers to high-level questions (e.g., “should I dodge this flying rock to the left or the right?”) from very complex intermediate systems (e.g., trillions of photons hitting the eye), which involves throwing out a lot of information.609
• Prof. Jared Kaplan noted that FLOPs erase bits, and in general, he expects order one bit-erasures per operation in computational systems. You generally don’t do a lot of complicated things with a single bit before erasing it (though there are some exceptions to this). His intuition about this was informed by his understanding of simple operations you can do with small amounts of information.610

If one imagines erasing lots of bits as the “default,” then you can also argue that the brain would need to be unrealistically energy-efficient (see next section) in order to justify any overheads incurred by transitioning to more reversible forms of computation.611 Dr. Paul Christiano noted, though, that if evolution had access to computational mechanisms capable of implementing useful, logically-reversible operations, brains may have evolved a reliance on them from the start.612

We can also look at models of neural computation to see what bit-erasures they imply. There is some risk, here, of rendering the limit method uninformative (e.g., if you’ve already decided how the brain computes, you can just estimate required FLOP/s directly).613 But it could still be helpful. For example:

• Some kinds of logical irreversibility may apply to large swaths of hypotheses about how the brain computes (e.g., hypotheses on which the membrane potential, which is routinely reset, carries task-relevant information).
• Some specific hypotheses (e.g., each neuron is equivalent to X-type of very large neural network) might imply bit-erasures incompatible with Landauer’s bound.
• If the brain is erasing lots of bits in one context, this might indicate that it does so elsewhere too, or everywhere.

Of course, it’s a further step from “the brain is probably erasing lots of logical bits” to “FLOP/s required to replicate the brain’s task-performance ÷ bit-erasures per second in the brain 1,” just as it’s a further step from “the old bridge was probably built using lots of mortar” to “bricks I’ll need ÷ pounds of mortar used for the old bridge 1.” One needs claims like:

1. A minimal, computationally useful operation in the brain probably erases at least one logical bit, on average.
2. One FLOP is probably enough to capture what matters about such an operation, on average.

Prof. Kaplan and Dr. Riedel both seemed to expect something like (1) and (2) to be true, and they seem fairly plausible to me as well. But the positive algorithmic arguments just listed don’t themselves seem to me obviously decisive.

#### Hardware arguments

Another class of arguments appeals to the energy dissipated by the brain’s computational mechanisms. After all, for required FLOPs per logical bit-erasure to be >1, it would need to be the case that required FLOPs per ~0.69kT of energy dissipation is >1 as well.

For example, in combination with (2) above, we might argue instead for:

1*. A minimal, computationally useful operation in the brain probably dissipates at least 0.69kT, on average.

One possibly instructive comparison is with the field of reversible computing, which aspires to build computers that dissipate arbitrarily small amounts of energy per operation.614 This requires logically reversible algorithms (since otherwise, Landauer’s principle will set a minimum energy cost per operation), but it also requires extremely non-dissipative hardware – indeed, hardware that is close to thermodynamically reversible (e.g., its operation creates negligible amounts of overall thermodynamic entropy).

Useful, scalable hardware of this kind would need to be really fancy. As Dr. Michael Frank puts it, it would require “a level of device engineering that’s so precise and sophisticated that it will make today’s top-of-the-line device technologies seem as crude in comparison, to future eyes, as the practice of chiseling stone tablets looks to us today.”615 According to Dr. Frank, the biggest current challenge centers on the trade-off between energy dissipation and processing speed.616 Dr. Christiano also mentioned challenges imposed by an inability to expend energy in order to actively set relevant physical variables into particular states: the computation needs to work for whatever state different physical variables happen to end up in.617

For context, the energy dissipation per logical bit-erasure in current digital computers appears to be ~1e5-1e6 worse than Landauer’s limit, and progress is expected to asymptote between 1e3 and 1e5.618 A V100 GPU, at 1e14 FLOP/s and 300W, requires ~1e9 0.69kT per FLOP (assuming room temperature).619 So in order to perform the logically-reversible equivalent of a FLOP for less than 0.69kT, you’d need a roughly billion-fold increase in energy efficiency.

Of course, biological systems have strong incentives to reduce energy costs.620 And some computational processes in biology are extremely efficient.621 But relative to a standard of 0.69kT per operation, the brain’s mechanisms generally appear highly dissipative.622 For example:

• Laughlin et al. (1998) suggest that synapses and cells use ~1e5-1e8kT per bit “observed” (though I don’t have a clear sense of what the relevant notion of observation implies).623
• A typical cortical spike dissipates around 1e10-1e11kT.624 Prof. David Wolpert noted that this process involves very complicated physical machinery, which he expects to be very far from theoretical limits of efficiency, being used to propagate a single bit.625
• Dr. Riedel mentioned that the nerves conveying a signal to kick your leg burn much more than 0.69kT per bit required to say how much to move the muscle.626
• A single molecule of ATP (the brain’s main energy currency) releases ~25kT,627 and Dr. Christiano was very confident that the brain would need at least 10 ATPs to get computational mileage equivalent to a FLOP.628 At a rough maximum of ~2e20 ATPs per second,629 this would suggest <2e19 FLOP/s.

Of course, the relevant highly-non-dissipative information-processing could be hiding somewhere we can’t see, and/or occurring in a way we don’t understand. But various experts also mentioned more general features of the brain that make it poorly suited to this, including:

• The size of its components.630
• Its warm temperature.631
• The need to boost signals in order to contend with classical noise.632
• Its reliance on diffusion to propagate information.633
• The extreme difficulty of building reversible computers in general.634

All of this seems to me like fairly strong evidence for something like 1*.

Note, though, that Landauer’s principle isn’t playing a very direct role here. We had intended to proceed from an estimate of the brain’s energy budget, to an upper bound on its logical bit-erasures (via Landauer’s principle), to an upper bound on the FLOP/s required to match its task performance. But hardware arguments skip the middle step, and just argue directly that you don’t need more than one FLOP per 0.69kT used by the brain. I think that this is probably true, but absent this middle step, 0.69kT doesn’t seem like a clearly privileged number to focus on.

### Overall weight for the limit method

Overall, it seems very unlikely to me that more than ~7e21 FLOP/s is required to match the brain’s task-performance. This is centrally because various experts I spoke to seemed confident about claims in the vicinity of (1), (1*), and (2) above; partly because those claims seem plausible to me as well; and partly because other methods generally seem to point to lower numbers.635

Indeed, lower numbers (e.g., 1e21 – ~ the maximum 8-bit irreversible FLOP/s a computer running on 20W at 310 Kelvin could perform, and 1e20 – the maximum number of required FLOP/s, assuming at least one ATP per required FLOP) seem likely to me to be overkill as well.636

That said, this doesn’t seem like a case of a hard physical limit imposing a clean upper bound. Even equipped with an application of the relevant limit to the brain (various aspects of this still confuse me – see endnote), further argument is required.637 And indeed, the arguments that seem most persuasive to me (e.g., hardware arguments) don’t seem to rely very directly on the limit itself. Still, we should take whatever evidence we can get.

## The communication method

Let’s briefly discuss a final method (the “communication method”), which attempts to use the communication bandwidth in the brain as evidence about its computational capacity. I haven’t explored this much, but I think it might well be worth exploring.

Communication bandwidth, here, refers to the speed with which a computational system can send different amounts of information different distances.638 This is distinct from the operations per second that a system can perform (computation), but it’s just as hard a constraint on what the system can do.

Estimating the communication bandwidth in the brain is a worthy project in its own right. But it also might help with computation estimates. This is partly because the marginal value of additional computation and communication are related (e.g., too little communication and your computational units sit idle; too few computational units and it becomes less useful to move information around).

Can we turn this into a FLOP/s estimate? The basic form of the argument would be roughly:

1. The profile of communication bandwidth in the brain is X.
2. If the profile of the communication bandwidth in the brain is X, then Y FLOP/s is probably enough to match its task performance.

I’ll discuss each premise in turn.

### Communication in the brain

One approach to estimating communication in the brain would be to identify all of the mechanisms involved in it, together with the rates at which they can send different amounts of information different distances.

• Axons are clearly a central mechanism here, and one in which a sizeable portion of the brain’s energy and volume have been invested.639 There is a large literature on estimating the information communicated by action potentials.640
• Dendrites also seem important, though generally at shorter distances (and at sufficiently short distances, distinctions between communication and computation may blur).641
• Other mechanisms (e.g. glia, neuromodulation, ephaptic effects, blood flow – I’m less sure about gap junctions) are plausibly low-bandwidth relative to axons and dendrites.642 If so, this would simplify the estimate. And the resources invested in axons and dendrites would make it seem somewhat strange if the brain has other, superior forms of communication available.643

Dr. Paul Christiano suggests a rough estimate of ~10 bits per spike for axon communication, and uses this to generate the bounds of ~1e9 bytes/s of long-distance communication across the brain, 1e11 bytes/s of short-distance communication (where each neuron could access ~1e7 nearby neurons), and larger amounts of very short-distance communication.644

Another approach would be to draw analogies with metrics used to assess the communication capabilities of human computers. AI Impacts, for example, recommends the traversed edges per second (TEPS) metric, which measures the time required to perform a certain kind of search through a random graph.645 They treat neurons as vertices on the graph, synapses as edges, and spikes through synapses as traversals of edges, yielding an overall estimate of ~2e13-6e14 TEPS (the same as their estimate of the number of spikes through synapses).646

I haven’t investigated either of these estimates in detail. But they’re instructive examples.

### From communication to FLOP/s

How do we move from a communication profile for the brain, to an estimate of the FLOP/s sufficient to match its task performance? There are a number of possibilities.

One simple argument runs as follows: if you have two computers comparable on one dimension important to performance (e.g., communication), but you can’t measure how they compare on some other dimension (e.g., computation), then other things equal, your median guess should be that they are comparable on this other dimension as well.647 Here, the assumption would be that the known dimension reflects the overall skill of the engineer, which was presumably applied to the unknown dimension as well.648 As an analogy: if all we know is that Bob’s cheesecake crusts are about as good as Maria’s, the best median guess is that they’re comparable cheesecake chefs, and hence that his cheesecake filling is about as good as hers as well.

Of course, we know much about brains and computers unrelated to how their communication compares. But those drawn to simple a priori arguments, perhaps this sort of approach can be useful.

Using Dr. Christiano’s estimates, discussed above, one can imagine comparing a V100 GPU to the brain as follows:649

Metric V100 Human brain
Short-distance communication 1e12 bytes/s of memory bandwidth 1e11 bytes/s to nearby neurons? (not vetted)650
Long-distance communication 3e11 bytes/s of off-chip bandwidth 1e9 bytes/s across the brain? (not vetted)651
Computation 1e14 FLOP/s ?
Figure 18: Comparing the brain to a V100.

On these estimates, the V100’s communication is at least comparable to the brain’s (indeed, it’s superior by between 10 and 300x). Naively, then, perhaps its computation is comparable (indeed, superior) as well.652 This would suggest 1e14 FLOP/s or less for the brain.

That said, it seems like a full version of this argument would include other available modes of comparison as well (continuing the analogy above: if you also know that that Maria’s jelly cheesecake toppings are much worse than Bob’s, you should take this into account too). For example, if we assume that synapse weights are the central means of storing memory in the brain,653 we might get:

Metric V100 Human brain
Memory 3e10 bytes on chip 1e14-1e15 synapses,654 each storing >5 bits?655
Power consumption 300W 20W656
Figure 19: Comparing the brain to a V100, continued.

So the overall comparison here becomes more complicated. V100 power consumption is >10x worse, and comparable memory, on this naive memory estimate for the brain, would require a cluster of ~3000-30,000 V100s, suggesting a corresponding increase to the FLOP/s attributed to the brain (memory access across the cluster would become more complex as well, and overall energy costs would increase).657

A related approach involves attempting to identify a systematic relationship between communication and computation in human computers – a relationship that might reflect trade-offs and constraints applicable to the brain as well.658 Thus, for example, AI Impacts examines the ratio of TEPS to FLOP/s in eight top supercomputers, and finds a fairly consistent ~500-600 FLOP/s per TEPS.659 Scaling up from their TEPS estimate for the brain, they get ~1e16-3e17 FLOP/s.660

A more sophisticated version of this approach would involve specifying a production function governing the returns on investment in marginal communication vs. computation.661 This function might allow evaluation of different hypothesized combinations of communication and computation in the brain. Thus, for example, the hypothesis that the brain performs the equivalent of 1e20 FLOP/s, but has the communication profile listed in the table above, might face the objection that it assigns apparently sub-optimal design choices to evolution: e.g., in such a world, the brain would have been better served re-allocating resources invested in computation (energy, volume, etc.) to communication instead.

And even if the brain were performing the equivalent of 1e20 FLOP/s (perhaps because it has access to some very efficient means of computing), such a production function might also indicate a lower FLOP/s budget sufficient, in combination with more communication than the brain can mobilize, to match the brain’s task performance overall (since there may be diminishing returns to more computation, given a fixed amount of communication).662

These are all just initial gestures at possible approaches, and efforts in this vein face a number of issues and objections, including:

• Changes over time to the ratio of communication to computation in human-engineered computers.663
• Differences in the constraints and trade-offs faced by human designers and evolution.

I haven’t investigated the estimates above very much, so I don’t put much weight on them. But I think approaches in this vicinity may well be helpful.

## Conclusion

I’ve discussed four different methods of generating FLOP/s budgets big enough to perform tasks as well as the human brain. Here’s a summary of the main estimates, along with the evidence/evaluation discussed:

Estimate Description ~FLOP/s Summary of evidence/evaluation
Mechanistic method low ~1 FLOP per spike through synapse; neuron models with costs ≤ Izhikevich spiking models run with 1 ms time-steps. 1e13-1e15 Simple model, and the default in the literature; some arguments suggest that models in this vein could be made adequate for task-performance without major increases in FLOP/s; these arguments are far from conclusive, but they seem plausible to me, and to some experts (others are more skeptical).
Mechanistic method high ~100 FLOPs per spike through synapse; neuron models with costs greater than Izhikevich models run with 1 ms time-steps, but less than single-compartment Hodgkin-Huxley run with 0.1 ms timesteps. 1e15-1e17 It also seems plausible to me that FLOP/s budgets for a fairly brain-like task-functional model would need to push into this range in order to cover e.g. learning, synaptic conductances, and dendritic computation (learning seems like an especially salient candidate here).
Mechanistic method very high Budgets suggested by more complex models – e.g., detailed biophysical models, large DNN neuron models, very FLOPs-intensive learning rules. >1e17 I don’t see much strong positive evidence that you need this much, even for fairly brain-like models, but it’s possible, and might be suggested by higher temporal resolutions, FLOP/s intensive DNN models of neuron behavior, estimates based on time-steps per variable, greater biophysical detail, larger FLOPs budgets for processes like dendritic computation/learning, and/or higher estimates of parameters like firing rate or synapse count.
Scaling up the DNN from Beniaguev et al. (2020) Example of an estimate >1e17 FLOP/s. Uses the FLOP/s for a DNN-reduction of a detailed biophysical model of a cortical neuron, scaled up by 1e11 neurons. 1e21 I think that this is an interesting example of positive evidence for very high mechanistic method estimates, as Beniaguev et al. (2020) found it necessary to use a very large model in order to get a good fit. But I don’t give this result on its own a lot of weight, partly because their model focuses on predicting membrane potential and individual spikes very precisely, and smaller models may prove adequate on further investigation.
Mechanistic method very low Models that don’t attempt to model every individual neuron/synapse. <1e13 It seems plausible to me that something in this range is enough, even for fairly brain-like models. Neurons display noise, redundancy, and low-dimensional behavior that suggest that modeling individual neurons/synapses might be overkill; mechanistic method estimates based on low-level components (e.g. transistors) substantially overestimate FLOP/s capacity in computers we actually understand; emulation imposes overheads; and the brain’s design reflects evolutionary constraints that could allow further simplification.
Functional method estimate based on Moravec’s retina estimate, scaled up to whole brain Assumes 1e9 calculations per second for the retina (100 calculations per edge/motion detection per, 10 edge/motion detections per second per cell, 1e6 cells); scaled up by 1e3-1e6 (the range suggested by portion of mass, volume, neurons, synapses, and energy). 1e12-1e15 (assuming 1 calculation ~= 1 FLOP) The retina does a lot of things other than edge and motion detection (e.g., it anticipates motion, it can signal that a predicted stimulus is absent, it can adapt to different lighting conditions, it can suppress vision during saccades); and there are lots of differences between the retina and the brain as a whole. But the estimate, while incomplete in its coverage of retinal function, might be instructive regardless, as a ballpark for some central retinal operations (I haven’t vetted the numbers Moravec uses for edge/motion detection, but Prof. Barak Pearlmutter expected them to be accurate).664
Functional method estimate based on DNN models of the retina, scaled up to the whole brain Estimates of retina FLOP/s implied by the models in Batty et al. (2017) (1e14 FLOP/s) and Maheswaranathan et al. (2019) (1e13 FLOP/s), scaled up to the brain as a whole using the same 1e3-1e6 range above. 1e16-1e20 I think this is some weak evidence for numbers higher than 1e17, and the models themselves are still far from full replications of retinal computation. However, I’m very uncertain about what it looks like to scale these models up to the retinas as a whole. And it also seems plausible to me that these models use many more FLOP/s than required to do what the retina does. For example, their costs reflect implementation choices and model sizes that haven’t yet been shown necessary, and Moravec’s estimate (even if incomplete) is much lower.
Low end functional method estimate based on the visual cortex Treats a 10 Hz EfficientNet-B2 image classifier, scaled up by 10x, as equivalent to 10% of the visual cortex’s information-processing capacity, then scales up to the whole brain based on portion of neurons (portion of synapses, volume, mass, and energy consumption might be larger, if the majority of these are in the cortex). 1e13-1e14 In general, I hold these estimates lightly, as I feel very uncertain about what the visual cortex is doing overall and how to compare it to DNN image classifiers, as well as about the scale-up in model size that will be required to reach image classification performance as generalizable across data sets and robust to adversarial examples as human performance is (the high-end correction for this used here - 1000x - is basically just pulled out of thin air, and could be too low). That said, I do think that, to the extent it makes sense at all to estimate the % of the visual cortex’s information-processing capacity mobilized in performing a task analogous to image classification, the number should be macroscopic enough to explain the interesting parallels between the feature detection in image classifiers and in the visual cortex (see Section 3.2 for discussion). 1% of V1 seems to me reasonably conservative in this regard, especially given that CNNs trained on image classification end up as state of the art predictors of neural activity in V1 (as well as elsewhere in the visual cortex). So I take these estimates as some weak evidence that the mechanistic method estimates I take most seriously (e.g., 1e13-1e17) aren’t way too low.
Middle-range functional method estimate based on visual cortex Same as previous, but scales up 10 Hz EfficientNet-B2 by 100x, and treats it as equivalent to 1% of the visual cortex’s information-processing capacity. 1e15-1e16
High end functional method estimate based on visual cortex Same as previous, but scales up 10 Hz EfficientNet-B2 by 1000x instead, and treats it as equivalent to 1% of V1’s information-processing capacity. 3e16-3e17
Limit method low end Maximum 8-bit, irreversible FLOP/s that a computer running on 20W at body temperature can perform, assuming current digital multiplier implementations (~500 bit-erasures per 8-bit multiply). 1e19 I don’t think that a robust version of the limit method should assume that the brain’s operations are analogous to standard, irreversible FLOP/s (and especially not FLOP/s in digital computers, given that there may be more energy-efficient analog implementations available – see Sarpeshkar (1998)). But it does seem broadly plausible to me that a minimal, computationally useful operation in the brain erases at least one logical bit, and very plausible that it dissipates at least 0.69kT (indeed, my best guess would be that it dissipates much more than that, given that cortical spikes dissipate 1e10-1e11kT; a single ATP releases ~25kT; the brain is noisy, warm, and reliant on comparatively large components, etc.). And it seems plausible, as well, that a FLOP is enough to replicate the equivalent of a minimal, computationally useful operation in the brain. Various experts (though not all) also seemed quite confident about claims in this vicinity. So overall, I do think it very unlikely that required FLOP/s exceeds e.g. 1e21. However, I don’t think this is a case of a physical limit imposing a clean upper bound. Rather, it seems like one set of arguments amongst others. Indeed, the arguments that seem strongest to me (e.g., arguments that appeal to the energy dissipated by the brain’s mechanisms) don’t seem to rely directly on Landauer’s principle at all.
Limit method middle Maximum 8-bit, irreversible FLOP/s that a computer running in 20W at body temperature can perform, assuming no intermediate bit-erasures (just a transformation from two n-bit inputs to one n-bit output). 1e21
Limit method high Maximum FLOP/s, assuming at least one logical bit-erasure, or at least 0.69kT
dissipation, per required FLOP.
7e21
ATPs Maximum FLOP/s, assuming at least one ATP used per required FLOP. 1e20
Communication method estimate based on comparison with V100 Estimates brain communication capacity, compares it to a V100, and infers on the basis of the comparability/inferiority of the brain’s communication to a V100s communication, perhaps it’s computational capacity is comparable/inferior as well. ≤1e14 I haven’t vetted these estimates much and so don’t put much weight on them. The main general question is whether the relationship between communication and computation in human-engineered computers provides much evidence about what to expect that relationship to be in the brain. Initial objections to comparisons to a V100, even granting the communication estimates for the brain that it’s based on, might center on complications introduced by also including memory and energy consumption in the comparison. Initial objections to relying on TEPS-FLOP/s ratios might involve the possibility that there are meaningfully more relevant “edges” in the brain than synapses, and/or “vertices” than neurons. Still, I think that approaches in this broad vicinity may well prove helpful on further investigation.
Communication method estimate based on TEPS to FLOP/s extrapolation Estimates brain TEPS via an analogy between spikes through synapses and traversals of an edge in a graph; then extrapolates to FLOP/s based on observed relationship between TEPS and FLOP/s in a small number of human-engineered computers. 1e16-3e17 FLOP/s
Figure 20: Summary and description of the main estimates discussed in the report.

Here are the main numbers plotted together:

None of these numbers are direct estimates of the minimum possible FLOP/s budget. Rather, they are different attempts to use the brain – the only physical system we know of that performs these tasks, but far from the only possible such system – to generate some kind of adequately (but not arbitrarily) large budget. If a given method is successful, it shows that a given number of FLOP/s is enough, and hence, that the minimum is less than that. But it doesn’t, on its own, indicate how much less.

Can we do anything to estimate the minimum directly, perhaps by including some sort of adjustment to one or more of these numbers? Maybe, but it’s a can of worms that I don’t want to open here, as addressing the question of where we should expect the theoretical limits of algorithmic efficiency to lie relative to these numbers (or, put another way, how many FLOP/s we should expect superintelligent aliens to use, if they were charged with replicating human-level task-performance using FLOPs) seems like a further, difficult investigation (though Dr. Paul Christiano expected the brain to be performing at least some tasks in close to maximally efficient ways, using a substantial portion of its resources – see endnote).665

Overall, I think it more likely than not that 1e15 FLOP/s is enough to perform tasks as well as the human brain (given the right software, which may be very hard to create). And I think it unlikely (<10%) that more than 1e21 FLOP/s is required. That said, as emphasized above:

• The numbers above are just very loose, back-of-the-envelope estimates.
• I am not a neuroscientist, and there is no consensus on this topic in neuroscience (or elsewhere).
• Basically all of my best-guesses are based on a mix of (a) shallow investigation of messy, unsettled science, and (b) a limited, non-representative sampling of expert opinion.

More specific probabilities require answering questions about the theoretical limits of algorithmic efficiency – questions that I haven’t investigated and that I don’t want to overshadow the evidence actually surveyed in the report. In the appendix, I discuss a few narrower conceptions of the brain’s FLOP/s capacity, and offer a few more specific probabilities there, keyed to one particular type of brain model. My current best-guess median for the FLOP/s required to run that particular type of model is around 1015 (recall that none of these numbers are estimates of the FLOP/s uniquely “equivalent” to the brain).

As can be seen from the figure above, the FLOP/s capacities of current computers (e.g., a V100 at ~1e14 FLOP/s for ~$10,000, the Fugaku supercomputer at ~4e17 FLOP/s for ~$1 billion) cover the estimates I find most plausible.666 However:

• Task-performance requires resources other than FLOP/s (for example, memory and memory bandwidth).
• Performing tasks on a particular machine can introduce further overheads and complications.
• Most importantly, matching the human brain’s task-performance requires actually creating sufficiently capable and computationally efficient AI systems, and this could be extremely (even prohibitively) difficult in practice even with computers that could run such systems in theory. Indeed, as noted above, the FLOP/s required to run a system that does X can be available even while the resources (including data) required to train it remain substantially out of reach. And what sorts of task-performance will result from what sorts of training is itself a further, knotty question.667

So even if my best-guesses are correct, this does not imply that we’ll see AI systems as capable as the human brain anytime soon.

### Possible further investigations

Here are a few projects that others interested in this topic might pursue (this list also doubles as a catalogue of some of my central ongoing uncertainties).

Mechanistic method

• Investigate the literature on population-level modeling and/or neural manifolds, and evaluate what sorts of FLOP/s estimates it might imply.
• Investigate the best-understood neural circuits (for example, Prof. Eve Marder mentioned some circuits in leeches, C. elegans, flies, and electric fish), and what evidence they provide about the computational models adequate for task-performance.668
• Follow up on the work in Beniaguev et al. (2020), testing different hypotheses about the size of deep neural networks required to fit neuron behavior with different levels of accuracy.
• Investigate the computational requirements and biological plausibility of different proposed learning rules in the brain in more depth.
• Investigate more deeply different possible hypotheses about molecular-level intracellular signaling processes taking place in the brain, and the FLOP/s they might imply.
• Investigate the FLOP/s implications of non-binary forms of axon signaling in more detail.

Functional method

• Following up on work by e.g. Batty et al. (2017) and Maheswaranathan et al. (2019), try to gather more data about the minimal artificial neural network models adequate to predict retinal spike trains across trials at different degrees of accuracy (including higher degrees of accuracy than these models currently achieve).
• Create a version of Moravec’s retina estimate that covers a wider range of computations that the retina performs, but which still focuses on high-level tasks rather than spike trains.
• Investigate the literature on comparisons between the feature detection in DNNs and in the visual cortex, and try to generate better quantitative estimates of the overlap and the functional method FLOP/s it would imply.
• Based on existing image classification results, try to extrapolate to the model size required to achieve human-level robustness to adversarial examples and/or generalization across image classification data sets.
• Investigate various other types of possible functional methods (for example, estimates based on ML systems performing speech recognition).

Limit method

• Investigate and evaluate more fleshed-out versions of algorithmic arguments.
• Look for and evaluate examples in biology where the limit method might give the wrong answer: e.g., where a biological system is performing some sort of useful computation that would require more than a FLOP to replicate, but which dissipates less than 0.69kT.

Communication method

• Estimate the communication bandwidth available in the brain at different distances.
• Investigate the trade-offs and constraints governing the relationship between communication and computation in human-engineered computers across different tasks, and evaluate the extent to which these would generalize to the brain.

General

• Gather more standardized, representative data about expert opinion on this topic.
• Investigate what evidence work on brain-computer interfaces might provide.
• Investigate and evaluate different methods of estimating the memory and/or number of parameters in the brain – especially ones that go beyond just counting synapses. What would e.g., neural manifolds, different models of state retention in neurons, models of biological neurons as multi-layer neural networks, dynamical models of synapses, etc., imply about memory/parameters?
• (Ambitious) Simulate a simple organism like C. elegans at a level of detail adequate to replicate behavioral responses and internal circuit dynamics across a wide range of contexts, then see how much the simulation can be simplified.

## Appendix: Concepts of brain FLOP/s

It is reasonably common for people to talk about the brain’s computation/task-performance in terms of metrics like FLOP/s. It is much less common for them to say what they mean.

When I first started this project, I thought that there might be some sort of clear and consensus way of understanding this kind of talk that I just hadn’t been exposed to. I now think this much less likely. Rather, I think that there are a variety of importantly different concepts in this vicinity, each implying different types of conceptual ambiguity, empirical uncertainty, and relevant evidence. These concepts are sufficiently inter-related that it can be easy to slip back and forth between them, or to treat them as equivalent. But if offering estimates, or making arguments about e.g. AI timelines using such estimates, it matters which you have in mind.

I’ll group these concepts into four categories:

1. FLOP/s required for task-performance, with no further constraints.
2. FLOP/s required for task-performance + brain-like-ness constraints (e.g., constraints on the similarity between the task-functional model and the brain’s internal dynamics).
3. FLOP/s required for task-performance + findability constraints (e.g., constraints on what sorts of processes would be able to create/identify the task-functional model in question).
4. Other analogies with human-engineered computers.

I find it useful, in thinking about these concepts, to keep the following questions in mind:

• Single answer: Does this concept identify a single, well-defined number of FLOP/s?
• Non-arbitrariness. Does it involve a highly arbitrary point of focus?
• One-FLOP-per-FLOP: To the extent that this concept purports to represent the brain’s FLOP/s capacity, does an analogous concept, applied to a human-engineered computer, identify the number of FLOP/s that computer actually performs? E.g., applied to a V100, does it pick out 1e14 FLOP/s?669
• Relationship to the literature: To what extent do estimates offered in the literature on this topic (mechanistic method, functional method, etc.) bear on the FLOP/s this concept refers to?
• Relevance to AI timelines: How relevant is this number of FLOP/s to when we should expect humans to develop AI systems that match human-level performance?

This appendix briefly discusses some of the pros and cons of these concepts in light of such questions, and it offers some probabilities keyed to one in particular.

### No constraints

This report has focused on the evidence the brain provides about the FLOP/s sufficient for task-performance, with no further constraints on the models/algorithms employed in performing the tasks. I chose this point of focus centrally because:

• Its breadth makes room for a wide variety of brain-related sources of evidence to be relevant.
• It avoids the disadvantages and controversies implied by further constraints (see below).
• It makes the discussion in the report more likely to be helpful to people with different assumptions and reasons for interest in the topic.

However, it has two main disadvantages:

• As noted in the report, evidence that X FLOP/s is sufficient is only indirect evidence about the minimum FLOP/s required; and the overall probability that X is sufficient depends, not just on evidence from the brain/current AI systems, but on further questions about where the theoretical limits of algorithmic efficiency are likely to lie. That said, as noted earlier, Dr. Paul Christiano expected there to be at least some tasks such (a) the brain’s methods of performing them are close to maximally efficient, and (b) these methods use most of the brain’s resources.670 I haven’t investigated this, but if true, it would reduce the force of this disadvantage.
• The relevance of in principle FLOP/s requirements to AI timelines is fairly indirect. If you know that Y type of task-performance is impossible without X FLOP/s, then you know that you won’t see Y until X FLOP/s are available. But once X FLOP/s are available (as I think they probably are), the question of when you’ll see Y is still wide open. You know that superintelligent aliens could do it in theory, if forced to use only the FLOP/s your computers make available. But on its own, this gives you very little indication of when humans will do it in practice.

In light of these disadvantages, let’s consider a few narrower points of focus.

### Brain-like-ness

One option is to require that models/algorithms employed in matching the brain’s task-performance exhibit some kind of resemblance to its internal dynamics as well. Call such requirements “brain-like-ness constraints.”

Such constraints restrict the set of task-functional models under consideration, and hence, to some extent, the relevance of questions about the theoretical limits of algorithmic efficiency. And they may suggest a certain type of “findability,” without building it into the definition of the models/algorithms under consideration. The brain, after all, is the product of evolution – a search and selection process whose power may be amenable to informative comparison with what we should expect the human research community to achieve.

But brain-likeness constraints also have disadvantages. Notably:

• From the perspective of AI timelines, it doesn’t matter whether the AI systems in question are brain-like.
• Functional method estimates are based on human-engineered systems that aren’t designed to meet any particular brain-like-ness constraints.
• It’s difficult to define brain-like-ness constraints in a manner that picks out a single, privileged number of FLOP/s, without making seemingly-arbitrary choices about the type of brain-like-ness in question and/or losing the One-FLOP-per-FLOP criterion above.

This last problem seems especially salient to me. Here are some examples where it comes up.

Brain simulations

Consider the question: what’s the minimum number of FLOP/s sufficient to simulate the brain? At a minimum, it depends on what you want the simulation to do (e.g., serve as a model for drug development? teach us how the brain works? perform a given type of task?). But even if we focus on replicating task-performance, there still isn’t a single answer, because we have not specified the level of brain-like-ness required to count as a simulation of the brain, assuming task-performance stays fixed.671 Simulating individual molecules is presumably not required. Is replicating the division of work between hemispheres, but doing everything within the hemispheres in a maximally efficient but completely non-brain-like-way, sufficient?672 If so, we bring back many of the questions about the theoretical limits of algorithmic efficiency we were aiming to avoid. If not, where’s the line in between? We haven’t said.

“Reasonably brain-like” models

A similar problem arises if we employ a vaguer standard – requiring, for example, that the algorithm in question be “reasonably brain-like.” What counts? Are birds reasonably plane-like? Are the units of a DNN reasonably neuron-like? Some vagueness is inevitable, but this is, perhaps, too much.

Just picking a constraint

One way to avoid this would be to just pick a precisely-specified type of brain-likeness to require. For example, we might require that the simulation feature neuron-like units (defined with suitable precision), a brain-like connectome, communication via binary spikes, brain-like average firing rates, but not e.g. individual ion channels, protein dynamics, membrane potential fluctuations, etc. But why these and not others? Absent a principled answer, the choice seems arbitrary.

The brain’s algorithm

Perhaps we might appeal to the FLOP/s required to reimplement what I will call “the brain’s algorithm.” The idea, here, would be to assume that there is a single, privileged description of how the brain performs the tasks that it performs – a description that allows us to pick out a single, privileged number of FLOP/s required to perform those tasks in that way.

We can imagine appealing, here, to influential work by David Marr, who distinguished between three different levels of understanding applicable to an information-processing system:

1. The computational level: the overall task that the system in question is trying to solve, together with the reason it is trying to solve this task.
2. The algorithmic level: how the task-relevant inputs and outputs are represented in the system, together with the intermediate steps of the input-output transformation.
3. The implementation level: how these representations and this algorithm are physically implemented.673

The report focused on level 1. But suppose we ask, instead: how many FLOP/s are required to replicate level 2? Again, the same problem arises: which departures from brain-like-ness are compatible with reimplementing the brain’s algorithm, and which are not (assuming high-level task performance remains unaffected regardless)? I have yet to hear a criterion that seems to me an adequate answer.674

Note that this problem arises even if we assume clean separations between implementation and algorithmic levels in the brain – a substantive assumption, and one that may be more applicable in the context of human-engineered computers than biological systems.675 For even in human-engineered computers, there are multiple algorithmic levels. Consider someone playing Donkey Kong on an MOS 6502. How many FLOP/s do you need to reimplement the “algorithmic level” of the MOS 6502, or to play Donkey Kong “the way the MOS 6502 does it”? I don’t think there’s a single answer. Do we need to emulate individual transistors, or are logic gates enough? Can we implement the adders, or the ALU, or the high-level architecture, in a different way? A full description of how the system performs the task involves all these levels of abstraction simultaneously. Given a description of an algorithm (e.g., a set of states and rules for transitioning between them), we can talk about the operations required to implement it.676 But given an actual physical system operating on multiple levels of abstraction, it’s much less clear what talk about the algorithm it’s implementing refers to.677

The lowest algorithmic level

Perhaps we could focus on the lowest algorithmic level, assuming this is well-defined (or, put another way, on replicating all the algorithmic levels, assuming that the lowest structures all the rest)? One problem with this is that even if we knew that a given type of brain simulation – for example, a connectome-like network of Izhikevich spiking neurons – could be made task-functional, we wouldn’t yet know whether it captured the level in question. Are ion channels above or below the lowest algorithmic level? To many brain modelers, these questions don’t matter: if you can leave something out without affecting the behavior you care about, all the better. But focusing on the lowest-possible algorithmic level brings to the fore abstract questions about where this level lies. And it’s not clear, at least to me, how to answer them.678

Another problem with focusing on the lowest algorithmic level is, to the extent that we want a FLOP/s estimate that would be to the brain what 1e14 FLOP/s is to a V100, we’ll do poorly on the One-FLOP-per-FLOP criterion above: e.g., if we assume that the lowest algorithmic level in a V100 is at the level of transistors, we’ll end up budgeting many more FLOP/s for a transistor-level simulation than the 1e14 FLOP/s the V100 actually performs.679

The highest algorithmic level

What about the highest algorithmic level? As with the lowest algorithmic level, it’s unclear where this highest level lies, and very high-level descriptions of the brain’s dynamics (analogous, e.g., to the “processor architecture” portion of the diagram above) may leave a lot of room for intuitively non-brain-like forms of efficiency (recall the “simulation” of the brain’s hemispheres discussed above). And it’s not clear that this standard passes the “one-FLOP-per-FLOP” test either: if a V100 performing some task is inefficient at some lower level of algorithmic description, then the maximally efficient way of performing that task in a manner that satisfies some higher level of description may use fewer FLOP/s than the V100 performs.

Nothing that doesn’t map to the brain

Nick Beckstead suggests a brain-like-ness constraint on which the algorithm used to match the brain’s task performance must be such that (a) all of its algorithmic states map onto brain states, and (b) the transitions between these algorithmic states mirror the transitions between the corresponding brain states.680 Such a constraint rules out replicating the division of work between hemispheres, but doing everything else in a maximally efficient way, because the maximally efficient way will presumably involve algorithmic states that don’t map onto brain states.

This constraint requires specifying the necessary accuracy of the mapping from algorithmic states to brain states (though note that defining task-performance at all requires something like this).681 I also worry that whether a given algorithm satisfies this constraint or not will end up depending on which operations are treated as basic (and hence immune from the requirement that the state-transitions involved in implementing them map onto the brain’s).682 And it’s not clear to me that this definition will capture One-FLOP-per-FLOP, since it seems to require a very high degree of emulation accuracy. That said, I think something in this vicinity might turn out to work.

More generally, though, brain-like-ness seems only indirectly relevant to what we ultimately care about, which is task-performance itself. Can findability constraints do better?

### Findability

Findability constraints restrict attention to the FLOP/s required to run task-functional systems that could be identified or created via a specific type of process. Examples include task-functional systems that:

1. humans will in fact create in the future (or, perhaps, the first such systems);
2. humans would/could create, given access to a specific set of resources and/or data;
3. would/could be identified via a specific type of training procedure – for example, a procedure akin to those used in machine learning today;
4. could/would be found via a specified type of evolution-like search process, akin to the one that “found” the biological brain;
5. could be created by an engineer “as good as evolution” at engineering.683

The central benefit of all such constraints is that they are keyed directly to what it takes to actually create a task-functional system, rather than what systems could exist in principle. This makes them more informative for the purposes of thinking about when such systems might in fact be created by humans.

But it’s also a disadvantage, as estimates involving findability constraints require answering many additional, knotty questions about what types of systems are what kinds of findable (e.g., what sorts of research programs or training methods could result in what sorts of task performance; what types of resources and data these programs/methods would require; what would in fact result from various types of counterfactual “evolution-like” search processes, etc.).

Findability constraints related to evolution-like search processes/engineering efforts (e.g., (d) and (e) above) are also difficult to define precisely, and they are quite alien to mainstream neuroscientific discourse. This makes them difficult to solicit expert opinion about, and harder to evaluate using evidence of the type surveyed in the report.

My favorite of these constraints is probably the FLOP/s that will be used by the first human-built systems to perform these tasks, since this is the most directly relevant to AI timelines. I see functional method estimates as especially relevant here, and mechanistic/limit method estimates as less so.

### Other computer analogies

There are a few other options as well, which appeal to various other analogies with human-engineered computers.

Operations per second

For example, we can imagine asking: how many operations per second does the brain perform? One problem here is that “operations” does not have a generic meaning. An operation is just an input-output relationship, implemented as part of a larger computation, and treated as basic for the purpose of a certain kind of analysis.684 The brain implements many different such relationships at different levels of abstraction: for example, it implements many more “ion-channel opening/closing” operations per second than it does “spikes through synapses” operations.685 Estimates that focus on the latter, then, need to say why they do so. You can’t just pick a thing to count, and count it.

More importantly, our ultimate interest is in systems that run on FLOP/s, that perform tasks at human-levels. To be relevant to this, then, we also need to know how many FLOP/s are sufficient to replicate one of the operations in question; and we need some reason to think that, so replicated, the resulting FLOP/s budget overall would be enough for task-performance. This amounts to something closely akin to the mechanistic method, and the same questions about the required degree of brain-like-ness apply.

FLOP/s it performs

What if we just asked directly: how many FLOP/s does the brain perform? Again, we need to know what is meant.

• One possibility is that we have in mind one of the other questions above: e.g., how many FLOP/s do you need to perform some set of tasks that the brain performs, perhaps with some kind of implicit brain-like-ness constraint. This raises the problems discussed in 7.1 and 7.2 above.
• Another possibility is that we are asking more literally: how many times per second does the brain’s biophysics implement e.g. an addition, subtraction, multiplication, or division operation of a given level of precision? In some places, we may be able to identify such implementation – for example, if synaptic transmission implements an addition operation via the postsynaptic membrane potential. In other places, though, the task-relevant dynamics in the brain may not map directly to basic arithmetic; rather, they may be more complicated, and require multiple FLOPs to capture. If we include these FLOPs (as we should, if we want the question to be relevant to the hardware requirements for advanced AI systems), we’re back to something closely akin to the mechanistic method, and to the same questions about brain-like-ness.

Usefulness limits

I’ll consider one final option, which seems to me (a) promising and (b) somewhat difficult to think about.

Suppose you were confronted with a computer performing various tasks, programmed by a programmer of unclear skill, using operations quite dissimilar from FLOP/s. You want some way of quantifying this computer’s computational capacity in FLOP/s. How would you do it?

As discussed above, using the minimum FLOP/s sufficient to perform any of the tasks the computer is currently programmed to perform seems dicey: this depends on where the theoretical limits of algorithmic efficiency lie, relative to algorithms the computer is running. But suppose we ask, instead, about the minimum FLOP/s sufficient to perform any useful task that the computer could in principle be programmed to perform, given arbitrary programming skill. An arbitrarily skillful programmer, after all, would presumably employ maximally efficient algorithms to use this computer to its fullest capacity.

Applied to a computer actually performing FLOP/s, this approach does well on the “One-FLOP-per-FLOP” criterion. That is, even an arbitrarily skillful programmer still cannot wring more FLOP/s out of a V100 than the computer actually performs, assuming this programmer is restricted to the computational mechanisms intended by the system’s designers. So the minimum FLOP/s sufficient to do any of the tasks that this programmer could use a V100 to perform would presumably be 1e14.

And it also fits well with what we’re intuitively doing when we ask about a system’s computational capacity: that is, we’re asking how useful this system can be for computational tasks. For instance, if a task requires 1e17 FLOP/s, can I do it with this machine? This approach gives the answers you would get if the machine actually performed FLOP/s itself.

Can we apply this approach to the brain? The main conceptual challenge, I think, is defining what sorts of interventions would count as “programming” the brain.686

• One option would be a restriction to external stimulation like e.g. talking, reading, etc. The tasks in question would be the set of tasks that any human could in principle be trained to perform, given arbitrary training time/arbitrarily skilled trainers. This would be limited by the brain’s existing methods of learning.
• Another option would be to allow direct intervention on biophysical variables in the brain. Here, the main problem would be putting limits on which variables can be intervened on, and by how much. Intuitively, we want to disallow completely remoulding the brain into a fundamentally different device, or “use” of mechanisms and variables that the brain does not currently “use” to store or process information. I think it possible that this sort of restriction can be formulated with reasonable precision, but I haven’t tried.

One might also object that this approach will focus attention on tasks that are overall much more difficult than the ones that we generally have in mind when we’re thinking about human-level task performance.687 I think that this is very likely true, but this seems quite compatible with using it as a concept of the brain’s FLOP/s capacity, as it seems fine (indeed, inuitive) if this concept indicates the limitations on the brain’s task performance imposed by hardware constraints alone, as opposed to other ways the system is sub-optimal.

### Summing up

Here is a summary of the various concepts I’ve discussed:

Minimum FLOP/s sufficient to match the brain’s task-performance Simple; broad; focuses directly on task-performance. Existing brains and AI systems provide only indirect evidence about the theoretical limits of algorithmic efficiency; questionably relevant to the FLOP/s we should expect human engineers to actually use.
Minimum FLOP/s sufficient to run a task-functional model that meets some brain-like-ness constraint, such as being a:

• “simulation of the brain”
• “reasonably brain-like model”
• model with X-very specific type of brain-like-ness
• model that captures “the algorithmic level”
• … “the lowest algorithmic level”
• … “the highest algorithmic level’
• model with no states/transitions that don’t map to the brain
Restricted space of models makes theoretical limits of algorithmic efficiency somewhat less relevant, and neuroscientific evidence more directly relevant; connection to evolution may indicate a type of findability (without needing to include such findability in the definition). Non-arbitrary brain-like-ness constraints are difficult to define with precision adequate to pick out a single number of FLOP/s; the systems we ultimately care about don’t need to be any particular degree of brain-like; functional method estimates are not based on systems designed to be brain-like; analogous standards, applied to a human-engineered computer, struggle to identify the FLOP/s that computer actually performs; the connection between evolutionary find-ability and specific computational models of the brain is often unclear.
Minimum FLOP/s sufficient to run a task-functional model that meets some findability constraint, such as being:

• the first such model humans will in fact create
• creatable by humans using X-type of training/resources/data etc.
• findable by X-type of hypothetical, evolution-like process
• creatable by an engineer “as good as evolution” at engineering
More directly relevant to the FLOP/s costs of models that we might expect humans to create, as opposed to ones that could exist in principle. “First model humans will in fact create” seems especially relevant (and functional method estimates may provide some purchase on it). Implicating of difficult further questions about which models are what kinds of findable; findability constraints based on evolutionary hypotheticals/evolution-level engineers are also difficult to define precisely, and they are fairly alien from mainstream neuroscientific discourse – a fact which makes them difficult to solicit expert opinion about and/or evaluate using evidence of the type surveyed in the report.
Other computer analogies:

• “Operations per second in the brain”
• “FLOP/s the brain performs”
• “Minimum FLOP/s sufficient to perform any task the brain could be programmed to perform”
Variable. Focusing on the tasks that the brain can be “programmed” to perform does fairly well on One-FLOP-per-FLOP, and it fits well with what we might want a notion of “FLOP/s capacity” to do, while also side-stepping questions about the degree of algorithmic inefficiency in the brain. In order to retain relevance to task-functional systems running on FLOP/s, “operations per second in the brain” and “FLOP/s the brain performs” seem to me to collapse back into something like the mechanistic method, and to correspondingly difficult questions about the theoretical limits of algorithmic efficiency, and/or brain-like-ness. Focusing on the tasks that the brain can be programmed to perform requires defining what interventions count as “programming” as opposed to reshaping – e.g., distinguishing between hardware and software, which is hard in the brain.
Figure 22: Concepts of “brain FLOP/s”

All these options have pros and cons. I don’t find any of them particularly satisfying, or obviously privileged as a way of thinking about the FLOP/s “equivalent” to the human brain. I’ve tried, in the body of the report, to use a broad framing; to avoid getting too bogged down in conceptual issues; and to survey evidence relevant to many narrower points of focus.

That said, it may be useful to offer some specific (though loose) probabilities for at least one of these. The point of focus I feel most familiar with is the FLOP/s required to run a task-functional model that satisfies a certain type of (somewhat arbitrary and ill-specified) brain-like-ness constraint, so I’ll offer some probabilities for that, keyed to the different mechanistic method ranges discussed above.

Best-guess probabilities for the minimum FLOP/s sufficient to run a task-functional model that satisfies the following conditions:

1. It includes units and connections between units corresponding to each neuron and synapse in the human brain (these units can have further internal structure, and the model can include other things as well).688
2. The functional role of these units and connections in task-performance is roughly similar to the functional role of the corresponding neurons and synapses in the brain.689

Caveats:

• These are rough subjective probabilities offered about unsettled science. Hold them lightly.690
• (2) is admittedly imprecise. My hope is that these numbers can be a helpful supplement to the more specific evidence surveyed in the report, but those who think the question ill-posed are free to ignore.691
• This is not an estimate of the “FLOP/s equivalent to the brain.” It’s an estimate of “the FLOP/s required to run a specific type of model of the brain.” See Sections 7.1-7.4 on why I think the concept of “the FLOP/s equivalent to the brain” is underspecified.
• I also think it very plausible that modeling every neuron/synapse is in some sense overkill (see Section 2.4.2) above), even in the context of various types of brain-like-ness constraints; and even more so without them.
• I assume access to “sparse FLOP/s,” as discussed in Section 2.1.1.2.2.
FLOP/s range Best-guess probability Central considerations I have in mind
<1e13 ~15% This is less than the estimate I’ve used for the spikes through synapses per second in the brain, so this range requires either that (a) this estimate is too high, or (b) satisfying the conditions above requires less than 1 FLOP per spike through synapse. (a) seems possible, as these parameters seem fairly unknown and I wouldn’t be that surprised if e.g. the average firing rate was <0.1 Hz, especially given the estimates in Lennie (2003). And (b) seems quite possible as well: a single FLOP might cover multiple spikes (for example, if what matters is a firing rate encoded in multiple spikes), and in general, it might well be possible to simplify what matters about the interactions between neurons in ways that aren’t salient to me (though note that simplifications that summarize groups of neurons are ruled out by the definition of the models in question).

This sort of range also requires <100 FLOP/s per neuron for firing decisions, which, assuming at least 1 FLOP per firing decision, means you have to be computing firing decisions less than 100 times per second. My naive guess would’ve been that you need to do it more frequently, if a neuron is operating on e.g. 1 ms timescales, but I don’t have a great sense of the constraints here, and Sarpeshkar (2010) and Dr. Paul Christiano both seemed to think it possible to compute firing decisions less than once per timestep (see Section 2.1.2.5).

And finally, this sort of range requires that the FLOP/s required to capture the contributions of all the other processes described in the mechanistic method section (e.g., dendritic computation, learning, alternative signaling mechanisms, etc.) are <1 FLOP per spike through synapse and <100 FLOP/s per neuron. Learning seems to me like the strongest contender for requiring more than this, but maybe it’s in the noise due to slower timescales, and/or only a small factor (e.g., 2× for something akin to gradient descent methods) on top of a very low-end baseline.

So overall, it doesn’t seem like this range is ruled out, even assuming that we’re modeling individual neurons and synapses. But it requires that the FLOPs costs of everything be on the low side. And my very vague impression that many experts (even those sympathetic to the adequacy of comparatively simple models) would think this range too low. That said, it also covers possible levels of simplification that current theories/models do not countenance. And it seems generally reasonable, in contexts with this level of uncertainty, to keep error bars (in both directions) wide.

1e13-1e15 ~30% This is the range that emerges from the most common type of methodology in the literature, which budgets one operation per spike through synapse, and seems to assume that (i) operations like firing decisions, that scale with the number of neurons (~1e11) rather than number of synapses (~1e14-1e15), are in the noise, and (ii) so is everything else (including learning, alternative signaling mechanisms, and so on).

As I discuss in Section 2.1.2.5, I think that assumption (i) is less solid if we budget FLOPs at synapses based on spike rates rather than timesteps, since the FLOPs costs of processes in a neuron could scale with timesteps per neuron per second, and timesteps are plausibly a few orders of magnitude more frequent than spikes, on average. Still, this range covers all neuron models with FLOP/s costs less than an Izhikevich spiking neuron model run with 1 ms timesteps (~1e15 FLOP/s for 1e11 neurons) – a set that includes many models in the integrate-and-fire family (run at similar temporal resolutions). So it still seems like a decent default budget for fairly simple models of neuron/synapse dynamics.

Dendritic computation and learning seem like prominent processes missing from such a basic model, so this range requires that these don’t push us beyond 1e15 FLOP/s. If we would end up on the low end of this range (or below) absent those processes, this would leave at least one or two orders of magnitude for them to add, which seems like a reasonable amount of cushion to me, given the considerations surveyed in Sections 2.1.2.2 and 2.2. That said, my best guess would be that we need at least a few FLOPs per spike through synapse to cover short-term synaptic plasticity, so there would need to be less than ~3e14 spikes through synapses per second to leave room for this. And most basic type of integrate-and-fire neuron model already puts us at ~5e14 FLOP/s (assuming 1 ms timesteps), so this doesn’t leave much room for increases from dendritic computation.692

Overall, this range represents a simple default model that seems fairly plausible to me, despite not budgeting explicitly for these other complexities; and various experts appear to find this type of simple default persuasive.693

1e15-1e17 ~30%. This range is similar to the last, but with an extra factor of 100x budgeted to cover various possible complexities that came up in my research. Specifically, assuming the number of spikes through synapses falls in the range I’ve used (1e13-1e15), it covers 100-10,000 FLOPs per spike through synapse (this would cover Sarpeshkar’s (2010) 50 FLOPs per spike through synapse for synaptic filtering and learning; along with various models of learning discussed in Section 2.2.2) as well as 1e4-1e6 FLOP/s per neuron (this would cover, on the top end, single-compartment Hodgkin-Huxley models run with 0.1 ms timesteps – a level of modeling detail/complexity that I expect many computational neuroscientists to consider unnecessary).

Overall, this range seems very plausibly adequate to me, and various experts I engaged with seemed to agree.694 I’m much less confident that it’s required, but as mentioned above, my best guess is that you need at least a few FLOPs per spike through synapse to cover short-term synaptic plasticity, and plausibly more for more complex forms of learning; and it seems plausible to me that ultimately, FLOPs budgets for firing decisions (including dendritic computation) are somewhere between Izhikevich spiking neurons and Hodgkin-Huxley models. But as discussed above, lower ranges seem plausible as well.

1e17-1e21 ~20% As I noted in the report, I don’t see a lot of strong positive evidence that budgets this high are required. The most salient considerations for me are (a) the large FLOP/s costs of various DNN models of neuron behavior discussed in the report, which could indicate types complexity that lower budgets do not countenance, and (b) if you budget at least one FLOP per timestep per synapse (as opposed to per spike through synapse), along with <1 ms timesteps, and>1e14 synapses, then you get above 1e17 FLOP/s, and it seems possible that sufficiently important and unsimplifiable changes are taking place at synapses this frequently (for example, changes involved in learning). Some experts also seem to treat “time-steps per second per variable” as a default method of generating FLOP/s estimates (and there may be many variables per synapse – see e.g. Benna and Fusi (2016)).

Beyond this, the other central pushes in this direction I feel involve (a) the general costliness of low-level modeling of biological and chemical processes; (b) the possibility that learning and dendritic computation introduce more complexity than 1e17 FLOP/s budgets for; (c) the fact that this range covers four orders of magnitude; (d) the possibility of some other type of unknown error or mistake, not currently on my radar, that pushes required FLOP/s into this range, and (e) an expectation that a decent number of experts would give estimates in this range as well.

>1e21 ~5% Numbers this high start to push past the upper bounds discussed in the limit method section. These bounds don’t seem airtight to me, but I feel reasonably persuaded by the hardware arguments discussed in Section 4.2.2 (e.g., I expect the brain to be dissipating at least a few kT per FLOP required to meet the conditions above, and to use at least 1 ATP, of which it has a maximum of ~1e20/s available). I also don’t see a lot of positive reason to go this high (though the DNN models I mentioned are one exception to this); other methods generally point to lower numbers; and some experts I spoke to were very confident that numbers in this range are substantial overkill. That said, I also put macroscopic probability on the possibility that these experts and arguments (possibly together with the broader paradigms they assume) are misguided in some way; that the conditions above, rightly understood, somehow end up requiring very large FLOP/s budgets (though this last one feels more like uncertainty about the concepts at stake in the question than uncertainty about the answer); and/or that the task-relevant causal structure in the brain is just intrinsically very difficult to replicate using FLOP/s (possibly because it draws on analog physical primitives, continuous/very fine-grained temporal dynamics, and/or complex biochemical interactions that are cheap for the brain, but very expensive to capture with FLOP/s). And in general, long tails seem appropriate in contexts with this level of uncertainty.
Figure 23: Best guess probabilities for the minimum FLOP/s sufficient to run a task-functional model that satisfies conditions (1) and (2) above.

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## Endnotes

• 1. The names “mechanistic method” and “functional method” were suggested by our technical advisor Dr. Dario Amodei, though he uses a somewhat more specific conception of the mechanistic method. Sandberg and Bostrom (2008) also distinguish between “straightforward multiplicate estimates” and those that are based on “analogy or constraints” (p. 84, Appendix A).
• 2. Here I am using “software” in a way that includes trained models in addition to hand-coded programs. Some forms of hardware (including neuromorphic hardware – see Mead (1989)) complicate traditional distinctions between hardware and software, but the broader consideration at stake here – e.g., that task-performance requires organizing available computational power in the right way – remains applicable.
• 3. Though it also seems easier, in general, to show that X is enough, than that X is strictly required – an asymmetry present throughout the report.
• 4. The probabilities reported here should be interpreted as subjective levels of confidence or “credences,” not as claims about objective frequencies, statistics, or “propensities” (see Peterson (2009), Chapter 7, for discussion of various alternative interpretations of probability judgments). One way of defining these credences is via preferences over lotteries - a definition I find useful (though not fully satisfactory). On such a definition, “I think it more likely than not” means that, for example, if I had the option to win $10,000 if 1015 FLOP/s is sufficient, in principle, to match human-level task-performance, or the option to win$10,000 if 1015 FLOP/s is not sufficient, I would choose the former option. Skepticism about my answer should go in proportion to confidence that 1e15 FLOP/s is not sufficient (e.g., those who disagree should prefer the latter option rather than the former), rather than with dissatisfaction with the evidence available either way (I too am quite dissatisfied in this regard), or disinclination to take real-world bets (why turn down a free chance at $10,000?). That said, for various reasons, I don’t find this definition of subjective probability judgments fully satisfactory (in particular, it transforms probabilistic claims about the world into true/false claims about one’s betting behavior– and it’s not clear exactly what sort of betting behavior is implied, or what consistency in such behavior assumed), so I offer it more as a gesture at a way of soliciting subjective credences than as an endorsed definition. See Peterson (2009), section 7.5, for discussion of lotteries of this type in the context of the literature on decision-theory. See also this blog post by Andrew Critch for more informal discussion; and see Muehlhauser (2017a), section 2, for discussion of some complexities involved in using these probabilities in practice. • 5. I focus on this model in particular because I think it fits best with the mechanistic method evidence I’ve thought about most and take most seriously. Offering specific probabilities keyed to the minimum FLOP/s sufficient for task-performance, by contrast, requires answering further questions about the theoretical limits of algorithmic efficiency that I haven’t investigated. • 6. See here for V100 prices (currently ~$8,799); and here the $1 billion Fugaku pricetag: “The six-year budget for the system and related technology development totaled about$1 billion, compared with the $600 million price tags for the biggest planned U.S. systems.” Fugaku FLOP/s performance is listed here, at around ~4×1017 FLOP/s-5×1017 FLOP/s. Google’s TPU supercomputer, which recently broke records in training ML systems, can also do ~4×1017 FLOP/s, though I’m not sure the costs. See Kumar (2020): “In total, this system delivers over 430 PFLOPs of peak performance.” The A100, for ~$200,000, can do 5×1015 FLOP/s – see Mehar (2020). NVIDIA’s newest SuperPOD can deliver ~7×1017 of AI performance – see Paikeday (2020).
• 7. See discussion in Section 1.3 below.
• 8. Selection effects include: expertise related to an issue relevant to the report, willingness to talk with me about the subject, recommendation by one of the other experts I spoke with as a possible source of helpful input, and connection (sometimes a few steps removed) with the professional and social communities that intersect at Open Philanthropy.
• 9. See Poldrack et al. (2017); Vul and Pashler (2017); Uttal (2012); Button et al. (2013); Szucs and P.A. loannidis (2017); and Carp (2012). And see also Muehlhauser (2017b), Appendix Z.8, for discussion of his reasons for default skepticism of published studies. My thanks to Luke Muehlhauser for suggesting this type of consideration and these references.
• 10. This effort is itself part of a project at Open Philanthropy currently called Worldview Investigations, which aims to investigate key questions informing our grant-making.
• 11. See, for example, Moravec (1998), chapter 2; and Kurzweil (2005), chapter 3. See this list from AI Impacts for related forecasts.
• 12. See, for example, Malcolm (2000); Lanier (2000) (“Belief # 5”); Russell (2019) (p. 78). AI Impacts offers a framework that I find helpful, which uses indifference curves indicating which combinations hardware and software capability yield the same overall task-performance. This framework (see especially Figure 3) makes clear that the first human-level AI systems could use much more or much less hardware than the amount “equivalent” to the human brain (at least assuming that this amount is not the absolute minimum) – though see figure 4 for a scenario in which brain-equivalent hardware is a better basis for forecasts.
• 13. Moravec argues here that “under current circumstances, I think computer power is the pacing factor for AI” (see his second reply to Robin Hanson). Kurzweil (2005) devotes all of Chapter 4 to the question of software.
• 14. For example: a ResNet-152 uses ~1e10 FLOP to classify an image, but took ~1e19 FLOP (a billion times more) to train, according to Hernandez and Amodei (2018) (see appendix, though see also Hernandez and Brown (2020) for discussion of decreasing training costs for vision models over time).
• 15. Silver et al. (2017): “Over the course of training, 4.9 million games of self-play were generated” (see “Empirical analysis of AlphaGo Zero training”). A bigger version of the model trained on 29 million games. See Kaplan et al. (2020) and Hestness et al. (2017) for more on the scaling properties for training in deep learning.
• 16. The question of what sorts of task-performance will result from what sorts of training is centrally important in this context, and I am not here assuming any particular answers to it.
• 17. The fact that training a model requires running it a lot makes this clear. But there are also more complex relationships between e.g. model size and amount of training data. See Kaplan et al. (2020) and Hestness et al. (2017).
• 18. See e.g. Dongerra et al. (2003): “the performance of a computer is a complicated issue, a function of many interrelated quantities. These quantities include the application, the algorithm, the size of the problem, the high-level language, the implementation, the human level of effort used to optimize the program, the compiler’s ability to optimize, the age of the compiler, the operating system, the architecture of the computer and the hardware characteristics” (p. 805); Moravec (1988): “Any particular formula for estimating power may be grossly misled by an unlucky or diabolic counterexample. For instance, if a computer’s power were defined simply by how many additions per second it could do, an otherwise useless special circuit made of an array of fast adders, and nothing else, costing a few hundred dollars, could outperform a