Efficient coding explains neural response homeostasis and stimulus-specific adaptation

Reviewer #1 (Public review):

This work derives a general theory of optimal gain modulation in neural populations. It demonstrates that population homeostasis is a consequence of optimal modulation for information maximization with noisy neurons. The developed theory is then applied to the distributed distributional code (DDC) model of the primary visual cortex to demonstrate that homeostatic DDCs can account for stimulus specific adaptation.

Strengths:

The theory of gain modulation proposed in the paper is rigorous and the analysis is thorough. It does address the issue in an interesting, general setting. The proposed approach separates the question of which bits of sensory information are transmitted (as defined by a specific computation and tuning curve shapes) and how well are they transmitted (as defined by the tuning curve gain optimized to combat noise). This separation permits the application of the developed theory to different neural systems.

Weaknesses:

The manuscript effectively consits of two parts: a general theory of optimal gain modulation and a DDC model of the visual cortex. From my perspective it is not entirely clear which components of the developed theory and the model it is applied to are essential to explain the experimental phenomena in the visual cortex (Fig. 12). This "separation" into two parts makes this work, in my view, somewhat diffused.

Overall, I think this is an interesting contribution and I assess it positively. It has the potential of deepening our understanding of efficient neural representations beyond sensory periphery.

Reviewer #2 (Public review):

Summary:

Using the theory of efficient coding, the authors study how neural gains may be adjusted to optimize information transmission by noisy neural populations while minimizing metabolic cost, under the assumption that other aspects of neural activity (i.e. tuning) are determined by the computation performed by the network.

The manuscript first presents mathematical results for the general case where the computational goals of the neural population are not specified (the computation is implicit in the assumed tuning curves). It then develops the theory for a specific probabilistic coding scheme. The general theory provides an explanation for firing rate homeostasis at the level of neural clusters with firing rate heterogeneity within clusters. The specific application further explains stimulus-specific adaptation in visual cortex.

The mathematical derivations, simulations and application to visual cortex data are solid as far as I can tell.

This remains a highly technical manuscript although the authors have improved the clarity of presentation of the general theory (which is the bulk of the work presented) and better motivated/explained modeling assumptions and choices. In the second part, the manuscript focuses on a specific code (homeostatic DDC) showing that this can be implemented by divisive normalization and can explain stimulus-specific adaptation.

Strengths:

The problem of efficient coding is a long-standing and important one. This manuscript contributes to that field by proposing a theory of efficient coding through gain adjustments, independent of the computational goals of the system. The main assumption, and insight, is that computational goals and efficiency can be in some sense factorized: tuning curve shapes are determined by the computational goal, whereas gains can be adjusted to optimize transmission of information.

One key result is a normative explanation for firing rate homeostasis at the level of neural clusters (groups of neurons that perform a similar computation) with firing rate heterogeneity within each cluster. Both phenomena are widely observed, and reconciling them under one theory is important.

The mathematical derivations are thorough. Although the model of neural activity is artificial, the authors make sure to include many aspects of cortical physiology, while also keeping the models quite general.

Section 2.5 derives the conditions in which homeostasis would be near-optimal in cortex, which appear to be consistent with many empirical observations in V1. This indicates that homeostasis in V1 might be indeed a close to optimal solution to code efficiently in the face of noise.

The application to the data of Benucci et al 2013 is the first to offer a normative explanation of stimulus-specific adaptation in V1.

The novelty and significance of the work are presented clearly in the newly extended Introduction and Discussion.

Weaknesses:

The manuscript remains hard to read. The general theory occupies most of the manuscript, as needed to convey it fully. But as a result the second part on homeostatic DDC and adaptation is somewhat underdeveloped and risks having less visibility than it might deserve.

The paper Benucci et al 2013 shows that homeostasis holds for some stimulus distributions, but not others i.e. when the 'adapter' is present too often. This manuscript, like the Benucci paper, discards those datasets. But from a theoretical standpoint, it seems important to consider why that would be the case, and if it can be predicted by the theory proposed here. The authors now acknowledge this limitation in the Discussion.

Author response:

The following is the authors’ response to the original reviews.

Reviewer #1(Public Review):

Major comments:

(1) Interpretation of key results and relationship between different parts of the manuscript. The manuscript begins with an information-transmission ansatz which is described as ”independent of the computational goal” (e.g. p. 17). While information theory indeed is not concerned with what quantity is being encoded (e.g. whether it is sensory periphery or hippocampus), the goal of the studied system is to *transmit* the largest amount of bits about the input in the presence of noise. In my view, this does not make the proposed framework ”independent of the computational goal”. Furthermore, the derived theory is then applied to a DDC model which proposes a very specific solution to inference problems. The relationship between information transmission and inference is deep and nuanced. Because the writing is very dense, it is quite hard to understand how the information transmission framework developed in the first part applies to the inference problem. How does the neural coding diagram in Figure 3 map onto the inference diagram in Figure 10? How does the problem of information transmission under constraints from the first part of the manuscript become an inference problem with DDCs? I am certain that authors have good answers to these questions - but they should be explained much better.

We are very thankful to the reviewer for highlighting the potential confusion surrounding these issues, in particular the relationship between the two halves of the paper – which was previously exacerbated by the length of the paper. We have now added further explanations at different points within the manuscript to better disentangle these issues and clarify our key assumptions. We have also significantly cut the length of the paper by moving more technical discussions to the Methods or Appendices. We will summarise these changes here and also clarify the rationale for our approach and point out potential disagreements with the reviewer.

Key to our approach is that we indeed do not assume the entire goal of the studied neural system (whether part of the sensory system or not) is to transmit the largest amount of information about the stimulus input (in the presence of noise). In fact, general computations, including the inference of latent causes of inputs, often require filtering out or ignoring some information in the sensory input. It is thus not plausible that tuning curves in general (i.e. in an arbitrary part of the nervous system) are optimised solely with regards to the criterion of information transmission. Accordingly we do not assume they are entirely optimised for that purpose. However, we do make a key assumption or hypothesis (which like any hypothesis might turn out to be partly or entirely wrong): that (1) a minimal feature of the tuning curve (its scale or gain) is entirely free to be optimised for the aim of information transmission (or more precisely the goal of combating the detrimental effect of neural noise on coding fidelity), (2) other aspects of the population tuning curve structure (i.e. the shape of individual tuning curves and their arrangement across the population) are determined by (other) computational goals beyond efficient coding. (Conceptually, this is akin to the modularization between indispensible error correction and general computations in a digital computer, and the need for the former to be performed in a manner that is agnostic as to the computations performed.) We have added two paragraphs in the manuscript which present the above rationale and our key hypothesis or assumption. The first of these was added to the (second paragraph of the) Introduction section, and the second is a new paragraph following Eq. 1 (which is about the gain-shape decomposition of the tuning curves, and the optimisation of the former based on efficient coding) of Results.

Our paper can be divided into two parts. In the first part, we develop a general, computationally agnostic (in the above sense, just as in the digital computer example), efficient coding theory. In the second part, we apply that theory to a specific form of computation, namely the DDC framework for Bayesian inference. The latter theory now determines the tuning curve shapes. When combined with the results of the first part (which dictate the tuning curve scale or gain according to efficient coding theory), this “homeostatic DDC” model makes full predictions for the tuning curves (i.e., both scale and shape) and how they should adapt to stimulus statistics.

So to summarise, it is not the case that the problem of information transmission (or rather mitigating the effect noise on coding fidelity under metabolic constraints), dealt with in the first part, has become a problem of Bayesian inference. But rather, the dictates of efficient coding for optimal gains for coding fidelity (under constraints) have been applied to and combined with a computational theory of inference.

We have added new expository text before and after Eq. 17 in Sec. 2.7 (at the beginning of the second part of the paper on homeostatic DDCs) to again make the connection with the first part and the rationale for its combination with the original DDC framework more clear.

With the changes outlined above, we believe and hope the connection between the two parts (which we agree with the reviewer, was indeed rather obscure previously) has been adequately clarified.

(2) Clarity of writing for an interdisciplinary audience. I do not believe that in its current form, the manuscript is accessible to a broader, interdisciplinary audience such as eLife readers. The writing is very dense and technical, which I believe unnecessarily obscures the key results of this study.

We thank the reviewer for this comment. We have taken several steps to improve the accessibility of this work for an interdisciplinary audience. Firstly, several sections containing dense, mathematical writing have now been moved into appendices or the Methods section, out from the main text; in their place we have made efforts to convey the core of the results, and to providing intuitions, without going into unnecessary technical detail. Secondly, we have added additional figures to help illustrate key concepts or assumptions (see Fig. 1B clarifying the conceptual approach to efficient coding and homeostatic adaptation, and Fig. 8A describing the clustered population). Lastly, we have made sure to refer back to the names of symbols more often, so as to make the analysis easier to follow for a reader with an experimental background.

(3) Positioning within the context of the field and relationship to prior work. While the proposed theory is interesting and timely, the manuscript omits multiple closely related results which in my view should be discussed in relationship to the current work. In particular, a number of recent studies propose normative criteria for gain modulation in populations: • Duong, L., Simoncelli, E., Chklovskii, D. and Lipshutz, D., 2024. Adaptive whitening with fast gain modulation and slow synaptic plasticity. Advances in Neural Information Processing Systems

Tring, E., Dipoppa, M. and Ringach, D.L., 2023. A power law describes the magnitude of adaptation in neural populations of primary visual cortex. Nature Communications, 14(1), p.8366.

Ml ynarski, W. and Tkaˇcik, G., 2022. Efficient coding theory of dynamic attentional modulation. PLoS Biology

Haimerl, C., Ruff, D.A., Cohen, M.R., Savin, C. and Simoncelli, E.P., 2023. Targeted V1 co-modulation supports task-adaptive sensory decisions. Nature Communications • The Ganguli and Simoncelli framework has been extended to a multivariate case and analyzed for a generalized class of error measures:

Yerxa, T.E., Kee, E., DeWeese, M.R. and Cooper, E.A., 2020. Efficient sensory coding of multidimensional stimuli. PLoS Computational Biology

Wang, Z., Stocker, A.A. and Lee, D.D., 2016. Efficient neural codes that minimize LP reconstruction error. Neural Computation, 28(12),

We thank the reviewer again for bringing these works to our attention. For each, we explain whether we chose to include them in our Discussion section, and why.

(1) Duong et al. (2024): We decided not to discuss this manuscript, as our assessment is that it is very relevant to our work. That study starts with the assumption that the goal of the sensory system under study is to whiten the signal covariance matrix, which is not the assumption we start with. A mechanistic ingredient (but not the only one) in their approach is gain modulation. However, in their case it is the gains of computationally auxiliary inhibitory neurons that is modulated and not (as in our case) the gain the (excitatory) coding neurons (i.e. those which encode information about the stimulus and whose response covariance is whitened). These key distinction make the connection with our work quite loose and we did not discuss this work.

(2) Tring et al. (2023): We have added a discussion of the results of this paper and its relationship to the results of our work and that of Benucci et al. This appears in the 7th paragraph of the Discussion. This study is indeed highly relevant to our paper, as it essentially replicates the Benucci et al. experiment, this time in awake mice (rather than anesthetised cats). However, in contrast to the resul‘ts of Benucci et al., Tring et al. do not find firing rate homeostasis in mouse V1. A second, remarkable finding of Tring et al. is that adaptation mainly changes the scale of the population response vector, and only minimally affects its direction. While Tring et al. do not portray it as such, this behaviour amounts to pure stimulus-specific adaptation without the neuron-specific factor found in the Benucci et al. results (see Eq. 24 of our manuscript). As we discuss in our manuscript, when our homeostatic DDC model is based on an ideal-observer generative model, it also displays pure stimulus-specific adaptation with no neuronal factor. Our final model for Benucci’s data did contain a neural factor, because we used a non-ideal observer DDC (in particular, we assumed a smoother prior distribution over orientations compared to the distribution used in the experiment - which has a very sharp peak – as it is more natural given the inductive biases we expect in the brain). The resultant neural factor suppresses the tuning curves tuned to the adaptor stimulus. Interestingly, when gain adaptation is incomplete, and happens to a weaker degree compared to what is necessary for firing rate homeostasis, an additional neural factor emerges that is greater than one for neurons tuned to the adaptor stimulus. These two multiplicative neural factors can approximately cancel each other; such a theory would thus predict both deviation from homeostasis and approximately pure stimulus-specific adaptation. We plan to explore this possibility in future work.

(3) Ml ynarski and Tkaˇcik (2022): We are now citing and discussing this work in the Discussion (penultimate paragraph), in the context of a possible future direction, namely extending our framework to cover the dynamics of adaptation (via a dynamic efficient gain modulation and dynamic inference). We have noted there that Mlynarski have used such a framework (which while similar has key technical differences with our approach) based on a task-dependent efficient coding objective to model top-down attentional modulation. By contrast, we have studied bottom-up and task-independent adaptation, and it would be interesting to extend our framework and develop a model to make predictions for the temporal dynamics of such adaptation.

(4) Haimerl et al. (2023): We have elected not to include this work within our discussion either, as we do not believe it is sufficiently relevant to our work to warrant inclusion. Although this paper also considers gain modulation of neural activity, the setting and the aims of the theoretical work and the empirical phenomena it is applied to are very different from our case in various ways. Most importantly, this paper is not offering a normative account of gain modulation; rather, gain modulation is used as a mechanism for enabling fast adaptive readouts of task relevant information.

(5) Yerxa et al. (2020): We have now included a discussion of this paper in our Discussion section. Note that, even though this study generalises the Ganguli and Simoncelli framework to higher diemsnions, just like that paper it still places strict requirements (which are arguably even more stringent in higher dimensions) on the form of the tuning curves in the population, viz. that there exists a differentiable transform of the stimulus space which renders these unimodal curves completely homogeneous (i.e., of the same shape, and placed regularly and with uniform density).

(6) Wang et al. (2016): We have included this paper in our discussion as well. As above, this paper does not consider general tuning curves, and places the same constraint on their shape and arrangement as in Ganguli and Simoncelli paper.

More detailed comments and feedback:

(1) I believe that this work offers the possibility to address an important question about novelty responses in the cortex (e.g. Homann et al, 2021 PNAS). Are they encoding novelty per-se, or are they inefficient responses of a not-yet-adapted population? Perhaps it’s worth speculating about.

We are not sure why the relatively large responses to “novel” or odd-ball stimuli should be considered inefficient or unadapted: in the context in which those stimuli are infrequent odd-balls (and thus novel or surprising when occurring), efficient coding theory would indeed typically predict a large response compared to the (relatively suppressed) responses to frequently occurring stimuli. Of course, if the statistics change and the odd-ball stimulus now becomes frequent, adaptation should occur and would be expected to suppress responses to this stimulus. As to the question of whether (large) responses to infrequent stimuli can or should be characterised as novelty responses: this is partly an interpretational or semantic issue – unless it is grounded in knowledge of how downstream populations use this type of coding in V1, which could then provide a basis for solidly linking them to detection of novelty per se. In short, our theory, could be applied to Homann et al.’s data, but we consider that beyond the scope of the current paper.

(2) Clustering in populations - typically in efficient coding studies, tuning curve distributions are a consequence of input statistics, constraints, and optimality criteria. Here the authors introduce randomly perturbed curves for each cluster - how to interpret that in light of the efficient coding theory? This links to a more general aspect of this work - it does not specify how to find optimal tuning curves, just how to modulate them (already addressed in the discussion).

We begin by addressing the reviewer’s more general concern regarding the fact that our theory does not address the problem of finding optimal tuning curves, only that of modulating them optimally. As we expound within the updated version of the paper (see the newly expanded 3rd paragraph in Sec. 2.1 and the expanded 2nd paragraph in Introduction), it is not plausible that the sole function of sensory systems, and neural circuits more generally, is the transmission of information. There are many other computational tasks which must be performed by the system, such as the inference of the latent causes of sensory inputs. For many such tasks, it is not even desirable to have complete transmission of information about the external stimulus, since a substantial portion of that information is not important for the task at hand, and must be discarded. For example, such discarding of information is the basis of invariant representations that occur, e.g., in higher visual areas. So we recognise that tuning curve shapes are in general dictated and shaped by computational goals beyond transmission of information or error correction. As such, we have remained agnostic as to the computational goals of neural systems and therefore the shape of the tuning curve. We have made the assumption and adopted the postulate that those computational goals determine the shape of the tuning curves, leaving the gains to be adjuted freely for the purpose of mitigating the effect noise on coding fidelity (this is similar to how error correction is done in computers independendently of the computations performed). by assuming that those computational goals are captured adequately by the shape of tuning curves, this leaves us free to optimise the gains of those curves for purely information theoretic objectives. Finally, we note that the case where the tuning curve shapes are additionally optimised for information transmission is a special case of our more general approach. For further discussion, see the updated version of our introduction.

We now turn to our choice to model clusters using random perturbations. This is, of course, a toy model for clustering tuning curves within a population. With this toy model we are attempting to capture the important aspects of tuning curve clusters within the population while not over-complicating the simulations. Within any neural population, there will be tuning curves that are similar; however, such curves will inevitably be heterogeneous, as opposed to completely identical. Thus, when we cluster together similar curves there will be an “average” cluster tuning curve (found by, e.g., normalising all individual curves and taking the average), which all other tuning curves within the cluster are deviations from. The random perturbations we apply are our attempt to capture these deviations. However, note that the perturbations are not fully random, but instead have an “effective dimensionality” which we vary over. By giving the perturbations an effective dimensionality, we aim to capture the fact that deviations from the average cluster tuning curve may not be fully random, and may display some structure.

(3) Figure 8 - where do Hz come from as physical units? As I understand there are no physical units in simulations.

We have clarified this within the figure caption. The within-cluster optimisation problem requires maximising a quadratic program subject to a constraint on the total mean spike count of the cluster. The objective for the quadratic program is however mathematically homogeneous. So we can scale the variables and parameters in a consistent to be in units of Hz – i.e., turn them into mean firing rates, instead of mean spike counts, with an assumption on the length of the coding time interval. We fix this cluster firing rate to be k × 5 Hz, so that the average single-neuron firing rate is 5 Hz (based on empirical estimates – see our Sec. 2.5). This agrees with our choice of µ in our simulations (i.e., µ = 10) if we assume a coding interval of 0.1 seconds.

(4) Inference with DDCs in changing environments. To perform efficient inference in a dynamically changing environment (as considered here), an ideal observer needs some form of posterior-prior updating. Where does that enter here?

A shortcoming of our theory, in its current form, is that it applies only to the system in “steady-state”, without specifying the dynamics of how adaptation temporlly evolves (we assume the enrivonment has periods of relative stability that are of relatively long duration compared to the dynamical timescales of adaptation, and consider the properties of the well-adapted steady state population). Thus our efficient coding theory (which predicts homeostatic adaptation under the outlined conditions) is silent on the time-course over which homeostasis occurs. Likewise, the DDC theory (in its original formulation in Vertes & Sahani) is silent on dynamic updating of posteriors and considers only static inference with a fixed internal model. We have now discuss a new future directoin in the Discussion (where we cite the work of Mlynarski and Tkacik) to point out that our theory can in principle be extended (based on dynamic inference and efficient coding) to account for the dynamics of attention, but this is beyond the scope of the current work.

(5) Page 6 - ”We did this in such a way that, for all , the correlation matrices, (), were derived from covariance matrices with a 1/n power-law eigenspectrum (i.e., the ranked eigenvalues of the covariance matrix fall off inversely with their rank), in line with the findings of Stringer et al. (2019) in the primary visual cortex.” This is a very specific assumption, taken from a study of a specific brain region - how does it relate to the generality of the approach?

Our efficient coding framework has been formulated without relying on any specific assumptions about the form of the (signal or noise) correlation matrices in cortex. The homeostatic solution to this efficient coding problem, however, emerges under certain conditions. But, as we demonstrate in our discussion of the analytic solutions to our efficient coding objective and the conditions necessary for the validity of the homeostatic solution, we expect homeostasis to arise whenever the signal geometry is sufficiently high-dimensional (among other conditions). By this we mean that the fall-off of the eigenvalues of the signal correlation matrix must be sufficiently slow. Thus, a fall-off in the eigenvalue spectrum slower than 1/n would favor homeostasis even more than our results. If the fall off was faster, then whether or not (and to what degree) firing rate homeostasis becomes suboptimal depends on factors such as the fastness of the fall-off and also the size of the population. Thus (1) rate homeostasis does not require the specific 1/n spectrum, but that spectrum is consistent with the conditions for optimality of rate homeostasis, (2) in our simulations we had to make a specific choice, and relying on empirical observations in V1 was of course a well-justified choice (moreover, as far as we are aware, there have been no other studies that have characterised the spectrum of the signal covariance matrix in response to natural stimuli, based on large population recordings).

Reviewer #2 (Public Review):

Strengths:

The problem of efficient coding is a long-standing and important one. This manuscript contributes to that field by proposing a theory of efficient coding through gain adjustments, independent of the computational goals of the system. The main result is a normative explanation for firing rate homeostasis at the level of neural clusters (groups of neurons that perform a similar computation) with firing rate heterogeneity within each cluster. Both phenomena are widely observed, and reconciling them under one theory is important.

The mathematical derivations are thorough as far as I can tell. Although the model of neural activity is artificial, the authors make sure to include many aspects of cortical physiology, while also keeping the models quite general.

Section 2.5 derives the conditions in which homeostasis would be near-optimal in the cortex, which appear to be consistent with many empirical observations in V1. This indicates that homeostasis in V1 might be indeed close to the optimal solution to code efficiently in the face of noise.

The application to the data of Benucci et al 2013 is the first to offer a normative explanation of stimulus-specific and neuron-specific adaptation in V1.

We thank the reviewer for these assessments.

Weaknesses:

The novelty and significance of the work are not presented clearly. The relation to other theoretical work, particularly Ganguli and Simoncelli and other efficient coding theories, is explained in the Discussion but perhaps would be better placed in the Introduction, to motivate some of the many choices of the mathematical models used here.

We thank the reviewer for this comment; we have updated our introduction to make clearer the relationship between this work and previous works within efficient coding theory. Please see the expanded 2nd paragraph of Introduction which gives a short account of previous efficient coding theories and now situates our work and differentiates it more clearly from past work.

The manuscript is very hard to read as is, it almost feels like this could be two different papers. The first half seems like a standalone document, detailing the general theory with interesting results on homeostasis and optimal coding. The second half, from Section 2.7 on, presents a series of specific applications that appear somewhat disconnected, are not very clearly motivated nor pursued in-depth, and require ad-hoc assumptions.

We thank the reviewer for this suggestion. The reviewer is right to note that our paper contains both the exposition of a general efficient coding theory framework in addition to applications of that framework. Following your advice we have implemented the following changes. (1) significantly shortened or entirely moved some of the less central results in the second half of Results, to the Methods or appendices (this includes the entire former section 2.7 and significant shortening of the section on implementation of Bayes ratio coding by divisive normalisation). (2) We have added a new figure (Fig 1B) and two long pieces of text to the (2nd paragraph of) Introduction, after Eq. (1), and in Sec. 2.7 (introducing homeostatic DDCs) to more clearly explain and clarify the assumptions underlying our efficient coding theory, and its connection with the second half of the Results (i.e. application to DDC theory of Bayesian inference), and better motivate why we consider the homeostatic DDC.

For instance, it is unclear if the main significant finding is the role of homeostasis in the general theory or the demonstration that homeostatic DDC with Bayes Ratio coding captures V1 adaptation phenomena. It would be helpful to clarify if this is being proposed as a new/better computational model of V1 compared to other existing models.

We see the central contribution of our work as not just that homeostasis arises as a result of an efficient coding objective, but also that this homeostasis is sufficient to explain V1 adaptation phenomena - in particular, stimulus specific adaptation (SSA) - when paired with an existing theory of neural representation, the DDC (itself applied to orientation coding in V1). Homeostatic adaptation alone does not explain SSA; nor do DDCs. However, when the two are combined they provide an explanation for SSA. This finding is significant, as it unifies two forms of adaptation (SSA and homeostatic adaptation) whose relationship was not previously appreciated. Our field does not currently have a standard model of V1, and we do not claim to have provided one either; rather, different models have captured different phenomena in V1, and we have done so for homeostatic SSA in V1.

Early on in the manuscript (Section 2.1), the theory is presented as general in terms of the stimulus dimensionality and brain area, but then it is only demonstrated for orientation coding in V1.

The efficient coding theory developed in Section 2 is indeed general throughout, we make no assumptions regarding the shape of the tuning curves or the dimensionality of the stimulus. Further, our demonstrations of the efficient coding theory through numerical simulations - make assumptions only about the form of the signal and noise covariance matrices. When we later turn our attention away from the general case, our choice to focus on orientation coding in V1 was motivated by empirical results demonstrating a co-occurrence of neural homeostasis and stimulus specific adaptation in V1.

The manuscript relies on a specific response noise model, with arbitrary tuning curves. Using a population model with arbitrary tuning curves and noise covariance matrix, as the basis for a study of coding optimality, is problematic because not all combinations of tuning curves and covariances are achievable by neural circuits (e.g. https://pubmed.ncbi.nlm.nih.gov/27145916/ )

First, to clarify, our theory allows for complete generality of neural tuning curve shapes, and assumes a broad family of noise models (which, while not completely arbitrary, includes cases of biological relevance and/or models commonly used in the theoretical literature). Within this class of noise covariance models, we have shown numerical results for different values for different parameters of the noise covariance model, but more importantly, have analytically outlined the general properties and requirements on noise strength and structure (and its relationship to tuning curves and signal structure) under which homeostatic adaptation would be optimal. Regarding the point that not all combinations of tuning curves and noise covariances occur in biology or are achievable by neural circuits: (1) If we are guessing correctly the specific point of the reviewer’s reference to the review paper by Kohn et al. 2016, we have in fact prominently discussed the case of information limiting noise which corresponds to a specific relationship between signal structure (as determined by tuning curves) and noise structure (as specified by the noise covariance matrix). Our family of noise models include that biologically relevant case and we have indeed paid it particular attention in our simulations and discussions (see discussion of Fig. 7 in Sec. 2.3, and that of aligned noise in Sec. 2.5). (2) As for the more general or abstract point that not all combinations of noise covariance and tuning curve structures are achievable by neural circuits, we can make the following comments. First, in lieu of a full theoretical or empirical understanding of the achievable combinations (which does not exist), we have outlined conditions for homeostatic adaptations under a broad class of noise models and arbitrary tuning curves. If some combinations within this class are not realised in biology, that does not invalidate the theoretical results, as the latter have been derived under more general conditions, which nevertheless include combinations that do occur in biology and are achievable by neural circuits (which, as pointed out, include the important case of aligned noise and signal structure – as reviewed in Kohn et al.– to which we have paid particular attention).

The paper Benucci et al 2013 shows that homeostasis holds for some stimulus distributions, but not others i.e. when the ’adapter’ is present too often. This manuscript, like the Benucci paper, discards those datasets. But from a theoretical standpoint, it seems important to consider why that would be the case, and if it can be predicted by the theory proposed here.

The theory we provide predicts that, under certain (specified) conditions, we ought to see deviation from exact homeostatic results; indeed, we provide a first order approximation to the optimal gains in this case which quantifies such deviations when they are small. However, unfortunately the form of this deviation depends on a precise choice of stimulus statistics (e.g. the signal correlation matrix, the noise correlation matrix averaged over all stimulus space, and other stimulus statistics), in contrasts to the universality of the homeostatic solution, when it is a valid approximation. In our model of Benucci et al.’s experiment, we restrict to a simple one-dimensional stimulus space (corresponding to orientated gratings), without specifying neural responses to all stimuli; as such, we are not immediately able to make predictions about whether the homeostatic failure can be predicted using the specific form of deviation from homeostasis. However, we acknowledge that this is a weakness of our analysis, and that a more complete investigation would address this question. For reasons of space, we elected not to pursue this further. We have added a paragraph to our Discussion (8th paragraph) explaining this.

Reviewer#1 (Recommendations for the authors):

(1) To make the article more accessible I would suggest the following:

(a) Include a few more illustrations or diagrams that demonstrate key concepts: adaptationof an entire population, clustering within a population, different sources of noise, inference with homeostatic DDCs, etc.

We thank the reviewer for this suggestion - we have added an additional figure in (Figure 8, Panel A) to explain the concept of clustering within a population. We also added a new panel to Figure 1 (Figure 1B) which we hope will clarify the conceptual postulate underlying our efficient coding framework and its link to the second half of the paper.

(b) Within the text refer to names of quantities much more often, rather than relying onlyon mathematical symbols (e.g. w,r,Ω, etc).

We thank the reviewer for the suggestion; we have updated the text accordingly and believe this has improved the clarity of the exposition.

(2) It is hard to distill which components of the considered theory are crucial to reproducing the experimental observations in Figure 12. Is it the homeostatic modulation, efficient coding, DDCs, or any combination of those or all of them necessary to reproduce the experiment? I believe this could be explained much better, also with an audience of experimentalists in mind.

We have updated the text to provide additional clarity on this matter (see the pointers to these changes and additions in the revised manuscript, given above in response to your first comment). In particular, reproducing the experimental results requires combining DDCs with homeostatic modulation – with the latter a consequence of our efficient coding theory, and not an independent ingredient or assumption.

(3) It would be good to comment on how sensitive the results are to the assumptions made, parameter values, etc. For example: do conclusions depend on statistics of neural responses in simulated environments? Do they generalize for different values of the constraint µ? This could be addressed in the discussion / supplementary material.

This issue is already discussed extensively within the text - see Sec. 2.4, Analytical insight on the optimality of homeostasis, and Sec. 2.5, Conditions for the validity of the homeostatic solution to hold in cortex. In these sections, we outline that - provided a certain parameter combination is small - we expect the homeostatic result to hold. Accordingly, we anticipate that our numerical results will generalise to any settings in which that parameter combination remains small.

(4) How many neurons/units were used for simulations?

We apologies for omitting this detail; we used 10,000 units for our simulations. We have edited both the main text and the methods section to reflect this.

(5) Typos etc: a) Figure 5 caption - the order of panels B and C is switched. b) Figure 6A - I suggest adding a colorbar.

Thank you. We have relabelled the panels B and C in the appropriate figures so that the ordering in the figure caption is correct. We feel that a colourbar in figure 6A would be unnecessary, since we are only trying to convey the concept of uniform correlations, rather than any particular value for the correlations; as such we have elected not to add a colourbar. We have, however, added a more explicit explanation of this cartoon matrix in the figure caption, by referring to the colors of diagonal vs off-diagonal elements.

Reviewer#2 (Recommendations for the authors):

The text on page 10, with the perturbation analysis, could be moved to a supplement, leaving here only the intuition.

We thank the reviewer for this suggestion; we have moved much of the argument into the appendix so as to not distract the reader with unnecessary technical details.

Text before eq. 12 “...in cluster a maximize the objective...” should be ‘minimize’?

The cluster objective as written is indeed maximised, as stated in the text. Note that, in the revised manuscript, this argument has been moved to an appendix to reduce the density of mathematics in the main text.

Top of page 25 “S₀ and S₀” should be “S₀ and S₁”?

Thank you, we have corrected the manuscript accordingly.