Title: Adaptive whitening with fast gain modulation and slow synaptic plasticity
Venue: NeurIPS 2023
Accepted: yes
Keywords: neuroscience

adaptation

whitening

efficient coding

recurrent neural network

gain modulation

synaptic plasticity

local learning rules
Abstract: Neurons in early sensory areas rapidly adapt to changing sensory statistics, both by normalizing the variance of their individual responses and by reducing correlations between their responses. Together, these transformations may be viewed as an adaptive form of statistical whitening. Existing mechanistic models of adaptive whitening exclusively use either synaptic plasticity or gain modulation as the biological substrate for adaptation; however, on their own, each of these models has significant limitations. In this work, we unify these approaches in a normative multi-timescale mechanistic model that adaptively whitens its responses with complementary computational roles for synaptic plasticity and gain modulation. Gains are modified on a fast timescale to adapt to the current statistical context, whereas synapses are modified on a slow timescale to match structural properties of the input statistics that are invariant across contexts. Our model is derived from a novel multi-timescale whitening objective that factorizes the inverse whitening matrix into basis vectors, which correspond to synaptic weights, and a diagonal matrix, which corresponds to neuronal gains. We test our model on synthetic and natural datasets and find that the synapses learn optimal configurations over long timescales that enable adaptive whitening on short timescales using gain modulation.

Adaptive whitening with fast gain modulation and slow synaptic plasticity

Abstract

Neurons in early sensory areas rapidly adapt to changing sensory statistics, both by normalizing the variance of their individual responses and by reducing correlations between their responses. Together, these transformations may be viewed as an adaptive form of statistical whitening. Existing mechanistic models of adaptive whitening exclusively use either synaptic plasticity or gain modulation as the biological substrate for adaptation; however, on their own, each of these models has significant limitations. In this work, we unify these approaches in a normative multi-timescale mechanistic model that adaptively whitens its responses with complementary computational roles for synaptic plasticity and gain modulation. Gains are modified on a fast timescale to adapt to the current statistical context, whereas synapses are modified on a slow timescale to match structural properties of the input statistics that are invariant across contexts. Our model is derived from a novel multi-timescale whitening objective that factorizes the inverse whitening matrix into basis vectors, which correspond to synaptic weights, and a diagonal matrix, which corresponds to neuronal gains. We test our model on synthetic and natural datasets and find that the synapses learn optimal configurations over long timescales that enable adaptive whitening on short timescales using gain modulation.

1 Introduction

Individual neurons in early sensory areas rapidly adapt to changing sensory statistics by normalizing the variance of their responses [1-3]. At the population level, neurons also adapt by reducing correlations between their responses [4; 5]. These adjustments enable the neurons to maximize the information that they transmit by utilizing their entire dynamic range and reducing redundancies in their representations [6-9]. A natural normative interpretation of these transformations is adaptive whitening , a context-dependent linear transformation of the sensory inputs yielding responses that have unit variance and are uncorrelated.

Decorrelation of the neural responses requires coordination between neurons and the neural mechanisms underlying such coordination are not known. Since neurons communicate via synaptic connections, it is perhaps unsurprising that most existing mechanistic models of adaptive whitening decorrelate neural responses by modifying the strength of these connections [10-16]. However, long-term synaptic plasticity is generally associated with long-term learning and memory [17], and thus may not be a suitable biological substrate for adaptive whitening (though short-term synaptic

S1

r1

context 2

context2

context1

context1

n1

S2

r2

Adapted within contexts: < g

Inputs (s1, $2)

Adapted across contexts: · WT o -W

Responses (r1,r2)

plasticity has been reported [18]). On the other hand, there is extensive neuroscience literature on rapid and reversible gain modulation [19-26]. Motivated by this, Duong et al. [27] proposed a mechanistic model of adaptive whitening in a neural circuit with fixed synaptic connections that adapts exclusively by modifying the gains of interneurons that mediate communication between the primary neurons. They demonstrate that an appropriate choice of the fixed synaptic weights can both accelerate adaptation and significantly reduce the number of interneurons that the circuit requires. However, it remains unclear how the circuit learns such an optimal synaptic configuration, which would seem to require synaptic plasticity.

In this study, we combine the learning and adaptation of synapses and gains, respectively, in a unified mechanistic neural circuit model that adaptively whitens its inputs over multiple timescales (Fig. 1). Our main contributions are as follows:

We introduce a novel adaptive whitening objective in which the (inverse) whitening matrix is factorized into a synaptic weight matrix that is optimized across contexts and a diagonal (gain) matrix that is optimized within each statistical context.

With this objective, we derive a multi-timescale online algorithm for adaptive whitening that can be implemented in a neural circuit comprised of primary neurons and an auxiliary population of interneurons with slow synaptic plasticity and fast gain modulation (Fig. 1).

We test our algorithm on synthetic and natural datasets, and demonstrate that the synapses learn optimal configurations over long timescales that enable the circuit to adaptively whiten its responses on short timescales exclusively using gain modulation.

Beyond the biological setting, multi-timescale learning and adaptation may also prove important in machine learning systems. For example, Mohan et al. [28] introduced 'gain-tuning', in which the gains of channels in a deep denoising neural network (with pre-trained synaptic weights) are adjusted to improve performance on samples with out-of-distribution noise corruption or signal properties. The normative multi-timescale framework developed here offers a new approach to continual learning and test-time adaptation problems such as this.

2 Adaptive symmetric whitening

Consider a neural population with N primary neurons (Fig. 1). The stimulus inputs to the primary neurons are represented by a random N -dimensional vector s whose distribution p ( s | c ) depends

on a latent context variable c . The stimulus inputs s can be inputs to peripheral sensory neurons (e.g., the rates at which photons are absorbed by N cones) or inputs to neurons in an early sensory area (e.g., glomerulus inputs to N mitral cells in the olfactory bulb). Context variables can include location (e.g., a forest or a meadow) or time (e.g., season or time of day). For simplicity, we assume the context-dependent inputs are centered; that is, E s ∼ p ( s | c ) [ s ] = 0 , where E s ∼ p ( s | c ) [ · ] denotes the expectation over the conditional distribution p ( s | c ) and 0 denotes the vector of zeros. See Appx. A for a consolidated list of notation used throughout this work.

The goal of adaptive whitening is to linearly transform the inputs s so that, conditioned on the context variable c , the N -dimensional neural responses r have identity covariance matrix; that is,

where F c is a context-dependent N × N whitening matrix. Whitening is not a unique transformationleft multiplication of the whitening matrix F c by any N × N orthogonal matrix results in another whitening matrix. We focus on symmetric whitening, also referred to as Zero-phase Components Analysis (ZCA) whitening or Mahalanobis whitening, in which the whitening matrix for context c is uniquely defined as

where we assume C ss ( c ) is positive definite for all contexts c . This is the unique whitening transformation that minimizes the mean-squared difference between the inputs and the outputs [29].

To derive an algorithm that learns the symmetric whitening matrix F c , we express F c as the solution to an appropriate optimization problem, which is similar to the optimization problem in [15, top of page 6]. For a context c , we can write the inverse symmetric whitening matrix M c := F -1 c as the unique optimal solution to the minimization problem

where S N ++ denotes the set of N × N positive definite matrices. 1 This follows from the fact that f c ( M ) is strictly convex with its unique minimum achieved at M c , where f c ( M c ) = 2Tr( M c ) (Appx. B.1). Existing recurrent neural circuit models of adaptive whitening solve the minimization problem in Eq. 2 by choosing a matrix factorization of M c and then optimizing the components [12; 13; 15; 27].

3 Adaptive whitening in neural circuits: a matrix factorization perspective

Here, we review two adaptive whitening objectives, which we then unify into a single objective that adaptively whitens responses across multiple timescales.

3.1 Objective for adaptive whitening via synaptic plasticity

Pehlevan and Chklovskii [12] proposed a recurrent neural circuit model that whitens neural responses by adjusting the synaptic weights between the N primary neurons and K ≥ N auxiliary interneurons according to a Hebbian update rule. Their circuit can be derived by factorizing the context-dependent matrix M c into a symmetric product M c = W c W ⊤ c for some context-dependent N × K matrix W c [15]. Substituting this factorization into Eq. 2 results in the synaptic plasticity objective in Table 1. In the recurrent circuit implementation, W ⊤ c denotes the feedforward weight matrix of synapses connecting primary neurons to interneurons and the matrix -W c denotes the feedback weight matrix of synapses connecting interneurons to primary neurons. Importantly, under this formulation, the circuit may reconfigure both the synaptic connections and synaptic strengths each time the context c changes, which runs counter to the prevailing view that synaptic plasticity implements long-term learning and memory [17].

3.2 Objective for adaptive whitening via gain modulation

Duong et al. [27] proposed a neural circuit model with fixed synapses that whitens the N primary responses by adjusting the multiplicative gains in a set of K auxiliary interneurons. To derive a neural

circuit with gain modulation, they considered a novel diagonalization of the inverse whitening matrix, M c = I N + W fixdiag ( g c ) W ⊤ fix , where W fix is an arbitrary, but fixed N × K matrix of synaptic weights (with K ≥ K N := N ( N + 1) / 2 ) and g c is an adaptive, context-dependent real-valued K -dimensional vector of gains. Note that unlike the conventional eigen-decomposition, the number of elements along the diagonal matrix is significantly larger than the dimensionality of the input space. Substituting this factorization into Eq. 2 results in the gain modulation objective in Table 1. As in the synaptic plasticity model, W ⊤ fix denotes the weight matrix of synapses connecting primary neurons to interneurons while -W fix connects interneurons to primary neurons. In contrast to the synaptic plasticity model, the interneuron outputs are modulated by context-dependent multiplicative gains, g c , that are adaptively adjusted to whiten the circuit responses.

Duong et al. [27] demonstrate that an appropriate choice of the fixed synaptic weight matrix can both accelerate adaptation and significantly reduce the number of interneurons in the circuit. In particular, the gain modulation circuit can whiten any input distribution provided the gains vector g c has dimension K ≥ K N (the number of degrees of freedom in an N × N symmetric covariance matrix). However, in practice, the circuit need only adapt to input distributions corresponding to natural input statistics [7; 30-32]. For example, the statistics of natural images are approximately translationinvariant, which significantly reduces the degrees of freedom in their covariance matrices, from O ( N 2 ) to O ( N ) . Therefore, while the space of all possible correlation structures is K N -dimensional, the set of natural statistics likely has far fewer degrees of freedom and an optimal selection of the weight matrix W fix can potentially offer dramatic reductions in the number of interneurons K required to adapt. As an example, Duong et al. [27] specify a weight matrix for performing 'local' whitening with O ( N ) interneurons when the input correlations are spatially-localized (e.g., as in natural images). However, they do not prescribe a method for learning a (synaptic) weight matrix that is optimal across the set of natural input statistics.

3.3 Unified objective for adaptive whitening via synaptic plasticity and gain modulation

We unify and generalize the two disparate adaptive whitening approaches [12; 27] in a single multitimescale nested objective in which gains g are optimized within each context and synaptic weights W are optimized across contexts. In particular, we optimize, with respect to W , the expectation of the objective from [27] (for some fixed K ≥ 1 ) over the distribution of contexts p ( c ) :

where we have also generalized the objective from [27] by including a fixed multiplicative factor α ≥ 0 in front of the identity matrix I N , and we have relaxed the requirement that K ≥ K N .

What is an optimal solution of Eq. 3? Since the convex function f c is uniquely minimized at M c , a sufficient condition for the optimality of a synaptic weight matrix W is that for each context c , there is a gain vector g c such that α I N + W diag ( g c ) W ⊤ = M c . Importantly, under such a synaptic configuration, the function f c can attain its minimum exclusively by adjusting the gains vector g . In the space of covariance matrices, we can express the statement as

where F ( W ) contains the set of covariance matrices that can be whitened with fixed synapses W and adaptive gains g . Fig. 2 provides an intuitive Venn diagram comparing a non-optimal synaptic configuration W 0 and an optimal synaptic configuration W T .

F(WT)

natural covariance

natural covariance

matrices

slow synaptic plasticity

matrices

F(Wo)

W。 → WT

Css(c)·

Cs(c)

fastgain modulation

fastgainmodulation

%→g

% →gc

cone of positive definite matrices

cone of positive definite matrices

4 Multi-timescale adaptive whitening algorithm and circuit implementation

In this section, we derive an online algorithm for optimizing the multi-timescale objective in Eq. 3, then map the algorithm onto a neural circuit with fast gain modulation and slow synaptic plasticity. Direct optimization of Eq. 3 results in an offline (or full batch) algorithm that requires the network to have access to the covariance matrices C ss ( c ) , Appx. C. Therefore, to derive an online algorithm that includes neural dynamics, we first add neural responses r to the objective, which introduces a third timescale to the objective. We then derive a multi-timescale gradient-based algorithm for optimizing the objective.

Adding neural responses to the objective. First, observe that we can write f c ( M ) , for M ∈ S N ++ , in terms of the neural responses r :

To see this, maximize over r to obtain r = M -1 s and then use the definition of C ss ( c ) from Eq. 1 (Appx. B.2). Substituting this expression for f c , with M = α I N + W diag ( g ) W ⊤ , into Eq. 3, dropping the constant term α I N term and using the cyclic property of the trace operator results in the following objective with 3 nested optimizations (Appx. B.2):

The inner-most optimization over r corresponds to neural responses and will lead to recurrent neural dynamics. The outer 2 optimizations correspond to the optimizations over the gains g and synaptic weights W from Eq. 3.

To solve Eq. 5 in the online setting, we assume there is a timescale separation between neural dynamics and the gain/weight updates. This allows us to perform the optimization over r before optimizing g and W concurrently. This is biologically sensible: neural responses (e.g., action potential firing) operate on a much faster timescale than gain modulation and synaptic plasticity [25; 33].

Recurrent neural dynamics. At each iteration, the circuit receives a stimulus s . We maximize ℓ ( W , g , r , s ) with respect to r by iterating the following gradient-ascent steps that correspond to repeated timesteps of the recurrent circuit (Fig. 1) until the responses equilibrate:

where η r > 0 is a small constant, z i = w ⊤ i r denotes the weighted input to the i th interneuron, n i = g i z i denotes the gain-modulated output of the i th interneuron. For each i , synaptic weights, w i , connect the primary neurons to the i th interneuron and symmetric weights, -w i , connect the i th interneuron to the primary neurons. From Eq. 6, we see that the neural responses are driven by feedforward stimulus inputs s , recurrent weighted feedback from the interneurons -Wn , and a leak term -α r .

Fast gain modulation and slow synaptic plasticity. After the neural activities equilibrate, we minimize ℓ ( W , g , r , s ) by taking concurrent gradient-descent steps

where η g and η w are the respective learning rates for the gains and synaptic weights. By choosing η g ≫ η w , we can ensure that the gains are updated at a faster timescale than the synaptic weights.

The update to the i th interneuron's gain g i depends on the difference between the online estimate of the variance of its input, z 2 i , and the squared-norm of the i th synaptic weight vector, ∥ w i ∥ 2 , quantities that are both locally available to the i th interneuron. Using the fact that z i = w ⊤ i r , we can rewrite the gain update as ∆ g i = η g [ w ⊤ i ( rr ⊤ -I N ) w i ] . From this expression, we see that the gains equilibrate when the marginal variance of the responses along the direction w i is 1, for i = 1 , . . . , K .

The update to the ( i, j ) th synaptic weight w ij is proportional to the difference between r i n j and w ij g j , which depends only on variables that are available in the pre- and postsynaptic neurons. Since r i n j is the product of the pre- and postynaptic activities, we refer to this update as Hebbian . In Appx. E.2, we decouple the feedforward weights w ⊤ i and feedback weights -w i and provide conditions under which the symmetry asymptotically holds.

Multi-timescale online algorithm. Combining the neural dynamics, gain modulation and synaptic plasticity yields our online multi-timescale adaptive whitening algorithm, which we express in vector-matrix form with ' ◦ ' denoting the Hadamard (elementwise) product of two vectors:

Alg. 1 is naturally viewed as a unification and generalization of previously proposed neural circuit models for adaptation. When α = 0 and the gains g are constant (e.g., η g = 0 ) and identically equal to the vector of ones 1 (so that n t = z t ), we recover the synaptic plasticity algorithm from [15]. Similarly, when α = 1 and the synaptic weights W are fixed (e.g., η w = 0 ), we recover the gain modulation algorithm from [27].

A)

B)

(smoothed)

2.5

C)

10

Wo

2.0

8

WT

5

1

5

1.5

Error

6

1.0

4

1

1

Error

0.5

2

1

0.0

0

S1

S1

101

102

103

0

10002000300040005000

Contextsample

Stimulussample

D)

Wo

E)

1.0

WT

WT

Density

Wo

S

2

v

0.5

S

S1

S1

1

0.0

10-4

10-3

10-2

10-1

100

W-VIlF

S1

5 Numerical experiments

We test Alg. 1 on stimuli s 1 , s 2 , . . . drawn from slowly fluctuating latent contexts c 1 , c 2 , . . . ; that is, s t ∼ p ( s | c t ) and c t = c t -1 with high probability. 2 To measure performance, we evaluate the operator norm on the difference between the expected response covariance and the identity matrix:

Geometrically, this 'worst-case' error measures the maximal Euclidean distance between the ellipsoid corresponding to M -1 t C ss ( c t ) M -1 t and the ( N -1) -sphere along all possible axes. To compare two synaptic weight matrices A , B ∈ R N × K , we evaluate ∥ ˆ A -B ∥ F , where ˆ A = AP and P is the permutation matrix (with possible sign flips) that minimizes the error.

5.1 Synthetic dataset

To validate our model, we first consider a 2-dimensional synthetic dataset in which an optimal solution is known. Suppose that each context-dependent inverse whitening matrix is of the form M c = I N + VΛ ( c ) V ⊤ , where V is a fixed 2 × 2 matrix and Λ ( c ) = diag ( λ 1 ( c ) , λ 2 ( c )) is a context-dependent diagonal matrix. Then, in the case α = 1 and K = 2 , an optimal solution of the objective in Eq. 3 is when the column vectors of W align with the column vectors of V .

To generate this dataset, we chose the column vectors of V uniformly from the unit circle, so they are not generally orthogonal. For each context c = 1 , . . . , 64 , we assume the diagonal entries of Λ ( c ) are sparse and i.i.d.: with probability 1/2, λ i ( c ) is set to zero and with probability 1/2, λ i ( c ) is chosen uniformly from the interval [0 , 4] . Example covariance matrices from different contexts are shown in Fig. 3A (note that they do not share a common eigen-decomposition). Finally, for each context, we generate 1E3 i.i.d. samples with context-dependent distribution s ∼ N ( 0 , M 2 c ) .

We test Alg. 1 with α = 1 , K = 2 , η w = 1E-5, and η g = 5E-2 on these sequences of synthetic inputs with the column vectors of W 0 chosen uniformly from the unit circle. The model successfully learns to whiten the different contexts, as indicated by the decreasing whitening error with the number of contexts presented (Fig. 3B). At the end of training, the synaptic weight matrix W T is optimized such that the circuit can adapt to changing contexts exclusively by adjusting its gains. This is evidenced by the fact that when the context changes, there is a brief spike in error as the gains adapt to the new context (Fig. 3C, red line). By contrast, the error remains high in many of the contexts when using

A)

C)

E)

W|modulated g|Ptest

101

Density

4

WT|modulated g|Ptrain

Overcomplete

WT1

fixed g|Ptest

2

Wo|modulated g|Ptest

Error

100

Train

0

0

1

2

3

4

5

Test

Error

10-1.

D)

0

N

2N

#ofinterneuronsK

F)

B)

5

30

W-Q*IIF

4

Count

20

W。-Q*IlF

Error

3

Q(c)-Q*IIF

2

10

1

0

101

103

0.10

0.12 0.14 0.16 0.18

0.20

Contextsample

Basiserror

the initial random synaptic weight matrix W 0 (Fig. 3C, blue line). In particular, the synapses learn (across contexts) an optimal configuration in the sense that the column vectors of W learn to align with the column vectors of V over the course of training (Fig. 3DE).

5.2 Natural images dataset

By hand-crafting a particular set of synaptic weights, Duong et al. [27] showed that their adaptive whitening network can approximately whiten a dataset of natural image patches with O ( N ) gainmodulating interneurons instead of O ( N 2 ) . Here, we show that our model can exploit spatial structure across natural scenes to learn an optimal set of synaptic weights by testing our algorithm on 56 highresolution natural images [34] (Fig. 4A, top). For each image, which corresponds to a separate context c , 5 × 5 pixel image patches are randomly sampled and vectorized to generate context-dependent samples s ∈ R 25 with covariance matrix C ss ( c ) ∈ S 25 ++ (Fig. 4A, bottom). We train our algorithm in the offline setting where we have direct access to the context-dependent covariance matrices (Appx. C, Alg. 2, α = 1 , J = 50 , η g = 5E-1, η w = 5E-2) with K = N = 25 and random W 0 ∈ O (25) on a training set of 50 of the images, presented uniformly at random 1E3 total times. We find that the model successfully learns a basis that enables adaptive whitening across different visual contexts via gain modulation, as shown by the decreasing training error (Eq. 9) in Fig. 4B.

How does the network learn to leverage statistical structure that is consistent across contexts? We test the circuit with fixed synaptic weights W T and modulated (adaptive) gains g on stimuli from the held-out images (Fig. 4C, red distribution shows the smoothed error over 2E3 random initializations W 0 ). The circuit performs as well on the held-out images as on the training images (Fig. 4C, red versus blue distributions). In addition, the circuit with learned synaptic weights W T and modulated gains g outperforms the circuit with learned synaptic weights W T and fixed gains (Fig. 4C, green distribution), and significantly outperforms the circuit with random synaptic weights W 0 and modulated gains (Fig. 4C, purple distribution). Together, these results suggest that the circuit

learns features W T that enable the circuit to adaptively whiten across statistical contexts exclusively using gain modulation, and that gain modulation is crucial to the circuit's ability to adaptively whiten. In Fig. 4D, we visualize the learned filters (columns of W T ), and find that they are approximately equal to the 2D discrete cosine transform (DCT, Appx. D), an orthogonal basis that is known to approximate the eigenvectors of natural image patch covariances [35; 36].

To examine the dependence of performance on the number of interneurons K , we train the algorithm with K = 1 , . . . , 2 N and report the final error in Fig. 4E. There is a steady drop in error as K ranges from 1 to N , at which point there is a (discontinuous) drop in error followed by a continued, but more gradual decay in both training and test images error as K ranges from N to 2 N (the overcomplete regime). To understand this behavior, note that the covariance matrices of image patches approximately share an eigen-decomposition [36]. To see this, let Q ( c ) denote the orthogonal matrix of eigenvectors corresponding to the context-dependent covariance matrix C ss ( c ) . As shown in Fig. 4F (green histogram), there is a small, but non-negligible, difference between the eigenvectors Q ( c ) and the eigenvectors Q ∗ of the average covariance matrix E c ∼ p ( c ) [ C ss ( c )] . When K = N , the column vectors of W T learn to align with Q ∗ (as shown in Fig. 4F, blue histogram), and the circuit approximately adaptively whitens the context-dependent stimulus inputs via gain modulation. As K ranges from 1 to N , W T progressively learns the eigenvectors of Q ∗ (Appx. D). Since W T achieves a full set of eigenvectors at K = N , this results in a large drop in error when measured using the operator norm. Finally, as mentioned, there is a non-negligible difference between the eigenvectors Q ( c ) and the eigenvectors Q ∗ . Therefore, increasing the number of interneurons from N to 2 N allows the circuit to learn an overcomplete set of basis vectors W T to account for the small deviations between Q ( c ) and Q ∗ , resulting in improved whitening error (Appx. D).

6 Discussion

We've derived an adaptive statistical whitening circuit from a novel objective (Eq. 3) in which the (inverse) whitening matrix is factorized into components that are optimized at different timescales. This model draws inspiration from the extensive neuroscience literature on rapid gain modulation [25] and long-term synaptic plasticity [17], and concretely proposes complementary roles for these computations: synaptic plasticity facilitates learning features that are invariant across statistical contexts while gain modulation facilitates adaptation within a statistical context. Experimental support for this will come from detailed understanding of natural sensory statistics across statistical contexts and estimates of (changes in) synaptic connectivity from wiring diagrams (e.g., [37]) or neural activities (e.g., [38]).

Our circuit uses local learning rules for the gain and synaptic weight updates that could potentially be implemented in low-power neuromorphic hardware [39] and incorporated into existing mechanistic models of neural circuits with whitened or decorrelated responses [40-45]. However, there are aspects of our circuit that are not biologically realistic. For example, we do not sign-constrain the gains or synaptic weight matrices, so our circuit can violate Dale's law. In addition, the feedforward synaptic weights W ⊤ and feedback weights -W are constrained to be symmetric. In Appx. E, we describe modifications of our model that can make it more biologically realistic. Additionally, while we focus on the potential joint function of gain modulation and synaptic plasticity in adaptation, short-term synaptic plasticity, which operates on similar timescales as gain modulation, has also been reported [18]. Theoretical studies suggest that short-term synaptic plasticity is useful in multi-timescale learning tasks [46-48] and it may also contribute to multi-timescale adaptive whitening. Ultimately, support for different adaptation mechanisms will be adjudicated by experimental observations.

Our work may also be relevant beyond the biological setting. Decorrelation and whitening transformations are common preprocessing steps in statistical and machine learning methods [49-53], and are useful for preventing representational collapse in recent self-supervised learning methods [54-57]. Therefore, our online multi-timescale algorithm may be useful for developing adaptive self-supervised learning algorithms. In addition, our work is related to the general problem of online meta-learning [58; 59]; that is, learning methods that can rapidly adapt to new tasks. Our solution-which is closely related to mechanisms of test-time feature gain modulation developed for machine learning models for denoising [28], compression [60; 61], and classification [62]-suggests a general approach to meta-learning inspired by neuroscience: structural properties of the tasks (contexts) are encoded in synaptic weights and adaptation to the current task (context) is achieved by adjusting the gains of individual neurons.

References

Naama Brenner, William Bialek, and Rob de Ruyter Van Steveninck. Adaptive rescaling maximizes information transmission. Neuron , 26(3):695-702, 2000.

Adrienne L Fairhall, Geoffrey D Lewen, and William Bialek. Efficiency and ambiguity in an adaptive neural code. Nature , 412:787-792, 2001.

Katherine I Nagel and Allison J. Doupe. Temporal processing and adaptation in the songbird auditory forebrain. Neuron , 51(6):845-859, 2006.

James R Muller, Andrew B Metha, John Krauskopf, and Peter Lennie. Rapid adaptation in visual cortex to the structure of images. Science , 285(5432):1405-1408, 1999.

Andrea Benucci, Aman B Saleem, and Matteo Carandini. Adaptation maintains population homeostasis in primary visual cortex. Nature Neuroscience , 16(6):724-729, 2013.

Fred Attneave. Some informational aspects of visual perception. Psychological Review , 61(3): 183-193, 1954.

H B Barlow. Possible Principles Underlying the Transformations of Sensory Messages. In Sensory Communication , pages 216-234. The MIT Press, 1961.

Simon Laughlin. A Simple Coding Procedure Enhances a Neuron's Information Capacity. Zeitschrift fur Naturforschung C, Journal of Biosciences , pages 910-2, 1981.

H B Barlow and P Foldiak. Adaptation and decorrelation in the cortex. In The Computing Neuron , pages 54-72. Addison-Wesley, 1989.

Stuart D Wick, Martin T Wiechert, Rainer W Friedrich, and Hermann Riecke. Pattern orthogonalization via channel decorrelation by adaptive networks. Journal of Computational Neuroscience , 28(1):29-45, 2010.

Paul D King, Joel Zylberberg, and Michael R DeWeese. Inhibitory interneurons decorrelate excitatory cells to drive sparse code formation in a spiking model of V1. Journal of Neuroscience , 33(13):5475-5485, 2013.

Cengiz Pehlevan and Dmitri B Chklovskii. A normative theory of adaptive dimensionality reduction in neural networks. Advances in Neural Information Processing Systems , 28, 2015.

Cengiz Pehlevan, Anirvan M Sengupta, and Dmitri B Chklovskii. Why do similarity matching objectives lead to Hebbian/anti-Hebbian networks? Neural Computation , 30(1):84-124, 2018.

Nikolai M Chapochnikov, Cengiz Pehlevan, and Dmitri B Chklovskii. Normative and mechanistic model of an adaptive circuit for efficient encoding and feature extraction. Proceedings of the National Academy of Sciences , 120(29):e2117484120, 2023.

David Lipshutz, Cengiz Pehlevan, and Dmitri B Chklovskii. Interneurons accelerate learning dynamics in recurrent neural networks for statistical adaptation. In The Eleventh International Conference on Learning Representations , 2023.

Zachary M Westrick, David J Heeger, and Michael S Landy. Pattern adaptation and normalization reweighting. Journal of Neuroscience , 36(38):9805-9816, 2016.

SP Martin, Paul D Grimwood, and Richard GM Morris. Synaptic plasticity and memory: an evaluation of the hypothesis. Annual Review of Neuroscience , 23(1):649-711, 2000.

Robert S Zucker and Wade G Regehr. Short-term synaptic plasticity. Annual Review of Physiology , 64(1):355-405, 2002.

Larry F Abbott, JA Varela, Kamal Sen, and SB Nelson. Synaptic depression and cortical gain control. Science , 275(5297):221-224, 1997.

Emilio Salinas and Peter Thier. Gain modulation: a major computational principle of the central nervous system. Neuron , 27(1):15-21, 2000.

Odelia Schwartz and Eero P Simoncelli. Natural signal statistics and sensory gain control. Nature Neuroscience , 4(8):819-825, 2001.

Frances S Chance, Larry F Abbott, and Alex D Reyes. Gain modulation from background synaptic input. Neuron , 35(4):773-782, 2002.

Yousheng Shu, Andrea Hasenstaub, Mathilde Badoual, Thierry Bal, and David A McCormick. Barrages of synaptic activity control the gain and sensitivity of cortical neurons. Journal of Neuroscience , 23(32):10388-10401, 2003.

Pierre-Olivier Polack, Jonathan Friedman, and Peyman Golshani. Cellular mechanisms of brain state-dependent gain modulation in visual cortex. Nature Neuroscience , 16(9):1331-1339, 2013.

Katie A Ferguson and Jessica A Cardin. Mechanisms underlying gain modulation in the cortex. Nature Reviews Neuroscience , 21(2):80-92, 2020.

David Wyrick and Luca Mazzucato. State-dependent regulation of cortical processing speed via gain modulation. Journal of Neuroscience , 41(18):3988-4005, 2021.

Lyndon R Duong, David Lipshutz, David J Heeger, Dmitri B Chklovskii, and Eero P Simoncelli. Adaptive whitening in neural populations with gain-modulating interneurons. Proceedings of the 40th International Conference on Machine Learning, PMLR , 202:8902-8921, 2023.

Sreyas Mohan, Joshua L Vincent, Ramon Manzorro, Peter Crozier, Carlos Fernandez-Granda, and Eero Simoncelli. Adaptive denoising via gaintuning. Advances in Neural Information Processing Systems , 34:23727-23740, 2021.

Yonina C Eldar and Alan V Oppenheim. MMSE whitening and subspace whitening. IEEE Transactions on Information Theory , 49(7):1846-1851, 2003.

Eero P Simoncelli and Bruno A Olshausen. Natural image statistics and neural representation. Annual Review of Neuroscience , 24(1):1193-1216, 2001.

Deep Ganguli and Eero P Simoncelli. Efficient sensory encoding and Bayesian inference with heterogeneous neural populations. Neural computation , 26(10):2103-2134, 2014.

Wiktor F Młynarski and Ann M. Hermundstad. Efficient and adaptive sensory codes. Nature Neuroscience , 24(7):998-1009, 2021.

Xiao-Jing Wang, Yinghui Liu, Maria V Sanchez-Vives, and David A McCormick. Adaptation and temporal decorrelation by single neurons in the primary visual cortex. Journal of Neurophysiology , 89(6):3279-3293, 2003.

JH van Hateren and A van der Schaaf. Independent component filters of natural images compared with simple cells in primary visual cortex. Proceedings: Biological Sciences , 265 (1394):359-366, 1998.

Nasir Ahmed, T Natarajan, and Kamisetty R Rao. Discrete cosine transform. IEEE Transactions on Computers , 100(1):90-93, 1974.

David Bull and Fan Zhang. Intelligent Image and Video Compression: Communicating Pictures . Academic Press, London, 2021.

Adrian A Wanner and Rainer W Friedrich. Whitening of odor representations by the wiring diagram of the olfactory bulb. Nature Neuroscience , 23(3):433-442, 2020.

Scott Linderman, Christopher H Stock, and Ryan P Adams. A framework for studying synaptic plasticity with neural spike train data. Advances in Neural Information Processing Systems , 27, 2014.

Mike Davies, Narayan Srinivasa, Tsung-Han Lin, Gautham Chinya, Yongqiang Cao, Sri Harsha Choday, Georgios Dimou, Prasad Joshi, Nabil Imam, Shweta Jain, Yuyun Liao, Chit-Kwan Lin, Andrew Lines, Ruokun Liu, Deepak Mathaikutty, Steven McCoy, Arnab Paul, Jonathan Tse, Guruguhanathan Venkataramanan, Yi-Hsin Weng, Andreas Wild, Yoonseok Yang, and Hong Wang. Loihi: A neuromorphic manycore processor with on-chip learning. IEEE Micro , 38(1): 82-99, 2018.

Cristina Savin, Prashant Joshi, and Jochen Triesch. Independent component analysis in spiking neurons. PLoS Computational Biology , 6(4):e1000757, 2010.

Siavash Golkar, David Lipshutz, Yanis Bahroun, Anirvan Sengupta, and Dmitri Chklovskii. A simple normative network approximates local non-Hebbian learning in the cortex. Advances in Neural Information Processing Systems , 33:7283-7295, 2020.

David Lipshutz, Yanis Bahroun, Siavash Golkar, Anirvan M Sengupta, and Dmitri B Chklovskii. A biologically plausible neural network for multichannel canonical correlation analysis. Neural Computation , 33(9):2309-2352, 2021.

David Lipshutz, Cengiz Pehlevan, and Dmitri B Chklovskii. Biologically plausible single-layer networks for nonnegative independent component analysis. Biological Cybernetics , 116(5-6): 557-568, 2022.

Manu Srinath Halvagal and Friedemann Zenke. The combination of Hebbian and predictive plasticity learns invariant object representations in deep sensory networks. Nature Neuroscience , 2023.

Siavash Golkar, Tiberiu Tesileanu, Yanis Bahroun, Anirvan Sengupta, and Dmitri Chklovskii. Constrained predictive coding as a biologically plausible model of the cortical hierarchy. Advances in Neural Information Processing Systems , 35:14155-14169, 2022.

Misha Tsodyks, Klaus Pawelzik, and Henry Markram. Neural networks with dynamic synapses. Neural Computation , 10(4):821-835, 1998.

Nicolas Y Masse, Guangyu R Yang, H Francis Song, Xiao-Jing Wang, and David J Freedman. Circuit mechanisms for the maintenance and manipulation of information in working memory. Nature Neuroscience , 22(7):1159-1167, 2019.

Kyle Aitken and Stefan Mihalas. Neural population dynamics of computing with synaptic modulations. Elife , 12:e83035, 2023.

Bruno Olshausen and David Field. Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature , 381:607-609, 1996.

Anthony J. Bell and Terrence J. Sejnowski. The 'independent components' of natural scenes are edge filters. Vision Research , 37:3327-3338, 1996.

Aapo Hyvärinen and Erkki Oja. Independent component analysis: algorithms and applications. Neural Networks , 13(4-5):411-430, 2000.

Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. ImageNet classification with deep convolutional neural networks. In Advances in Neural Information Processing Systems , 2009.

Adam Coates, Andrew Ng, and Honglak Lee. An analysis of single-layer networks in unsupervised feature learning. In Proceedings of the fourteenth international conference on artificial intelligence and statistics , pages 215-223. JMLR Workshop and Conference Proceedings, 2011.

Aleksandr Ermolov, Aliaksandr Siarohin, Enver Sangineto, and Nicu Sebe. Whitening for self-supervised representation learning. In International Conference on Machine Learning , pages 3015-3024. PMLR, 2021.

Jure Zbontar, Li Jing, Ishan Misra, Yann LeCun, and Stéphane Deny. Barlow twins: Selfsupervised learning via redundancy reduction. In International Conference on Machine Learning , pages 12310-12320. PMLR, 2021.

Tianyu Hua, Wenxiao Wang, Zihui Xue, Sucheng Ren, Yue Wang, and Hang Zhao. On feature decorrelation in self-supervised learning. In Proceedings of the IEEE/CVF International Conference on Computer Vision , pages 9598-9608, 2021.

Adrien Bardes, Jean Ponce, and Yann LeCun. VICReg: Variance-invariance-covariance regularization for self-supervised learning. International Conference on Learning Representations , 2022.

Sebastian Thrun and Lorien Pratt. Learning to Learn . Springer Science & Business Media, 2012.

Chelsea Finn, Aravind Rajeswaran, Sham Kakade, and Sergey Levine. Online meta-learning. In International Conference on Machine Learning , pages 1920-1930. PMLR, 2019.

Johannes Ballé, Philip A Chou, David Minnen, Saurabh Singh, Nick Johnston, Eirikur Agustsson, Sung Jin Hwang, and George Toderici. Nonlinear transform coding. IEEE Journal of Selected Topics in Signal Processing , 15(2):339-353, 2020.

Lyndon R Duong, Bohan Li, Cheng Chen, and Jingning Han. Multi-rate adaptive transform coding for video compression. In ICASSP 2023-2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) , pages 1-5. IEEE, 2023.

Dequan Wang, Evan Shelhamer, Shaoteng Liu, Bruno Olshausen, and Trevor Darrell. Tent: Fully test-time adaptation by entropy minimization. arXiv preprint arXiv:2006.10726 , 2020.

John F Kolen and Jordan B Pollack. Backpropagation without weight transport. In Proceedings of 1994 IEEE International Conference on Neural Networks (ICNN'94) , volume 3, pages 1375-1380. IEEE, 1994.

A Notation

For N,K ≥ 2 , let R N denote N -dimensional Euclidean space equipped with the usual Euclidean norm ∥ · ∥ and let R N + denote the non-negative orthant. Let 0 = [0 , . . . , 0] ⊤ and 1 = [1 , . . . , 1] ⊤ respectively denote the vectors of zeros and ones, whose dimensions should be clear from context.

Let R N × K denote the set of N × K real-valued matrices. Let ∥ · ∥ F denote the Frobenius norm and ∥ · ∥ op denote the operator norm. Let O ( N ) denote the set of N × N orthogonal matrices. Let S N (resp. S N ++ ) denote the set of N × N symmetric (resp. positive definite) matrices. Let I N denote the N × N identity matrix.

Given vectors u , v ∈ R N , let u ◦ v = [ u 1 v 1 , . . . , u N v N ] ⊤ ∈ R N denote the Hadamard (elementwise) product of u and v . Let diag ( u ) denote the N × N diagonal matrix whose ( i, i ) th entry is u i . Given a matrix M ∈ R N × N , we also let diag ( M ) denote the N -dimensional vector whose i th entry is M ii .

B Calculation details

B.1 Minimum of the objective

Here we show that the minimum of f c , defined in equation 2, is achieved at M c = C 1 / 2 ss ( c ) . Differentiating f c ( M ) with respect to M yields

Setting the gradient to zero and solving for M yields M = C 1 / 2 ss ( c ) . Substituting into f c yields f c ( M c ) = Tr( M -1 / 2 c C ss ( c ) + M c ) = 2 Tr( M c ) .

B.2 Adding neural responses

Here we show that equation 4 holds. First, note that the trace term in equation 4 is strictly concave with respect to r (assuming M is positive definite) and setting the derivative equal to zero yields

Therefore, the maximum in equation 4 is achieved at r = M -1 s . Substituting into equation 4 with this form for r , we get

where the second equality uses the linearity of the expectation and trace operators as well as the formula C ss ( c ) := E s ∼ p ( s | c ) [ ss ⊤ ] . This completes the proof that equation 4 holds. Next, using this expression for f c , we have

Substituting into equation 3 and dropping the constant αN term results in equation 5.

C Offline multi-timescale adaptive whitening algorithm

We consider an algorithm where we directly optimize the objective in equation 3. In this case, we assume that the input to the algorithm is a sequence of covariance matrices C ss (1) , C ss (2) , . . . . Within each context c = 1 , 2 , . . . , we take J ≥ 1 concurrent gradient descent steps with respect to g and W :

where we assume η g ≫ η w > 0 as in Algorithm 1 and the gradient of f c ( M ) with respect to M is given in equation 10. These updates for g and W can also be obtained by averaging the corresponding updates in equations 7 and 8 over the conditional distribution p ( s | c ) . This results in Algorithm 2.

Algorithm 2: Offline multi-timescale adaptive whitening 1: Input: Covariance matrices C ss (1) , C ss (2) , . . . 2: Initialize: W ∈ R N × K ; g ∈ R K ; α ≥ 0 ; J ≥ 1 ; η g ≫ η w > 0 3: for c = 1 , 2 , . . . do 4: for j = 1 , . . . , J do 5: M ← α I N + W diag ( g ) W ⊤ 6: g ← g -η g diag [ W ⊤ ∇ M f c ( M ) W ] 7: W ← W -η w ∇ M f c ( M ) W diag ( g ) 8: end for 9: end for

D Adaptive whitening of natural images

In this section, we elaborate on the converged structure of W T using natural image patches. To better visualize the relationship between the learned columns of W and sinusoidal basis functions (e.g.

A)

B)

C)

PCA

Learned

W

W

+

WWW

WW

WW

Q

0

0

0

10

10

10

10

DCT), we focus on 1-dimensional image patches (rows of pixels). The results are similar with 2D image patches.

It is well known that eigenvectors of natural images are well-approximated by sinusoidal basis functions [e.g. the DCT; 35; 36]. Using the same images from the main text [34], we generated 56 contexts by sampling 16 × 1 pixel patches from separate images, with 2E4 samples each. We train Algorithm 2 with K = N = 16 , η w = 5E -2 , and random W 0 ∈ O (16) on a training set of X of the images, presented uniformly at random T = 1E5 times. Fig 5A,B shows that W T approximates the principal components of the aggregated context-dependent covariance, E c ∼ p ( c ) [ C ss ( c )] , which are closely aligned with the DCT. To show that this structure is inherent in the spatial statistics of natural images, we generated control contexts, C ss ( c ) , by forming covariance matrices with matching eigenspectra, but each with random and distinct eigenvectors. This destroys the structure induced by natural image statistics. Consequently, the learned vectors in W T are no longer sinusoidal (Fig 5C). As a result, whitening error with W T is much higher on the training set, with 0 . 3 ± 0 . 02 error (mean ± standard error over 10 random initializations; Eq. 9) on natural image contexts and 2 . 7 ± 0 . 1 on the control contexts. While for the natural images, a basis approximating the DCT was sufficient to adaptively whiten all contexts in the ensemble, this is not the case for the generated control contexts.

Finally, we find that as K increases from K = 1 to K = 16 , the basis vectors in W T progressively learn higher frequency components of the DCT (Fig. 6). This is a sensible solution, due to the ℓ 2 reconstruction error of our objective, and the 1 /f spectral content of natural image statistics. With more flexibility, as K increases past N (i.e. the overcomplete regime), the network continues to improve its whitening error (Fig. 7A) by learning a basis, W T , that can account for withincontext information that is insufficiently captured by the DCT (Fig. 7B). Taken together, our model successfully learns a basis W T that exploits the spatial structure present in natural images.

E Modifications for increased biological realism

In this section, we modify Algorithm 1 to be more biologically realistic.

E.1 Enforcing unit norm basis vectors

In our algorithm, there is no constraint on the magnitude of the column vectors of W . We can enforce a unit norm (here measured using the Euclidean norm) constraint by adding Lagrange multipliers to the objective in equation 3:

K=1

K=2

K=4

K=5

K=6

1

16

1

16

1

16

1

16

1

16

1

16

1

16

1

16

K=9

K=12

K=13

K=14

K=15

K=16

1

16

1

16

1

16

1

16

1

16

1

16

1

16

1

16

A)

B)

K=19

K=20

101

Overcomplete

1

16

32

1

16

32

1

16

32

1

16

32

K=21

K=22

K=23

K=24

Error

100

16

32

1

16

32

1

16

32

1

16

32

K=26

K=27

K=28

Train error

Testerror

16

32

1

16

32

1

16

32

1

16

32

K=N

K=29

K=30

K=31

K=32

10

20

30

K

1632

1

1632

1

16321

1632

where

Taking partial derivatives with respect to w i and m i results in the updates:

Furthermore, since the weights are constrained to have unit norm, we can replace ∥ w i ∥ 2 with 1 in the gain update:

E.2 Decoupling the feedforward and feedback weights

We replace the primary neuron-to-interneuron weight matrix W ⊤ (resp. interneuron-to-primary neuron weight matrix -W ) with W rn (resp. -W nr ). In this case, the update rules are

A)

B)

2.0

Error (smoothed)

1.0

Wo

0.8

1.5 -

WT

0.6

Error

1.0

0.4

0.5-

0.2

0.0

104

105

106

0

10002000300040005000

Stimulus sample

Stimulussample

C)

D)

0.12

Wo

WT

/M-"MIl

0.10

v

S

0.08

900

0.04

S1

103

104

105

106

Stimulus sample

Let W rn,t and W nr,t denote the values of the weights W rn and W nr , respectively, after t = 0 , 1 , . . . iterates. Then for all t = 0 , 1 , . . . ,

Thus, if g i + m i ∈ (0 , 2 η -1 w ) for all i (e.g., by enforcing non-negative g i , m i and choosing η w > 0 sufficiently small), then the difference decays exponentially in t and the feedforward and feedback weights are asymptotically symmetric. This result is similar to that of Kolen and Pollack [63], where the authors trained a network with weight decay to show that weight symmetry naturally arises to induce bidirectional function synchronization during learning.

E.3 Sign-constraining the synaptic weights and gains

The synaptic weight matrix W and gains vector g are not sign-constrained in Algorithm 1, which is not consistent with biological evidence. We can modify the algorithm to enforce the sign constraints by rectifying the weights and gains at each step. Here [ · ] + denote the elementwise rectification operation. This results in the updates

E.4 Online algorithm with improved biological realism

Combining these modifications yields our more biologically realistic multi-timescale online algorithm, Algorithm 3.

We test Algorithm 3 on a similar synthetic data setup to what was used in section 5.1, except that we sample the column vectors of V from the intersection of the unit circle with the nonnegative quadrant, Figure 8.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a normative principle for the symmetric whitening problem, i.e., a batch optimization problem which is provable to attain symmetric whitening is used to derive an online adaptive algorithm. The proposed framework unifies the previous works, and the resulting algorithm maps into single layer network with interneurons and local learning rules. The mathematical findings of this normative approach is also supported with the evidences from neuroscience. The proposed approached is tested with synthetic data and natural images to demonstrate the performance. Potential improvements for more biologically realistic method is also discussed.\n\nSTRENGTHS:\nThe paper provides a neural network solution to the whitening problem which satisfies two important constraints for biological plausibility: i) the network operates in an online manner, and ii) the weight and gain updates are local. The proposed online algorithm is illustrated to obtain whitening with both synthetic and natural datasets.\n\nWEAKNESSES:\nThe writing is lacking in some parts, and some important details are missing in the paper. More numerical experiments should be presented to assess the introduced framework. For exact details on the weaknesses, please see my comments below and questions section.\n\n   * Line 83, \"... where $f_c(\\mathbf{M}_c) = 2 \\mathbf{M}_c$\" should be changed to \"... where $f_c(\\mathbf{M}_c) = \\text{Tr}(2 \\mathbf{M}_c)$\" since $f_c$ is a scalar function of matrices. Moreover, even though it is a simple proof, it would be valuable how this unique minimum is achieved. \n    \n   * The intermediate step (stated between line 150 and line 153) to obtain Equation (4) should be written \texplicitly in the paper (can be in the appendix).\n    \n   * The decoupling the feedforward and feedback weights (in Appendix D.2) is very similar to the idea presented in [i] since the assumption $g_i + m_i \\in (0, 2 \\eta_w^{-1})$ corresponds to weight decaying. The paper should cite this reference.\n\n    [i]J.F. Kolen and J.B. Pollack. Backpropagation without weight transport. In Proceedings of 1994 IEEE International Conference on Neural Networks (ICNN’94), volume 3, pages 1375–1380 vol.3, 1994. doi: 10.1109/ICNN.1994.374486.\n    \n   * For the experiment with synthetic data, the experiment only considers the whitening matrix of the form $\\mathbf{M}_c = \\mathbf{I}_N + \\mathbf{V} \\mathbf{\\Lambda}(c)\\mathbf{V}^T$. An experiment with more general form of whitening matrix is required to assess the performance of the proposed algorithm.\n\n   * Hyperparameters (i.e., $\\eta_w, \\eta_g, \\eta_r$) are not provided in the paper (except that the authors provide $\\eta_w$ for Algorithm 2 in Appendix C). It would be beneficial for the paper to include a table that provides the hyperparameters used in each experiment\n\n   * The differences and novelties compared to previous works by Pehlevan and Chklovskii [11] and Duong et al. [18] are discussed quite well, but the comparison in the numerical experiments is missing.\n\n   * A single Python script is shared for the proposed method, although it does not include the experiment code. The authors mentioned in the paper that the full code would be shared upon publication. However, the current code does not provide a comprehensive understanding of the conducted experiments.\n\nQUESTIONS:\n* The numerical experiments presented in Appendix C uses Algorithm 2, but the biological plausibility of it is questionable due to matrix inversion and batch learning. Did you experiment with this setup using Algorithm 1?\n\n* The authors propose Algorithm 3 as more biologically plausible algorithm compared to the Algorithm 1, by complying to the Dale's law and decoupling the feedforward and feedback weights for asymmetry (weight transport problem). However, no numerical experiment has been demonstrated for this proposal. Did you test this algorithm?\n\n* The value of $\\alpha$ seems to be also an hyperparameter for the proposed framework. Did you analyze the effect of this choice?\n\nLIMITATIONS:\nI think that the limitations are adequately discussed in the paper.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-06-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper gives an algorithm for learning weights of a neural network over long time scales, which allow interneurons to decorrelate the responses of excitatory neurons by modulating their gains over short time scales. On both synthetic and natural image datasets, this algorithm is shown to be effective and to generalize to new contexts.\n\nSTRENGTHS:\n* This paper makes progress on a fundamental question in neuroscience: How a population of neurons can coordinate their responses to stimuli to encode it. This work should also be of interest in machine learning, as it has obvious applications to transfer learning.\n* Overall, the paper is clear and easy to understand.\n* The learning algorithm is online and local.\n* The technical evaluations are fairly thorough and support the paper's claims.\n\nWEAKNESSES:\nSome weaknesses are inherited from the Duong et al. model which this paper is building on, specifically related to its biological plausibility:\n* All neurons are linear.\n* Dale's law is not observed.\n* Forward and backward weights are mirrored.\n* In general, more interneurons than excitatory neurons are required (in the worst case, as many as $\\Omega(n^2)$ interneurons for $n$ excitatory neurons), which does not match the ratios observed in the brain. Experimentally, the algorithm fails gracefully when fewer interneurons than required are available.\nA more realistic version of the model which addresses the first two issues is provided in the appendix. However, given that the scope of this work is primarily modeling a phenomenon in neuroscience, I think the paper would be stronger if this biologically plausible version were featured and evaluated more prominently.\n\nQUESTIONS:\n* Line 83: Missing a trace on the RHS of the equation?\n* I would appreciate a more in-depth evaluation, perhaps on a synthetic dataset, of what happens when there are fewer interneurons than required.\n* Did you consider adding an activation function, at least to the interneurons? I could imagine that there is a similar learning algorithm which allows for fewer nonlinear interneurons.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors produce a mechanistic model that combines synaptic plasticity and gain modulation to adaptively whiten responses. This model is constructed from an objective for learning a whitening transformation and then considering matrix factorization.  Simple factorization introduces interneurons and optimization via gradient descent can be mapped to synaptic plasticity which adapts the weights to a particular context.  Another decomposition that enforces a particular diagonalization leads whitening via gain modulation when the weights of the interneurons are fixed.  The contribution of this paper is to combine these two mechanisms into a single cost function and network instantiation in which the gain optimization produces a whitening for a given context, and the synaptic plasticity adapts to properties useful over multiple contexts.  The describe an implementation of this optimization in a recurrent neural network and consider different time scales, in which gain adaptation happens quickly within a context and synaptic plasticity happens over a longer timescale.  The authors test this framework on simulated data that matches the decomposition assumptions and show that the method is successful.  They further test it on natural images and show that the learned connectivity successfully adapts to new contexts.\n\nSTRENGTHS:\nThis is a very nice paper that connects the computational problem of whitening with multiple neural mechanisms into a single normative framework.  While it still has issues with full biological plausibility (as noted by the authors) this is an interesting step in thinking about this computation and the roles of these neural mechanisms.  The presentation is very clear (Figure 2 is particularly nice).\n\nWEAKNESSES:\nThere are no significant weaknesses in this paper.  To the authors’ credit, every question I had that I thought was in the category of “necessary for publication” they addressed.  I’ll leave everything else to “Questions”, below.\n\nThe biggest weakness I see regards the generality of the matrix decomposition.  The extent to which the authors test or address this appears on par with other approaches in the field, though.  I have some questions below on this topic.\n\nQUESTIONS:\nCan you make any statements about how general the matrix decomposition in Section 3.3 is?  It seems very specific.  You demonstrate numerically that this decomposition works for your experiment.  When do you expect it to fail?  Can you characterize or quantify the differences in context necessary before this decomposition no longer works?  \n\nRelated to this, it looks like you’re using images from the van Hateren database.  Can you try the held out image experiments with images from very different contexts and origins, since those images all have clear similarities in large scale properties?  This might help find the limits of this algorithm/decomposition.  \n\nI was going to ask about breaking the symmetry on the weights, which you answered in an appendix.  They will asymptotically align, but can you say something about this timescale?  Is there a restriction on when the alignment has to happen with respect to the gain modulation and synaptic plasticity that is meaningful or prohibitive?\n\nYou provide the error as a function of number of interneurons for your natural image experiment.  Can you compute some normalization so that there is an idea of what errors are meaningful (maybe the Frobenius norm of C)?  Your interneurons are not necessarily the “interneurons” people talk about in real circuits, but it makes one imagine nonetheless that you will have far fewer interneurons than primary neurons.  Is this a functional problem given the amount of error you see?  Maybe it’s “good enough” even with few interneurons?\n\nLIMITATIONS:\nThe authors do not discuss societal impact.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2023-07-07"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a neural circuit model that combines fast gain modulation and slow synaptic plasticity to adaptively white sensory inputs. It appears that this paper is a combination of the studies of ref. 11 and 18.\n\nSTRENGTHS:\n### Originality \nThe strength of this paper is that it combines the fast gain modulation and slow synaptic plasticity in a whitening neural circuit, and addresses the shortcomings of earlier models with either gain modulation or synaptic plasticity. This is the most conceptual and technical advance of this study. \n\n### Clarity\nOverall, the paper is well written. But see my comments and questions on some parts I am unclear about.\n\nWEAKNESSES:\nI don't have major concerns about this study from the math point of view. Nonetheless, I have a significant concern from the neuroscience point of view.\n\nThe derived updating rule of gain (Eq. after line 174) depends on the current weight. The neural \"gain\" is typically adjusted by inhibitory interneurons, and thus I am concerned about how interneurons \"sense\" the synaptic weights. If there are neurobiological studies supporting this updating rule, please cite the reference in the paper, otherwise, it is better to discuss it at the end.\n\nQUESTIONS:\n- Line 91: ref. 11 requires $K \\geq N$, i.e., the number of I neurons is larger than the E neurons. Does this requirement also need in this paper? Is there a way to make the number of I neurons less than the E neurons? This is also the case in the cortex.\n- It seems that the $f_c(M)$ function was used before but the writing still looks abrupt to me about why to propose an objective function with such a form.\n- The paragraph starting at line 90. It is better to first explain the decomposition $M_c = W_c W_c^T$ regards to the weights between E and I neurons. Otherwise, people can easily regard $W_c$ as the feedforward weights.\n\nLIMITATIONS:\nI realize the framework assumes the synaptic weights between excitatory (E) and inhibitory (I) neurons are symmetric with each other by just differing a sign. Is there a way to break this symmetry? Will the symmetry breaking of E-I weights facilitate optimization?", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-10"}