Title: Neural Sampling in Hierarchical Exponential-family Energy-based Models
Venue: NeurIPS 2023
Accepted: yes
Keywords: Baysian brain

sampling-based inference

energy-based models

local learning

exponential-family
Abstract: Bayesian brain theory suggests that the brain employs generative models to understand the external world. The sampling-based perspective posits that the brain infers the posterior distribution through samples of stochastic neuronal responses. Additionally, the brain continually updates its generative model to approach the true distribution of the external world. In this study, we introduce the Hierarchical Exponential-family Energy-based (HEE) model, which captures the dynamics of inference and learning. In the HEE model, we decompose the partition function into individual layers and leverage a group of neurons with shorter time constants to sample the gradient of the decomposed normalization term. This allows our model to estimate the partition function and perform inference simultaneously, circumventing the negative phase encountered in conventional energy-based models (EBMs). As a result, the learning process is localized both in time and space, and the model is easy to converge. To match the brain's rapid computation, we demonstrate that neural adaptation can serve as a momentum term, significantly accelerating the inference process. On natural image datasets, our model exhibits representations akin to those observed in the biological visual system. Furthermore, for the machine learning community, our model can generate observations through joint or marginal generation. We show that marginal generation outperforms joint generation and achieves performance on par with other EBMs.

Neural Sampling in Hierarchical Exponential-family Energy-based Models

Abstract

Bayesian brain theory suggests that the brain employs generative models to understand the external world. The sampling-based perspective posits that the brain infers the posterior distribution through samples of stochastic neuronal responses. Additionally, the brain continually updates its generative model to approach the true distribution of the external world. In this study, we introduce the Hierarchical Exponential-family Energy-based (HEE) model, which captures the dynamics of inference and learning. In the HEE model, we decompose the partition function into individual layers and leverage a group of neurons with shorter time constants to sample the gradient of the decomposed normalization term. This allows our model to estimate the partition function and perform inference simultaneously, circumventing the negative phase encountered in conventional energy-based models (EBMs). As a result, the learning process is localized both in time and space, and the model is easy to converge. To match the brain's rapid computation, we demonstrate that neural adaptation can serve as a momentum term, significantly accelerating the inference process. On natural image datasets, our model exhibits representations akin to those observed in the biological visual system. Furthermore, for the machine learning community, our model can generate observations through joint or marginal generation. We show that marginal generation outperforms joint generation and achieves performance on par with other EBMs.

1 Introduction

Human behavioral studies [1, 2, 3] and animal neurophysiological studies [4, 5] have suggested that the brain performs statistically optimal Bayesian inference to interpret the external world [6, 7, 8, 9]. One promising theory for implementing Bayesian inference in the brain is to interpret the variability of neural responses as Monte Carlo sampling of the posterior distribution [10]. This perspective naturally accounts for the irregular firing patterns and other response properties observed in sensory cortex neurons [11, 12, 13]. Numerous sampling-based models [10, 14, 15, 16, 17, 12, 13, 18] have been proposed to elucidate neural dynamics and the underlying mechanisms of Bayesian inference.

To facilitate the brain's ability to derive meaningful representations from sensory input, it must continually update its generative model to approximate the true distribution of the external world [19]. Previous approaches often neglect this critical learning process. They either maintain fixed parameters for their generative model [15, 17, 16], or they employ biologically implausible methods like the variational approach with backpropagation (BP) for training [9]. Is there a generative model that integrates sampling-based inference with the capability to learn locally in both time and space?

Energy-based models (EBMs) [20] provide a framework for inference with sampling method and learning with spatially localized rules. The reason for non-locality in time is the need to estimate the partition function. To estimate the global partition function, the network has to perform a topdown pass to obtain a negative sample. This process is referred to as the negative phase. During this pass, it disrupts the neural network's stored inference results, which is essentially the same reason for the temporal non-locality as in the case of BP. Recently, predictive coding networks (PCNs) [21, 22, 23, 24] use the Gaussian distribution to avoid the negative phase since the partition function of Gaussian is constant. But further study [25] reveals that the Gaussian assumption is restrictive when dealing with complex probability distributions. Moreover, setting aside biological constraints, estimating the partition function itself poses a significant challenge. In the field of machine learning, various sampling methods, such as amortized generation [26] and implicit generation [27], have been proposed to tackle this issue. However, both of these methods involve the use of the negative phase, which is known for being challenging to converge. Generative adversarial networks (GANs) address this issue by utilizing a discriminative model, but GANs are notoriously difficult to train in practice.

Besides, the inference dynamic of Gibbs sampling [28, 20] or Langevin sampling [29] in EBMs essentially perform random walks in local regions rather than the whole posterior space, which is too slow to be compatible with brain functions [16]. Therefore, it is crucial to investigate whether neural circuits in the brain have the capacity of realizing sampling-based inference rapidly.

Summary of the work. In Sec.2, we propose that our brain holds an EBM as the intrinsic generative model to interpret the external world. The neural dynamics employ a sampling-based method for Bayesian inference. Simultaneously, the learning dynamic aims to minimize the discrepancy between the observed distribution of intrinsic model and the real world. In Section 3, we introduce the Hierarchical Exponential-family Energy-based (HEE) model, which allocates the partition function across each layer. This allocation shifts the estimation of the total sample space required for calculating the partition function from a product of individual layer spaces to a sum of layer spaces. This approach enhances the convergence of our model. Furthermore, we efficiently sample the normalization term of the exponential-family in each layer using a group of neurons with fast dynamics. This localizes the learning process in both time and space. In Sec.4, we find that incorporating noisy adaptation, a generic feature of neuronal responses, into the inference dynamics effectively yields a second-order Langevin dynamic. In Sec.5, we validate the capabilities of the HEE model using 2D synthetic datasets and FashionMNIST [30]. Then, we incorporate receptive field as the biological constrains to the HEE model training on CIFAR10 [31]. We show that the HEE model can achieve a performance comparable with previous EBMs [27]. We also investigate the neural representation of semantic information, including orientation, color and category, which exhibit similarities to biological visual systems. And the neural adaptation can trigger neural phenomena including oscillations and transient which are widely observed in biological systems. In Sec.6, we discuss several related theories and models.

Main Contributions. We propose a hierarchical EBM whose learning process is localized both in time and space, which could potentially serve as a mechanism for the brain to utilize changes in synaptic strength for learning. And our brain-inspired EBM also presents a technique for estimating the partition function, which is a challenging problem within the machine learning community.

2 The intrinsic generative model

In this section, we propose that our brain holds energy-based models (EBMs) as the intrinsic generative model consisting of two components: inference and learning. Inference is believed to be carried out through neural sampling [10, 20], while learning is accomplished through long-term synaptic plasticity [32].

Let x be the observation received by our brain, and let z be the latent variable represented by neurons. The joint distribution of the EBMs is written as p θ ( x , z ) = p θ ( x | z ) p θ ( z ) , where θ are stored in the connection weights of neurons (Fig.1A). The EBMs aims to minimize the difference between the intrinsic marginal distribution p θ ( x ) and the true distribution of the external world p true ( x ) , which is described by the Kullback-Leibler divergence,

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The neural system can adopts the gradient based learning method such as gradient decent. And the gradients is calculated as (see SI for detailed proof),

The second expectation of the above equation requires the posterior of a given observation x , which is calculated as p θ ( z | x ) = p θ ( x , z ) /p θ ( x ) . Practically, the posterior is intractable for the denominator p θ ( x ) requires complex integral. Variational inference is usually used to solve this problem (such as VAE [33]). And the EBMs adopts the sampling method to avoid complex calculations. By leveraging the relationship ∇ z ln p θ ( z | x ) = ∇ z ln p θ ( x , z ) ,the samples of posterior can be offered by the neural dynamic (Langevin sampling),

where ξ is Gaussian white noise and τ z is the time constant. The stationary distribution of the above dynamic is our target distribution p θ ( z | x ) . The sampling algorithm, along with the joint distribution, determines the connections of neurons and their inference dynamic (Fig.1B).

For the learning dynamic, the brain receives the observations from the real world continuously ( E x ∼ p true ( x ) ), and the neural dynamic mentioned above can produce samples of their posterior simultaneously ( E z ∼ p θ ( z | x ) ). The parameters can be updated according to the gradient, which is often implemented by Hebbian learning rules(Fig.1B),

where τ θ is the time constant of synapses.

For neuroscience, this is the end of the story. Nevertheless, EBMs can also generate observations which the machine learning society follow with interest. In the generation process, the observation is not fixed but undergoes the Langevin sampling,

The latent variable z can follow the inference dynamic Eq.(3). In this case, the observation x and latent variable z together follow the joint distribution p θ ( x , z ) . Thus, the observation x can produce samples following p θ ( x ) , which is called joint generation (Fig.1C). However, in order to get the marginal distribution p θ ( x ) , the latent variable z just needs to follow the prior distribution p θ ( z ) rather than reaching the posterior. In this case, the generation dynamic of latent variable is written as,

And the stationary distribution of Eq.(5) performed by observation x still equals to p θ ( x ) , which is called marginal generation (Fig.1D). In Sec.5, we will see that the marginal generation performs better than joint generation. Here is an informal understanding. The process of sampling ( x , z ) can be understood as searching for a specific pair. We assume that the size of the x -space is O ( m ) and the size of the z -space is O ( n ) . In joint generation, the search is conducted simultaneously in both the x -space and z -space, resulting in a required search space of O ( n ∗ m ) . In marginal generation, the process involves initially searching in the z -space according to p θ ( z ) . Once z is found, it is fixed. This step's search space size is O ( n ) . Then, x is searched based on p θ ( x | z ) in the x -space. This step's search space size is O ( m ) , leading to a combined required search space size of O ( n + m ) .

3 Exponential-family energy-based model

In this section, we provide a neural implementation of the HEE model and outline the specific dynamics involved in inference, learning, and generation, as discussed in Section 2. Approximating the target distribution p true ( x ) which is diverse and complex requires a good representation ability of the model. Exponential families include many of the most common distributions (such as normal, Poisson, gamma distribution and so on). Moreover, exponential families can be easily parameterized, allowing for generalization and flexibility in modeling various types of distributions.

Let x 0 ∈ R n 0 be the observation (such as an image) received by our brain. And there are L layers of neurons representing the latent variables x 1: L = { x 1 , x 2 , ..., x L } , x l ∈ R n l . The joint distribution is a Markov chain (Fig.2A) starting with p ( x L ) = exp [ η T L ϕ ( x L ) + g ( x L ) -A ( η L ) ] ,

The natural parameter η l ∈ R n l is a function of x l +1 with parameters θ l ∈ R n l × n l +1 , which is written as η l = θ l f ( x l +1 ) , where f ( · ) is the activation function. And η L is constant. The sufficient statistic ϕ ( x l ) ∈ R n l and the base measure g ( x l ) ∈ R is the function of x l . A ( η l ) ∈ R is the normalize term (log-partition function) to make sure the sum of the probability equals to 1 . In order to get the inference dynamic, we substitute the joint distribution Eq.(7) into the Langevin dynamic Eq.(3) obtaining,

where f ′ ( x l ) , ϕ ′ ( x l ) ∈ R n l × n l are diagonal matrices. The derivative of log-partition A ′ ( η l -1 ) is intractable for it needs complex integral. Here, we use a group of interneurons ε l -1 ∈ R n l -1 to represent the term ϕ ( x l -1 ) -A ′ ( η l -1 ) . It can be proved that A ′ ( η l -1 ) = E x l -1 ∼ p θ ( x l -1 | x l ) [ ϕ ( x l -1 )] (See SI for detailed proof). Thus, in order to calculate A ′ ( η l -1 ) , the interneurons need to produce samples x l -1 following the distribution p θ ( x l -1 | x l ) in a short time compared with the inference dynamic. Therefore, the dynamic of ε l -1 can be written as,

By setting the time constant τ u ≪ τ z 1 , we can ensure that u l -1 converges much faster than x l , and the stationary distribution of u l -1 corresponds to p θ ( u l -1 | x l ) . This leads to ε l -1 = ϕ ( x l -1 ) -A ′ ( η l -1 ) .

Now, we can rewrite the inference dynamic Eq.(8) into,

The first term on the right side of the dynamic equation indicates that neurons x l receive feedback from interneurons ε l -1 , which provides likelihood information p θ ( x l -1 | x l ) . The second term shows

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that neurons x l receive prior knowledge p θ ( x l | x l +1 ) from interneurons ε l through a feedforward loop. The term g ′ ( x l ) controls the self-connections within layer l (Fig.2B). ε l -1 is also called the error term in PCNs. Here, we show that the predictions in PCNs essentially estimate the log-partition.

Then, we can obtain the learning dynamic by substituting the joint distribution Eq.(7) into the gradient decent dynamic Eq.(4),

The derivatives of the log-partition function are stored in the interneurons ε l . As a result, the synaptic changes are determined solely by local neurons, adhering to Hebbian rules.

After inference and learning, our model can also generate observations. During joint generation, the dynamics of neurons x 1: L follow the same principles as the inference dynamics described by Eq.(10). By substituting the joint distribution Eq.(7) into the Langevin dynamic Eq.(5), we can generate new samples from the model by,

And for marginal generation, we can substitute the prior distribution described in Eq.(7) into Eq.(6) obtaining the dynamic of neurons x 1: L ,

4 Neural adaptation accelerate the sampling process

Langevin sampling essentially performs random walks in local regions rather than the whole posterior space [37], because the drift term ∇ z ln p θ ( x , z ) in Eq.(3) will vanish near the local minima and only noise term remains (Fig.3A). Sampling the entire posterior space is a time-consuming process and does not align with the brain's ability to perform tasks quickly. Therefore, it is important to implement a faster sampling algorithms for our model.

In this section, we show that by including noisy adaptation, the network is able to speed up the inference dynamic significantly. Adaptation is a common phenomenon observed in neural systems,

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where negative feedback mechanisms are employed to suppress neuronal responses when they reach high levels. Here, we show that the neural adaptation v can introduce an auxiliary variable to implement a faster sampling algorithm, which is called second-order Langevin dynamics (SLD),

where m> 0 controls the adaptation strength. The auxiliary variable v can keep the network moving while reaching the local minima, which essentially serves as the momentum term (Fig.3B). When there is no adaptation ( m = 0 ), the above dynamic will degenerate to the Langevin sampling (LS) described in Eq.(3). And the stationary distribution of the above dynamic remains to be p θ ( z | x ) (See SI for proof). By substituting the joint distribution of exponential family Eq.(7) into the above dynamic, we can obtain the SLD inference dynamic,

Here we exemplify the neural adaptation with spike frequency adaptation (SFA) [38]. In the neural system, the adaptation current v l accumulates the noise term ξ l coming from the ion concentrations, release of neural transmitters, activation/inactivation of ion channels and so on, which is described by Eq.(17). The adaptation current induces suppression on neurons x l , acting as momentum variables to accelerate the sampling process (Eq.(16)).

We conduct further investigation to elucidate the precise mechanism by which noisy adaptation facilitates the acceleration of the sampling process in the HEE model. Considering that the energy function -ln p θ ( x 1: L | x ) is non-convex, the sampling process can be divided into two parts. In the first part, the network needs to find a local minima and samples near the local minima, which takes a certain amount of time called recurrence time T rec (Fig.3C). In the second part, the network needs to leave the local minima and find a new one, which takes a certain amount of time called the escape time T esc . Typically, we have T rec ≪ T esc . The total time to get stationary distribution can be approximated by T = T esc + T rec . It can be proved that [39] the recurrence time is bounded by T rec = O (1 /λ 1 ( H J )) . λ 1 ( H J ) is the smallest eigenvalue of the matrix H J ,

where H is the Hessien matrix of the energy function -ln p θ ( x 1: L | x 0 ) . The smallest eigenvalue of H J is calculated as λ 1 ( H J ) = min { λ 1 ( H ) /τ z , m/ (2 τ v ) } . Thus, in the case m> 2 λ 1 ( H ) τ v /τ z , the SLD can reduce the recurrence time T rec to accelerate the sampling process. And the escape

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time is bounded by T esc = O ( √ 1 / det( H J ) ) . The determinant of H J is calculated as det( H J ) = det( H )( mτ/ 2 τ v ) ∑ n l . Thus, det( H J ) is monotonically increaseing with m , indicating that the larger m is, the shorter the time it takes to escape from the local minima.

The analysis presented above demonstrates that the convergence speed of the inference dynamic is determined by the values of λ 1 ( H ) and det( H ) . This valuable insight can be leveraged to guide the design of the network architecture, enabling the creation of more efficient and effective models. Specifically, with a fixed total number of neurons ∑ l n l , increasing the number of layers L results in a deeper network, while decreasing the number of layers results in a wider network. Practically, we show that det( H ) will decrease with layers L while λ 1 will increase (Fig.3D), which indicates that there is a trade-off between the recurrence time T rec and the escaping time T esc with different layers L (See SI for detailed setting and analysis).

5 Experiment

In this section, we firstly validate the capability of HEE model for approaching complex distribution by examining the quality of generation. Then, we show that our model demonstrates similarity in the representation of natural images to the biological visual system. And adaptation can induce oscillatory behavior and transient overshoots in neurons during the inference phase.

5.1 Generation

Firstly, we conducted experiments using three variations of the HEE model, each with different ϕ ( x ) functions and sampling methods (Tabel 1), to evaluate their capabilities. The experiments were performed on both 2D synthetic datasets and the FashionMINST . We use the fully connected architecture, i.e., θ l has no zero elements. The results (second and third column) show that HEE with linear statistic (HEE-L) struggles to capture the complex distribution. Moreover, we theoretically prove that HEE-L can only approach unimodal distributions (See SI for detailed proof). For HEE-NL, some modes are missing while using the joint generation. And when the spacing between modes is large, there is an issue of non-uniformity among different modes. And the marginal generation converge much faster than the joint generation. In FashionMNIST , it takes less time for marginal method to get the generation of high-quality images.

Then, we employ the HEE-NL-A with layers L = 10 on the CIFAR10 unconditional. The sparse connection is employed as a method to mimic the receptive field behavior found in biological systems. We quantitatively evaluate image quality of HEE-NL-A with Inception score [40] and FID score [41] in

Tabel 2. Overall, we achieve a performance comparable with the previous EBMs. And the generation quality of the marginal method is better than joint method, which agrees with the previous results [42].

5.2 Inference

We further use the HEE-NL-A trained on CIFAR10 to explore the relationship between the latent features and semantic information, including orientation, color and category.

Orientation: Simple cells [43] and complex cells [44] are the most prominent and widely observed neurons in the biological visual system that exhibit tuning to orientation. They are found in the primary visual cortex (V1) of numerous animal species [45, 46]. We present the model with gabor images of different orientations commonly used in experiments (Fig.6A) and compute the mean and variance of the neural responses for each neuron. Then, we use a Gaussian curve with bandwidth limited from 20 ◦ to 90 ◦ and a two-modes Gaussian curve to fit the simple cell and complex cell, respectively (Fig.6A&B). We find that the proportion of simple cells and complex cells remains relatively consistent across each layer and both decrease with increasing layers in our model (Fig.6C).

End stopping: In the HEE model, interneurons essentially represent the error term in the PCNs. We have observed the phenomenon of 'end stopping' in interneurons (Fig.6D), which aligns with the end-stopping behavior observed in error neurons in the PCNs [21].

Color: Recent study shows that there is a hierarchical representation for chromatic processing across the ventral pathway of macaque [47]. We present our model using reshaped natural images [48] and employ principal component analysis to demonstrate that the middle layer's neural representation's most informative dimensions carry chromatic information (Fig.6E).

Category: Visual object recognition is believed to be solved by the brain hierarchically [49]. A recent study [50] demonstrate that the inferotemporal cortex, situated in the deeper layer of the visual pathway, is capable of constructing a linear map of the object space. For each layer in our model, we employed a linear support vector machine (SVM) to classify the ten labels of the CIFAR10 . The SVM was trained using the average neural responses as features. Fig.6F illustrates the projection of the features in the last layer onto the SVM weights corresponding to the "cat" and "dog" labels. Furthermore, we observe that the classification accuracy improves as we move up the layers of our model (Fig.6G), which is consistent with findings in both biological visual systems [50] and the artificial neural networks [51].

Phenomena: Oscillations [52] and transients [53] are two kinds of spatial-temporal dynamic features in neural systems, which play a crucial role during the sampling process [13]. Here, we show that by adjusting the adaptation strength m , the oscillation frequency of the HEE model can span within the

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6 Discussion

The present study investigates the sampling-based inference and learning dynamic within the framework of an intrinsic generative model. We introduce the HEE model as a neural implementation that utilizes neural dynamics and Hebbian learning. Additionally, we demonstrate that the inclusion of

neural adaptation can significantly accelerate the sampling process and give rise to various dynamic phenomena throughout the network. In this section, we will discuss several related theories and models.

Probabilistic Population Code (PPC) [55, 56] is another theory that explains how the brain perform Bayesian inference, in which neural responses are interpreted as the parameters of the probability distributions. We adopted the idea [57] that PPC theory incorporates two generative models. In the framework of PPC, the experimenter presents the subject with observation x (gabor image) based on the semantic information s (orientation), which actually defines an external generative model from s to x . And the subjects holds an intrinsic generative model with latent variable z represented by neural response to interpret the observation x . PPC theory integrates two generative models into a single generative model, in which the neural response z is regarded as the observation generated from the semantic information s . We propose that the learning dynamic occurs exclusively within the intrinsic generative model, as the brain is not aware of the external generative model.

Energy-based models (EBMs) [20, 58] When EBMs were initially proposed [20], they had latent variables corresponding to neurons. Later, to enhance the model's expressive power, hierarchical structures were introduced [59, 60]. Such EBMs could typically ensure local learning in space. As artificial neural networks have become increasingly powerful, it has been observed that for generative tasks, there's no need to explicitly introduce neurons as latent variables within EBMs. Instead, one can directly employ a neural network to represent the energy [27]. Training such EBMs often involves utilizing BP. The distinction between these EBMs and traditional EBMs is akin to the difference between dynamic systems and recurrent neural networks.

Regardless of whether it's the traditional EBMs or the new type of EBMs, both involve the challenge of estimating the partition function. This difficulty arises from the fact that as the depth of the energy function increases, the total sample space required multiplies the space for each layer. In the case of HEE, we allocate the partition function across each layer. As a result, the total sample space required is the sum of the spaces for each layer. This significantly reduces the required sample space.

Predictive coding networks (PCNs) [21, 22] The interneurons in the HEE model serve a similar role to the prediction error in PCNs. And our theoretical analysis shows that the predictions in PCNs essentially represent the decomposed log-partition function. And PCNs don't stress the samplingbased inference, which requires them to approximate the energy function using variation inference by delta function. Sampling-based inference can assist the network in exploring the posterior probability space, leading to a more accurate estimation of the energy function. Additionally, it can account for the observed neural variability in experiments.

Diffusion models (DDPMs) [61, 62] The HEE model and DDPMs share the same joint distribution and both exhibit a hierarchical Markov structure, which may contribute to the HEE model's strong expressive potential. The marginal generation is also called latent space MCMC [63, 42], which is similar to the generation process of DDPMs. While DDPMs unfolds the Markov chain over time, the HEE model unfold it between layers of neurons. However, in order to reach a better performance, DDPMs use a fixed diffusion process as the inference dynamic, which may not be adopted by our brain since the latent variables in our brain carry semantic information (such as simple cells [43]).

Speed up sampling [17, 38] In previous work, inhibition neurons were used to serve as momentum terms to accelerate sampling [17]. However, this approach required a one-to-one correspondence between inhibition neurons and excitatory neurons. In our approach, we consider the adaptive properties inherent in each neuron itself to serve as momentum, naturally resolving the one-to-one correspondence issue. Furthermore, our consideration extends to sampling in a non-convex energy space, which differs from the prior focus solely on convex space convergence properties [38].

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces the hierarchical exponential family energy based model as a biologically plausible mechanism for the brain to interpret the external world. The authors first describe the learning/inference and the generation process of the model, the dynamics is local and biologically plausible through introducing a set of fast interneurons for the log-partition functions. They further show that adding adaptation mechanism can accelerate the sampling process. Finally, the authors present extensive numerical results to show that the model achieves good generation quality, and exhibit representations similar to those in the biological visual systems.\n\nSTRENGTHS:\n1. This paper presents a nice combination of theoretical analysis and extensive numerical results to demonstrate that their model exhibits multiple desired properties (including the acceleration effect of the adaptation mechanism, generation quality and representations). \n2. The authors show multiple interesting aspects of their models that are very relevant to the neuroscience community, including the acceleration effect of neural adaptation, and the similarity between the representations in biological visual system and the model, thereby providing new insights on the possible advantages/origins of well-known neural mechanisms/representations in the context of Bayesian inference in the brain.\n\nWEAKNESSES:\nThe paper is very well written in general, and establishes nice connections between their findings and neural representations in the visual system. However, the comparison with neuroscience is limited to well-known experimental findings, it would be nice if the authors can add a discussion to highlight several possible experimental predictions of their model.\n\nQUESTIONS:\n1. Eq4: It is a little confusing, before the second equality there is the KL divergence which is defined by Eq.2 with averaging over $z$ and $x$, whereas after the equality it is actually $p_{\\theta}(x,z)$ for a sample of $\\{x,z\\}$. Maybe it would be better to introduce separate notations for the random variables $x$ and $z$ (perhaps capital letters) and the realizations of $x$ and $z$.\n2. Line 99-100: If I understand correctly, both joint generation and marginal generation can generate samples following  $p_{\\theta}(x)$? Why say “However, in order to get the marginal distribution $p_{\\theta}(x)$” here?\n3. What could the fast interneurons $\\epsilon_l$ correspond to in the visual circuitry? \n4. Is there an intuitive explanation why the marginal method always seem to perform better than the joint method for generation? \n5. Fig 6D: As the proportion of both simple and complex cells decrease with the number of layers, do you see neurons with more complicated receptive fields (objects, etc.)?\n\nLIMITATIONS:\nYes the authors have addressed the limitations.", "TITLE": "Official Review", "RECOMMENDATION": 8, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2023-06-24"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a new model for sampling-based neural inference. The model posits that the brain attempts to match the marginal distribution $p(\\mathbf{x})$ of an internal generative model $p(\\mathbf{x}|\\mathbf{z})$ (based on a neural representation $\\mathbf{z}$) to the observed distribution of sensory inputs. In the model, sampling occurs via Langevin dynamics. The contributions of the paper are 1) use of a hierarchical exponential family model, 2) introduction of a second latent represenation $\\mathbf{u}$ (putatively identified with interneuron dynamics) responsible for estimating the demeaned residual $\\phi(\\mathbf{x}) - A'(\\boldsymbol{\\eta})$, and 3) use of a second-order Langevin sampling scheme. Experiments show that the resulting generative model can perform on toy data sets and there are further claims about relation to neural data in the experiments.\n\nThis is a potentially interesting idea that I found to be muddled by lack of justification of several modeling choices and unclear exposition, particularly in discussing the rationale and results of the experiments.\n\nSTRENGTHS:\n- The neural sampling framework has been an interesting line of inquiry over the previous decade, particularly in thinking about how the brain might implement Bayesian inference, and this work furthers that approach.\n- I found the use of a hierarchical model and the second latent representation to be interesting technical twists that are potentially powerful.\n- The paper attempts to align both the additional latents and the second-order Langevin dynamics with known features of neural physiology.\n\nWEAKNESSES:\n- The paper is written as if its main contribution will be to theoretical neuroscience, but the experiments present evaluations of the generative model on toy data sets, and the experiments underlying Figure 6 have an unclear rationale.\n- Overall, the paper has some deficits in presentation. For instance, section 2 is somewhat confusingly written. There is some parallel material that might be merged and better motivation given to the proposed choices. Why Langevin dynamics? Why not some other sampling method? Similarly, I didn't find the diagrams in Figure 1 particularly helpful in giving an intuition for the math. \n- Likewise, the rationale behind most of the experiments is not clearly explained in the text, particularly, those underlying Figure 6. Captions are too sparse to explicate what the results mean, and the discussion in text only barely explains the relevant neuroscience background. I have no idea what analysis was done for Figures 6A and 6B, nor why. Figures 5 and 6E are impossibly small.\n- ll. 120-23: This is a pretty compressed discussion of previously proposed approaches and perhaps difficult to follow for a non-expert reader.\n- More generally, identifications of model constructs with neuroscience findings ($\\mathbf{u}$ with interneuron activity, $\\mathbf{v}$ with spike-rate adaptation) are simply stated without any real justification. I'm not saying these choices are impossible to justify, but no real arguments that these are plausible assumptions are given. Again, if this is supposed to be a generative model loosely based on neuroscience, that can be fine, but if it's a theoretical neuroscience model, it's not clear these choices have been fully thought-through, and the experiments should ideally be focused on demonstrating that the model can perform like the brain in some perceptual task.\n\nQUESTIONS:\n- ll. 152-156: It's clear from (17) that $\\mathbf{v}$ represents a a noise term with temporal autocorrelation. In typical neural models, this is assumed semi-empirically as the result of temporal correlations among a large number of inputs to a given system. Is there an intuition the authors can give in this text for why it is better to identify (17) with spike frequency adaption as opposed to the existence of autocorrelated noise in synaptic inputs?\n- It's not very clear from the results presented what benefits hierarchy in processing brings. Can the authors show in some more convincing way which posterior features hierarchy specifically helps to capture?\n\nLIMITATIONS:\n- The model in the paper assumes a feedforward generative model rather than a recurrent model, as in most brain circuits. \n- The assumption $\\tau_u \\ll \\tau_z$ (line 128) is a strong one, and one presumes is justified by the relatively higher firing rates of cortical interneurons compared with principal neurons. But can this be justified from firing rates alone? Is there an argument that needs to be made about relative equilibration times of neuronal voltage dynamics for these two types of cells to substantiate this?\n- It's clear from the material in the supplement that some versions of the model cannot deal with multimodal posteriors, and even the second-order Langevin dynamics proposed will have difficulty mixing well in high-dimensional latent spaces.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2023-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this article the authors introduce a new model, called the ‘’Hierarchical Exponential family Energy’ (HEE) that is biologically plausible and captures the dynamic of inference and learning in the brain. The model introduces multiple layers to decompose the EBM normalizing constant in a bio-plausible fashion, and leverages a neural adaption mechanism to make the sampling process compatible with ‘biological time’. In a series of experiments on synthetic dataset, Fashion MNIST and CIFAR10, the authors demonstrate the generation abilities of the HEE model. In addition, the authors demonstrate that the representation elicited by the HEE are similar to those observed in biological systems.\n\nSTRENGTHS:\nThis article offers an interesting and valuable perspective on Energy Based Models (EBM) taking inspiration from neuroscience. The proposed solution to the intractable normalizing constant is original and elegant, and its link to predictive coding is interesting. Overall, there are plenty of good ideas in these article.\n\nWEAKNESSES:\nMy biggest concerns are related to the experimental part:  (1) ablation studies that would validate the mathematical choice of the authors are missing , and (2) in general the experiments lack careful description. In addition, (3) the last part concerning the similarities between biological and HEE representation lack better comparison with other models. See more detailed comment below:\n\n\n1 ) One of the innovation of this article lies in the decomposition of the intractable partition function using multiple layers. Intuitively I do see the advantage of the proposed method compared to the bufferization of the amortization method used in standard EBM. But I was expected a comparative experiment between the 3 different methods to confirm the advantage of the proposed method. The comparison with the IEBM of [1] is not carefully enough conducted to conclude to the superiority of the proposed method : the parametrization seems to be different (CNN versus FullyConnected + number of parameters) and other improvement are introduced (e.g. neural adaptation) that might biased the comparison. In general I would suggest more ablation experiments that would test the benefits of all the HEE components separately.\n\n2) The networks are poorly described in the experimental part. What is the exact parametrization of the network (dimension of the fully connected network) ? What is the exact sampling procedure you use in the HEE : Do you first wait for a stable point in the sampling of the latent variable x_l before sampling the next one ? Or do you do one step of sampling for all the variables until you reach a global stable point ? Such implementation details might lead to very different equilibrium state and should be at least discussed. Also, how many sampling steps are you using ? Or are you forcing a stoping criterion (based on the prediction error ?) to decide when to stop the sampling process ?\n\n3) The experimental results in section 5.2 lacks comparison with other similar model. There is nothing showing the results you show are not  general to all networks (and not specific to HEE). Could you compare the HEE representations with those obtained for other EBM (or even other generative model) ? The idea would be to demonstrate that an EBM compatible (in terms of implementation) with biology would produce representation more aligned with biology.\n\nQUESTIONS:\n* You did not include any comparison with other EBM for Fashion MNIST or the synthetic dataset. Is there a reason for that ? The synthetic dataset would have offered the perfect opportunity to conduct the comparison described previously .\n\n* One of the major difficulty in the IEBM [1], is the lack of stability of the training procedure (partly due to the mode traversing issue in the negative phase). Have you faced the same issue with the HEE ?\n\n* Line 200 : I don’t understand the sentence : \"The dropout technique is employed as a method to mimic the receptive field behavior found in biological systems. » Could you further explain ?\n\n* Have you tried some convolutional architecture ? Don’t you think it would improve the results ?\n\n* The difference between HEE-NL-A and HEE-NL is only the number of layers ? This should be clarified.\n\n* In Table 1, could you define LS and SLD ?\n\n* In Fig5, the samples seems be visually lower quality than those of the IEBM [1], but the FID and IS are better. Do you have an explanation ? In particular, in the samples you present, the textures seem to be rather uniform…\n\n* What is the impact of the number of layers in the quality of the generation ??\n\n\nTypo : \nLine 65 : EMB —> EBM \nLine 135: PCN is used but not defined yet.\n\n[1] Du, Yilun, and Igor Mordatch. \"Implicit generation and generalization in energy-based models.\" arXiv preprint arXiv:1903.08689 (2019).\n\nLIMITATIONS:\nLimitations have been properly addressed by the authors", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this submission, the authors introduce a biologically plausible method to train a hierarchical energy-based model as well as perform inference over it, via Langevin sampling. This accomplishes a goal that has long been sought in computational neuroscience, namely, neurally plausible methods for inference over a hierarchical generative model of the world.\n\nPreviously, the main problem facing inference in hierarchical energy-based models is the difficulty of evaluating the partition function. The authors address this issue by modeling the partition function with Langevin sampling, but with a faster timescale than the Langevin sampling used for inference more generally. This appears to work well empirically. Also underlying their model's impressive generation performance is the use of a momentum term in the Langevin sampling, similar to Hamiltonian Monte Carlo, which they associate with short-term adaptation in the brain. Learning in this system is Hebbian and biologically plausible, and additionally the authors show some rough parallels with biological phenomena, such that the inferred activations of selected neurons have orientation- and hue-specific tuning curves.\n\nSTRENGTHS:\nI was impressed with the quality of samples generated by this hierarchical EBM, and all the more so because the learning process is so simple and Hebbian in nature. The model aligns well with a wide literature in computational neuroscience, including hierarchical predictive coding networks, and it extends previous EBM models in an important way by enabling a pure sampling-based inference procedure. The model combines multiple previous ideas into a single framework, and the approach feels natural.\n\nWEAKNESSES:\nAlthough generation was impressive, this is also a proposal for Bayesian inference, and there were no serious benchmarks quantifying inference. Some thought should be put into evaluating known benchmarks for inference over hierarchical energy-based models. Many examples are possible, e.g. hierarchical gaussian mixture models, etc. \n\nClarity was a major issue. This was difficult to read. If this is going to be widely read, it should to be rewritten. I’ll describe specific problems below.\n\nFirst, there was not sufficient context in the surrounding literature. The introduction is rather short about EBMs. Otherwise it is difficult to know about this work’s specific contribution as many of these ideas have separately been introduced before (H-EBMs, Hamiltonian MC, etc.).  Also, for example, section 2 should summarize previous hierarchal EBMs, as currently it reads like the authors’ original proposal (which it is not). The related work section should be greatly expanded.\n\nThere were not also enough details about the experimental methods. I would certainly not be able to reproduce these figures from the manuscript. If there is not enough room, at least please put more details in the supplementary figures. \n\nFor clarity, I would recommend adding more detail about the biological circuit proposal. It is difficult for me to imagine the connectivity of the inhibitory neurons. (Do they need to project from one area to another, for example?) A figure with biology in mind would help. More details about biology should be added, as right now it is rather vague as to how this would be implemented in the brain. \n\nIn addition, many sentences state a controversial hypothesis as a known fact. These should be weakened considerably. For example,\nLine 46: “In this study, we show that our brain holds an intrinsic energy-based model”. ‘Show’ implies a certain conclusion. Better would be ‘hypothesize’ or ‘propose’.\nLine 70: “We demonstrate that our brain holds an intrinsic energy-based model”. Again, ‘demonstrate’ is conclusive. You are demonstrating that the brain *might* use such a model.\nLine 1: “The brain engages in probabilistic inference” This is just a hypothesis, although a popular one. Neural networks can appear to be Bayes-optimal even though they are not at all probabilistic internally. See Orhan, A. Emin, and Wei Ji Ma. \"Efficient probabilistic inference in generic neural networks trained with non-probabilistic feedback.\" Nature communications 8.1 (2017): 138.\n\n\n**Minor comments:**\n\nThe word ‘society’ is used when I think the authors want ‘community’.\n\nIn general the sentences are often awkwardly phrased. I recommend asking for editing help, perhaps using a large language model for grammar.\n\nSection 4. This seems to be Hamiltonian Monte Carlo. Is that correct? If so, this should be cited (e.g. Neal (2010)) as well as previous HMC in the brain proposals, like the Aitchison and Lengyel (2016) paper cited elsewhere. If it is incorrect, at least please cite SLD.\n\nLine 31: Many have argued that variational methods are biological plausible. See for example:\nD. Rezende and W. Gerstner, “Stochastic variational learning in recurrent spiking networks,” Frontiers in Computational Neuroscience, vol. 8, p. 38, 2014, doi: 10.3389/fncom.2014.00038.\n\nLine 84: “avoid complex calculations”. Could you be more specific?\nLine 121-124. This sentence is unclear and needs to be unpacked. What is amortized generation method, etc.\n\nQUESTIONS:\n- How many long does convergence take to produce the samples shown in the manuscript?\n - What is the connectivity of the interneurons? What cell types might these be? What evidence is there for their faster time constants?\n - I don't fully understand the circuit diagram for biological neurons. Could a schematic be drawn with individual neurons, complete with a table of the key properties of those neurons in this theory?\n\nLIMITATIONS:\nThere was not much mention of any limitations of this method. What are the least biologically plausible aspects? What aspects need to be experimentally confirmed?", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-05"}