SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Xingsi Dong1,2,3
dxs19980605@pku.edu.cn
Si  $\mathbf{W}\mathbf{u}^{1,2,3}$
1. PKU-Tsinghua Center for Life Sciences, Academy for Advanced Interdisciplinary Studies.
2. School of Psychological and Cognitive Sciences,
Beijing Key Laboratory of Behavior and Mental Health, Peking University.
3. IDG/McGovern Institute for Brain Research. Center of Quantitative Biology, Peking University.
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.
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 top-down 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  $\mathbf{x}$  be the observation received by our brain, and let  $\mathbf{z}$  be the latent variable represented by neurons. The joint distribution of the EBMs is written as  $p_{\theta}(\mathbf{x},\mathbf{z}) = p_{\theta}(\mathbf{x}|\mathbf{z})p_{\theta}(\mathbf{z})$ , where  $\theta$  are stored in the connection weights of neurons (Fig.1A). The EBMs aims to minimize the difference between the intrinsic marginal distribution  $p_{\theta}(\mathbf{x})$  and the true distribution of the external world  $p_{\mathrm{true}}(\mathbf{x})$ , which is described by the Kullback-Leibler divergence,
$$
\min  _ {\theta} D _ {\mathrm {K L}} \left[ p _ {\text {t r u e}} (\mathbf {x}) \| p _ {\theta} (\mathbf {x}) \right]. \tag {1}
$$
Figure 1: (A) The directed graphical model of the energy-based model (EBM). (B) The latent variable  $\mathbf{z}$  receives the likelihood information  $p_{\theta}(\mathbf{x}|\mathbf{z})$  from the observation  $\mathbf{x}$  and combines it with the prior knowledge  $p_{\theta}(\mathbf{z})$  to perform the inference dynamic. And the connected weights  $\theta$  changes following the learning dynamic (dashed line). (C) The latent variable  $\mathbf{z}$  performs inference dynamic and the observation  $\mathbf{x}$  performs generation dynamic, which leads to the distribution of  $\mathbf{x}, \mathbf{z} \sim p_{\theta}(\mathbf{x}, \mathbf{z})$ . And the connection weights  $\theta$  are fixed (solid line). (D) The latent variable doesn't receive likelihood information from observation and  $\mathbf{z} \sim p_{\theta}(\mathbf{z}), \mathbf{x} \sim p_{\theta}(\mathbf{x})$  which is called marginal generation.
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),
$$
\nabla_ {\theta} D _ {\mathrm {K L}} \left[ p _ {\text {t r u e}} (\mathbf {x}) \| p _ {\theta} (\mathbf {x}) \right] = - \mathbb {E} _ {\mathbf {x} \sim p _ {\text {t r u e}} (\mathbf {x})} \mathbb {E} _ {\mathbf {z} \sim p _ {\theta} (\mathbf {z} | \mathbf {x})} \left[ \nabla_ {\theta} \ln p _ {\theta} (\mathbf {x}, \mathbf {z}) \right]. \tag {2}
$$
The second expectation of the above equation requires the posterior of a given observation  $\mathbf{x}$ , which is calculated as  $p_{\theta}(\mathbf{z}|\mathbf{x}) = p_{\theta}(\mathbf{x},\mathbf{z}) / p_{\theta}(\mathbf{x})$ . Practically, the posterior is intractable for the denominator  $p_{\theta}(\mathbf{x})$  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  $\nabla_{\mathbf{z}}\ln p_{\theta}(\mathbf{z}|\mathbf{x}) = \nabla_{\mathbf{z}}\ln p_{\theta}(\mathbf{x},\mathbf{z})$ , the samples of posterior can be offered by the neural dynamic (Langevin sampling),
$$
\tau_ {z} \frac {\mathrm {d} \mathbf {z}}{\mathrm {d} t} = \nabla_ {\mathbf {z}} \ln p _ {\theta} (\mathbf {x}, \mathbf {z}) + \sqrt {2 \tau_ {z}} \boldsymbol {\xi}, \tag {3}
$$
where  $\xi$  is Gaussian white noise and  $\tau_z$  is the time constant. The stationary distribution of the above dynamic is our target distribution  $p_{\theta}(\mathbf{z}|\mathbf{x})$ . 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  $(\mathbb{E}_{\mathbf{x}\sim p_{\mathrm{true}}(\mathbf{x})})$ , and the neural dynamic mentioned above can produce samples of their posterior simultaneously  $(\mathbb{E}_{\mathbf{z}\sim p_{\theta}(\mathbf{z}|\mathbf{x})})$ . The parameters can be updated according to the gradient, which is often implemented by Hebbian learning rules(Fig.1B),
$$
\tau_ {\theta} \frac {\mathrm {d} \theta}{\mathrm {d} t} = - \nabla_ {\theta} D _ {\mathrm {K L}} [ p _ {\text {t r u e}} (\mathbf {x}) \| p _ {\theta} (\mathbf {x}) ] = \nabla_ {\theta} \ln p _ {\theta} (\mathbf {x}, \mathbf {z}), \tag {4}
$$
where  $\tau_{\theta}$  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,
$$
\tau_ {x} \frac {\mathrm {d} \mathbf {x}}{\mathrm {d} t} = \nabla_ {\mathbf {x}} \ln p _ {\theta} (\mathbf {x}, \mathbf {z}) + \sqrt {2 \tau_ {x}} \boldsymbol {\xi}. \tag {5}
$$
The latent variable  $\mathbf{z}$  can follow the inference dynamic Eq.(3). In this case, the observation  $\mathbf{x}$  and latent variable  $\mathbf{z}$  together follow the joint distribution  $p_{\theta}(\mathbf{x},\mathbf{z})$ . Thus, the observation  $\mathbf{x}$  can produce samples following  $p_{\theta}(\mathbf{x})$ , which is called joint generation (Fig.1C). However, in order to get the marginal distribution  $p_{\theta}(\mathbf{x})$ , the latent variable  $\mathbf{z}$  just needs to follow the prior distribution  $p_{\theta}(\mathbf{z})$  rather than reaching the posterior. In this case, the generation dynamic of latent variable is written as,
$$
\tau_ {z} \frac {\mathrm {d} \mathbf {z}}{\mathrm {d} t} = \nabla_ {\mathbf {z}} \ln p _ {\theta} (\mathbf {z}) + \sqrt {2 \tau_ {z}} \boldsymbol {\xi}. \tag {6}
$$
And the stationary distribution of Eq.(5) performed by observation  $\mathbf{x}$  still equals to  $p_{\theta}(\mathbf{x})$ , 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  $(\mathbf{x},\mathbf{z})$  can be understood as searching for a specific pair. We assume that the size of the  $\mathbf{x}$ -space is  $O(m)$  and the size of the  $\mathbf{z}$ -space is  $O(n)$ . In joint generation, the search is conducted simultaneously in both the  $\mathbf{x}$ -space and  $\mathbf{z}$ -space, resulting in a required search space of  $O(n*m)$ . In marginal generation, the process involves initially searching in the  $\mathbf{z}$ -space according to  $p_{\theta}(\mathbf{z})$ . Once  $\mathbf{z}$  is found, it is fixed. This step's search space size is  $O(n)$ . Then,  $\mathbf{x}$  is searched based on  $p_{\theta}(\mathbf{x}|\mathbf{z})$  in the  $\mathbf{x}$ -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  $p_{\mathrm{true}}(\mathbf{x})$  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  $\mathbf{x}_0\in \mathbb{R}^{n_0}$  be the observation (such as an image) received by our brain. And there are  $L$  layers of neurons representing the latent variables  $\mathbf{x}_{1:L} = \{\mathbf{x}_1,\mathbf{x}_2,\dots,\mathbf{x}_L\}$ ,  $\mathbf{x}_l\in \mathbb{R}^{n_l}$ . The joint distribution is a Markov chain (Fig.2A) starting with  $p(\mathbf{x}_L) = \exp \left[\pmb {\eta}_L^T\phi (\mathbf{x}_L) + g(\mathbf{x}_L) - A(\pmb {\eta}_L)\right]$ ,
$$
p _ {\theta} \left(\mathbf {x} _ {0: L}\right) = p \left(\mathbf {x} _ {l}\right) \prod_ {l = 0} ^ {L - 1} p _ {\theta} \left(\mathbf {x} _ {l} \mid \mathbf {x} _ {l + 1}\right), \quad p _ {\theta} \left(\mathbf {x} _ {l} \mid \mathbf {x} _ {l + 1}\right) = \exp \left[ \boldsymbol {\eta} _ {l} ^ {T} \phi \left(\mathbf {x} _ {l}\right) + g \left(\mathbf {x} _ {l}\right) - A \left(\boldsymbol {\eta} _ {l}\right) \right]. \tag {7}
$$
The natural parameter  $\pmb{\eta}_l\in \mathbb{R}^{n_l}$  is a function of  $\mathbf{x}_{l + 1}$  with parameters  $\theta_{l}\in \mathbb{R}^{n_{l}\times n_{l + 1}}$  , which is written as  $\pmb {\eta}_l = \theta_lf(\mathbf{x}_{l + 1})$  , where  $f(\cdot)$  is the activation function. And  $\pmb{\eta}_{L}$  is constant. The sufficient statistic  $\phi (\mathbf{x}_l)\in \mathbb{R}^{n_l}$  and the base measure  $g(\mathbf{x}_l)\in \mathbb{R}$  is the function of  $\mathbf{x}_l$  .  $A(\pmb {\eta}_l)\in \mathbb{R}$  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,
$$
\tau_ {z} \frac {\mathrm {d} \mathbf {x} _ {l}}{\mathrm {d} t} = f ^ {\prime} (\mathbf {x} _ {l}) \theta_ {l - 1} ^ {T} \left[ \phi (\mathbf {x} _ {l - 1}) - A ^ {\prime} (\boldsymbol {\eta} _ {l - 1}) \right] + \phi^ {\prime} (\mathbf {x} _ {l}) \boldsymbol {\eta} _ {l} + g ^ {\prime} (\mathbf {x} _ {l}) + \sqrt {2 \tau_ {z}} \xi_ {l}, \tag {8}
$$
where  $f^{\prime}(\mathbf{x}_l), \phi^{\prime}(\mathbf{x}_l) \in \mathbb{R}^{n_l \times n_l}$  are diagonal matrices. The derivative of log-partition  $A^{\prime}(\boldsymbol{\eta}_{l-1})$  is intractable for it needs complex integral. Here, we use a group of interneurons  $\varepsilon_{l-1} \in \mathbb{R}^{n_{l-1}}$  to represent the term  $\phi(\mathbf{x}_{l-1}) - A^{\prime}(\boldsymbol{\eta}_{l-1})$ . It can be proved that  $A^{\prime}(\boldsymbol{\eta}_{l-1}) = E_{\mathbf{x}_{l-1} \sim p_{\theta}(\mathbf{x}_{l-1} | \mathbf{x}_l)}[\phi(\mathbf{x}_{l-1})]$  (See SI for detailed proof). Thus, in order to calculate  $A^{\prime}(\boldsymbol{\eta}_{l-1})$ , the interneurons need to produce samples  $\mathbf{x}_{l-1}$  following the distribution  $p_{\theta}(\mathbf{x}_{l-1} | \mathbf{x}_l)$  in a short time compared with the inference dynamic. Therefore, the dynamic of  $\varepsilon_{l-1}$  can be written as,
$$
\varepsilon_ {l - 1} = \phi (\mathbf {x} _ {l - 1}) - \phi (\mathbf {u} _ {l - 1}), \quad \tau_ {u} \frac {\mathrm {d} \mathbf {u} _ {l - 1}}{\mathrm {d} t} = \phi^ {\prime} (\mathbf {u} _ {l - 1}) \boldsymbol {\eta} _ {l - 1} + g ^ {\prime} (\mathbf {u} _ {l - 1}) + \sqrt {2 \tau_ {u}} \xi_ {u}. \tag {9}
$$
By setting the time constant  $\tau_{u} \ll \tau_{z}^{1}$ , we can ensure that  $\mathbf{u}_{l-1}$  converges much faster than  $\mathbf{x}_l$ , and the stationary distribution of  $\mathbf{u}_{l-1}$  corresponds to  $p_{\theta}(\mathbf{u}_{l-1}|\mathbf{x}_l)$ . This leads to  $\varepsilon_{l-1} = \phi(\mathbf{x}_{l-1}) - A'(\boldsymbol{\eta}_{l-1})$ . Now, we can rewrite the inference dynamic Eq.(8) into,
$$
\tau_ {z} \frac {\mathrm {d} \mathbf {x} _ {l}}{\mathrm {d} t} = f ^ {\prime} (\mathbf {x} _ {l}) \theta_ {l - 1} ^ {T} \varepsilon_ {l - 1} + \phi^ {\prime} (\mathbf {x} _ {l}) \boldsymbol {\eta} _ {l} + g ^ {\prime} (\mathbf {x} _ {l}) + \sqrt {2 \tau_ {z}} \xi_ {l}. \tag {10}
$$
The first term on the right side of the dynamic equation indicates that neurons  $\mathbf{x}_l$  receive feedback from interneurons  $\varepsilon_{l-1}$ , which provides likelihood information  $p_\theta(\mathbf{x}_{l-1}|\mathbf{x}_l)$ . The second term shows
There are various types of interneurons that target on pyramidal cells, comprising approximately  $10 - 20\%$  of the overall neuron population in the cerebral cortex [34]. The interneurons in the HEE model bear the closest resemblance to the Large Basket Cell or Nest Basket Cell [35], which collectively constitute around  $50\%$  of interneurons. Their electrophysiological characteristics include fast spiking, non-accommodating, and non-adapting behaviors. These interneurons have also been identified in the visual cortex of ferrets [36], displaying short-duration action potentials (approximately  $0.5\mathrm{ms}$  at half height). This suggests that these neurons have shorter time constants compared to pyramidal cells.
(A) Probabilistic model
Figure 2: (A) The directed graphical model of the hierarchical exponential-family energy-based (HEE) model. (B) The inference and learning dynamic of HEE model. The red arrows represent the likelihood information and the black arrows represent the prior information. Neurons  $\mathbf{x}_l$  receive the likelihood information from  $\varepsilon_{l-1}$  and receive prior information  $\pmb{\eta}_l$  from  $\varepsilon_l$ . The interneurons  $\varepsilon_{l-1}$  receive the natural parameter  $\pmb{\eta}_{l-1}$  from neurons  $\mathbf{x}_l$  and perform the Langevin sampling dynamic to approximate  $A'(\pmb{\eta}_{l-1})$ . Then, the interneurons compare it with the sufficient statistic  $\phi(\mathbf{x}_{l-1})$  received from  $\mathbf{x}_{l-1}$  to calculate the value of  $\varepsilon_{l-1}$ . The dashed line shows that the connection weights  $\theta$  will perform gradient decent in the learning dynamic. (C) The neural connection diagram between layer  $l$  and layer  $l+1$ .
(B) Inference & learning
that neurons  $\mathbf{x}_l$  receive prior knowledge  $p_{\theta}(\mathbf{x}_l|\mathbf{x}_{l + 1})$  from interneurons  $\varepsilon_{l}$  through a feedforward loop. The term  $g^{\prime}(\mathbf{x}_l)$  controls the self-connections within layer  $l$  (Fig.2B).  $\varepsilon_{l - 1}$  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),
$$
\tau_ {\theta} \frac {\mathrm {d} \theta_ {l}}{\mathrm {d} t} = \left[ \phi (\mathbf {x} _ {l}) - A ^ {\prime} (\boldsymbol {\eta} _ {l}) \right] f (\mathbf {x} _ {l + 1}) ^ {T} = \varepsilon_ {l} f (\mathbf {x} _ {l + 1}) ^ {T}. \tag {11}
$$
The derivatives of the log-partition function are stored in the interneurons  $\varepsilon_{l}$ . 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  $\mathbf{x}_{1:L}$  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,
$$
\tau_ {x} \frac {\mathrm {d} \mathbf {x} _ {0}}{\mathrm {d} t} = \phi^ {\prime} (\mathbf {x} _ {0}) \boldsymbol {\eta} _ {0} + g ^ {\prime} (\mathbf {x} _ {0}) + \sqrt {2 \tau_ {x}} \xi_ {0} \tag {12}
$$
And for marginal generation, we can substitute the prior distribution described in Eq.(7) into Eq.(6) obtaining the dynamic of neurons  $\mathbf{x}_{1:L}$ ,
$$
\tau_ {z} \frac {\mathrm {d} \mathbf {x} _ {l}}{\mathrm {d} t} = \phi^ {\prime} (\mathbf {x} _ {l}) \boldsymbol {\eta} _ {l} + g ^ {\prime} (\mathbf {x} _ {l}) + \sqrt {2 \tau_ {z}} \xi_ {l}. \tag {13}
$$
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  $\nabla_{\mathbf{z}}\ln p_{\theta}(\mathbf{x},\mathbf{z})$  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,
Figure 3: (A)(B) The sampling trace of Langevin dynamic and second-order Langevin dynamic of a mixture of 2D Gaussian distribution. (C) Illustration of the sampling process. States with lower energy indicates a higher probability, which needs to be stay longer. And the network needs to cross the energy barrier to reach a new local minima for the non-convex of the energy function. (D)  $\lambda_{1}$  and  $\operatorname{det}(H)$  increases and decreases, respectively, with the changes of layers  $L$  and the total number of the neurons fixed ( $\sum_{l} n_{l} = 10000$ ).
where negative feedback mechanisms are employed to suppress neuronal responses when they reach high levels. Here, we show that the neural adaptation  $\mathbf{v}$  can introduce an auxiliary variable to implement a faster sampling algorithm, which is called second-order Langevin dynamics (SLD),
$$
\tau_ {z} \frac {\mathrm {d} \mathbf {z}}{\mathrm {d} t} = \nabla_ {\mathbf {z}} \ln p _ {\theta} (\mathbf {z} | \mathbf {x}) - \mathbf {v} + \sqrt {2 \tau_ {z}} \xi , \tag {14}
$$