Neuronal Circuits Underlying Persistent Representations Despite Time Varying Activity

This week in computational neuroscience journal club we discussed the paper

Neuronal Circuits Underlying Persistent Representations Despite Time Varying Activity
Shaul Druckmann and Dmitri B. Chklovskii
Current Biology, 22(22):2095–2103, 2012.

The main idea of this work was to demonstrate that the fact that non-trivial temporal dynamics occur in neural circuits, this activity does not prevent such circuits from persistently representing a constant stimulus. The main example used (although other neural regions, such as V1 are discussed) is working memory in pre-frontal cortex. The authors cite previous literature that has shown that even when a stimulus is `stored’ in memory, there is still significant time-varying neural activity. While the authors make no indication that the represented stimulus must be preserved (i.e. temporal dynamics are due to temporally changing stimulus representations), they do present a viable architecture that describes how the temporal dynamics do not effect the stimulus stored for the working memory task.

At a high level, the authors use the high redundancy of neural circuits to introduce a network that can retain a stimulus in the space of allowable features while still permitting neural activity to continue changing the null-space of the feature matrix. This paper ties together some very nice concepts from linear algebra and network dynamics to present a nice and tidy `proof-by-construction’ of the hypothesis that evolving networks can retain constant stimuli information. To tie this paper in a broader setting, the network dynamics here are similar to those used in network encoding models (e.g. the locally competitive algorithm for sparse coding) and the idea of non-trivial activity in the null space of a neural representation ties in nicely to later work discussing preparatory activity in motion tasks.

The formal mathematics of the paper are concerned with neural representations based on the linear generative model. This model asserts that a stimulus s\in\mathbb{R}^M can be composed of a linear sum of $K$ dictionary elements d_i \in\mathbb{R}^K

s = \sum_{i=1}^N d_i a_i = Da

where the coefficients a_i represent amount of each d_i needed to construct s. In the network, a_i is also the activity of neuron i, meaning that activity in neuron i directly relates to the stimulus encoded by the network.  If the number of dictionary elements K is the same as the dimension of the stimulus M, then there is no possible way for the neural activity a to deviate from the unique solution a = D^{-1}s without changing the stimulus encoded by the system. The authors note, however, that in many neural systems, the neural code is highly redundant (i.e. K >> M), indicating that the matrix D has a large null-space and thus there are an infinite number of ways that a can change while still faithfully encoding the stimulus.

With the idea of allowing neural activity to vary within the nullspace, the authors turn to an analysis of a specific neural network evolution equation where the change in neural activity decays is defined by a leaky-integrator-type system

\tau \frac{d}{dt} a = -a + La

where \tau is the network time constant, and L is the network connectivity matrix, i.e. L_{i,j} indicates how activity in neuron j influences neuron i. The authors do not specify a particular input method, however it seems that it is assumed that at t=0 the neural activity encodes the stimulus. To keep the stimulus constant, the authors observe that setting

\frac{d}{dt}s = D\frac{d}{dt}a = 0

results in the feature vector recombination (FEVER) rule

D = DL

This rule is the primary focus of attention of the paper. The FEVER rule is a mathematically simple rule that lays out the necessary condition for a linear network (and some classes of non-linear networks) to have time-varying activity while encoding the same stimulus. Note that this concept is in stark contrast to much of the encoding literature (particularly for vision) which utilizes over-complete neural codes to encode stimuli more efficiently. Instead this work does not try to constrain the encoding any more than the encoding must represent the stimulus. The remainder of this paper is concerned with how such networks can be found given a feature dictionary D and the resulting properties of the connectivity matrix L. Specifically the problem of finding L given D is highly under-determined. In fact there are K^2 unknowns and only M equations. As we already have K>M, this problem requires fairly strong regularization in order to yield a good solution. Two methods are proposed to solve for L. The first method seeks a sparse connectivity matrix by solving the \ell_1-regularized least squares (LASSO) optimization

\widehat{L} = \arg\min_{L} \|D - DL\|_F^2 + \lambda\sum_{i,j} |L_{i,j}|

where \lambda trades off between the sparsity of L and how strictly we adhere to the FEVER rule. This optimization, however, could result in neurons that both excite and inhibit other neurons. By Dale’s law, this should not be possible, so an alternate optimization is proposed. Setting L = E-N where E is an excitatory matrix and N is an inhibition matrix, the authors make the assumption that there are dispersed, yet sparse excitation connections (E is sparse) and that there are few, widely connected, inhibitory neurons (N is low-rank). Thus these two matrices can be solved using a “sparse + low-rank” optimization program

\{\widehat{E},\widehat{N}\} = \arg\min_{E,N} \|D - D(E-N)\|_F^2 + \lambda_1\sum_{i,j} |E_{i,j}| + \lambda_2\|N\|_{\ast}

where \lambda_1,\lambda_2 trade off between the sparsity of E, rank of N and adherence to the FEVER rule. This optimization is surprisingly well defined and allows for the determination of L from D. Before implementing this learning, however, the authors note two more aspects of the model. First, by using a modified FEVER rule \alpha D = DL, a memory element can be built into the network. Second, certain classes of nonlinear networks can use the FEVER rule to preserve stimulus encoding. Specifically, networks that evolve as

\tau \frac{d}{dt} a = -f(a) + Lf(a)

with a point-wise nonlinearity f(\cdot) will have ds/dt = 0 for a linear generative model. This property only holds, however, since the nonlinearity comes before the linear summation L. If the nonlinearity came afterwards, the derivative of f(\cdot) would have to be accounted for, complicating the calculations. The authors use these two properties to introduce realistic properties, such as fading memory and spiking activity, into the FEVER network.

With the network architecture set and an inference method for L, the authors then analyze the resulting properties of the connectivity L. Specifically, the authors make note of the eigenvalues and engenvectors L as well as the occurrence of different motifs (connectivity patters). The authors also make indirect observations by looking at the resulting time-traces resulting from the dynamics under the FEVER network.

In terms of the eigenvalues – arguably the most important aspect of a linear dynamical system – the first major observation is that L must have at least M eigenvalues at one. This necessitates from the fact that in the FEVER condition, L must preserve all the stimulus dimensions. As all other eigenvalues lie in the null-space of D, the authors do not prove stability directly, but instead demonstrate empirically that the FEVER networks do not result in large eigenvalues. It most likely helps that sparsity constraints attempt to reduce the overall magnitude of the network connections, reducing the chance that eigenvalues not necessary for maintaining the stimulus will be bias towards smaller values (more explicitly so in the low-rank segment of the Dale’s FEVER inference). As a side note, the authors do present, however, in the supplementary information a proof that inhibitory connections are required for stability.

For the eigenvectors, the authors mostly aim to show that there are mostly sparse connections between neurons (i.e. FEVER networks are not fully connected). Branching from this measure, the total number of single connections, pairs of connected neurons, and motifs of triads of connected neurons are compared to the similar counts in rat V1, layer V. The authors show that the above-chance occurrence of these connectivity patters match the above-chance occurrences in rat V1.

For the indirect relation to biological systems, the authors also show how the temporal evolution in time can match the behavior of pre-frontal cortex in working memory tasks. Specifically, a spiking FEVER network is driven to a given stimulus, and then allowed to evolve naturally. The authors observe that the PSTH of subsets of neurons match the qualitative behaviors observed in biological studies: ramp up, ramp down, and time-invariant.

The authors use this accumulation of evidence to as a steps to the conclusion that time-varying activity and persistent stimulus encoding are not mutually exclusive. All in all I believe the following quote from the paper summarizes the paper quite nicely:

“Our study does not refute the idea of explicit coding of time in working memory but rather shows that time-varying activity does not necessarily imply that the underlying network stimulus representations explicitly encodes time-varying properties”

 

 

 

 

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Fast approximate inference for directed graphical model: a Bayesian auto-encoder

In this week’s lab meting, I presented the following paper from Max Welling’s group:

Auto-Encoding Variational Bayes
Diederik P. Kingma, Max Welling
arXiv, 2013.

The paper proposed an efficient inference and learning method for directed probabilistic
models with continuous latent variables (with intractable posterior distributions), for use with large datasets. The directed graphical model under consideration is as follows,Screen Shot 2016-01-13 at 4.06.02 PM

The dataset is \mathbf{X}=\{\mathbf{x}^{(i)}\}_{i=1}^N consisting of N i.i.d. samples of some continuous or discrete variable \mathbf{x}. \mathbf{z} is an unobserved continuous random variable generating the data (solid lines: p_\mathbf{\theta}(\mathbf{z})p_\mathbf{\theta}(\mathbf{x}|\mathbf{z})), where \mathbf{\theta} is the parameter set involved in the generative model. The ultimate task is to learn both \mathbf{\theta} and \mathbf{z}. A general method to solve such a problem is to marginalize out \mathbf{z} to get the marginal likelihood p_\mathbf{\theta}(\mathbf{x})=\int p_\mathbf{\theta}(\mathbf{z})p_\mathbf{\theta}(\mathbf{x}|\mathbf{z})d\mathbf{z}, and maximize this likelihood  to learn \mathbf{\theta}. However, in many application cases, e.g. a neural network with a nonlinear hidden layer, the integral is intractable. In order to overcome this intractability, sampling-based methods, e.g. Monte Carlo EM, are introduced. But when the dataset is large, batch optimization is too costly and sampling loop per datapoint is very expensive. Therefore, the paper introduced a stochastic variational inference and learning algorithm that scales to large datasets and, under some mild differentiability conditions, even works in the intractable case.

First, they defined a recognition model q_\Phi(\mathbf{z}|\mathbf{x}): an approximation to the intractable true posterior p_\mathbf{\theta}(\mathbf{z}|\mathbf{x}), which is interpreted as a probabilistic encoder (dash line in the directed graph), and correspondingly, p_\mathbf{\theta}(\mathbf{x}|\mathbf{z}) is the probabilistic decoder. Given the recognition model, the variational lower bound \mathcal{L}(\mathbf{\theta},\Phi;\mathbf{x}^{(i)}) is defined as

\mbox{log}p_\mathbf{\theta}(\mathbf{x}^{(i)})\ge\mathcal{L}(\mathbf{\theta},\Phi;\mathbf{x}^{(i)})=-D_{KL}(q_\Phi(\mathbf{z}|\mathbf{x}^{(i)})||p_\mathbf{\theta}(\mathbf{z}))+\mathbb{E}_{q_\Phi(\mathbf{z}|\mathbf{x}^{(i)})}\left[\mbox{log}p_\mathbf{\theta}(\mathbf{x}^{(i)}|\mathbf{z})\right]

In the paper’s setting,

p_\mathbf{\theta}(\mathbf{z})=\mathcal{N}(\mathbf{0},\mathbf{I})q_\Phi(\mathbf{z}|\mathbf{x})=\mathcal{N}(\mathbf{\mu},\mathbf{\sigma}^2)

Therefore, D_{KL}(q_\Phi(\mathbf{z}|\mathbf{x}^{i})||p_\mathbf{\theta}(\mathbf{z})) has an analytical form. The major tricky term is the expectation which usually doesn’t have any closed solution. The usual Monte Carlo estimator for this type of problem exhibits very high variance and is not capable to take derivatives w.r.t. \Phi. Given such a problem, the paper proposed a reparameterization trick of the expectation term yields a lower bound estimator that can be straightforwardly optimized using standard stochastic gradient methods.

The key reparameterization trick constructs samples \mathbf{z}\sim q_\Phi(\mathbf{z}|\mathbf{x}) in two steps:

  1. \mathbf{\epsilon} \sim p(\mathbf{\epsilon}) (random seed independent of \Phi)
  2.  \mathbf{z}=g(\Phi,\mathbf{\epsilon},\mathbf{x}) (differentiable perturbation)

such that \mathbf{z}\sim q_\Phi(\mathbf{z}|\mathbf{x}) (the correct distribution). This yields an estimator which typically has less variance than the generic estimator:

\mathcal{L}(\mathbf{\theta},\Phi;\mathbf{x}^{(i)})=-D_{KL}(q_\Phi(\mathbf{z}|\mathbf{x}^{(i)})||p_\mathbf{\theta}(\mathbf{z}))+\frac{1}{L}\sum_{l=1}^L\mbox{log}p_\mathbf{\theta}(\mathbf{x}^{(i)}|\mathbf{z}^{(i,l)})

where \mathbf{z}^{(i,l)}=g(\Phi,\mathbf{\epsilon}^{(i,l)},\mathbf{x}^{(i)}) and \mathbf{\epsilon}^{(l)}\sim p(\mathbf{\epsilon})

A connection with auto-encoders becomes clear when looking at the objective function. The first term is the KL divergence of the approximate posterior from the prior acts as a regularizer, while the second term is a an expected negative reconstruction error.

In the experiment, they set p_\mathbf{\theta}(\mathbf{x}|\mathbf{z}) to be a Bernoulli or Gaussian MLP, depending on the type of data they are modeling. They presented the comparisons of their method to the wake-sleep algorithm and Monte Carlo EM on MNIST and Frey Face datasets.

Overall, I think their contributions are two-fold. First, the reparameterization of the variational lower bound yields a lower bound estimator that can be straightforwardly optimized using standard stochastic gradient methods. Second, they showed that for i.i.d. datasets with continuous latent variables per datapoint, posterior inference can be made especially efficient by fitting an approximate inference model (also called a recognition model) to the intractable posterior using the proposed lower bound estimator. The stochastic gradient method helps to parallelize the algorithm so as to improve the efficiency in largescale dataset.

 

 

Binary Neurons can have Short Temporal Memory

This (belated) post is about the paper:

Randomly connected networks have short temporal memory,
Wallace, Hamid, & Latham, Neural Computation  (2013),

which I presented a few weeks ago at the Pillow lab group meeting. This paper analyzes the abilities of randomly connected networks  of binary neurons to store memories of network inputs. Network memory is  valuable quantity to bound; long memory indicates that a network is more likely to be able to perform complex operations on streaming inputs. Thus, the ability to recall past inputs provides a proxy for being able to operate on those inputs.  The overall result seems to stand in contrast to much of the literature because it predicts a very short memory (on the order of the logarithm of the number of nodes). The authors mention that this difference in the result is due to their use of more densely connected networks. There seem to be additional differences, though, between this papers’s network construction and those analyzed in related work.

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Inferring synaptic plasticity rules from spike counts

In last week’s computational & theoretical neuroscience journal club I presented the following paper from Nicolas Brunel’s group:

Inferring learning rules from distributions of firing rates in cortical neurons.
Lim, McKee, Woloszyn, Amit, Freedman, Sheinberg, & Brunel.
Nature Neuroscience (2015).

The paper seeks to explain experience-dependent changes in IT cortical responses in terms of an underlying synaptic plasticity rule. Continue reading

Exact solutions to the nonlinear dynamics of learning in deep linear neural networks

This week in lab meeting we discussed:

Exact solutions to the nonlinear dynamics of learning in deep linear neural networks Andrew M. Saxe, James L. McClelland, Surya Ganguli. arxiv (2013).

This work aims to start analyzing a gnawing question in machine learning: How do deep neural networks actually work? Continue reading

Deep Exponential Families

In this week’s lab meeting, I presented:

Deep Exponential Families
Rajesh Ranganath, Linpeng Tang, Laurent Charlin and David Blei.
http://arxiv.org/abs/1411.2581

This paper describes a class of latent variable models inspired by deep neural net and hierarchical generative model, called Deep Exponential Families (DEFs). DEFs stack multiple layers of exponential families and connects them with certain link functions to capture the hierarchy of dependencies.

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