Learning in Wilson-Cowan model for metapopulation

TL;DR

The paper formulates a metapopulation Wilson–Cowan neural mass network with $\,\mathcal{N}$ nodes and inter-node coupling via $\mathbf{A}$, embedding $K$ stable attractors in the kernel of $\mathbf{A}$ to serve as class memories. By enforcing linear stability and training non-embedded eigenvectors, eigenvalues, and a global gain $\gamma$, the model learns to map inputs to planted attractors, functioning as a classifier. Extensive experiments on MNIST, Fashion-MNIST, CIFAR-10, TF-FLOWERS, and IMDB (with BERT) show high classification accuracy, with stability guiding the learning process and enabling an invertible forward–backward dynamic. While performance approaches but does not surpass current state-of-the-art deep learning models, the approach demonstrates a biologically inspired mechanism for learning and memory, with potential extensions toward more plausible learning rules and neurodynamic validation. The work provides a bridge between neural mass dynamics and supervised learning, highlighting the utility of planted attractors and spectral structure in brain-inspired computation.

Abstract

The Wilson-Cowan model for metapopulation, a Neural Mass Network Model, treats different subcortical regions of the brain as connected nodes, with connections representing various types of structural, functional, or effective neuronal connectivity between these regions. Each region comprises interacting populations of excitatory and inhibitory cells, consistent with the standard Wilson-Cowan model. By incorporating stable attractors into such a metapopulation model's dynamics, we transform it into a learning algorithm capable of achieving high image and text classification accuracy. We test it on MNIST and Fashion MNIST, in combination with convolutional neural networks, on CIFAR-10 and TF-FLOWERS, and, in combination with a transformer architecture (BERT), on IMDB, always showing high classification accuracy. These numerical evaluations illustrate that minimal modifications to the Wilson-Cowan model for metapopulation can reveal unique and previously unobserved dynamics.

Figures (10)

• Figure 1: Cartoon of the Neural Mass Network Model. This figure illustrates a cartoon of the Neural Mass Network Model. a) Nodes represent distinct brain regions. b) Within each region, or node of the network, there are interacting subpopulations of excitatory and inhibitory cells, in accord with the standard Wilson-Cowan model. c) Standard Wilson-Cowan model.
• Figure 2: Cartoon of the bistable profile. This figure illustrates a cartoon of the bistable profile of the standard Wilson-Cowan model we adopt in each node of the network.
• Figure 3: Stability before training. The figure shows the maximum value of the real part of the eigenvalues $s^{(\pm)}$ as a function of the parameter $\lambda_i$. The two curves, with different colors, identify respectively the two different steady states. The parameters here employed read $\omega_{II}=1$, $\omega_{IE}=0.0$, $\omega_{EI}=2$, $\omega_{EE}=7.2$$\alpha_E=1.5$, $\alpha_I=0.4$, $h_E=-1.2$, $h_I=0.1$, $\gamma=0.25$, $\beta_E=3.7$, $f^{(1)}_{E}=0.25$, $f^{(2)}_{E}=0.65$, $\beta_I=1$, $f^{(1)}_{I}=0.5$, $f^{(2)}_{I}=0.5$, $\mathcal{N}=784$.
• Figure 4: Building the model. The figure illustrates the composition of the dataset and the method for introducing stable attractors into the dynamics. Panel a) displays the labels, images, and targets used in our model. The first row shows the labels of the MNIST image ($\mathcal{N}=784$) examples found in the second row. The third row presents the ten target images we created to serve as attractors for the dynamics. Panel b) shows the creation of an eigenvector of matrix $\mathbf{A}$. The image is composed solely of the excitatory stationary parts of the two fixed points in the Wilson-Cowan model. Each black pixel corresponds to a $\overline{x}^{(1)}$, while each white pixel corresponds to a $\overline{x}^{(2)}$. In general, to create $K$ target vectors for a dataset, we generate $K$ vectors of size $\mathcal{N}$, where $\frac{\mathcal{N}}{K+2}$ components are set to $\overline{x}^{(1)}$ and the remaining components to $\overline{x}^{(2)}$. Each unique target vector is constructed by setting the components from $k\frac{\mathcal{N}}{K+2}$ to $((k+1)\frac{\mathcal{N}}{K+2})-1$ to $\overline{x}^{(1)}$, with all other components set to $\overline{x}^{(2)}$, with $k=0,\dots,K-1$ that identifies the label of a single class. Panel c) illustrates the positions of the fixed eigenvectors and eigenvalues (in red) and the trainable ones (in green). Recall that the matrix $\mathbf{A}$ is diagonalizable by definition.
• Figure 5: Dynamics of the model. The figure illustrates a schematic representation of our Neural Mass Network Model. Panel a) displays an image from the MNIST ($\mathcal{N}=784$) dataset along with its associated target. The image features a black pixel with a red border, which we track throughout its dynamics. Panel b) shows this dynamic process. The equations are highlighted in red to emphasize that we are focusing on that single pixel. This pixel is depicted as a red sphere moving within a double well potential. Our model, we recall, is configured to exhibit a bistable profile. Initially, the vector $\vec{\zeta}_i$ is set to represent the black pixel for both subpopulations of neurons. As the dynamics evolve according to our model's equations, the pixel explores the bistable profile. It does not stop at the closest minimum; instead, it reaches the correct minimum designated for classification, thanks to the learned coupling in the matrix $\mathbf{A}$. Panel c) illustrates the overall dynamics of the entire image. Starting from an initial condition—the image we want to classify—the system uses the equations in \ref{['eq_stoc_vec']} to achieve the final state, which is our stable attractor. The precision cutoff for this analysis has been set to the seventh decimal digit.