Statistical Physics and Representations in Real and Artificial Neural Networks

TL;DR

This paper explores neural representations in biological and artificial systems, analyzing models of spatial maps in the hippocampus and data representation algorithms like PCA and RBMs using statistical physics tools.

Contribution

It extends the Hopfield model to store multiple spatial maps and applies statistical physics to analyze PCA and RBMs in machine learning.

Findings

Extended Hopfield model for multiple spatial maps

Decoding spatial representations with an effective Ising model

Statistical physics analysis of PCA and RBMs

Abstract

This document presents the material of two lectures on statistical physics and neural representations, delivered by one of us (R.M.) at the Fundamental Problems in Statistical Physics XIV summer school in July 2017. In a first part, we consider the neural representations of space (maps) in the hippocampus. We introduce an extension of the Hopfield model, able to store multiple spatial maps as continuous, finite-dimensional attractors. The phase diagram and dynamical properties of the model are analyzed. We then show how spatial representations can be dynamically decoded using an effective Ising model capturing the correlation structure in the neural data, and compare applications to data obtained from hippocampal multi-electrode recordings and by (sub)sampling our attractor model. In a second part, we focus on the problem of learning data representations in machine learning, in particular with artificial neural networks. We start by introducing data representations through some illustrations. We then analyze two important algorithms, Principal Component Analysis and Restricted Boltzmann Machines, with tools from statistical physics.