The Complexity of Dynamics in Small Neural Circuits.

Diego Fasoli,Anna Cattani,Stefano Panzeri,Bard Ermentrout

doi:10.1371/journal.pcbi.1004992

Abstract

Mean-field approximations are a powerful tool for studying large neural networks. However, they do not describe well the behavior of networks composed of a small number of neurons. In this case, major differences between the mean-field approximation and the real behavior of the network can arise. Yet, many interesting problems in neuroscience involve the study of mesoscopic networks composed of a few tens of neurons. Nonetheless, mathematical methods that correctly describe networks of small size are still rare, and this prevents us to make progress in understanding neural dynamics at these intermediate scales. Here we develop a novel systematic analysis of the dynamics of arbitrarily small networks composed of homogeneous populations of excitatory and inhibitory firing-rate neurons. We study the local bifurcations of their neural activity with an approach that is largely analytically tractable, and we numerically determine the global bifurcations. We find that for strong inhibition these networks give rise to very complex dynamics, caused by the formation of multiple branching solutions of the neural dynamics equations that emerge through spontaneous symmetry-breaking. This qualitative change of the neural dynamics is a finite-size effect of the network, that reveals qualitative and previously unexplored differences between mesoscopic cortical circuits and their mean-field approximation. The most important consequence of spontaneous symmetry-breaking is the ability of mesoscopic networks to regulate their degree of functional heterogeneity, which is thought to help reducing the detrimental effect of noise correlations on cortical information processing.

Highlights

The brain is a complex system organized at multiple spatial scales, and the concerted interactions between these multiple scales of organization are probably crucial for the emergence of its computational power [24]
We proved the emergence of complex dynamics in small neural circuits, characterized by strong finite-size effects, which cannot be accounted for by the mean-field approximation and that are much less likely to occur in large fully-connected neural systems
Through a detailed numerical and analytical study of the bifurcations, that small homogeneous neural networks undergo spontaneous symmetry-breaking through branching-point bifurcations, which leads to the formation of strongly heterogeneous activity in the inhibitory population

Summary

Introduction

The brain is a complex system organized at multiple spatial scales, and the concerted interactions between these multiple scales of organization are probably crucial for the emergence of its computational power [24]. There is an intermediate, mesoscopic level of organization between the macroscopic and microscopic one: neurons are organized in microcircuits, whose size can vary from several thousands of cells as in cortical columns, to a few tens of cells as in micro-columns [45,46,47] This mesoscopic level of investigation has received considerable attention in recent years both from the theoretical [25,26,27,28,29] and experimental [21, 22] point of view, and it is often seen as a middle ground that is fundamental to link single neuron activity to behavior [23]. For this reason, finding an appropriate mathematical description of the brain at the mesoscopic scale is of fundamental importance for unveiling its emergent properties

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