Balanced Dynamics in Strongly Coupled Networks

TL;DR

This paper explores the mathematical conditions under which strongly coupled networks of interacting agents, such as neuronal systems, can maintain a balanced state of excitation and inhibition, challenging traditional scaling assumptions.

Contribution

It formulates a general mathematical conjecture for balanced behaviors in stochastic interacting systems and provides a complete proof in a specific neuronal network model.

Findings

Numerical and theoretical exploration of the conjecture in neuronal models

Complete proof of asymptotic behavior in a one-dimensional chemically-coupled neuron model

Application of desingularization techniques from PDEs to mean-field limits

Abstract

Many mathematical models of interacting agents assume that individual interactions scale down in proportion to the network size, ensuring that the combined input received from the network does not diverge. In theoretical neuroscience, Sompolinsky and Van Vreeswijk proposed in 1996 that, should these scalings be violated (and under appropriate conditions), the system may not diverge but rather approach a balanced state where the inputs to each neuron compensate each other (in neuroscience, where inhibitory currents compensate the excitatory ones). We come back to this observation and formulate here a mathematical conjecture for the occurrence of such behaviors in general stochastic systems of interacting agents. From a mathematical viewpoint, this conjecture can be viewed as a double-limit problem in the space of probability measures, which we discuss in detail, as it provides several possible mathematical avenues for proving this result. We provide some numerical and theoretical explorations of the conjecture in classical models of neuronal networks. Moreover, we provide a complete proof of an asymptotic result consistent with one of the double-limit problems in a one-dimensional model with separable coupling inspired by models of chemically-coupled neurons. This proof relies on asymptotic methods, and particularly desingularization techniques used in some PDEs, that we apply here to the mean-field limit of the network as the coupling is made to diverge. From the applications viewpoint, this theory provides an alternative, minimalistic explanation for the widely observed balance of excitation and inhibition in the cerebral cortex not requiring the assumption of the existence of complex regulatory mechanisms.