Abstract
The authors present a theory of balanced neural networks, which identifies a type of network state with nonlinear response properties.
Highlights
Networks composed of a large number of interacting units are ubiquitous in physical, biological, financial, and ecological systems [1,2,3,4]
We mathematically demonstrate how these biologically realistic neural networks lead to membrane potentials undergoing fractional, Lévy diffusion instead of the normal, Brownian diffusion arising in homogeneous networks
We further demonstrate that in this fractional state, the neural response is maximized as a function of structural connectivity, indicating that the heterogeneity of neural connectivity has an impact on neural dynamics
Summary
Networks composed of a large number of interacting units are ubiquitous in physical, biological, financial, and ecological systems [1,2,3,4]. It has been empirically found that neural firing activity has super-Poisson dynamics [7], with great heterogeneity and fluctuations occurring at multiple scales [24,25,26], and that membrane potentials fluctuate far from the firing threshold [27] Some of these properties are not necessarily incompatible with Gaussian input statistics: non-Gaussian, lognormal distributions of synaptic strengths lead to the classical, normal diffusion theory in the large network limit. Based on balanced spiking neural networks with heterogeneous connection strengths, our analysis reveals that synaptic inputs in such heterogeneous networks possess heavy-tailed, Lévy fluctuations, a type of fluctuation typical for nonequilibrium systems with many interacting units [29,30,31] This leads to a relation between complex neural dynamics and fractional diffusion formalisms developed for studying nonequilibrium physical systems [32,33,34,35,36,37]; correspondingly, we refer to this as the fractional diffusion theory of balanced neural networks. Our fractional diffusion framework provides a unified account of a variety of the key features of neural dynamics, but can be applied to understand how complex dynamics emerge from other nonequilibrium systems with large numbers of interacting units
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