Fractional diffusion theory of balanced heterogeneous neural networks

Abstract

Interactions of large numbers of spiking neurons give rise to complex neural dynamics with fluctuations occurring at multiple scales. Understanding the dynamical mechanisms underlying such complex neural dynamics is a long-standing topic of interest in neuroscience, statistical physics and nonlinear dynamics. Conventionally, fluctuating neural dynamics are formulated as balanced, uncorrelated excitatory and inhibitory inputs with Gaussian properties. Yet heterogeneous, non-Gaussian properties have been widely observed in both neural connections and neural dynamics. Based on balanced neural networks with heterogeneous, non-Gaussian features, our analysis reveals that synaptic inputs possess power-law fluctuations in the limit of large network size, leading to a remarkable relation between complex neural dynamics and the fractional diffusion formalisms of nonequilibrium physical systems. We derive a fractional Fokker-Planck equation with analytically tractable boundary conditions for the network, uniquely accounting for the leapovers caused by the fluctuations of spiking activity. This body of formalisms represents a fractional diffusion theory of heterogeneous neural networks and results in an exact description of the network activity states. In particular, the fractional diffusion theory identifies a dynamical state at which the neural response is maximized as a function of structural connectivity. This state is then implemented in a balanced spiking neural network and displays rich, nonlinear response properties, providing a unified account of a variety of experimental findings on neural dynamics at the individual neuron and network levels, including fluctuations of membrane potentials and population firing rates. Our theory and its network implementations provide a framework for investigating complex neural dynamics emerging from large networks of spiking neurons and their functional roles in neural processing.