Fractional diffusion theory of balanced heterogeneous neural networks

Pulin Gong

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.