Path-integral methods for analyzing the effects of fluctuations in stochastic hybrid neural networks

Abstract

We consider applications of path-integral methods to the analysis of a stochastic hybrid model representing a network of synaptically coupled spiking neuronal populations. The state of each local population is described in terms of two stochastic variables, a continuous synaptic variable and a discrete activity variable. The synaptic variables evolve according to piecewise-deterministic dynamics describing, at the population level, synapses driven by spiking activity. The dynamical equations for the synaptic currents are only valid between jumps in spiking activity, and the latter are described by a jump Markov process whose transition rates depend on the synaptic variables. We assume a separation of time scales between fast spiking dynamics with time constant [Formula: see text] and slower synaptic dynamics with time constant τ. This naturally introduces a small positive parameter [Formula: see text], which can be used to develop various asymptotic expansions of the corresponding path-integral representation of the stochastic dynamics. First, we derive a variational principle for maximum-likelihood paths of escape from a metastable state (large deviations in the small noise limit [Formula: see text]). We then show how the path integral provides an efficient method for obtaining a diffusion approximation of the hybrid system for small ϵ. The resulting Langevin equation can be used to analyze the effects of fluctuations within the basin of attraction of a metastable state, that is, ignoring the effects of large deviations. We illustrate this by using the Langevin approximation to analyze the effects of intrinsic noise on pattern formation in a spatially structured hybrid network. In particular, we show how noise enlarges the parameter regime over which patterns occur, in an analogous fashion to PDEs. Finally, we carry out a [Formula: see text]-loop expansion of the path integral, and use this to derive corrections to voltage-based mean-field equations, analogous to the modified activity-based equations generated from a neural master equation.

Keywords: Large deviations; Path-integrals; Stochastic hybrid systems; Stochastic neural networks.

Fig. 1

a Deterministic trajectories of a multistable dynamical system. The subset Ω is contained within the basin of attraction of a fixed point ${\mathbf{u}}_s$. The boundary of the basin of attraction consists of separatrices, which are also solution trajectories. Paths that start in a neighborhood of a separatrix are attracted by different fixed points, depending whether they begin on the left or right of the separatrix. b Random trajectories of the stochastic system. Escape from the domain Ω occurs when a random trajectory hits the boundary ∂Ω

Fig. 2

Stochastic pattern formation in a scalar neural network. a Plot of Fourier transformed weight distribution as a function of wavenumber k for various values of gain $\mu =f^{\prime }(u_0)$: $\mu =0.12,0.15,0.18,0.2,0.22$. b Sketch of corresponding power spectra $S(k,0)$, showing the peak in the spectrum at the critical wavenumber $k_c$ for $\mu <{\mu }_c$. Parameter values are ${\sigma }^2=4$, $A=0.5$, $F_0=20$, $\eta =2$, $\kappa =0$