Large Deviations for Nonlocal Stochastic Neural Fields

Abstract

We study the effect of additive noise on integro-differential neural field equations. In particular, we analyze an Amari-type model driven by a Q-Wiener process, and focus on noise-induced transitions and escape. We argue that proving a sharp Kramers’ law for neural fields poses substantial difficulties, but that one may transfer techniques from stochastic partial differential equations to establish a large deviation principle (LDP). Then we demonstrate that an efficient finite-dimensional approximation of the stochastic neural field equation can be achieved using a Galerkin method and that the resulting finite-dimensional rate function for the LDP can have a multiscale structure in certain cases. These results form the starting point for an efficient practical computation of the LDP. Our approach also provides the technical basis for further rigorous study of noise-induced transitions in neural fields based on Galerkin approximations.Mathematics Subject Classification (2000): 60F10, 60H15, 65M60, 92C20.

High-Frequency Resonance in the Gerbil Medial Superior Olive

“…The nonlinearity of voltage responses to large amplitude stimuli biases the calculation of sine- by power spectrum analysis towards higher frequencies. We therefore used maximum difference between local voltage peaks and the adjacent minima, or Max-min, divided by the peak-to-peak sinusoidal current as a measurement at each stimulus frequency instead for large amplitude stimuli [28]. This was used exclusively when determining the effect of different stimulus amplitudes on subthreshold resonant properties.…”

“…I Na and I KHT were removed altogether as their contribution was negligible when frozen), to study how activation of I KLT influenced the voltage responses at different sinusoidal stimulus amplitudes. This has previously been implemented successfully by Rotstein [28] to explain subthreshold resonance in linear and quadratic models. We plotted V - w 4 trajectories that represent the final cycle of a response to a sinusoidal stimulus at a particular frequency and stimulus amplitude.…”

A gradient flow formulation for the stochastic Amari neural field model

“…The basic theory for (1.1), including existence, regularity, Galerkin approximations and large deviations of (1.1), has been covered in the work by Kuehn and Riedler [44] for bounded domains, while unbounded domains are considered by Faugeras and Inglis [28]. The influence of noise on traveling waves in stochastic neural fields has been studied intensively in recent years [15,37,40,42,43,47,57].…”

“…where X is a suitable function space, e.g., a Hilbert, Banach or even metric space [3], and F : X → R is a functional, which often has the natural interpretation of an energy, entropy, or some other physical notion. Of course, to have such a gradient structure in the (stochastic) neural field case would not only be interesting for Kramers' law as suggested by Kuehn and Riedler [44] but also open up a general area of techniques, which has been extremely successful for other differential equations [38,56]. Furthermore, if we can characterize the types of kernels for which gradient structures exist, it would give us an understanding, if and when the brain might be working in two different regimes such as energy-decay versus complex non-equilibrium pattern formation.…”

Macroscopic coherent structures in a stochastic neural network: from interface dynamics to coarse-grained bifurcation analysis

“…Recent studies have analysed neural fields with additive noise [38,28,46], multiplicative noise [15], or noisy firing thresholds [7], albeit these models are still mostly phenomenological. Even though several papers derive continuum neural fields from firing rate functions with Heaviside distributions [11,12].…”