Power spectrum and diffusion of the Amari neural field

TL;DR

This paper analyzes the power spectrum and diffusion properties of the Amari neural field model, deriving analytical formulas and exploring the effects of nonlinearity and reaction-diffusion dynamics.

Contribution

It provides an analytical derivation of the power spectrum for the Amari neural field and links linearized dynamics to diffusion and reaction-diffusion equations.

Findings

Derived an explicit formula for the neural field's power spectrum.

Established the equivalence of the linearized Amari equation to a diffusion equation at large wavelengths.

Showed that weak nonlinearity leads to a reaction-diffusion equation with analytical solutions.

Abstract

We study the power spectrum of a space-time dependent neural field which describes the average membrane potential of neurons in a single layer. This neural field is modelled by a dissipative integro-differential equation, the so-called Amari equation. By considering a small perturbation with respect to a stationary and uniform configuration of the neural field we derive a linearized equation which is solved for a generic external stimulus by using the Fourier transform into wavevector-freqency domain, finding an analytical formula for the power spectrum of the neural field. In addition, after proving that for large wavelengths the linearized Amari equation is equivalent to a diffusion equation which admits space-time dependent analytical solutions, we take into account the nonlinearity of the Amari equation. We find that for large wavelengths a weak nonlinearity in the Amari equation gives rise to a reaction-diffusion equation which can be formally derived from a neural action functional by introducing a dual neural field. For some initial conditions, we discuss analytical solutions of this reaction-diffusion equation.