Power spectrum and diffusion of the Amari's neural field

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 elegant analytical formula for the power spectrum of the neural field. In the case of an istantaneous and localized external stimulus, we show that for large frequencies $\omega$ the power spectrum scales as $1/\omega^2$, in agreement with experimental data of electroencephalography. 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. For some initial conditions, we discuss analytical solutions of this reaction-diffusion equation.