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 integrodifferential 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. We find a simple analytical formula for the power spectrum of the neural field in the case of a istantaneous and localized external stimulus. In the regime of small wavenumber and large frequency we show that the power spectrum scales as 1/ω, in good agreement with several data of electroencephalography and functional magnetic resonance imaging of brain neocortex. Finally, we prove that for large wavelengths the linearized Amari equation is equivalent to a diffusion equation, which can be formally derived from a neural action functional by introducing a dual neural field.