A new numerical study of space–time fractional advection–reaction–diffusion equation with Rabotnov fractional‐exponential kernel

2020 ◽  
Author(s):  
Sachin Kumar ◽  
Bashir Ahmad
Keyword(s):  
Diffusion Equation ◽  
Numerical Study ◽  
Reaction Diffusion ◽  
Space Time ◽  
Reaction Diffusion Equation ◽  
Exponential Kernel

scholarly journals Power Spectrum and Diffusion of the Amari Neural Field

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 134
Author(s):  
Luca Salasnich
Keyword(s):  
Power Spectrum ◽  
Diffusion Equation ◽  
Analytical Solutions ◽  
Initial Conditions ◽  
Analytical Formula ◽  
Reaction Diffusion ◽  
Space Time ◽  
Neural Field ◽  
Time Dependent ◽  
Reaction Diffusion Equation

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.

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