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Symmetry 2019, 11(2), 134; https://doi.org/10.3390/sym11020134

Power Spectrum and Diffusion of the Amari Neural Field

1
Dipartimento di Fisica e Astronomia “Galileo Galilei” and CNISM, Università di Padova, Via Marzolo 8, 35131 Padova, Italy
2
Istituto Nazionale di Ottica (INO) del Consiglio Nazionale delle Ricerche (CNR), Via Nello Carrara 1, I-50019 Sesto Fiorentino, Italy
Received: 29 December 2018 / Revised: 21 January 2019 / Accepted: 22 January 2019 / Published: 26 January 2019
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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. View Full-Text
Keywords: Neural field theory; Amari equation; power spectrum; reaction-diffusion Neural field theory; Amari equation; power spectrum; reaction-diffusion
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Salasnich, L. Power Spectrum and Diffusion of the Amari Neural Field. Symmetry 2019, 11, 134.

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