Existence and stability of periodic solutions in a neural field equation

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Abstract

We study the existence and linear stability of stationary periodic solutions to a neural field model, an intergo-differential equation of the Hammerstein type. Under the assumption that the activation function is a discontinuous step function and the kernel is decaying sufficiently fast, we formulate necessary and sufficient conditions for the existence of a special class of solutions that we call 1-bump periodic solutions. We then analyze the stability of these solutions by studying the spectrum of the Frechet derivative of the corresponding Hammerstein operator. We prove that the spectrum of this operator agrees up to zero with the spectrum of a block Laurent operator. We show that the non-zero spectrum consists of only eigenvalues and obtain an analytical expression for the eigenvalues and the eigenfunctions. The results are illustrated by multiple examples.

About the authors

Karina Kolodina

Norwegian University of Life Sciences

Email: karina.a.kolodina@gmail.com
PhD in Applied Mathematics P.O. Box 5003, №-1432 ˚As 5003, Norway

Vadim Kostrykin

Johannes Gutenberg University of Mainz

Doctor of Mathematics, Professor Staudingerweg 9, 55099 Mainz, Germany

Anna Oleynik

University of Bergen

Email: anna.oleynik@uib.no
Doctor of Applied Mathematics P.O. Box 7803, №-5020 Bergen 7803, Norway

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