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Russian Universities Reports. Mathematics, 2021, Volume 26, Issue 135, Pages 271–295
DOI: https://doi.org/10.20310/2686-9667-2021-26-135-271-295
(Mi vtamu231)
 

This article is cited in 1 scientific paper (total in 1 paper)

Scientific articles

Existence and stability of periodic solutions in a neural field equation

K. Kolodinaa, V. V. Kostrykinb, A. Oleynikc

a Norwegian University of Life Sciences
b Johannes Gutenberg University of Mainz
c University of Bergen
References:
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.
Keywords: nonlinear integral equations, sigmoid type nonlinearities, neural field model, periodic solutions, block Laurent operators.
Funding agency Grant number
Research Council of Norway 239070
This research work was supported by the Norwegian University of Life Sciences and The Research Council of Norway (projects no. 239070).
Received: 23.06.2021
Document Type: Article
UDC: 517.98
Language: English
Citation: K. Kolodina, V. V. Kostrykin, A. Oleynik, “Existence and stability of periodic solutions in a neural field equation”, Russian Universities Reports. Mathematics, 26:135 (2021), 271–295
Citation in format AMSBIB
\Bibitem{KolKosOle21}
\by K.~Kolodina, V.~V.~Kostrykin, A.~Oleynik
\paper Existence and stability of periodic solutions in a neural field equation
\jour Russian Universities Reports. Mathematics
\yr 2021
\vol 26
\issue 135
\pages 271--295
\mathnet{http://mi.mathnet.ru/vtamu231}
\crossref{https://doi.org/10.20310/2686-9667-2021-26-135-271-295}
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  • https://www.mathnet.ru/eng/vtamu/v26/i135/p271
  • This publication is cited in the following 1 articles:
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    Russian Universities Reports. Mathematics
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