6533b82afe1ef96bd128b814

RESEARCH PRODUCT

Existence and stability of periodic solutions in a neural field equation

Karina KolodinaAnna OleynikVadim Kostrykin

subject

Operator (physics)Mathematical analysisSpectrum (functional analysis)Fréchet derivativeGeneral MedicineEigenfunctionFunctional Analysis (math.FA)Mathematics - Functional AnalysisMathematics - Analysis of PDEsKernel (statistics)Step functionFOS: MathematicsEigenvalues and eigenvectorsAnalysis of PDEs (math.AP)Linear stabilityMathematics

description

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.

yearjournalcountryeditionlanguage
2017-12-27
http://arxiv.org/abs/1712.09688