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Abstract <p> We consider a model of a circular network of neurons where the functioning of each neuron is described by an equation with two delays. The model under study is a modification considered in the paper of Glyzin et al., where the model of a solitary neuron is based on of the equation with one delay—the Hutchinson equation. We construct discrete traveling waves, i.e., a periodic solution of the system such that all components coincide with the same function shifted by a quantity that is multiple of a certain parameter. To find this solution, we study an auxiliary differential-difference equation of the Volterra type with three delays. For this equation, for any natural <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2677_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(m\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2677_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(n\)</EquationSource> </InlineEquation>, we establish the existence of a periodic solution that contains <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2677_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(m\)</EquationSource> </InlineEquation> packets, each of which contains <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2677_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(n\)</EquationSource> </InlineEquation> bursts per period. </p>

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Cycles with the embedded bursting effect in a circle of neural oscillators

  • 期刊论文

  • 发表日期：

    2025年8月23日

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  • I. D. Voronov,
  • M. M. Preobrazhenskaia,
  • I. V. Teplyashin

摘要

Abstract

We consider a model of a circular network of neurons where the functioning of each neuron is described by an equation with two delays. The model under study is a modification considered in the paper of Glyzin et al., where the model of a solitary neuron is based on of the equation with one delay—the Hutchinson equation. We construct discrete traveling waves, i.e., a periodic solution of the system such that all components coincide with the same function shifted by a quantity that is multiple of a certain parameter. To find this solution, we study an auxiliary differential-difference equation of the Volterra type with three delays. For this equation, for any natural \(m\) and \(n\) , we establish the existence of a periodic solution that contains \(m\) packets, each of which contains \(n\) bursts per period.