mathematicsВестник российских университетов. МатематикаRussian Universities Reports. Mathematics2686-96672782-3342Тамбовский государственный университет имени Г.Р. Державина10.20310/2686-9667-2022-27-140-318-327mathematics-158Research ArticleНАУЧНЫЕ СТАТЬИORIGINAL ARTICLESО существовании и устойчивости решений типа «кольцо» уравнений нейронного поля Амари с периодической микроструктурой и функцией активации ХевисайдаOn existence and stability of ring solutions to Amari neural field equation with periodic microstructure and Heaviside activation functionhttps://orcid.org/0000-0002-2194-1497АтманияРашидAtmaniaRachidatmania.rachid@gmail.comhttps://orcid.org/0000-0002-7286-9456БурлаковЕвгений ОлеговичBurlakovEvgenii O.eb_@bk.ruhttps://orcid.org/0000-0001-5845-5591МальковИван НиколаевичMalkovIvan N.i.n.malkov@yandex.ruФГАОУ ВО «Тюменский государственный университет»РоссияTyumen State UniversityRussian FederationФГАОУ ВО «Тюменский государственный университет»; ФГБОУ ВО «Тамбовский государственный университет им. Г.Р. Державина»РоссияTyumen State University; Derzhavin Tambov State UniversityRussian Federation20222612202227140318327Copyright © Атмания Р., Бурлаков Е.О., Мальков И.Н., 20222022Атмания Р., Бурлаков Е.О., Мальков И.Н.Atmania R., Burlakov E.O., Malkov I.N.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://mathematics.elpub.ru/jour/article/view/158

В работе изучены существование и устойчивость решений типа «кольцо» двумерного уравнения нейронного поля Амари с периодической микроструктурой и функцией активации типа Хевисайда. Получены результаты, отражающие зависимость внутреннего и внешнего радиусов колец от порога активации нейронной среды и степени ее неоднородности. Сформулированы необходимое условие существования и достаточное условие отсутствия радиально распространяющихся из эпицентра бегущих волн, как в однородной нейронной среде, так и при слабо выраженной микроструктуре нейронной среды. Результаты исследования проиллюстрированы примером, основанном на выборе одной из типично используемых в математической нейробиологии функций межнейронной связи.

In the present research, existence and stability of ring solutions to two-dimensional Amari neural field equation with periodic microstructure and Heaviside activation function are studied. Results on dependence of the inner and the outer radii of the ring solutions are obtained. Necessary conditions for existence and sufficient conditions for non-existence of radial travelling waves are formulated for homogeneous neural medium and neural media with mild periodic microstructure. Theoretical results obtained are illustrated with a concrete example based on a connectivity function commonly used in the neuroscience community.

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The authors declare that there are no conflicts of interest present.