{"title":"Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation.","authors":"Khashayar Pakdaman, Benoît Perthame, Delphine Salort","doi":"10.1186/2190-8567-4-14","DOIUrl":null,"url":null,"abstract":"<p><p>Motivated by a model for neural networks with adaptation and fatigue, we study a conservative fragmentation equation that describes the density probability of neurons with an elapsed time s after its last discharge. In the linear setting, we extend an argument by Laurençot and Perthame to prove exponential decay to the steady state. This extension allows us to handle coefficients that have a large variation rather than constant coefficients. In another extension of the argument, we treat a weakly nonlinear case and prove total desynchronization in the network. For greater nonlinearities, we present a numerical study of the impact of the fragmentation term on the appearance of synchronization of neurons in the network using two \"extreme\" cases. Mathematics Subject Classification (2000)2010: 35B40, 35F20, 35R09, 92B20. </p>","PeriodicalId":54271,"journal":{"name":"Journal of Mathematical Neuroscience","volume":"4 ","pages":"14"},"PeriodicalIF":2.3000,"publicationDate":"2014-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1186/2190-8567-4-14","citationCount":"53","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Neuroscience","FirstCategoryId":"3","ListUrlMain":"https://doi.org/10.1186/2190-8567-4-14","RegionNum":4,"RegionCategory":"医学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2014/1/1 0:00:00","PubModel":"eCollection","JCR":"Q1","JCRName":"Neuroscience","Score":null,"Total":0}
引用次数: 53
Abstract
Motivated by a model for neural networks with adaptation and fatigue, we study a conservative fragmentation equation that describes the density probability of neurons with an elapsed time s after its last discharge. In the linear setting, we extend an argument by Laurençot and Perthame to prove exponential decay to the steady state. This extension allows us to handle coefficients that have a large variation rather than constant coefficients. In another extension of the argument, we treat a weakly nonlinear case and prove total desynchronization in the network. For greater nonlinearities, we present a numerical study of the impact of the fragmentation term on the appearance of synchronization of neurons in the network using two "extreme" cases. Mathematics Subject Classification (2000)2010: 35B40, 35F20, 35R09, 92B20.
期刊介绍：
The Journal of Mathematical Neuroscience (JMN) publishes research articles on the mathematical modeling and analysis of all areas of neuroscience, i.e., the study of the nervous system and its dysfunctions. The focus is on using mathematics as the primary tool for elucidating the fundamental mechanisms responsible for experimentally observed behaviours in neuroscience at all relevant scales, from the molecular world to that of cognition. The aim is to publish work that uses advanced mathematical techniques to illuminate these questions.
It publishes full length original papers, rapid communications and review articles. Papers that combine theoretical results supported by convincing numerical experiments are especially encouraged.
Papers that introduce and help develop those new pieces of mathematical theory which are likely to be relevant to future studies of the nervous system in general and the human brain in particular are also welcome.
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