Blow-up, steady states and long time behaviour of excitatory-inhibitory nonlinear neuron models

María J. Cáceres, Ricarda Schneider

    Research output: Contribution to journalArticleResearchpeer-review

    8 Citations (Scopus)

    Abstract

    © American Institute of Mathematical Sciences. Excitatory and inhibitory nonlinear noisy leaky integrate and fire models are often used to describe neural networks. Recently, new mathematical results have provided a better understanding of them. It has been proved that a fully excitatory network can blow-up in finite time, while a fully inhibitory network has a global in time solution for any initial data. A general description of the steady states of a purely excitatory or inhibitory network has been also given. We extend this study to the system composed of an excitatory population and an inhibitory one. We prove that this system can also blow-up infinite time and analyse its steady states and long time behaviour. Besides, we illustrate our analytical description with some numerical results. The main tools used to reach our aims are: the control of an exponential moment for the blow-up results, a more complicate strategy than that considered in [5] for studying the number of steady states, entropy methods combined with Poincaré inequalities for the long time behaviour and, finally, high order numerical schemes together with parallel computation techniques in order to obtain our numerical results.
    Original languageEnglish
    Pages (from-to)587-612
    JournalKinetic and Related Models
    Volume10
    Issue number3
    DOIs
    Publication statusPublished - 1 Jan 2017

    Keywords

    • Blow-up
    • Entropy
    • Leaky integrate and fire models
    • Long time behaviour
    • Neural networks
    • Noise
    • Steady states

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