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# Dynamics of a structured neuron population

Title | Dynamics of a structured neuron population |

Publication Type | Journal Article |

Year of Publication | 2010 |

Authors | Pakdaman, K, Perthame, B, Salort, D |

Journal | Nonlinearity |

Volume | 23 |

Pagination | 55 |

Abstract | We study the dynamics of assemblies of interacting neurons. For large fully connected networks, the dynamics of the system can be described by a partial differential equation reminiscent of age-structure models used in mathematical ecology, where the 'age' of a neuron represents the time elapsed since its last discharge. The nonlinearity arises from the connectivity J of the network. We prove some mathematical properties of the model that are directly related to qualitative properties. On the one hand, we prove that it is well-posed and that it admits stationary states which, depending upon the connectivity, can be unique or not. On the other hand, we study the long time behaviour of solutions; both for small and large J , we prove the relaxation to the steady state describing asynchronous firing of the neurons. In the middle range, numerical experiments show that periodic solutions appear expressing re-synchronization of the network and asynchronous firing. |

URL | http://stacks.iop.org/0951-7715/23/i=1/a=003 |