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.

中文翻译：

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非线性破碎方程中神经元网络和大时间渐近的适应和疲劳模型。

受具有适应和疲劳的神经网络模型的启发，我们研究了一个保守的碎片方程，该方程描述了神经元在最后一次放电后经过时间 s 的密度概率。在线性设置中，我们扩展了 Laurençot 和 Perthame 的论点，以证明指数衰减到稳态。这种扩展允许我们处理具有较大变化的系数而不是常数系数。在论证的另一个扩展中，我们处理弱非线性情况并证明网络中的完全不同步。对于更大的非线性，我们使用两个“极端”情况对碎片项对网络中神经元同步外观的影响进行了数值研究。数学学科分类（2000）2010：35B40、35F20、35R09、92B20。