Qualitative properties of solutions for the noisy integrate and fire model in computational neuroscience
Qualitative properties of solutions for the noisy integrate and fire model in computational neuroscience
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DOI：
10.1088/0951-7715/28/9/3365
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发表时间：
2015-08
期刊：
影响因子：
1.7
通讯作者：
José Antonio Carrillo;B. Perthame;Delphine Salort;D. Smets
中科院分区：
文献类型：
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作者：
José Antonio Carrillo;B. Perthame;Delphine Salort;D. Smets
Noisy Integrate-and-Fire 方程是一个标准的非线性 Fokker-Planck 方程，用于描述同质神经网络的活动，其特征为连通性 b（每个神经元通过突触权重与所有其他神经元相连）； b > 0 描述了兴奋性网络，b 太高，解不可能永远存在。在本文中，我们展示了发射率的先验统一时间界限，以丢弃爆炸场景，并且，对于小连通性，我们证明了解决方案的长期行为的定性属性。这些方法一方面基于导致 L2 估计的相对熵和庞加莱不等式，另一方面基于“通用超解”的概念和抛物线正则化效应以获得 L∞ ?> 边界。
The Noisy Integrate-and-Fire equation is a standard non-linear Fokker–Planck equation used to describe the activity of a homogeneous neural network characterized by its connectivity b (each neuron connected to all others through synaptic weights); b > 0 describes excitatory networks and b is too high, solutions cannot exist for all times. In this paper, we show a priori uniform bounds in time on the firing rate to discard the scenario of blow-up, and, for small connectivity, we prove qualitative properties on the long time behavior of solutions. The methods are based on the one hand on relative entropy and Poincaré inequalities leading to L2 estimates and on the other hand, on the notion of ‘universal super-solution’ and parabolic regularizing effects to obtain L∞ ?> bounds.