A constructive mean-field analysis of multi-population neural networks with random synaptic weights and stochastic inputs.
A constructive mean-field analysis of multi-population neural networks with random synaptic weights and stochastic inputs.
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DOI：
10.3389/neuro.10.001.2009
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发表时间：
2009
影响因子：
3.2
通讯作者：
Cessac B
中科院分区：
文献类型：
--
作者：
Faugeras O;Touboul J;Cessac B
我们处理的问题，桥接两个尺度之间的差距在神经元建模。在第一个（微观）尺度上，神经元被单独考虑，它们的行为由随机微分方程描述，该方程控制它们的膜电位的时间变化。它们通过突触连接耦合，突触连接作用于它们产生的活动，这是它们的膜电位的非线性函数。在第二（介观）尺度上，相互作用的神经元群体由类似的方程单独描述。描述动力学和稳态平均场行为的方程被认为是一组随机过程的函数方程。使用这个新的观点，使我们能够证明，这些方程是适定的任何有限的时间间隔，并提供了一个建设性的方法，有效地计算其唯一的解决方案。证明了该方法收敛于唯一解，并给出了其复杂性和收敛速度的特征。我们还提供了部分结果的平稳问题的无限时间间隔。这些结果为詹森和里特的神经质量模型提供了一些新的线索：它们的动力学似乎是我们分析中出现的更丰富动力学的粗略近似。我们的数值实验证实，我们提出的框架和数值方法，我们从它提供了一个新的和强大的工具，在不同尺度上的神经行为的探索。
We deal with the problem of bridging the gap between two scales in neuronal modeling. At the first (microscopic) scale, neurons are considered individually and their behavior described by stochastic differential equations that govern the time variations of their membrane potentials. They are coupled by synaptic connections acting on their resulting activity, a nonlinear function of their membrane potential. At the second (mesoscopic) scale, interacting populations of neurons are described individually by similar equations. The equations describing the dynamical and the stationary mean-field behaviors are considered as functional equations on a set of stochastic processes. Using this new point of view allows us to prove that these equations are well-posed on any finite time interval and to provide a constructive method for effectively computing their unique solution. This method is proved to converge to the unique solution and we characterize its complexity and convergence rate. We also provide partial results for the stationary problem on infinite time intervals. These results shed some new light on such neural mass models as the one of Jansen and Rit: their dynamics appears as a coarse approximation of the much richer dynamics that emerges from our analysis. Our numerical experiments confirm that the framework we propose and the numerical methods we derive from it provide a new and powerful tool for the exploration of neural behaviors at different scales.