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积分发放型非光滑动力系统的李雅普诺夫指数谱

Spectrum of Lyapunov exponents of non-smooth dynamical systems of integrate-and-fire type.

作者信息

Zhou Douglas, Sun Yi, Rangan Aaditya V, Cai David

机构信息

Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA.

出版信息

J Comput Neurosci. 2010 Apr;28(2):229-45. doi: 10.1007/s10827-009-0201-3. Epub 2009 Dec 9.

DOI:10.1007/s10827-009-0201-3
PMID:20012178
Abstract

We discuss how to characterize long-time dynamics of non-smooth dynamical systems, such as integrate-and-fire (I&F) like neuronal network, using Lyapunov exponents and present a stable numerical method for the accurate evaluation of the spectrum of Lyapunov exponents for this large class of dynamics. These dynamics contain (i) jump conditions as in the firing-reset dynamics and (ii) degeneracy such as in the refractory period in which voltage-like variables of the network collapse to a single constant value. Using the networks of linear I&F neurons, exponential I&F neurons, and I&F neurons with adaptive threshold, we illustrate our method and discuss the rich dynamics of these networks.

摘要

我们讨论如何使用李雅普诺夫指数来刻画非光滑动力系统的长时间动力学，例如类积分发放（I&F）神经网络，并提出一种稳定的数值方法，用于精确评估这类动力学的李雅普诺夫指数谱。这些动力学包含：（i）如发放重置动力学中的跳跃条件；（ii）简并性，例如在不应期，网络中类似电压的变量会坍缩为单个恒定值。我们使用线性I&F神经元网络、指数I&F神经元网络以及具有自适应阈值的I&F神经元网络来说明我们的方法，并讨论这些网络丰富的动力学。

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