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Journal of Computational Neuroscience

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01-10-2013

Lyapunov exponents computation for hybrid neurons

Authors: Federico Bizzarri, Angelo Brambilla, Giancarlo Storti Gajani

Published in: Journal of Computational Neuroscience | Issue 2/2013

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Abstract

Lyapunov exponents are a basic and powerful tool to characterise the long-term behaviour of dynamical systems. The computation of Lyapunov exponents for continuous time dynamical systems is straightforward whenever they are ruled by vector fields that are sufficiently smooth to admit a variational model. Hybrid neurons do not belong to this wide class of systems since they are intrinsically non-smooth owing to the impact and sometimes switching model used to describe the integrate-and-fire (I&F) mechanism. In this paper we show how a variational model can be defined also for this class of neurons by resorting to saltation matrices. This extension allows the computation of Lyapunov exponent spectrum of hybrid neurons and of networks made up of them through a standard numerical approach even in the case of neurons firing synchronously.

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Whenever only a switching event occurs at the boundary \(\Sigma \), condition (4) can be strengthened as follows if one wants to prevent sliding motion (Di Bernardo et al. 2008) on it:

\(\displaystyle {\left [ \eta ^{\mathrm {T}}(x,t) f_1(x(t)) \right ]\cdot \left [ \eta ^{\mathrm {T}}(x,t) f_2(x(t)) \right ] > 0.}\)

The hybrid systems considered in this work fulfill both assumptions.

It is obvious that the unitary entry in each \(e_j\) could make this vector everything but a small perturbation of \(x_0\). On the other side, it is worth noting that the solution of the linearized problem can be scaled preserving its properties. This is exactly the case when one is interested in Lyapunov exponents evaluation by resorting to the variational model solution: the initial condition of (7) is usually set to \(\epsilon \mathbb {1}_N\) with \(\epsilon \ll 1\).

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Title: Lyapunov exponents computation for hybrid neurons
Authors: Federico Bizzarri
Angelo Brambilla
Giancarlo Storti Gajani
Publication date: 01-10-2013
Publisher: Springer US
Published in: Journal of Computational Neuroscience / Issue 2/2013
Print ISSN: 0929-5313
Electronic ISSN: 1573-6873
DOI: https://doi.org/10.1007/s10827-013-0448-6

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