Lyapunov exponents computation for hybrid neurons

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 m...

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Published in:	Journal of computational neuroscience Vol. 35; no. 2; pp. 201 - 212
Main Authors:	Bizzarri, Federico, Brambilla, Angelo, Storti Gajani, Giancarlo
Format:	Journal Article
Language:	English
Published:	Boston Springer US 01.10.2013 Springer Nature B.V
Subjects:	Algorithms Biomedical and Life Sciences Biomedicine Brain - physiology Computer Simulation Electrophysiological Phenomena - physiology Human Genetics Models, Neurological Nerve Net Neural Networks (Computer) Neurology Neurons - physiology Neurosciences Theory of Computation Variational model Hybrid model Neuron networks Integrate and fire neuron Lyapunov exponent Saltation Matrix
ISSN:	0929-5313, 1573-6873, 1573-6873
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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.
AbstractList	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.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. 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. 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.[PUBLICATION ABSTRACT]
Author	Bizzarri, Federico Brambilla, Angelo Storti Gajani, Giancarlo
Author_xml	– sequence: 1 givenname: Federico surname: Bizzarri fullname: Bizzarri, Federico email: federico.bizzarri@polimi.it organization: Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano – sequence: 2 givenname: Angelo surname: Brambilla fullname: Brambilla, Angelo organization: Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano – sequence: 3 givenname: Giancarlo surname: Storti Gajani fullname: Storti Gajani, Giancarlo organization: Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano
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Cites_doi	10.1007/978-1-4612-3486-9 10.1016/0960-0779(94)00170-U 10.1017/CBO9780511815706 10.1007/s10827-009-0201-3 10.1145/1089014.1089020 10.1109/TCSI.2008.916443 10.1007/978-1-4757-3978-7 10.1016/S0021-8928(98)00087-2 10.1109/TCSII.2011.2111570 10.1109/TCSI.2012.2188953 10.1007/s10827-007-0038-6 10.1007/b137198 10.1063/1.2975967 10.1109/81.828574 10.1016/j.matcom.2010.10.012 10.1109/TCSI.2011.2167273 10.1016/j.physd.2011.05.012 10.1142/S0218127403008090 10.1016/0001-8708(70)90023-X 10.3938/jkps.57.1363 10.1162/089976606775093882 10.1016/0167-2789(85)90011-9 10.1152/jn.00240.2010 10.1109/TNN.2003.820440 10.1109/ECCTD.2011.6043387 10.1109/ECCTD.2011.6043647
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Keywords	Variational model Hybrid model Neuron networks Integrate and fire neuron Lyapunov exponent Saltation Matrix
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Snippet	Lyapunov exponents are a basic and powerful tool to characterise the long-term behaviour of dynamical systems. The computation of Lyapunov exponents for...
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SubjectTerms	Algorithms Biomedical and Life Sciences Biomedicine Brain - physiology Computer Simulation Electrophysiological Phenomena - physiology Human Genetics Models, Neurological Nerve Net Neural Networks (Computer) Neurology Neurons - physiology Neurosciences Theory of Computation
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Title	Lyapunov exponents computation for hybrid neurons
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