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Networks of piecewise linear neural mass models

Published online by Cambridge University Press:  20 February 2018

S. COOMBES ,
Y. M. LAI ,
M. ŞAYLI  and
R. THUL
S. COOMBES
Affiliation:
Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK emails: stephen.coombes@nottingham.ac.uk, Yi.Lai1@nottingham.ac.uk, Mustafa.Sayli@nottingham.ac.uk, ruediger.thul@nottingham.ac.uk
Y. M. LAI
Affiliation:
Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK emails: stephen.coombes@nottingham.ac.uk, Yi.Lai1@nottingham.ac.uk, Mustafa.Sayli@nottingham.ac.uk, ruediger.thul@nottingham.ac.uk
M. ŞAYLI
Affiliation:
Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK emails: stephen.coombes@nottingham.ac.uk, Yi.Lai1@nottingham.ac.uk, Mustafa.Sayli@nottingham.ac.uk, ruediger.thul@nottingham.ac.uk
R. THUL
Affiliation:
Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK emails: stephen.coombes@nottingham.ac.uk, Yi.Lai1@nottingham.ac.uk, Mustafa.Sayli@nottingham.ac.uk, ruediger.thul@nottingham.ac.uk
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Abstract

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Neural mass models are ubiquitous in large-scale brain modelling. At the node level, they are written in terms of a set of ordinary differential equations with a non-linearity that is typically a sigmoidal shape. Using structural data from brain atlases, they may be connected into a network to investigate the emergence of functional dynamic states, such as synchrony. With the simple restriction of the classic sigmoidal non-linearity to a piecewise linear caricature, we show that the famous Wilson–Cowan neural mass model can be explicitly analysed at both the node and network level. The construction of periodic orbits at the node level is achieved by patching together matrix exponential solutions, and stability is determined using Floquet theory. For networks with interactions described by circulant matrices, we show that the stability of the synchronous state can be determined in terms of a low-dimensional Floquet problem parameterised by the eigenvalues of the interaction matrix. Moreover, this network Floquet problem is readily solved using linear algebra to predict the onset of spatio-temporal network patterns arising from a synchronous instability. We further consider the case of a discontinuous choice for the node non-linearity, namely the replacement of the sigmoid by a Heaviside non-linearity. This gives rise to a continuous-time switching network. At the node level, this allows for the existence of unstable sliding periodic orbits, which we explicitly construct. The stability of a periodic orbit is now treated with a modification of Floquet theory to treat the evolution of small perturbations through switching manifolds via the use of saltation matrices. At the network level, the stability analysis of the synchronous state is considerably more challenging. Here, we report on the use of ideas originally developed for the study of Glass networks to treat the stability of periodic network states in neural mass models with discontinuous interactions.

Keywords

General applied mathematicssynchronisationnon-smooth equationscomplex networksneural networks
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Special Issue 5: Theory and applications of nonsmooth dynamical systems , October 2018 , pp. 869 - 890
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © Cambridge University Press 2018

Footnotes

†

This work was supported by the Engineering and Physical Sciences Research Council [grant number EP/P007031/1]. M. Şayli was supported by a grant from TÜBITAK.

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