Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T08:49:33.688Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected], [email protected], [email protected], [email protected]
Y. M. LAI
Affiliation:
Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK emails: [email protected], [email protected], [email protected], [email protected]
M. ŞAYLI
Affiliation:
Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK emails: [email protected], [email protected], [email protected], [email protected]
R. THUL
Affiliation:
Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK emails: [email protected], [email protected], [email protected], [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

References

[1] Ahmadizadeh, S., Nesic, D., Grayden, D. B. & Freestone, D. R. (2015) Analytic synchronization conditions for a network of Wilson and Cowan oscillators. In: IEEE 54th Annual Conference on Decision and Control.Google Scholar
[2] Amari, S. (1977) Dynamics of pattern formation in lateral-inhibition type neural fields. Biol. Cybern. 27, 77–87.Google Scholar
[3] Amari, S.-I. (2014) Heaviside world: Excitation and self-organization of neural fields. In: Neural Field Theory, Springer, Berlin.Google Scholar
[4] Assenza, S., Gutiérrez, R., Gómez-Gardenes, J., Latora, V. & Boccaletti, S. (2011) Emergence of structural patterns out of synchronization in networks with competitive interactions. Sci. Rep. 1, 99.Google Scholar
[5] Breakspear, M. (2017) Dynamic models of large-scale brain activity. Nat. Neurosci. 20, 340–352.Google Scholar
[6] Campbell, S. & Wang, D. (1996) Synchronization and desynchronization in a network of locally coupled Wilson-Cowan oscillators. IEEE Trans. Neural Netw. 7, 541–554.Google Scholar
[7] Colombo, A., di Bernardo, M., Hogan, S. J. & Jeffrey, M. R. (2012) Bifurcations of piecewise smooth flows: Perspectives, methodologies and open problems. Physica D 241, 1845–1860.Google Scholar
[8] Coombes, S. (2005) Waves, bumps, and patterns in neural field theories. Biol. Cybern. 93, 91–108.Google Scholar
[9] Coombes, S., beimGraben, P. Graben, P., Potthast, R. & Wright, J. (editors) (2014) Neural Fields: Theory and Applications, Springer, Berlin.Google Scholar
[10] Coombes, S. & Thul, R. (2016) Synchrony in networks of coupled non-smooth dynamical systems: Extending the master stability function. Eur. J. Appl. Math. 27, 904–922.Google Scholar
[11] Cowan, J. D. (2004) History of concepts and techniques. Intell. Syst. 3, 375–400.Google Scholar
[12] Cowan, J. D., Neuman, J. & van Drongelen, W. (2016) Wilson-Cowan equations for neocortical dynamics. J. Math. Neurosci. 6 (1), 2016.Google Scholar
[13] Deco, G., Jirsa, V. K., McIntosh, A. R., Sporns, O. & Kötter, R. (2009) Key role of coupling, delay, and noise in resting brain fluctuations. Proc. Natl. Acad. Sci. USA 106, 10302–10307.Google Scholar
[14] Destexhe, A. & Sejnowski, T. J. (2009) The Wilson–Cowan model, 36 years later. Biol. Cybern. 101, 1–2.Google Scholar
[15] di Bernardo, M., Budd, C., Champneys, A. R. & Kowalczyk, P. (2008) Piecewise-Smooth Dynamical Systems: Theory and Applications, Applied Mathematical Sciences, Springer, London.Google Scholar
[16] Dörfler, F. & Bullo, F. (2014) Synchronization in complex networks of phase oscillators: A survey. Automatica 50, 1539–1564.Google Scholar
[17] Edwards, R. (2000) Analysis of continuous-time switching networks. Physica D 146, 165–199.Google Scholar
[18] Edwards, R. & Glass, L. (2000) Combinatorial explosion in model gene networks. Chaos 10, 691–704.Google Scholar
[19] Ermentrout, G. B. (1991) An adaptive model for synchrony in the firefly Pteroptyx malaccae. J. Math. Biol. 29, 571–585.Google Scholar
[20] Filippov, A. F. (1988) Differential Equations with Discontinuous Righthand Sides, Kluwer Academic Publishers, Norwell.Google Scholar
[21] Glass, L. (1975) Classification of biological networks by their qualitative dynamics. J. Theor. Biol. 54, 85–107.Google Scholar
[22] Glass, L. & Kauffman, S. A. (1971) The logical analysis of continuous non-linear biochemical control networks. J. Theor. Biol. 39, 103–129.Google Scholar
[23] Goel, P. & Ermentrout, B. (2002) Synchrony, stability, and firing patterns in pulse-coupled oscillators. Physica D 163, 191–216.Google Scholar
[24] Hansel, D. & Sompolinsky, H. (1998) Modeling feauture selectivity in local cortical circuits. In: Methods in Neuronal Modeling, 2nd ed., MIT Press, Cambridge Massachusetts.Google Scholar
[25] Harris, J. & Ermentrout, B. (2015) Bifurcations in the Wilson–Cowan equations with nonsmooth firing rate. SIAM J. Appl. Dyn. Syst. 14, 43–72.Google Scholar
[26] Jansen, B. H. & Rit, V. G. (1995) Electroencephalogram and visual evoked potential generation in a mathematical model of coupled cortical columns. Biol. Cybern. 73, 357–366.Google Scholar
[27] Jeffrey, M. R. (2015) Smoothing tautologies, hidden dynamics, and sigmoid asymptotics for piecewise smooth systems. Chaos 25, 103125.Google Scholar
[28] Kielblock, H., Kirst, C. & Timme, M. (2011) Breakdown of order preservation in symmetric oscillator networks with pulse-coupling. Chaos 21, 025113.Google Scholar
[29] Kilpatrick, Z. P. & Bressloff, P. C. (2010) Effects of synaptic depression and adaptation on spatiotemporal dynamics of an excitatory neuronal network. Physica D 239, 547–560.Google Scholar
[30] Kuramoto, Y. (1984) Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Heidelberg.Google Scholar
[31] Laing, C. R. & Chow, C. C. (2002) A spiking neuron model for binocular rivalry. J. Comput. Neurosci. 12, 39–53.Google Scholar
[32] Lea-Carnall, C. A., Montemurro, M. A., Trujillo-Barreto, N. J., Parkes, L. M. & El-Deredy, W. (2016) Cortical resonance frequencies emerge from network size and connectivity. PLoS Comput. Biol. 12, 1–19.Google Scholar
[33] Leine, R. & Nijmeijer, H. (2004) Dynamics and Bifurcations of Non-Smooth Mechanical Systems, Lecture Notes in Applied and Computational Mechanics book series, Vol. 18, Springer-Verlag, Berlin.Google Scholar
[34] Liley, D. T. J., Cadusch, P. J. & Dafilis, M. P. (2002) A spatially continuous mean field theory of electrocortical activity. Network 13, 67–113.Google Scholar
[35] McCleney, Z. T. & Kilpatrick, Z. P. (2016) Entrainment in up and down states of neural populations: Non-smooth and stochastic models. J. Math. Biol. 73, 1131–1160.Google Scholar
[36] Meijer, H. G. E., Eissa, T. L., Kiewiet, B., Neuman, J. F., Schevon, C. A., Emerson, R. G., Goodman, R. R., McKhann, G. M., Marcuccilli, C. J., Tryba, A. K., Cowan, J. D., vanGils, S. A. Gils, S. A. & vanDrongelen, W. Drongelen, W. (2015) Modeling focal epileptic activity in the Wilson–Cowan model with depolarization block. J. Math. Neurosci. 5, 7.Google Scholar
[37] Nicks, R., Chambon, L. & Coombes, S. (2018) Clusters in nonsmooth oscillator networks. Phys. Rev. E, submitted.Google Scholar
[38] Onslow, A. C. E., Jones, M. W. & Bogacz, R. (2014) A canonical circuit for generating phase-amplitude coupling. PLoS One 9, 1–15.Google Scholar
[39] Pecora, L. M. & Carroll, T. L. (1998) Master stability functions for synchronized coupled systems. Phys. Rev. Lett. 80, 2109–2112.Google Scholar
[40] Pecora, L. M., Sorrentino, F., Hagerstrom, A. M., Murphy, T. E. & Roy, R. (2014) Cluster synchronization and isolated desynchronization in complex networks with symmetries. Nat. Commun. 5, 4079.Google Scholar
[41] Sanz-Leon, P., Knock, S. A., Spiegler, A. & Jirsa, V. K. (2015) Mathematical framework for large-scale brain network modeling in The Virtual Brain. NeuroImage 111, 385–430.Google Scholar
[42] Senthilkumar, D. V. & Lakshmanan, M. M. (2005) Bifurcations and chaos in time delayed piecewise linear dynamical systems. Int. J. Bifurc. Chaos 15, 2895–2912.Google Scholar
[43] Sorrentino, F., Pecora, L. M., Hagerstrom, A. M., Murphy, T. E. & Roy, R. (2016) Complete characterization of the stability of cluster synchronization in complex dynamical networks. Sci. Adv. 2, e1501737–e1501737.Google Scholar
[44] Timme, M., Wolf, F. & Geisel, T. (2002) Coexistence of regular and irregular dynamics in complex networks of pulse-coupled oscillators. Phys. Rev. Lett. 89, 258701.Google Scholar
[45] Ueta, T. & Chen, G. (2003) On synchronization and control of coupled Wilson Cowan neural oscillators. Int. J. Bifurc. Chaos 13, 163–175.Google Scholar
[46] Valdes-Sosa, P. A., Sanchez-Bornot, J. M., Sotero, R. C., Iturria-Medina, Y., Aleman-Gomez, Y., Bosch-Bayard, J., Carbonell, F. & Ozaki, T. (2009) Model driven EEG/fMRI fusion of brain oscillations. Hum. Brain Mapp. 30, 2701–2721.Google Scholar
[47] Wilson, H. R. & Cowan, J. D. (1972) Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J. 12, 1–24.Google Scholar
[48] Wilson, H. R. & Cowan, J. D. (1973) A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik 13, 55–80.Google Scholar
[49] Woolrich, M. W. & Stephan, K. E. (2013) Biophysical network models and the human connectome. NeuroImage 80, 330–338.Google Scholar
[50] Zetterberg, L. H., Kristiansson, L. & Mossberg, K. (1978) Performance of a model for a local neuron population. Biol. Cybern. 31, 15–26.Google Scholar