×

The role of network structure and time delay in a metapopulation Wilson-Cowan model. (English) Zbl 1416.92007

Summary: We study the dynamics of a network Wilson-Cowan model (a system of connected Wilson-Cowan oscillators) for interacting excitatory and inhibitory neuron populations with time delays. Each node in this model corresponds to a population of neurons, including excitatory and inhibitory subpopulations, and hence it can be viewed as a metapopulation model. It is known that information transfer within each cortical area is not instantaneous, and therefore we consider a system of delay differential equations with two different kinds of discrete delays. We account for the time delay in information propagation between individual excitatory and inhibitory subpopulations at each node via intra-node time delays, and we account for time delay in information propagation between neuron populations at different nodes with inter-node time delays. The biologically relevant resting state solutions are oscillatory (stable limit cycles). After determining the influence of the coupling parameters between nodes, the intra-node delays, and the inter-node delays on the dynamics of the two coupled Wilson-Cowan oscillators, we then explore a variety of larger networks of 16 and 100 nodes, in order to determine how the network topology will influence time delayed Wilson-Cowan dynamics. We find that network structure can regularize or deregularize the dynamics, with networks of higher mean degree permitting stable limit cycles and networks with smaller mean degree yielding less regular dynamics (which may range from chaotic solutions, to solutions for which limit cycles collapse into steady states, which are biologically undesirable compared with the preferred stable limit cycles). Furthermore, heterogeneity in the degree distribution of the network (resulting from networks with nodes of varying degree) can result in asynchronous dynamics, even if at each node the local dynamics are that of a limit cycle, in contrast to the synchronization of dynamics between nodes seen when the degree of all nodes is equal. This suggests that homogeneous and well-connected networks permit robust limit cycles under time-delayed Wilson-Cowan dynamics, whereas heterogeneous or poorly connected networks may fail to provide such desirable dynamics, a phenomena akin to structural loss of neuron connections in neurodegenerative diseases.

MSC:

92B20 Neural networks for/in biological studies, artificial life and related topics
05C90 Applications of graph theory

Software:

dde23
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ashwin, P.; Coombes, S.; Nicks, R., Mathematical frameworks for oscillatory network dynamics in neuroscience, J. Math. Neurosci., 6, 1, 2 (2016) · Zbl 1356.92015
[2] Ben-Ari, Y., The GABA excitatory/inhibitory developmental sequence: a personal journey, Neuroscience, 279, 1, 187-219 (2014)
[3] Buckner, R. L.; Sepulcre, J.; Talukdar, T.; Krienen, F. M.; Liu, H.; Hedden, T.; Andrews-Hanna, J. R.; Sperling, R. A.; Johnson, K. A., Cortical hubs revealed by intrinsic functional connectivity: mapping, assessment of stability, and relation to Alzheimer’s disease, J. Neurosci., 29, 6, 1860-1873 (2009)
[4] Byrne, A.; Brookes, M. K.; Coombes, S., A mean field model for movement induced changes in the beta rhythm, J. Comput. Neurosci., 43, 2, 143-158 (2017) · Zbl 1402.92078
[5] Campbell, S.; Wang, D., Synchronization and desynchronization in a network of locally coupled Wilson-Cowan oscillators, IEEE Trans. Neural Netw., 7, 3, 541-554 (1996)
[6] Ching, S.; Brown, E. N.; Kramer, M. A., Distributed control in a mean-field cortical network model: implications for seizure suppression, Phys. Rev. E, 86, 2, 021920 (2012)
[7] Coombes, S.; Graben, P.; Potthast, R.; Wright, J., Neural fields. Theory and Applications. (2014), Springer Verlag
[8] Coombes, S.; Laing, C., Delays in activity-based neural networks, Philos. Trans. R. Soc.London A Math. Phys. Eng. Sci., 367, 1891, 1117-1129 (2009) · Zbl 1185.92003
[9] Daffertshofer, A.; van Wijk, B., On the influence of amplitude on the connectivity between phases, Front. Neuroinform., 5, 6 (2011)
[11] Deco, G.; Jirsa, V.; McIntosh, A. R.; Sporns, O.; Kötter, R., Key role of coupling, delay, and noise in resting brain fluctuations, Proc.Natl. Acad. Sci., 106, 25, 10302-10307 (2009)
[12] Dijkstra, K.; van Gils, S. A.; Janssens, S.; Kuznetsov, Y. A.; Visser, S., Pitchfork-Hopf bifurcations in 1D neural field models with transmission delays, Phys. D, 297, 88-101 (2015) · Zbl 1392.37041
[13] Galán, R. F., On how network architecture determines the dominant patterns of spontaneous neural activity, PLoS ONE, 3, 5, e2148 (2008)
[16] Haidar, I.; Pasillas-Lépine, W.; Chaillet, A.; Panteley, E.; Palfi, S.; Senova, S., Closed-loop firing rate regulation of two interacting excitatory and inhibitory neural populations of the basal ganglia, Biol. Cybern., 110, 1, 55-71 (2016) · Zbl 1345.92040
[17] Van den Heuvel, M. P.; Mandl, R. C.; Stam, C. J.; Kahn, R. S.; Pol, H. E.H., Aberrant frontal and temporal complex network structure in schizophrenia: a graph theoretical analysis, J. Neurosci., 30, 47, 15915-15926 (2010)
[20] Jiang, C.; Guo, S.; He, Y., Dynamics in time-delay recurrently coupled oscillators, Int. J. Bifurcation Chaos, 21, 03, 775-788 (2011) · Zbl 1215.34101
[21] Jiang, Y.; Guo, S., Linear stability and Hopf bifurcation in a delayed two-coupled oscillator with excitatory-to-inhibitory connection, Nonlinear Anal. Real World Appl., 11, 3, 2001-2015 (2010) · Zbl 1205.34107
[22] Kiewiet, B., On the effect of a gaussian firing rate function and propagation delays on the dynamics of a network of Wilson-Cowan populations (2014), Faculty of Electrical Engineering, Mathematics and Computer Science, University of Twente, Master thesis
[23] König, P.; Schillen, T. B., Stimulus-dependent assembly formation of oscillatory responses: i. Synchronization, Neural Comput., 3, 2, 155-166 (1991)
[24] Malagarriga, D.; Villa, A. E.; Garcia-Ojalvo, J.; Pons, A. J., Mesoscopic segregation of excitation and inhibition in a brain network model, PLoS Comput. Biol., 11, 2, e1004007 (2015)
[25] Malyutina, E.; Wyller, J.; Ponosov, A., Two bump solutions of a homogenized Wilson-Cowan model with periodic microstructure, Phys. D, 271, 19-31 (2014) · Zbl 1298.35228
[26] Meijer, H. G.; 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., Modeling focal epileptic activity in the Wilson-Cowan model with depolarization block, J. Math. Neurosci. (JMN), 5, 1, 7 (2015) · Zbl 1357.92013
[27] Minati, L.; Varotto, G.; D’Incerti, L.; Panzica, F.; Chan, D., From brain topography to brain topology: relevance of graph theory to functional neuroscience, Neuroreport, 24, 10, 536-543 (2013)
[28] Muldoon, S. F.; Pasqualetti, F.; Gu, S.; Cieslak, M.; Grafton, S. T.; Vettel, J. M.; Bassett, D. S., Stimulation-based control of dynamic brain networks, PLoS Comput. Biol., 12, 9, e1005076 (2016)
[29] Newman, M., Networks: An introduction (2010), Oxford University Press
[30] Pasillas-Lépine, W., Delay-induced oscillations in Wilson and Cowan’s model: an analysis of the subthalamo-pallidal feedback loop in healthy and parkinsonian subjects, Biol. Cybern., 107, 3, 289-308 (2013) · Zbl 1267.92034
[33] Qiao, Y.; Meng, Y.; Duan, L.; Fang, F.; Miao, J., Qualitative analysis and application of locally coupled neural oscillator network, Neural Comput. Appl., 21, 7, 1551-1562 (2012)
[34] Rădulescu, A.; Verduzco-Flores, S., Nonlinear network dynamics under perturbations of the underlying graph, Chaos, 25, 1, 013116 (2015) · Zbl 1345.34062
[35] Roxin, A.; Montbrió, E., How effective delays shape oscillatory dynamics in neuronal networks, Phys. D, 240, 3, 323-345 (2011) · Zbl 1213.37127
[37] Rădulescu, A., Neural network spectral robustness under perturbations of the underlying graph, Neural Comput., 28, 1, 1-44 (2016) · Zbl 1414.92043
[38] Sanz-Leon, P.; Knock, S. A.; Spiegler, A.; Jirsa, V. K., Mathematical framework for large-scale brain network modeling in the virtual brain, Neuroimage, 111, 385-430 (2015)
[39] Schillen, T. B.; König, P., Stimulus-dependent assembly formation of oscillatory responses: ii. Desynchronization, Neural Comput., 3, 2, 167-178 (1991)
[40] Seeley, W. W.; Crawford, R. K.; Zhou, J.; Miller, B. L.; Greicius, M. D., Neurodegenerative diseases target large-scale human brain networks, Neuron, 62, 1, 42-52 (2009)
[41] Shampine, L. F.; Thompson, S., Solving DDEs in Matlab, Appl. Numer. Math., 37, 4, 441-458 (2014) · Zbl 0983.65079
[42] Sharp, D. J.; Scott, G.; Leech, R., Network dysfunction after traumatic brain injury, Nat. Rev. Neurol., 10, 3, 156 (2014)
[44] Sporns, O., Networks of the Brain (2010), MIT press
[45] Sreenivasan, V.; Menon, S. N.; Sinha, S., Emergence of coupling-induced oscillations and broken symmetries in heterogeneously driven nonlinear reaction networks, Sci. Rep., 7, 1, 1594 (2017)
[46] Tang, E., Bassett, D. S., 2017. Control of dynamics in brain networks. arXiv:1701.01531; Tang, E., Bassett, D. S., 2017. Control of dynamics in brain networks. arXiv:1701.01531
[47] Touboul, J., Mean-field equations for stochastic firing-rate neural fields with delays: derivation and noise-induced transitions, Phys. D, 241, 15, 1223-1244 (2012) · Zbl 1317.92021
[48] Tu, C., Rocha, R. P., Corbetta, M., Zampieri, S., Zorzi, M., Suweis, S., 2017. Warnings and caveats in brain controllability. arXiv:1705.08261; Tu, C., Rocha, R. P., Corbetta, M., Zampieri, S., Zorzi, M., Suweis, S., 2017. Warnings and caveats in brain controllability. arXiv:1705.08261
[49] Veltz, R., Interplay between synaptic delays and propagation delays in neural field equations, SIAM J. Appl. Dyn. Syst., 12, 3, 1566-1612 (2013) · Zbl 1284.34116
[50] Wang, D., Emergent synchrony in locally coupled neural oscillators, IEEE Trans. Neural Netw., 6, 4, 941-948 (1995)
[51] Wang, D., The time dimension for scene analysis, IEEE Trans. Neural Netw., 16, 6, 1401-1426 (2005)
[52] Wang, L.; Peng, J.; Jin, Y.; Ma, J., Synchronous dynamics and bifurcation analysis in two delay coupled oscillators with recurrent inhibitory loops, J. Nonlinear Sci., 23, 2, 283-302 (2013) · Zbl 1318.34102
[54] Xiao, K.; Guo, S., Synchronization for two coupled oscillators with inhibitory connection, Math. Methods Appl. Sci., 33, 7, 892-903 (2010) · Zbl 1187.92007
[55] Zhang, P.; Guo, S.; He, Y., Dynamics of a delayed two-coupled oscillator with excitatory-to-excitatory connection, Appl. Math. Comput., 216, 2, 631-646 (2010) · Zbl 1210.34117
[56] Zhao, L.; Cupertino, T. H.; Bertini Jr, J. R., Chaotic synchronization in general network topology for scene segmentation, Neurocomputing, 71, 16-18, 3360-3366 (2008)
[57] Zhao, L.; Macau, E. E.; Omar, N., Scene segmentation of the chaotic oscillator network, Int. J. Bifurcation Chaos, 10, 07, 1697-1708 (2000)
[58] Zhou, J.; Gennatas, E. D.; Kramer, J. H.; Miller, B. L.; Seeley, W. W., Predicting regional neurodegeneration from the healthy brain functional connectome, Neuron, 73, 6, 1216-1227 (2012)
[59] Zhou, J.; Seeley, W. W., Network dysfunction in Alzheimer’s disease and frontotemporal dementia: implications for psychiatry, Biol. Psychiatry, 75, 7, 565-573 (2014)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.