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Dynamics of a neuron-glia system: the occurrence of seizures and the influence of electroconvulsive stimuli. A mathematical and numerical study. (English) Zbl 1451.92070
Summary: In this paper, we investigate the dynamics of a neuron-glia cell system and the underlying mechanism for the occurrence of seizures. For our mathematical and numerical investigation of the cell model we will use bifurcation analysis and some computational methods. It turns out that an increase of the potassium concentration in the reservoir is one trigger for seizures and is related to a torus bifurcation. In addition, we will study potassium dynamics of the model by considering a reduced version and we will show how both mechanisms are linked to each other. Moreover, the reduction of the potassium leak current will also induce seizures. Our study will show that an enhancement of the extracellular potassium concentration, which influences the Nernst potential of the potassium current, may lead to seizures. Furthermore, we will show that an external forcing term (e.g. electroshocks as unidirectional rectangular pulses also known as electroconvulsive therapy) will establish seizures similar to the unforced system with the increased extracellular potassium concentration. To this end, we describe the unidirectional rectangular pulses as an autonomous system of ordinary differential equations. These approaches will explain the appearance of seizures in the cellular model. Moreover, seizures, as they are measured by electroencephalography (EEG), spread on the macro-scale (cm). Therefore, we extend the cell model with a suitable homogenised monodomain model, propose a set of (numerical) experiment to complement the bifurcation analysis performed on the single-cell model. Based on these experiments, we introduce a bidomain model for a more realistic modelling of white and grey matter of the brain. Performing similar (numerical) experiment as for the monodomain model leads to a suitable comparison of both models. The individual cell model, with its seizures explained in terms of a torus bifurcation, extends directly to corresponding results in both the monodomain and bidomain models where the neural firing spreads almost synchronous through the domain as fast traveling waves, for physiologically relevant parameters.

92C20 Neural biology
92C32 Pathology, pathophysiology
92C55 Biomedical imaging and signal processing
35B32 Bifurcations in context of PDEs
Full Text: DOI
[1] Atherton, LA; Prince, LY; Tsaneva-Atanasova, K., Bifurcation analysis of a two-compartment hippocampal pyramidal cell model, Journal of Computational Neuroscience, 41, 1, 91-106 (2016)
[2] Barreto, E.; Cressman, JR, Ion concentration dynamics as a mechanism for neuronal bursting, Journal of Biological Physics, 37, 3, 361-373 (2011)
[3] Breakspear, M., Dynamic models of large-scale brain activity, Nature neuroscience, 20, 3, 340 (2017)
[4] Cressman, JR; Ullah, G.; Ziburkus, J.; Schiff, SJ; Barreto, E., The influence of sodium and potassium dynamics on excitability, seizures, and the stability of persistent states: I. Single neuron dynamics, Journal of Computational Neuroscience, 26, 2, 159-170 (2009)
[5] Cressman, JR; Ullah, G.; Ziburkus, J.; Schiff, SJ; Barreto, E., Erratum to: The influence of sodium and potassium dynamics on excitability, seizures, and the stability of persistent states: I. single neuron dynamics, Journal of Computational Neuroscience, 30, 3, 781-781 (2011)
[6] Cymbalyuk, G.; Shilnikov, A., Coexistence of tonic spiking oscillations in a leech neuron model, Journal of Computational Neuroscience, 18, 3, 255-263 (2005)
[7] Desroches, M.; Guckenheimer, J.; Krauskopf, B.; Kuehn, C.; Osinga, HM; Wechselberger, M., Mixed-mode oscillations with multiple time scales, SIAM Review, 54, 2, 211-288 (2012) · Zbl 1250.34001
[8] Dhooge, A.; Govaerts, W.; Kuznetsov, YA, Matcont: A matlab package for numerical bifurcation analysis of odes, ACM Transactions on Mathematical Software, 29, 2, 141-164 (2003) · Zbl 1070.65574
[9] Dhooge, A.; Govaerts, W.; Kuznetsov, YA; Meijer, HGE; Sautois, B., New features of the software matcont for bifurcation analysis of dynamical systems, Mathematical and Computer Modelling of Dynamical Systems, 14, 2, 147-175 (2008) · Zbl 1158.34302
[10] Dougherty, E.T., Turner, J.C., & Vogel, F. (2014). Multiscale coupling of transcranial direct current stimulation to neuron electrodynamics: modeling the influence of the transcranial electric field on neuronal depolarization. Computational and Mathematical Methods in Medicine 2014. · Zbl 1423.92023
[11] Du, M.; Li, J.; Wang, R.; Wu, Y., The influence of potassium concentration on epileptic seizures in a coupled neuronal model in the hippocampus, Cognitive Neurodynamics, 10, 5, 405-414 (2016)
[12] Rognes, M.E., Farrell, P.E., Funke, S.W., Hake, J.E., & Maleckar, M.M.C. (2017). cbcbeat: an adjoint-enabled framework for computational cardiac electrophysiology. The Journal of Open Source Software 2.
[13] Erhardt, AH, Bifurcation analysis of a certain Hodgkin-Huxley model depending on multiple bifurcation parameters, Mathematics, 6, 6, 1-15 (2018) · Zbl 1407.37120
[14] Erhardt, AH, Early afterdepolarisations induced by an enhancement in the calcium current, Processes, 7, 1, 1-16 (2019)
[15] Frey, R.; Heiden, A.; Scharfetter, J.; Schreinzer, D.; Blasbichler, T.; Tauscher, J.; Felleiter, P.; Kasper, S., Inverse relation between stimulus intensity and seizure duration: implications for ect procedure, The Journal of ECT, 17, 2, 102-108 (2001)
[16] Fröhlich, F.; Bazhenov, M.; Timofeev, I.; Steriade, M.; Sejnowski, TJ, Slow state transitions of sustained neural oscillations by activity-dependent modulation of intrinsic excitability, Journal of Neuroscience, 26, 23, 6153-6162 (2006)
[17] Fröhlich, F.; Bazhenov, M.; Sejnowski, TJ, Pathological effect of homeostatic synaptic scaling on network dynamics in diseases of the cortex, Journal of Neuroscience, 28, 7, 1709-1720 (2008)
[18] Girish, K.; Gangadhar, B.; Janakiramaiah, N.; Lalla, RK, Seizure threshold in ect: effect of stimulus pulse frequency, The Journal of ECT, 19, 3, 133-135 (2003)
[19] González, OC; Krishnan, GP; Timofeev, I.; Bazhenov, M., Ionic and synaptic mechanisms of seizure generation and epileptogenesis, Neurobiology of Disease, 130, 104485 (2019)
[20] Govaerts, W.; Kuznetsov, YA; Dhooge, A., Numerical continuation of bifurcations of limit cycles in matlab, SIAM Journal on Scientific Computing, 27, 1, 231-252 (2005) · Zbl 1087.65118
[21] Gutkin, BS; Ermentrout, GB, Dynamics of membrane excitability determine interspike interval variability: A link between spike generation mechanisms and cortical spike train statistics, Neural Computation, 10, 5, 1047-1065 (1998)
[22] Hodgkin, AL; Huxley, AF, A quantitative description of membrane current and its application to conduction and excitation in nerve, Journal of Physiology, 117, 4, 500-544 (1952)
[23] Hübel, N.; Andrew, RD; Ullah, G., Large extracellular space leads to neuronal susceptibility to ischemic injury in a na+/k+ pumps-dependent manner, Journal of Computational Neuroscience, 40, 2, 177-192 (2016)
[24] Izhikevich, EM, Neural excitability, spiking and bursting, Internat J Bifur Chaos, 10, 6, 1171-1266 (2000) · Zbl 1090.92505
[25] Jirsa, VK; Stacey, WC; Quilichini, PP; Ivanov, AI; Bernard, C., On the nature of seizure dynamics, Brain: A Journal of Neurology, 137, 8, 2210-2230 (2014)
[26] Jirsa, VK; Proix, T.; Perdikis, D.; Woodman, MM; Wang, H.; Gonzalez-Martinez, J.; Bernard, C.; Bénar, C.; Guye, M.; Chauvel, P., The virtual epileptic patient: individualized whole-brain models of epilepsy spread, NeuroImage, 145, 377-388 (2017)
[27] Ju, H.; Neiman, AB; Shilnikov, AL, Bottom-up approach to torus bifurcation in neuron models, Chaos: An Interdisciplinary Journal of Nonlinear Science, 28, 10, 106317 (2018)
[28] Kager, H.; Wadman, WJ; Somjen, GG, Seizure-like afterdischarges simulated in a model neuron, Journal of Computational Neuroscience, 22, 2, 105-128 (2007)
[29] Keener, JP; Sneyd, J., Mathematical physiology (1998), New York: Springer, New York · Zbl 0913.92009
[30] Krishnan, GP; Bazhenov, M., Ionic dynamics mediate spontaneous termination of seizures and postictal depression state, Journal of Neuroscience, 31, 24, 8870-8882 (2011)
[31] Krishnan, GP; Filatov, G.; Shilnikov, A.; Bazhenov, M., Electrogenic properties of the na+/k+ atpase control transitions between normal and pathological brain states, Journal of Neurophysiology, 113, 9, 3356-3374 (2015)
[32] Krupa, M.; Gielen, S.; Gutkin, B., Adaptation and shunting inhibition leads to pyramidal/interneuron gamma with sparse firing of pyramidal cells, Journal of Computational Neuroscience, 37, 2, 357-376 (2014) · Zbl 1409.92044
[33] Kuehn, C., Multiple time scale dynamics, applied mathematical sciences, Vol. 191 (2015), New York: Springer, New York
[34] Kuznetsov, YA, Elements of applied bifurcation theory (1998), New York: Springer, New York · Zbl 0914.58025
[35] Lee, WH; Lisanby, SH; Laine, AF; Peterchev, AV, Comparison of electric field strength and spatial distribution of electroconvulsive therapy and magnetic seizure therapy in a realistic human head model, European Psychiatry, 36, 55-64 (2016)
[36] Lopes, MA; Junges, L.; Tait, L.; Terry, JR; Abela, E.; Richardson, MP; Goodfellow, M., Computational modelling in source space from scalp eeg to inform presurgical evaluation of epilepsy, Clinical Neurophysiology, 131, 1, 225-234 (2020)
[37] Lopez-Rincon, A.; Cantu, C.; Etcheverry, G.; Soto, R.; Shimoda, S., Function based brain modeling and simulation of an ischemic region in post-stroke patients using the bidomain, Journal of Neuroscience Methods, 331, 108464 (2020)
[38] Mori, Y., A multidomain model for ionic electrodiffusion and osmosis with an application to cortical spreading depression, Physica D: Nonlinear Phenomena, 308, 94-108 (2015) · Zbl 1365.92021
[39] Olmi, S.; Petkoski, S.; Guye, M.; Bartolomei, F.; Jirsa, V., Controlling seizure propagation in large-scale brain networks, PLOS Computational Biology, 15, 2, 1-23 (2019)
[40] Østby, I., Øyehaug, L., Einevoll, G.T., Nagelhus, E.A., Plahte, E., Zeuthen, T., & et al. (2009). Astrocytic mechanisms explaining neural-activity-induced shrinkage of extraneuronal space. PLoS Computational Biology 5 (1).
[41] Øyehaug, L.; Østby, I.; Lloyd, CM; Omholt, SW; Einevoll, GT, Dependence of spontaneous neuronal firing and depolarisation block on astroglial membrane transport mechanisms, Journal of Computational Neuroscience, 32, 1, 147-165 (2012)
[42] Peterchev, AV; Rosa, MA; Deng, ZD; Prudic, J.; Lisanby, SH, Electroconvulsive therapy stimulus parameters, Journal of ECT, 26, 3, 159-174 (2010)
[43] Rotstein, HG; Oppermann, T.; White, JA; Kopell, N., The dynamic structure underlying subthreshold oscillatory activity and the onset of spikes in a model of medial entorhinal cortex stellate cells, Journal of Computational Neuroscience, 21, 3, 271-292 (2006)
[44] Rubin, J.; Wechselberger, M., Giant squid-hidden canard: The 3d geometry of the Hodgkin-Huxley model, Biological Cybernetics, 97, 1, 5-32 (2007) · Zbl 1125.92015
[45] Rubin, JE; Terman, D., High frequency stimulation of the subthalamic nucleus eliminates pathological thalamic rhythmicity in a computational model, Journal of Computational Neuroscience, 16, 3, 211-235 (2004)
[46] Sanz-Leon, P.; Knock, SA; Spiegler, A.; Jirsa, VK, Mathematical framework for large-scale brain network modeling in the virtual brain, NeuroImage, 111, 385-430 (2015)
[47] Shilnikov, AL, Complete dynamical analysis of a neuron model, Nonlinear Dynamics, 68, 3, 305-328 (2012) · Zbl 1254.37058
[48] Shilnikov, LP; Shilnikov, AL; Turaev, DV; Chua, LO, Methods of qualitative theory in nonlinear dynamics. Part I, Vol. 4 (1998), Singapore: World Scientific, Singapore
[49] Shilnikov, LP; Shilnikov, AL; Turaev, DV; Chua, LO, Methods of qualitative theory in nonlinear dynamics. Part II, Vol. 5 (2001), Singapore: World Scientific, Singapore
[50] Somjen, G.G., Kager, H., & Wadman, W.J. (2008a). Calcium sensitive non-selective cation current promotes seizure-like discharges and spreading depression in a model neuron. Journal of Computational Neuroscience, 26(1), 139.
[51] Somjen, G.G., Kager, H., & Wadman, W.J. (2008b). Computer simulations of neuron-glia interactions mediated by ion flux. Journal of Computational Neuroscience, 25(2), 349-365.
[52] Sundnes, J.; Lines, GT; Nielsen, BF; Mardal, KA; Tveito, A., Computing the electrical activity in the heart (2006), Berlin: Springer, Berlin · Zbl 1182.92020
[53] Tsaneva-Atanasova, K.; Shuttleworth, TJ; Yule, DI; Thompson, JL; Sneyd, J., Calcium oscillations and membrane transport: The importance of two time scales, Multiscale Model Simul, 3, 2, 245-264 (2005) · Zbl 1079.34038
[54] Tsaneva-Atanasova, K.; Osinga, HM; Rieb, T.; Sherman, A., Full system bifurcation analysis of endocrine bursting models, J Theoret Biol, 264, 1133-1146 (2010) · Zbl 1406.92122
[55] Tsumoto, K.; Kitajima, H.; Yoshinaga, T.; Aihara, K.; Kawakami, H., Bifurcations in Morris-Lecar neuron model, Neurocomputing, 69, 4, 293-316 (2006)
[56] Ullah, G., Cressman, Jr J.R., Barreto, E., & Schiff, S.J. (2009). The influence of sodium and potassium dynamics on excitability, seizures, and the stability of persistent states: Ii. network and glial dynamics. Journal of Computational Neuroscience, 26(2), 171-183.
[57] Wang, Y.; Rubin, JE, Multiple timescale mixed bursting dynamics in a respiratory neuron model, Journal of Computational Neuroscience, 41, 3, 245-268 (2016) · Zbl 1382.92099
[58] Wei, Y., Ullah, G., Ingram, J., & Schiff, S.J. (2014a). Oxygen and seizure dynamics: Ii. computational modeling. Journal of Neurophysiology, 112(2), 213-223.
[59] Wei, Y., Ullah, G., & Schiff, S.J. (2014b). Unification of neuronal spikes, seizures, and spreading depression. Journal of Neuroscience, 34(35), 11733-11743.
[60] Y Ho, EC; Truccolo, W., Interaction between synaptic inhibition and glial-potassium dynamics leads to diverse seizure transition modes in biophysical models of human focal seizures, Journal of Computational Neuroscience, 41, 2, 225-244 (2016) · Zbl 1382.92155
[61] Yao, W.; Huang, H.; Miura, RM, A continuum neuronal model for the instigation and propagation of cortical spreading depression, Bulletin of Mathematical Biology, 73, 11, 2773-2790 (2011) · Zbl 1334.92085
[62] Ying, W.; Henriquez, CS, Hybrid finite element method for describing the electrical response of biological cells to applied fields, IEEE Transactions on Biomedical Engineering, 54, 4, 611-620 (2007)
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