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Section DYNAMICS IN LIFE SCIENCES, NEUROSCIENCE APPLICATIONS WORKSHOP
Extreme Events in Bursting Neurons
Arindam Mishra1,2, Suman Saha1, Hilda Cerdeira3, Syamal K. Dana1*
1 CSIR-Indian Institute of Chemical Biology, Jadavpur, Kolkata 700032, India;
2 Department of Physics, Jadavpur University, Kolkata 700032, India;
3 Instituto de Física Teórica, UNESP, Universidade Estadual Paulista, Rua Dr. Bento Teobaldo Ferraz 271, Bloco II, Barra Funda, 01140-070 Sao Paulo, Brazil.
* Presenting e-mail: syamaldana@gmail.com
Coexisting coherent and noncoherent subpopulations in networks of identical oscillators hitherto called as chimera states [1-3] is intriguing, however, existence of such pattern is observed in many systems under both nonlocal coupling and global coupling. We reported [4] such chimera-like states in globally coupled oscillators under the influence of both attractive and repulsive coupling, which is a common type of coupling in biological networks, in other words, there they are called as excitatory and inhibitory coupling. We further extended this work to an ensemble of bursting neurons taking the Hindamarsh-model as the dynamical units, however, using purely repulsive global coupling. We find clear evidence of chimera-like states in a parameter space. Surprisingly, in the noncoherent subgroup of oscillators, we observe signatures of extreme event as intermittent extreme value amplitude of the dynamical variable in the nearest neighbors of dynamical units. The temporal dynamics of the dynamical units in the noncohrent subgroup separately does not reveal signature of the extreme events, however, local mean-field as a microscopic dynamics clearly shows large excursion from their average value that follows a log-tail Gaussian distribution.
To understand the mechanism of the coexisting extreme events, we took a simple two dynamical units' model of the Hindmarsh-Rose system under pure repulsive coupling which shows presence of extreme events. Two coupled bursting neurons establish antiphase (out-of-phase) synchronization when the repulsive coupling strength crosses a threshold. Near the threshold, the coupled dynamics shows intermittent jumps from the antiphase synchronization manifold as large excursions. The large excursion amplitudes in a long run of the dynamical variable show dragon-king [5] like probability distribution with a long tail. For a network of large number oscillators, the coherent subpopulation is in regular dynamical state while the noncoherent population is in a chaotic mode where the temporal dynamics of some of the oscillators intermittently jumps to large values. This is reflected as large excursion in their mean field dynamics that follows non-Gaussian statistics.
Acknowledgements
A.M. is supported by the University Grant Commission (UGC), India. S.S and S.K.D. are supported by the CSIR (India) Emeritus Scientist Scheme. S.K.D. also acknowledges support by the FAPESP (Grant No.2011/11973-4), Sao Paulo, Brazil and the Indo-Russian programme supported by DST-RFBR.
References
1. Y. Kuramoto, Battogtokh, Nonlin. Phen. Complex Syst 5, 380 (2002).
2. D.M. Abrams and S.H. Strogate, Phys.Rev.Letts. 93, 174102 (2004).
3. G. C. Sethia and A. Sen, Phys. Rev. Letts. 112, 144101 (2014).
4. A.Mishra, C.R.Hens, M.Bose, P.K.Roy, S.K.Dana, Phys. Rev. E 92, 062920 (2015); C. R. Hens, A.Mishra , P.K.Roy , A, Sen and S. K. Dana, Pramana -J. Phys. 84 (2), 229 (2015)
5. H.L. D. de S. Cavalcante, M. O., D. Sornette, E. Ott, D. J. Gauthier, Phys. Rev. Letts. 111, 198701 (2013).
6. A.Mishra, S. Saha, H. Cerdeira, S. K. Dana, (in preparation).
Semi-Analytical Approach to Criteria for Ignition of Excitation Waves
B. Bezekci, I. Idris, R. D. Simitev, V. N. Biktashev*
College of Engineering, Mathematics and Physical Sciences, University of Exeter. * Presenting e-mail: v.n.biktashev@exeter.ac.uk
We consider the problem of ignition of propagating waves in one-dimensional bistable or excitable systems by an
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Section DYNAMICS IN LIFE SCIENCES, NEUROSCIENCE APPLICATIONS WORKSHOP
instantaneous spatially extended stimulus. Earlier we proposed a method (Idris and Biktashev, PRL, vol 101, 2008, 244101) for analytical description of the threshold conditions based on an approximation of the (center-)stable manifold of a certain critical solution. Here we generalize this method to address a wider class of excitable systems, such as multicomponent reaction-diffusion systems and systems with non-self-adjoint linearized operators, including systems with moving critical fronts and pulses. We also explore an extension of this method from a linear to a quadratic approximation of the (center-)stable manifold, resulting in some cases in a significant increase in accuracy. The applicability of the approach is demonstrated on five test problems ranging from archetypal examples such as the Zeldovich--Frank-Kamenetsky equation to near realistic examples such as the Beeler-Reuter model of cardiac excitation. While the method is analytical in nature, it is recognised that essential ingredients of the theory can be calculated explicitly only in exceptional cases, so we also describe methods suitable for calculating these ingredients numerically. rameter (See equation 1). In many experimental setups, a model that allows for couplings which include higher harmonics is needed, such as electrochemical oscillators and ^-Josephson junctions. Moreover, coupling terms can be nonlinear functions of the order parameters. It has also been shown that such a model is microscopically equivalent to a fully connected hypernetwork where interactions are via triplets.
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Neuroprotective Role of Gap Junctions in a Neuron Astrocyte Network Model
David Terman*
Dept. of Mathematics, Ohio State University, USA. * Presenting e-mail: terman.1@osu.edu
A detailed biophysical model for a neuron/astrocyte network is developed in order to explore mechanisms responsible for the initiation and propagation of cortical spreading depolarizations and the role of astrocytes in maintaining ion homeostasis, thereby preventing these pathological waves. Simulations of the model illustrate how properties of spreading depolarizations, such as wave-speed and duration of depolarization, depend on several factors, including the neuron and astrocyte Na-K ATPase pump strengths. In particular, we consider the neuroprotective role of astrocyte gap junction coupling. The model demonstrates that a syncytium of electrically coupled astrocytes can maintain a physiological membrane potential in the presence of an elevated extracellular potassium concentration and efficiently distribute the excess potassium across the syncytium. This provides an effective neuroprotective mechanism for delaying or preventing the initiation of spreading depolarizations.
Spike-Adding in Parabolic Bursting: the Role of Folded-Saddle Canards
Mathieu Desroches*
Inria, Sophia Antipolis - Méditerranée, France. * Presenting e-mail: mathieu.desroches@inria.fr
In this talk I will present a new approach to studying parabolic bursting. Looking at classical parabolic bursters such as the Plant model from the perspective of slow-fast dynamics, reveals that the number of spikes per burst may vary upon parameter changes. However the spike-adding process occurs in an explosive fashion that involves special solutions called canards. This spike-adding canard explosion phenomenon can be analysed by using tools from geometric singular perturbation theory in conjunction with numerical bifurcation techniques. The bifurcation structure persists across all considered parabolic bursters, namely the Plant model and the Baer-Rinzel-Carillo phase model. That is, spikes within the burst are incremented via the crossing of an excitability threshold given by a particular type of canard orbit, namely the true canard of a folded-saddle singularity. Using these findings, a new polynomial approximation of the Plant model is constructed, which retains all the key elements for parabolic bursting including spike-adding transitions organized by folded-saddle canards. Finally, I will briefly explain the presence of canard-mediated spike-adding transitions in planar phase models of parabolic bursting, namely the theta model (or Atoll model) by Ermentrout and Kopell.
