Stochastic Hierarchical Systems: Excitable Dynamics

Leonhardt, Helmar; Zaks, Michael A.; Falcke, Martin; Schimansky‐Geier, Lutz

doi:10.1007/s10867-008-9112-1

Cited by 2 publications

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References 63 publications

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“…Two-and three-state models have been applied to neuronal systems [6,7] and recently to language dynamics [8]. Synchronization behavior, phase transitions and reaction to time delayed feedback [3,4,[9][10][11][12][13][14] as well as excitability [15,16] are general aspects that apply to a wide range of natural phenomena.…”

Section: Introductionmentioning

confidence: 99%

Tristable and multiple bistable activity in complex random binary networks of two-state units

2017

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Abstract.We study complex networks of stochastic two-state units. Our aim is to model discrete stochastic excitable dynamics with a rest and an excited state. Both states are assumed to possess different waiting time distributions. The rest state is treated as an activation process with an exponentially distributed life time, whereas the latter in the excited state shall have a constant mean which may originate from any distribution. The activation rate of any single unit is determined by its neighbors according to a random complex network structure. In order to treat this problem in an analytical way, we use a heterogeneous mean-field approximation yielding a set of equations generally valid for uncorrelated random networks. Based on this derivation we focus on random binary networks where the network is solely comprised of nodes with either of two degrees. The ratio between the two degrees is shown to be a crucial parameter. Dependent on the composition of the network the steady states show the usual transition from disorder to homogeneously ordered bistability as well as new scenarios that include inhomogeneous ordered and disordered bistability as well as tristability. The various steady states differ in their spiking activity expressed by a state dependent spiking rate. Numerical simulations agree with analytic results of the heterogeneous mean-field approximation.

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Section: Introductionmentioning

confidence: 99%

Tristable and multiple bistable activity in complex random binary networks of two-state units

2017

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Understanding cardiac alternans: A piecewise linear modeling framework

Thul

Coombes

2010

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Cardiac alternans is a beat-to-beat alternation in action potential duration ͑APD͒ and intracellular calcium ͑Ca 2+ ͒ cycling seen in cardiac myocytes under rapid pacing that is believed to be a precursor to fibrillation. The cellular mechanisms of these rhythms and the coupling between cellular Ca 2+ and voltage dynamics have been extensively studied leading to the development of a class of physiologically detailed models. These have been shown numerically to reproduce many of the features of myocyte response to pacing, including alternans, and have been analyzed mathematically using various approximation techniques that allow for the formulation of a low dimensional map to describe the evolution of APDs. The seminal work by Shiferaw and Karma is of particular interest in this regard ͓Shiferaw, Y. and Karma, A., "Turing instability mediated by voltage and calcium diffusion in paced cardiac cells," Proc. Natl. Acad. Sci. U.S.A. 103, 5670-5675 ͑2006͔͒. Here, we establish that the key dynamical behaviors of the Shiferaw-Karma model are arranged around a set of switches. These are shown to be the main elements for organizing the nonlinear behavior of the model. Exploiting this observation, we show that a piecewise linear caricature of the Shiferaw-Karma model, with a set of appropriate switching manifolds, can be constructed that preserves the physiological interpretation of the original model while being amenable to a systematic mathematical analysis. In illustration of this point, we formulate the dynamics of Ca 2+ cycling ͑in response to pacing͒ and compute the properties of periodic orbits in terms of a stroboscopic map that can be constructed without approximation. Using this, we show that alternans emerge via a period-doubling instability and track this bifurcation in terms of physiologically important parameters. We also show that when coupled to a spatially extended model for Ca 2+ transport, the model supports spatially varying patterns of alternans. We analyze the onset of this instability with a generalization of the master stability approach to accommodate the nonsmooth nature of our system. © 2010 American Institute of Physics. ͓doi:10.1063/1.3518362͔Beat-to-beat patterns of alternating activity seen in whole heart electrocardiograms are often taken as a sign of an abnormal rhythm that may lead to sudden cardiac death. The initiation and maintenance of such pathological cardiac alternans can be linked to dynamics at the single cell level. Here, it also manifests itself as a beat-to-beat alternation, although this time in the duration of action potentials. These electrical spikes of activity ultimately control the contraction of heart muscle fibers. Because of the known link between cellular alternans and cardiac arrhythmias, understanding the mechanism of their generation is vital for developing preventative clinical treatments. Experiments have shown that cellular alternans arise from an interplay of membrane voltage dynamics and intracellular Ca 2+ cycling. However, these studies have not been able t...

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Stochastic Hierarchical Systems: Excitable Dynamics

Cited by 2 publications

References 63 publications

Tristable and multiple bistable activity in complex random binary networks of two-state units

Tristable and multiple bistable activity in complex random binary networks of two-state units

Understanding cardiac alternans: A piecewise linear modeling framework

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