Dynamics of a neuron model in different two-dimensional parameter-spaces

Rech, Paulo C.

doi:10.1016/j.physleta.2011.02.037

Cited by 39 publications

(21 citation statements)

References 9 publications

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“…To some extent, also raising the injected current I leads to this spike adding, but this also typically increases frequency of the bursts. Other combinations of parameters, e.g., µ and I, lead to similar results [19,40,47]. One of the most widely used approaches to analyse period-adding in the HR and related models is so-called slow-fast analysis [11,20,43,44,45,48].…”

Section: Spike Adding and Slow-fast Analysismentioning

confidence: 95%

Codimension-Two Homoclinic Bifurcations Underlying Spike Adding in the Hindmarsh--Rose Burster

Linaro¹,

Champneys²,

Desroches³

et al. 2012

SIAM J. Appl. Dyn. Syst.

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The Hindmarsh–Rose model of neural action potential is revisited from the point of view of global bifurcation analysis, with the singular perturbation parameter held fixed. Of particular concern is a parameter regime where lobe-shaped regions of irregular bursting undergo a transition to stripeshaped regions of periodic bursting. The boundary of each stripe represents a fold bifurcation that causes a smooth spike adding transition where the number of spikes in each burst is increased by one.\ud It is shown via numerical path-following that the lobe-to-stripe transition is organized by a sequence of codimension-one and -two homoclinic bifurcations. Specifically, each of a sequence of homoclinic bifurcation curves in the parameter plane is found to undergo a sharp turn, due to interaction between a two-dimensional unstable manifold and the one-dimensional slow manifold that persists from the singular limit. Local analysis using approximate Poincare' maps shows that each turning point induces an inclination-flip bifurcation that gives birth to the fold curve that organizes the spike adding transition. Implications of this mechanism for other excitable systems are discussed

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Section: Spike Adding and Slow-fast Analysismentioning

confidence: 95%

Codimension-Two Homoclinic Bifurcations Underlying Spike Adding in the Hindmarsh--Rose Burster

Linaro¹,

Champneys²,

Desroches³

et al. 2012

SIAM J. Appl. Dyn. Syst.

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“…4(c) in ref[24]) in a study by Rech. However, this chaotic region was not mentioned in either of these papers [12], [24]. Two examples of the chaotic firing lying between period-1 and period-2 firings within a bifurcation scenario have been mentioned in our previous experimental investigations on a neural pacemaker [31], [32].…”

Section: Introductionmentioning

confidence: 90%

“…1 in ref[24]) and in the lower-left corner (parameter space) of another figure (Fig. 4(c) in ref[24]) in a study by Rech. However, this chaotic region was not mentioned in either of these papers [12], [24].…”

Section: Introductionmentioning

confidence: 94%

Biological Experimental Observations of an Unnoticed Chaos as Simulated by the Hindmarsh-Rose Model

2013

PLoS ONE

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An unnoticed chaotic firing pattern, lying between period-1 and period-2 firing patterns, has received little attention over the past 20 years since it was first simulated in the Hindmarsh-Rose (HR) model. In the present study, the rat sciatic nerve model of chronic constriction injury (CCI) was used as an experimental neural pacemaker to investigate the transition regularities of spontaneous firing patterns. Chaotic firing lying between period-1 and period-2 firings was observed located in four bifurcation scenarios in different, isolated neural pacemakers. These bifurcation scenarios were induced by decreasing extracellular calcium concentrations. The behaviors after period-2 firing pattern in the four scenarios were period-doubling bifurcation not to chaos, period-doubling bifurcation to chaos, period-adding sequences with chaotic firings, and period-adding sequences with stochastic firings. The deterministic structure of the chaotic firing pattern was identified by the first return map of interspike intervals and a short-term prediction using nonlinear prediction. The experimental observations closely match those simulated in a two-dimensional parameter space using the HR model, providing strong evidences of the existence of chaotic firing lying between period-1 and period-2 firing patterns in the actual nervous system. The results also present relationships in the parameter space between this chaotic firing and other firing patterns, such as the chaotic firings that appear after period-2 firing pattern located within the well-known comb-shaped region, periodic firing patterns and stochastic firing patterns, as predicted by the HR model. We hope that this study can focus attention on and help to further the understanding of the unnoticed chaotic neural firing pattern.

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“…Block-structured dynamics was observed as the pervading bifurcation structure when bursting dynamics occurs. 10,11,23,37,42 Chaos also showed up as a comb-shaped zone inside a periodic area, with the comb teeth associated to the block structured dynamics. 23 BSD-t: block structured dynamics with triangular bursts, BSD-r: block structured dynamics with rectangular bursts and chaos, BSD-s1b: block structured dynamic with square waves displaying one burster, BSD-s2b, block structured dynamics with square waves displaying two bursters, s1b-r: smooth transition between BSD-s2b and BSD-r. Other labels are selfexplanatory.…”

Section: Discussionmentioning

confidence: 99%

Nonlinear dynamics of the membrane potential of a bursting pacemaker cell

González-Miranda

2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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This article presents the results of an exploration of one two-parameter space of the Chay model of a cell excitable membrane. There are two main regions: a peripheral one, where the system dynamics will relax to an equilibrium point, and a central one where the expected dynamics is oscillatory. In the second region, we observe a variety of self-sustained oscillations including periodic oscillation, as well as bursting dynamics of different types. These oscillatory dynamics can be observed as periodic oscillations with different periodicities, and in some cases, as chaotic dynamics. These results, when displayed in bifurcation diagrams, result in complex bifurcation structures, which have been suggested as relevant to understand biological cell signaling.

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Dynamics of a neuron model in different two-dimensional parameter-spaces

Cited by 39 publications

References 9 publications

Codimension-Two Homoclinic Bifurcations Underlying Spike Adding in the Hindmarsh--Rose Burster

Codimension-Two Homoclinic Bifurcations Underlying Spike Adding in the Hindmarsh--Rose Burster

Biological Experimental Observations of an Unnoticed Chaos as Simulated by the Hindmarsh-Rose Model

Nonlinear dynamics of the membrane potential of a bursting pacemaker cell

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