Voltage Interval Mappings for an Elliptic Bursting Model

Wojcik, Jeremy; Shilnikov, Andrey

doi:10.1007/978-3-319-09864-7_9

Cited by 13 publications

(14 citation statements)

References 38 publications

(56 reference statements)

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“…Incorporating slow dynamics, it can exhibit a wide range of different firing patterns in certain parameter regimes. The FHR model was introduced by FitzHugh and Rinzel [28][29][30][31] and is also known as the elliptic bursting model 31 . It is described by the system of ordinary differential equations…”

Section: The Excitable Fhr Modelmentioning

confidence: 99%

“…The decrease of a and c gives rise to longer intervals between consecutive bursting activities, with the system showing relatively fixed times in burst duration. Additionally, with the increase of a, the interburst intervals become shorter and the oscillations change to tonic spiking 28,31 . In our work, we have used δ = 0.08, a = 0.7, b = 0.8, µ = 0.002, c = −0.775 and we will be varying I.…”

Section: The Excitable Fhr Modelmentioning

confidence: 99%

“…The excitable FHR model studied here is an extended version of the FitzHugh-Nagumo (FHN) model 27 . Excitable, biophysical media represented by the slow-fast dynamics of electrically coupled FHR neurons [28][29][30][31] give rise to wave propagation when the system becomes excited above a threshold, called a traveling wave, that travels through the nerve cells. The responses propagate along the axons, allowing the consideration of the system as a spatially distributed neural medium.…”

Section: Introductionmentioning

confidence: 99%

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Spatiotemporal characteristics in systems of diffusively coupled excitable slow–fast FitzHugh–Rinzel dynamical neurons

Mondal

Sharma

et al. 2021

Chaos: An Interdisciplinary Journal of Nonlinear Science

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In this paper, we study an excitable, biophysical system that supports wave propagation of nerve impulses. We consider a slow-fast, FitzHugh-Rinzel neuron model where only the membrane voltage interacts diffusively, giving rise to the formation of spatiotemporal patterns. We focus on local, nonlinear excitations and diverse neural responses in an excitable 1-and 2-dimensional configuration of diffusively coupled FitzHugh-Rinzel neurons. The study of the emerging spatiotemporal patterns is essential in understanding the working mechanism in different brain areas. We derive analytically the coefficients of the amplitude equations in the vicinity of Hopf bifurcations and characterize various patterns, including spirals exhibiting complex geometric substructures. Further, we derive analytically the condition for the development of antispirals in the neighborhood of the bifurcation point. The emergence of broken target waves can be observed to form spiral-like profiles. The spatial dynamics of the excitable system exhibits 2-and multi-arm spirals for small diffusive couplings. Our results reveal a multitude of neural excitabilities and possible conditions for the emergence of spiral-wave formation. Finally, we show that the coupled excitable systems with different firing characteristics, participate in a collective behavior that may contribute significantly to irregular neural dynamics.

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Section: The Excitable Fhr Modelmentioning

confidence: 99%

Section: The Excitable Fhr Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Spatiotemporal characteristics in systems of diffusively coupled excitable slow–fast FitzHugh–Rinzel dynamical neurons

Mondal

Sharma

et al. 2021

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…Synchronization conditions in a network in which the central element modeling the pacemaking neuron is linked to the group of uncoupled neurons were presented. Using the Poincaré return mappings, the authors of [14] were able to examine in detail the bifurcations that underlie the complex activity transitions between: tonic spiking and bursting, bursting and mixed-mode oscillations, and finally, mixed-mode oscillations and quiescence in the FR model. This paper is devoted to clarifying the analytical properties of the FR model.…”

Section: Introductionmentioning

confidence: 99%

Analytical Properties and Solutions of the FitzHugh – Rinzel Model

Zemlyanukhin¹,

Бочкарев²

2019

Nelin. Dinam.

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The FitzHugh-Rinzel model is considered, which differs from the famous FitzHugh-Nagumo model by the presence of an additional superslow dependent variable. Analytical properties of this model are studied. The original system of equations is transformed into a third-order nonlinear ordinary differential equation. It is shown that, in the general case, the equation does not pass the Painlevé test, and the general solution cannot be represented by Laurent series. Using the singular manifold method in terms of the Schwarzian derivative, an exact particular solution in the form of a kink is constructed, and restrictions on the coefficients of the equation necessary for the existence of such a solution are revealed. An asymptotic solution is obtained that shows good agreement with the numerical one. This solution can be used to verify the results in a numerical study of the FitzHugh-Rinzel model.

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“…A whole CPG can be also multifunctional if it governs more than one behavior, in contrast to a dedicated CPG which is arranged for the purpose of a single locomotion. Modeling studies, mathematical and computational, have proven to be useful for gaining insights into operational principles of CPGs [8][9][10][11][12][13], in particular, multifunctional ones [14][15][16]. This study has been inspired by recent experimental studies of neuronal activity patterns recorded from the identified interneurons of a [dedicated] CPG governing the swim pattern of a sea slug Melibe leonina [17][18][19] and possible consequences of understanding neurological phenomena in general.…”

Section: Introductionmentioning

confidence: 99%

Toward robust phase-locking in Melibe swim central pattern generator models

Jalil

Allen

Youker

et al. 2013

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Small groups of interneurons, abbreviated by CPG for central pattern generators, are arranged into neural networks to generate a variety of core bursting rhythms with specific phase-locked states, on distinct time scales, that govern vital motor behaviors in invertebrates such as chewing, swimming, etc. These movements in lower level animals mimic motions of organs in higher animals due to evolutionarily conserved mechanisms. Hence, various neurological diseases can be linked to abnormal movement of body parts that are regulated by a malfunctioning CPG. In this paper, we, being inspired by recent experimental studies of neuronal activity patterns recorded from a swimming motion CPG of the sea slug Melibe leonina, examine a mathematical model of a 4-cell network that can plausibly and stably underlie the observed bursting rhythm. We develop a dynamical systems framework for explaining the existence and robustness of phase-locked states in activity patterns produced by the modeled CPGs. The proposed tools can be used for identifying core components for other CPG networks with reliable bursting outcomes and specific phase relationships between the interneurons. Our findings can be employed for identifying or implementing the conditions for normal and pathological functioning of basic CPGs of animals and artificially intelligent prosthetics that can regulate various movements. Repetitive behaviors are often associated hypothetically with the phenomenon of rhythmogenesis in small networks that are able autonomously to generate or continue, after induction a variety of activity patterns without further external input, abrupt or not. The goal of this modeling study is to identify decisive components of a biologically based CPG that has been linked to a specific motion in a lower order animal, sea slug Melibe leonina, which produces a specific bursting pattern, as well as to identify components that ensure the pattern's robustness. Due to the recurrent nature of such bursting patterns of self-sustained activity we employ Poincare return maps defined on phases and phase-lags between burst initiations in the interneurons to study quantitative and qualitative properties of CPG rhythms and corresponding attractors.The proposed approach is specifically tailored for various studies of neural networks in neuroscience, computational and experimental. Development of such tools and our understanding of such CPGs can be applied to gain insight into governing principles of neurological phenomena in higher order animals and can aid in treating anomalies associated with neurological disorders.

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Voltage Interval Mappings for an Elliptic Bursting Model

Cited by 13 publications

References 38 publications

Spatiotemporal characteristics in systems of diffusively coupled excitable slow–fast FitzHugh–Rinzel dynamical neurons

Spatiotemporal characteristics in systems of diffusively coupled excitable slow–fast FitzHugh–Rinzel dynamical neurons

Analytical Properties and Solutions of the FitzHugh – Rinzel Model

Toward robust phase-locking in Melibe swim central pattern generator models

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