Voltage interval mappings for activity transitions in neuron models for elliptic bursters

Wojcik, Jeremy; Shilnikov, Andrey

doi:10.1016/j.physd.2011.04.003

Cited by 31 publications

(26 citation statements)

References 46 publications

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“…Further decrease of

b_{1}

eventually leads to fixed point dynamics (Figure 3e). The transition from bursting to quiescence is a separate topic and is beyond the scope of this article, but similar transitions have been studied in other models [10,14,15]. …”

Section: Torus Canards In the Hindmarsh-rose Systemmentioning

confidence: 99%

A showcase of torus canards in neuronal bursters

Burke

Desroches

Barry

et al. 2012

The Journal of Mathematical Neuroscience

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Rapid action potential generation - spiking - and alternating intervals of spiking and quiescence - bursting - are two dynamic patterns commonly observed in neuronal activity. In computational models of neuronal systems, the transition from spiking to bursting often exhibits complex bifurcation structure. One type of transition involves the torus canard, which we show arises in a broad array of well-known computational neuronal models with three different classes of bursting dynamics: sub-Hopf/fold cycle bursting, circle/fold cycle bursting, and fold/fold cycle bursting. The essential features that these models share are multiple time scales leading naturally to decomposition into slow and fast systems, a saddle-node of periodic orbits in the fast system, and a torus bifurcation in the full system. We show that the transition from spiking to bursting in each model system is given by an explosion of torus canards. Based on these examples, as well as on emerging theory, we propose that torus canards are a common dynamic phenomenon separating the regimes of spiking and bursting activity.

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“…Further decrease of

b_{1}

Section: Torus Canards In the Hindmarsh-rose Systemmentioning

confidence: 99%

A showcase of torus canards in neuronal bursters

Burke

Desroches

Barry

et al. 2012

The Journal of Mathematical Neuroscience

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“…The disappearance of the tonic spiking orbit, TS, accords with the onset of regular bursting in the mapping and in the model (1). In the mapping, a bursting orbit is comprised of iterates on the tonic spiking and quiescent sections separated by the unstable threshold fixed point, UP 1 , of the mapping in Fig.…”

Section: From Bursting To Mixed-mode Oscillations and Quiescencementioning

confidence: 58%

“…This allows for not only the description of bifurcations post factum, but to predict the changes in the dynamics of the model under consideration before they actually occur. For additional analysis on elliptic bursters including torus formation, we refer the reader to [1].…”

Section: Discussionmentioning

confidence: 99%

“…Varying the parameter, c, we are able to obtain a dense family that covers all behaviors, bifurcations and transitions of the model (1). A family of the mappings for the parameter, c, varied within the range [−1, −0.55] is shown in Fig.…”

Section: Voltage Interval Mappingsmentioning

confidence: 99%

“…In this section we discuss a quantitative property of the interval mappings for the dynamics of the model (1). In particular, we carry out the examination of complex dynamics with the use of calculus-based and calculus-free tools such as Lyapunov exponents and kneading invariants for the symbolic description of MMOs.…”

Section: Quantitative Features Of Mappings: Kneadingsmentioning

confidence: 99%

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Voltage Interval Mappings for an Elliptic Bursting Model

Wojcik

Shilnikov

2015

Nonlinear Systems and Complexity

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We employed Poincaré return mappings for a parameter interval to an exemplary elliptic bursting model, the FitzHugh-Nagumo-Rinzel model. Using the interval mappings, we 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 FitzHugh-Nagumo-Rinzel model. We illustrate the wealth of information, qualitative and quantitative, that was derived from the Poincaré mappings, for the neuronal models and for similar (electro) chemical systems.

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