Origin of Bursting through Homoclinic Spike Adding in a Neuron Model

Channell, Paul J.; Cymbalyuk, Gennady; Shilnikov, Andrey

doi:10.1103/physrevlett.98.134101

Cited by 94 publications

(75 citation statements)

References 27 publications

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“…A novel practical method for constructing a complete family of onto mappings for membrane potentials of slow-fast neuronal models was proposed in Channell et al (2007a, b) following an idea suggested originally in Shilnikov (1993). Using this approach we have studied spike adding transitions in a leech heart interneuron model and revealed that this phenomenon is associated with homoclinic bifurcations of a saddle periodic orbit (Channell et al 2007a). …”

Section: Introductionmentioning

confidence: 96%

“…They work exceptionally well in the most low-order mathematical and realistic models far from bifurcations of bursting. However, at activity transitions, the bursting behavior can become drastically complex and even chaotic (Terman 1991(Terman , 1992Holden and Fan 1992;Wang 1993;Belykh et al 2000;Feudel et al 2000;Deng and Hines 2002;Elson et al 2002;Shilnikov and Cymbalyuk 2004;Cymbalyuk and Shilnikov 2005;Channell et al 2007a) due to reciprocal interaction between the slow and fast dynamics, which lead to the emergence of novel dynamical phenomena and bifurcations that can only occur in the entire system. Recently a novel method which identifies effective low-dimensional local models to study the transition between tonic and bursting regimes was developed in Clewley et al (2009).…”

Section: Introductionmentioning

confidence: 98%

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Variability of bursting patterns in a neuron model in the presence of noise

et al. 2009

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Spiking and bursting patterns of neurons are characterized by a high degree of variability. A single neuron can demonstrate endogenously various bursting patterns, changing in response to external disturbances due to synapses, or to intrinsic factors such as channel noise. We argue that in a model of the leech heart interneuron existing variations of bursting patterns are significantly enhanced by a small noise. In the absence of noise this model shows periodic bursting with fixed numbers of interspikes for most parameter values. As the parameter of activation kinetics of a slow potassium current is shifted to more hyperpolarized values of the membrane potential, the model undergoes a sequence of incremental spike adding transitions accumulating towards a periodic tonic spiking activity. Within a narrow parameter window around every spike adding transition, spike alteration of bursting is deterministically chaotic due to homoclinic bifurcations of a saddle periodic orbit. We have found that near these transitions the interneuron model becomes extremely sensitive to small random perturbations that cause a wide expansion and overlapping of the chaotic windows. The chaotic behavior is characterized by positive values of the largest Lyapunov exponent, and of the Shannon entropy of probability distribution of spike numbers per burst. The windows of chaotic dynamics resemble the Arnold tongues being plotted in the parameter plane, where the noise intensity serves as a second control parameter. We determine the critical noise intensities above which the interneuron model generates only irregular bursting within the overlapped windows.

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Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 98%

Variability of bursting patterns in a neuron model in the presence of noise

et al. 2009

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“…Starting from that seminal mathematical model, a lot of variant models describing different kinds of neuron cells in numerous animals have been proposed in the literature. For instance, a reduced model [2][3][4] of the bursting of leech heart neuron is specified by the following three equations derived through the Hodgkin-Huxley gated variable formalism:…”

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confidence: 99%

Symbolic dynamical unfolding of spike-adding bifurcations in chaotic neuron models

Barrio¹,

Lefranc²,

Martínez³

et al. 2015

EPL

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PACS 05.45.Ac -Low-dimensional chaos PACS 05.45.Pq -Numerical simulations of chaotic systems PACS 87.19.ll -Models of single neurons and networks Abstract -We characterize the systematic changes in the topological structure of chaotic attractors that occur as spike-adding and homoclinic bifurcations are encountered in the slow-fast dynamics of neuron models. This phenomenon is detailed in the simple Hindmarsh-Rose neuron model, where we show that the unstable periodic orbits appearing after each spike-adding bifurcation are associated with specific symbolic sequences in the canonical symbolic encoding of the dynamics of the system. This allows us to understand how these bifurcations modify the internal structure of the chaotic attractors.The wide-range assessment of brain and behaviors is one of the pivotal challenges of this century. To understand how an incredibly sophisticated system such as the brain per se functions dynamically, it is imperative to study the dynamics of its constitutive elements -neurons. Therefore, the design of mathematical models for neurons has arisen as a trending topic in science for a few decades, since Hodgkin and Huxley developed the first model of action potentials in the neuron membrane [1]. Starting from that seminal mathematical model, a lot of variant models describing different kinds of neuron cells in numerous animals have been proposed in the literature. For instance, a reduced model [2-4] of the bursting of leech heart neuron is specified by the following three equations derived through the Hodgkin-Huxley gated variable formalism: As control parameters of the system are varied, its dynamics undergoes bifurcations such as classified by dynamical systems theory that match qualitative metamorphoses in terms specific to neuroscience. A broad range of non-stationary activity types can be observed, which includes regular and irregular tonic spiking, bursting and mixed-mode oscillations and combinations of them, as well as oscillatory transients toward quiescent states. In terms of dynamical system theory, these behaviors correspond to stable periodic and deterministically chaotic orbits in the phase space of the model. Figure 1 represents a twoparameter sweep of the leech model in the (τ K2 , I app )-plane, where the number of spikes per period, measured with the spike-counting method (SPC) [5], is color-coded. Banded structures correlated with zones of chaotic behavior appear clearly in this diagram. They are associated with spike-adding bifurcations, where the number of spikes is incremented by one, as indicated in the figure.Spike-adding bifurcations are special bifurcations that are common in fast-slow systems. They lead to the appearance of extra spikes (turns) in the fast manifold region and are quite important in that they progressively p-1

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“…A functionally driven reduction of all currents involved in this feedback could be expected to allow meaningful quantification of the degree to which release plays a role in comparison to escape. It may also facilitate phase plane and bifurcation analysis of the hybrid dynamics for these feedback loops, for instance along the lines of [1,11,19,24,26].…”

Section: Insight Into the Escape-release Mechanismmentioning

confidence: 99%

Inferring and quantifying the role of an intrinsic current in a mechanism for a half-center bursting oscillation

Clewley

2011

J Biol Phys

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This paper illustrates an informatic technique for inferring and quantifying the dynamic role of a single intrinsic current in a mechanism of neural bursting activity. We analyze the patterns of the most dominant currents in a model of half-center oscillation in the leech heartbeat central pattern generator. We find that the patterns of dominance change substantially over a cycle, allowing different local reductions to be applied to the model. The result is a hybrid dynamical systems model, which is a piecewise representation of the mechanism combining multiple vector fields and discrete state changes. The simulation of such a model tests explicit hypotheses about the mechanism and is a novel way to retain both mathematical clarity and scientific detail in answering mechanistic questions about a complex model. Several insights into the central mechanism of "escape-release" in the model are elucidated by this analysis and compared with previous studies. The broader application and extension of this technique is also discussed.

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Origin of Bursting through Homoclinic Spike Adding in a Neuron Model

Cited by 94 publications

References 27 publications

Variability of bursting patterns in a neuron model in the presence of noise

Variability of bursting patterns in a neuron model in the presence of noise

Symbolic dynamical unfolding of spike-adding bifurcations in chaotic neuron models

Inferring and quantifying the role of an intrinsic current in a mechanism for a half-center bursting oscillation

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