2,986 research outputs found
Entrainment and chaos in a pulse-driven Hodgkin-Huxley oscillator
The Hodgkin-Huxley model describes action potential generation in certain
types of neurons and is a standard model for conductance-based, excitable
cells. Following the early work of Winfree and Best, this paper explores the
response of a spontaneously spiking Hodgkin-Huxley neuron model to a periodic
pulsatile drive. The response as a function of drive period and amplitude is
systematically characterized. A wide range of qualitatively distinct responses
are found, including entrainment to the input pulse train and persistent chaos.
These observations are consistent with a theory of kicked oscillators developed
by Qiudong Wang and Lai-Sang Young. In addition to general features predicted
by Wang-Young theory, it is found that most combinations of drive period and
amplitude lead to entrainment instead of chaos. This preference for entrainment
over chaos is explained by the structure of the Hodgkin-Huxley phase resetting
curve.Comment: Minor revisions; modified Fig. 3; added reference
In-phase and anti-phase synchronization in noisy Hodgkin-Huxley neurons
We numerically investigate the influence of intrinsic channel noise on the
dynamical response of delay-coupling in neuronal systems. The stochastic
dynamics of the spiking is modeled within a stochastic modification of the
standard Hodgkin-Huxley model wherein the delay-coupling accounts for the
finite propagation time of an action potential along the neuronal axon. We
quantify this delay-coupling of the Pyragas-type in terms of the difference
between corresponding presynaptic and postsynaptic membrane potentials. For an
elementary neuronal network consisting of two coupled neurons we detect
characteristic stochastic synchronization patterns which exhibit multiple
phase-flip bifurcations: The phase-flip bifurcations occur in form of alternate
transitions from an in-phase spiking activity towards an anti-phase spiking
activity. Interestingly, these phase-flips remain robust in strong channel
noise and in turn cause a striking stabilization of the spiking frequency
Chaos at the border of criticality
The present paper points out to a novel scenario for formation of chaotic
attractors in a class of models of excitable cell membranes near an
Andronov-Hopf bifurcation (AHB). The mechanism underlying chaotic dynamics
admits a simple and visual description in terms of the families of
one-dimensional first-return maps, which are constructed using the combination
of asymptotic and numerical techniques. The bifurcation structure of the
continuous system (specifically, the proximity to a degenerate AHB) endows the
Poincare map with distinct qualitative features such as unimodality and the
presence of the boundary layer, where the map is strongly expanding. This
structure of the map in turn explains the bifurcation scenarios in the
continuous system including chaotic mixed-mode oscillations near the border
between the regions of sub- and supercritical AHB. The proposed mechanism
yields the statistical properties of the mixed-mode oscillations in this
regime. The statistics predicted by the analysis of the Poincare map and those
observed in the numerical experiments of the continuous system show a very good
agreement.Comment: Chaos: An Interdisciplinary Journal of Nonlinear Science
(tentatively, Sept 2008
Instability of synchronized motion in nonlocally coupled neural oscillators
We study nonlocally coupled Hodgkin-Huxley equations with excitatory and
inhibitory synaptic coupling. We investigate the linear stability of the
synchronized solution, and find numerically various nonuniform oscillatory
states such as chimera states, wavy states, clustering states, and
spatiotemporal chaos as a result of the instability.Comment: 8 pages, 9 figure
Mean Field description of and propagation of chaos in recurrent multipopulation networks of Hodgkin-Huxley and Fitzhugh-Nagumo neurons
We derive the mean-field equations arising as the limit of a network of
interacting spiking neurons, as the number of neurons goes to infinity. The
neurons belong to a fixed number of populations and are represented either by
the Hodgkin-Huxley model or by one of its simplified version, the
Fitzhugh-Nagumo model. The synapses between neurons are either electrical or
chemical. The network is assumed to be fully connected. The maximum
conductances vary randomly. Under the condition that all neurons initial
conditions are drawn independently from the same law that depends only on the
population they belong to, we prove that a propagation of chaos phenomenon
takes places, namely that in the mean-field limit, any finite number of neurons
become independent and, within each population, have the same probability
distribution. This probability distribution is solution of a set of implicit
equations, either nonlinear stochastic differential equations resembling the
McKean-Vlasov equations, or non-local partial differential equations resembling
the McKean-Vlasov-Fokker- Planck equations. We prove the well-posedness of
these equations, i.e. the existence and uniqueness of a solution. We also show
the results of some preliminary numerical experiments that indicate that the
mean-field equations are a good representation of the mean activity of a finite
size network, even for modest sizes. These experiment also indicate that the
McKean-Vlasov-Fokker- Planck equations may be a good way to understand the
mean-field dynamics through, e.g., a bifurcation analysis.Comment: 55 pages, 9 figure
Multistable jittering in oscillators with pulsatile delayed feedback
Oscillatory systems with time-delayed pulsatile feedback appear in various
applied and theoretical research areas, and received a growing interest in the
last years. For such systems, we report a remarkable scenario of
destabilization of a periodic regular spiking regime. In the bifurcation point
numerous regimes with non-equal interspike intervals emerge simultaneously. We
show that this bifurcation is triggered by the steepness of the oscillator's
phase resetting curve and that the number of the emerging, so-called
"jittering" regimes grows exponentially with the delay value. Although this
appears as highly degenerate from a dynamical systems viewpoint, the
"multi-jitter" bifurcation occurs robustly in a large class of systems. We
observe it not only in a paradigmatic phase-reduced model, but also in a
simulated Hodgkin-Huxley neuron model and in an experiment with an electronic
circuit
Clarification and complement to "Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons"
In this note, we clarify the well-posedness of the limit equations to the
mean-field -neuron models proposed in Baladron et al. and we prove the
associated propagation of chaos property. We also complete the modeling issue
in Baladron et al. by discussing the well-posedness of the stochastic
differential equations which govern the behavior of the ion channels and the
amount of available neurotransmitters
Spontaneous spiking in an autaptic Hodgkin-Huxley set up
The effect of intrinsic channel noise is investigated for the dynamic
response of a neuronal cell with a delayed feedback loop. The loop is based on
the so-called autapse phenomenon in which dendrites establish not only
connections to neighboring cells but as well to its own axon. The biophysical
modeling is achieved in terms of a stochastic Hodgkin-Huxley model containing
such a built in delayed feedback. The fluctuations stem from intrinsic channel
noise, being caused by the stochastic nature of the gating dynamics of ion
channels. The influence of the delayed stimulus is systematically analyzed with
respect to the coupling parameter and the delay time in terms of the interspike
interval histograms and the average interspike interval. The delayed feedback
manifests itself in the occurrence of bursting and a rich multimodal interspike
interval distribution, exhibiting a delay-induced reduction of the spontaneous
spiking activity at characteristic frequencies. Moreover, a specific
frequency-locking mechanism is detected for the mean interspike interval.Comment: 8 pages, 10 figure
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