Crisis of interspike intervals in Hodgkin–Huxley model

doi:10.1016/j.chaos.2005.04.062

Chaos, Solitons & Fractals

Volume 27, Issue 4, February 2006, Pages 952-958

https://doi.org/10.1016/j.chaos.2005.04.062 Get rights and content

Abstract

The bifurcations of the chaotic attractor in a Hodgkin–Huxley (H–H) model under stimulation of periodic signal is presented in this work, where the frequency of signal is taken as the controlling parameter. The chaotic behavior is realized over a wide range of frequency and is visualized by using interspike intervals (ISIs). Many kinds of abrupt undergoing changes of the ISIs are observed in different frequency regions, such as boundary crisis, interior crisis and merging crisis displaying alternately along with the changes of external signal frequency. And there are logistic-like bifurcation behaviors, e.g., periodic windows and fractal structures in ISIs dynamics. The saddle-node bifurcations resulting in collapses of chaos to period-6 orbit in dynamics of ISIs are identified.

Introduction

There has been a continuing argument on neuronal spike trains which play an important role in encoding and decoding the neuronal information. As we all know, the neural systems have strong nonlinear characters and usually able to display different dynamics according to system parameters or external inputs in ISIs sequence. When these parameters are slightly modified, the system’s dynamics usually experience also little modification, except when these changes occur in the vicinity of a critical point. In that case an abrupt qualitative change or transition in the dynamics occurs. These transitions, for example, may be from periodic to chaotic, from chaotic to chaotic, or in their inverse [1].

The role of chaotic activities in the brain is growingly interested in neuroscience and chaotic theory. For instance, the numerical evidence and theoretical reasoning has proved that there is a chaos–chaos transition in the brain, in which the change of the attractor size is sudden but continuous, different from general discontinues chaos–chaos transitions, and which occurs in the Hindmarsh–Rose model of a neuron. This transition corresponds to different neural dynamics, i.e. the chaotic dynamics of bursting or spiking dynamics [2]. The crisis of the thermally sensitive neuron resulted from homoclinic bifurcation of a saddle-focus fixed point which is embedded in the chaotic attractors, also been studied in Ref. [3]. Similarly, the external shifted stimulus current will induce the chaos collapsing to a period-3 orbit in the dynamics of a quadratic logistic map neuron [4]. Lee and Farhat [5], [6] suggested that the bifurcating neuron’s bistability and associative memory is related to attractor-merging crisis; and Xie et al. [7] introduced periodic orbit theory to characterize the dynamical behavior of aperiodic firing neurons, and considered that bifurcations, crises and sensitive dependence of chaotic motions on control parameters can be the underlying mechanisms.

There are many chaotic activities that have been observed in experimental studies of electroencephalogram (EEG) signals and ISIs sequence. For example, Xie et al. [8] have studied variations of chaos during an epileptic seizure; Freeman and Skarda [9] observed the olfactory system of the rabbit that the nervous activity of the olfactory system switches from a chaotic to a periodic state whenever a familiar odor is detected. This experimental observation stimulated their reflection on the role of chaos in perception processes and led them to postulate that chaos can serve as the ground state of a perception process, i.e. an elevated state that has quick transition routes to many periodic states [5]; and onset of the explosion occurs of ISIs and at least two periodic windows have been observed in cases of increasing and decreasing temperature of the saline bath for crayfish caudal photoreceptor cell [3].

The study of transitions between different dynamic behaviors in nonlinear systems is an issue of major interest for the theory of nonlinear dynamics and chaos [1], [10], [11]. That has also been widely interested in by biophysicist. The observation of bifurcations and crisis in this work is relevant both to the theory of nonlinear dynamics and chaos, and to biophysics, also, particularly to neurobiology. Chaos–chaos transitions will help us to understand how the neural system is able to give quick responses to the different external or internal stimulus, i.e., rapid switches between different neuronal dynamic activities.

Section snippets

The Hodgkin–Huxley (H–H) model

As is well known, the H–H model equations have been derived from a squid giant axon. These equations can describe the spiking behavior and refractoriness of real neuron very well [12], [13], so that this kind of model is employed in this work. The H–H model for the action potential of a space-clamped squid axon is defined by the four-dimensional vector field [14] $\{\begin{matrix} \dot{u} = I_{ext} - [120 m^{3} h (u + 115) + 36 n^{4} (u - 12) + 0.3 (u + 10.6)], \\ \dot{m} = (1 - m) Ψ (\frac{u + 25}{10}) - m (4 \exp (\frac{u}{18})), \\ \dot{n} = 0.1 (1 - n) Ψ (\frac{u + 10}{10}) - n (0.125 \exp (\frac{u}{80})), \\ \dot{h} = 0.07 (1 - h) Ψ (\frac{u}{20}) - h (\frac{1}{1 + \exp (\frac{u + 30}{10})}), \end{matrix}$

Bifurcations and crisis of ISIs in the H–H model

The H–H model has been simulated numerically in the absence of noise, using the ISIs as a state variable. The ISIs are registered by the membrane potential crossing a threshold (at 60 mV) with positive derivative (Poincaré surface of section). With controlled frequency ranges f ∈ [0.01, 10]f₀, the general bifurcation diagram of ISIs for the H–H model under periodic stimulus is shown in Fig. 1. The abrupt and separate increase of the duration of ISIs occurs in the region f ∈ [2.21, 3.468]f₀ (cf. Fig. 1

Saddle-node bifurcation

Bifurcation diagrams of ISIs of H–H model shown in Fig. 1, Fig. 2 suggest that saddle-node, period doubling and other common basic bifurcations underlie ISIs of H–H neuronal dynamics as logistic map. In this work, we aim at one of the numerous bifurcation processes collapse of chaos to a period-3 orbit in the H–H spiking dynamics, which emerges at f = 3.4711f₀.

Seen from bifurcation diagrams Figs. 2b and 4a, the period-3 orbit is embedded in two chaotic attractors, respectively, and their shape of

Physiological evidence and summary

The behaviors of bifurcation and crisis stated in the work presented have been partly testified by Feudel et al. as in work [3], they offer qualitative evidence of the bifurcations and crises of ISIs obtained from extracellular recordings of the discharges from of caudal photoreceptor cell embedded in the crayfish sixth or terminal ganglion. With the initial period-doubling bifurcation included, periodic windows and onset of the explosion have been observed with decreasing temperature (cf. Fig.

Acknowledgement

We are grateful to the National Natural Science Foundation of China for its support (Grant No. 10432010).

References (16)

M. Lysetskiy et al.
Bifurcating neuron: Computation and learning ykola
Neural Networks
(2004)
G. Lee et al.
The bifurcating neuron network 1
Neural Networks
(2001)
G. Lee et al.
The bifurcating neuron network 2: An analogy associative memory
Neural Networks
(2002)
Y. Xie et al.
Dynamical mechanisms for sensitive response of aperiodic firing cells to external stimulation
Chaos, Solitons & Fractals
(2004)
W. Freeman et al.
Spatial EEG patterns, nonlinear dynamics and perception: Neo-Sherringtonian view
Brain Res Rev
(1985)
L. Hong et al.
Crises and chaotic transients studied by the generalized cell mapping digraph method
Phys Lett A
(1999)
W.Y. Jin et al.
Rate of afferent stimulus dependent synchronization and coding in coupled neurons system
Chaos, Solitons & Fractals
(2004)
E. Ott
Chaos in dynamical systems
(2002)

There are more references available in the full text version of this article.

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Citation Excerpt :
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Crisis of interspike intervals in Hodgkin–Huxley model

Abstract

Introduction

Section snippets

The Hodgkin–Huxley (H–H) model

Bifurcations and crisis of ISIs in the H–H model

Saddle-node bifurcation

Physiological evidence and summary

Acknowledgement

Neural Networks

Neural Networks

Neural Networks

Chaos, Solitons & Fractals

Brain Res Rev

Phys Lett A

Chaos, Solitons & Fractals

Chaos in dynamical systems