Two-parameter bifurcations in the Hodgkin-Huxley equations for muscle fibers

Terada, Kazuko; Tanaka, Hisaaki; Yoshizawa, Shuji

doi:10.1002/(sici)1520-6440(200006)83:6<86::aid-ecjc10>3.0.co;2-c

Cited by 12 publications

(4 citation statements)

References 23 publications

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“…The local and even global phenomena observed in the model (6)–(7) have been pointed out in single cell models [29] , [30] , [83] , [84] . The similarity between the dynamics of population firing described here, and the membrane potential dynamics in single cells, is more substantial than phase plane analogies for occasional parameter values and is currently under rigorous examination.…”

Section: Discussionmentioning

confidence: 99%

Mechanisms Explaining Transitions between Tonic and Phasic Firing in Neuronal Populations as Predicted by a Low Dimensional Firing Rate Model

Rǎdulescu

2010

PLoS ONE

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Several firing patterns experimentally observed in neural populations have been successfully correlated to animal behavior. Population bursting, hereby regarded as a period of high firing rate followed by a period of quiescence, is typically observed in groups of neurons during behavior. Biophysical membrane-potential models of single cell bursting involve at least three equations. Extending such models to study the collective behavior of neural populations involves thousands of equations and can be very expensive computationally. For this reason, low dimensional population models that capture biophysical aspects of networks are needed. The present paper uses a firing-rate model to study mechanisms that trigger and stop transitions between tonic and phasic population firing. These mechanisms are captured through a two-dimensional system, which can potentially be extended to include interactions between different areas of the nervous system with a small number of equations. The typical behavior of midbrain dopaminergic neurons in the rodent is used as an example to illustrate and interpret our results. The model presented here can be used as a building block to study interactions between networks of neurons. This theoretical approach may help contextualize and understand the factors involved in regulating burst firing in populations and how it may modulate distinct aspects of behavior.

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Section: Discussionmentioning

confidence: 99%

Mechanisms Explaining Transitions between Tonic and Phasic Firing in Neuronal Populations as Predicted by a Low Dimensional Firing Rate Model

Rǎdulescu

2010

PLoS ONE

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“…Recently, it is pointed out [104,105] that the ions channel-based neuronal model should take into account the size effect of nerve fibres. Moreover, codim-2 bifurcation analysis has been explored to reveal the complex dynamical behavior of some neuron models in [106][107][108]. Bertram [109] studied some bursting oscillations in the Chay-Cook model by using two-parameter bifurcation diagrams.…”

Section: One-or Two-parameter Bifurcation Analysis For Neuronal Firinmentioning

confidence: 99%

Dynamics of firing patterns, synchronization and resonances in neuronal electrical activities: experiments and analysis

Yang

et al. 2008

Acta Mech Sin

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Recent advances in the experimental and theoretical study of dynamics of neuronal electrical firing activities are reviewed. Firstly, some experimental phenomena of neuronal irregular firing patterns, especially chaotic and stochastic firing patterns, are presented, and practical nonlinear time analysis methods are introduced to distinguish deterministic and stochastic mechanism in time series. Secondly, the dynamics of electrical firing activities in a single neuron is concerned, namely, fast-slow dynamics analysis for classification and mechanism of various bursting patterns, one-or two-parameter bifurcation analysis for transitions of firing patterns, and stochastic dynamics of firing activities (stochastic and coherence resonances, integer multiple and other firing patterns induced by noise, etc.).types of synchronization of coupled neurons with electrical and chemical synapses are discussed. As noise and time delay are inevitable in nervous systems, it is found that noise and time delay may induce or enhance synchronization and change firing patterns of coupled neurons. Noise-induced resonance and spatiotemporal patterns in coupled neuronal networks are also demonstrated. Finally, some prospects are presented for future research. In consequence, the idea and methods of nonlinear dynamics are of great significance in exploration of dynamic processes and physiological functions of nervous systems.

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“…Fukai and his fellows examined the structure of the model's bifurcations produced by i ext and one of the other parameters [21]. Taking leakage conductance g l and sodium channel effective bias voltage

{\bar{V}}_{m}

as parameters, Terada et al analyzed bifurcation in Hodgkin-Huxley model for muscles of frogs [22]. …”

Section: Introductionmentioning

confidence: 99%

Effects of Maximal Sodium and Potassium Conductance on the Stability of Hodgkin-Huxley Model

Zhang

Wang

Yuan

et al. 2014

Computational and Mathematical Methods in Medicine

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Hodgkin-Huxley (HH) equation is the first cell computing model in the world and pioneered the use of model to study electrophysiological problems. The model consists of four differential equations which are based on the experimental data of ion channels. Maximal conductance is an important characteristic of different channels. In this study, mathematical method is used to investigate the importance of maximal sodium conductance g-Na and maximal potassium conductance g-K. Applying stability theory, and taking g-Na and g-K as variables, we analyze the stability and bifurcations of the model. Bifurcations are found when the variables change, and bifurcation points and boundary are also calculated. There is only one bifurcation point when g-Na is the variable, while there are two points when g-K is variable. The (g-Na, g-K) plane is partitioned into two regions and the upper bifurcation boundary is similar to a line when both g-Na and g-K are variables. Numerical simulations illustrate the validity of the analysis. The results obtained could be helpful in studying relevant diseases caused by maximal conductance anomaly.

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Two-parameter bifurcations in the Hodgkin-Huxley equations for muscle fibers

Cited by 12 publications

References 23 publications

Mechanisms Explaining Transitions between Tonic and Phasic Firing in Neuronal Populations as Predicted by a Low Dimensional Firing Rate Model

Mechanisms Explaining Transitions between Tonic and Phasic Firing in Neuronal Populations as Predicted by a Low Dimensional Firing Rate Model

Dynamics of firing patterns, synchronization and resonances in neuronal electrical activities: experiments and analysis

Effects of Maximal Sodium and Potassium Conductance on the Stability of Hodgkin-Huxley Model

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