Bifurcation of the Hodgkin and Huxley equations: A new twist

Guckenheimer, John; Labouriau, Isabel S.

doi:10.1007/bf02460693

Cited by 86 publications

(49 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…While improved models for the membrane potential of the squid axon [3] have been formulated, the Hodgkin-Huxley model remains the paradigm for conductance-based models of neural systems. From a mathematical viewpoint, varied properties of the dynamics of the HodgkinHuxley vector field have been studied [13,8,11,14,16,19,5]. Nonetheless, we remain far from a comprehensive understanding of the dynamics displayed by this vector field.…”

Section: Introduction the Hodgkin-huxley Modelmentioning

confidence: 99%

Chaos in the Hodgkin--Huxley Model

Guckenheimer¹,

Oliva²

2002

SIAM J. Appl. Dyn. Syst.

144

View full text Add to dashboard Cite

Abstract. The Hodgkin-Huxley model was developed to characterize the action potential of a squid axon. It has served as an archetype for compartmental models of the electrophysiology of biological membranes. Thus the dynamics of the Hodgkin-Huxley model have been extensively studied both with a view to their biological implications and as a test bed for numerical methods that can be applied to more complex models. This note demonstrates previously unobserved dynamics in the Hodgkin-Huxley model, namely, the existence of chaotic solutions in the model with its original parameters. The solutions are found by displaying rectangles in a cross-section whose images under the return map produce a Smale horseshoe. The chaotic solutions are highly unstable, but they are significant as they lie in the basin boundary that establishes the threshold of the system.

show abstract

Section: Introduction the Hodgkin-huxley Modelmentioning

confidence: 99%

Chaos in the Hodgkin--Huxley Model

Guckenheimer¹,

Oliva²

2002

SIAM J. Appl. Dyn. Syst.

144

View full text Add to dashboard Cite

show abstract

“…Neurons activity results from complex and nonlinear mechanisms (Hodgkin & Huxley, 1952;Cronin, 1987;Dayan & Abbott, 2001;Gerstner & Kistler, 2002b), leading to a wide variety of dynamical behaviours (Cronin, 1987;Guckenheimer & Labouriau, 1993). This activity is revealed by the emission of action potentials or "spikes".…”

mentioning

confidence: 99%

How Gibbs Distributions May Naturally Arise from Synaptic Adaptation Mechanisms. A Model-Based Argumentation

et al. 2009

View full text Add to dashboard Cite

This paper addresses two questions in the context of neuronal networks dynamics, using methods from dynamical systems theory and statistical physics: (i) How to characterize the statistical properties of sequences of action potentials ("spike trains") produced by neuronal networks ? and; (ii) what are the effects of synaptic plasticity on these statistics ? We introduce a framework in which spike trains are associated to a coding of membrane potential trajectories, and actually, constitute a symbolic coding in important explicit examples (the so-called gIF models). On this basis, we use the thermodynamic formalism from ergodic theory to show how Gibbs distributions are natural probability measures to describe the statistics of spike trains, given the empirical averages of prescribed quantities. As a second result, we show that Gibbs distributions naturally arise when considering "slow" synaptic plasticity rules where the characteristic time for synapse adaptation is quite longer than the characteristic time for neurons dynamics.

show abstract

“…Action potential initiation is driven by activation of sodium channels and prevented by activation of potassium channels. For a stability analysis and a complete two-parameter bifurcation analysis of the Hodgkin-Huxley model see Guckenheimer and Labouriau (1993) and Hassard (1978).…”

Section: Appendix a Computational Methods For Channel Noisementioning

confidence: 99%

Characterizing spiking in noisy type II neurons

Boďová

Paydarfar

Forger³

2015

Journal of Theoretical Biology

View full text Add to dashboard Cite

Understanding the dynamics of noisy neurons remains an important challenge in neuroscience. Here, we describe a simple probabilistic model that accurately describes the firing behavior in a large class (type II) of neurons. To demonstrate the usefulness of this model, we show how it accurately predicts the interspike interval (ISI) distributions, bursting patterns and mean firing rates found by: (1) simulations of the classic Hodgkin-Huxley model with channel noise, (2) experimental data from squid giant axon with a noisy input current and (3) experimental data on noisy firing from a neuron within the suprachiasmatic nucleus (SCN). This simple model has 6 parameters, however, in some cases, two of these parameters are coupled and only 5 parameters account for much of the known behavior. From these parameters, many properties of spiking can be found through simple calculation. Thus, we show how the complex effects of noise can be understood through a simple and general probabilistic model.

show abstract

Bifurcation of the Hodgkin and Huxley equations: A new twist

Cited by 86 publications

References 14 publications

Chaos in the Hodgkin--Huxley Model

Chaos in the Hodgkin--Huxley Model

How Gibbs Distributions May Naturally Arise from Synaptic Adaptation Mechanisms. A Model-Based Argumentation

Characterizing spiking in noisy type II neurons

Contact Info

Product

Resources

About