The bifurcation of periodic solutions in the Hodgkin-Huxley equations

Troy, William C.

doi:10.1090/qam/472116

Cited by 47 publications

(12 citation statements)

References 11 publications

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“…As a result, this bifurcation has been scrutinized numerically and analytically by many researches (see e.g. [Hassard, 1978;Troy, 1978;Rinzel & Miller, 1980;Hassard et al, 1981;Holden et al, 1991;Bedrov et al, 1992]). A remarkable historical fact is that many important neuroscience properties, such as all-or-none response, threshold, and integration, have been introduced or illustrated using classical HodgkinHuxley model despite the fact that the model does not exhibit any of these properties, as we see below.…”

Section: Class 2 Excitable Systems Near An Andronov Hopf Bifurcationmentioning

confidence: 99%

Neural Excitability, Spiking and Bursting

Izhikevich

2000

Int. J. Bifurcation Chaos

1,866

1,581

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Bifurcation mechanisms involved in the generation of action potentials (spikes) by neurons are reviewed here. We show how the type of bifurcation determines the neuro-computational properties of the cells. For example, when the rest state is near a saddle-node bifurcation, the cell can fire all-or-none spikes with an arbitrary low frequency, it has a well-defined threshold manifold, and it acts as an integrator ; i.e. the higher the frequency of incoming pulses, the sooner it fires. In contrast, when the rest state is near an Andronov-Hopf bifurcation, the cell fires in a certain frequency range, its spikes are not all-or-none, it does not have a well-defined threshold manifold, it can fire in response to an inhibitory pulse, and it acts as a resonator ; i.e. it responds preferentially to a certain (resonant) frequency of the input. Increasing the input frequency may actually delay or terminate its firing.We also describe the phenomenon of neural bursting, and we use geometric bifurcation theory to extend the existing classification of bursters, including many new types. We discuss how the type of burster defines its neuro-computational properties, and we show that different bursters can interact, synchronize and process information differently.

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Section: Class 2 Excitable Systems Near An Andronov Hopf Bifurcationmentioning

confidence: 99%

Neural Excitability, Spiking and Bursting

Izhikevich

2000

Int. J. Bifurcation Chaos

1,866

1,581

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“…The comparison between the two ®gures highlights that the main dierence between the diagrams of the deterministic and the stochastic systems resides at the level of the transition between the two regimes (low-to-high current and lowto-high noise). While the bifurcation of the deterministic system is well-understood (Troy 1978;Rinzel and Miller 1980), this is hardly the case for the noisy system because the mathematical analysis of such noise-induced transitions is still under active development (Arnold 1998). …”

Section: Noise Induced Transition In the Hodgkin-huxley Modelmentioning

confidence: 99%

Noise-induced transition in excitable neuron models

Tanabe

Pakdaman

2001

Biological Cybernetics

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We studied the influence of noisy stimulation on the Hodgkin-Huxley neuron model. Rather than examining the noise-related variability of the discharge times of the model--as has been done previously--our study focused on the effect of noise on the stationary distributions of the membrane potential and gating variables of the model. We observed that a gradual increase in the noise intensity did not result in a gradual change of the distributions. Instead, we could identify a critical intermediate noise range in which the shapes of the distributions underwent a drastic qualitative change. Namely, they moved from narrow unimodal Gaussian-like shapes associated with low noise intensities to ones that spread widely at large noise intensities. In particular, for the membrane potential and the sodium activation variable, the distributions changed from unimodal to bimodal. Thus, our investigation revealed a noise-induced transition in the Hodgkin-Huxley model. In order to further characterize this phenomenon, we considered a reduced one-dimensional model of an excitable system, namely the active rotator. For this model, our analysis indicated that the noise-induced transition is associated with a deterministic bifurcation of approximate equations governing the dynamics of the mean and variance of the state variable. Finally, we shed light on the possible functional importance of this noise-induced transition in neuronal coding by determining its effect on the spike timing precision in models of neuronal ensembles.

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“…So far, the original HH equations (Plant 1976;Rinzel 1978;Hassard 1978;Troy 1978;Rinzel and Miller 1980;Bedrov et al 1992;Guckenheimer and Labouriau 1993;Fukai et al 2000a,b) and various kinds of HH-type equations (Chay and Rinzel 1985;Alexander and Cai 1991;Canavier et al 1993;Av-Ron 1994;Bertram 1994;Rush and Rinzel 1995;Schweighofer et al 1999) have been analyzed numerically and/or analytically. The importance of diering time scales has already been demonstrated in the production of bursting oscillations in many types of cells.…”

Section: Introductionmentioning

confidence: 99%

Complex nonlinear dynamics of the Hodgkin–Huxley equations induced by time scale changes

Nabetani

Kumagai

2001

Biol Cybern

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The Hodgkin-Huxley equations with a slight modification are investigated, in which the inactivation process (h) of sodium channels or the activation process of potassium channels (n) is slowed down. We show that the equations produce a variety of action potential waveforms ranging from a plateau potential, such as in heart muscle cells, to chaotic bursting firings. When h is slowed down--differently from the case of n variable being slow --chaotic bursting oscillations are observed for a wide range of parameter values although both variables cause a decrease in the membrane potential. The underlying nonlinear dynamics of various action potentials are analyzed using bifurcation theory and a so-called slow-fast decomposition analysis. It is shown that a simple topological property of the equilibrium curves of slow and fast subsystems is essential to the production of chaotic oscillations, and this is the cause of the large difference in global firing characteristics between the h-slow and n-slow cases.

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The bifurcation of periodic solutions in the Hodgkin-Huxley equations

Cited by 47 publications

References 11 publications

Neural Excitability, Spiking and Bursting

Neural Excitability, Spiking and Bursting

Noise-induced transition in excitable neuron models

Complex nonlinear dynamics of the Hodgkin–Huxley equations induced by time scale changes

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