Bifurcation of periodic solutions of the Hodgkin-Huxley model for the squid giant axon

Hassard, Brian

doi:10.1016/0022-5193(78)90168-6

Cited by 160 publications

(98 citation statements)

References 4 publications

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“…For the squid giant axon parameters, the corresponding, non-driven dynamics possesses a single fixed point and does not exhibit a spiking activity in the absence of external stimulus, I ext (t) = 0. However, if a constant stimulus, I ext , is applied, the fixed point loses its stability with increasing strength I ext upon I ext P I 2 % 9.763 lA/cm 2 [24,25]. For such a super-threshold current strength, the membrane exhibits a periodic spiking activity which reflects the presence of a stable limit cycle, see Fig.…”

Section: The Hodgkin-huxley Modelmentioning

confidence: 99%

Intrinsic coherence resonance in excitable membrane patches

Schmid¹,

Hänggi²

2007

Mathematical Biosciences

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The influence of intrinsic channel noise on the spiking activity of excitable membrane patches is studied by use of a stochastic generalization of the Hodgkin-Huxley model. Internal noise stemming from the stochastic dynamics of individual ion channels does affect the electric properties of the cell-membrane patches. There exists an optimal size of the membrane patch for which the internal noise alone can cause a nearly regular spontaneous generation of action potentials. We consider the influence of intrinsic channel noise in presence of a constant and an oscillatory current driving for both, the mean interspike interval and the phenomenon of coherence resonance for neuronal spiking. Given small membrane patches, implying that channel noise dominates the excitable dynamics, we find the phenomenon of intrinsic coherence resonance. In this case, the relatively regular spiking behavior becomes essentially independent of an applied stimulus. We observed, however, the occurrence of a skipping of supra-threshold input events due to channel noise for intermediate patch sizes. This effect consequently reduces the overall coherence of the spiking.

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Section: The Hodgkin-huxley Modelmentioning

confidence: 99%

Intrinsic coherence resonance in excitable membrane patches

Schmid¹,

Hänggi²

2007

Mathematical Biosciences

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“…the so-called rest state for I i;ext < I 1 % 6:26 lA=cm 2 , (ii) a stable spiking solution for I i;ext > I 2 % 9:763 lA=cm 2 and (iii) a bistable regime for which the stable rest state and a stable oscillatory spiking solution coexist, i.e. for I 1 < I i;ext < I 2 [32][33][34][35][36]. In particular, for I i;ext ¼ 0 the membrane potential is V rest ¼ À65:0 mV.…”

Section: Biophysical Model Setupmentioning

confidence: 99%

In-phase and anti-phase synchronization in noisy Hodgkin–Huxley neurons

Hänggi

Schmid

2013

Mathematical Biosciences

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a b s t r a c tWe numerically investigate the influence of intrinsic channel noise on the dynamical response of delaycoupling 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 antiphase spiking activity. Interestingly, these phase-flips remain robust for strong channel noise and in turn cause a striking stabilization of the spiking frequency.

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“…Several authors have been attracted by this geometric aspect of the center manifold and have treated a partial differential equation which exhibits regular Hopf bifurcation by first reducing it to an ordinary differential equation in the center manifold. As examples of this approach we mention the investigation of periodic solutions in the HodgkinHuxley model [15] and the periodic motion of a tube carrying a fluid [43]. Also, the approach of the Hopf bifurcation in [5,30] is along this line.…”

mentioning

confidence: 99%

“…where the /" are homogeneous polynomials of degree n, and then equating corresponding powers of ux we obtain a unique formal Taylor expansion of a; for a discussion of this procedure as well as the numerical implementation, see [15,16,23]. The number N can be taken as high as the differentiability of nt permits (i = 1,2).…”

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confidence: 99%

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Properties of center manifolds

Sijbrand¹

1985

Trans. Amer. Math. Soc.

137

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Abstract.The center manifold has a number of puzzling properties associated with the basic questions of existence, uniqueness, differentiability and analyticity which may cloud its profitable application in e.g. bifurcation theory. This paper aims to deal with some of these subtle properties.Regarding existence and uniqueness, it is shown that the cut-off function appearing in the usual existence proof is responsible for the selection of a single center manifold, thereby hiding the inherent nonuniqueness. Conditions are given for different center manifolds at an equilibrium point to have a nonempty intersection. This intersection will include at least the families of stationary and periodic solutions crossing through the equilibrium. In the case of nonuniqueness the difference between any two center manifolds will be less than c, exp(c2A_1) with c, and c2 constants, which explains why the formal Taylor expansions of different center manifolds are the same, while the expansions do not converge.The differentiability of a center manifold will in certain cases decrease when moving out of the origin and a simple example shows how the differentiability may be lost.Center manifolds are usually not analytic; however, an analytic manifold may exist which contains all periodic solutions of a certain type but which may otherwise not be invariant. Using this manifold, a new and very simple proof of the Lyapunov subcenter theorem is given.

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Bifurcation of periodic solutions of the Hodgkin-Huxley model for the squid giant axon

Cited by 160 publications

References 4 publications

Intrinsic coherence resonance in excitable membrane patches

Intrinsic coherence resonance in excitable membrane patches

In-phase and anti-phase synchronization in noisy Hodgkin–Huxley neurons

Properties of center manifolds

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