Perturbation analysis of an approximation to the Hodgkin-Huxley theory

Casten, Richard G.; Cohen, Hirsh; Lagerstrom, P. A.

doi:10.1090/qam/445095

Cited by 149 publications

(58 citation statements)

References 12 publications

(3 reference statements)

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“…(1) and (2) admit solutions only in the form of traveling waves (autowaves) [2,3,5,8,10,12,28,29], and when ǫ ≪ 1 and α > ∼ 1 (KN systems) they admit only solutions in the form of static patterns, the simplest of which are AS in the form of solitary spots and stripes of high or low values of the activator surrounded by the "sea" of low or high values of the activator, respectively ("hot" and "cold" AS) [11,12]. Notice that because of the monostability of the considered system the radius of a spot or the width of a stripe cannot be greater than certain value of order one [11,12].…”

Section: Scenarios Of Pattern Formation: Results Of the Simulationsmentioning

confidence: 99%

Scenarios of domain pattern formation in a reaction-diffusion system

Muratov

Осипов²

1996

Phys. Rev. E

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We performed an extensive numerical study of a two-dimensional reaction-diffusion system of the activator-inhibitor type in which domain patterns can form. We showed that both multidomain and labyrinthine patterns may form spontaneously as a result of Turing instability. In the stable homogeneous system with the fast inhibitor one can excite both localized and extended patterns by applying a localized stimulus. Depending on the parameters and the excitation level of the system stripes, spots, wriggled stripes, or labyrinthine patterns form. The labyrinthine patterns may be both connected and disconnected. In the the stable homogeneous system with the slow inhibitor one can excite self-replicating spots, breathing patterns, autowaves and turbulence. The parameter regions in which different types of patterns are realized are explained on the basis of the asymptotic theory of instabilities for patterns with sharp interfaces developed by us in Phys. Rev. E. 53, 3101 (1996). The dynamics of the patterns observed in our simulations is very similar to that of the patterns forming in the ferrocyanide-iodate-sulfite reaction. PACS number(s): 05.70. Ln, 82.20.Mj, 47.54.+r

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Section: Scenarios Of Pattern Formation: Results Of the Simulationsmentioning

confidence: 99%

Scenarios of domain pattern formation in a reaction-diffusion system

Muratov

Осипов²

1996

Phys. Rev. E

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“…To do so, it is convenient to approximate the traveling pulse solution by a periodic orbit of large period so that we can use certain theoretical results obtained for adjoint solutions to such periodic solutions. Previous work has established that the singular perturbation construction of traveling pulses to a reaction-diffusion analogue of (3.1) on the real line, in the absence of stimulation, generalizes directly to give the existence of a periodic solution on a finite spatial domain with periodic boundary conditions [4]. Indeed, the argument for the existence of traveling pulses in [20] shows how to generalize the construction in [4] to the unstimulated form of (3.1), and the extension to the periodic case follows immediately.…”

Section: Behavior Of the Solution To The Adjoint Equationmentioning

confidence: 99%

Stimulus-Driven Traveling Solutions in Continuum Neuronal Models with a General Smooth Firing Rate Function

Ermentrout

Jalics

Rubin

2010

SIAM J. Appl. Math.

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Abstract. We examine the existence of traveling wave solutions for a continuum neuronal network modeled by integro-differential equations. First, we consider a scalar field model with a general smooth firing rate function and a spatiotemporally varying stimulus. We prove that a traveling front solution that is locked to the stimulus exists for a certain interval of stimulus speeds. Next, we include a slow adaptation equation and obtain a formula, which involves a certain adjoint solution, for the stimulus speeds that induce locked traveling pulse solutions. Further, we use singular perturbation analysis to characterize an approximation to the adjoint solution that we compare to a numerically computed adjoint. Numerical simulations are used to illustrate the traveling fronts and pulses that we study and to make comparisons with our analytically computed bounds for stimulus-locked wave behavior.

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“…In [5], a similar symmetry argument is used to obtain an appropriate trailing back solution to the Fitzhugh-Nagumo equations. More generally, however, it may happen that, as V increases along the g + (V ) branch, the cusp of the sigmoid is encountered before a drop point is found for which the back speed matches that of the front.…”

Section: Singular Perturbation Constructionmentioning

confidence: 99%

“…Examined extensively in the form of the reaction-diffusion equations, traveling fronts (see [37], [22], [21]), pulses (see [5], [30], [29]), standing patterns (see [16], [53]), and other spatial structures [60], [36] (see [55] for review) have each been analytically demonstrated to exist as solutions.…”

Section: Singular Perturbation Constructionmentioning

confidence: 99%

Spatially Structured Activity in Synaptically Coupled Neuronal Networks: I. Traveling Fronts and Pulses

Pinto¹,

Ermentrout²

2001

SIAM J. Appl. Math.

290

407

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Abstract. We consider traveling front and pulse solutions to a system of integro-differential equations used to describe the activity of synaptically coupled neuronal networks in a single spatial dimension. Our first goal is to establish a series of direct links between the abstract nature of the equations and their interpretation in terms of experimental findings in the cortex and other brain regions. This is accomplished first by presenting a biophysically motivated derivation of the system and then by establishing a framework for comparison between numerical and experimental measures of activity propagation speed. Our second goal is to establish the existence of traveling pulse solutions using more rigorous methods. Two techniques are presented. The first, a shooting argument, reduces the problem from finding a specific solution to an integro-differential equation system to finding any solution to an ODE system. The second, a singular perturbation argument, provides a construction of traveling pulse solutions under more general conditions.

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Perturbation analysis of an approximation to the Hodgkin-Huxley theory

Cited by 149 publications

References 12 publications

Scenarios of domain pattern formation in a reaction-diffusion system

Scenarios of domain pattern formation in a reaction-diffusion system

Stimulus-Driven Traveling Solutions in Continuum Neuronal Models with a General Smooth Firing Rate Function

Spatially Structured Activity in Synaptically Coupled Neuronal Networks: I. Traveling Fronts and Pulses

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