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Wei-Ping Wang

Bio: Wei-Ping Wang is an academic researcher from University of North Carolina at Chapel Hill. The author has contributed to research in topics: Impulse (physics). The author has an hindex of 1, co-authored 1 publications receiving 31 citations.

Papers
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Journal ArticleDOI
TL;DR: In this article, it was shown that an n-ple impulse solution resembling the superposition of n unstable solitary impulses has at most 2n - 1, and at least n, unstable modes: exactly n unstable modes corresponding to the amplitudes and the rest of them corresponding to spacings.
Abstract: We study McKean's caricature of a nerve conduction equation where H is the Heaviside function. It is proved that an n-ple impulse solution resembling the superposition of n unstable solitary impulses has at most 2n - 1, and at least n, unstable modes: exactly n unstable modes corresponding to the amplitudes and the rest of them corresponding to the spacings. The n amplitude modes always exist. We prove also that for an n-ple impulse solution resembling the superposition of n stable solitary impulses, there are at most n - 1 unstable modes and all of them are of spacing type.

32 citations


Cited by
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MonographDOI
21 Oct 1994
TL;DR: In this article, the authors present a wave propagation model for combustion problems and asymptotics of the speed of combustion waves with complex kinetics, as well as approximate analytical methods in combustion problems.
Abstract: Part I. Stationary waves: Scalar equation Leray-Schauder degree Existence of waves Structure of the spectrum Stability and approach to a wave Part II. Bifurcation of waves: Bifurcation of nonstationary modes of wave propagation Mathematical proofs Part III. Waves in chemical kinetics and combustion: Waves in chemical kinetics Combustion waves with complex kinetics Estimates and asymptotics of the speed of combustion waves Asymptotic and approximate analytical methods in combustion problems (supplement) Bibliography.

880 citations

Journal ArticleDOI
TL;DR: This work focuses on planar piecewise linear models that can mimic the firing response of several different cell types, capable of producing realistic action potential shapes, that can be used as the basis for understanding dynamics at the network level.
Abstract: The presence of gap junction coupling among neurons of the central nervous systems has been appreciated for some time now. In recent years there has been an upsurge of interest from the mathematical community in understanding the contribution of these direct electrical connections between cells to large-scale brain rhythms. Here we analyze a class of exactly soluble single neuron models, capable of producing realistic action potential shapes, that can be used as the basis for understanding dynamics at the network level. This work focuses on planar piece-wise linear models that can mimic the firing response of several different cell types. Under constant current injection the periodic response and phase response curve (PRC) is calculated in closed form. A simple formula for the stability of a periodic orbit is found using Floquet theory. From the calculated PRC and the periodic orbit a phase interaction function is constructed that allows the investigation of phase-locked network states using the theory of weakly coupled oscillators. For large networks with global gap junction connectivity we develop a theory of strong coupling instabilities of the homogeneous, synchronous and splay state. For a piece-wise linear caricature of the Morris-Lecar model, with oscillations arising from a homoclinic bifurcation, we show that large amplitude oscillations in the mean membrane potential are organized around such unstable orbits.

139 citations

Journal ArticleDOI
TL;DR: This paper reviews some of the more popular spiking models of this class and describes the types of spiking pattern that they can generate, and develops a novel technique that can handle the nonsmooth reset of the model upon spiking.

88 citations

Journal ArticleDOI
TL;DR: Within the context of Lienard equations, the FitzHugh--Nagumo model with an idealized nonlinearity is presented with an analytical expression for the transient regime corresponding to the emission of a finite number of action potentials (or spikes) and for the asymptotic regime correspondingto the existence of a limit cycle.
Abstract: Within the context of Lienard equations, we present the FitzHugh--Nagumo model with an idealized nonlinearity. We give an analytical expression (i) for the transient regime corresponding to the emission of a finite number of action potentials (or spikes), and (ii) for the asymptotic regime corresponding to the existence of a limit cycle. We carry out a global analysis to study periodic solutions, the existence of which is linked to the solutions of a system of transcendental equations. The periodic solutions are obtained with the help of the harmonic balance method or as limit behavior of the transient regime. We show how the appearance of periodic solutions corresponds either to a fold limit cycle bifurcation or to a Hopf bifurcation at infinity. The resultsobtained are in agreement with local analysis methods, i.e., the Melnikov method and the averaging method. The generalization of the model leads us to formulate two conjectures concerning the number of limit cycles for the piecewise linear Lienard equ...

69 citations

Journal ArticleDOI
TL;DR: This work investigates the mechanism of abrupt transition between small- and large amplitude oscillations in fast-slow piecewise-linear (PWL) models of FitzHugh–Nagumo (FHN) type, and describes the mechanism that leads to the abrupt, canard-like transition between subthreshold oscillations and spikes.
Abstract: We investigate the mechanism of abrupt transition between small- and large amplitude oscillations in fast-slow piecewise-linear (PWL) models of FitzHugh–Nagumo (FHN) type. In the context of neuroscience, these oscillatory regimes correspond to subthreshold oscillations and action potentials (spikes), respectively. The minimal model that shows such phenomena has a cubic-like nullcline (for the fast equation) with two or more linear pieces in the middle branch and one piece in the left and right branches. Simpler models with only one linear piece in the middle branch or a discontinuity between the left and right branches (McKean model) show a single oscillatory mode. As the number of linear pieces increases, PWL models of FHN type approach smooth FHN-type models. For the minimal model we investigate the bifurcation structure; we describe the mechanism that leads to the abrupt, canard-like transition between subthreshold oscillations and spikes; and we provide an analytical way of predicting the amplitude re...

65 citations