883 research outputs found
Spatially Discrete FitzHugh-Nagumo Equations
This is the published version, also available here: http://dx.doi.org/10.1137/S003613990343687X.We consider pulse and front solutions to a spatially discrete FitzHugh--Nagumo equation that contains terms to represent both depolarization and hyperpolarization of the nerve axon. We demonstrate a technique for deriving candidate solutions for the McKean nonlinearity and present and apply solvability conditions necessary for existence. Our equation contains both spatially continuous and discrete diffusion terms
Pulse propagation in discrete systems of coupled excitable cells
Propagation of pulses in myelinated fibers may be described by appropriate
solutions of spatially discrete FitzHugh-Nagumo systems. In these systems,
propagation failure may occur if either the coupling between nodes is not
strong enough or the recovery is too fast. We give an asymptotic construction
of pulses for spatially discrete FitzHugh-Nagumo systems which agrees well with
numerical simulations and discuss evolution of initial data into pulses and
pulse generation at a boundary. Formulas for the speed and length of pulses are
also obtained.Comment: 16 pages, 10 figures, to appear in SIAM J. Appl. Mat
Standing Waves Of Spatially Discrete Fitzhugh-nagumo Equations
We study a system of spatially discrete FitzHugh-Nagumo equations, which are nonlinear differential-difference equations on an infinite one-dimensional lattice. These equations are used as a model of impulse propagation in nerve cells. We employ McKean\u27s caricature of the cubic as our nonlinearity, which allows us to reduce the nonlinear problem into a linear inhomogeneous problem. We find exact solutions for standing waves, which are steady states of the system. We derive formulas for all 1-pulse solutions. We determine the range of parameter values that allow for the existence of standing waves. We use numerical methods to demonstrate the stability of our solutions and to investigate the relationship between the existence of standing waves and propagation failure of traveling waves
Wave trains, self-oscillations and synchronization in discrete media
We study wave propagation in networks of coupled cells which can behave as
excitable or self-oscillatory media. For excitable media, an asymptotic
construction of wave trains is presented. This construction predicts their
shape and speed, as well as the critical coupling and the critical separation
of time scales for propagation failure. It describes stable wave train
generation by repeated firing at a boundary. In self-oscillatory media, wave
trains persist but synchronization phenomena arise. An equation describing the
evolution of the oscillator phases is derived.Comment: to appear in Physica D: Nonlinear Phenomen
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