Dynamics of one- and two-dimensional fronts in a bistable equation with time-delayed global feedback : propagation failure and control mechanisms

We study the evolution of fronts in a bistable equation with time-delayed global feedback in the fast reaction and slow diffusion regime. This equation generalizes the Hodgkin-Grafstein and Allen-Cahn equations. We derive a nonlinear equation governing the motion of fronts, which includes a term with delay. In the one-dimensional case this equation is linear. We study the motion of one- and two-dimensional fronts, finding a much richer dynamics than for the previously studied cases (without time-delayed global feedback). We explain the mechanism by which localized fronts created by inhibitory global coupling loose stability in a Hopf bifurcation as the delay time increases. We show that for certain delay times, the prevailing phase is different from that corresponding to the system in the absence of global coupling. Numerical simulations of the partial differential equation are in agreement with the analytical predictions

Low-dimensional models of single neurons: A review

The classical Hodgkin-Huxley (HH) point-neuron model of action potential generation is four-dimensional. It consists of four ordinary differential equations describing the dynamics of the membrane potential and three gating variables associated to a transient sodium and a delayed-rectifier potassium ionic currents. Conductance-based models of HH type are higher-dimensional extensions of the classical HH model. They include a number of supplementary state variables associated with other ionic current types, and are able to describe additional phenomena such as sub-threshold oscillations, mixed-mode oscillations (subthreshold oscillations interspersed with spikes), clustering and bursting. In this manuscript we discuss biophysically plausible and phenomenological reduced models that preserve the biophysical and/or dynamic description of models of HH type and the ability to produce complex phenomena, but the number of effective dimensions (state variables) is lower. We describe several representative models. We also describe systematic and heuristic methods of deriving reduced models from models of HH type

Front motion for phase transitions in systems with memory

We consider the Allen-Cahn equations with memory (a partial integro-differential convolution equation). The prototype kernels are exponentially decreasing functions of time and they reduce the integrodifferential equation to a hyperbolic one, the damped Klein-Gordon equation. By means of a formal asymptotic analysis we show that to the leading order and under suitable assumptions on the kernels, the integro-differential equation behave like a hyperbolic partial differential equation obtained by considering prototype kernels: the evolution of fronts is governed by the extended, damped Born-Infeld equation. We also apply our method to a system of partial integro-differential equations which generalize the classical phase field equations with a non-conserved order parameter and describe the process of phase transitions where memory effects are present

Ramped-induced states in a parametrically driven Ginzburg-Landau equation

We introduce a parametrically driven Ginzburg-Landau (GL) model, which admits a gradient representation, and is subcritical in the absence of the parametric drive (PD). In the case when PD acts uniformly in space, this model has a stable kink solution. A nontrivial situation takes places when PD is itself subject to a kink-like spatial modulation, so that it selects real and imaginary constant solutions at +infinity and -infinity. In this situation, we find stationary solutions numerically, and also analytically for a particular case. They seem to be of two different types, viz., a pair of kinks in the real and imaginary components, or the same with an extra kink inserted into each component, but we show that both belong to a single continuous family of solutions. The family is parametrized by the coordinate of a point at which the extra kinks are inserted. Solutions with more than one kink inserted into each component do not exist. Simulations show that the former solution is always stable, and the latter one is, in a certain sense, neutrally stable, as there is a special type of small perturbations that remain virtually constant in time, rather than decaying or growing (they eventually decay, but extremely slowly).Comment: A latex text file and 8 ps files with figures. Physics Letters A, in pres