Class of Exactly Solvable Models of Excitable Media

Abstract

We introduce a class of exactly solvable reaction diffusion models of excitable media with nondiffusive control kinetics and the source term in the diffusion equation depending only parametrically on the control variable. A pulse solution can be found in the entire domain without any use of singular perturbation theory. We reduce the nonlinear eigenvalue problem for a steadily propagating onedimensional pulse to a set of transcendental equations which can be compactly solved analytically within any power of the smallness parameter´. [S0031-9007 (98)06442-4] PACS numbers: 82.40.Ck, 87.10. + eExcitation waves in excitable media are commonly described by a set of reaction diffusion equations (RDEs) [1]. A minimum set of RDEs consists of a diffusion equation for a variable u with a nonlinear source f͑u, y͒ depending also on a control variable y that obeys a local kinetic equation ᠨ y ´g͑u, y͒ where a smallness parameter´represents the ratio of the time scales on which u and y vary. The most basic features of a reaction diffusion (RD) system are revealed through analysis of steadily propagating plane excitation waves. For´ø 1, a propagating excitation usually occurs on two different time scales and hence can be viewed as a sequence of three processes (i) formation of the front in which the system passes to a new metastable state, (ii) a slow formation of the top of the wave, and (iii) the formation of the wave back where the medium returns to its ground state. In the singular limit,´ 0, these three stages are uncoupled [1,2]. In the first and third stages the RDEs describe trigger waves representing the wave's front and back, respectively. In the second stage evolving in the slow time´t diffusion effects are small. Despite the fact that the trigger wave solutions of the RDEs are well known [1-3] only a few recent studies consider coupled singular solutions for 1D plane wave dynamics [4] and 2D curved waves kinematics [5,6]. However, further progress faces the difficulty that the processes on the wave's front appear to depend on those on the wave's back. This influence appears crucial for evolution and stability of complex wave patterns. Most of the studies of excitation waves have been done in the singular limit in which the wave's structure is not resolved and its intrinsic governing parameters remain indefinite. It has recently been shown that it is these parameters that determine the most important dynamical instabilities, such as the transition to meandering of a spiral wave and detachment of the wave's tip from an unexcitable wall (see [7]). In this Letter we introduce a class of models of excitable media with parametric dependence of the RDEs source term on the slow control variable. For such models the exact solution for a steadily propagating wave can be found in the entire domain in a closed form. In contrast to existing models in which resolution of the wave structure has proven elusive, the proposed class of models allows one to explicitly find the wavelength in a wide range of propagation velocities includ...

Critical scale of propagation influences dynamics of waves in a model of excitable medium

“…Steady state restitution curves for two values of excitation threshold v r = 0.25 (upper curve) and v r = 0.31 (lower curve). Insert A shows the analytically determined dependence of T h on excitation threshold v r for stable solitary pulse [ 29 ]. Critical level of = 0.359 beyond which no stable solitary pulse exists is indicated by the dashed line.…”

“…Insert A portrays the analytical dependence of T h on excitation threshold v r for a stable solitary pulse [ 29 ]. It also shows the critical level of excitation threshold ( = 0.359, vertical dashed line) beyond which no stable solitary pulse exists.…”

“…Basic equations that describe a class of exactly solvable models for excitable media have been defined in [ 29 , 30 ]. Here we introduce a modification of this analytical model by adjusting the excitation threshold, v r , in response to changes in frequency of external pacing.…”

“…In contrast to the complexity of the majority of existing reaction-diffusion models, the CSC model employs just a few fundamental parameters and is analytically solvable. It offers a rigorous criterion for determining the stability of excitation wave propagation in an excitable cable [ 29 , 30 ].…”

“…We demonstrate that in a wide range of excitation thresholds, v r , and pacing rates, T m , regardless of the particular values of slopes of restitution curves and rates of adaptation of v r (more or less memory), the loss of stability of waves in an excitable cable of finite length is determined by proximity of the minimal level of the recovery variable, v , at the foot of the action potential and the critical excitation threshold of the solitary pulse computed analytically in [ 29 ].…”