Asymptotic Analysis and Analytical Solutions of a Model of Cardiac Excitation

Abstract

We describe an asymptotic approach to gated ionic models of single-cell cardiac excitability. It has a form essentially different from the Tikhonov fast-slow form assumed in standard asymptotic reductions of excitable systems. This is of interest since the standard approaches have been previously found inadequate to describe phenomena such as the dissipation of cardiac wave fronts and the shape of action potential at repolarization. The proposed asymptotic description overcomes these deficiencies by allowing, among other non-Tikhonov features, that a dynamical variable may change its character from fast to slow within a single solution. The general asymptotic approach is best demonstrated on an example which should be both simple and generic. The classical model of Purkinje fibers (Noble in J. Physiol. 160:317-352, 1962) has the simplest functional form of all cardiac models but according to the current understanding it assigns a physiologically incorrect role to the Na current. This leads us to suggest an "Archetypal Model" with the simplicity of the Noble model but with a structure more typical to contemporary cardiac models. We demonstrate that the Archetypal Model admits a complete asymptotic solution in quadratures. To validate our asymptotic approach, we proceed to consider an exactly solvable "caricature" of the Archetypal Model and demonstrate that the asymptotic of its exact solution coincides with the solutions obtained by substituting the "caricature" right-hand sides into the asymptotic solution of the generic Archetypal Model. This is necessary, because, unlike in standard asymptotic descriptions, no general results exist which can guarantee the proximity of the non-Tikhonov asymptotic solutions to the solutions of the corresponding detailed ionic model.

Predicting critical ignition in slow-fast excitable models

“…Continuing the rest state for increasing current forcing J connects to a Hopf bifurcation, from which a family of periodic orbits emanate. Continuing this family of periodic orbits to large periods with c = 0 followed by decreasing current forcing yields an unforced periodic orbit of the autonomous system, (20), equivalently a traveling wave solution of (1) with periodic boundary conditions on a domain x ∈ [0, L). To compute asymptotic traveling wave solutions of (1) we continue the unforced periodic orbit in the (L, c)-plane.…”

“…The periodic critical solution is computed on an adaptive collocation grid using Auto [6], at large γ and interpolated onto a Chebyshev grid of size M × DA representing M Chebyshev modes and a dealiasing factor DA ≥ 1. A nonlinear boundary value problem is constructed which corresponds to equations (20) and projection boundary conditions. The projection boundary conditions require the eigenvectors of the Jacobian evaluated at the rest state.…”

“…Straightforward extension of this study would be to slow-fast systems with more than one fast and/or more than one slow component with similar, Tikhonov type occurrence of the small parameter. A more intriguing possibility is about asymptotics in case of non-Tikhonov slow-fast models, of the kind discussed in [19][20][21]. The difference from systems with the Tikhonov structure is that the asymptotic limit of the critical solution is not a stationary "nucleus", but a moving front.…”

Cardiac cell modelling: Observations from the heart of the cardiac physiome project

“…However, to reproduce the morphology of spike-and-dome action potentials, an additional variable is required Fenton 2004, Bueno-Orovio et al 2008). These models can be obtained by asymptotic approaches different from the standard fast-slow reductions used in simplified models (Biktashev et al 2008). …”

Nonlinear Dynamics in Cardiology

“…In order to lower model complexity and/or computational costs (in particular for three-dimensional tissue simulations), reduced models with fewer variables and parameters have also been developed, e.g., Refs. (46, 47). …”