Bifurcations in a modulation equation for alternans in a cardiac fiber

Abstract

Abstract. While alternans in a single cardiac cell appears through a simple period-doubling bifurcation, in extended tissue the exact nature of the bifurcation is unclear. In particular, the phase of alternans can exhibit wave-like spatial dependence, either stationary or travelling, which is known as discordant alternans. We study these phenomena in simple cardiac models through a modulation equation proposed by Echebarria-Karma. As shown in our previous paper, the zero solution of their equation may lose stability, as the pacing rate is increased, through either a Hopf or steady-state bifurcation. Which bifurcation occurs first depends on parameters in the equation, and for one critical case both modes bifurcate together at a degenerate (codimension 2) bifurcation. For parameters close to the degenerate case, we investigate the competition between modes, both numerically and analytically. We find that at sufficiently rapid pacing (but assuming a 1:1 response is maintained), steady patterns always emerge as the only stable solution. However, in the parameter range where Hopf bifurcation occurs first, the evolution from periodic solution (just after the bifurcation) to the eventual standing wave solution occurs through an interesting series of secondary bifurcations.Mathematics Subject Classification. 35B32, 92C30.

Spatiotemporal dynamics of calcium-driven cardiac alternans

“…To date, our basic theoretical understanding of SDA is well developed primarily for the case where alternans is "voltagedriven" [1,[22][23][24][25], i.e., originate from an instability of the V m dynamics. For a one-dimensional cable of length L, the V m dynamics is governed by the well-known cable equation…”

“…where Λ is a lengthscale related to the slope of the CV restitution curve, w is a short lengthscale ∼ D V /cv(D * ), τ bcl is the pacing period or basic cycle length (BCL), and σ and χ can be expressed in terms of derivatives of the APD restitution curve evaluated at the fixed point, and σ measures the distance from the bifurcation point. Analysis of this amplitude equation has yielded a fundamental understanding of the formation of SDA in terms of a linear instability forming periodic wave patterns [22][23][24][25]. Depending on the relative magnitude of ξ, w and Λ, SDA formation is associated with a bifurcation to standing waves with stationary nodes and a wavelength λ s ∼ (wΛ) 1/2 or travel-ing waves with moving nodes and λ s ∼ (ξ 2 Λ) 1/3 [22,23].…”

Analytical and numerical bifurcation analysis of a forest ecosystem model with human interaction

“…not just temporal numerical simulations for the verification of the analytical findings, which do not suffice to identify and trace branches in the parameter space of bifurcation points such as limit points of limit cycles), thus to study the dynamics in a combined and complete way (see e.g. [5,10,14,20,32,33,36,37]).…”