Traveling Waves in a Simplified Model of Calcium Dynamics

Abstract

We analyze traveling wave propagation in a simplified model of intracellular calcium dynamics. Despite its simplicity, the model is thought to capture fundamental features of wave propagation in calcium models. We explore aspects of the dynamics of traveling front, pulse, and periodic wave solutions as $J$, a parameter in our model, is varied. We focus on the closed-cell version of the model, which corresponds to a singular limit of the full (open-cell) model. We use our results about the closed-cell model to make conjectures about the nature of wave solutions in the open-cell version of the model. A comparison between the properties of wave solutions of the calcium model and wave solutions of the FitzHugh--Nagumo equations reveals that the calcium model is an excitable system essentially different from the FitzHugh--Nagumo equations. Our analysis suggests that there are two regimes in which the closed-cell model has traveling fronts. In the regime with lower values of $J$ there are two families of traveling fronts, each parametrized by $J$: one with wave speed $s_F(J)$ and one with wave speed $s_B(J)$. For $J$ such that $s_F(J)>s_B(J)$, there is a unique (up to a translation) traveling pulse with wave speed $s_P(J)\in(s_B(J),s_F(J))$, while for $J$ such that $s_F(J)\leq s_B(J)$ there is no traveling pulse. In the regime with higher values of $J$ there are analogous families of traveling fronts and pulses, but in this case the traveling pulses exist when $s_F(J)<s_B(J)$. The stability of the traveling fronts and pulses identified is investigated using the Evans function. The traveling front with wave speed $s_F(J)$ is always stable, while for the traveling front with wave speed $s_B(J)$, we can find numerically a Hopf bifurcation at $s=s_B(J_{\mathrm{HP}})$ dividing the curve $s=s_B(J)$ into stable and unstable sections. The traveling pulse is unstable when it exists. The existence of periodic waves is also investigated.