Traveling waves in a simplified model of calcium dynamics

Je Chiang Tsai, Wenjun Zhang, Vivien Kirk, James Sneyd

Research output: Contribution to journalArticle

8 Citations (Scopus)

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 sF (J) and one with wave speed sB(J). For J such that sF (J) > sB(J), there is a unique (up to a translation) traveling pulse with wave speed sP (J) € sB(J), sF (J)), while for J such that sF (J) = sB(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 sF (J) > sB(J). The stability of the traveling fronts and pulses identified is investigated using the Evans function. The traveling front with wave speed sF (J) is always stable, while for the traveling front with wave speed sB(J), we can find numerically a Hopf bifurcation at s = sB(JHP) dividing the curve s = sB(J) into stable and unstable sections. The traveling pulse is unstable when it exists. The existence of periodic waves is also investigated.

Original languageEnglish
Pages (from-to)1149-1199
Number of pages51
JournalSIAM Journal on Applied Dynamical Systems
Volume11
Issue number4
DOIs
Publication statusPublished - 2012 Dec 26

Fingerprint

Calcium
Traveling Wave
Travelling Fronts
Wave Speed
Cell
FitzHugh-Nagumo Equations
Model
Closed
Wave propagation
Wave Propagation
Unstable
Evans Function
Excitable Systems
Periodic Wave Solution
Singular Limit
Periodic Wave
Hopf bifurcation
Hopf Bifurcation
Simplicity
Curve

Keywords

  • Calcium dynamics
  • FitzHugh-Nagumo
  • Reaction-diffusion system
  • Stability
  • Traveling waves

ASJC Scopus subject areas

  • Analysis
  • Modelling and Simulation

Cite this

Traveling waves in a simplified model of calcium dynamics. / Tsai, Je Chiang; Zhang, Wenjun; Kirk, Vivien; Sneyd, James.

In: SIAM Journal on Applied Dynamical Systems, Vol. 11, No. 4, 26.12.2012, p. 1149-1199.

Research output: Contribution to journalArticle

Tsai, Je Chiang ; Zhang, Wenjun ; Kirk, Vivien ; Sneyd, James. / Traveling waves in a simplified model of calcium dynamics. In: SIAM Journal on Applied Dynamical Systems. 2012 ; Vol. 11, No. 4. pp. 1149-1199.
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AB - 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 sF (J) and one with wave speed sB(J). For J such that sF (J) > sB(J), there is a unique (up to a translation) traveling pulse with wave speed sP (J) € sB(J), sF (J)), while for J such that sF (J) = sB(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 sF (J) > sB(J). The stability of the traveling fronts and pulses identified is investigated using the Evans function. The traveling front with wave speed sF (J) is always stable, while for the traveling front with wave speed sB(J), we can find numerically a Hopf bifurcation at s = sB(JHP) dividing the curve s = sB(J) into stable and unstable sections. The traveling pulse is unstable when it exists. The existence of periodic waves is also investigated.

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