### Abstract

In many physiologically important excitable systems, such as intracellular calcium dynamics, the diffusing variable is highly buffered. In addition, all physiological buffered excitable systems contain multiple buffers, with different affinities. It is thus important to understand theproperties of wave solutions in excitable systems with multiple buffers, and to understand how multiple buffers interact. Under the assumption that buffering acts on a fast time scale, we derivea criterion for the existence of a traveling pulse with positive wave speed in the buffered FitzHugh-Nagumo model, a prototypical excitable system. This condition suggests that there exists a criticalexcitability corresponding to the excitability parameter ac such that, for systems with excitability above this critical excitability (the excitability parameter a ε (0, ac)), buffers cannot prevent the propagation of traveling pulses with positive wave speed, provided that the parameter ε Lt; 1. Further, buffers can speed up wave propagation if the diffusivity of the buffer increases. On the other hand, for systems with excitability below this critical excitability (the excitability parameter a ε (ac, 1/2)), we can find a critical dissociation constant (K = K(a)) such that buffers can be classified intotwo types: weak buffers (K ε (K(a), ∞)) and strong buffers (K ε (0, K(a))). It turns out thatthe wave properties are strongly affected by competition between strong buffers and weak buffers. Weak buffers not only can help the existence of waves, but also can speed up wave propagation iftheir diffusivity increases. In contrast, strong buffers can eliminate calcium waves if the product oftheir diffusivity and total concentration exceeds some critical value. Moreover, as the diffusivity ofthe strong buffer increases to some critical value, the waves slow down to zero. Finally, adding asufficiently large amount of buffer, either strong or weak, can eliminate the wave.

Original language | English |
---|---|

Pages (from-to) | 1606-1636 |

Number of pages | 31 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 71 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2011 Nov 14 |

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### Keywords

- Buffers
- Calcium
- FitzHugh-Nagumo equations
- Traveling pulse

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

*SIAM Journal on Applied Mathematics*,

*71*(5), 1606-1636. https://doi.org/10.1137/110820348

**Traveling waves in the buffered fitzhugh-nagumomodel.** / Tsai, Je Chiang; Sneyd, James.

Research output: Contribution to journal › Article

*SIAM Journal on Applied Mathematics*, vol. 71, no. 5, pp. 1606-1636. https://doi.org/10.1137/110820348

}

TY - JOUR

T1 - Traveling waves in the buffered fitzhugh-nagumomodel

AU - Tsai, Je Chiang

AU - Sneyd, James

PY - 2011/11/14

Y1 - 2011/11/14

N2 - In many physiologically important excitable systems, such as intracellular calcium dynamics, the diffusing variable is highly buffered. In addition, all physiological buffered excitable systems contain multiple buffers, with different affinities. It is thus important to understand theproperties of wave solutions in excitable systems with multiple buffers, and to understand how multiple buffers interact. Under the assumption that buffering acts on a fast time scale, we derivea criterion for the existence of a traveling pulse with positive wave speed in the buffered FitzHugh-Nagumo model, a prototypical excitable system. This condition suggests that there exists a criticalexcitability corresponding to the excitability parameter ac such that, for systems with excitability above this critical excitability (the excitability parameter a ε (0, ac)), buffers cannot prevent the propagation of traveling pulses with positive wave speed, provided that the parameter ε Lt; 1. Further, buffers can speed up wave propagation if the diffusivity of the buffer increases. On the other hand, for systems with excitability below this critical excitability (the excitability parameter a ε (ac, 1/2)), we can find a critical dissociation constant (K = K(a)) such that buffers can be classified intotwo types: weak buffers (K ε (K(a), ∞)) and strong buffers (K ε (0, K(a))). It turns out thatthe wave properties are strongly affected by competition between strong buffers and weak buffers. Weak buffers not only can help the existence of waves, but also can speed up wave propagation iftheir diffusivity increases. In contrast, strong buffers can eliminate calcium waves if the product oftheir diffusivity and total concentration exceeds some critical value. Moreover, as the diffusivity ofthe strong buffer increases to some critical value, the waves slow down to zero. Finally, adding asufficiently large amount of buffer, either strong or weak, can eliminate the wave.

AB - In many physiologically important excitable systems, such as intracellular calcium dynamics, the diffusing variable is highly buffered. In addition, all physiological buffered excitable systems contain multiple buffers, with different affinities. It is thus important to understand theproperties of wave solutions in excitable systems with multiple buffers, and to understand how multiple buffers interact. Under the assumption that buffering acts on a fast time scale, we derivea criterion for the existence of a traveling pulse with positive wave speed in the buffered FitzHugh-Nagumo model, a prototypical excitable system. This condition suggests that there exists a criticalexcitability corresponding to the excitability parameter ac such that, for systems with excitability above this critical excitability (the excitability parameter a ε (0, ac)), buffers cannot prevent the propagation of traveling pulses with positive wave speed, provided that the parameter ε Lt; 1. Further, buffers can speed up wave propagation if the diffusivity of the buffer increases. On the other hand, for systems with excitability below this critical excitability (the excitability parameter a ε (ac, 1/2)), we can find a critical dissociation constant (K = K(a)) such that buffers can be classified intotwo types: weak buffers (K ε (K(a), ∞)) and strong buffers (K ε (0, K(a))). It turns out thatthe wave properties are strongly affected by competition between strong buffers and weak buffers. Weak buffers not only can help the existence of waves, but also can speed up wave propagation iftheir diffusivity increases. In contrast, strong buffers can eliminate calcium waves if the product oftheir diffusivity and total concentration exceeds some critical value. Moreover, as the diffusivity ofthe strong buffer increases to some critical value, the waves slow down to zero. Finally, adding asufficiently large amount of buffer, either strong or weak, can eliminate the wave.

KW - Buffers

KW - Calcium

KW - FitzHugh-Nagumo equations

KW - Traveling pulse

UR - http://www.scopus.com/inward/record.url?scp=80755130502&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80755130502&partnerID=8YFLogxK

U2 - 10.1137/110820348

DO - 10.1137/110820348

M3 - Article

AN - SCOPUS:80755130502

VL - 71

SP - 1606

EP - 1636

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 5

ER -