### Abstract

The PitzHugh-Nagumo equations have been used as a caricature of the Hodgkin-Huxley equations of neuron firing and to capture, qualitatively, the general properties of an excitable membrane. In this paper, we utilize a modified version of the FitzHugh-Nagumo equations to model the spatial propagation of neuron firing; we assume that this propagation is (at least, partially) caused by the cross-diffusion connection between the potential and recovery variables. We show that the cross-diffusion version of the model, besides giving rise to the typical fast traveling wave solution exhibited in the original "diffusion" FitzHugh-Nagumo equations, additionally gives rise to a slow traveling wave solution. We analyze all possible traveling wave solutions of the model and show that there exists a threshold of the cross-diffusion coefficient (for a given speed of propagation), which bounds the area where "normal" impulse propagation is possible.

Original language | English (US) |
---|---|

Pages (from-to) | 239-260 |

Number of pages | 22 |

Journal | Mathematical Biosciences and Engineering |

Volume | 5 |

Issue number | 2 |

State | Published - Apr 2008 |

### Fingerprint

### Keywords

- Cross-diffusion
- Fitzhugh
- Traveling wave solutions

### ASJC Scopus subject areas

- Applied Mathematics
- Modeling and Simulation
- Computational Mathematics
- Agricultural and Biological Sciences(all)
- Medicine(all)

### Cite this

*Mathematical Biosciences and Engineering*,

*5*(2), 239-260.

**"Traveling wave" solutions of Fitzhugh model with cross-diffusion.** / Berezovskaya, Faina; Camacho, Erika; Wirkus, Stephen; Karev, Georgy.

Research output: Contribution to journal › Article

*Mathematical Biosciences and Engineering*, vol. 5, no. 2, pp. 239-260.

}

TY - JOUR

T1 - "Traveling wave" solutions of Fitzhugh model with cross-diffusion

AU - Berezovskaya, Faina

AU - Camacho, Erika

AU - Wirkus, Stephen

AU - Karev, Georgy

PY - 2008/4

Y1 - 2008/4

N2 - The PitzHugh-Nagumo equations have been used as a caricature of the Hodgkin-Huxley equations of neuron firing and to capture, qualitatively, the general properties of an excitable membrane. In this paper, we utilize a modified version of the FitzHugh-Nagumo equations to model the spatial propagation of neuron firing; we assume that this propagation is (at least, partially) caused by the cross-diffusion connection between the potential and recovery variables. We show that the cross-diffusion version of the model, besides giving rise to the typical fast traveling wave solution exhibited in the original "diffusion" FitzHugh-Nagumo equations, additionally gives rise to a slow traveling wave solution. We analyze all possible traveling wave solutions of the model and show that there exists a threshold of the cross-diffusion coefficient (for a given speed of propagation), which bounds the area where "normal" impulse propagation is possible.

AB - The PitzHugh-Nagumo equations have been used as a caricature of the Hodgkin-Huxley equations of neuron firing and to capture, qualitatively, the general properties of an excitable membrane. In this paper, we utilize a modified version of the FitzHugh-Nagumo equations to model the spatial propagation of neuron firing; we assume that this propagation is (at least, partially) caused by the cross-diffusion connection between the potential and recovery variables. We show that the cross-diffusion version of the model, besides giving rise to the typical fast traveling wave solution exhibited in the original "diffusion" FitzHugh-Nagumo equations, additionally gives rise to a slow traveling wave solution. We analyze all possible traveling wave solutions of the model and show that there exists a threshold of the cross-diffusion coefficient (for a given speed of propagation), which bounds the area where "normal" impulse propagation is possible.

KW - Cross-diffusion

KW - Fitzhugh

KW - Traveling wave solutions

UR - http://www.scopus.com/inward/record.url?scp=52049102278&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=52049102278&partnerID=8YFLogxK

M3 - Article

VL - 5

SP - 239

EP - 260

JO - Mathematical Biosciences and Engineering

JF - Mathematical Biosciences and Engineering

SN - 1547-1063

IS - 2

ER -