### Abstract

Original language | Unknown |
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Title of host publication | Springer Proceedings in Mathematics and Statistics |

Pages | 551-559 |

ISBN (Electronic) | 978-1-4614-7333-6 |

DOIs | |

Publication status | Published - 1 Jan 2013 |

Event | International Conference on Differential and Difference Equations and Applications - Duration: 1 Jan 2011 → … |

### Conference

Conference | International Conference on Differential and Difference Equations and Applications |
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Period | 1/01/11 → … |

### Cite this

*Springer Proceedings in Mathematics and Statistics*(pp. 551-559) https://doi.org/10.1007/978-1-4614-7333-6_50

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*Springer Proceedings in Mathematics and Statistics.*pp. 551-559, International Conference on Differential and Difference Equations and Applications, 1/01/11. https://doi.org/10.1007/978-1-4614-7333-6_50

**Existence of Nonoscillatory Solutions of the Discrete FitzHugh-Nagumo Systems.** / Pedro, Ana Maria Manteigas.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Existence of Nonoscillatory Solutions of the Discrete FitzHugh-Nagumo Systems

AU - Pedro, Ana Maria Manteigas

PY - 2013/1/1

Y1 - 2013/1/1

N2 - In this work, we are concerned with a system of two functional differential equations of mixed type (with delays and advances), known as the discrete Fitzhugh-Nagumo equations, which arises in the modelling of impulse propagation in a myelinated axon: C dv dt (t) = 1 R (v(t +τ)−2v(t) +v(t −τ)) + f(v(t))−w(t) dw dt = σv(t)−γw(t). (1) In the case γ = σ = 0, this system reduces to a single equation, which is well studied in the literature. In this case it is known that for each set of the equation parameters (within certain constraints), there exists a value of τ (delay) for which the considered equation has a monotone solution v satisfying certain conditions at infinity. The main goal of the present work is to show that for sufficiently small values of the coefficients in the second equation of the system (1), this system has a solution (v,w) whose first component satisfies certain boundary conditions and has similar properties to the ones of v, in the case of a single equation. With this purpose we linearize the original system as t → −∞ and t → ∞ and analyse the corresponding characteristic equations. We study the existence of nonoscillatory solutions, based on the number and nature of the roots of these equations.

AB - In this work, we are concerned with a system of two functional differential equations of mixed type (with delays and advances), known as the discrete Fitzhugh-Nagumo equations, which arises in the modelling of impulse propagation in a myelinated axon: C dv dt (t) = 1 R (v(t +τ)−2v(t) +v(t −τ)) + f(v(t))−w(t) dw dt = σv(t)−γw(t). (1) In the case γ = σ = 0, this system reduces to a single equation, which is well studied in the literature. In this case it is known that for each set of the equation parameters (within certain constraints), there exists a value of τ (delay) for which the considered equation has a monotone solution v satisfying certain conditions at infinity. The main goal of the present work is to show that for sufficiently small values of the coefficients in the second equation of the system (1), this system has a solution (v,w) whose first component satisfies certain boundary conditions and has similar properties to the ones of v, in the case of a single equation. With this purpose we linearize the original system as t → −∞ and t → ∞ and analyse the corresponding characteristic equations. We study the existence of nonoscillatory solutions, based on the number and nature of the roots of these equations.

KW - Non-oscillatory solutions

KW - Discrete Fitzhugh-Nagumo equations

KW - Mixed-type functional-differential equations

U2 - 10.1007/978-1-4614-7333-6_50

DO - 10.1007/978-1-4614-7333-6_50

M3 - Conference contribution

SN - 978-4624-7332-9

SP - 551

EP - 559

BT - Springer Proceedings in Mathematics and Statistics

ER -