### Abstract

Propagation can be encumbered in an excitable cable in which intrinsic properties change abruptly. A sudden increase in diameter or a decrease in conductivity or excitability can lead to propagation block or delay in propagation with or without reflection. We study such transient phenomena from a geometric point of view. A simple two-cell caricature, with one cell enlarged to mimic diameter increase, is developed and analyzed. Our analysis indicates that reflected waves may result from the existence of an unstable periodic orbit. As the inhomogeneity parameter is varied, this unstable cycle is nearer to and then farther from the initial state that mimics an incoming wave. This fact leads to a variety of complicated reflected waves. Correspondingly, we find numerically complex sequences of reflected-transmitted waves in biophysically more realistic cable analogues. The unstable periodic orbit in the cable appears to be related to a one-dimensional spiral wave described by Kopell and Howard [Stud. Appl. Math., 64 (1981), pp. 1-56]. Finally, we argue that reflection phenomena occur more robustly when excitability is due to saddle-type threshold behavior (type I excitability in the sense of Rinzel and Ermentrout [in Methods in Neuronal Modeling: From Synapses to Networks, C. Koch and I. Segev, eds., MIT Press, Cambridge, MA, 1989]).

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

Pages (from-to) | 1107-1128 |

Number of pages | 22 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 56 |

Issue number | 4 |

State | Published - Aug 1996 |

### Fingerprint

### Keywords

- Cable equations
- Echo waves
- Excitable media

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*SIAM Journal on Applied Mathematics*,

*56*(4), 1107-1128.

**Reflected waves in an inhomogeneous excitable medium.** / Ermentrout, G. Bard; Rinzel, John.

Research output: Contribution to journal › Article

*SIAM Journal on Applied Mathematics*, vol. 56, no. 4, pp. 1107-1128.

}

TY - JOUR

T1 - Reflected waves in an inhomogeneous excitable medium

AU - Ermentrout, G. Bard

AU - Rinzel, John

PY - 1996/8

Y1 - 1996/8

N2 - Propagation can be encumbered in an excitable cable in which intrinsic properties change abruptly. A sudden increase in diameter or a decrease in conductivity or excitability can lead to propagation block or delay in propagation with or without reflection. We study such transient phenomena from a geometric point of view. A simple two-cell caricature, with one cell enlarged to mimic diameter increase, is developed and analyzed. Our analysis indicates that reflected waves may result from the existence of an unstable periodic orbit. As the inhomogeneity parameter is varied, this unstable cycle is nearer to and then farther from the initial state that mimics an incoming wave. This fact leads to a variety of complicated reflected waves. Correspondingly, we find numerically complex sequences of reflected-transmitted waves in biophysically more realistic cable analogues. The unstable periodic orbit in the cable appears to be related to a one-dimensional spiral wave described by Kopell and Howard [Stud. Appl. Math., 64 (1981), pp. 1-56]. Finally, we argue that reflection phenomena occur more robustly when excitability is due to saddle-type threshold behavior (type I excitability in the sense of Rinzel and Ermentrout [in Methods in Neuronal Modeling: From Synapses to Networks, C. Koch and I. Segev, eds., MIT Press, Cambridge, MA, 1989]).

AB - Propagation can be encumbered in an excitable cable in which intrinsic properties change abruptly. A sudden increase in diameter or a decrease in conductivity or excitability can lead to propagation block or delay in propagation with or without reflection. We study such transient phenomena from a geometric point of view. A simple two-cell caricature, with one cell enlarged to mimic diameter increase, is developed and analyzed. Our analysis indicates that reflected waves may result from the existence of an unstable periodic orbit. As the inhomogeneity parameter is varied, this unstable cycle is nearer to and then farther from the initial state that mimics an incoming wave. This fact leads to a variety of complicated reflected waves. Correspondingly, we find numerically complex sequences of reflected-transmitted waves in biophysically more realistic cable analogues. The unstable periodic orbit in the cable appears to be related to a one-dimensional spiral wave described by Kopell and Howard [Stud. Appl. Math., 64 (1981), pp. 1-56]. Finally, we argue that reflection phenomena occur more robustly when excitability is due to saddle-type threshold behavior (type I excitability in the sense of Rinzel and Ermentrout [in Methods in Neuronal Modeling: From Synapses to Networks, C. Koch and I. Segev, eds., MIT Press, Cambridge, MA, 1989]).

KW - Cable equations

KW - Echo waves

KW - Excitable media

UR - http://www.scopus.com/inward/record.url?scp=0030216813&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030216813&partnerID=8YFLogxK

M3 - Article

VL - 56

SP - 1107

EP - 1128

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 4

ER -