Nonlinear Wave Processes in Excitable Media pp 213-231 | Cite as

# The Dynamics of Forced Excitable Systems

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## Abstract

An excitable system is one whose dynamics have the following properties: (i) there is a rest point or a steady state that is globally attracting relative to some large set in phase space, and (ii) there is a region in state space that can be idealized as a surface of codimension one that locally partitions the phase space into two sets *D* and *A.* the rest point lies in *D* (the decaying **set)** and all orbits through initial points in *D* return to the rest point without any substantial growth in any of the state variables. Thus an impulsive perturbation of the rest point that leaves the state in *V* decays without significant growth, and the responses are called **subthreshold**. By contrast, perturbations that carry the dynamics into *A* (the **amplifying set)** can lead to a large change in one or more of the state variables, even though the system eventually returns to the rest state. The surface that locally separates the amplifying and decaying sets is called the **threshold** surface, and perturbations that carry the state into *A* are called **superthreshold**. Excitable dynamics occur in many biological processes, including activation of contraction in cardiac tissue, nerve conduction, and cell signalling in development. In the Fitzhugh-Nagumo equations [10], the Hodgkin- Huxley equations [15], models of the cellular slime mold *Dictyostelium discoideum* [22, 20, 21], the Field-Noyes model of the Zhabotinskii-Belousov reaction [9] and other models, the parameters can be chosen so that the dynamics are excitable. Frequently these systems also have periodic solutions for parameters close to those that produce excitable behaviour.

## Keywords

Periodic Solution Rotation Number Dictyostelium Discoideum Singular Limit Excitable System## Preview

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## References

- [1]Alexander, J.C., Doedel, E.J. & Othmer, H.G. On the resonance structure in a forced excitable system. To appear in
*SIAM J. Appl. Math.*, 1990.Google Scholar - [2]Alexander, J.C., Doedel, E.J. & Othmer, H.G. (1989). Resonance and phase-locking in excitable systems.
*Lects. on Math. Life Sci.***21**, 1–37.MathSciNetGoogle Scholar - [3]Aronson, D., Doedel, E.J. & Othmer, H.G. (1986). Bistable behavior in coupled oscillators. In
*Nonlinear Oscillations in Biology and Chemistry*, pp. 221–231, Othmer, H.G. (ed.). Lecture Notes in Biomathematics 66, Springer-Verlag.CrossRefGoogle Scholar - [4]Aronson, D., Doedel, E.J. & Othmer, H.G. (1987). An analytical and numerical study of the bifurcations in a system of linearly-coupled oscillators.
*Physica***25D**, 20–104.zbMATHMathSciNetGoogle Scholar - [5]Builder, G. & Roberts, N.F. (1939). The synchronization of a simple relaxation oscillator.
*AW A Tech. Rev.***4**, 164–180.Google Scholar - [6]Chialvo, D.R. & Jalife, J. (1987). Non-linear dynamics of cardiac excitation and impulse propagation.
*Nature***330**, 749–752.ADSCrossRefGoogle Scholar - [7]DeYoung, G. & Othmer, H.G. Resonance in oscillatory and excitable systems. To appear in
*Ann. N.Y. Acad. Sci.*, 1990.Google Scholar - [8]Doedel, E.J. (1981). AUTO: A program for the automatic bifurcation and analysis of autonomous systems. In
*Proc. 10th Manitoba Conf. Num. Anal. and Comp.*, pp. 265–284.Google Scholar - [9]Field, R.J. & Noyes, R.M. (1974). Oscillations in chemical systems, IV. Limit cycle behavior in a model of a real chemical reaction.
*J. Chem. Physics***60**, 1877–1884.ADSCrossRefGoogle Scholar - [10]Fitzhugh, R. (1969). Mathematical models of excitation and propagation in nerve. In
*Biological Engineering*, pp. 1–85, Schwan, H.P. (ed.). McGraw-Hill.Google Scholar - [11]Glass, L., Guevara, M.R., Belair, J. & Shrier, A. (1984). Global bifurcations of a periodically forced biological oscillator.
*Phys. Rev.***A29**, 1348–1357.ADSCrossRefMathSciNetGoogle Scholar - [12]Glass, L. & Belair, J. (1986). Continuation of Arnold tongues in mathematical models of periodically forced biological oscillators. In
*Nonlinear Oscillations in Biology and Chemistry*, pp. 232–243, Othmer, H.G. (ed.). Lecture Notes in Biomathematics 66.CrossRefGoogle Scholar - [13]Guttman, R. , Feldman, L. & Jakobsson, E. (1980). Frequency entrainment of squid axon membrane.
*J. Membrane Biol.***56**, 9–18.CrossRefGoogle Scholar - [14]Hartman, P. (1964).
*Ordinary Differential Equations*. John Wiley and Sons.zbMATHGoogle Scholar - [15]Hodgkin, A.L. & Huxley, A.F. (1952). A quantitative description of membrane current and application to conduction and excitation in nerve.
*J. Physiol.***117**, 500–544.Google Scholar - [16]Holden, A.V. (1976). The response of excitable membrane models to a cyclic input.
*Biol. Cybernetics***21**, 1–7.CrossRefGoogle Scholar - [17]Hudson, J.L., Lamba, P. & Mankin, J.C. (1986). Experiments on low-amplitude forcing of a chemical oscillator.
*J. Phys. Chem.***90**, 3430–3434.CrossRefGoogle Scholar - [18]Matsumoto, G., Aihara, K., Hanyu, Y., Takahashi, N., Yoshizawa, S. & Nagumo, J.-I. (1987). Chaos and phase locking in normal squid axons.
*Phys. Rev. Lett. A***123**(4), 162–166.ADSCrossRefGoogle Scholar - [19]Markevich, N.I. & Sel’kov, E.E. (1989). Parametric resonance and amplification in excitable membranes. The Hodgkin-Huxley model.
*J. Theor. Biol.***140**, 27–38.CrossRefMathSciNetGoogle Scholar - [20]Monk, P.B. & Othmer, H.G. (1989). Cyclic AMP oscillations in suspensions of
*Dictyostelium discoideum. Phil. Trans. R. Soc. Lond*.**323**(1215), 185–224.ADSCrossRefGoogle Scholar - [21]Monk, P.B. & Othmer, H.G. Wave propagation in aggregation fields of the cellular slime mode
*Dictyostelium discoideum.*Submitted.Google Scholar - [22]Othmer, H.G. & Monk, P.B. (1988). Concentration waves in aggregation fields of a cellular slime mold. In
*Biomathematics and Related Computational Problems*, pp. 381–398, Ricciardi, L. (ed.). Kluwer Academic Publishers: Dordrecht.CrossRefGoogle Scholar - [23]Nagumo, J.-I. & Sato, S. (1972). On a response characteristic of a mathematical neuron model.
*Kybernetik***10**, 155–164.CrossRefzbMATHGoogle Scholar - [24]Sato, S., Hatta, M. & Nagumo, J.-I. (1974). Response characteristics of a neuron model to a periodic input.
*Kybernetik***16**, 1–8.CrossRefzbMATHGoogle Scholar