Waves in a simple, excitable or oscillatory, reaction-diffusion model

G. Bard Ermentrout, John Rinzel

Research output: Contribution to journalArticlepeer-review


A simple one variable caricature for oscillating and excitable reaction-diffusion systems is introduced. It is shown that as a parameter, λ, varies the system dynamics change from oscillatory (λ > λ0) to excitable (λ < λ0) and the frequency of the oscillation vanishes as {Mathematical expression} for λ ↘ λ0. When such dynamics are coupled by continuous diffusion in a ring geometry (1-space dimension), propagating wave trains may be found. On an infinite ring excitable devices lead to unique solitary waves which are analogous to "pulse" waves. A solvable example is presented, illustrating properties of dispersion, excitability, and waves. Finally it is shown that the caricature arises in a natural way from more general excitable/oscillatory systems.

Original languageEnglish (US)
Pages (from-to)269-294
Number of pages26
JournalJournal Of Mathematical Biology
Issue number3
StatePublished - Mar 1981


  • Excitability
  • Oscillation
  • Pulse waves
  • Rings
  • Waves

ASJC Scopus subject areas

  • Modeling and Simulation
  • Agricultural and Biological Sciences (miscellaneous)
  • Applied Mathematics


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