Modelling wave propagation across a series of gaps Gavin R. Armstrong,* a Annette F. Taylor, a Stephen K. Scott a and Vilmos Ga´spa´r b a Department of Chemistry, University of Leeds, Leeds, UK LS2 9JT. E-mail: G.Armstrong@chem.leeds.ac.uk b Department of Chemistry, University of Debrecen, P. O. Box 7 4010 Debrecen, Hungary Received 27th April 2004, Accepted 30th July 2004 First published as an Advance Article on the web 18th August 2004 Wave propagation across a series of gaps in a one-dimensional excitable medium is simulated using the Oregonator model of the Belousov–Zhabotinsky reaction. In agreement with recent experiments, we observe features such as the critical gap width W cr , critical spacing between gaps S cr and frequency transformation of the passage of a train of waves across a gap with width W r W cr . The role of activator kinetics in the gap is studied and the effect of excitability (through variation of parameters f and e) on the fraction of waves which successfully cross the domain is determined. We also find that the probability of a wave successfully propagating through the entire domain decreases with increasing number of gaps, and the profile of the activator species is examined for evidence of a ‘‘weakening’’ effect in a multiple gap system. 1. Introduction Excitable media such as the heart muscle or nerve tissue are capable of supporting chemical/electrical waves of activity. The Belousov–Zhabotinsky reaction has been extensively used as a chemical model to investigate generic dynamical features of excitable media. 1 Current studies focus on the propagation of waves in heterogeneous media, for which the excitability is a function of spatial coordinate, a situation which is more likely to occur in biological systems than one of uniform excitabil- ity. 2,3 Heterogeneity may contribute to wave break-up and spiral formation in the heart, a phenomenon associated with cardiac arrhythmia and ventricular fibrillation. 4,5 Multiple mechanisms for this dangerous heart condition have been proposed, but a deeper understanding of the effect of hetero- geneity on wave propagation is required. 6,7 Experiments and simulations have demonstrated that waves are capable of propagating across a single gap in the medium i.e. a region of low or no excitability. 8–10 Passage of the wave is thought to involve passive diffusion of an activator species across the gap and a supercritical threshold of activator is required to initiate activity on the other side of the gap. There exists a critical gap width (W cr ) for which waves passing across a larger gap will fail, a phenomenon known as wave block. 11 Moreover, a gap close in width but less than W cr can act as a frequency transformer, such that only a fraction of waves f n in a wave train will cross the gap, for every wave that enters. Thus we define a firing number f n ¼ m/n, where m is the number of waves transmitted across a gap (or an array of gaps) and n is the number of incident waves. 12,13 This phenomenon has also been observed in experiments on chemical wave propagation through capillary tubes and can be explained by the existence of a refractory period following excitation, i.e. a period of time during which the excitable medium recovers its excitability. 14 The threshold of activator required to stimulate excitation is much higher during the recovery time. Recent experiments investigated the passage of a wave across a series of gaps in the excitable domain. 8 The excitable domain was constructed by using a DeskJet printer to print the reaction catalyst onto a membrane thereby fixing the catalyst in a defined pattern. This pattern had a fixed-frequency wave source and several lanes of catalyst interrupted by gaps. The experiments confirmed the existence of a critical spacing between gaps S cr for successful propagation of a wave across the entire system. A ‘‘weakening’’ effect was also observed, whereby for systems with S and W close to the critical values, waves failed to propagate farther than seven gaps. In this paper, we simulate the propagation of waves across a series of gaps in a one-dimensional excitable domain using the two-variable Oregonator model, 15 which is generally used to model the Belousov–Zhabotinsky reaction. This model also serves as a paradigm for biological excitable media. We determine the fraction of waves successfully propagating across the entire domain and how this depends on the gap width, spacing between gaps and the activator kinetics in the gap. We also examine the concentration profile of the activator species to investigate a possible ‘‘weakening’’ of the wave by the existence of multiple gaps. 2. Model Numerical simulations were performed using the two-variable Oregonator model, based on the kinetic scheme of Field, Ko¨ro¨s and Noyes. 16 We applied the scalings of Tyson and Fife 17 and assumed diffusion only of the activator species (u) in one spatial dimension (x) only: @u @t ¼ @ 2 u @x 2 þ 1 e uð1  uÞ fv ðu  qÞ ðu þ qÞ   ð1Þ @v @t ¼ u  v ð2Þ where u and v are the dimensionless concentrations of the auto- catalyst HBrO 2 and oxidized form of the metal catalyst M ox respectively and the parameters e, f and q can be related to rate constants and initial concentrations. The values of the para- meters were chosen to reflect typical experimental conditions: e ¼ 0.02–0.07, f ¼ 2.4–3.1 and q ¼ 8  10 4 . The excitability of the medium can be considered to decrease as e is increased from 0.02 to 0.07 and as f is increased from 2.4 to 3.1. The equations were integrated using XPPAUT 18 in a 1-d spatial domain. The integration method used was ‘‘cvode’’ with di- mensionless timestep dt ¼ 0.0005, and tolerance ¼ 1  10 7 . Time was scaled by t ¼ tk c B, where t is dimensional time, k c is a rate constant and B relates to initial malonic acid RESEARCH PAPER PCCP www.rsc.org/pccp DOI: 10.1039/b406301e Phys. Chem. Chem. Phys., 2004, 6 , 4677–4681 4677 This journal is & The Owner Societies 2004