A model illustrating amplification of perturbations in an excitable medium

Abstract

The oscillatory Belousov–Zhabotinskii reaction can be modelled approximately by five irreversible steps: A + Y → X (M1), X + Y → P (M2), B + X → 2X + Z (M3), 2X → Q (M4), Z →ƒY. (M5). These equations are based on the chemical equalities X = HBrO₂, Y = Br^–, Z = 2Ce(IV), and A = B = BrO^–₃. If the rate constants k_M1 to k_M4 are assigned by experimental estimates from oxybromine chemistry, the kinetic behaviour of the model depends critically upon the remaining parameters k_M5 and ƒ. When ƒ does not differ too greatly from unity, and when k_M5 is not too large, the steady state is unstable to perturbation and the system oscillates by describing a limit cycle trajectory.

When ƒ and k_M5 lie outside the range of instability, the steady state is stable to very small perturbations. However, the steady state may still be excitable so that perturbation of the control intermediate Y by a few percent will instigate a single excursion during which concentrations of X, Y, and Z change by factors of about 10⁵ before the system returns to the original steady state. This ability of a small perturbation of the steady state to trigger a major response by the system is just the type of behaviour necessary to explain the initiation of a trigger-wave by a heterogeneous “pacemaker” as has been observed by Winfree. The same type of excitability of a steady state has important implications for the understanding of biochemical control mechanisms.