PhysicsLettersA 162 (1992) 157—161 • PHYSICS LETTERS A North-Holland Analog-to-frequency transduction by nonlinear excitable media Arkady Rovinsky’ and Michael Menzinger Department of Chemistry, University of Toronto, Toronto, Ontario, Canada M5S JAI Received 30 September 1991; revised manuscript received 18 November 1991; accepted for publication 22 November 1991 Communicated by A.R. Bishop Spatially extended excitable media may, through an appropriate constant force on the dynamical variables (boundary condi- tion), be made to generate periodic travelling waves whose frequency encodes the constant external stimulus. It is shown that to operate as an analog-to-frequency converter over a wide dynamical frequency range, it is necessary that the system’s recovery to its fixed point be nonlinearly slowed down. 1. Introduction (R~=Smax/Smjn and RT= Tmax/Tmin=Wmax/U)min; T= a) Nonlinear dynamical systems can achieve analog- Nonlinear dynamical systems are distinguished by to-frequency conversion in two different ways. First, their great variety of dynamical response that in- the input signal may act on the parameters of the sys- cludes oscillations, multistability, excitability, signal tern. Changing a parameter changes the system itself: propagation and pattern formation. In this Letter we this may cause a bifurcation from a stable fixed point demonstrate an additional function that is of great to a limit cycle and it may furthermore alter the sys- importance in nature but which has received so far tern’s internal time scale and hence its frequency. only little attention. It is the ability of nonlinear dy- In the second case, the stimulus constitutes a con- namical systems to react to a constant perturbation stant boundary condition imposed on a dynamical with the frequency of its oscillatory response, similar variable of a spatially extended excitable medium. to the voltage-to-frequency conversion of commer- This is characteristic for sensory neurons where a cial electronic devices, constant “generator potential” produced by a recep- This function is of great importance in biological tor is encoded, at the junction of spatially distin- systems, where sensory neurons universally trans- guished transduction regions, by a train of action po- duce external stimuli into sequences of action po- tentials [21. Boundary conditions do not act on the tentials. Usually, the frequency of a pulse train w(S) system’s parameters nor do they affect its intrinsic emitted by a sensory neuron is a rising function of time scale. Therefore, it is not obvious ad hoc whether the stimulus intensity S and at least in some cases a spatially extended excitable medium can respond [11 it obeys a power law to constant stimulation applied locally by emitting w(S) =k(S—S 0) m. (1) pulse trains over a wide range of frequency. A few previous studies have demonstrated the possibility For a sensory organ to be biologically useful, the of pulse generation under a constant stimulation at transduction process must cover reasonably wide dy- the boundary (Dirichiet’s boundary condition). namical ranges of both stimulus and frequency However, the range of frequency response never ex- ceeded a factor of about two [3,4] — far too narrow to be significant for sensory transduction. On leave from: Institute of Theoretical and Experimental Bio- physics of the Academy of Sciences, Pusehino, Moscow Re- The aim of this paper is then to find the general gion 142292, USSR. conditions under which a spatially extended excit- 0375-9601/92/S 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved. 157