Rotational Waves in Rings of Electrochemical Oscillators

Chemical and biological systems exhibit network properties dependent upon the behavior of individual units and the interactions between units.[1]  The presence or absence of interactions and the strength of the interactions determine the topology of the network and in turn the network properties.  From a mathematical perspective, a variety of non-rotating and rotating waves can exist for a ring of locally coupled units[2] and the non-rotating solutions are the dominant outcome when coupling is the only mean of interactions regardless of the initial conditions.[3] Therefore, engineering is required to achieve solutions aside from the dominant non-rotating state. 

The formation of waves is examined in oscillatory nickel dissolution where units of individual oscillators interact through external resistance which controls the strength of interaction.  An experimental technique using a locally coupled twenty nickel electrode ring is presented in which through a combination of temporary alterations in topology and the application of global feedback provides all possible rotational solutions. 

There are four rotational waves which can propagate about the ring and are distinguished by the velocity and the number of waves.  The propagation rate is inversely proportional to the number of waves in the rotation (m) for all rotations.  Solutions with no rotation have an m value of zero while rotational solutions are obtained for m=1,2,3,4.  Larger m values (m ≥ 5) are shown to not be stable in the network.  The results thus indicate that while network plasticity is thought to be significant in the operation of neural systems, it can also play a role in pattern selection of chemical systems.

References:

[1] J. L. Hudson, I. Z. Kiss, H. Kori, C. G. Rusin, Science 2007, 316, 1886-1889.

[2] G. B. Ermentrout, J. Math. Biol. 1985, 23, 55-74.

[3] D. A. Wiley, S. H. Strogatz, M. Girvan, Chaos 2006, 16, 1054-1500.