Effects of asymmetric coupling and self-coupling on metastable dynamical transient rotating waves in a ring of sigmoidal neurons

• Metastable dynamical transient rotating waves in a ring neural network are studied.
• A kinematical equation for a change in bump length of rotating waves is derived.
• Conditions for the stabilization and pinning of rotating waves are obtained.
• An exponential growth rate of the duration of transient rotating waves is obtained.
• The growth rate depends on the strength of asymmetric coupling and self-coupling.

Transient rotating waves in a ring of sigmoidal neurons with asymmetric bidirectional coupling and self-coupling were studied. When a pair of stable steady states and an unstable traveling wave coexisted, rotating waves propagating in a ring were generated in transients. The pinning (propagation failure) of the traveling wave occurred in the presence of asymmetric coupling and self-coupling, and its conditions were obtained. A kinematical equation for the propagation of wave fronts of the traveling and rotating waves was then derived for a large output gain of neurons. The kinematical equation showed that the duration of transient rotating waves increases exponentially with the number of neurons as that in a ring of unidirectionally coupled neurons (metastable dynamical transients). However, the exponential growth rate depended on the asymmetry of bidirectional coupling and the strength of self-coupling. The rate was equal to the propagation time of the traveling wave (a reciprocal of the propagation speed), and it increased near pinned regions. Then transient rotating waves could show metastable dynamics (extremely long duration) even in a ring of a small number of neurons.