Metastable dynamical patterns and their stabilization in arrays of bidirectionally coupled sigmoidal neurons

Effects of self-coupling and asymmetric output on metastable dynamical transient firing patterns in arrays of neurons with bidirectional inhibitory coupling

Metastable dynamical transient patterns in arrays of bidirectionally coupled neurons with self-coupling and asymmetric output were studied. First, an array of asymmetric sigmoidal neurons with symmetric inhibitory bidirectional coupling and self-coupling was considered and the bifurcations of its steady solutions were shown. Metastable dynamical transient spatially nonuniform states existed in the presence of a pair of spatially symmetric stable solutions as well as unstable spatially nonuniform solutions in a restricted range of the output gain of a neuron. The duration of the transients increased exponentially with the number of neurons up to the maximum number at which the spatially nonuniform steady solutions were stabilized. The range of the output gain for which they existed reduced as asymmetry in a sigmoidal output function of a neuron increased, while the existence range expanded as the strength of inhibitory self-coupling increased. Next, arrays of spiking neuron models with slow synaptic inhibitory bidirectional coupling and self-coupling were considered with computer simulation. In an array of Class 1 Hindmarsh-Rose type models, in which each neuron showed a graded firing rate, metastable dynamical transient firing patterns were observed in the presence of inhibitory self-coupling. This agreed with the condition for the existence of metastable dynamical transients in an array of sigmoidal neurons. In an array of Class 2 Bonhoeffer-van der Pol models, in which each neuron had a clear threshold between firing and resting, long-lasting transient firing patterns with bursting and irregular motion were observed.

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

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.