Bifurcation and stabilization of oscillations in ring neural networks with inertia

Effects of inertia on oscillations in ring networks of unidirectionally coupled sigmoidal neurons are studied. It is known that ring neural networks without inertia are bistable and the duration of transient oscillations increases exponentially with the number of neurons. In this paper, a kinematical description of traveling waves in the oscillations in the networks is extended to networks with inertia. When the inertia is below a critical value and the state of each neuron is over-damped, properties of the networks are qualitatively the same as those without inertia. The duration of the transient oscillations then increases with inertia, and the increasing rate of the logarithm of the duration becomes more than double. When the inertia exceeds a critical value and the state of each neuron becomes under-damped, properties of the networks qualitatively change. The periodic solution is then stabilized through the pitchfork bifurcation as inertia increases. More bifurcations occur so that various periodic solutions are generated, and the stability of the periodic solutions changes alternately. Further, stable oscillations generated with inertia are observed in an experiment on an analog circuit.