Sustained oscillations generated by mutually inhibiting neurons with adaptation

Autonomic oscillatory activities exist in almost every living thing and most of them are produced by rhythmic activities of the corresponding neural systems (locomotion, respiration, heart beat, etc.). This paper mathematically discusses sustained oscillations generated by mutual inhibition of the neurons which are represented by a continuous-variable model with a kind of fatigue or adaptation effect. If the neural network has no stable stationary state for constant input stimuli, it will generate and sustain some oscillation for any initial state and for any disturbance. Some sufficient conditions for that are given to three types of neural networks: lateral inhibition networks of linearly arrayed neurons, symmetric inhibition networks and cyclic inhibition networks. The result suggests that the adaptation of the neurons plays a very important role for the appearance of the oscillations. Some computer simulations of rhythic activities are also presented for cyclic inhibition networks consisting of a few neurons.