Slow–fast dynamics of tri-neuron Hopfield neural network with two timescales

• The transition of slow–fast trajectory is clarified.
• The existence of relaxation oscillation is proved.
• The approximate expression of relaxation oscillation and its period are obtained.

The slow-fast dynamics of a tri-neuron Hopfield neural network with two timescales is stated in present paper. On the basis of geometric singular perturbation theory, the transition of the solution trajectory is illuminated, and the existence of the relaxation oscillation with rapid movement process alternating with slow movement process is proved. It is indicated the characteristic of the relaxation oscillation is dependent on the structure of the slow manifold. Moreover, the approximate expression of the relaxation oscillation and its period are obtained analytically. Case studies are given to demonstrate the validity of theoretical results.