Limit Cycle of a Single-Neuron System With Smooth Continuous and Binary-Value Activation Functions and Its Circuitry Design

In this brief, we investigate the limit cycle of a single-neuron system with smooth continuous and binary-value activation functions and its circuit design. By transforming the system into Liénard-type and using Poincaré–Bendixson theorem as well as the symmetry of these systems, we obtain the existence conditions of limit cycle of the system. Then, by comparing the integral value of the differential of positive definite function along two assumed limit cycles, we prove that the system cannot produce two coexisting limit cycles, which means that the system has at most one limit cycle. In addition, according to the two specific functions, i.e., smooth continuous and binary-value activation functions of the system, we give the numerical simulation and realize the circuit design of the single-neuron system by using Multisim modeling, respectively. The waveform diagram and phase diagram of the numerical simulation and circuit simulation are obtained. By comparing the results of numerical and circuit simulation, the effectiveness of our mathematical analysis and the feasibility of circuit design are better illustrated.