Effects of external signals on neural oscillator stability

Abstract

Neural oscillators, which are a mathematical model of a central pattern generator, have been used to investigate human gait and control mobile robots. A typical neural oscillator uses two neurons that mutually inhibit each other’s activity. The exact orbitally-stable conditions of the neural activity of neural oscillators without external signals have been reported. However, the behavior of neural oscillators with external signals is unclear because their neural activity depends on the external signals, which have many types. In this study, for simplicity, external signals were regarded as a sinusoidal wave with a period T_ex and an amplitude A_ex. The connectivity a_ij, ratio for time constants τ_z/τ_x, and fatigue coefficient b were changed for neurons i and j, while T_ex and A_ex were changed for external signals. The orbit-stability of the output signals from a neuron was decided based on the transient time (≦ 3 s) and the duration (≧ 30 s). The period T_out and the amplitude A_out of the output signals were evaluated. T_out had discrete values of T_ex, 2T_ex, or 0.5T_ex or was non-orbitally-stable (value of 0). When A_ex was greater than or equal to u_i, the neural oscillator became synchronized. For a small T_ex, some combinations of τ_z/τ_x and a_ij values led to instability. For a large T_ex, some combinations of b and a_ij values led to instability. In contrast to T_out, the amplitude A_out of the output signal showed continuous changes depending on τ_z/τ_x, b, and a_ij. The amplitude A_out could be expressed as the sum of u_i and A_ex. A_out was not significantly affected by τ_z/τ_x but decreased with decreasing b.