Frequency Switches at Transition Temperature in Voltage-Gated Ion Channel Dynamics of Neural Oscillators

Understanding of the mechanisms of neural phase transitions is crucial for clarifying cognitive processes in the brain. We investigate a neural oscillator that undergoes different bifurcation transitions from the big saddle homoclinic orbit type to the saddle node on an invariant circle type, and the saddle node on an invariant circle type to the small saddle homoclinic orbit type. The bifurcation transitions are accompanied by an increase in thermodynamic temperature that affects the voltage-gated ion channel in the neural oscillator. We show that nonlinear and thermodynamical mechanisms are responsible for different switches of the frequency in the neural oscillator. We report a dynamical role of the phase response curve in switches of the frequency, in terms of slopes of frequency-temperature curve at each bifurcation transition. Adopting the transition state theory of voltage-gated ion channel dynamics, we confirm that switches of the frequency occur in the first-order phase transition temperature states and exhibit different features of their potential energy derivatives in the ion channel. Each bifurcation transition also creates a discontinuity in the Arrhenius plot used to compute the time constant of the ion channel.