Hopf bifurcations in electrochemical, neuronal, and semiconductor systems analysis by impedance spectroscopy

Spontaneous oscillations in a variety of systems, including neurons, electrochemical, and semiconductor devices, occur as a consequence of Hopf bifurcation in which the system makes a sudden transition to an unstable dynamical state by the smooth change of a parameter. We review the linear stability analysis of oscillatory systems that operate by current–voltage control using the method of impedance spectroscopy. Based on a general minimal model that contains a fast-destabilizing variable and a slow stabilizing variable, a set of characteristic frequencies that determine the shape of the spectra and the associated dynamical regimes are derived. We apply this method to several self-sustained rhythmic oscillations in the FitzHugh–Nagumo neuron, the Koper–Sluyters electrocatalytic system, and potentiostatic oscillations of a semiconductor device. There is a deep and physically grounded analogy between different oscillating systems: neurons, electrochemical, and semiconductor devices, as they are controlled by similar fundamental processes unified in the equivalent circuit representation. The unique impedance spectroscopic criteria for widely different variables and materials across several fields provide insight into the dynamical properties and enable the investigation of new systems such as artificial neurons for neuromorphic computation.