Analyzing the Resonant Behavior of a Single Neuron at the Subthreshold Level

Different brain rhythms are often signatures of different behavioral and perceptual states of the brain. As resonance at neurons has a close connection with brain rhythms, characterizing and analyzing the neuronal resonance promote the understanding of brain rhythms and dynamical organization of the brain. In this paper, we study the resonant behaviors of both the four-dimensional (4D) Hodgkin-Huxley model and a reduced-order (2D) model at the subthreshold level. We analyze the frequency responses of a neuron and characterize resonance through investigating the neuron's transfer functions, frequency response functions, and root locus plots. The 2D model, especially, allows us to visualize the state space, perform phase plane analysis, and derive a closed-form formula for the resonant frequency. These analyses will be generalized to study resonance at the spiking regime and network level in future work.