Abstract. 
Many neuron types exhibit preferred frequency responses to
subthreshold oscillatory input currents reflected in an voltage
amplitude peak (resonance) and a zero phase-shift (phasonance or
phase-resonance). This phenomena may occur in the absence of intrinsic
oscillations in the corresponding autonomous system. The dynamics
principles that govern the generation of resonance and the effect of
the biophysical parameters on the resonant properties are not well
understood.

We propose a framework to analyze the role of different ionic currents
and their interactions in shaping the properties of the impedance
amplitude and phase profiles (graphs of these quantities as a function
of the input frequency) in linearized and quadratic biophysical
models. We adapt the classical phase-plane analysis approach to
account for the dynamic effects of oscillatory inputs and develop a
tool, the envelope-plane diagrams, that capture the role that
conductances and time scales play in amplifying the voltage response
at the resonant frequency band as compared to smaller and larger
frequencies. We use envelope-plane diagrams in our analysis to explain
why the resonance phenomena do not necessarily arise from the presence
of imaginary eigenvalues at rest, but rather it emerges from the
interplay of the intrinsic and input time scales. This interaction is
based mostly on transient effects. We further explain why an increase
in the time scale separation causes an amplification of the voltage
response in addition to shifting the resonant and phase-resonant
frequencies. We extend this approach to explain the effects of
nonlinearities on both resonance and phase-resonance.

We demonstrate that nonlinearities in the voltage equation cause
amplifications of the voltage response and shifts in the resonant and
phase-resonant frequencies that are not predicted by the corresponding
linearized model. The differences between the nonlinear response and
the linear prediction increase with increasing levels of the time
scale separation between the voltage and the gating variable, and they
almost disappear when both equations evolve at comparable rates. In
contrast, voltage responses are almost insensitive to nonlinearities
located in the gating variable equation. The method we develop
provides a framework for the investigation of the preferred frequency
responses in three-dimensional and nonlinear neuronal models as well
as simple models of coupled neurons.