  This article develops a fundamental insight into the behavior of neuronal
membranes, focusing on their responses to stimuli measured with power spectra
in the frequency domain. It explores the use of linear and nonlinear (quadratic
sinusoidal analysis) approaches to characterize neuronal function. It further
delves into the random theory of internal noise of biological neurons and the
use of stochastic Markov models to investigate these fluctuations. The text
also discusses the origin of conductance noise and compares different power
spectra for interpreting this noise. Importantly, it introduces a novel
sequential chemical state model, named p2, which is more general than the
Hodgkin-Huxley formulation, so that the probability for an ion channel to be
open does not imply exponentiation. In particular, it is demonstrated that the
p2 (without exponentiation) and n4 (with exponentiation) models can produce
similar neuronal responses. A striking relationship is also shown between
fluctuation and quadratic power spectra, suggesting that voltage-dependent
random mechanisms can have a significant impact on deterministic nonlinear
responses, themselves known to have a crucial role in the generation of action
potentials in biological neural networks.
