Voltage-dependent ion channels as operators

Question

Does it make sense (or is even the standard approach) to model mathematically a voltage-dependent ion channel not as a function, that maps a voltage to a conductance ($f:\mathbb{R}\rightarrow \mathbb{R}$), but as an operator (in the sense of functional analysis), that maps a voltage profile (= function of time = element of a vector space) to a conductance profile (= function of time = element of another vector space)?

Are there references to the literature?

[Considered as operators it would be interesting to know in which ranges (of the vector space of voltage profiles) the channels behave linearly (in mathematical terms: are linear) - so we were done with knowing their behaviour for only a couple of base vectors (= profiles), i.e. square signals of equal height, but different widths. We would learn about rise times and time delays. In other ranges of course they behave quite non-linearily, esp. when the voltage profile crosses the threshold.]

Answer 1

Yes, they are nonlinear filters, which are operators on temporal signals. I have not seen a single ion channel analyzed as nonlinear filter though. Most of the time, a system containing ion channels (such as some neuron's membrane) is analyzed as such. Papers describing the biophysics of neurons and their function as filtering devices has been one of the topics in computational neuroscience.

• Gerstner, W. and Kistler, W. M. (2002). Spiking Neuron Models. Cambridge University Press
• Marmarelis, V. Z. (2004). Nonlinear dynamic modeling of physiological systems.