# Voltage-dependent ion channels as operators

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.]