  Recent experiments have demonstrated that the timescale of adaptation of
single neurons and ion channel populations to stimuli slows down as the length
of stimulation increases; in fact, no upper bound on temporal time-scales seems
to exist in such systems. Furthermore, patch clamp experiments on single ion
channels have hinted at the existence of large, mostly unobservable,
inactivation state spaces within a single ion channel. This raises the question
of the relation between this multitude of inactivation states and the observed
behavior. In this work we propose a minimal model for ion channel dynamics
which does not assume any specific structure of the inactivation state space.
The model is simple enough to render an analytical study possible. This leads
to a clear and concise explanation of the experimentally observed exponential
history-dependent relaxation in sodium channels in a voltage clamp setting, and
shows that their recovery rate from slow inactivation must be voltage
dependent. Furthermore, we predict that history-dependent relaxation cannot be
created by overly sparse spiking activity. While the model was created with ion
channel populations in mind, its simplicity and genericalness render it a good
starting point for modeling similar effects in other systems, and for scaling
up to higher levels such as single neurons which are also known to exhibit
multiple time scales.
