  The classical Hodgkin--Huxley (HH) model neglects the time-dependence of ion
concentrations in spiking dynamics. The dynamics is therefore limited to a time
scale of milliseconds, which is determined by the membrane capacitance
multiplied by the resistance of the ion channels, and by the gating time
constants. We study slow dynamics in an extended HH framework that includes
time-dependent ion concentrations, pumps, and buffers. Fluxes across the
neuronal membrane change intra- and extracellular ion concentrations, whereby
the latter can also change through contact to reservoirs in the surroundings.
Ion gain and loss of the system is identified as a bifurcation parameter whose
essential importance was not realized in earlier studies. Our systematic study
of the bifurcation structure and thus the phase space structure helps to
understand activation and inhibition of a new excitability in ion homeostasis
which emerges in such extended models. Also modulatory mechanisms that regulate
the spiking rate can be explained by bifurcations. The dynamics on three
distinct slow times scales is determined by the cell volume-to-surface-area
ratio and the membrane permeability (seconds), the buffer time constants (tens
of seconds), and the slower backward buffering (minutes to hours). The
modulatory dynamics and the newly emerging excitable dynamics corresponds to
pathological conditions observed in epileptiform burst activity, and spreading
depression in migraine aura and stroke, respectively.
