Table 1.
Parameters for Hodgkin–Huxley model.
Table 2.
Model parameters for ion–based model only.
Figure 1.
Upper panels: Membrane and Nernst potentials, lower panel: Ion concentrations vs time.
(a) Response of the model to a sec long sodium current pulse with amplitude 150
(marked by the black star). The pulse causes voltage spiking that stops in a strongly depolarized state (see blow–up inset). The membrane potential
takes a final value of about
(upper panel). The ion gradients, i.e., the differences between intra– and extracellular ion concentrations, reduce drastically during the stimulation and slowly adjust to a new fixed point after a couple of hundreds of seconds (lower panel). (b) Switching off the ion pump for
(indicated by the light grey interval) causes similar dynamics. The membrane depolarization and dissipation of ion gradients is a bit slower than for (a). After the pump is switched on again the system attains the same fixed point as in (a).
Figure 2.
Bifurcation diagram of the ion–based model.
Bifurcations are marked by red circles, the physiological equilibrium by a green square. Following the z–shaped fixed point characteristic from below there are two saddle–node bifurcations (limit point, LP) at
and
, and three subcritical Hopf bifurcations (HB) at
,
and
. The limit cycles created in HB1, HB2 and HB3 disappear in homoclinic bifurcations (HOM) at
,
and
, respectively. The second LP and the second HB together with the HOM of limit cycles occur in a very narrow parameter range (see blow–up inset). The number of stable (
) and unstable (
) directions of the fixed point is indicated by the
–tuples. There is bistability of a physiological state and a depolarized state with largely reduced ion concentration gradients between
and
.
Table 3.
Buffering parameters.
Figure 3.
Upper panels: Membrane and Nernst potentials, lower panel: Ion concentrations vs time.
(a) Stimulation with the same, but earlier applied, current pulse as in Fig. 1(a). Due to the additional potassium regulation the system returns to the physiological equilibrium after an approximately 60 sec lasting FES and subsequent hyperpolarization. (b) Similar dynamics as in (a) is observed for a temporary pump switch–off like in Fig. 1(b).
Figure 4.
Projection of the trajectories corresponding to Fig. 1a and Fig. 3a.
The shaded regions indicate unphysiological –combinations that imply negative
(lower left region) or negative
(upper region in left plot). Stable and unstable fixed points are marked by solid and open circles. (a) In the bistable case an initial stimulation (dashed line) leads to large subsequent changes in ion concentrations that terminate in the second fixed point of the system. (b) The excitable motion starts very similar to case (a), but after reaching the extremal concentration values the system slowly returns to its initial state.
Table 4.
Membrane permeabilities for GHK current.
Figure 5.
Bifurcation diagrams of fixed points for different models.
The effects of chloride and active ion channels are compared for each of the four possible pump (A vs B) and current model (Nernst vs GHK) combinations. The physiological equilibrium for normal pump rates (
and
) is marked by a green square. The model from fig. 2 is marked by a star. The value is the same with and without chloride or active channels. Insets show the bifurcation diagrams for low pump rates (
). Fixed point lines for models without active ion channels are shaded (see insets). Note the different scales on the main figures, insets are for the same range in each panel.
Figure 6.
Overview of the parameter regimes for bistability, polarized and depolarized stability for different models (1–8).
The change from the monostable depolarized regime to bistability (red to orange) defines the minimal physiological pump rate, i.e., the pump rate required for the existence of a polarized fixed point. The line separating the bistable from the monostable polarized regime (orange to green) defines the minimal recovery pump rate, i.e., the pump rate required to return from the depolarized fixed point to the polarized equilibrium. The model from Sec. Model is marked by a star.
Figure 7.
Bifurcation diagram of the fixed points for the Kager et al. model[17].
The physiological equilibrium is at
, the minimal physiological pump rate is
, and the recovery rate is
. The limit cycle emanating from the HB undergoes four saddle–node bifurcations of limit cycles (indicated by the stability changes, but not explicitly labled) before it disappears in a homoclinic bifurcation (HOM). Like in Fig. 5 the fixed point line for the corresponding leak–only model is shaded. Its value at
indicates the Donnan equilibrium.
Figure 8.
Two–parameter continuations of the fixed point bifurcations of Fig. 2.
In the left plot the dimensionless surface size parameter is varied, in the right plot the extracellular volume fraction
is changed. The insets show the LP1 curves that mark the minimal physiological pump rate. The pump rates for which the system is bistable range from the LP1 to the HB3. The HB3 pump rate is required to repolarize a neuron that is in the depolarized equilibrium. The parameter
(left plot plus inset) almost does not change the stability of the system, but
(right plot) reduces the recovery pump rate significantly. The inset shows that the minimal physiological pump rate is much less affected. In each plot and inset the standard parameter value is indicated by the light–blue vertical line.