Figure 1.
Neural tissue as a composite system with walls and surroundings.
The ion–based model describes a system, comprising extracellular and intracellular compartments separated by a membrane, and the surroundings of the system. The latter provides an energy source and, if the system is not closed, also an ion reservoir.
Table 1.
Parameters for ion–based model.
Figure 2.
Bifurcation diagram of the reduced model for as the bifurcation parameter (purely transmembrane dynamics) showing (a) the membrane potential of fixed points (FP) and limit cycles (LC), and (b) potassium concentrations. The fixed point continuation yields the black curves. Solid sections are fully stable, dashed sections are unstable. The stability of the fixed point changes in HBs and LPs. The initial physiological condition is marked by a black square. The limit cycle is represented by the extremal values of the dynamical variables during one oscillation. The continuation yields the green lines with the same stability convention for solid and dashed sections. The stability of the limit cycle changes either in a LP
or in a period–doubling bifurcation (PD). In (b) the maximal and minimal extracellular potassium concentration of the limit cycle never differs by more than
mM. The values can hence not be distinguished on the scale of this figure and therefore only the maximal value is drawn. The bifurcations are marked by full circles and labelled by the type, i.e., HB, LP or
, and a counter (cf. also the insets with blow–ups, in particular the rightmost one showing
and
on a very small horizontal scale). The vertical and diagonal arrows labelled ‘m’ and ‘r’ indicate the direction of extracellular potassium changes due to ion fluxes across the membrane (‘m’) and changes only due to
, i.e., because of ion exchange with a reservoir (‘r’). Note that along the horizontal directions only the ICS potassium concentration changes by a precise mixture of fluxes across the membrane and ion exchange with a reservoir.
Figure 3.
Bifurcation diagram of the model from Kager et al. (cf. last paragraph of Sect. Models). Like in Fig. 2 panel (a) shows the membrane potential and panel (b) shows the extracellular potassium concentration of the invariant sets, i.e., fixed points and limit cycles. The line style convention (solid for stable, dashed for unstable) and bifurcation labels are the same as in Fig. 2. Note the similar shape to Fig. 2, but also the different scale of the two figures.
Figure 4.
Fixed point continuations for a range of impermeant intracellular chloride concentrations in (a), (b) the
–plane and (c), (d) the
–plane. The black curves are the stable FES branches that lose their stability in Hopf bifurcations (black circles). Starting from the leftmost fixed point curves the fixed
values are 8, 12, 16, 20, 24, 28 and 32 mM for the reduced model and 9, 13, 17, 21, 25, 29 and 33 mM for the detailed model. The Hopf bifurcations for different chloride concentrations lead to the blue Hopf line. As a reference the fixed point curves from Figs. 2 and 3 are also included in the diagram and drawn in grey.
Figure 5.
Time series for single SD excursions in (a), (c) the reduced and in (b), (d) the detailed model. In the reduced model SD is triggered by an interruption of the pump activity for about 10 sec (shaded region). In the detailed model the extracellular potassium concentration is increased by mM after 20 sec (vertical line). In (a) and (b) the time series of the membrane potentials (black lines) are shown. Nernst potentials for all ion species are included to the diagrams as a reference. Ion dynamics are shown in (c) and (d) where extracellular ion concentrations are in lighter color.
Figure 6.
Phase space plots of the simulations in Fig. 5. As in Fig. 4 panels (a) and (b) contain plots of the membrane potentials, in panels (c) and (d) extracellular potassium is shown. (a) and (c) are for the reduced model, (b) and (d) for the detailed model. The trajectories of the reduced model are represented as red curves, those of the detailed model are magenta. The sections of the trajectories that belong to times before and during the stimulation are dashed. The fixed point curves from Fig. 4 are added on the plots as shaded lines whereas the fixed point continuations for the unbuffered models with dynamical chloride are slightly darker. The pair of arrows in the extracellular potassium plots indicates the direction of pure transmembrane (vertical) and pure buffering dynamics (diagonal).
Figure 7.
Time series for three types of oscillatory dynamics in the bath coupled reduced model. In the left panels (a), (c) and (e) the membrane potential and the three Nernst potentials are shown. Ion concentrations are shown in the right panels (b), (d) and (f). The color code is as in Fig. 5. (a) and (b), (c) and (d), and (e) and (f) are simulations for ,
and
, respectively. The dynamics is typical for (a) and (b) seizure–like activity, (c) and (d) tonic firing, (e) and (f) periodic SD. Note the different time scales of SLA, tonic firing and period SD and also the different oscillation amplitudes in the ionic variables.
Figure 8.
Bifurcation diagram of the bath coupled reduced model for –variation. Color and line style conventions for fixed points and limit cycles are Figs. 2 and 3: black and green lines are fixed point and limit cycles, solid and dashed line styles mean stable and unstable sections. Stable solution on invariant tori are blue. They were obtained by direct simulations. The fixed point changes stability in HBs and LPs. The bifurcation types limit cycle undergoes are
, period–doubling (PD) and torus bifurcation (TR). Some physiologically irrelevant unstable limit cycles are omitted (cf. text). Panel (a) shows the membrane potential, panel (b) shows the extracellular potassium concentration. (b) does not contain the limit cycle, because it can hardly be distinguished from the fixed point line.
Figure 9.
Different representations of the bifurcation diagram of Fig. 8. Panel (a) shows the extracellular sodium concentration and includes an inset around TR4 and PD. Panel (b) presents the potassium gain/loss.
Figure 10.
Phase space plots of the simulations (a) for SLA and (b) periodic SDs from Fig. 7. Only extracellular potassium is shown. The limit cycle and fixed point curves from Figs. 2 and 4 are superimposed to the plots as shaded lines whereas the limit cycle and fixed point from Fig. 2 (dynamical chloride) are darker. The limit cycle and fixed point are not graphically distinguished, but comparison with Fig. 2 should avoid confusion.
Figure 11.
Fundamental bifurcation diagram in the slowest–scale dynamics, the potassium ion gain or loss through reservoirs (i.e., the bifurcation parameter). The unit of the bifurcation parameter was chosen such that it denotes the ion concentration with respect to the extracellular volume. The actual extracellular potassium concentration is the order parameter. Shown are the stable branches and
(see Sec. Results) and the directions (arrows) of two paths of ‘pure’ flux condition: fluxes exclusively across the membrane and fluxes exclusively from (or to) reservoirs. A horizontal path is caused by a particular mixture of these fluxes that induces potassium ion concentration changes exclusively to the intracellular compartment. Ionic excitability can be understood as a cyclic process in this diagram (see text).