Figure 1.
Type 1 and Type 2 dynamics in the hippocampal neuron model.
The time-course of the membrane voltage with increasing steady current for low and high K channel densities. (A) = 20 µm/s and
= 5 µm/s. The onset frequency is infinitely small. (B)
= 20 µm/s and
= 5 µm/s. The onset frequency is 30 Hz. Note the damped oscillation with stimulation at 114 mA/m2.
Figure 2.
Oscillation maps for the hippocampal neuron model.
(A) Regions in the −
plane associated with different threshold dynamics. Oscillations occur within the area defined by the continuous line. Double-limit cycle bifurcations in the A2 region, Andronov-Hopf bifurcations (together with double-limit cycle bifurcations) in the B region and saddle-node bifurcations in the C1 region. The bold dashed line indicates the border for channel densities associated with three stationary potentials. The map is a projection of a curved plane in the
−
−Istim space (on which the oscillation starts) to the
−
plane. (B) The corresponding three-dimensional map, showing the volume associated with oscillations in the
−
−Istim space. Oscillations occur in the volume defined by blue and green surfaces. The green surface area represents double-limit cycle bifurcations and the blue area saddle-node bifurcations (SNICs).
Figure 3.
Bifurcation diagrams for the hippocampal neuron model.
(A) A saddle node bifurcation in region C1. There are three stationary voltages in the Istim range of −40 to +50 mA/m2. The oscillations occur when the stable stationary potential Vs1 merges with a saddle point voltage Vs2. Type 1 threshold dynamics is generated if the limit cycle involves the merged point, i.e. a saddle-node bifurcation on an invariant circle (SNIC). = 20 µm/s,
= 2 µm/s. (B) Subcritical Andronov-Hopf and double-limit cycle bifurcations in region B,
= 20 µm/s,
= 10 µm/s. The oscillations emerge at Istim = 84 mA/m2, thus when the corresponding stationary point/voltage still is stable. The loss of stability is due to a double-limit cycle bifurcation, characterized in the variable space by the simultaneous appearance of two limit cycles of opposite stability, one yielding stable and persistent oscillations. This bifurcation is not detectable by the Jacobian matrix of the stationary point; instead the bifurcation depends on the global properties of the variable space. The local Andronov-Hopf bifurcation (also named degenerate Andronov-Hopf bifurcation because of the way the limit cycles collapse onto the equilibrium point [21], [29]) occurs at Istim = 92 mA/m2. There is also an additional Andronov-Hopf bifurcation at higher Istim (524 mA/m2, now shown) that terminates the oscillations. (C) For higher values of
(region A2) these two Andronov-Hopf points collide and disappear (the non-transversal Andronov-Hopf bifurcation), after which no Andronov-Hopf points are present
= 20 µm/s,
= 20 µm/s.
Figure 4.
Prerequisites for three stationary potentials (defining region C1).
Steady-state currents versus membrane voltage for the hippocampal neuron model. Calculated from Equation 17. The Na channel density is varied while other parameters are maintained constant to demonstrate the requirement of a high Na channel density to obtain three stationary potentials. Inward currents are shown as positive. (A) = 30 µm/s and
= 5 µm/s. (B)
= 11 µm/s and
= 5 µm/s.
Figure 5.
Type 2 dynamics within region C1 for the hippocampal neuron model.
The time-course of the membrane voltage with increasing steady current. = 40 µm/s and
= 15 µm/s. The onset frequency is 8 Hz.
Figure 6.
A Andronov-Hopf bifurcation within region C1.
Schematic bifurcation diagram showing a subcritical Andronov-Hopf bifurcation within the range of three stationary potentials. The distance between the Andronov-Hopf bifurcation and the coalescence of Vs1 and Vs2 has been extrapolated.
Figure 7.
Revised oscillation maps for the hippocampal neuron model.
Regions associated with oscillations in the −
plane, showing the existence of Type 2 dynamics within region C1. (A) Onset frequencies. (B) Oscillation map for comparison with the map of Fig. 2, showing the subregions C1a and C1b. The border between C1a and C1b closely follows the Bogdanov-Takens bifurcation curve (see Table 1).
Table 1.
Characterization of Regions in the Channel-density Plane of the Models.
Figure 8.
Bifurcation curves and the three-root solution space for the hippocampal neuron model.
Istim− diagrams at
= 40 µm/s. The thick continuous line defines the region associated with three-root solutions of Equation 17. The thin continuous line is the Andronov-Hopf bifurcation curve and the hatched line, defining the oscillation limit, is the double limit cycle bifurcation curve. (A) An overall perspective. (B) A detailed view of the cusp of the three-root solution space to describe the two subregions, defined by the stability of the stationary potentials. The Bogdanov-Takens bifurcation point is marked.
Table 2.
Kinetic Parameter Values for the Models.
Figure 9.
Exclusively Type 2 dynamics in the myelinated axon model.
The time-course of the membrane voltage with increasing steady current for low and high K channel densities. (A) = 300 µm/s and
= 0 µm/s. The onset frequency is 59Hz. (B)
= 300 µm/s and
= 40 µm/s. The onset frequency is 139Hz.
Figure 10.
Bifurcation curves and the three-root solution space for the myelinated axon model.
Istim− diagrams at
= 200 µm/s. The thick continuous line defines the region associated with three-root solutions of Equation 14. The thin continuous line is the Andronov-Hopf bifurcation curve and the hatched line is the double limit cycle bifurcation curve. (A) An overall perspective. (B) A detailed view of the cusp of the three-root solution space to describe the subregions.
Figure 11.
Exclusively Type 2 dynamics in the squid axon model.
The time-course of the membrane voltage with increasing steady current for low and high K channel densities. (A) = 1200 S/m2 and
= 50 S/m2. Onset frequency is 22 Hz. (B)
= 1200 S/m2 and
= 360 S/m2 (values used by Hodgkin and Huxley in their original study from 1952 [23]). The onset frequency is 52 Hz.
Figure 12.
Bifurcation curves and the three-root solution space for the myelinated axon model.
Istim− diagrams at
= 1200 S/m2. The thick continuous line defines the region associated with three-root solutions of Equation 14. The thin continuous line is the Andronov-Hopf bifurcation curve and the hatched line is the double limit cycle bifurcation curve. (A) An overall perspective. (B) A detailed view of the cusp of the three-root solution space to describe the three subregions. The Bogdanov-Takens bifurcation point is marked.
Figure 13.
Oscillation maps for the axon models.
Regions associated with oscillations in the −
or
−
plane. (A) The frog myelinated axon model. (B) The squid giant axon model. As seen there is no C1a region in any of the maps and consequently both axon models lack Type 1 dynamics. Note also that the myelinated axon model (A) allows oscillations for
= 0 (no K channels). (C) Onset frequency in the myelinated axon model. (D) Onset frequency in the squid axon model. Circles indicates the original values used by Hodgkin and Huxley for the model of the axon of Loligo forbesi [23] and Frankenhaeuser and Huxley for model of the sciatic nerve of Xenpus leavis [22].
Table 3.
Parameter Values for the Models.
Table 4.
Characterization of Stationary Points Based on the Eigenvalues from Equation 15.