Fig. 16. (a) Representative sub- (green) and supra-threshold orbits (red) and (b) their temporal evolution
Neuronal Dynamics and Brain Connectivity 31 Bifurcations and Neuronal Firing in Planar Models
A further examination of the equation for the ^-nullcline (37) shows that the synaptic current is a purely additive term. It hence acts to translate this nullcline in the vertical direction, with no influence on its shape and no influence on the n-nullcline. In Fig. 17, a close-up of the nullclines is shown for values of I = 0, 2, 4.51 and 6. As I is increased from 0 to 2 (dot-dashed), we see an upward shift of the ^-nullcline so that the saddle and node fixed points are closer together in phase space. At I = 4.5 (dashed), the nullclines are tangent and the fixed points have hence collided. At I = 6 (dotted) there are no nullcline intersections: hence their collision has led to their mutual annihilation!
This is exactly the "saddle-node" bifurcation defined at Fig. 12. In the present setting, the synaptic input I functions as the bifurcation parameter. However, in addition to the structure of Fig. 12, an additional "global" feature of the phase space in the current system requires consideration. When the fixed points collide, the short heterocline is abolished, but the long heterocline7 remains (Fig. 17b). Indeed even when I > 4.51 this orbit is still an invariant of the dynamics. However, with no fixed point along its domain, it is now a continuously looping limit cycle.
Figure 18 shows the limit cycle attractor (red) and its temporal dynamics for I = 4.75 (top row) and I = 6 (bottom row). Note that although the phase space portraits look similar, the frequency of the dynamics increases substantially with the increase in synaptic current.
This can be understood as a consequence of the bifurcation. Just after the bifurcation, although the nullclines do not intersect, the limit cycle must pass
Fig. 17. Saddle-node bifurcation in the planar system. (a) Nullclines near fixed points for I = 0, 2, 4.51, 6. Red circle denotes "saddle-node" fixed point (b) Homo-clinic orbit for the system when I = 4.51
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