Primera parte

For KOM systems with two varying connections, g31 and g32, asymmetric transitions occur away

from system symmetry for all ranges of Iapp. As described in Chapters 3 and 4, and in the pairwise-

biased systems, these transitions occur almost exclusively via saddle-node or heteroclinic saddle- node bifurcations. In Figure 5.8, like the previous example, a system in which a pair of heteroclinic saddle-node bifurcations occurs is seen, as the traveling wave fixed points, near (∆12, ∆13) = (1/3,

2/3) and (2/3, 1/3), each collide with one of the black-purple saddles and obliterate one another with increasing g31 = g32 coupling. Unlike in the preceding case, this leads to the disappearance of

both black and purple FP attractors with formation of a heteroclinic loop between the incoming separatrices of the remaining red saddles as trajectories are trapped between, with the red pacemaker continuing to coexist. Additional increases in g31=g32 coupling lead to additional

movement of the remaining two saddles that ultimately results in the switching of trajectories to the other side of the incoming separatrices, as seen in Figure 5.5, and acquisition of the entire initial condition space by the red pacemaker rhythm. This type of behavior could represent transitions in plasticity where some conditions can lead to seemingly erratic rhythmic behavior before settling back to a stability, and parallel brief shocks to a system in which transient seizure or abrupt and sporadic changes in behavior might be explained through network transition.

Figure 5.8 Loss of TWs with heteroclinic loops in KOM systems

As in the previous example, a pair of heteroclinic saddle-node bifurcations occur as the TW fixed points, near (∆12, ∆13) = (1/3, 2/3) and (2/3, 1/3), collide with the black-purple saddles and

obliterate one another with increasing g31 = g32 coupling. Unlike Figure 5.7, this leads to

disappearance of both black and purple FP attractors with formation of a heteroclinic loop

between the incoming separatrices of the remaining red saddles. Additional increases in coupling lead to further shifting of the remaining saddles that results in switching of trajectories to the other side of the incoming separatrices and acquisition of all IC-space by the red PM.

Parameters: Iapp = 0.5687, gij = 0.001 except g31 = g32 = 0.0012162, 0.0013514, and 0.0014865.

In Figure 5.9, another example of acquisition of the entire initial condition space by the red pacemaker is observed via simultaneous heteroclinic saddle-node bifurcations. In this example, pacemaker rhythms within the escape mechanism ranges of Iapp > 0.55 we can observe asymmetric

shifts in the acquisition of both the blue and green PM basins of attraction by the red attractor, as collision of the repellors, near (∆12, ∆13) = (1/3, 2/3) and (2/3, 1/3), with the both the left red-to-

blue and lower blue-to-green saddles leads to division of the traces within the original blue and green basins, passing near each of the regions of fleeting CTW and CCTW rhythmicity, by the red and green, and red and blue, basins respectively. Further increases in g31 = g32 synaptic coupling

strength lead to ultimate destruction of both the blue and green fixed point nodes via collision with the saddles near (∆12, ∆13) = (1/3, 3/5) and (2/3, 1/5) through saddle-node bifurcation.

Figure 5.9 PM basin acquisition via heteroclinic SN bifurcations

Another example of acquisition of the entire IC-space by the red PM via simultaneous

heteroclinic saddle-node bifurcations. Here, PM rhythms within the escape range of Iapp > 0.55

observe asymmetric shifts in acquisition of the blue and green PM basins of attraction by the red attractor, as collision of the repellors, near (∆12, ∆13) = (1/3, 2/3) and (2/3, 1/3), with the left red-

blue and lower blue-green saddles leads to splitting of traces passing near TW rhythmicity within the original blue and green basins by the red and green, and red and blue, basins respectively (B). Further increases in g31 = g32 coupling strength lead to the destruction of the blue and green FP

nodes through collision with the saddles near (∆12, ∆13) = (1/3, 3/5) and (2/3, 1/5) via saddle-

node bifurcation (C). Parameters: Iapp = 0.5858, gij = 0.001 except g31 = g32 = 0.00094595,

0.0010811, and 0.0012162.

In Figure 5.10, the system begins with coexistence of both a red pacemaker attractor and an unstable invariant circle (or repelling river in this case), near ∆13 ≈ 0.75. The saddles then split and

the red PM basin of attraction becomes blocked to all interior initial conditions by the incoming red saddle separatrices. Heteroclinic connection in the system now passes through all five of the traditional (∆12, ∆13)-space fixed-point locations and rhythms will observe phase-slip in which

transient portions may appear to briefly pass through regions in which traces mimic green PM, purple CCTW, red PM, blue PM, and black CTW alignment repetitively. With increased g31=g32

coupling, the red PM collides with the saddle and all initial condition space now converges to this periodic phase-slip behavior.

Figure 5.10 Heteroclinic connection between SN separatrices

Beginning in a system with coexistence of a red PM attractor and an unstable invariant circle (or repelling river), near ∆13 ≈ 0.75, saddles split and the red PM basin of attraction of red PM is

blocked to all interior ICs by the incoming saddle separatrices (B). Heteroclinic connection in the system now passes through all the traditional (∆12, ∆13) FP locations and rhythms undergo phase-

slip, in which transient portions may appear to briefly mimic green PM → purple CCTW, red PM → blue PM → black CTW alignment repetitively. With increased g31 = g32 coupling, the red

PM node collides with the saddle and all IC-space now converges to this periodic phase-slip behavior. Parameters: Iapp = 0.5943, gij = 0.001 except g31 = g32 = 0.0044595, 0.005, 0.0052703,

and 0.0055405.