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Previously, examination of the king-of-the-mountain system revealed broad regions of purely pacemaker behavior within all Iapp ranges. This did not distinguish, however, between differences

in 1, 2, or 3 PM behaviors which are expected to dominate in one direction or the other with changes in two connections, g31 = g32. Regions of purely TW behavior were observed primarily

within oscillatory ranges between the release and escape cases, 0.45 < Iapp < 0.55, being largest at

patterns most often occurred briefly for connectivity above symmetry, with a small region of such behavior around symmetry at near-knee proximity of the nullcline within the release case. Additional phase-slip behavior was observed for Iapp below full symmetry, and additional in

extreme escape cases in the upper right quadrant with strong g31-g32 coupling. This connectivity

framework is again dominated by the various saddle-node bifurcation, with Andronov-Hopf and pitchfork bifurcations only occurring at symmetry in some cases with vertical transitions in Iapp.

Detailed examination in Figure 4.19 of traveling wave pattern formation separately from other patterns emphasizes a narrow range of dominance of these patterns, as seen in Figure 3.11, with TWs occurring primarily around full system symmetry, g31 = g32 = gij = 0.01, and in or near fully-

oscillatory systems with 0.45 < Iapp < 0.55. Most of the bi-parametric (g31=g32, Iapp)-space for which

these patterns exist is dominated exclusively by both traveling waves coexisting (lighter green region in the bifurcation diagram). This region extends upward and downward along Iapp near full

symmetry. A few singular cases of mono-TW behavior exist (darker green region in the bifurcation diagram), with such cases being dominated by clockwise patterns, for g31 = g32< 0.01, and counter-

clockwise patterns, for g31 = g32 > 0.01. Arrows in the figure indicate presence of saddle-node

bifurcations in which TW patterns are created or destroyed, via either the creation of both a FP- node and a saddle from an unstable point or collision of an FP-node and a saddle eliminating a rhythm and forming a repelling unstable point, and occur at every line in the system in this figure. Heteroclinic saddle-node bifurcations occur with the appearance of both traveling waves from purely phase-slip behavior, discussed later, on the right side of the light green region where g31 = g32 < 0.01. Andronov-Hopf bifurcation may occur through vertical transitions in Iapp at system

symmetry in asymmetric systems and would be observed here as a direct vertical transition from two traveling waves to none. As can be seen, this type of bifurcation occurs only at symmetry for

the lower or upper bounds of the escape and release case values of Iapp, respectively, or at near-

knee proximity of the nullclines for the release case, and are indicated by o marks.

Figure 4.19 Traveling wave formation within KOM networks

Detailed examination of TW pattern formation separately from all other behavior emphasizes dominance of these patterns around full system symmetry, g31 = g32 = gij = 0.01, and in or near

fully-oscillatory systems, with 0.45 < Iapp < 0.55. A singular exception exists in a small region of

symmetry at near-knee proximity of the nullcline within the release case. Most bi-parametric (g31=g32, Iapp)-space for which these exist is dominated by both TWs (lighter green region),

extending upward and downward along Iapp near full symmetry. Singular cases of mono-TW

behavior exist (darker green regions), with such cases being dominated by either CTW, for g31 = g32< 0.01, or CCTW, for g31 = g32> 0.01. Most TW-related bifurcations in the KOM motif are

standard SN bifurcations (indicated by arrows), with birth or destruction of both an FP-node and a saddle, a singular line of heteroclinic SN bifurcations occurring along the left edge where g31 = g32< 0.01. AH bifurcation occurs through vertical transitions at symmetry at the bounds of 2-TW

Detailed examination in Figure 4.20 of pacemaker formation separately emphasizes dominance of these patterns across the bi-parametric (g31=g32, Iapp)-space, with a narrow range extending upward

and downward along Iapp near full symmetry (g31 = g32 = gij = 0.01), line not shown here but visible

in Figure 3.11, in which all three pacemakers exist (bright blue region in the bifurcation diagram).

Figure 4.20 Pacemaker formation within KOM networks

Detailed examination of PM formation separately emphasizes dominance of these patterns across the bi-parametric (g31=g32, Iapp)-space, with a narrow range extending upward and downward

along Iapp near full symmetry (g31 = g32 = gij = 0.01), line not shown here but visible in Figure

3.11, in which all three PMs exist (bright blue). This vertical line also indicates the transition point at which 1-PM rhythms (light blue) gain or lose the red PM, at values of g31 = g32< 0.01 or g31 = g32> 0.01, respectively, at the outer limits. A small region of 2-PM behavior, in which only

the blue and green PMs occur, exists within the release case for g31 = g32 < 0.01. PM-related

bifurcations (key areas noted by arrows) occurring in the KOM motif are evenly split between standard and heteroclinic SN for both release and escape, and exclusively standard SN for oscillatory ranges between. Transitions from 1-3, 3-1, or 1-none are all heteroclinic. Heteroclinic SN bifurcation from a single red PM to phase-slip is discussed in Figure 5.17. Axes: Iapp = [0.39,

This vertical line also indicates the transition point at which singular-pacemaker rhythms

(light blue region in the bifurcation diagram) gain or lose the red pacemaker, at values of g31 = g32

< 0.01 or g31 = g32 > 0.01, respectively, at the outer limits. A small region of two-pacemaker

behavior, in which only the blue and green pacemakers occur, exists within the release case, for

g31 = g32< 0.01, and is indicative of inherent pacemaker rhythm formation in release case systems

at low synaptic coupling. Pacemaker-related bifurcations (noted by arrows in the bifurcation diagram) occurring in the king-of-the-mountain motif are undergone via a combination of both standard saddle-node and heteroclinic saddle-node bifurcations for both the release and escape case ranges of Iapp, Iapp < 0.45 and Iapp > 0.55, respectively, and exclusively through standard

saddle-node bifurcation for oscillatory ranges in between. Transitions from one-to-three, three-to- one, or one-to-no pacemakers are all heteroclinic in this system. Heteroclinic saddle-node bifurcation from a single red pacemaker is observed in the upper right quadrant for extreme escape cases with very strong coupling and is discussed in further detail in Figure 5.17. Although emphasis here is primarily on changes due to g31 = g32, pitchfork bifurcation may occur through

vertical transitions in Iapp at system symmetry in asymmetric systems and would be observed here

as a direct vertical transition from all three pacemaker rhythms to none. An example of this bifurcation, denoted an x in this figure, and in Figure 4.22, occurs only at the lower bounds of the release case at symmetry with near-knee proximity of the nullclines.

Detailed examination in Figure 4.21 of phase-slip formation separately from other patterns indicates that this transition occurs much more broadly for all oscillatory and escape case ranges of Iapp > 0.43 away from full symmetry (g31 = g32 = gij = 0.01), line not shown here but visible in

Figure 3.11. Before this line, one and two traveling wave patterns appear within oscillatory ranges of Iapp, while red pacemaker behavior appears from phase-slip within escape ranges, for g31 = g32

< 0.01. Beyond the line of symmetry, there is a narrow region (light gray middle region in the bifurcation diagram) for which one or both traveling wave patterns are lost via heteroclinic saddle- node bifurcation. In addition, within the escape case (right gray region in the bifurcation diagram) where the red pacemaker is ultimately lost via a heteroclinic bifurcation, in which the FP-node and saddles collide to form a heteroclinic loop, and phase-slip behavior is observed (examples of this

Figure 4.21 Phase-slip within KOM networks

Detailed examination of PS pattern formation separately from other behaviors indicates more extensive phase-slip away from full symmetry (g31 = g32 = gij = 0.01), line not shown here but

visible in Figure 3.11, for all oscillatory and escape case ranges of Iapp > 0.43. This vertical line

also indicates the transition point at which 1- or 2-TW patterns appear within oscillatory ranges of Iapp, or red PM behavior appears from PS within escape ranges, for g31 = g32 < 0.01. Above g31

= g32 > 0.01, there is a narrow region (light gray middle region) for which 1- or 2-TW patterns

disappear and within the escape case (right region) where the red PM is lost and PS behavior is observed. All PS-related bifurcations occurring in the KOM motif are heteroclinic SN

can be seen and are discussed further in Figures 3.10 and 5.16. All phase-slip bifurcations occurring in the king-of-the-mountain motif are created via heteroclinic SN bifurcations, indicated by arrows in the figure.

Figure 4.22 Detailed bifurcation transitions within KOM networks

Detailed overlay of bifurcations within the KOM asymmetric motif indicates extensive standard and heteroclinic SN bifurcation behavior. Below system symmetry (g31 = g32 < gij = 0.01),

heteroclinic SN bifurcations are characteristic for movement from PS to either PM or TW rhythms, but have no occurrence within the release case, Iapp < 0.45. Above system symmetry,

extensive occurrence of heteroclinic SN bifurcation is observed in direct transitions from 3-to-1 PM regimes within both release and escape, and in the appearance or disappearance of PS in the escape case. No such behavior is observed for the oscillatory ranges in between. The existence of a single pitchfork and three AH bifurcations occurs only at system symmetry with vertical transitions of Iapp, and is denoted by x and o marks, respectively. Where possible, all transitions

are clearly denoted but for complex cases around full symmetry, g31 = g32 = gij = 0.01, within

Combining all three detailed analyses of the three key rhythm types, seen in Figure 4.22, provides a comprehensive view of the occurrence of the bifurcations described. Within the king-of-the- mountain asymmetric motif, only standard saddle-node and heteroclinic saddle-node bifurcations occur with horizontal transitions in g31 = g32. Three cases of Andronov-Hopf bifurcation are

observed for both the release and escape cases at full symmetry (g31 = g32 = gij = 0.01), occurring

with vertical transitions in Iapp taking the system outside the lowermost and uppermost region of

two traveling wave rhythms, respectively, and again for only the extreme release case at near-knee proximity of the nullclines (shown with o marks here and in Figure 4.19). A singular case of pitchfork bifurcation is also observed in the extreme near-knee release case (marked by an x here and in Figure 4.20).