Relación familia – escuela

For pairwise-biased systems with two varying connections, g31 and g13, asymmetric transitions in

pattern generation and switching are again readily obtained away from system symmetry for all ranges of Iapp. As described in Chapters 3 and 4, these transitions occur almost exclusively via

saddle-node or heteroclinic saddle-node bifurcations. As in the example in Figure 5.1, here we observe a system in which a blue pacemaker rhythm coexists with an invariant circle near (∆12, ∆13) = (2/3, 1/3). As in Figure 5.5, additional asymmetry is introduced here, here with g13held at

half the strength of system symmetry, at g13=0.005, but with g31 synaptic coupling decreasing in

strength from symmetry at g31=gij=0.001 in a mono-biased fashion. With decreasing g31 coupling,

the blue PM basin of attraction is consumed by the purple invariant circle through heteroclinic saddle-node bifurcation in saddle-node to invariant circle, or SNIC-like, behavior. At a critical value of g31, a heteroclinic loop between the saddles occurs and a system with two different

coexisting invariant circles exists, one with s-shaped phase-slip behavior passing through regions that once characterized the red PM, black CTW, and blue PM patterns. This phase-slip pattern then consumes the basin of attraction of the purple invariant circle as its node and remaining saddle collide and eliminate one another. This behavior then converges to stereotypical phase-slip, like that existing in typical pairwise-biased asymmetric systems described in Chapter 4. All initial condition space solutions converge and remain on this phase-slip path, with regular repeating phase shifts passing through regions of all what used to be the standard rhythm patterns (transient seemingly red PM patterns shifting to black CTW, then through blue PM, and purple CCTW rhythms before returning to red PM behavior, and repeating continuously) laying between the position of the remaining saddle near the traditional green PM FP location near (∆12, ∆13) = (1/2,

E’, along with three traces at the lower right showing shifts in seeming PM to TW or vice versa slipping. Traces correspond to the transitions indicated by the arrows in the phase-basin representation in E’.

Figure 5.6 Invariant circle multiplicity in pairwise-biased systems

A system in which a blue PM coexists with an invariant circle near (∆12, ∆13) = (2/3, 1/3).

Additional asymmetry exists, with g13connectivity weak, at g13 = 0.005, and g31 decreasing in

strength from symmetry beginning at g31 = gij = 0.001. With decreasing g31 coupling, the blue

PM basin of attraction is consumed by the purple invariant circle via heteroclinic saddle-node bifurcation in SNIC-like behavior. At a critical value of g31, a heteroclinic loop between saddles

forms and two different invariant circles coexist, one purple SNIC-like case and one with s- shaped phase-slip passing through regions that once characterized the red PM, black CTW, and blue PM FP locations. This phase-slip then consumes the purple invariant circle basin as its node and remaining saddle collide. This converges to stereotypical phase-slip existing in typical pairwise-biased asymmetric systems, as g31 = g13. All solutions converge to this path, with

repeating phase shifts passing through regions of all 4 standard rhythm patterns (red PM → black CTW → blue PM → purple CCTW → red PM) occurring between the remaining saddle near (∆12, ∆13) = (1/2, 0) and (1/2, 1). Examples of shifts in phase-lag can be seen in E’ and the three

traces at the lower right, indicated by the arrows in the phase-basin representation shown in E’. Parameters: Iapp = 0.413, gij = 0.001 except g13 = 0.0005 and g31 = 0.001, 0.0009, 0.0008, 0.0007,

In Figure 5.7, an example is observed of a pairwise-biased system in which an unstable invariant circle, or ‘river’, along the ∆12-axis repels all trajectories away and toward one of the two traveling

wave fixed points, near (∆12, ∆13) = (1/3, 2/3) and (2/3, 1/3). With increasing g13=g31 coupling

strength, the two remaining TW FPs each approach one of the remaining black-purple saddles,

Figure 5.7 Heteroclinic SN bifurcation in pairwise-biased systems

Example of a pairwise-biased system with an unstable invariant circle, or ‘river’, along the ∆12-

axis which repels all trajectories away and toward one of the two TW fixed points (A), near (∆12, ∆13) = (1/3, 2/3) and (2/3, 1/3). With increasing g13 = g31 connectivity, each of the remaining FPs

approach one of the two remaining black-purple saddles, ultimately colliding and eliminating each other in two simultaneous heteroclinic saddle-node bifurcations. All solutions converge to this invariant circle with left-moving phase-slip behavior, with cells 1 and 3 remaining in quasi- antiphase, ∆13 ≈ 0.5, while cell 2 fires with shorter period slips continuously leftward in traces of

bursting activity. Parameters: Iapp = 0.5006, gij = 0.001 except g13 = g31 = 0.001, 0.0023649,

ultimately colliding and eliminating each other in two simultaneous heteroclinic saddle-node bifurcations. All initial condition space solutions converge to this invariant circle with left-moving phase-slip behavior, cells 1 and 3 remaining in quasi-antiphase, ∆13 ≈ 0.5, with oscillatory left- and

right-ward jitter, while cell 2 fires with shorter period and appears to slip continuously leftward in observation of individual traces of bursting activity in this system.