Mechanism of quasi-periodic lag jitter in bursting rhythms by a neuronal network

Abstract

We study a heteroclinic bifurcation leading to the onset of robust phase-lag jittering in bursting rhythms generated by a neuronal circuit. We show that the jitter phenomenon is associated with the occurrence of a stable invariant curve emerging through a torus bifurcation in 2D return maps for phase lags between three constituent bursters. To study biologically plausible and phenomenological models of rhythmic neuronal networks we have further developed parallel computational techniques for parameter continuations of all possible fixed points and invariant curves of such return maps. The method is based on a "fine" brute-force analysis of the large data set generated by the computational techniques.

Gait Transitions in a Phase Oscillator Model of an Insect Central Pattern Generator

“…Recent studies of different 3-cell ion-channel bursting CPG networks [25,26,27] share some common features with the current paper. Without explicitly addressing insect locomotion, or using phase reduction theory, the authors numerically extract Poincaré maps defined on 2-dimensional tori which have multiple stable fixed points corresponding to orbits with specific phase differences.…”

“…Note that the estimated coupling strengths in only the second row of Table 3 approximately satisfy the balance equation (26) and also c 1 ≈ c 2 ≈ c 3 . Hence, as our analysis predicts, the system has 4 fixed points: a sink corresponding to a tripod gait, a source and 2 saddle points.…”

“…which both are zero by Equations (26). Therefore, (θ 1 1 , θ 1 2 ) is a fixed point of Equations (24).…”

“…Proposition 2. If the coupling strengths c i satisfy Equations (26) and (27), then for any η ∈ [0, T /6], Equations (20) admit the following stable T -periodic solutions…”

“…In Proposition 1, we provided sufficient conditions for the stability of tetrapod gaits when the coupling strengths satisfy the balance condition, Equation (26).…”