A mathematical model for the role of dopamine-D2 self-regulation in the production of ultradian rhythms
Fig 4
Bifurcation analysis of the core DUSR model.
A: Bifurcation diagrams for nine parameters with top local period sensitivity. Black curves indicate the stable (solid) and unstable (dotted) steady states for DAex. Coloured curves identify the limit-cycle oscillatory behaviours (max, min of the oscillation; period in colours). Stable DA equilibrium is replaced with self-sustained oscillations marked at the supercritical Hopf bifurcation point (red circle) and re-emerges from a saddle-node bifurcation (yellow star). An infinite period solution appears at the moment of the saddle-node bifurcation, forming a saddle-node on invariant cycle (SNIC) bifurcation and is characterized by heightened sensitivity of both the presence and the period of the DA oscillation to parameter values. B: The corresponding ultradian period at parameter values within the stable limit-cycle range. The majority of the period flexibility occurs within ±2 h of the nominal period (4.0 h) when shifting individual parameters within their oscillatory range. DAex timecourses for a long period (12 h) and a short period (3 h) are shown above and below the parameter legend.