Quantifying the Relative Contributions of Divisive and Subtractive Feedback to Rhythm Generation
Figure 8
Alternate estimation of the relative contributions of each process using phase plane analysis.
A. Representation of the system in the a,s-plane. The system trajectory is shown as a thick black curve with arrows at the transitions between activity phases. The trajectory follows the dynamic nullcline (thin black S-shaped curve) which moves left during the silent phase and reaches the thick gray nullcline on the left at episode onset. At onset, the trajectory reaches the low knee (s(t) = sk(t)). During the active phase, the dynamic nullcline moves to the right toward the thick gray nullcline on the right. It reaches it at episode termination, as the trajectory reaches the high knee. Note that the lower portion of the nullcline is much more sensitive to θ than the higher portion. LK, low knee; HK, high knee of the a-nullcline. B. Variations of CAP and CSP (calculated using the phase plane approximation illustrated in A) with w (for τθ/τs = 1 and θ0 = 0). There is good agreement with the computational method based on small perturbations in the time constants (compare with Figure 6B). C. For large values of θ0, such that high activity episodes require that θ be close to 0, the computational calculation of the relative contributions (left panel) and the phase plane estimation (right panel) can disagree. In the case shown, τθ/τs = 0.1 (w = g = 1), so the phase plane method estimates that θ should control both active and silent phase (right panel). The disagreement with the computed CAP and CSP (left panel) is a result of the geometric argument used to estimate the contributions neglecting the fact that the speed of variation of s and θ can slow down dramatically when approaching their asymptotic values (see text).