Fig. 4.

Closed-loop fast subsystem undergoes bifurcations differently than the open-loop fast subsystem. A, top: bifurcation diagram of open-loop fast subsystem $\left(\dot{V}\text{,}\hspace{0.17em}\dot{n}\right)$ with bifurcation parameter h and g_tonic = 0.3 nS. Black curve $\mathcal{S}$ shows stable (thick lines) and unstable (thin lines) fixed points of the fast subsystem. Solid black dots indicate saddle-node (SN), Hopf (HB), and homoclinic (HC) bifurcations of the fast subsystem. Blue trace is the bursting trajectory from the open-loop system projected onto the h-V plane. Bottom: zoomed-in view of image at top, also showing the h-nullcline (dashed gray line). Open gray dot is an unstable fixed point of the full CPG subsystem $\left(\dot{V}\text{,}\hspace{0.17em}\dot{n}\text{,}\hspace{0.17em}\dot{h}\right)$; the bursting trajectory circumnavigates this unstable fixed point. Additional unstable fixed points located at (h = 0.20, V = −39) and (h = 0.02, V = −24) are not shown. B–D, top: bifurcation diagrams of closed-loop fast subsystem during silent phase (B), at the onset of spiking (C), and at the termination of spiking (D). Black trace is the closed-loop bursting trajectory, and gray curves show how $\mathcal{S}$ shifts as g_tonic varies during closed-loop bursting (the locations shown correspond to the points labeled B–D in Fig. 3). B: cyan dot shows the location of the trajectory at the minimum g_tonic value (0.12 nS) observed during closed-loop bursting. Lower portion of $\mathcal{S}$ and corresponding SN point are shifted to the right relative to the open-loop system, and the CPG is not spiking. Cyan arrow indicates that $\mathcal{S}$ will move to the left as the trajectory evolves and g_tonic increases through the remainder of the silent phase of the burst. C: green dot shows the location of trajectory at the maximum g_tonic value (0.22 nS) observed during closed-loop bursting. Lower portion of $\mathcal{S}$ and SN point are shifted to the left relative to B, and the CPG is about to start spiking. Green arrow indicates that $\mathcal{S}$ will move to the right as the trajectory evolves and g_tonic decreases, during the spiking phase of the burst. D: magenta dot shows the location of the trajectory at g_tonic = 0.22, which is near the HC bifurcation that terminates spiking. Lower portion of $\mathcal{S}$ is shifted to the left relative to B and to the right relative to C. Magenta arrow indicates that $\mathcal{S}$ will continue to move to the right until reaching the minimum g_tonic configuration shown in B. B–D, bottom: solid gray dots are stable fixed points of the full CPG subsystem $\left(\dot{V}\text{,}\hspace{0.17em}\dot{n}\text{,}\hspace{0.17em}\dot{h}\right)$. The trajectory does not circumnavigate these fixed points but exhibits bursting due to the movement of $\mathcal{S}$, the fast subsystem’s steady-state curve.