For

*MCN*1 rhythmic, fixed points can be lost in two ways: through a saddle-node bifurcation as

*s* slowly changes due to

*s*_{1} or when

*s*_{2} changes on the fast timescale. In all figures, the dashed vertical line is

*V*_{L} =

*V*_{T}. (A.) On the left branches of the nullclines,

*s*_{2} = 1 and

*s*_{1} moves the

*V*_{L} nullcline to the right resulting in a saddle-node bifurcation of the fixed point at ○ once

. The trajectory is forced to jump to the stable fixed point on the right branches of the nullclines (shown by ●). A similar transition can also occur when

*s*_{AB}_{→}_{I} = 1 and

*s*_{1} reaches

. (B.) When

*s*_{2} jumps from

*s*_{2}_{min} to 1, the

*V*_{L} nullcline jumps from the left (dotted nullcline) to the right (solid nullcline). Thus the fixed point on the left branches of the nullclines is instantaneously lost because

and the trajectory will jump to the fixed point on the right branches of the nullclines (shown by ●). A similar transition can also occur when

*s*_{AB}_{→}_{I} = 1 and

*s*_{2} jumps to 1 such that

. (C.) When

*MCN*1 is rhythmic and

*s*_{AB}_{→}_{I}(

*t*) oscillates between 0 and 1, the fixed point can be lost in another way. While

*s*_{2} = 1 and

*s*_{AB}_{→}_{I}(

*t*) jumps from 0 to 1 (the position of the

*V*_{I} nullcline jumps from the upward dotted nullcline to the lower solid nullcline), the fixed point on the left branches of the nullclines (shown by ○) is instantaneously lost because

and the solution trajectory is forced to jump to the fixed point on the right branches of the nullclines (shown by ●). (D.) On the right branches of the nullclines for

*MCN*1 tonic, when

*s*_{AB}_{→}_{I} jumps from 1 to 0 (the position of the

*V*_{I} nullcline jumps from the lower dotted nullcline to the upward solid nullcline), the fixed point (shown by ○) is instantaneously lost because

and the solution trajectory must jump to the stable fixed point on the left branches of the nullclines (shown by ●).

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