Figure 7

Capture due to a change in the topology of the phase portrait. A capture results from a trajectory being recruited into a new basin of attraction due to the movement of a separatrix. In this example, the relevant separatrix is created and caused to move by a preceding bifurcation event, which leads to the appearance of a new attractor state, resulting in a change of phase space topology. Upper panels show (quasi-)potential surfaces, lower panels phase portraits as in Figure 3C. The progress of time is shown through increasingly dark shading, and by the arrow at the bottom of the figure. (A) The system starts off in the bistable regime and the initial conditions place the trajectory in the basin of the high x, low y attractor (dark blue). (B) Changes in the values of the auto-activation thresholds (a and c, see equation 1) cause the system to undergo a subcritical pitchfork bifurcation and enter the tristable regime (see also Figure 2). At the time of the bifurcation, the trajectory is still attracted towards the dark blue attractor. (C) As auto-activation thresholds are further increased, the two separatrices surrounding the new attractor state (shown in light blue) separate from each other, enlarging the corresponding basin of attraction. A capture event takes place as the separatrix ‘overtakes’ the trajectory, recruiting it into the new basin of attraction. The system will now converge towards the light blue attractor. This change in basins of attraction is represented by the colour coding of the trajectory on the phase portrait. (D) As auto-activation thresholds are further increased, the system will transition from the tristable into the monostable regime (see also Figures 2 and 4). This causes the dark blue attractors and their basins to disappear altogether, but does not influence the direction of the trajectory anymore, which will eventually converge to the light blue attractor at low x and y. See also Additional file 4, Supporting Movie S4.