Illustration of limit cycle repair. The left column indicates four networks with logical update rules. In each network, a node x is marked in red to indicate f
= OFF when the network is damaged. Black, solid arrows indicate positive or mixed regulation and red, dashed arrows indicate negative regulation, as described in the accompanying Boolean update rules. The right column shows a portion of the synchronous state transition networks corresponding to each network on the left. States with a grey border correspond to a limit cycle of the undamaged network, a, with grey arrows indicating the procession among the states. States with a grey fill correspond to an attractor of the damaged network, with solid black arrows indicating the procession among the states. All other states act as transient states in the damaged network. Dashed black arrows indicate the modification of a state due to the damage (i.e. the turning off of the red node). Blue arrows with unfilled arrowheads indicate desired transitions, to be achieved through interaction modification, that stabilize the damage-equivalent attractor a
d. (a-b) An example of a limit cycle where the network damage reduces the size of the repairable limit cycle (see Methods). Panels (c-d), (e-f), and (g-h) show limit cycles that cannot be repaired with the methodology presented in this report, illustrating cases 1, 2, and 3 (see Methods).