(a) Positive feedback loop with self-activation coupled to a negative feedback loop with species and , as considered as bistable frustrated unit in Sec. III. (b) Repressilator (upper) loop of three mutually repressive units , coupled to a second loop of two repressive and two activating units for modeling cell-to-cell communication in a concrete context.
Trajectories in phase space of and for different values of and fixed values of , , and : (a) excitable behavior for ; [(b) and (c)] limit cycle behavior for and 95, respectively; (d) excitable behavior for . The dashed lines are the nullclines, their intersection indicates the location of the fixed-point.
Zoom into the phase space trajectories close to the fixed-point for two values of : (a) , (c) . The trajectories plotted as full lines, start close to the fixed-point and directly evolve into the fixed-point, the thick dashed trajectories start away from the fixed-point and make a long excursion in phase space, before they relax to the fixed-point. The nullclines are indicated as thin dashed lines. Parts (b) and (d) show the spikes in , corresponding to the excitatory excursions in phase space of (a) and (c), respectively.
Complex eigenvalues and for one BFU. The negative imaginary part is suppressed. Dotted-dashed line: , dashed line: , full line: positive part of . The meaning of the various -values is explained in the text.
Hysteresis in fixed-point and limit cycle behavior in the vicinity of subcritical Hopf bifurcations for two regimes of : [(a) and (c)] for small , [(b) and (d)] for large , where (a) and (b) show the period of the limit cycle, while (c) and (d) display their perimeters. The red (black) trajectories (points) result from initial conditions far away from (close to) the fixed-point, respectively. For a tiny interval in , we observe a slowly growing spiral trajectory outward with angular velocity , as predicted by the . These frequency values are indicated by the encircled points in (a).
Frequency for four noise intensities as a function of : (a) for internal multiplicative noise in , (b) for external multiplicative noise in . In (a) the oscillatory regime is extended for increasing noise intensities, the maximal frequencies are the same. In (b) range of limit cycles and maximal frequency are almost the same.
Composition of a single BFU (a), coupled BFUs [(b)–(d)] which are frustrated (f) or not (u), which act as repressors or activators on their neighbors.
Frustrated plaquette in the oscillatory regime of for with multistable behavior of three coexisting states, which are reached for different initial conditions. Multistability is manifested in three patterns of phase-locked motion of spikes in as a function of time: (a) phases of nodes 1 and 4 almost coincide, those of nodes 2 and 3 differ, corresponding to pattern (i), (b) all four phases are different, pattern (ii), (c) two of the four phases coincide, those of oppositely located nodes, pattern (iii). Simulations were run with the Runge–Kutta-4 algorithm and step size .
Frustrated plaquette in the excitatory regime of for with multistable behavior of three coexisting states, which are reached for different initial conditions. Multistability is manifest in one fixed-point solution, not displayed, and two patterns of phase-locked motion of spikes in as a function of time: (a) all four phases are different, (b) phases of oppositely located nodes agree. Simulations were run with Runge–Kutta-4 and .
Unfrustrated plaquette in the oscillatory regime of , otherwise , , and , for various values of . Left panels: amplitude of as a function of time for the four oscillators for four couplings: (a) , (b) , (c) , (d) ; right panels: corresponding average values of vs to illustrate incoherent (e) and coherent, i.e., phase locked motion with closed limit cycles [(f)–(h)], respectively. The amplitude of the activated node 2 (blue dotted line) is qualitatively different from the other repressed nodes for larger couplings [(c) and (d) ].
Stability properties of states: stands for decay into a state of the same phase locked pattern, for decay into a different pattern.
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