^{1,a)}and Hildegard Meyer-Ortmanns

^{1,b)}

### Abstract

We study the role of frustration in excitable systems that allow for oscillations either by construction or in an induced way. We first generalize the notion of frustration to systems whose dynamical equations do not derive from a Hamiltonian. Their couplings can be directed or undirected; they should come in pairs of opposing effects like attractive and repulsive, or activating and repressive, ferromagnetic and antiferromagnetic. As examples we then consider bistable frustrated units as elementary building blocks of our motifs of coupled units. Frustration can be implemented in these systems in various ways: on the level of a single unit via the coupling of a self-loop of positive feedback to a negative feedback loop, on the level of coupled units via the topology or via the type of coupling which may be repressive or activating. In comparison to systems without frustration, we analyze the impact of frustration on the type and number of attractors and observe a considerable enrichment of phase space, ranging from stable fixed-point behavior over different patterns of coexisting options for phase-locked motion to chaotic behavior. In particular we find multistable behavior even for the smallest motifs as long as they are frustrated. Therefore we confirm an enrichment of phase space here for excitable systems with their many applications in biological systems, a phenomenon that is familiar from frustrated spin systems and less known from frustrated phase oscillators. So the enrichment of phase space seems to be a generic effect of frustration in dynamical systems. For a certain range of parameters our systems may be realized in cell tissues. Our results point therefore on a possible generic origin for dynamical behavior that is flexible and functionally stable at the same time, since frustrated systems provide alternative paths for the same set of parameters and at the same “energy costs.”

The concept of frustration is a very generic one. In physics it is usually discussed in the context of Hamiltonian dynamics: what is curvature in general relativity is field strength in gauge theories and frustration in spin systems. Particularly from spin glasses it is known that the effect of frustration is to make the energy landscape rough with many local minima, to lead to glass-like behavior in relaxation times to stationary states. In this paper we want to study the effect of frustration on dynamical systems without an associated Hamiltonian, systems which belong to the broad class of excitable media. We present a qualitative criterion for frustration in terms of attractive and repulsive couplings when they are undirected as in case of coupled phase oscillators, and repressive and activating couplings when they are directed, as in case of excitable media. This criterion allows to decide whether a given link in a loop of coupled interacting units is frustrated or not. We then study the phase space of the most simple motifs which can be realized with and without frustration. We confirm the conjecture that the phase space of the frustrated realizations is considerably more versatile in the type and number of possible attractors as compared to realizations without frustration. We see fixed-point behavior with and without large excursions in phase space, various patterns of coexisting phase-locked motion in case of synchronized oscillations, and certain indications of chaotic behavior. The versatility gets lost when the frustration is reduced either by the number or by the type of couplings. Our application considered in this paper is so-called bistable frustrated units which we identify as excitable elements. They contain both individually an element of frustration and are coupled with frustration. These units can be realized in genetic systems, at least for a certain range of parameters (but we consider these units for parameters independent of their realizability in genetic systems). Flexible dynamical behavior along with versatile functions is, of course, well-known from biological systems. It is here our aim to illustrate how this rich and flexible dynamical behavior can be obtained by means of relatively simple building blocks, assigned to nodes and links of networks (here a network of spatially organized genes), once the main ingredient is frustration. Frustration can be realized via the very number (even or odd) and the very combination of couplings (repressive or activating). We expect in general that an appropriate degree of frustration makes a system flexible and stable in their functionality if it is not too low and at the same time stable against noise if it is not too high. Our results correspond to a step toward confirming this conjecture.

We would like to thank Sandeep Krishna for useful discussions during his visit at Jacobs University in an early stage of this work.

I. INTRODUCTION

II. QUALITATIVE CRITERIA FOR FRUSTRATION

III. COUPLED FRUSTRATED BISTABLE UNITS

A. A single BFU revisited

B. BFU under the influence of noise

C. Interacting BFUs

D. Frustration induced by the topology: A plaquette of BFUs

1. BFUs in the oscillatory regime

2. BFUs in the excitatory regime

3. The unfrustrated plaquette

4. The unfrustrated triangle

5. The frustrated triangle

IV. OUTLOOK AND CONCLUSIONS

### Key Topics

- Oscillators
- 13.0
- Attractors
- 11.0
- Bifurcations
- 11.0
- Anatomy
- 9.0
- Genetic networks
- 7.0

## Figures

(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.

(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.

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.

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.

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).

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.

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.

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 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 .

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) ].

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) ].

## Tables

Stability properties of states: stands for decay into a state of the same phase locked pattern, for decay into a different pattern.

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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