Heteroclinic networks for brain dynamics.

Heteroclinic networks are a mathematical concept in dynamic systems theory that is suited to describe metastable states and switching events in brain dynamics. The framework is sensitive to external input and, at the same time, reproducible and robust against perturbations. Solutions of the corresponding differential equations are spatiotemporal patterns that are supposed to encode information both in space and time coordinates. We focus on the concept of winnerless competition as realized in generalized Lotka-Volterra equations and report on results for binding and chunking dynamics, synchronization on spatial grids, and entrainment to heteroclinic motion. We summarize proposals of how to design heteroclinic networks as desired in view of reproducing experimental observations from neuronal networks and discuss the subtle role of noise. The review is on a phenomenological level with possible applications to brain dynamics, while we refer to the literature for a rigorous mathematical treatment. We conclude with promising perspectives for future research.

On relaxation times of heteroclinic dynamics.

Heteroclinic dynamics provide a suitable framework for describing transient dynamics such as cognitive processes in the brain. It is appreciated for being well reproducible and at the same time highly sensitive to external input. It is supposed to capture features of switching statistics between metastable states in the brain. Beyond the high sensitivity, a further desirable feature of these dynamics is to enable a fast adaptation to new external input. In view of this, we analyze relaxation times of heteroclinic motion toward a new resting state, when oscillations in heteroclinic networks are arrested by a quench of a bifurcation parameter from a parameter regime of oscillations to a regime of equilibrium states. As it turns out, the relaxation is underdamped and depends on the nesting of the attractor space, the size of the attractor's basin of attraction, the depth of the quench, and the level of noise. In the case of coupled heteroclinic units, it depends on the coupling strength, the coupling type, and synchronization between different units. Depending on how these factors are combined, finite relaxation times may support or impede a fast switching to new external input. Our results also shed some light on the discussion of how the stability of a system changes with its complexity.

Controlling the Mean Time to Extinction in Populations of Bacteria.

Populations of ecological systems generally have demographic fluctuations due to birth and death processes. At the same time, they are exposed to changing environments. We studied populations composed of two phenotypes of bacteria and analyzed the impact that both types of fluctuations have on the mean time to extinction of the entire population if extinction is the final fate. Our results are based on Gillespie simulations and on the WKB approach applied to classical stochastic systems, here in certain limiting cases. As a function of the frequency of environmental changes, we observe a non-monotonic dependence of the mean time to extinction. Its dependencies on other system parameters are also explored. This allows the control of the mean time to extinction to be as large or as small as possible, depending on whether extinction should be avoided or is desired from the perspective of bacteria or the perspective of hosts to which the bacteria are deleterious.

Coupled heteroclinic networks in disguise.

We consider diffusively coupled heteroclinic networks, ranging from two coupled heteroclinic cycles to small numbers of heteroclinic networks, each composed of two connected heteroclinic cycles. In these systems, we analyze patterns of synchronization as a function of the coupling strength. We find synchronized limit cycles, slowing-down states, as well as quasiperiodic motion of rotating tori solutions, transient chaos, and chaos, in general along with multistable behavior. This means that coupled heteroclinic networks easily come in disguise even when they constitute the main building blocks of the dynamics. The generated spatial patterns are rotating waves with on-site limit cycles and perturbed traveling waves from on-site quasiperiodic behavior. The bifurcation diagrams of these simple systems are in general quite intricate.

State estimation of power flows for smart grids via belief propagation.

Belief propagation is an algorithm that is known from statistical physics and computer science. It provides an efficient way of calculating marginals that involve large sums of products which are efficiently rearranged into nested products of sums to approximate the marginals. It allows a reliable estimation of the state and its variance of power grids that is needed for the control and forecast of power grid management. At prototypical examples of IEEE grids we show that belief propagation not only scales linearly with the grid size for the state estimation itself but also facilitates and accelerates the retrieval of missing data and allows an optimized positioning of measurement units. Based on belief propagation, we give a criterion for how to assess whether other algorithms, using only local information, are adequate for state estimation for a given grid. We also demonstrate how belief propagation can be utilized for coarse-graining power grids toward representations that reduce the computational effort when the coarse-grained version is integrated into a larger grid. It provides a criterion for partitioning power grids into areas in order to minimize the error of flow estimates between different areas.

Dynamically generated hierarchies in games of competition.

Spatial many-species predator-prey systems have been shown to yield very rich space-time patterns. This observation begs the question whether there exist universal mechanisms for generating this type of emerging complex patterns in nonequilibrium systems. In this work we investigate the possibility of dynamically generated hierarchies in predator-prey systems. We analyze a nine-species model with competing interactions and show that the studied situation results in the spontaneous formation of spirals within spirals. The parameter dependence of these intriguing nested spirals is elucidated. This is achieved through the numerical investigation of various quantities (correlation lengths, densities of empty sites, Fourier analysis of species densities, interface fluctuations) that allows us to gain a rather complete understanding of the spatial arrangements and the temporal evolution of the system. A possible generalization of the interaction scheme yielding dynamically generated hierarchies is discussed. As cyclic interactions occur spontaneously in systems with competing strategies, the mechanism discussed in this work should contribute to our understanding of various social and biological systems.

Long-period clocks from short-period oscillators.

We analyze repulsively coupled Kuramoto oscillators, which are exposed to a distribution of natural frequencies. This source of disorder leads to closed orbits of repetitive temporary patterns of phase-locked motion, providing clocks on macroscopic time scales. The periods can be orders of magnitude longer than the periods of individual oscillators. By construction, the attractor space is quite rich. This may cause long transients until the deterministic trajectories find their stationary orbits. The smaller the width of the distribution about the common natural frequency, the longer are the emerging time scales on average. Among the long-period orbits, we find self-similar sequences of temporary phase-locked motion on different time scales. The ratio of time scales is determined by the ratio of widths of the distributions. The results illustrate a mechanism for how simple systems can provide rather flexible dynamics, with a variety of periods even without external entrainment.

Islanding the power grid on the transmission level: less connections for more security.

Islanding is known as a management procedure of the power system that is implemented at the distribution level to preserve sensible loads from outages and to guarantee the continuity in electricity supply, when a high amount of distributed generation occurs. In this paper we study islanding on the level of the transmission grid and shall show that it is a suitable measure to enhance energy security and grid resilience. We consider the German and Italian transmission grids. We remove links either randomly to mimic random failure events, or according to a topological characteristic, their so-called betweenness centrality, to mimic an intentional attack and test whether the resulting fragments are self-sustainable. We test this option via the tool of optimized DC power flow equations. When transmission lines are removed according to their betweenness centrality, the resulting islands have a higher chance of being dynamically self-sustainable than for a random removal. Less connections may even increase the grid's stability. These facts should be taken into account in the design of future power grids.

Physical aging of classical oscillators.

Aging is a familiar phenomenon from glassy systems like spin glasses and materials with slow relaxation processes, breaking of time-translation invariance, and dynamical scaling. We study aging in active rotators and Kuramoto oscillators that are coupled with frustrated bonds. The induced multiplicity of attractors of fixed-point or limit-cycle solutions leads to a rough potential landscape. When the system is exposed to noise, the oscillator phases migrate through this landscape and generate a multitude of different escape times from one metastable state to the next. When the system is quenched from the regime of a unique fixed point toward the regime of multistable limit-cycle solutions, the autocorrelation functions depend on the waiting time after the quench and show dynamical scaling. In this way we uncover a common mechanism behind aging in quite different realizations.

Networks of coupled circuits: from a versatile toggle switch to collective coherent behavior.

We study the versatile performance of networks of coupled circuits. Each of these circuits is composed of a positive and a negative feedback loop in a motif that is frequently found in genetic and neural networks. When two of these circuits are coupled with mutual repression, the system can function as a toggle switch. The variety of its states can be controlled by two parameters as we demonstrate by a detailed bifurcation analysis. In the bistable regimes, switches between the coexisting attractors can be induced by noise. When we couple larger sets of these units, we numerically observe collective coherent modes of individual fixed-point and limit-cycle behavior. It is there that the monotonic change of a single bifurcation parameter allows one to control the onset and arrest of the synchronized oscillations. This mechanism may play a role in biological applications, in particular, in connection with the segmentation clock. While tuning the bifurcation parameter, also a variety of transient patterns emerges upon approaching the stationary states, in particular, a self-organized pacemaker in a completely uniformly equipped ensemble, so that the symmetry breaking happens dynamically.

Caveats in modeling a common motif in genetic circuits.

From a coarse-grained perspective, the motif of a self-activating species, activating a second species that acts as its own repressor, is widely found in biological systems, in particular in genetic systems with inherent oscillatory behavior. Here we consider a specific realization of this motif as a genetic circuit, termed the bistable frustrated unit, in which genes are described as directly producing proteins. Upon an improved resolution in time, we focus on the effect that inherent time scales on the underlying scale can have on the bifurcation patterns on a coarser scale. Time scales are set by the binding and unbinding rates of the transcription factors to the promoter regions of the genes. Depending on the ratio of these rates to the decay times of both proteins, the appropriate averaging procedure for obtaining a coarse-grained description changes and leads to sets of deterministic equations, which considerably differ in their bifurcation structure. In particular, the desired intermediate range of regular limit cycles fades away when the binding rates of genes are not fast as compared to the decay time of the proteins. Our analysis illustrates that the common topology of the widely found motif alone does not imply universal features in the dynamics.

Noise as control parameter in networks of excitable media: Role of the network topology.

We analyze coupled FitzHugh-Nagumo oscillators on various network topologies, in particular random diluted and scale-free topologies, under the influence of noise. Similarly to globally coupled excitable units, noise acts as control parameter: changing monotonically its strength, the collective dynamical behavior varies from stable equilibrium solutions to coherent firing of a large fraction, and for even stronger noise to incoherent firing leading to chaotic behavior of the excitable elements. For strong noise the system is less sensitive to the network topology. The specific topology enters via the degree of nodes and determines the average number of spikes. Apart from bifurcation regions, it is the ratio of noise intensity to size that determines the dynamical behavior of average values. Specific behavior such as limit cycles may then be realized for strong noise and large systems or for low noise and small systems. Within bifurcation regions, the actual values of noise intensity and system-size matter independently. Here we analyze in more detail phase portraits of small systems. For a given noise intensity and network topology we have studied the regularity of signals as a function of time. We observe the phenomenon of system-size resonance for a whole interval of noise intensities as long as the degree distribution is homogeneous, so that no fine tuning of the noise is needed. Therefore it is plausible that natural systems make actually use of noise when noise is unavoidably present.

On the role of frustration in excitable systems.

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."

Universality class of triad dynamics on a triangular lattice.

We consider triad dynamics as it was recently considered by Antal [Phys. Rev. E 72, 036121 (2005)] as an approach to social balance. Here we generalize the topology from all-to-all to the regular one of a two-dimensional triangular lattice. The driving force in this dynamics is the reduction of frustrated triads in order to reach a balanced state. The dynamics is parametrized by a so-called propensity parameter p that determines the tendency of negative links to become positive. As a function of p we find a phase transition between different kinds of absorbing states. The phases differ by the existence of an infinitely connected (percolated) cluster of negative links that forms whenever pp(c). From a finite-size scaling analysis we numerically determine the static critical exponents beta and nu(perpendicular) together with gamma, tau, sigma, and the dynamical critical exponents nu(parallel) and delta. The exponents satisfy the hyperscaling relations. We also determine the fractal dimension d(f) that satisfies a hyperscaling relation as well. The transition of triad dynamics between different absorbing states belongs to a universality class with different critical exponents. We generalize the triad dynamics to four-cycle dynamics on a square lattice. In this case, again there is a transition between different absorbing states, going along with the formation of an infinite cluster of negative links, but the usual scaling and hyperscaling relations are violated.

Social balance as a satisfiability problem of computer science.

Reduction of frustration was the driving force in an approach to social balance as it was recently considered by Antal [T. Antal, P. L. Krapivsky, and S. Redner, Phys. Rev. E 72, 036121 (2005)]. We generalize their triad dynamics to k-cycle dynamics for arbitrary integer k. We derive the phase structure, determine the stationary solutions, and calculate the time it takes to reach a frozen state. The main difference in the phase structure as a function of k is related to k being even or odd. As a second generalization we dilute the all-to-all coupling as considered by Antal to a random network with connection probability w<1. Interestingly, this model can be mapped to a satisfiability problem of computer science. The phase of social balance in our original interpretation then becomes the phase of satisfaction of all logical clauses in the satisfiability problem. In common to the cases we study, the ideal solution without any frustration always exists, but the question actually is as to whether this solution can be found by means of a local stochastic algorithm within a finite time. The answer depends on the choice of parameters. After establishing the mapping between the two classes of models, we generalize the social-balance problem to a diluted network topology for which the satisfiability problem is usually studied. On the other hand, in connection with the satisfiability problem we generalize the random local algorithm to a p-random local algorithm, including a parameter p that corresponds to the propensity parameter in the social balance problem. The qualitative effect of the inclusion of this parameter is a bias towards the optimal solution and a reduction of the needed simulation time.

Reentrant synchronization and pattern formation in pacemaker-entrained Kuramoto oscillators.

We study phase entrainment of Kuramoto oscillators under different conditions on the interaction range and the natural frequencies. In the first part the oscillators are entrained by a pacemaker acting like an impurity or a defect. We analytically derive the entrainment frequency for arbitrary interaction range and the entrainment threshold for all-to-all couplings. For intermediate couplings our numerical results show a reentrance of the synchronization transition as a function of the coupling range. The origin of this reentrance can be traced back to the normalization of the coupling strength. In the second part we consider a system of oscillators with an initial gradient in their natural frequencies, extended over a one-dimensional chain or a two-dimensional lattice. Here it is the oscillator with the highest natural frequency that becomes the pacemaker of the ensemble, sending out circular waves in oscillator-phase space. No asymmetric coupling between the oscillators is needed for this dynamical induction of the pacemaker property nor need it be distinguished by a gap in the natural frequency.

Entrainment of coupled oscillators on regular networks by pacemakers.

We study Kuramoto oscillators, driven by one pacemaker, on d-dimensional regular topologies with nearest neighbor interactions. We derive the analytical expressions for the common frequency in the case of phase-locked motion and for the critical frequency of the pacemaker, placed at an arbitrary position in the lattice, so that above the critical frequency no phase-locked motion is possible. We show that the mere change in topology from an open chain to a ring induces synchronization for a certain range of pacemaker frequencies and couplings, while keeping the other parameters fixed. Moreover, we demonstrate numerically that the critical frequency of the pacemaker decreases as a power of the linear size of the lattice with an exponent equal to the dimension of the system. This leads in particular to the conclusion that for infinite-dimensional topologies the critical frequency for having entrainment decreases exponentially with increasing size of the system, or, more generally, with increasing depth of the network, that is, the average distance of the oscillators from the pacemaker.

Self-similar scale-free networks and disassortativity.

Self-similar networks with scale-free degree distribution have recently attracted much attention, since these apparently incompatible properties were reconciled in [C. Song, S. Havlin, and H. A. Makse, Nature 433, 392 (2005)] by an appropriate box-counting method that enters the measurement of the fractal dimension. We study two genetic regulatory networks (Saccharomyces cerevisiae [N. M. Luscombe, M. M. Babu, H. Yu, M. Snyder, S. Teichmann, and M. Gerstein, Nature 431, 308 (2004)] and Escherichia coli [http://www.ccg.unam.mx/Computational_Genomics/regulondb/DataSets/RegulonNetDataSets.html and http://www.gbf.de/SystemsBiology]) and show their self-similar and scale-free features, in extension to the datasets studied by [C. Song, S. Havlin, and H. A. Makse, Nature 433, 392 (2005)]. Moreover, by a number of numerical results we support the conjecture that self-similar scale-free networks are not assortative. From our simulations so far these networks seem to be disassortative instead. We also find that the qualitative feature of disassortativity is scale-invariant under renormalization, but it appears as an intrinsic feature of the renormalization prescription, as even assortative networks become disassortative after a sufficient number of renormalization steps.