Synchronization in complex networks of phase oscillators: A survey <s> Phase Balancing, Splay State, and Patterns <s> A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector q0 and frequency ω0 of the instability. Type Is systems (ω0=0, q0≠0) are stationary in time and periodic in space; type IIIo systems (ω0≠0, q0=0) are periodic in time and uniform in space; and type Io systems (ω0≠0, q0≠0) are periodic in both space and time. Near a continuous (or supercritical) instability, the dynamics may be accurately described via "amplitude equations," whose form is universal for each type of instability. The specifics of each system enter only through the nonuniversal coefficients. Far from the instability threshold a different universal description known as the "phase equation" may be derived, but it is restricted to slow distortions of an ideal pattern. For many systems appropriate starting equations are either not known or too complicated to analyze conveniently. It is thus useful to introduce phenomenological order-parameter models, which lead to the correct amplitude equations near threshold, and which may be solved analytically or numerically in the nonlinear regime away from the instability. The above theoretical methods are useful in analyzing "real pattern effects" such as the influence of external boundaries, or the formation and dynamics of defects in ideal structures. An important element in nonequilibrium systems is the appearance of deterministic chaos. A greal deal is known about systems with a small number of degrees of freedom displaying "temporal chaos," where the structure of the phase space can be analyzed in detail. For spatially extended systems with many degrees of freedom, on the other hand, one is dealing with spatiotemporal chaos and appropriate methods of analysis need to be developed. In addition to the general features of nonequilibrium pattern formation discussed above, detailed reviews of theoretical and experimental work on many specific systems are presented. These include Rayleigh-Benard convection in a pure fluid, convection in binary-fluid mixtures, electrohydrodynamic convection in nematic liquid crystals, Taylor-Couette flow between rotating cylinders, parametric surface waves, patterns in certain open flow systems, oscillatory chemical reactions, static and dynamic patterns in biological media, crystallization fronts, and patterns in nonlinear optics. A concluding section summarizes what has and has not been accomplished, and attempts to assess the prospects for the future. <s> BIB001 </s> Synchronization in complex networks of phase oscillators: A survey <s> Phase Balancing, Splay State, and Patterns <s> The emergence of a unified cognitive moment relies on the coordination of scattered mosaics of functionally specialized brain regions. Here we review the mechanisms of large-scale integration that counterbalance the distributed anatomical and functional organization of brain activity to enable the emergence of coherent behaviour and cognition. Although the mechanisms involved in large-scale integration are still largely unknown, we argue that the most plausible candidate is the formation of dynamic links mediated by synchrony over multiple frequency bands. <s> BIB002 </s> Synchronization in complex networks of phase oscillators: A survey <s> Phase Balancing, Splay State, and Patterns <s> Abstract. The coordinated reset of neural subpopula-tions is introduced as an eﬀectively desynchronizingstimulation technique. For this, short sequences of high-frequency pulse trains are administered at diﬀerent sitesin a coordinated way. Desynchronization is easilymaintained by performing a coordinated reset withdemand-controlled timing or by periodically adminis-tering resetting high-frequency pulse trains of demand-controlled length. Unlike previously developed methods,this novel approach is robust against variations of modelparameters and does not require time-consuming cali-bration. The novel technique is suggested to be used fordemand-controlled deep brain stimulation in patientssuﬀering from Parkinson’s disease or essential tremor. Itmight even be applicable to diseases with intermittentlyemerging synchronized neural oscillations like epilepsy.1 IntroductionIn several neurological diseases like Parkinson’s disease(PD) or essential tremor, brain function is severelyimpaired by synchronization processes (Elble and Koller1990). Parkinsonian resting tremor appears to be causedby a population of neurons located in the thalamus andthe basal ganglia. These neurons ﬁre in a synchronizedand intrinsically periodical manner at a frequencysimilar to that of the tremor, regardless of any feedbacksignals (Llina´s and Jahnsen 1982; Pare et al. 1990; Lenzet al. 1994). In contrast, under physiological conditionsthese neurons ﬁre incoherently (Nini et al. 1995). Inpatients with PD, this cluster acts like a pacemaker andactivates premotor areas and the motor cortex (Albertset al. 1969; Volkmann et al. 1996), where the lattersynchronize their oscillatory activity (Tass et al. 1998).In patients with advanced PD or essential tremor whono longer respond to drug therapy, depth electrodes arechronically implanted in target areas like the thalamicventralis intermedius nucleus or the subthalamic nucleus(Benabid et al. 1991; Blond et al. 1992). Electrical deepbrain stimulation (DBS) is performed by administering apermanent high-frequency (HF) (> 100 Hz) periodicpulse train via the depth electrodes (Benabid et al. 1991;Blond et al. 1992). HF DBS has been developed empiri-cally, and its mechanism is not yet fully understood.Permanent HF stimulation basically mimics the eﬀect oftissue lesioning by suppressing neuronal ﬁring, which inturn suppresses the peripheral tremor (Wielepp et al.2001).HFDBSisreversibleandhasamuchlowerrateofside eﬀects than lesioning with thermocoagulation(Schuurman et al. 2000). Nevertheless, due to currentspread, for example, HF DBS may lead to severe sideeﬀects like dysarthria, dysesthesia, or cerebellar ataxia.For this reason stimulation techniques have beendeveloped that aim at desynchronizing the pacemaker’spathologically synchronized ﬁring in a demand-con-trolled way instead of by simply suppressing the neu-ronal ﬁring (Tass 2001a,b,c, 2002a,b,d). These methodsshare one particular feature: each stimulus consists oftwo qualitatively diﬀerent stimuli. The ﬁrst stimulus isstronger and resets (restarts) the cluster, whereas thesecond, weaker stimulus is a single pulse that is admin-istered after a constant time delay and desynchronizes byhitting the cluster in a vulnerable state. The goal of thereset is to control the dynamics of the cluster byrestarting the cluster in a stereotypical way. The resetmay be achieved by means of a strong single pulse (Tass2001a,c, 2002d), a HF pulse train (Tass 2001b), or a low-frequency pulse train (Tass 2002a,b).A desynchronizing single pulse is eﬀective only if ithits the population very precisely at a vulnerable phase(Tass 1999). Hence the composite stimulation techniquesonly work with an eﬀective reset that may require highstimulation intensities, and minor variations of criticalstimulation parameters (especially the delay betweenresetting stimulus and desynchronizing single pulse) mayabolish the desynchronizing eﬀect (Tass 2001c, 2002b).Thus these methods are not robust against larger vari-ations of the model parameters (frequencies of the <s> BIB003 </s> Synchronization in complex networks of phase oscillators: A survey <s> Phase Balancing, Splay State, and Patterns <s> Coupled oscillator models use a single phase variable to approximate the voltage oscillation of each neuron during repetitivefiring where the behavior of the model depends on the connectivityand the interaction function chosen to describe the coupling. Weintroduce a network model consisting of a continuum of theseoscillators that includes the effects of spatially decaying coupling and axonal delay. We derive equations for determining the stability of solutions and analyze the network behavior for two different interaction functions. The first is a sine function, and the second is derived from a compartmental model of a pyramidal cell.In both cases, the system of coupled neural oscillators can undergo a bifurcation from synchronous oscillations to waves.The change in qualitative behavior is due to the axonal delay,which causes distant connections to encourage a phase shift between cells. We suggest that this mechanism could contribute to the behavior observed in several neurobiological systems. <s> BIB004 </s> Synchronization in complex networks of phase oscillators: A survey <s> Phase Balancing, Splay State, and Patterns <s> Recently, many experiments and analyses with biped robots have been carried out. Steady walking of a biped robot implies a stable limit cycle in the state space of the robot. In the design of a locomotion control system, there are primarily three problems associated with achieving such a stable limit cycle: the design of the motion of each limb, interlimb coordination, and posture control. In addition to these problems, when environmental conditions change or disturbances are added to the robot, there is the added problem of obtaining robust walking against them. In this paper we attempt to solve these problems and propose a locomotion control system for a biped robot to achieve robust walking by the robot using nonlinear oscillators, each of which has a stable limit cycle. The nominal trajectories of each limb's joints are designed by the phases of the oscillators, and the interlimb coordination is designed by the phase relation between the oscillators. The phases of the oscillators are reset and the nominal trajectories are modified using sensory feedbacks that depend on the posture and motion of the robot to achieve stable and robust walking. We verify the effectiveness of the proposed locomotion control system, analyzing the dynamic properties of the walking motion by numerical simulations and hardware experiments. <s> BIB005 </s> Synchronization in complex networks of phase oscillators: A survey <s> Phase Balancing, Splay State, and Patterns <s> We present a system of coupled nonlinear oscillators to be used as programmable central pattern generators, and apply it to control the locomotion of a humanoid robot. Central pattern generators are biological neural networks that can produce coordinated multidimensional rhythmic signals, under the control of simple input signals. They are found both in vertebrate and invertebrate animals for the control of locomotion. In this article, we present a novel system composed of coupled adaptive nonlinear oscillators that can learn arbitrary rhythmic signals in a supervised learning framework. Using adaptive rules implemented as differential equations, parameters such as intrinsic frequencies, amplitudes, and coupling weights are automatically adjusted to replicate a teaching signal. Once the teaching signal is removed, the trajectories remain embedded as the limit cycle of the dynamical system. An interesting aspect of this approach is that the learning is completely embedded into the dynamical system, and does not require external optimization algorithms. We use our system to encapsulate rhythmic trajectories for biped locomotion with a simulated humanoid robot, and demonstrate how it can be used to do online trajectory generation. The system can modulate the speed of locomotion, and even allow the reversal of direction (i.e. walking backwards). The integration of sensory feedback allows the online modulation of trajectories such as to increase the basin of stability of the gaits, and therefore the range of speeds that can be produced <s> BIB006 </s> Synchronization in complex networks of phase oscillators: A survey <s> Phase Balancing, Splay State, and Patterns <s> This paper proposes a design methodology to stabilize isolated relative equilibria in a model of all-to-all coupled identical particles moving in the plane at unit speed. Isolated relative equilibria correspond to either parallel motion of all particles with fixed relative spacing or to circular motion of all particles with fixed relative phases. The stabilizing feedbacks derive from Lyapunov functions that prove exponential stability and suggest almost global convergence properties. The results of the paper provide a low-order parametric family of stabilizable collectives that offer a set of primitives for the design of higher-level tasks at the group level <s> BIB007 </s> Synchronization in complex networks of phase oscillators: A survey <s> Phase Balancing, Splay State, and Patterns <s> This paper proposes a design methodology to stabilize relative equilibria in a model of identical, steered particles moving in the plane at unit speed. Relative equilibria either correspond to parallel motion of all particles with fixed relative spacing or to circular motion of all particles around the same circle. Particles exchange relative information according to a communication graph that can be undirected or directed and time-invariant or time-varying. The emphasis of this paper is to show how previous results assuming all-to-all communication can be extended to a general communication framework. <s> BIB008 </s> Synchronization in complex networks of phase oscillators: A survey <s> Phase Balancing, Splay State, and Patterns <s> Coordinated Control and Estimation for Multi-agent Systems: Theory and Practice Daniel J. Klein Chair of the Supervisory Committee: Professor Kristi A. Morgansen Aeronautics & Astronautics The problems studied in this thesis center on coordinated control and estimation tasks for multi-agent systems. The work begins technically, with a Kuramoto-inspired study of coordinated phasor centroid control for a group ofN homogeneous phase integrators. A main result is the development of a controller which guarantees asymptotic stability of the phasor centroid to a known reference vector. An extensions of the splay state to non-balanced group states is then explored. The work then moves on to a group of N heterogeneous agents wherein some agents act as leaders to the others, which obey a standard sinusoidal protocol. One interesting point here is that unlike in the related controlled linear agreement problem, symmetry of the communication topology does not determine uncontrollability. While these results are derived in continuous time, the next contributions shows that many of the desirable coordination properties can be preserved in discrete time. The work then moves to a more applied focus. A simulation study of coordinated target tracking is conducted, considering questions of control, estimation, and communication for distributed systems. Finally, the design and development of a computer vision system which enables a demonstration of some of the theory-oriented work on the University of Washington’s Underwater Fin-actuated Autonomous Vehicle testbed is discussed. <s> BIB009 </s> Synchronization in complex networks of phase oscillators: A survey <s> Phase Balancing, Splay State, and Patterns <s> This paper considers the integration of communication and control with respect to the task of coordinated heading control for a group of N vehicles. The heading control employed on each vehicle is a discretization of the well-known Kuramoto model of nonlinearly coupled oscillators. Stability for both all-to-all and random one-to-all broadcasts is shown to be dependent on the coupling strength, K, and the time discretization, DeltaT. For a desired system performance, DeltaT imposes a tight deadline by which the state information (M bits) must be propagated through the communication network. Routing optimization with respect to minimizing energy consumption is formulated considering the DeltaT deadline. Multi-hopping is optimal if time permits, but one-to-all broadcasting is more energy efficient when M/DeltaT is sufficiently large. <s> BIB010 </s> Synchronization in complex networks of phase oscillators: A survey <s> Phase Balancing, Splay State, and Patterns <s> The problem of controlling locomotion is an area in which neuroscience and robotics can fruitfully interact. In this article, I will review research carried out on locomotor central pattern generators (CPGs), i.e. neural circuits capable of producing coordinated patterns of high-dimensional rhythmic output signals while receiving only simple, low-dimensional, input signals. The review will first cover neurobiological observations concerning locomotor CPGs and their numerical modelling, with a special focus on vertebrates. It will then cover how CPG models implemented as neural networks or systems of coupled oscillators can be used in robotics for controlling the locomotion of articulated robots. The review also presents how robots can be used as scientific tools to obtain a better understanding of the functioning of biological CPGs. Finally, various methods for designing CPGs to control specific modes of locomotion will be briefly reviewed. In this process, I will discuss different types of CPG models, the pros and cons of using CPGs with robots, and the pros and cons of using robots as scientific tools. Open research topics both in biology and in robotics will also be discussed. <s> BIB011 </s> Synchronization in complex networks of phase oscillators: A survey <s> Phase Balancing, Splay State, and Patterns <s> We consider the problem of desynchronizing a network of synchronized, globally (all-to-all) coupled neurons using an input to a single neuron. This is done by applying the discrete time dynamic programming method to reduced phase models for neural populations. This technique numerically minimizes a certain cost function over the whole state space, and is applied to a Kuramoto model and a reduced phase model for Hodgkin-Huxley neurons with electrotonic coupling. We evaluate the effectiveness of control inputs obtained by averaging over results obtained for different coupling strengths. We also investigate the applicability of this method to Hodgkin-Huxley models driven by multiplicative stimuli. <s> BIB012 </s> Synchronization in complex networks of phase oscillators: A survey <s> Phase Balancing, Splay State, and Patterns <s> Motivated by neuroscience applications, and in particular by the deep brain stimulation treatment for Parkinson’s disease, we have recently derived a simplified model of an interconnected neuronal population under the effect of its mean-field proportional feedback. In this paper, we rely on that model to propose conditions under which proportional mean-field feedback achieves either oscillation inhibition or desynchronization. More precisely, we show that for small natural frequencies, this scalar control signal induces an inhibition of the collective oscillation. For the closed-loop system, this situation corresponds to a fixed point which is shown to be almost globally asymptotically stable in the fictitious case of zero natural frequencies and all-to-all coupling and feedback. In the case of an odd number of oscillators, this property is shown to be robust to small natural frequencies and heterogencities in both the coupling and feedback topology. On the contrary, for large natural frequencies, we show that scalar proportional mean-field feedback is able to induce desynchronization. After having recalled a formal definition for desynchronization, we show how it can be induced in a network of originally synchronized oscillators. <s> BIB013 </s> Synchronization in complex networks of phase oscillators: A survey <s> Phase Balancing, Splay State, and Patterns <s> We discuss spatially distributed networks that exhibit a diffusive coupling structure, common in biomolecular networks and multi-agent systems. We first review conditions that guarantee spatial homogeneity of the solutions of these systems, referred to as “synchrony.” We next point to structural system properties that allow diffusion-driven instability - a phenomenon critical to pattern formation in biology - and show that an analogous instability mechanism exists in multi-agent systems. The results reviewed in the paper also demonstrate the role played by the Laplacian eigenvalues in determining the dynamical properties of diffusively coupled systems. We conclude with a discussion of how these eigenvalues can be assigned with a design of node and edge weights of a graph, and present a formation control example. <s> BIB014
Applications in neuroscience BIB004 BIB002  , deep-brain stimulation BIB003 BIB012 BIB013 , vehicle coordination BIB007 BIB008 BIB009 BIB010 , and central pattern generators for locomotion purposes BIB011 BIB005 BIB006 motivate the study of coherent behaviors with synchronized frequencies where the phases are not cohesive, but rather dispersed in appropriate patterns. Whereas the phase-synchronized state is characterized by the order parameter r achieving its maximal (unit) magnitude, we say that a solution θ : R ≥0 → T n to the coupled oscillator model (1) achieves phase balancing if all phases θ i (t) asymptotically converge to the set that is, asymptotically the oscillators are uniformly distributed over the unit circle S 1 so that their centroid converges to the origin; see Fig. 5(d) and 5(e) for illustrations. We refer to BIB007 for a geometric characterization of the balanced state. One balanced state of particular interest in neuroscience applications is the so-called splay state {θ ∈ T n | θ i = i · 2π/n + ϕ (mod 2π) , ϕ ∈ S 1 , i ∈ {1, . . . , n}} ⊆ Bal n corresponding to phases uniformly distributed around the unit circle S 1 with distances 2π/n, see Fig. 5 (e). Other highly-symmetric balanced states consist of multiple clusters of collocated oscillators, where the clusters themselves are arranged in splay state. In particular, if m is a divisor of n, we define a symmetric balanced (m, n)-pattern to be a symmetric arrangement of the n phases consisting of m clusters uniformly spaced on S 1 , where each cluster contains n/m synchronized phases. Fig. 9 illustrates all symmetric balanced (m, n)-patterns for n = 12. Notice that, for any n ∈ N, there are at (4, 12) (6, 12) (12, 12) (1, 12) (2, 12) (3, 12) least two symmetric patterns: the (1, n)-pattern (i.e., the phase-synchronized state) and the (n, n)-pattern (i.e., the splay state). Arbitrary (m, n)-patterns can be stabilized, for example, by using coupling functions with higher harmonics, such as sin(m(θ i −θ j )); see BIB008 . The topic of symmetric phase balancing is similar in spirit to pattern formation BIB001 BIB014 , where phase synchronization corresponds to the "flat" solution with uniform phases and balanced (m, n)-patterns correspond to spatially non-uniform solutions with "higher-order modes."