collective phase description of coupled oscillators with general network structure we develop a collective phase description for a population of nonidentical limit cycle oscillators with any network structure undergoing fully phase locked collective oscillations the whole network dynamics can be described by a single collective phase variable we derive a general formula for the collective phase sensitivity which quantifies the phase response of the whole network to weak external perturbations applied to the constituent oscillators moreover we consider weakly interacting multiple networks and develop an effective phase coupling description for them several examples are given to illustrate our theory

carleman approximation by conformal minimal immersions and directed holomorphic curves abstract let r be an open riemann surface in this paper we prove that every continuous function m r n n 3 defined on a divergent jordan arc m r can be approximated in the carleman sense by conformal minimal immersions thus providing a new generalisation of carleman s theorem in fact we prove that this result remains true for null curves and many other classes of directed holomorphic immersions for which the directing variety satisfies a certain flexibility property furthermore the constructed immersions may be chosen to be complete or proper under natural assumptions on the variety and the continuous map as a consequence we give an approximate solution to a plateau problem for divergent jordan curves in the euclidean spaces

control and synchronization of neuron ensembles synchronization of oscillations is a phenomenon prevalent in natural social and engineering systems controlling synchronization of oscillating systems is motivated by a wide range of applications from surgical treatment of neurological diseases to the design of neurocomputers in this paper we study the control of an ensemble of uncoupled neuron oscillators described by phase models we examine controllability of such a neuron ensemble for various phase models and furthermore study the related optimal control problems in particular by employing pontryagin s maximum principle we analytically derive optimal controls for spiking single and two neuron systems and analyze the applicability of the latter to an ensemble system finally we present a robust computational method for optimal control of spiking neurons based on pseudospectral approximations the methodology developed here is universal to the control of general nonlinear phase oscillators

synchronization and transient stability in power networks and nonuniform kuramoto oscillators motivated by recent interest for multiagent systems and smart grid architectures we discuss the synchronization problem for the network reduced model of a power system with nontrivial transfer conductances our key insight is to exploit the relationship between the power network model and a first order model of coupled oscillators assuming overdamped generators possibly due to local excitation controllers a singular perturbation analysis shows the equivalence between the classic swing equations and a nonuniform kuramoto model here nonuniform kuramoto oscillators are characterized by multiple time constants nonhomogeneous coupling and nonuniform phase shifts extending methods from transient stability synchronization theory and consensus protocols we establish sufficient conditions for synchronization of nonuniform kuramoto oscillators these conditions reduce to necessary and sufficient tests for the standard kuramoto model combining our singular perturbation and kuramoto analyses we derive

type i membranes phase resetting curves and synchrony type i membrane oscillators such as the connor model connor et al 1977 and the morris lecar model morris and lecar 1981 admit very low frequency oscillations near the critical applied current hansel et al 1995 have numerically shown that synchrony is difficult to achieve with these models and that the phase resetting curve is strictly positive we use singular perturbation methods and averaging to show that this is a general property of type i membrane models we show in a limited sense that so called type ii resetting occurs with models that obtain rhythmicity via a hopf bifurcation we also show the differences between synapses that act rapidly and those that act slowly and derive a canonical form for the phase interactions

variable structure control using system decomposition an approach to the design of large variable structure systems subject to control bounds is introduced the method includes a switching hyperplane design based on generalized inverses and system decomposition to ensure reaching the hyperplane and achieving a sliding condition the control is switched between local equivalent control and bounded corrective control the design of the corrective control component is completed using system decomposition into smaller subsystems an estimate of the domain of attraction corresponding to the bounded control is obtained and used to select the appropriate controller bounds the method is illustrated using a fifth order numerical example

analysis and design of variable structure systems using a geometric approach multivariable variable structure systems in the sliding mode are studies using a geometric approach the properties of system order reduction and disturbance rejection are proved using projector theory new design methods for the choice of switching hyperplanes are derived for the closed loop eigenvalue eigenvector assignment problem

synchronization limit of weakly forced nonlinear oscillators nonlinear oscillators exhibit synchronization injection locking to external periodic forcings which underlies the mutual synchronization in networks of nonlinear oscillators despite its history of synchronization and the practical importance of injection locking to date there are many important open problems of an efficient injection locking for a given oscillator in this work i elucidate a hidden mechanism governing the synchronization limit under weak forcings which is related to a widely known inequality holder s inequality this mechanism enables us to understand how and why the efficient injectionlocking is realized a general theory of synchronization limit is constructed where the maximization of the synchronization range or the stability of synchronization for general forcings including pulse trains and a fundamental limit of general m n phase locking are clarified systematically these synchronization limits and their utility are systematically verified in the hodgkin huxley neuron model as an example s online supplementary data available from stacks iop org jpa 47 402002 mmedia

constrained charge balanced minimum power controls for spiking neuron oscillators abstract in this paper we study the optimal control of phase models for spiking neuron oscillators we focus on the design of minimum power current stimuli that elicit spikes of neurons at specified times subject to the charge balance constraint such charge balanced controls in practice are used to suppress undesirable side effects occurred due to the accumulation of electric charge resulting from the applied external stimuli we derive charge balanced minimum power controls for a general phase model using the maximum principle where the cases with unbounded and bounded control amplitude are examined the latter is of fundamental importance because phase models are valid under weak forcing the developed optimal control strategies are applied to both mathematically ideal and experimentally observed phase models to demonstrate their applicability including the widely studied hodgkin huxley model

on the phase reduction and response dynamics of neural oscillator populations weundertake a probabilistic analysis of the response of repetitively firing neural populations to simple pulselike stimuli recalling and extending results from the literature we compute phase response curves prcs valid near bifurcations to periodic firing for hindmarsh rose hodgkin huxley fitzhugh nagumo and morris lecar models encompassing the four generic codimension one bifurcations phase density equations are then used to analyze the role of the bifurcation and the resulting prc in responses to stimuli in particular we explore the interplay among stimulus duration baseline firing frequency and population level response patterns we interpret the results in terms of the signal processing measure of gain and discuss further applications and experimentally testable predictions

estimation of sliding mode domains of uncertain variable structure systems with bounded controllers the problem of estimating regions of asymptotic stability ras of uncertain dynamical systems with bounded controllers and the sliding model requirement is discussed the notion of combining different lyapunov functions is utilized different lyapunov functions are found and for each lyapunov function an estimate of the ras is found different regions are combined in order to find an improved ras the combined region is not in general a convex region and cannot be found analytically by using just a single lyapunov function a class of uncertain linear time invariant multivariable control systems is considered and a transformation for decoupling the system into two subsystems is used the transformation simplifies the ras estimation problem stability domains are estimated for a linear variable structure control system with sliding mode performance in the case of a discontinuous bounded controller

dynamics of limit cycle oscillators subject to general noise the phase description is a powerful tool for analyzing noisy limit cycle oscillators the method however has found only limited applications so far because the present theory is applicable only to gaussian noise while noise in the real world often has non gaussian statistics here we provide the phase reduction method for limit cycle oscillators subject to general colored and non gaussian noise including a heavy tailed one we derive quantifiers like mean frequency diffusion constant and the lyapunov exponent to confirm consistency of the results applying our results we additionally study a resonance between the phase and noise

optimal control of neuronal activity we investigate the optimal control of neuronal spiking activity for neurons receiving a class of random synaptic inputs characterized by a positive parameter alpha optimal control signals and optimal variances are found exactly for the diffusion process approximating an integrate and fire model when synaptic inputs are sub poisson alpha 0 5 inputs the optimal control is smooth and unique the optimal variance obtained in the current paper sets the lowest possible bound in controlling the stochasticity of neuronal activity we also discuss how to implement the optimal control signal for certain model neurons

design of charge balanced time optimal stimuli for spiking neuron oscillators in this letter we investigate the fundamental limits on how the interspike time of a neuron oscillator can be perturbed by the application of a bounded external control input a current stimulus with zero net electric charge accumulation we use phase models to study the dynamics of neurons and derive charge balanced controls that achieve the minimum and maximum interspike times for a given bound on the control amplitude our derivation is valid for any arbitrary shape of the phase response curve and for any value of the given control amplitude bound in addition we characterize the change in the structures of the charge balanced time optimal controls with the allowable control amplitude we demonstrate the applicability of the derived optimal control laws by applying them to mathematically ideal and experimentally observed neuron phase models including the widely studied hodgkin huxley phase model and by verifying them with the corresponding original full state space models this work addresses a fundamental problem in the field of neural control and provides a theoretical investigation to the optimal control of oscillatory systems

estimation of asymptotic stability region and sliding domain of uncertain variable structure systems with bounded controllers this work is concerned with the problem of estimating the asymptotic stability region and sliding domain of uncertain variable structure systems with bounded controllers the properties of the decoupling theorem and strictly positive real function are used to incorporate with the lyapunov stability to estimate the stability region and the attractive region of the sliding mode it will be shown that the special state transformation used in previous work is no longer needed obviously our result is directly related to the original system states

simultaneous velocity consensus and shape control for a finite number of point agents on the unit circle in this paper we study a second order distributed control system for a finite number of point agents on the unit circle that achieves simultaneously velocity consensus and distance based formation shape control based on tools from riemannian geometry we propose a system of second order differential equations on the n dimensional torus that achieves these two goals we prove convergence of the trajectories to single closed geodesics on a torus and investigate the stability properties of the distributed algorithm

estimation of the asymptotic stability region of uncertain systems with bounded sliding mode controllers in this note the problem of estimating the asymptotic stability region asr of uncertain systems with bounded sliding mode controllers is considered to simplify the problem the authors use a state transformation and choose a suitable lyapunov function for the transformed control system using the lyapunov function the authors estimate the asr and show the exponential stability of the closed loop control system in the region also by an example the authors show that for a certain class of uncertain dynamical systems with bounded sliding mode controllers their method gives a more improved estimate of the asr

optimal entrainment with smooth pulse and square signals in weakly forced nonlinear oscillators abstract a physical limit of entrainability of nonlinear oscillators is considered for an external weak signal forcing this limit of entrainability is characterized by the optimization problem maximizing the width of the arnold tongue the frequency locking range versus forcing magnitude under certain practical constraints here we show a solution to this optimization problem thanks to a direct link to holder s inequality this solution defines an ideal forcing realizing the entrainment limit and as the result a fundamental limit of entrainment is clarified as follows for 1 1 entrainment we obtain i a construction of the global optimal forcing and a condition for its uniqueness in l p space with p 1 and ii a construction of the global optimal pulse like forcings in l 1 space and for m n entrainment m n some informations about the non existence of the ideal forcing iii in addition we establish definite algorithms for obtaining the global optimal forcings for 1 p and these pulse like forcings for p 1 these theoretical findings are verified by systematic extensive numerical calculations and simulations

optimal design of minimum power stimuli for phase models of neuron oscillators in this paper we study optimal control problems of spiking neurons whose dynamics are described by a phase model we design minimum power current stimuli controls that lead to targeted spiking times in particular we consider bounded control amplitude and characterize the range of possible spiking times determined by the bound which can be chosen sufficiently small within the range where the phase model is valid we show that for a given bound the corresponding feasible spiking times are optimally achieved by piecewise continuous controls we present analytic expressions with numerical simulations of the minimum power stimuli for several phase models we demonstrate the applicability of our method with an experimentally determined phase response curve

phase reduction of weakly perturbed limit cycle oscillations in time delay systems abstract the phase reduction method is applied to a general class of weakly perturbed time delay systems exhibiting periodic oscillations the adjoint equation with an appropriate initial condition for the infinitesimal phase response curve of a time delay system is derived the method is demonstrated numerically for the mackey glass equation as well as for a chaotic rossler system subject to a delayed feedback control dfc we show that the profile of the phase response curve of a periodic orbit stabilized by the dfc algorithm does not depend on the control matrix this property is universal and holds for any dynamical system subject to the dfc

optimizing mutual synchronization of rhythmic spatiotemporal patterns in reaction diffusion systems optimization of the stability of synchronized states between a pair of symmetrically coupled reaction diffusion systems exhibiting rhythmic spatiotemporal patterns is studied in the framework of the phase reduction theory the optimal linear filter that maximizes the linear stability of the in phase synchronized state is derived for the case in which the two systems are nonlocally coupled the optimal nonlinear interaction function that theoretically gives the largest linear stability of the in phase synchronized state is also derived the theory is illustrated by using typical rhythmic patterns in fitzhugh nagumo systems as examples

constrained minimum energy optimal control of the dissipative bloch equations abstract in this letter we apply optimal control theory to design minimum energy 2 and pulses for the bloch system in the presence of relaxation with constrained control amplitude we consider a commonly encountered case in which the transverse relaxation rate is much larger than the longitudinal one so that the latter can be neglected using pontryagin s maximum principle we derive optimal feedback laws which are characterized by the number of switches depending on the control bound and the coordinates of the desired final state

collective phase description of globally coupled excitable elements we develop a theory of collective phase description for globally coupled noisy excitable elements exhibiting macroscopic oscillations collective phase equations describing macroscopic rhythms of the system are derived from langevin type equations of globally coupled active rotators via a nonlinear fokker planck equation the theory is an extension of the conventional phase reduction method for ordinary limit cycles to limit cycle solutions in infinite dimensional dynamical systems such as the time periodic solutions to nonlinear fokker planck equations representing macroscopic rhythms we demonstrate that the type of the collective phase sensitivity function near the onset of collective oscillations crucially depends on the type of the bifurcation namely it is type i for the saddle node bifurcation and type ii for the hopf bifurcation