synchronization via active control of identical and non identical φ6 chaotic oscillators with external excitation abstract using the active control technique with lyapunov stability theory and the routh hurwitz criteria control functions are designed to achieve complete synchronization between two identical 6 van der pol oscillators 6 vdpos two identical 6 duffing oscillators 6 dos and two non identical 6 oscillators comprising 6 vdpo and 6 do for the triple well configuration of the 6 potential the coefficient matrix of the error dynamics between each pair of synchronized systems is chosen such that the number of active control functions reduces from two to one thereby significantly reducing controller complexity in the design the designed controllers enable the state variables of the response system to synchronize with those of the drive system in both the identical and non identical cases the results are validated using numerical simulations

a transition to sharp timing in stochastic leaky integrate and fire neurons driven by frozen noisy input the firing activity of intracellularly stimulated neurons in cortical slices has been demonstrated to be profoundly affected by the temporal structure of the injected current mainen sejnowski 1995 this suggests that the timing features of the neural response may be controlled as much by its own biophysical characteristics as by how a neuron is wired within a circuit modeling studies have shown that the interplay between internal noise and the fluctuations of the driving input controls the reliability and the precision of neuronal spiking cecchi et al 2000 tiesinga 2002 fellous rudolph destexhe sejnowski 2003 in order to investigate this interplay we focus on the stochastic leaky integrate and fire neuron and identify the holder exponent h of the integrated input as the key mathematical property dictating the regime of firing of a single unit neuron we have recently provided numerical evidence taillefumier magnasco 2013 for the existence of a phase transition when becomes less than the statistical holder exponent associated with internal gaussian white noise h 1 2 here we describe the theoretical and numerical framework devised for the study of a neuron that is periodically driven by frozen noisy inputs with exponent h 0 in doing so we account for the existence of a transition between two regimes of firing when h 1 2 and we show that spiking times have a continuous density when the holder exponent satisfies h 1 2 the transition at h 1 2 formally separates rate codes for which the neural firing probability varies smoothly from temporal codes for which the neuron fires at sharply defined times regardless of the intensity of internal noise

periodically forced leaky integrate and fire model the discharge pattern of periodically forced leaky integrate and fire models is studied while previous analyses have been mainly concerned with the response of this model to sinusoidal stimulation our results hold for arbitrary periodic inputs it is shown that for any periodic input the map representing the relation between input phases at consecutive discharge times can be restricted to a piecewise continuous orientation preserving circle map this implies that i the rotation number is well defined and independent of the initial condition and ii in the same way as for sinusoidal forcing other forms of periodic stimuli can evoke only one of four types of response namely phase locking quasiperiodic discharges nonchaotic aperiodic firing and termination of the discharge after a finite number of firings

fractal distribution of floaters on a fluid surface and the transition to chaos for random maps abstract the long time spatial distribution of particles floating on the surface of a confined fluid whose flow velocity has complicated time dependence is considered it is shown that this distribution can be either a fractal or else can clump at several or one discrete points the transition from the latter type of distribution to the former occurs when the lyapunov exponent characterizing the particle motion passes through zero from negative values to positive values the characteristic features of this type of transition are investigated using random maps it is shown that near the transition there are extremely intermittent temporal fluctuations in the particle cloud and their scaling with a parameter is elucidated

noise induced synchronization of uncoupled nonlinear systems we derive a recursion formulae of transition probability of the noise induced synchronization arising in a pair of identical uncoupled logistic maps linked by common noisy excitation only the formulae has a delta type stationary solution which represents the perfect synchronization with probability 1 the stationary solution maintains under chaotic bifurcation while the escape times to reach the perfect synchronization increase in the chaotic region the escape times analysis implies existence of lower dimensional dynamics around the perfect synchronization we also provide a physical implementation of the synchronization

modulus of continuity of the canonic brownian motion on the group of diffeomorphisms of the circle abstract an explicit modulus of holder continuity is given for the flow associated to the canonic brownian motion on the diffeomorphism group of the circle

calculus on manifolds of conformal maps and cft in conformal field theory cft on simply connected domains of the riemann sphere the natural conformal symmetries under self maps are extended in a certain way to local symmetries under general conformal maps and this is at the basis of the powerful techniques of cft conformal maps of simply connected domains naturally have the structure of an infinite dimensional groupoid which generalizes the finite dimensional group of self maps we put a topological structure on the space of conformal maps on simply connected domains which makes it into a topological groupoid further we almost extend this to a local manifold structure based on the infinite dimensional frechet topological vector space of holomorphic functions on a given domain a from this we develop the notion of conformal a differentiability at the identity our main conclusion is that quadratic differentials characterizing cotangent elements on the local manifold enjoy properties similar to those of the holomorphic stress energy tensor of cft these properties underpin the local symmetries of cft applying the general formalism to cft correlation functions we show that the stress energy tensor is exactly such a quadratic differential this is at the basis of constructing the stress energy tensor in conformal loop ensembles it also clarifies the relation between cardy s boundary conditions for cft on simply connected domains and the expression of the stress energy tensor in terms of metric variations

synchronization of self sustained oscillators by common white noise abstract we study the stability of self sustained oscillations under the influence of external noise for small noise amplitude a phase approximation for the langevin dynamics is valid a stationary distribution of the phase is used for an analytic calculation of the maximal lyapunov exponent we demonstrate that for small noise the exponent is negative which corresponds to synchronization of oscillators

correlation anti correlation transition by state dependent poisson noise noise may induce some degrees of order in many biological and complex systems we present an example of a noise dependent transition from correlation to anti correlation of the states of a bivariate system coupled by state dependent poisson noises by varying the degree of dependence of the rate of jumps on the other variable the system undergoes different degrees of correlation from independence to perfect correlation which in this case coincides with perfect synchronization in the limit of the deterministic periodic case

precision and reliability of periodically and quasiperiodically driven integrate and fire neurons neurons in the brain communicate via trains of all or none electric events known as spikes how the brain encodes information using spikes char22 the neural code char22 remains elusive here the robustness against noise of stimulus induced neural spike trains is studied in terms of attractors and bifurcations the dynamics of model neurons converges after a transient onto an attractor yielding a reproducible sequence of spike times at a bifurcation point the spike times on the attractor change discontinuously when a parameter is varied reliability the stability of the attractor against noise is reduced when the neuron operates close to a bifurcation point we determined using analytical spike time maps the attractor and bifurcation structure of an integrate and fire model neuron driven by a periodic or a quasiperiodic piecewise constant current and investigated the stability of attractors against noise the integrate and fire model neuron became mode locked to the periodic current with a rational winding number p q and produced p spikes per q cycles there were q attractors p q mode locking regions formed arnold tongues in the model reliability was the highest during 1 1 mode locking when there was only one attractor as was also observed in recent experiments the quasiperiodically driven neuron mode locked to either one of the two drive periods or to a linear combination of both of them mode locking regions were organized in arnold tongues and reliability was again highest when there was only one attractor these results show that neuronal reliability in response to the rhythmic drive generated by synchronized networks of neurons is profoundly influenced by the location of the arnold tongues in parameter space

class ii neurons display a higher degree of stochastic synchronization than class i neurons we describe the relationship between the shape of the phase resetting curve prc and the degree of stochastic synchronization observed between a pair of uncoupled general oscillators receiving partially correlated poisson inputs in addition to inputs from independent sources we use perturbation methods to derive an expression relating the shape of the prc to the probability density function pdf of the phase difference between the oscillators we compute various measures of the degree of synchrony and cross correlation from the pdf s and use the same to compare and contrast differently shaped prcs with respect to their ability to undergo stochastic synchronization since the shape of the prc depends on underlying dynamical details of the oscillator system we utilize the results obtained from the analysis of general oscillator systems to study specific models of neuronal oscillators it is shown that the degree of stochastic synchronization is controlled both by the firing rate of the neuron and the membership of the prc type i or type ii it is also shown that the circular variance for the integrate and fire neuron and the generalized order parameter for a hippocampal interneuron model have a nonlinear relationship to the input correlation

on the interspike intervals of periodically driven integrate and fire models abstract we analyze properties of the firing map which iterations give information about consecutive spikes for periodically driven linear integrate and fire models by considering locally integrable thus in general not continuous input functions we generalize some results of other authors in particular we prove theorems concerning continuous dependence of the firing map on the input in suitable function spaces using mathematical study of the displacement sequence of an orientation preserving circle homeomorphism we provide a complete description of regularity properties of the sequence of interspike intervals and behaviour of the interspike interval distribution our results allow to explain some facts concerning this distribution observed numerically by other authors these theoretical findings are illustrated by computational examples

dynamical phase transitions in periodically driven model neurons transitions between dynamical states in integrate and fire neuron models with periodic stimuli result from tangent or discontinuous bifurcations of a return map we study their characteristic scaling laws and show that discontinuous bifurcations exhibit a kind of phase transition intermediate between continuous and first order in the model independent spirit of our analysis we show that a six dimensional 6d gating variable model with an attracting limit cycle has similar phase transitions governed by a 1d return map this reduction to 1d map dynamics should extend to real neurons in a periodic current clamp setting

synchronous firing and higher order interactions in neuron pool the stochastic mechanism of synchronous firing in a population of neurons is studied from the point of view of information geometry higher order interactions of neurons which cannot be reduced to pairwise correlations are proved to exist in synchronous firing in a neuron pool where each neuron fires stochastically the probability distribution q r of the activity r which is the fraction of firing neurons in the pool is studied when q r has a widespread distribution in particular when q r has two peaks the neurons fire synchronously at one time and are quiescent at other times the mechanism of generating such a probability distribution is interesting because the activity r is concentrated on its mean value when each neuron fires independently because of the law of large numbers even when pairwise interactions or third order interactions exist the concentration is not resolved this shows that higher order interactions are necessary to generate widespread activity distributions we analyze a simple model in which neurons receive common overlapping inputs and prove that such a model can have a widespread distribution of activity generating higher order stochastic interactions

canonical brownian motion on the diffeomorphism group of the circle for infinitesimal data given on the group of diffeomorphism of the circle with respect to the metric h3 2 the associated brownian motion has been constructed by malliavin c r acad sci parist 329 1999 325 329 in this work we shall give another approach and prove the invariance of heat measures under the adjoint action of s1

the geometry of neuronal recruitment we address the question of whether or not a periodic train of excitatory synaptic inputs recruits an excitable cell such that it fires repeatedly or does not recruit a cell such that it fails to fire possibly after some transient in particular we study the scenarios of one or two inputs per period in the latter case the degree of synchrony of the inputs is a crucial factor in recruitment we establish rigorous geometric conditions that pinpoint the transition between recruitment and non recruitment as the degree of synchrony between input pairs or other input parameters is varied these conditions can be used to determine whether a particular temporal relation between periodic input pairs leads to recruitment or not and to prove in certain parameter regimes that recruitment can only occur when the inputs are sufficiently closely synchronized the concepts in this paper are derived for both the integrate and fire neuron and the theta neuron models in the former the location in phase space of the unique fixed point of a relevant two dimensional map determines firing while in the latter it is the existence or lack of existence of a fixed point of the map which does so these results are discussed in the context of recruitment of cells into localized activity patterns

riemannian geometry of diff s1 s1 abstract the main result of the paper is a computation of the ricci curvature of diff s 1 s 1 unlike earlier results on the subject we do not use the kahler structure symmetries to compute the ricci curvature but rather rely on classical finite dimensional results of nomizu et al on riemannian geometry of homogeneous spaces

the covariant measure of sle on the boundary we construct a natural measure supported on the intersection of a chordal sle curve with mathbb r in the range 4 8 the measure is a function of the sle path in question assuming that boundary measures transform in a d dimensional way where d is the hausdorff dimension of gamma cap mathbb r we show that the measure we construct is up to multiplicative constant the unique measure valued function of the sle path that satisfies the domain markov property

synchronization and desynchronization of self sustained oscillators by common noise we consider the effect of external noise on the dynamics of limit cycle oscillators the lyapunov exponent becomes negative under influence of small white noise what means synchronization of two or more identical systems subject to common noise we analytically study the effect of small nonidentities in the oscillators and in the noise and derive statistical characteristics of deviations from the perfect synchrony large white noise can lead to desynchronization of oscillators provided they are nonisochronous this is demonstrated for the van der pol duffing system

a dimension spectrum for sle boundary collisions we consider chordal sle curves for where the intersection of the curve with the boundary is a random fractal of almost sure hausdorff dimension we study the random sets of points at which the c

high order synchronization transitions and competition among arnold tongues in a rotator under harmonic forcing we consider a rotator whose equation of motion for the angle theta consists of the zeroth and first fourier modes numerical analysis based on the trailing of saddle node bifurcations is used to locate the n 1 arnold tongues where synchronization occurs several of them are wide enough for high order synchronization to be seen in passive observations by sweeping the system parameters within a certain range we find that the stronger the dependence of theta over on theta the wider the regions of synchronization use of a synchronization index reveals a vast number of very narrow n m arnold tongues a competition phenomenon among the tongues is observed in that they push and squeeze one another as some tongues widen others narrow two mechanisms for transitions between different n m synchronization states are considered slow variation of the driving frequency and the influence of low frequency noise on the rotator

dynamics and reliability of bistable neurons driven with time dependent stimuli the reliability of a spiking neuron depends on the frequency content of the driving input signal previous studies have shown that well above threshold regularly firing neurons generate reliable responses when the input signal resonates with the firing frequency of the cell instead well below threshold reliable responses are obtained when the input frequency resonates with the subthreshold oscillations of the neuron previous theories however provide no clear prediction for the input frequency giving rise to maximally reliable spiking at threshold which is probably the most relevant firing regime in mammalian cortex under physiological conditions in particular when the firing onset is governed by a subcritical hopf bifurcation the frequency of subthreshold oscillations often differs from the firing rate at threshold the predictions of previous studies hence cannot be smoothly merged at threshold here we explore the behavior of reliability in bistable neurons near threshold using three types of driving stimuli constant periodic and stochastic we find that the two natural frequencies of the system associated with the two coexisting attractors provide a rich variety of possible locking modes with the external signal reliability is determined by the sensitivity to noise of each locking mode and by the transition probabilities between modes noise increases the amount of spike time jitter and minimal jitter is obtained for input frequencies coinciding with the suprathreshold firing rate of the cell in addition noise may either enhance or inhibit transitions between the two attractors depending on the input frequency the dual role played by noise in bistable systems implies that reliability is determined by a delicate balance between spike time jitter and the rate of transitions between attractors

hilbert schmidt groups as infinite dimensional lie groups and their riemannian geometry abstract we describe the exponential map from an infinite dimensional lie algebra to an infinite dimensional group of operators on a hilbert space notions of differential geometry are introduced for these groups in particular the ricci curvature which is understood as the limit of the ricci curvature of finite dimensional groups is calculated we show that for some of these groups the ricci curvature is

a cameron martin type quasi invariance theorem for brownian motion on a compact riemannian manifold abstract let m be a compact manifold without boundary o be a fixed base point in m g be a riemannian metric on m and be a g compatible covariant derivative on tm the tangent space to m assume the torsion t of satisfies the skew symmetry condition g t x y y 0 for all vector fields x and y on m for example take to be the levi civita covariant derivative on m g also let denote the wiener measure on w0 m c 0 1 m 0 o and let h s denote stochastic parallel translation relative to along the path w0 m up to time s 0 1 given a c1 function h 0 1 t0m it is shown that the differential equation t h t h with initial condition 0 id w m w m has a solution r maps w m w m the measurable maps from w m to w m this function is a flow on w m i e for all t r t t furthermore t has the quasi invariance property the law t of t with respect to the wiener measure is equivalent to for all t r this result is used to prove an integration by parts formula for the h derivative hf defined by h f d dt 0 f t where f is a c2 cylinder function on w m

geometric investigations of a vorticity model equation abstract this article consists of a detailed geometric study of the one dimensional vorticity model equation t u x 2 u x 0 h u x t r x s 1 which is a particular case of the generalized constantin lax majda equation wunsch showed that this equation is the euler arnold equation on diff s 1 when the latter is endowed with the right invariant homogeneous h 1 2 metric in this article we prove that the exponential map of this riemannian metric is not fredholm and that the sectional curvature is locally unbounded furthermore we prove a beale kato majda type blow up criterion which we then use to demonstrate a link to our non fredholmness result finally we extend a blow up result of castro cordoba to the periodic case and to a much wider class of initial conditions using a new generalization of an inequality for hilbert transforms due to cordoba cordoba

mode locking and arnold tongues in integrate and fire neural oscillators an analysis of mode locked solutions that may arise in periodically forced integrate and fire if neural oscillators is introduced based upon a firing map formulation of the dynamics a q p mode locked solution is identified with a spike train in which p firing events occur in a period q ensuremath delta where ensuremath delta is the forcing period a linear stability analysis of the map of firing times around such solutions allows the determination of the arnold tongue structure for regions in parameter space where stable solutions exist the analysis is verified against direct numerical simulations for both a sinusoidally forced if system and one in which a periodic sequence of spikes is used to induce a biologically realistic synaptic input current this approach is extended to the case of two synaptically coupled if oscillators showing that mode locked states can exist for some self consistently determined common period of repetitive firing numerical simulations show that such solutions have a bursting structure where regions of spiking activity are interspersed with quiescent periods before repeating the influence of the synaptic current upon the arnold tongue structure is explored in the regime of weak coupling

characterization of stochastic bifurcations in a simple biological oscillator this study of the effect of noise on bifurcations in a simple biological oscillator with a periodically modulated threshold uses the first passage time problem of the ornstein uhlenbeck process with a periodic boundary to define the operator governing the transition of a threshold phase density stochastic phase locking is analyzed numerically by evaluating the evolution of the probability density function of the threshold phase a firing phase map in a noisy environment is extended to a stochastic kernel so that stochastic bifurcations can be investigated by spectral analysis of the kernel

integrate and fire models with an almost periodic input function abstract we investigate leaky integrate and fire models lif models for short driven by stepanov and almost periodic functions special attention is paid to the properties of the firing map and its displacement which give information about the spiking behavior of the considered system we provide conditions under which such maps are well defined and are uniformly continuous we show that the lif models with stepanov almost periodic inputs have uniformly almost periodic displacements we also show that in the case of almost periodic drives it may happen that the displacement map is uniformly continuous but is not almost periodic and thus cannot be stepanov or uniformly almost periodic by allowing discontinuous inputs we extend some previous results showing for example that the firing rate for the lif models with stepanov almost periodic input exists and is unique this is a starting point for the investigation of the dynamics of almost periodically driven integrate and fire systems

universality in three frequency resonances we investigate the hierarchical structure of three frequency resonances in nonlinear dynamical systems with three interacting frequencies we hypothesize an ordering of these resonances based on a generalization of the farey tree organization from two frequencies to three in experiments and numerical simulations we demonstrate that our hypothesis describes the hierarchies of three frequency resonances in representative dynamical systems we conjecture that this organization may be universal across a large class of three frequency systems

quantifying encoding redundancy induced by rate correlations in poisson neurons temporal correlations in neuronal spike trains are known to introduce redundancy to stimulus encoding however exact methods to describe how these correlations impact neural information transmission quantitatively are lacking here we provide a general measure for the information carried by correlated rate modulations only neglecting other spike correlations and use it to investigate the effect of rate correlations on encoding redundancy we derive it analytically by calculating the mutual information between a time correlated rate modulating signal and the resulting spikes of poisson neurons whereas this information is determined by spike autocorrelations only the redundancy in information encoding due to rate correlations depends on both the distribution and the autocorrelation of the rate histogram we further demonstrate that at very small signal strengths the information carried by rate correlated spikes becomes identical to that of independent spikes in effect measuring the signal modulation depth in contrast a vanishing signal correlation time maximizes information but does not generally yield the information of independent spikes overall our study sheds light on the role of signal induced temporal correlations for neural coding by providing insight into how signal features shape redundancy and by establishing mathematical links between existing methods

collisions and spirals of loewner traces we analyze loewner traces driven by functions asymptotic to 1 t we prove a stability result when 6 4 and show that 4 can lead to non locally connected hulls as a consequence we obtain a driving term t so that the hulls driven by t are generated by a continuous curve for all 0 with 6 4 but not when 4 so that the space of driving terms with continuous traces is not convex as a byproduct we obtain an explicit construction of the traces driven by 1 t and a conceptual proof of the corresponding results of kager nienhuis and kadanoff

divergence measures based on the shannon entropy a novel class of information theoretic divergence measures based on the shannon entropy is introduced unlike the well known kullback divergences the new measures do not require the condition of absolute continuity to be satisfied by the probability distributions involved more importantly their close relationship with the variational distance and the probability of misclassification error are established in terms of bounds these bounds are crucial in many applications of divergence measures the measures are also well characterized by the properties of nonnegativity finiteness semiboundedness and boundedness

transition to stochastic synchronization in spatially extended systems spatially extended dynamical systems namely coupled map lattices driven by additive spatio temporal noise are shown to exhibit stochastic synchronization in analogy with low dimensional systems synchronization can be achieved only if the maximum lyapunov exponent becomes negative for sufficiently large noise amplitude moreover noise can suppress also the nonlinear mechanism of information propagation which may be present in the spatially extended system an example of phase transition is observed when both the linear and the nonlinear mechanisms of information production disappear at the same critical value of the noise amplitude the corresponding critical properties cannot be estimated numerically with great accuracy but some general argument suggests that they could be ascribed to the kardar parisi zhang universality class conversely when the nonlinear mechanism prevails on the linear one another type of phase transition to stochastic synchronization occurs this one is shown to belong to the universality class of directed percolation

spike count distribution in a neuronal population under weak common stimulation we study the probability distribution of the number of synchronous action potentials spike count in a model network consisting of a homogeneous neural population that is driven by a common time dependent stimulus we derive two analytical approximations for the count statistics which are based on linear response theory and hold true for weak input correlations comparison to numerical simulations of populations of integrate and fire neurons in different parameter regimes reveals that our theory correctly predicts how much a weak common stimulus increases the probability of common firing and of common silence in the neural population

schramm s formula and the green s function for multiple sle we construct martingale observables for systems of multiple sle curves by applying screening techniques within the cft framework recently developed by kang and makarov extended to admit multiple sles we illustrate this approach by rigorously establishing explicit expressions for the green s function and schramm s formula in the case of two curves growing towards infinity in the special case when the curves are fused and start at the same point some of the formulas we prove were predicted in the physics literature

aspects of bifurcation theory for piecewise smooth continuous systems abstract systems that are not smooth can undergo bifurcations that are forbidden in smooth systems we review some of the phenomena that can occur for piecewise smooth continuous maps and flows when a fixed point or an equilibrium collides with a surface on which the system is not smooth much of our understanding of these cases relies on a reduction to piecewise linearity near the border collision we also review a number of codimension two bifurcations in which nonlinearity is important

evaluation of entrainment of a nonlinear neural oscillator to white noise the lyapunov exponent for a one dimensional neural oscillator model the theta neuron is computed for white noise forcing using the steady state solution to the associated fokker planck equation the latter is mildly singular due to the nature of the multiplicative input in agreement with previous results with similar models the exponent is negative for all forcing amplitudes but here it is shown to be small relative to that for periodic drive in a range of forcing strengths thus the synchronization of an ensemble of independent neurons receiving common but random input can be slow moreover this implies that aperiodic input may be suboptimal in some contexts for preserving the reliability of fine spike timing a potentially important component of the neural code