bifurcations stability and monotonicity properties of a delayed neural network model abstract a delay differential equation modelling an artificial neural network with two neurons is investigated a linear stability analysis provides parameter values yielding asymptotic stability of the stationary solutions these can lose stability through either a pitchfork or a hopf bifurcation which is shown to be supercritical at appropriate parameter values an interaction takes place between the pitchfork and hopf bifurcations conditions are also given for the set of initial conditions that converge to a stable stationary solution to be open and dense in the functional phase space analytic results are illustrated with numerical simulations

time delay effects on coupled limit cycle oscillators at hopf bifurcation abstract we present a detailed study of the effect of time delay on the collective dynamics of coupled limit cycle oscillators at hopf bifurcation for a simple model consisting of just two oscillators with a time delayed coupling the bifurcation diagram obtained by numerical and analytical solutions shows significant changes in the stability boundaries of the amplitude death phase locked and incoherent regions a novel result is the occurrence of amplitude death even in the absence of a frequency mismatch between the two oscillators similar results are obtained for an array of n oscillators with a delayed mean field coupling and the regions of such amplitude death in the parameter space of the coupling strength and time delay are quantified some general analytic results for the n thermodynamic limit are also obtained and the implications of the time delay effects for physical applications are discussed

strong and robustly strong polynomial time reducibilities to sparse sets reducibility defined by oracle strong nondeterministic machines is studied two definitions of relativized strength are presented and separated the corresponding reduction classes of the sparse sets are characterized in terms of nonuniform complexity classes an oracle restricted positive relativization of the probabilistic class zpp is developed

on some natural complete operators abstract an operator is a mapping from integer functions to integer functions it is known from a result by baker gill and solovay 1975 that there exists an operator computable in polynomial time by a nondeterministic oracle turing machine called an np operator but not computable in polynomial time by any deterministic oracle turing machine we investigate several natural operators which share similar properties we use the concept of completeness to give a precise classification of the complexity of these operators for example the question of finding maximum values of polynomial time computable functions can be formulated as an operator complete for the class of np operators

nonuniform proof systems a new framework to describe nonuniform and probabilistic complexity classes the concept of non uniform proof systems is introduced this notion allows a uniform description of non uniform complexity classes 10 probabilistic classes e g bpp 8 15 27 32 am 2 3 and language classes defined by simultaneous non uniform and nondeterministic time bounds non uniform proof systems provide a better understanding of many results concerning these classes particularly their connections to uniform complexity measures we give an uniform approach to lowness results 19 20 for various complexity classes for instance we show that co np poly np is contained in the third level of the low hierarchy and that np c np co np poly implies that the polynomial time hierarchy collapses to its second level see also 1 finally some evidence is given that the low hierarchy cannot be extended beyond its third level by the current techniques

disk envelopes of functions ii given a suslin function on a domainyin a banach spacexone can define two envelopes of related to the pluripotential theory the first one denoted byp is the supremum of all plurisubharmonic functions less than the value of the other one denoted byd at some pointzis equal to the infimum of integrals over the boundaries of all closed analytic disks centered atz in this paper we study the relationship between these envelopes we show thatd p for a large class of functions on domains in banach spaces wherep is the upper regularization ofp ifx cn then we prove thatd is always equal top

mean field approximation of two coupled populations of excitable units the analysis on stability and bifurcations in the macroscopic dynamics exhibited by the system of two coupled large populations composed of n stochastic excitable units each is performed by studying an approximate system obtained by replacing each population with the corresponding mean field model in the exact system one has the units within an ensemble communicating via the time delayed linear couplings whereas the interensemble terms involve the nonlinear time delayed interaction mediated by the appropriate global variables the aim is to demonstrate that the bifurcations affecting the stability of the stationary state of the original system governed by a set of 4n stochastic delay differential equations for the microscopic dynamics can accurately be reproduced by a flow containing just four deterministic delay differential equations which describe the evolution of the mean field based variables in particular the considered issues include determining the parameter domains where the stationary state is stable the scenarios for the onset and the time delay induced suppression of the collective mode as well as the parameter domains admitting bistability between the equilibrium and the oscillatory state we show how analytically tractable bifurcations occurring in the approximate model can be used to identify the characteristic mechanisms by which the stationary state is destabilized under different system configurations like those with symmetrical or asymmetrical interpopulation couplings

phase models with explicit time delays studying weakly connected oscillators leads to phase models it has been proven recently that weakly connected oscillators with delayed interactions do not lead to phase models with time delays even when the delay is of the same order of magnitude as the period of oscillation this has resulted in a fading interest in such models we prove here that if the interaction delay between weakly connected oscillators is much longer than the period of oscillation then the corresponding phase model does have an explicit time delay

relative extremal functions and characterization of pluripolar sets in complex manifolds we study a disc formula for the relative extremal function for a subset of a complex manifold and apply it to give a description of pluripolar sets and polynomial hulls

delay induced periodicity in a neural netlet of excitation and inhibition abstract the dynamical behaviour of a two neuron netlet of excitation and inhibition with a transmission delay is investigated it is shown that in the absence of delay the netlet relaxes to the trivial resting state if the delay is of sufficient magnitude the network is excited to a temporally periodic cyclic behaviour the analytical mechanism for the onset of cyclic behaviour is through a hopf type bifurcation approximate solusions to the periodic output of the netlet is calculated stability of the temporally periodic cycle is investigated it is shown that the bifurcation is supercritical a related discrete version of the continuous time system is formulated it is found that the discrete system also displays a cyclic behaviour results of a number of computer simulations are displayed graphically the article concludes with a brief neurobiological discussion

on small generators abstract yap 1983 shows that each set having small generators is in the class np poly which was introduced by karp and lipton 1980 we show here that the converse is also true i e each set in np poly has small generators this settles a question left open by yap 1983 further an argument by meyer stated in berman and hartmanis 1977 is generalized to show that a set has small generators if and only if for some sparse set s a np s

multistability for delayed neural networks via sequential contracting in this paper we explore a variety of new multistability scenarios in the general delayed neural network system geometric structure embedded in equations is exploited and incorporated into the analysis to elucidate the underlying dynamics criteria derived from different geometric configurations lead to disparate numbers of equilibria a new approach named sequential contracting is applied to conclude the global convergence to multiple equilibrium points of the system the formulation accommodates both smooth sigmoidal and piecewise linear activation functions several numerical examples illustrate the present analytic theory

mean field approximation for noisy delay coupled excitable neurons abstract mean field approximation of a large collection of fitzhugh nagumo excitable neurons with noise and all to all coupling with explicit time delays modelled by n 1 stochastic delay differential equations is derived the resulting approximation contains only two deterministic delay differential equations but provides excellent predictions concerning the stability and bifurcations of the averaged global variables of the exact large system

logarithmic advice classes abstract karp and lipton 1980 introduced the notion of nonuniform complexity classes where a certain amount of additional information the advice is given for free the advice only depends on the length of the input karp and lipton initiated the study of classes with either logarithmic or polynomial advice however later yap 1983 schoning 1984 balcazar 1987 and ko and schoning 1985 concentrated on the study of classes of the form c poly where c is p np or pspace and poly denotes a polynomial size advice this paper considers classes of the form c log as a main result it is shown that in the context of an np log computation log bounded advice is equivalent to a sparse oracle in np in contrast it has been shown that a poly bounded advice to an arbitrary sparse oracle set furthermore a general theorem is presented that generalizes karp and lipton s round robin tournament method

synchronization of noisy delayed feedback systems with delayed coupling synchronization of delayed coupled and stochastically perturbed systems with delayed nonlinear feedback is studied using as an example circular chains of three and four delayed coupled ikeda oscillators it is proved that in the case of multiplicative noise the exact synchronization in the mean occurs for sufficiently large coupling and an analytic estimate of the sufficient coupling is given the sufficiency condition is compared with numerical computations and typical effects of noise on the exact and some generalized types of synchronization are illustrated

short time delay limit of the self coupled fitzhugh nagumo system we analyze the fitzhugh nagumo equations subject to time delayed self feedback in the activator variable parameters are chosen such that the steady state is stable independent of the feedback gain and delay we demonstrate that stable large amplitude periodic oscillations can however coexist with a stable steady state even for small delays which is mathematically counterintuitive in order to explore how these solutions appear in the bifurcation diagram we propose three different strategies we first analyze the emergence of periodic solutions from hopf bifurcation points for small and show that a subcritical hopf bifurcation allows the coexistence of stable periodic and stable steady state solutions second we construct a periodic solution by using singular perturbation techniques appropriate for slow fast systems the theory assumes o 1 and its validity as 0 is investigated numerically by integrating the original equations third we develop an asymptotic theory where the delay is scaled with respect to the fast timescale of the activator variable the theory is applied to the fitzhugh nagumo equations with threshold nonlinearity and we show that the branch of periodic solutions emerges from a limit point of limit cycles

inphase and antiphase synchronization in a delay coupled system with applications to a delay coupled fitzhugh nagumo system a time delay is inevitable in the coupled system and is an essential property of the coupling which cannot be neglected in many realistic coupled systems in this paper we first study the existence of a hopf bifurcation induced by coupling time delay and then investigate the influence of coupling time delay on the patterns of hopf bifurcating periodic oscillations how the coupling time delay leads to complex scenarios of synchronized inphase or antiphase oscillations is analytically investigated as an example we study the delay coupled fitzhugh nagumo system we find conditional stability absolute stability and stability switches of the steady state provoked by the coupling time delay then we investigate the inphase and antiphase synchronized periodic solutions induced by delay and determine the direction and stability of these bifurcating periodic orbits by employing the center manifold reduction and normal form theory we find that in the region where stability switches occur there exist synchronization transitions i e synchronized dynamics can be switched from inphase antiphase to antiphase inphase and back to inphase antiphase and so on just by progressive increase of the coupling time delay

stability bifurcation and multistability in a system of two coupled neurons with multiple time delays a system of delay differential equations representing a model for a pair of neurons with time delayed connections between the neurons and time delayed feedback from each neuron to itself is studied conditions for the linear stability of the trivial solution of this system are represented in a parameter space consisting of the sum of the time delays between the elements and the product of the strengths of the connections between the elements it is shown that the trivial fixed point may lose stability via a pitchfork bifurcation a hopf bifurcation or one of three types of codimension two bifurcations multistability near these latter bifurcations is predicted using center manifold analysis and confirmed using numerical simulations

approximation of jensen measures by image measures under holomorphic functions and applications we show that jensen measures defined on c n or more generally on a complex banach space x can be approximated by the image of lebesgue measure on the torus under x valued polynomials defined on c we give similar characterizations for jensen measures in terms of analytic martingales and hardy martingales the results are applied to approximate plurisubharmonic martingales by hardy martingales which enables us to give a characterization of the analytic radon nikodym property of banach spaces in terms of convergence of plurisubharmonic martingales thus solving a problem of g a edgar

the existence and density of generalized complexity cores if c is a class of sets and a is not in c then an infinite set h is a proper hard core for a with respect to c if h a and for every c e c such that c a c c h is finite it is shown that if c is a countable class of sets of strings that is closed under finite union and finite variation then every infinite set not in c has a proper hard core with respect to c in addition the density of such generalized complexity cores is studied

period adding bifurcations and chaos in a periodically stimulated excitable neural relaxation oscillator the response of an excitable neuron to trains of electrical spikes is relevant to the understanding of the neural code in this paper we study a neurobiologically motivated relaxation oscillator with appropriately identified fast and slow coordinates that admits an explicit mathematical analysis an application of geometric singular perturbation theory shows the existence of an attracting invariant manifold which is used to construct the fenichel normal form for the system this facilitates the calculation of the response of the system to pulsatile stimulation and allows the construction of a so called extended isochronal map the isochronal map is shown to have a single discontinuity and be of a type that can admit three types of response mode locked quasiperiodic and chaotic the bifurcation structure of the system is seen to be extremely rich and supports period adding bifurcations separated by windows of both chaos and periodicity a bifurcation analysis of the isochronal map is presented in conjunction with a description of the various routes to chaos in this system

a novel approach to synchronization of nonlinearly coupled network systems with delays in this investigation a novel approach to establishing the global synchronization of coupled network systems is presented under this approach individual subsystems can be non autonomous and the coupling configuration is rather general the coupling terms can be non diffusive nonlinear time dependent asymmetric and with time delays with an iteration scheme the problem of synchronization is transformed into solving a corresponding linear system of algebraic equations subsequently delay dependent and delay independent criteria for global synchronization can be established we implement the present approach to analyze synchronization of the fitzhugh nagumo systems under delayed and nonlinear sigmoidal coupling two examples are presented to demonstrate new dynamical scenarios where oscillatory behavior and multistability emerge or are suppressed as the coupled neurons synchronize under the synchronization criterion in addition asynchrony induced by the coupling strength or coupling delay occurs while the synchronization criterion is violated

some consequences of non uniform conditions on uniform classes we obtain some results of the form if certain complexity classes satisfy a non uniform condition then some unlikely consequences follow more precisely 1 1 if the non uniform polynomial time hierarchy collapses at level i 0 i e ipoly ipoly then the meyer stockmeyer hierarchy collapses at level i 2 i e i 2 i 2 this strengthens a generalization of a result of karp and lipton 1980 2 2 if co np is conjunctively reducible to a sparse set then p np this generalizes a theorem of fortune 1979 3 3 if np is conjunctively and disjunctively reducible to a sparse np complete set then p np this is a partial generalization of a result of mahaney 1980 conjuctive and disjunctive reducibility were introduced by ladner lynch and selman 1975 4 4 if co np is reducible to a sparse set then np co np reducibility was introduced 5 by adleman and manders 1977

synchronization in relaxation oscillator networks with conduction delays we study locally coupled networks of relaxation oscillators with excitatory connections and conduction delays and propose a mechanism for achieving zero phase lag synchrony our mechanism is based on the observation that different rates of motion along different nullclines of the system can lead to synchrony in the presence of conduction delays we analyze the system of two coupled oscillators and derive phase compression rates this analysis indicates how to choose nullclines for individual relaxation oscillators in order to induce rapid synchrony the numerical simulations demonstrate that our analytical results extend to locally coupled networks with conduction delays and that these networks can attain rapid synchrony with appropriately chosen nullclines and initial conditions the robustness of the proposed mechanism is verified with respect to different nullclines variations in parameter values and initial conditions

dynamics of fitzhugh nagumo excitable systems with delayed coupling small lattices of n nearest neighbor coupled excitable fitzhugh nagumo systems with time delayed coupling are studied and compared with systems of fitzhugh nagumo oscillators with the same delayed coupling bifurcations of equilibria in an n 2 case are studied analytically and it is then numerically confirmed that the same bifurcations are relevant for the dynamics in the case ng2 bifurcations found include inverse and direct hopf and fold limit cycle bifurcations typical dynamics for different small time lags and coupling intensities could be excitable with a single globally stable equilibrium asymptotic oscillatory with symmetric limit cycle bistable with stable equilibrium and a symmetric limit cycle and again coherent oscillatory but nonsymmetric and phase shifted for an intermediate range of time lags inverse sub critical hopf and fold limit cycle bifurcations lead to the phenomenon of oscillator death the phenomenon does not occur in the case of fitzhugh nagumo oscillators with the same type of coupling

phase equations for relaxation oscillators we use the malkin theorem to derive phase equations for networks of weakly connected relaxation oscillators we find an explicit formula for the connection functions when the oscillators have one dimensional slow variables the functions are discontinuous in the relaxation limit mu rightarrow 0 which provides a simple alternative illustration to the major conclusion of the fast threshold modulation ftm theory by somers and kopell biological cybernetics 68 1993 pp 393 407 that synchronization of relaxation oscillators has properties that are quite different from those of smooth nonrelaxation oscillators we use bonhoeffer van der pol relaxation oscillators to illustrate the theory numerically