complementation in spaces of continuous functions on compact lines abstract we characterize order preserving continuous surjections between compact linearly ordered spaces which admit an averaging operator together with estimates of the norm of such an operator this result is used to the study of strengthenings of the separable complementation property in spaces of continuous functions on compact lines these properties include in particular continuous separable complementation property and existence of a projectional skeleton

dynamics of moments of fitzhugh nagumo neuronal models and stochastic bifurcations for the study of the behavior of noisy neuronal models rodriguez and tuckwell have introduced an elegant and systematic method which consists of replacing the system of stochastic differential equations with a system of deterministic equations representing the dynamics of the means variances and covariance of the state variables r rodriguez and h c tuckwell phys rev e 54 5585 1996 in this work we first report a modification of their method in the case of the fitzhugh nagumo model which enhances the accuracy of the approximation without including higher order moments this method is then combined with a self consistency argument in order to better characterize the behavior of the underlying stochastic processes through the computation of approximate auto and cross correlation functions of the state variables finally we argue that the moments equations can also reveal the existence of stochastic bifurcations i e qualitative changes in the dynamics of stochastic systems

statistical properties of stochastic nonlinear dynamical models of single spiking neurons and neural networks dynamical stochastic models of single neurons and neural networks often take the form of a system of n 2 coupled stochastic differential equations we consider such systems under the assumption that third and higher order central moments are relatively small in the general case a system of 1 n n13 generally nonlinear coupled ordinary differential equations holds for the approximate means variances and covariances for the general linear system the solutions of these equations give exact results this is illustrated in a simple case generally the moment equations can be solved numerically results are given for a spiking fitzhugh nagumo model neuron driven by a current with additive white noise differential equations are obtained for the means variances and covariances of the dynamical variables in a network of n connected spiking neurons in the presence of noise s1063 651x 96 03511 8 pacs number s 87 10 1e 05 40 1j 02 50 2r

order results for implicit runge kutta methods applied to stiff systems an extensive order analysis is given for the local and global errors of certain classes of runge kutta methods applied to general nonlinear stiff initial value problems it turns out that there are three order levels for linear problems y ay the classical orders apply for general nonlinear problems however two types of order reductions occur there a distinction has to be made between stiff problems where only f y is large and stiff problems where also some of the other derivatives f t f tt f yy cdots are large for these two classes of problems order results can be guaranteed which differ only by one power of the stepsize parameter h the order levels for both classes of problems are considerably lower than the respective conventional orders these order results are optimal in the sense that higher orders would contradict numerical observations made with particular initial value problems

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

numerical solution of nonlinear differential equations with algebraic constraints i convergence results for backward differentiation formulas in this paper we investigate the behavior of numerical ode methods for the solution of systems of differential equations coupled with algebraic constraints systems of this form arise frequently in the modelling of problems from physics and engineering we study some particular examples from electrical networks fluid dynamics and constrained mechanical systems we show that backward differentiation formulas converge with the expected order of accuracy for these systems

almost disjoint families of countable sets and separable complementation properties abstract we study the separable complementation property scp and its natural variations in banach spaces of continuous functions over compacta k a induced by almost disjoint families a of countable subsets of uncountable sets for these spaces we prove among other things that c k a has the controlled variant of the separable complementation property if and only if c k a is lindelof in the weak topology if and only if k a is monolithic we give an example of a for which c k a has the scp while k a is not monolithic and an example of a space c k a with controlled and continuous scp which has neither a projectional skeleton nor a projectional resolution of the identity finally we describe the structure of almost disjoint families of cardinality 1 which induce monolithic spaces of the form k a they can be obtained from countably many ladder systems and pairwise disjoint families by applying simple operations

the numerical solution of higher index differential algebraic equations by implicit methods this paper studies the order stability and convergence properties of implicit runge kutta irk methods applied to differential algebraic systems with index greater than one these methods do not in general attain the same order of accuracy for higher index differential algebraic systems as they do for index 1 systems or for purely differential systems necessary and sufficient conditions on the method coefficients are derived to ensure that the local and global errors of the method attain a given order of accuracy for higher index linear constant coefficient systems irk methods applied to nonlinear semi explicit index 2 systems are studied and a sufficient set of conditions is derived which ensures that a method is accurate to a given order for these systems finally some numerical experiments are presented that illustrate the theoretical results and demonstrate the effects of roundoff errors on the solution

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

dynamical mean field theory of noisy spiking neuron ensembles application to the hodgkin huxley model a dynamical mean field approximation dma previously proposed by the present author h hasegawa phys rev e 67 041903 2003 has been extended to ensembles described by a general noisy spiking neuron model ensembles of n unit neurons each of which is expressed by coupled k dimensional differential equations des are assumed to be subject to spatially correlated white noises the original mathrm kn dimensional stochastic des have been replaced by k k 2 dimensional deterministic des expressed in terms of means and the second order moments of local and global variables the fourth order contributions are taken into account by the gaussian decoupling approximation our dma has been applied to an ensemble of hodgkin huxley hh neurons k 4 for which effects of the noise the coupling strength and the ensemble size on the response to a single spike input have been investigated numerical results calculated by the dma theory are in good agreement with those obtained by direct simulations although the former computation is about a thousand times faster than the latter for a typical hh neuron ensemble with n 100

efficiently implementable algebraically stable runge kutta methods recent work by butcher bit 16 1976 pp 237 240 on the implementation of implicit runge kutta methods has shown that if his technique is used the most efficiently implementable methods are those whose runge kutta matrix has a single real eigenvalue in this paper therefore we analyze the algebraic stability of two families of methods with this property namely singly implicit and diagonally implicit methods

compact lines and the sobczyk property abstract we show that sobczyk s theorem holds for a new class of banach spaces namely spaces of continuous functions on linearly ordered compacta

augmented moment method for stochastic ensembles with delayed couplings ii fitzhugh nagumo model dynamics of fitzhugh nagumo fn neuron ensembles with time delayed couplings subject to white noises has been studied by using both direct simulations and a semianalytical augmented moment method amm which has been proposed in a preceding paper h hasegawa phys rev e 70 021911 2004 for n unit fn neuron ensembles amm transforms original 2n dimensional stochastic delay differential equations sddes to infinite dimensional deterministic des for means and correlation functions of local and global variables infinite order recursive des are terminated at the finite level m in the level m amm mathrm amm m yielding 8 m 1 dimensional deterministic des when a single spike is applied the oscillation may be induced if parameters of coupling strength delay noise intensity and or ensemble size are appropriate effects of these parameters on the emergence of the oscillation and on the synchronization in fn neuron ensembles have been studied the synchronization shows the fluctuation induced enhancement at the transition between nonoscillating and oscillating states results calculated by amm5 are in fairly good agreement with those obtained by direct simulations

order results for implicit runge kutta methods applied to differential algebraic systems in this paper we study the order stability and convergence properties of implicit runge kutta methods applied to a relatively simple class of nonlinear differential algebraic systems these methods often do not attain the same order of accuracy for differential algebraic systems as they do for purely differential systems we derive a set of order conditions which the method coefficients should satisfy in addition to the usual order conditions to ensure a given order of accuracy and we present results on the stability and convergence properties of these methods

augmented moment method for stochastic ensembles with delayed couplings i langevin model by employing a semianalytical dynamical mean field approximation theory previously proposed by the author h hasegawa phys rev e 67 041903 2003 we have developed an augmented moment method amm in order to discuss dynamics of an n unit ensemble described by langevin equations with delays in an amm original n dimensional stochastic delay differential equations sddes are transformed to infinite dimensional deterministic des for means and correlations of local as well as global variables infinite order des arising from the non markovian property of sdde are terminated at the finite level m in the level m amm mathrm amm m which yields 3 m dimensional deterministic des model calculations have been made for linear and nonlinear langevin models the stationary solution of amm for the linear langevin model with n 1 is nicely compared to the exact result in the nonlinear langevin ensemble the synchronization is shown to be enhanced near the transition point between the oscillating and nonoscillating states results calculated by amm6 are in good agreement with those obtained by direct simulations

on extensions of c0 valued operators abstract we study pairs of banach spaces x y with y x for which the thesis of sobczyk s theorem holds namely such that every bounded c 0 valued operator defined in y extends to x we are mainly concerned with the case when x is a c k space and y c l is a banach subalgebra of c k the main result of the article states that if k is a compact line and l is countable then every bounded c 0 valued operator defined in c l extends to c k

on the c0 extension property for compact lines abstract we present a characterization of the continuous increasing surjections k l between compact lines k and l for which the corresponding subalgebra c l has the c 0 extension property in c k a natural question arising in connection with this characterization is shown to be independent of the axioms of zfc

a note on monolithic scattered compacta abstract for a banach space e it is well known that a necessary condition for e to have the controlled separable complementation property cscp for short is that the dual unit ball b e be monolithic in the weak star topology we prove here that when x is a scattered first countable locally compact space then monolithicity of x turns out to be sufficient for c 0 x to enjoy the cscp

banach spaces with projectional skeletons a projectional skeleton in a banach space is a directed family of projections onto separable subspaces covering the entire space the class of banach spaces with projectional skeletons is strictly larger than the class of plichko spaces i e banach spaces with a countably norming markushevich basis we show that every space with a projectional skeleton has a projectional resolution of the identity and has a norming space with similar properties to spaces we characterize the existence of a projectional skeleton in terms of elementary substructures providing simple proofs of known results concerning weakly compactly generated spaces and plichko spaces we prove a preservation result for plichko banach spaces involving transfinite sequences of projections as a corollary we show that a banach space is plichko if and only if it has a commutative projectional skeleton

on weak extensible banach spaces abstract we study the stability properties of the class of weak extensible spaces introduced by wang zhao and qiang showing among other things that weak extensibility is equivalent to having a weak sequentially continuous dual ball in short w sc for duals of separable spaces or twisted sums of w sc spaces this shows that weak extensibility is not a 3 space property solving a question posed by wang zhao and qiang we also introduce a restricted form of weak extensibility called separable weak extensibility and show that separably weak extensible banach spaces have the gelfand phillips property although they are not necessarily w sc spaces

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