point process principal components analysis via geometric optimization there has been a fast growing demand for analysis tools for multivariate point process data driven by work in neural coding and more recently high frequency finance here we develop a true or exact as opposed to one based on time binning principal components analysis for preliminary processing of multivariate point processes we provide a maximum likelihood estimator an algorithm for maximization involving steepest ascent on two stiefel manifolds and novel constrained asymptotic analysis the method is illustrated with a simulation and compared with a binning approach

an algebraic method to solve the radial schrodinger equation we propose a method of numerical integration of differential equations of the type x 2y f x y 0 by approximating its solution with solutions of equations of the type x 2y ax 2 bx c y 0 this approximation is performed by segmentary approximation on an interval we apply the method to obtain approximate solutions of the radial schrodinger equation on a given interval and test it for two different potentials we conclude that our method gives a similar accuracy than the taylor method of higher order

rieffel deformation via crossed products we start from rieffel data a where a is a c algebra is an action of an abelian group on a and is a 2 cocycle on the dual group using landstad theory of crossed product we get a deformed c algebra a in the case of rn we obtain a very simple proof of invariance of k groups under the deformation in the general case we also get a very simple proof that nuclearity is preserved under the deformation we show how our approach leads to quantum groups and investigate their duality the general theory is illustrated by an example of the deformation of sl 2 c a description of it in terms of noncommutative coordinates is given

quantum symmetry groups of noncommutative spheres we show that the noncommutative spheres of connes and landi are quantum homogeneous spaces for certain compact quantum groups we give a general construction of homogeneous spaces which support noncommutative spin geometries

unitary events in multiple single neuron spiking activity ii nonstationary data in order to detect members of a functional group cell assembly in simultaneously recorded neuronal spiking activity we adopted the widely used operational definition that membership in a common assembly is expressed in near simultaneous spike activity unitary event analysis a statistical method to detect the significant occurrence of coincident spiking activity in stationary data was recently developed see the companion article in this issue the technique for the detection of unitary events is based on the assumption that the underlying processes are stationary in time this requirement however is usually not fulfilled in neuronal data here we describe a method that properly normalizes for changes of rate the unitary events by moving window analysis uemwa analysis for unitary events is performed separately in overlapping time segments by sliding a window of constant width along the data in each window stationarity is assumed performance and sensitivity are demonstrated by use of simulated spike trains of independently firing neurons into which coincident events are inserted if cortical neurons organize dynamically into functional groups the occurrence of near simultaneous spike activity should be time varying and related to behavior and stimuli uemwa also accounts for these potentially interesting nonstationarities and allows locating them in time the potential of the new method is illustrated by results from multiple single unit recordings from frontal and motor cortical areas in awake behaving monkey

cortical activity flips among quasi stationary states parallel recordings of spike trains of several single cortical neurons in behaving monkeys were analyzed as a hidden markov process the parallel spike trains were considered as a multivariate poisson process whose vector firing rates change with time as a consequence of this approach the complete recording can be segmented into a sequence of a few statistically discriminated hidden states whose dynamics are modeled as a first order markov chain the biological validity and benefits of this approach were examined in several independent ways i the statistical consistency of the segmentation and its correspondence to the behavior of the animals ii direct measurement of the collective flips of activity obtained by the model and iii the relation between the segmentation and the pair wise short term cross correlations between the recorded spike trains comparison with surrogate data was also carried out for each of the above examinations to assure their significance our results indicated the existence of well separated states of activity within which the firing rates were approximately stationary with our present data we could reliably discriminate six to eight such states the transitions between states were fast and were associated with concomitant changes of firing rates of several neurons different behavioral modes and stimuli were consistently reflected by different states of neural activity moreover the pair wise correlations between neurons varied considerably between the different states supporting the hypothesis that these distinct states were brought about by the cooperative action of many neurons

a flexible method to measure synchrony in neuronal firing neurons can transmit information about the characteristics of a stimulus through the spike rate of neurons and synchronization of the neurons various association measure can be used to describe how synchronous two spike trains are we propose a new measure of synchrony the conditional synchrony measure which is the probability of firing together given that at least one of the two neurons is active focus is on the specification of a flexible marginal model for multivariate correlated binary data together with a pseudolikelihood estimation approach to adequately and directly describe the measures of interest a joint model must allow different time and covariate dependent firing rates for each neuron and also must account for the association between them the association between neurons might depend on covariates as well

modified explicit runge kutta methods for the numerical solution of the schrodinger equation a new procedure for constructing runge kutta methods for an efficient integration of the radial schrodinger equation is developed in this paper the new modified runge kutta methods are of algebraically order four the asymptotic expressions of the local errors for large energies are discussed numerical results obtained for the widely used woods saxon potential show the efficiency of the new methods compared with other special optimized fourth order runge kutta methods the error analysis will be clearly confirmed by the resonance problem

unbounded elements affiliated with c algebras and non compact quantum groups the affiliation relation that allows to include unbounded elements operators into thec algebra framework is introduced investigated and applied to the quantum group theory the quantum deformation of the two fold covering of the group of motions of euclidean plane is constructed a remarkable radius quantization is discovered it is also shown that the quantumsu 1 1 group does not exist on thec algebra level for real value of the deformation parameter

damping and phase analysis for some methods for solving second order ordinary differential equations we consider numerical methods for initial value problems for second order systems of ordinary differential equations analysing them by applying them to the test equation we discuss conditions which ensure an oscillatory numerical solution and the desirability of such a property we also use a slightly more general test equation to derive conditions which ensure that the numerical forced oscillation is in phase with the true forced oscillation to illustrate the theory we consider the damping and phase properties of some frequently used methods

rieffel deformation of group coactions let g be a locally compact group gamma subset g an abelian subgroup and let be a continuous 2 cocycle on the dual group hat gamma let b be a c algebra and delta b in rm mor b b otimes rm c 0 g a continuous right coaction using rieffel deformation we can construct a quantum group rm c 0 g tilde psi otimes psi delta psi and the deformed c algebra b the aim of this paper is to present a construction of the continuous coaction delta b psi of the quantum group rm c 0 g tilde psi otimes psi delta psi on b the transition from the coaction b to its deformed counterpart delta b psi is nontrivial in the sense that delta b psi contains complete information about b in order to illustrate our construction we apply it to the action of the lorentz group on the minkowski space obtaining a c algebraic quantum minkowski space

operator equalities related to the quantum e 2 group the paper deals with normal operatorsr ands satisfying simple commutation relations that we encounter investigating the quantum deformation of thee 2 group we show thatr s admits a normal extension if and only ifr 1s satisfies a certain spectral condition a number of related formulae are derived in particular all the functionsf satisfying the character equationf r s f r f s are found

rieffel deformation of homogeneous spaces let g1 g be a closed subgroup of a locally compact group g and let x g g1 be the quotient space of left cosets let x c0 x x be the corresponding g c algebra where g c0 g suppose that is a closed abelian subgroup of g1 and let be a 2 cocycle on the dual group let g be the rieffel deformation of g using the results of the previous paper of the author we may construct g c algebra x the rieffel deformation of x on the other hand we may perform the rieffel deformation of the subgroup g1 obtaining the closed quantum subgroup g1 g which in turn by the results of s vaes leads to the g c algebra g g1 in this paper we show that g g1 x we also consider the case where g is not a subgroup of g1 for which we cannot construct the subgroup g1 then generically x cannot be identified with a quantum quotient what may be shown is that it is a g simple object in the category of g c algebras

multiple tests based on a gaussian approximation of the unitary events method with delayed coincidence count the unitary events ue method is one of the most popular and efficient methods used over the past decade to detect patterns of coincident joint spike activity among simultaneously recorded neurons the detection of coincidences is usually based on binned coincidence count grun 1996 which is known to be subject to loss in synchrony detection grun diesmann grammont riehle aertsen 1999 this defect has been corrected by the multiple shift coincidence count grun et al 1999 the statistical properties of this count have not been further investigated until this work the formula being more difficult to deal with than the original binned count first we propose a new notion of coincidence count the delayed coincidence count which is equal to the multiple shift coincidence count when discretized point processes are involved as models for the spike trains moreover it generalizes this notion to nondiscretized point processes allowing us to propose a new gaussian approximation of the count since unknown parameters are involved in the approximation we perform a plug in step where unknown parameters are replaced by estimated ones leading to a modification of the approximating distribution finally the method takes the multiplicity of the tests into account via a benjamini and hochberg approach benjamini hochberg 1995 to guarantee a prescribed control of the false discovery rate we compare our new method mtgaue multiple tests based on a gaussian approximation of the unitary events and the ue method proposed in grun et al 1999 over various simulations showing that mtgaue extends the validity of the previous method in particular mtgaue is able to detect both profusion and lack of coincidences with respect to the independence case and is robust to changes in the underlying model furthermore mtgaue is applied on real data

on a certain approach to quantum homogeneous spaces we propose a definition of a quantum homogeneous space of a locally compact quantum group we show that classically it reduces to the notion of homogeneous spaces giving rise to an operator algebraic characterization of the transitive group actions on the quantum level our definition goes beyond the quotient case providing a framework which besides the vaes quotient of a locally compact quantum group by its closed quantum subgroup our main motivation is also compatible with generically non quotient quantum homogeneous spaces of a compact quantum group studied by p podle as well as the rieffel deformation of g homogeneous spaces finally our definition rules out the paradoxical examples of the non compact quantum homogeneous spaces of a compact quantum group

correlations without synchrony peaks in spike train correlograms are usually taken as indicative of spike timing synchronization between neurons strictly speaking however a peak merely indicates that the two spike trains were not independent two biologically plausible ways of departing from independence that are capable of generating peaks very similar to spike timing peaks are described here covariations over trials in response latency and covariations over trials in neuronal excitability since peaks due to these interactions can be similar to spike timing peaks interpreting a correlogram may be a problem with ambiguous solutions what peak shapes do latency or excitability interactions generate when are they similar to spike timing peaks when can they be ruled out from having caused an observed correlogram peak these are the questions addressed here the previous article in this issue proposes quantitative methods to tell cases apart when latency or excitability covariations cannot be ruled out

surrogate data methods based on a shuffling of the trials for synchrony detection the centering issue we investigate several distribution free dependence detection procedures all based on a shuffling of the trials from a statistical point of view the mathematical justification of such procedures lies in the bootstrap principle and its approximation properties in particular we show that such a shuffling has mainly to be done on centered quantities that is quantities with zero mean under independence to construct correct p values meaning that the corresponding tests control their false positive fp rate thanks to this study we introduce a method named permutation ue which consists of a multiple testing procedure based on permutation of experimental trials and delayed coincidence count each involved single test of this procedure achieves the prescribed level so that the corresponding multiple testing procedure controls the false discovery rate fdr and this with as few assumptions as possible on the underneath distribution except independence and identical distribution across trials the mathematical meaning of this assumption is discussed and it is in particular argued that it does not mean what is commonly referred in neuroscience to as cross trials stationarity some simulations show moreover that permutation ue outperforms the trial shuffling of pipa and grun 2003 and the mtgaue method of tuleau malot eti al 2014 in terms of single levels and fdr for a comparable amount of false negatives application to real data is also provided

dependence of neuronal correlations on filter characteristics and marginal spike train statistics correlated neural activity has been observed at various signal levels e g spike count membrane potential local field potential eeg fmri bold most of these signals can be considered as superpositions of spike trains filtered by components of the neural system synapses membranes and the measurement process it is largely unknown how the spike train correlation structure is altered by this filtering and what the consequences for the dynamics of the system and for the interpretation of measured correlations are in this study we focus on linearly filtered spike trains and particularly consider correlations caused by overlapping presynaptic neuron populations we demonstrate that correlation functions and statistical second order measures like the variance the covariance and the correlation coefficient generally exhibit a complex dependence on the filter properties and the statistics of the presynaptic spike trains we point out that both contributions can play a significant role in modulating the interaction strength between neurons or neuron populations in many applications the coherence allows a filter independent quantification of correlated activity in different network models we discuss the estimation of network connectivity from the high frequency coherence of simultaneous intracellular recordings of pairs of neurons

the quantum square well with moving boundaries a numerical analysis we present some numerical discussions concerning the infinite square well in one dimension with moving boundaries our results show that if the speed of displacement is small objects of physical relevance like probability density averaged position or mean value of the energy have a smooth behavior on the contrary if this speed becomes large many irregularities arise which has a difficult qualitative explanation these irregularities manifest themselves as sharp bumps on the probability distribution or a chaotic shape on the averaged values of position and energy none of these patterns is the result of numerical errors and therefore we conclude that an unknown and very nontrivial effect is produced at high speeds of the moving wall

families of two step fourth order p stable methods for second oder differential equations this paper deals with a class of symmetric hybrid two step fourth order p stable methods for the numerical solution of special second order initial value problems such methods were proposed independently by cash 1 and chawla 3 and normally require three function evaluations per step the purpose of this paper is to point out that there are some values of the free parameters available in the methods proposed which can reduce this work we study two classes of such methods the first is the class of economical methods see definition 3 1 which reduce this work to two function evaluations per step and the second is the class of efficient methods see definition 3 2 which reduce this work with respect to implementation for nonlinear problems we report numerical experiments to illustrate the order acuracy and implementational aspects of these two classes of methods

can spike coordination be differentiated from rate covariation there has been a long and lively debate on whether rate covariance and temporal coordination of spikes regarded as potential origins for correlations in cortical spike signals fulfill different roles in the cortical code in this context studies that report spike coordination have often been criticized for ignoring fast nonstationarities which would result in wrongly assigned spike coordination the underlying hypothesis of this critique is that spike coordination is essentially identical to rate covariation only on a shorter timescale this study investigates the validity of this critique we provide a decomposition for the cross correlation function of doubly stochastic point processes where each of the components corresponds precisely to the concepts of dependence under investigation this allows us to correct the correlation function for rate effects which implies that spike coordination and rate covariation are statistically separable concepts of dependence furthermore we present direct and intuitive model implementations of the discussed concepts and illustrate that their difference is not a matter of timescale analysis of data generated by our models and analytical description of the relevant estimators reveals however that spike coordination dramatically influences the accuracy of rate covariance estimation as a consequence extreme parameter combinations can lead to situations where the concept of dependence cannot be identified empirically however for a wide range of parameters the concept of dependence underlying a given data set can be identified regardless of its timescale

improvements to the sensitivity of gravitational clustering for multiple neuron recordings we outline two improvements to the technique of gravitational clustering for detection of neuronal synchrony which are capable of improving the method s detection of weak synchrony with limited data the advantages of the enhancements are illustrated using data with known levels of synchrony and different interspike interval distributions the novel simulation method described can easily generate such test data an important dependence of the sensitivity of gravitational clustering to the interspike interval distribution of the analysed spike trains is described

non compact quantum groups associated with abelian subgroups letg be a lie group for any abelian subalgebra mathfrak h of the lie algebra g ofg and any r in mathfrak h lambda mathfrak h the difference of the left and right translates ofr gives a compatible poisson bracket ong we show how to construct the corresponding quantum group in thec algebra setting the main tool used is the general deformation quantization construction developed earlier by the author for actions of vector groups onc algebras

deformation of c algebras by cocycles on locally compact quantum groups given a c algebra a with a left action of a locally compact quantum group g on it and a unitary 2 cocycle on g we define a deformation a of a the construction behaves well under certain additional technical assumptions on the most important of which is regularity meaning that c0 g g is isomorphic to the algebra of compact operators on some hilbert space in particular then a is stably isomorphic to the iterated twisted crossed product g op g a also in good situations the c algebra a carries a left action of the deformed quantum group g and we have an isomorphism g a g a when g is a genuine locally compact group we show that the action of g on c0 g cr g is always integrable stronger assumptions of properness and saturation of the action imply regularity as an example we make a preliminary analysis of the cocycles on the duals of some solvable lie groups recently constructed by bieliavsky et al and discuss the relation of our construction to that of bieliavsky and gayral

information geometric measures for estimation of connection weight under correlated inputs the brain processes information in a highly parallel manner determination of the relationship between neural spikes and synaptic connections plays a key role in the analysis of electrophysiological data information geometry ig has been proposed as a powerful analysis tool for multiple spike data providing useful insights into the statistical interactions within a population of neurons previous work has demonstrated that ig measures can be used to infer the connection weight between two neurons in a neural network this property is useful in neuroscience because it provides a way to estimate learning induced changes in synaptic strengths from extracellular neuronal recordings a previous study has shown however that this property would hold only when inputs to neurons are not correlated since neurons in the brain often receive common inputs this would hinder the application of the ig method to real data we investigated the two neuron ig measures in higher order log linear models to overcome this limitation first we mathematically showed that the estimation of uniformly connected synaptic weight can be improved by taking into account higher order log linear models second we numerically showed that the estimation can be improved for more general asymmetrically connected networks considering the estimated number of the synaptic connections in the brain we showed that the two neuron ig measure calculated by the fourth or fifth order log linear model would provide an accurate estimation of connection strength within approximately a 10 error these studies suggest that the two neuron ig measure with higher order log linear expansion is a robust estimator of connection weight even under correlated inputs providing a useful analytical tool for real multineuronal spike data

information geometric measure for neural spikes this study introduces information geometric measures to analyze neural firing patterns by taking not only the second order but also higher order interactions among neurons into account information geometry provides useful tools and concepts for this purpose including the orthogonality of coordinate parameters and the pythagoras relation in the kullback leibler divergence based on this orthogonality we show a novel method for analyzing spike firing patterns by decomposing the interactions of neurons of various orders as a result purely pairwise triple wise and higher order interactions are singled out we also demonstrate the benefits of our proposal by using several examples

a noumerov type method with minimal phase lag for the integration of second order periodic initial value abstract in a recent paper 2 we gave a noumerov type method with minimal phase lag for the integration of second order initial value problems y f t y y t0 y0 y t0 y 0 however the method given there is implicit we show here the interesting result that if the noumerov type methods of 2 are made explicit with the help of the classical second order method then there exists a selection of the free parameter for which the resulting method has a considerably small frequency distortion of size 1 40320 h 6 and also a slightly larger interval of periodicity of size 2 75 than the phase lag of size 1 12096 h 6 and interval of periodicity of size 2 71 for the implicit method 2 more interestingly it turns out that noumerov made explicit of chawla 3 also has less frequency distortion than the implicit noumerov method we shall assume a familiarity with the notation and discussion given in 2

influence of external inputs and asymmetry of connections on information geometric measures involving up to ten neuronal interactions the investigation of neural interactions is crucial for understanding information processing in the brain recently an analysis method based on information geometry ig has gained increased attention and the property of the pairwise ig measure has been studied extensively in relation to the two neuron interaction however little is known about the property of ig measures involving more neuronal interactions in this study we systematically investigated the influence of external inputs and the asymmetry of connections on the ig measures in cases ranging from 1 neuron to 10 neuron interactions first the analytical relationship between the ig measures and external inputs was derived for a network of 10 neurons with uniform connections our results confirmed that the single and pairwise ig measures were good estimators of the mean background input and of the sum of the connection weights respectively for the ig measures involving 3 to 10 neuronal interactions we found that the influence of external inputs was highly nonlinear second by computer simulation we extended our analytical results to asymmetric connections for a network of 10 neurons the simulation showed that the behavior of the ig measures in relation to external inputs was similar to the analytical solution obtained for a uniformly connected network when the network size was increased to 1000 neurons the influence of external inputs almost disappeared this result suggests that all ig measures from 1 neuron to 10 neuron interactions are robust against the influence of external inputs in addition we investigated how the strength of asymmetry influenced the ig measures computer simulation of a 1000 neuron network showed that all the ig measures were robust against the modulation of the asymmetry of connections our results provide further support for an information geometric approach and will provide useful insights when these ig measures are applied to real experimental spike data

investigation of possible neural architectures underlying information geometric measures a novel analytical method based on information geometry was recently proposed and this method may provide useful insights into the statistical interactions within neural groups the link between informationgeometric measures and the structure of neural interactions has not yet been elucidated however because of the ill posed nature of the problem here possible neural architectures underlying information geometric measures are investigated using an isolated pair and an isolated triplet of model neurons by assuming the existence of equilibrium states we derive analytically the relationship between the information geometric parameters and these simple neural architectures for symmetric networks the first and second order information geometric parameters represent respectively the external input and the underlying connections between the neurons provided that the number of neurons used in the parameter estimation in the log linear model and the number of neurons in the network are the same for asymmetric networks however these parameters are dependent on both the intrinsic connections and the external inputs to each neuron in addition we derive the relation between the information geometric parameter corresponding to the two neuron interaction and a conventional cross correlation measure we also show that the information geometric parameters vary depending on the number of neurons assumed for parameter estimation in the log linear model this finding suggests a need to examine the information geometric method carefully a possible criterion for choosing an appropriate orthogonal coordinate is also discussed this article points out the importance of a model based approach and sheds light on the possible neural structure underlying the application of information geometry to neural network analysis