the interaction space of neural networks with sign constrained synapses the authors investigate the optimal storage capacity of attractor neural networks with sign constrained weights which are prescribed a priori the storage capacity is calculated by considering the fractional volume of weights which can store a set of random patterns as attractors for a given stability parameter it is found that this volume is independent of the particular distribution of signs gauge invariance and that the storage capacity of such constrained networks is exactly one half that of the unconstrained network with the corresponding value of the stability parameter

retrieval properties of a hopfield model with random asymmetric interactions the process of pattern retrieval in a hopfield model in which a random antisymmetric component is added to the otherwise symmetric synaptic matrix is studied by computer simulations the introduction of the antisymmetric component is found to increase the fraction of random inputs that converge to the memory states however the size of the basin of attraction of a memory state does not show any significant change when asymmetry is introduced in the synaptic matrix we show that this is due to the fact that the spurious fixed points which are destabilized by the introduction of asymmetry have very small basins of attraction the convergence time to spurious fixed point attractors increases faster than that for the memory states as the asymmetry parameter is increased the possibility of convergence to spurious fixed points is greatly reduced if a suitable upper limit is set for the convergence time this prescription works better if the synaptic matrix has an antisymmetric component

exchange relations and correlation functions for a quantum particle on the su2 group manifold the so4 invariant quantum dynamics of a point particle moving on the 3 sphere or equivalently of the relative motion of the spherical top is considered quantum exchange relations for different times are derived with an r matrix depending on the time difference and on the conserved angular momentum their implications for correlation functions are spelled out the chiral exchange relations of alekseev and faddeev commun math phys 141 413 422 1991 are also extended to different times

on the storage capacity of neural networks with sign constrained weights the authors derive the maximal capacity theorem for neural networks with sign constrained weights using a simple geometrical argument and mention its implications for multi layered networks with sign constrained weights

non abelian harmonic oscillators and chiral theories we show that a large class of physical theories which has been under intensive investigation recently share the same geometric features in their hamiltonian formulation these dynamical systems range from harmonic oscillations to wzw like models and to the kdv dynamics on diff0s1 to the same class belong also the hamiltonian systems on groups of maps the common feature of these models are the chiral equations of motion for so called chiral decomposition of the phase space

learning and retrieval in attractor neural networks above saturation the authors investigate neural networks in the range of parameters when the ground state energy is positive namely when a synaptic matrix which satisfies all the desired constraints cannot be found by the learning algorithm in particular they calculate the typical distribution functions of local stabilities obtained for a number of algorithms in this region these functions are used to investigate the retrieval properties as reflected by the size of the basins of attraction this is done analytically in sparsely connected networks and numerically in fully connected networks the main conclusion is that the retrieval behaviour of attractor neural networks can be improved by learning above saturation

braid matrices and structure constants for minimal conformal models using the feigin fuchs representation of minimal conformal models in a form introduced recently by one of us the braid group representation matrices describing the analytic continuation properties of conformal blocks are computed in a suitable normalization their matrix elements are shown to essentially factorize into pairs of boltzmann weights of critical rsos models in a certain limit of the spectral parameter these boltzmann weights are related to quantum groupr matrices by the vertex sos transformation we show that the crossing symmetry of the four point function in left right symmetric models follows from a quantum group relation also called crossing symmetry this observation gives a simple way to evaluate the structure constants

the quantum group structure of 2d gravity and minimal models i on the unit circle an infinite family of chiral operators is constructed whose exchange algebra is given by the universalr matrix of the quantum groupsl 2 q this establishes the precise connection between the chiral algebra of two dimensional gravity or minimal models and this quantum group the method is to relate the monodromy properties of the operator differential equations satisfied by the generalized vertex operators with the exchange algebra ofsl 2 q the formulae so derived which generalize an earlier particular case worked out by babelon are remarkably compact and may be entirely written in terms of q deformed factorials and binomial coefficients

optimal storage capacity of neural networks at finite temperatures gardner s analysis 1989 of the optimal storage capacity of neural networks is extended to study finite temperature effects the typical volume of the space of interactions is calculated for strongly diluted networks as a function of the storage ratio alpha temperature t and the tolerance parameter m from which the optimal storage capacity alpha c is obtained as a function of t and m at zero temperature it is found that alpha c 2 regardless of m while alpha c in general increases with the tolerance at finite temperatures the authors show how the best performance for given alpha and t is obtained which reveals a first order transition from high quality performance to a low quality one at low temperatures an approximate criterion for recalling which is valid near m 1 is also discussed

learning and retrieval in attractor neural networks with noise abstract a recent study on noiseless learning and retrieval in attractor neural networks above saturation by griniasty and gutfreund is extended to take account for imperfect learning by means of a temperature 1 t violations of the constraint imposed on the local stabilities are taken into account by various cost functions the distribution of local stabilities and the fraction of errors during the learning stage are analysed the retrieval dynamics for the sparsely connected network is studied showing high retrieval overlap in reduced retrieval regions for finite t

content addressability and learning in neural networks the content addressability of patterns stored in ising spin neural network models with symmetric interactions is studied numerical results from simulations on the icl distributed array processor dap involving systems with up to 2048 neurons are presented behaviour consistent with finite size scaling characteristic of a first order phase transition is shown to be exhibited by the basins of attraction of the stored patterns both in the case of the hopfield model and for systems using a local iterative learning algorithm designed to optimise the basins of attraction estimates are obtained for the critical minimum overlaps which an input pattern must have with a stored pattern in order to successfully retrieve it

stock market dynamics abstract we elucidate on several empirical statistical observations of stock market returns moreover we find that these properties are recurrent and are also present in invariant measures of low dimensional dynamical systems thus we propose that the returns are modeled by the first poincare return time of a low dimensional chaotic trajectory this modeling which captures the recurrent properties of the return fluctuations is able to predict well the evolution of the observed statistical quantities in addition it explains the reason for which stocks present simultaneously dynamical properties and high uncertainties in our analysis we use data from the s p 500 index and the brazilian stock telebras

retrieval phase diagrams for attractor neural networks with optimal interactions the authors consider the retrieval properties of attractor neural networks whose synaptic matrices have been constructed to maximise the number of patterns which can be stored in a perceptron satisfying certain constraints retrieval is studied in the absence as well as in the presence of fast noise temperature the discussion is restricted to dilute networks for which dynamical equations for the overlaps are available when the patterns are stored with a prescribed lower limit on the stability parameter on every site the full stability of the perceptron ensures the existence of an attractor with perfect retrieval it is found that the curve of critical storage capacity alpha against temperature t as a line of first order transitions for high values of alpha and becomes second order for low alpha at a point of a tricritical nature the phase diagram is compared with the dilute hopfield model it is found that at high synaptic noise levels the diluted hopfield net stores more effectively than the network trained for optimal perceptron storage when a given fraction of sites is allowed to violate the stability bound the solution of the perceptron learning problem does not ensure the existence of an attractor of finite overlap even in the absence of noise this case is studied separately for t 0 and for finite t

competitive attraction in neural networks with sign constrained weights the presence of many attractors in neural networks give rise to interesting competitive phenomena the authors consider dilute recurrent or attractor neural networks with sign constrained weights and storing uncorrelated patterns with maximal stability the dynamics of these networks is governed by the competitive effects of retrieval nonretrieval and uniform i e ferromagnetic attractors which result in basin encroaching shrinking splitting and wedging they have found the parameter regions in which each of these attractors exist the basins of attraction of the uniform attractors enlarge at the expense of the other attractors even when the weight signs are slightly imbalanced but can be compensated by the introduction of a dynamical threshold

sign constrained linear learning and diluting in neural networks for neural networks with predefined effects of the synapses excitatory or inhibitory the simplex algorithm is applied as a learning rule it is assumed that the given signs of the synapses can never be changed during learning the maximum possible dilution of synapses as a result of the learning is found at the maximum storage capacity of a model with only positive or randomly distributed signs for the case of infinitely many neurons a replica symmetric calculation of the free energy and the distribution of coupling strengths is presented the linear algorithm is also applied to networks with a more suitable choice of sign constraints with the result of a higher storage capacity

linear readout of object manifolds objects are represented in sensory systems by continuous manifolds due to sensitivity of neuronal responses to changes in physical features such as location orientation and intensity what makes certain sensory representations better suited for invariant decoding of objects by downstream networks we present a theory that characterizes the ability of a linear readout network the perceptron to classify objects from variable neural responses we show how the readout perceptron capacity depends on the dimensionality size and shape of the object manifolds in its input neural representation

the roles of stability and symmetry in the dynamics of neural networks the authors study the retrieval phase of spin glass like neural networks considering that the dynamics should depend only on gauge invariant quantities they propose that two such parameters characterising the symmetry of the neural net s connections and the stabilities of the patterns are responsible for most of the dynamical effects this is supported by a numerical study of the shape of the basins of attraction for a one pattern neural network model the effects of stability and symmetry on the short time dynamics of this model are studied analytically and the full dynamics for vanishing symmetry is shown to be exactly solvable

differential calculi on the quantum group glp q 2 the author presents a systematical study of left covariant differential calculus on the quantum group glp q 2 a two parameter deformation of the general linear group in particular the author explicitly constructs the most general bicovariant calculus it depends on a new parameter s the corresponding lie algebra of left coinvariant vector fields is a two parameter deformation of the classical lie algebra p and q appear only in the combination r pq in the limit p q to 1 one obtains non standard bicovariant differential calculi on the classical lie group

what can a neuron learn with spike timing dependent plasticity spiking neurons are very flexible computational modules which can implement with different values of their adjustable synaptic parameters an enormous variety of different transformations f from input spike trains to output spike trains we examine in this letter the question to what extent a spiking neuron with biologically realistic models for dynamic synapses can be taught via spike timing dependent plasticity stdp to implement a given transformation f we consider a supervised learning paradigm where during training the output of the neuron is clamped to the target signal teacher forcing the well known perceptron convergence theorem asserts the convergence of a simple supervised learning algorithm for drastically simplified neuron models mcculloch pitts neurons we show that in contrast to the perceptron convergence theorem no theoretical guarantee can be given for the convergence of stdp with teacher forcing that holds for arbitrary input spike patterns on the other hand we prove that average case versions of the perceptron convergence theorem hold for stdp in the case of uncorrelated and correlated poisson input spike trains and simple models for spiking neurons for a wide class of cross correlation functions of the input spike trains the resulting necessary and sufficient condition can be formulated in terms of linear separability analogously as the well known condition of learnability by perceptrons however the linear separability criterion has to be applied here to the columns of the correlation matrix of the poisson input we demonstrate through extensive computer simulations that the theoretically predicted convergence of stdp with teacher forcing also holds for more realistic models for neurons dynamic synapses and more general input distributions in addition we show through computer simulations that these positive learning results hold not only for the common interpretation of stdp where stdp changes the weights of synapses but also for a more realistic interpretation suggested by experimental data where stdp modulates the initial release probability of dynamic synapses

perceptron learning with sign constrained weights the authors study neural network models in which the synaptic efficacies are restricted to have a prescribed set of signs it is proved that such neural networks can learn a set of random patterns by a perceptron like algorithm which respects the synaptic restrictions at every step in particular it shows that learning can take place iteratively in a network which obeys dale s rule i e in which neurons are exclusively excitatory or inhibitory the learning algorithm as well as its convergence theorem are stated in perceptron language and it is proved that the algorithm converges under the same conditions as required for an unconstrained perceptron numerical experiments show that these necessary conditions can actually be met for relatively large sets of patterns to be learned they then argue that the results are invariant under the distribution of the signs due to gauge invariance for random patterns as a consequence the same sets of random patterns can be learned by networks which have any fixed distribution of synaptic signs ranging from fully inhibitory to fully excitatory

finite size effects and bounds for perceptron models ablncl in lhis paper we consider two main aspsln of the binary perccptron problem the maximal capacity when random paltem are stored model a and its generalhation ability model b we have extended previous numerical estimates of critical capacilies and studied thermal properties of systems of small sizes to lest recent replica predictions we have also considered some simpler versions of thest models the discrete spherical vemions can be salved exactly using gardner s replica calculation for the spherical model and are shown to give a rigorous upper bound and lower bound on the capacities of models a and b respectively iby vemions of models a and b are soivcd in detail and provide information which is useful for interpreting the bnile size eeeciects present in the numerical studies of models a and b

from statistical mechanics to neural networks and back abstract the neural network models which have attracted so much attention in the physics community consist of a large assembly of two state elements highly interconnected by ferromagnetic and antiferromagnetic interactions methods of statistical mechanics have been applied to these models bringing new discoveries and surprises to neural network science at the same time neural network models have posed new problems and incentives to statistical mechanics one of the basic problems is that of learning which in the context of these models means a choice of the interactions for which the network dynamics leads to specific attractors the main emphasis in this paper is on the essential developments and results on this problem derived from methods of statistical mechanics

categorization in the pseudo inverse neural network we investigate analytically the emergence of the categorization ability in the pseudo inverse attractor neural network more pointedly we consider the problem of learning an extensive number of concepts by storing a finite number of examples s of each concept we find that there is a critical value beyond which the categorization error as measured by the average fraction of unstable sites in the concepts decreases monotonically with s

scaling of swimming sequences in copepod behavior data analysis and simulation abstract we consider symbolic sequences of copepod behavior classified as break and slow swimming states successive residence times for each state have been recorded for 327 symbolic times series obtained from 52 different trajectories we performed a symbolic dynamics analysis of the succession of states and showed that the probability densities of the duration of each state obey power laws for large times we analyzed the transition between states and showed as expected the non markovian properties of this dynamics we further simulate a copepod trajectory using an alternate renewal process for the generation of the sequence time series and using a brownian motion for the copepod motion during swimming periods

quantum groups and wznw models an explanation of the appearance of quantum groups in chiral wznw models is given invariance of the theory under quantum group action is discussed

free electron gas in the apollonian network multifractal energy spectrum and its thermodynamic fingerprints we study the free electron gas in an apollonian network within the tight binding framework the scale free and small world character of the underlying lattice is known to result in a quite structured energy spectrum with deltalike singularities gaps and minibands after an exact numerical diagonalization of the corresponding adjacency matrix of the network with a finite number of generations we employ a scaling analysis of the moments of the density of states to characterize its multifractality and report the associated singularity spectrum the fractal nature of the energy spectrum is also shown to be reflected in the thermodynamic behavior by logarithmic modulations on the temperature dependence of the specific heat the absence of chiral symmetry of the apollonian network leads to distinct thermodynamic behaviors due to electrons and holes thermal excitations

the failure distribution in percolation models of breakdown the probability of finding a largest defect cluster of size n in a percolation network is calculated analytically using a new distribution function scaling equation from this result and the stress voltage enhancement at the tip of the most critical defect in the network the probability of failure in percolation models of breakdown is calculated for defect fractions less than the percolation point this distribution is found to be of the form exponential of an exponential numerical simulations on the two dimensional random fuse network confirm the new distribution function and convincingly distinguish between it and the weibull 1951 form most often used in the fitting of breakdown data

critical capacity of constrained perceptrons the authors calculate the critical capacity of a single layer perceptron with interactions ji constrained to take values on a general surface the critical capacity is expressed in terms of a geometrical property of the constraining surface

sign constrained synapses and biased patterns in neural networks the storage of biased patterns is examined in neural networks with sign constrained synapses every neuron has outgoing synapses which are either all inhibitory or all excitatory for random patterns stored in such networks it is known that the presence of a discrete gauge symmetry makes the maximal storage capacity independent of the proportion of excitatory neurons to inhibitory neurons when the stored patterns are biased however this discrete gauge symmetry is broken with the result that the maximal capacity depends on the proportion of excitatory neurons to inhibitory ones the dependence of the capacity on the fraction of excitatory neurons in the network f is calculated using the space of interactions approach it is found that the storage capacity is maximal at f 0 5 this result is true regardless of the particular value of the bias in the stored patterns the significance of this result in the neurophysiological context is discussed

universality in the space of interactions for network models by modifying the measure used to sum over coupling matrices the authors generalise gardner s 1988 calculation of the fractional interaction space volume and storage capacity of neural network models they also compute the local field distribution for the network the generalised measure allows one to consider networks with a wide variety of properties away from saturation but they find that the original results for saturated networks are universal for all well behaved measures other universality classes including those containing hebb matrices and pseudo inverse matrices are obtained by considering singular measures

learning only when necessary better memories of correlated patterns in networks with bounded synapses learning in a neuronal network is often thought of as a linear superposition of synaptic modifications induced by individual stimuli however since biological synapses are naturally bounded a linear superposition would cause fast forgetting of previously acquired memories here we show that this forgetting can be avoided by introducing additional constraints on the synaptic and neural dynamics we consider hebbian plasticity of excitatory synapses a synapse is modified only if the postsynaptic response does not match the desired output with this learning rule the original memory performances with unbounded weights are regained provided that 1 there is some global inhibition 2 the learning rate is small and 3 the neurons can discriminate small differences in the total synaptic input e g by making the neuronal threshold small compared to the total postsynaptic input we prove in the form of a generalized perceptron convergence theorem that under these constraints a neuron learns to classify any linearly separable set of patterns including a wide class of highly correlated random patterns during the learning process excitation becomes roughly balanced by inhibition and the neuron classifies the patterns on the basis of small differences around this balance the fact that synapses saturate has the additional benefit that nonlinearly separable patterns such as similar patterns with contradicting outputs eventually generate a subthreshold response and therefore silence neurons that cannot provide any information

perceptrons above saturation we study the storage of random patterns by a perceptron above its storage capacity alpha c i e in the region where perfect storage becomes impossible we determine the minimal fraction of learning errors and the distribution of stabilities for different learning rules in one step replica symmetry breaking thereby we not only extend the known replica symmetric results to values of the storage capacity beyond the at line but also show that depending on the learning rule replica symmetry may be globally unstable already well below the at line as an example for possible implications we compare the results for the typical basins of attraction of an extremely diluted attractor neural network as given by replica symmetry and one step replica symmetry breaking

fractal and multifractal properties of exit times and poincarerecurrences systems with chaotic dynamics possess anomalous statistical properties and their trajectories do not correspond to the gaussian process this property imposes description of such time characteristics as the distribution of exit times or poincar acute e recurrences by introducing a multi fractal time scale in order to satisfy the observed powerlike tails of the distributions we introduce a corresponding phase space time partitioning and spectral function for dimensions and make a connection between dimensions and transport exponent that defines the anomalous open quotes strange close quotes kinetics copyright ital 1997 ital the american physical society

topological representations of the quantum groupuq sl2 we define a topological action of the quantum groupuq sl2 on a space of homology cycles with twisted coefficients on the configuration space of the punctured disc this action commutes with the monodromy action of the braid groupoid which is given by ther matrix ofuq sl2

on non negative solutions to large systems of random linear equations abstract systems of random linear equations may or may not have solutions with all components being non negative the question is e g of relevance when the unknowns are concentrations or population sizes in the present paper we show that if such systems are large the transition between these two possibilities occurs at a sharp value of the ratio between the number of unknowns and the number of equations we analytically determine this threshold as a function of the statistical properties of the random parameters and show its agreement with numerical simulations we also make contact with two special cases that have been studied before the storage problem of a perceptron and the resource competition model of macarthur

fock representations and brst cohomology insl 2 current algebra we investigate the structure of the fock modules overa 1 1 introduced by wakimoto we show that irreducible highest weight modules arise as degree zero cohomology groups in a brst like complex of fock modules chiral primary fields are constructed as brst invariant operators acting on fock modules as a result we obtain a free field representation of correlation functions of thesu 2 wzw model on the plane and on the torus we also consider representations of fractional level arising in polyakov s 2d quantum gravity finally we give a geometrical borel weil like interpretation of the wakimoto construction

chiral zero modes of the su n wess zumino novikov witten model we define the chiral zero modes phase space of the g su n wess zumino novikov witten wznw model as an n 1 n 2 dimensional manifold q equipped with a symplectic form q involving a wess zumino term which depends on the monodromy m and is implicitly defined on an open dense neighbourhood of the group unit by this classical system exhibits a poisson lie symmetry that evolves upon quantization into a uq sln symmetry for q a primitive even root of 1 for each non degenerate constant solution of the classical yang baxter equation we write down explicitly a m satisfying equation and invert the form q thus computing the poisson bivector of the system the resulting poisson brackets pb appear as the classical counterpart of the exchange relations of the quantum matrix algebra studied previously in furlan et al 2000 preprint hep th 0003210 we argue that it is advantageous to equate the determinant d of the zero modes matrix aj to a pseudoinvariant under permutations q polynomial in the su n weights rather than to adopt the familiar convention d 1 a finite dimensional fock space operator realization of the factor algebra q h where h is an appropriate ideal in q for qh 1 is briefly discussed

hecke algebraic properties of dynamical r matrices application to related quantum matrix algebras the quantum dynamical yang baxter or gervais neveu felder equation defines an r matrix r p where p stands for a set of mutually commuting variables a family of sl n type solutions of this equation provides a new realization of the hecke algebra we define quantum antisymmetrizers introduce the notion of quantum determinant and compute the inverse quantum matrix for matrix algebras of the type r p a1a2 a1a2r it is pointed out that such a quantum matrix algebra arises in the operator realization of the chiral zero modes of the wznw model

on the capacity per synapse the optimal storage capacity of a perceptron with a finite fraction of sign constrained weights which are prescribed a priori is examined the storage capacity is calculated by considering the fractional volume of weights which can store a set of alpha n random patterns where n is the size of the input it is found that in the case where 1 s n weights are sign constrained the capacity is 1 s 2 alpha gc k where alpha gc k is the maximal storage capacity in gardner s case and k is a stability parameter

classical wakimoto realizations of chiral wznw bloch waves it is well known that the chiral wznw bloch waves satisfy a quadratic classical exchange algebra which implies the affine kac moody algebra for the corresponding currents here we obtain a direct derivation of the exchange algebra by inverting the symplectic form on the space of bloch waves and give a completely algorithmic construction of its generalized free field realizations that extend the classical wakimoto realizations of the current algebra

optimal basins of attraction in randomly sparse neural network models the size of the basin of attraction for randomly sparse neural networks with optimal interactions is calculated for all values of the storage ratio alpha p c 2 where p is the number of random uncorrelated patterns and c is the connectivity the basin of attraction is finite while for alpha 0 42 the basin of attraction is almost 100

on the structure of unitary conformal field theory ii representation theoretic approach two dimensional unitary rational conformal field theory is studied from the point of view of the representation theory of chiral algebras chiral algebras are equipped with a family of co multiplications which serve to define tensor product representations chiral vertices arise as clebsch gordan operators from tensor product representations to irreducible subrepresentations of a chiral algebra the algebra of chiral vertices is studied and shown to give rise to representations of the braid groups determined by yang baxter braid matrices chiral fusion is analyzed it is shown that the braid and fusion matrices determine invariants of knots and links connections between the representation theories of chiral algebras and of quantum groups are sketched finally it is shown how the local fields of a conformal field theory can be reconstructed from the chiral vertices of two chiral algebras

on the exchange matrix for wznw model the connection between the exchange algebra in thesu 2 wess zumino novikov witten model and the quantum groupsu 2 is discussed it is shown that on the quasiclassical level this connection has the simple interpretation in terms of the lie poisson action ofsu 2 on the chiral components of the fields in the wznw model

solving the strongly coupled 2d gravity 1 unitary truncation and quantum group structure strongly compled gravity theories with virasoro central charges equal to 7 13 and 19 are shown to enjoy striking properties at these values the subset of chiral operators with real virasoro weights acting on a subspacehphys is shown to be closed by fusion and braiding and to leave this subspace invariant moreover the representation of the virasoro algebra becomes unitary when it is restricted tohphys strongly coupled 2d gravity withcgrav 7 13 or 19 may thus be naturally truncated obtaining a consistent conformal theory this result is similar to the truncation that occurs forc 1 6 p p 2 pp withp andp integers where only a finite number of primary fields remains as is well known in rational theories the proof of this unitary truncation theorem already summarized in a recent letter is fully described here

quantum group structure in the fock space resolutions of widehat sl n representations we describe a complex of wakimoto type fock space modules for the affine kac moody algebra widehat sl n the intertwining operators that build the complex are obtained from contour integrals of so called screening operators we show that a quantum group structure underlies the algebra of screening operators this observation greatly facilitates the explicit determination of the intertwiners we conjecture that the complex provides a resolution of an irreducible highest weight module in terms of fock spaces

quasi quantum groups knots three manifolds and topological field theory we show how to construct starting from a quasi hopf algebra or quasiquantum group invariants of knots and links in some cases these invariants give rise to invariants of the three manifolds obtained by surgery along these links this happens for a finite dimensional quasi quantum group whose definition involves a finite groupg and a 3 cocycle which was first studied by dijkgraaf pasquier and roche we treat this example in more detail and argue that in this case the invariants agree with the partition function of the topological field theory of dijkgraaf and witten depending on the same datag

recurrence time analysis long term correlations and extreme events the recurrence times between extreme events have been the central point of statistical analyses in many different areas of science simultaneously the poincar e recurrence time has been extensively used to characterize nonlinear dynamical systems we compare the main properties of these statistical methods pointing out their consequences for the recurrence analysis performed in time series in particular we analyze the dependence of the mean recurrence time and of the recurrence time statistics on the probability density function on the interval whereto the recurrences are observed and on the temporal correlations of time series in the case of long term correlations we verify the validity of the stretched exponential distribution which is uniquely defined by the exponent ensuremath gamma at the same time showing that it is restricted to the class of linear long term correlated processes simple transformations are able to modify the correlations of time series leading to stretched exponentials recurrence time statistics with different ensuremath gamma which shows a lack of invariance under the change of observables

balanced excitation and inhibition are required for high capacity noise robust neuronal selectivity neurons and networks in the cerebral cortex must operate reliably despite multiple sources of noise to evaluate the impact of both input and output noise we determine the robustness of single neuron stimulus selective responses as well as the robustness of attractor states of networks of neurons performing memory tasks we find that robustness to output noise requires synaptic connections to be in a balanced regime in which excitation and inhibition are strong and largely cancel each other we evaluate the conditions required for this regime to exist and determine the properties of networks operating within it a plausible synaptic plasticity rule for learning that balances weight configurations is presented our theory predicts an optimal ratio of the number of excitatory and inhibitory synapses for maximizing the encoding capacity of balanced networks for given statistics of afferent activations previous work has shown that balanced networks amplify spatiotemporal variability and account for observed asynchronous irregular states here we present a distinct type of balanced network that amplifies small changes in the impinging signals and emerges automatically from learning to perform neuronal and network functions robustly

multivalley structure of attractor neural networks we numerically investigate the structure of the basins of attraction and the nature of the spurious attractors of the pseudo inverse and the optimal weights attractor neural networks we show that the number of attractors in the optimal weights model increases as the margin parameter increases and that the basins of attraction of the stored patterns are not significantly enlarged by moreover the number of attractors is smaller than in the pseudo inverse model

optimal storage properties of neural network models the authors calculate the number p alpha n of random n bit patterns that an optimal neural network can store allowing a given fraction f of bit errors and with the condition that each right bit is stabilised by a local field at least equal to a parameter k for each value of alpha and k there is a minimum fraction fmin of wrong bits they find a critical line alpha c k with alpha c 0 2 the minimum fraction of wrong bits vanishes for alpha alpha c k the calculations are done using a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is locally stable in a finite region of the k alpha plane including the line alpha c k but there is a line above which the solution becomes unstable and replica symmetry must be broken

learning algorithms with optimal stability in neural networks to ensure large basins of attraction in spin glass like neural networks of two state elements xi imu or 1 the authors propose to study learning rules with optimal stability delta where delta is the largest number satisfying delta or sigma j jij xi jmu xi imu mu 1 p i 1 n where n is the number of neurons and p is the number of patterns they motivate this proposal and provide optimal stability learning rules for two different choices of normalisation for the synaptic matrix jij in addition numerical work is presented which gives the value of the optimal stability for random uncorrelated patterns

the space of interactions in neural network models the typical fraction of the space of interactions between each pair of n ising spins which solve the problem of storing a given set of p random patterns as n bit spin configurations is considered the volume is calculated explicitly as a function of the storage ratio alpha p n of the value kappa 0 of the product of the spin and the magnetic field at each site and of the magnetisation m here m may vary between 0 no correlation and 1 completely correlated the capacity increases with the correlation between patterns from alpha 2 for correlated patterns with kappa 0 and tends to infinity as m tends to 1 the calculations use a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is shown to be locally stable a local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist

pattern selectivity in neural networks as a means of understanding basin structures the authors study the problem of learning and retrieving for a pair of correlated patterns within an extensive number of uncorrelated patterns for networks where learning may be treated as an optimization process with respect to an arbitrary cost function this formulation is then applied to several specific examples where they study the pattern selectivity of these systems i e their ability to differentiate correlated patterns and investigate the process of basin shrinking with increasing loading levels they define and discuss several different retrieval phases whose existence depends on the competitive interplay of the loading level and the pattern correlation discussion of asymptotic retrieval is restricted to dilute asymmetric networks

capacity of neural networks with discrete synaptic couplings the authors study the optimal storage capacity of neural networks with discrete local constraints on the synaptic couplings jij models with such constraints include those with binary couplings jij or 1 or jij 0 1 quantised couplings with larger synaptic range e g jij or 1 l or 2 l or 1 and in the limit continuous couplings confined to the hypercube mod jij mod or 1 box confinement they find that the optimal storage capacity alpha kappa is best determined by the vanishing of a suitably defined entropy as calculated in the replica symmetric approximation they also extend their results to cases with biased memories and make contact with sparse coding models

scaling features of texts images and time series in the given paper we consider the scaling features of long letter sequences like human writings discretized images and discretized 3nancial data using several approaches we show that the symbolic strings and time series being analyzed have a complex multiscale structure and demonstrate di5erent scalings for large and small 6uctuations we discuss complex phenomena in the scaling behavior of partition functions in the case of high frequency dax future data c 2001 elsevier science b v all rights reserved

deformed integrable σ models classical r matrices and classical exchange algebra on drinfel d doubles we describe a unifying framework for the systematic construction of integrable deformations of integrable models within the hamiltonian formalism it applies equally to both the yang baxter type as well as gauged wzw type deformations which were considered recently in the literature as a byproduct these two families of integrable deformations are shown to be poisson lie t dual of one another

relative stability in the dynamics of a two pattern neural net the authors investigate how patterns of different acquisition strengths influence each other s stability a neural network model with two strictly stable patterns stored on a noisy background is studied by a novel approximation of short time dynamics that singles out coherent contributions systematically and uses a gaussian approximation for incoherent sums the basin of attraction of the weaker pattern is found to shrink depending on the two acquisition strengths for a diluted hopfield type version of the model the weaker pattern may become unrecognisable

canonical quantization of the boundary wess zumino witten model we present an analysis of the canonical structure of the wess zumino witten theory with untwisted conformal boundary conditions the phase space of the boundary theory on a strip is shown to coincide with the phase space of the chern simons theory on a solid cylinder a disc times a line with two wilson lines this reveals a new aspect of the relation between two dimensional boundary conformal field theories and three dimensional topological theories a decomposition of the chern simons phase space on a punctured disc in terms of the one on a punctured sphere and of coadjoint orbits of the loop group easily lends itself to quantization it results in a description of the quantum boundary degrees of freedom in the wzw model by invariant tensors in a triple product of quantum group representations in the action on the space of states of the boundary theory the bulk primary fields of the wzw model are shown to combine the usual vertex operators of the current algebra with monodromy acting on the quantum group invariant tensors we present the details of this construction for the spin 1 2 fields in the su 2 wzw theory establishing their locality and computing their 1 point functions

characterizing cerebrovascular dynamics with the wavelet based multifractal formalism abstract using the wavelet transform modulus maxima wtmm approach we study the dynamics of cerebral blood flow cbf in rats aiming to reveal responses of macro and microcerebral circulations to changes in the peripheral blood pressure we show that the wavelet based multifractal formalism allows quantifying essentially different reactions in the cbf dynamics at the level of large and small cerebral vessels we conclude that unlike the macrocirculation that is nearly insensitive to increased peripheral blood pressure the microcirculation is characterized by essential changes of the cbf complexity

multiconnected neural network models a generalisation of the hopfield model which includes interactions between p 2 ising spins is considered the exact storage capacity behaves as np 1 2 p 1 ln n when the number of nodes n is large in the limit p to infinity the thermodynamics of the model can be solved exactly without using the replica method at zero temperature a solution which is completely correlated with the input pattern exists for alpha alpha c where alpha c to infinity as p to infinity and this solution has lower energy than the spin glass solution if alpha alpha 1 1 4 ln 2 where the number of patterns n 2 alpha p np 1 for finite values of p the correlation with the input pattern is not complete for p 3 and 4 approximate values of alpha c and alpha 1 are obtained and for p to infinity the replica symmetric approximation gives alpha c approximately p 4 ln p

basin of attraction of the optimal perceptron with biased patterns for an extremely diluted neutral network model designed by the optimal gardner rule that stores patterns with low level of activity the basin of attraction is studied analytically the recursion relation for the overlap with a stored pattern is two dimensional and yields a richer bifurcation scenario than in the case of patterns with symmetric statistics the bias in the patterns gives rise to ferromagnetic attractors which compete with the patterns

classical origin of quantum group symmetries in wess zumino witten conformal field theory we elucidate the way by which the quantum group symmetries of the wzw models arise within the canonical formalism of the classical field theory