noise in integrate and fire neurons from stochastic input to escape rates we analyze the effect of noise in integrate and fire neurons driven by time dependent input and compare the diffusion approximation for the membrane potential to escape noise it is shown that for time dependent subthreshold input diffusive noise can be replaced by escape noise with a hazard function that has a gaussian dependence on the distance between the noise free membrane voltage and threshold the approximation is improved if we add to the hazard function a probability current proportional to the derivative of the voltage stochastic resonance in response to periodic input occurs in both noise models and exhibits similar characteristics

neural coding higher order temporal patterns in the neurostatistics of cell assemblies recent advances in the technology of multiunit recordings make it possible to test hebb s hypothesis that neurons do not function in isolation but are organized in assemblies this has created the need for statistical approaches to detecting the presence of spatiotemporal patterns of more than two neurons in neuron spike train data we mention three possible measures for the presence of higher order patterns of neural activation coef ficients of log linear models connected cumulants and redundancies and present arguments in favor of the coefficients of loglinear models we present test statistics for detecting the presence of higher order interactions in spike train data by parameterizing these interactions in terms of coefficients of log linear models we also present a bayesian approach for inferring the existence or absence of interactions and estimating their strength the two methods the frequentist and the bayesian one are shown to be consistent in the sense that interactions that are detected by either method also tend to be detected by the other a heuristic for the analysis of temporal patterns is also proposed finally a bayesian test is presented that establishes stochastic differences between recorded segments of data the methods are applied to experimental data and synthetic data drawn from our statistical models our experimental data are drawn from multiunit recordings in the prefrontal cortex of behaving monkeys the somatosensory cortex of anesthetized rats and multiunit recordings in the visual cortex of behaving monkeys

optimal detection of sparse principal components in high dimension we perform a finite sample analysis of the detection levels for sparse principal components of a high dimensional covariance matrix our minimax optimal test is based on a sparse eigenvalue statistic alas computing this test is known to be np complete in general and we describe a computationally efficient alternative test using convex relaxations our relaxation is also proved to detect sparse principal components at near optimal detection levels and performs very well on simulated datasets ams 2000 subject classifications primary 62h25 secondary 62f04 90c22

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

cortical potential distributions and information processing the use of sets of spatiotemporal cortical potential distributions cpds as the basis for cognitive information processing results in a very large space of cognitive elements with natural metrics results obtained from current source density csd analysis suggest that in the cpd picture action potentials may make only a relatively minor contribution to the brain s code in order to establish if two cpds are close we consider standard metrics in spaces of continuous functions and these may be employed to ascertain if two stimuli will be identified as the same the correspondence between the electrical activity within brain regions including not only action potentials but all postsynaptic potentials psps and cpds is considered we examine the possibility of using the csd approach to find potential distributions using the descriptive approach in which precise sets of times are ascribed to the occurrence of action potentials and psps using metrics in the multidimensional space of paths of collections of point processes we show that closeness of cpds is implied by closeness of sets of spike times and psp times if a certain metric is used but not others we also set forth a dynamical model consisting of a system of reaction dif fusion equations for ionic concentrations coupled with nerve membrane potential equations and active transport systems making the approximation of a descriptive approach the correspondence between sets of spike times and psp times and cpds is obtained as with the csd method however since it is not possible to ascribe precise times to the occurrence of psps and action potentials the descriptive approach cannot be used to describe the configuration of electrical activity in cortical regions accurately we also discuss how the cpd framework relates to the binding problem and submillisecond timing

statistical clustering of temporal networks through a dynamic stochastic block model statistical node clustering in discrete time dynamic networks is an emerging field that raises many challenges here we explore statistical properties and frequentist inference in a model that combines a stochastic block model sbm for its static part with independent markov chains for the evolution of the nodes groups through time we model binary data as well as weighted dynamic random graphs with discrete or continuous edges values our approach motivated by the importance of controlling for label switching issues across the different time steps focuses on detecting groups characterized by a stable within group connectivity behavior we study identifiability of the model parameters propose an inference procedure based on a variational expectation maximization algorithm as well as a model selection criterion to select for the number of groups we carefully discuss our initialization strategy which plays an important role in the method and compare our procedure with existing ones on synthetic datasets we also illustrate our approach on dynamic contact networks one of encounters among high school students and two others on animal interactions an implementation of the method is available as a r package called dynsbm

latent space approaches to social network analysis network models are widely used to represent relational information among interacting units in studies of social networks recent emphasis has been placed on random graph models where the nodes usually represent individual social actors and the edges represent the presence of a specified relation between actors we develop a class of models where the probability of a relation between actors depends on the positions of individuals in an unobserved social space we make inference for the social space within maximum likelihood and bayesian frameworks and propose markov chain monte carlo procedures for making inference on latent positions and the effects of observed covariates we present analyses of three standard datasets from the social networks literature and compare the method to an alternative stochastic blockmodeling approach in addition to improving on model fit for these datasets our method provides a visual and interpretable model based spatial representation of social relationships and improv

statistical computational tradeoffs in planted problems and submatrix localization with a growing number of clusters and submatrices we consider two closely related problems planted clustering and submatrix localization in the planted clustering problem a random graph is generated based on an underlying cluster structure of the nodes the task is to recover these clusters given the graph the submatrix localization problem concerns locating hidden submatrices with elevated means inside a large real valued random matrix of particular interest is the setting where the number of clusters submatrices is allowed to grow unbounded with the problem size these formulations cover several classical models such as planted clique planted densest subgraph planted partition planted coloring and the stochastic block model which are widely used for studying community detection graph clustering and bi clustering for both problems we show that the space of the model parameters cluster submatrix size edge probabilities and the mean of the submatrices can be partitioned into four disjoint regions corresponding to decreasing statistical and computational complexities 1 the impossible regime where all algorithms fail 2 the hard regime where the computationally expensive maximum likelihood estimator mle succeeds 3 the easy regime where the polynomial time convexified mle succeeds 4 the simple regime where a local counting thresholding procedure succeeds moreover we show that each of these algorithms provably fails in the harder regimes our results establish the minimax recovery limits which are tight up to universal constants and hold even with a growing number of clusters submatrices and provide order wise stronger performance guarantees for polynomial time algorithms than previously known our study demonstrates the tradeoffs between statistical and computational considerations and suggests that the minimax limits may not be achievable by polynomial time algorithms

discrete and continuous time probabilistic models and algorithms for inferring neuronal up and down states up and down states the periodic fluctuations between increased and decreased spiking activity of a neuronal population are a fundamental feature of cortical circuits understanding up down state dynamics is important for understanding how these circuits represent and transmit information in the brain to date limited work has been done on characterizing the stochastic properties of up down state dynamics we present a set of markov and semi markov discrete and continuous time probability models for estimating up and down states from multiunit neural spiking activity we model multiunit neural spiking activity as a stochastic point process modulated by the hidden up and down states and the ensemble spiking history we estimate jointly the hidden states and the model parameters by maximum likelihood using an expectation maximization em algorithm and a monte carlo em algorithm that uses reversible jump markov chain monte carlo sampling in the e step we apply our models and algorithms in the analysis of both simulated multiunit spiking activity and actual multi unit spiking activity recorded from primary somatosensory cortex in a behaving rat during slow wave sleep our approach provides a statistical characterization of up down state dynamics that can serve as a basis for verifying and refining mechanistic descriptions of this process

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

response properties of an integrate and fire model that receives subthreshold inputs a computational technique is described for calculation of the interspike interval and poststimulus time histograms for the responses of an integrate and fire model to arbitrary inputs the effects of the model parameters on the response statistics were studied systematically specifically the probability distribution of the membrane potential was calculated as a function of time and the mean interspike interval and pst histogram were calculated for arbitrary inputs for stationary inputs the regularity of the output was studied in detail for various model parameters for nonstationary inputs the effects of the model parameters on the output synchronization index were explored the results show that enhanced synchronization in response to low frequency stimuli required a large number n 25 of weak inputs irregular responses and a linear input output rate relationship required strong but subthreshold inputs with a small time constant a model cell with mixed amplitude synaptic inputs can respond to stationary inputs irregularly and have enhanced synchronization to nonstationary inputs that are phase locked to low frequency inputs both of these response properties have been reported for some cells in the ventral cochlear nucleus in the auditory brainstem

large independent sets in general random intersection graphs we investigate the existence and efficient algorithmic construction of close to optimal independent sets in random models of intersection graphs in particular a we propose a new model for random intersection graphs g n m p which includes the model of m karonski e r scheinerman k b singer cohen on random intersection graphs the subgraph problem combinatorics probability and computing journal 8 1999 131 159 the uniform random intersection graph models as an important special case we also define an interesting variation of the model of random intersection graphs similar in spirit to random regular graphs b for this model we derive exact formulae for the mean and variance of the number of independent sets of size k for any k in the graph c we then propose and analyse three algorithms for the efficient construction of large independent sets in this model the first two are variations of the greedy technique while the third is a totally new algorithm our algorithms are analysed for the special case of uniform random intersection graphs our analyses show that these algorithms succeed in finding close to optimal independent sets for an interesting range of graph parameters

asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications in this paper we extend our previous work on the stochastic block model a commonly used generative model for social and biological networks and the problem of inferring functional groups or communities from the topology of the network we use the cavity method of statistical physics to obtain an asymptotically exact analysis of the phase diagram we describe in detail properties of the detectability undetectability phase transition and the easy hard phase transition for the community detection problem our analysis translates naturally into a belief propagation algorithm for inferring the group memberships of the nodes in an optimal way i e that maximizes the overlap with the underlying group memberships and learning the underlying parameters of the block model finally we apply the algorithm to two examples of real world networks and discuss its performance

achieving the bayes error rate in synchronization and block models by sdp robustly we study the statistical performance of semidefinite programming sdp relaxations for clustering under random graph models under the mathbb z 2 synchronization model censored block model cbm and stochastic block model sbm we show that sdp achieves an error rate of the form exp 1 o 1 bar n i here bar n is an appropriate multiple of the number of nodes and i is an information theoretic measure of the signal to noise ratio we provide matching lower bounds on the bayes error for each model and therefore demonstrate that the sdp approach is bayes optimal as a corollary our results imply that sdp achieves the optimal exact recovery threshold under each model furthermore we show that sdp is robust the above bound remains valid under semirandom versions of the models in which the observed graph is modified by a monotone adversary our proof is based on a novel primal dual analysis of sdp under a unified framework for all three models and the analysis shows that sdp tightly approximates a joint majority voting procedure

information limits for recovering a hidden community we study the problem of recovering a hidden community of cardinality k from an n times n symmetric data matrix a where for distinct indices i j a ij sim p if i j both belong to the community and a ij sim q otherwise for two known probability distributions p and q depending on n if p mathrm bern p and q mathrm bern q with p q it reduces to the problem of finding a densely connected k subgraph planted in a large erdos renyi graph if p mathcal n mu 1 and q mathcal n 0 1 with mu 0 it corresponds to the problem of locating a k times k principal submatrix of elevated means in a large gaussian random matrix we focus on two types of asymptotic recovery guarantees as n to infty 1 weak recovery expected number of classification errors is o k and 2 exact recovery probability of classifying all indices correctly converges to one under mild assumptions on p and q and allowing the community size to scale sublinearly with n we derive a set of sufficient conditions and a set of necessary conditions for recovery which are asymptotically tight with sharp constants the results hold in particular for the gaussian case and for the case of bounded log likelihood ratio including the bernoulli case whenever p q and 1 p 1 q are bounded away from zero and infinity previous work has shown that if weak recovery is achievable then exact recovery is achievable in linear additional time by a simple voting procedure we provide a converse showing the condition for the voting procedure to succeed is almost necessary for exact recovery

integrator or coincidence detector a novel measure based on the discrete reverse correlation to determine a neuron s operational mode in this letter we propose a definition of the operational mode of a neuron that is whether a neuron integrates over its input or detects coincidences we complete the range of possible operational modes by a new mode we call gap detection which means that a neuron responds to gaps in its stimulus we propose a measure consisting of two scalar values both ranging from 1 to 1 the neural drive which indicates whether its stimulus excites the neuron serves as background noise or inhibits it the neural mode which indicates whether the neuron s response is the result of integration over its input of coincidence detection or of gap detection with all three modes possible for all neural drive values this is a pure spike based measure and can be applied to measure the influence of either all or subset of a neuron s stimulus we derive the measure by decomposing the reverse correlation test it in several artificial and biological settings and compare it to other measures finding little or no correlation between them we relate the results of the measure to neural parameters and investigate the effect of time delay during spike generation our results suggest that a neuron can use several different modes simultaneously on different subsets of its stimulus to enable it to respond to its stimulus in a complex manner

shot noise in the leaky integrate and fire neuron we study the influence of noise on the transmission of temporal information by a leaky integrate and fire neuron using the theory of shot noise the model includes a finite number of synapses and has a membrane potential variance de facto modulated by the input signal the phenomenon of stochastic resonance in spiking neurons is analytically exhibited using an inhomogeneous poisson process model of the spike trains and links with the traditional ornstein uhlenbeck process obtained by a diffusion approximation are given it is shown that the modulated membrane potential variance inherent to the model gives better signal processing capabilities than the diffusion approximation

spotting neural spike patterns using an adversary background model the detection of a specific stochastic pattern embedded in an unknown background noise is a difficult pattern recognition problem encountered in many applications such as word spotting in speech a similar problem emerges when trying to detect a multineural spike pattern in a single electrical recording embedded in the complex cortical activity of a behaving animal solving this problem is crucial for the identification of neuronal code words with specific meaning the technical difficulty of this detection is due to the lack of a good statistical model for the background activity which rapidly changes with the recording conditions and activity of the animal this work introduces the use of an adversary background model this model assumes that the background knows the pattern sought up to a first order statistics and this knowledge creates a background composed of all the permutations of our pattern we show that this background model is tightly connected to the type based information theoretic approach furthermore we show that computing the likelihood ratio is actually decomposing the log likelihood distribution according to types of the empirical counts we demonstrate the application of this method for detection of the reward patterns in the basal ganglia of behaving monkeys yielding some unexpected biological results

phase transitions for detecting latent geometry in random graphs random graphs with latent geometric structure are popular models of social and biological networks with applications ranging from network user profiling to circuit design these graphs are also of purely theoretical interest within computer science probability and statistics a fundamental initial question regarding these models is when are these random graphs affected by their latent geometry and when are they indistinguishable from simpler models without latent structure such as the erd s renyi graph mathcal g n p we address this question for two of the most well studied models of random graphs with latent geometry the random intersection and random geometric graph our results are as follows a the random intersection graph is defined by sampling n random sets s 1 s 2 ldots s n by including each element of a set of size d in each s i independently with probability delta and including the edge i j if s i cap s j ne emptyset we prove that the random intersection graph converges in total variation to an erd s renyi graph if d tilde omega n 3 and does not if d o n 3 for both dense and sparse edge densities p this resolves an open problem in fill et al random struct algorithms 16 2 156 176 2000 rybarczyk random struct algorithms 38 1 2 205 234 2011 and kim et al random struct algorithms 52 4 662 679 2018 the same result was obtained simultaneously and independently by bubeck et al when random intersection graphs lose geometry manuscript 2019 b we strengthen the preceding argument to show that the matrix of random intersection sizes s i cap s j converges in total variation to a symmetric matrix with independent poisson entries this yields the first total variation convergence result for tau random intersection graphs where the edge i j is included if s i cap s j ge tau more precisely our results imply that if p is bounded away from 1 then the tau random intersection graph with edge density p converges to mathcal g n p if d omega tau 3 n 3 c the random geometric graph on mathbb s d 1 is defined by sampling x 1 x 2 dots x n uniformly at random from mathbb s d 1 and including the edge i j if vert x i x j vert 2 le tau a result of bubeck et al random struct algorithms 49 503 532 2016 showed that this model converges to mathcal g n p in total variation where p is chosen so that the models have matching edge densities as long as d omega n 3 it was conjectured in bubeck et al 2016 that this threshold decreases drastically for p small we make the first progress towards this conjecture by showing convergence if d tilde omega left min pn 3 p 2 n 7 2 right our proofs are a hybrid of combinatorial arguments direct couplings and applications of information inequalities previous upper bounds on the total variation distance between random graphs with latent geometry and mathcal g n p have typically not been both combinatorial and information theoretic while this interplay is essential to the sharpness of our bounds

extracting state transition dynamics from multiple spike trains using hidden markov models with correlated poisson distribution neural activity is nonstationary and varies across time hidden markov models hmms have been used to track the state transition among quasi stationary discrete neural states within this context an independent poisson model has been used for the output distribution of hmms hence the model is incapable of tracking the change in correlation without modulating the firing rate to achieve this we applied a multivariate poisson distribution with correlation terms for the output distribution of hmms we formulated a variational bayes vb inference for the model the vb could automatically determine the appropriate number of hidden states and correlation types while avoiding the overlearning problem we developed an efficient algorithm for computing posteriors using the recursive relationship of a multivariate poisson distribution we demonstrated the performance of our method on synthetic data and real spike trains recorded from a songbird

characteristic formulae for processes with divergence abstract characteristic formulae have been introduced by graf and sifakis to relate equational reasoning about proceses to reasoning in a modal logic and therefore to allow proofs about processes to be carried out in a logical framework this work which concerned finite processes and bisimulation like equivalences has later on been extended to finite state processes and further equivalences based upon an intuitionistic understanding of hennessy milner logic hml with mutual recursion we extend these results to cover bisimulation like preorders which are sensitive to liveness properties this demonstrates the expressive power of intuitionistically interpreted hml with mutual recursion and it builds the theoretical basis for a uniform and efficient method to automatically verify bisimulation like relations between processes by means of model checking

information theoretic bounds and phase transitions in clustering sparse pca and submatrix localization we study the problem of detecting a structured low rank signal matrix corrupted with additive gaussian noise this includes clustering in a gaussian mixture model sparse pca and submatrix localization each of these problems is conjectured to exhibit a sharp information theoretic threshold below which the signal is too weak for any algorithm to detect we derive upper and lower bounds on these thresholds by applying the first and second moment methods to the likelihood ratio between these planted models and null models where the signal matrix is zero for sparse pca and submatrix localization we determine this threshold exactly in the limit where the number of blocks is large or the signal matrix is very sparse for the clustering problem our bounds differ by a factor of sqrt 2 when the number of clusters is large moreover our upper bounds show that for each of these problems there is a significant regime where reliable detection is information theoretically possible but where known algorithms such as pca fail completely since the spectrum of the observed matrix is uninformative this regime is analogous to the conjectured hard but detectable regime for community detection in sparse graphs

typology of phase transitions in bayesian inference problems many inference problems undergo phase transitions as a function of the signal to noise ratio a prominent example of this phenomenon being found in the stochastic block model sbm that generates a random graph with a hidden community structure some of these phase transitions affect the information theoretic optimal but possibly computationally expensive estimation procedure others concern the behavior of efficient iterative algorithms a computational gap opens when the phase transitions for these two aspects do not coincide leading to a hard phase in which optimal inference is computationally challenging in this paper we refine this description in two ways from a qualitative perspective we emphasize the existence of more generic phase diagrams with a hybrid hard phase in which it is computationally easy to reach a nontrivial inference accuracy but computationally hard to match the information theoretic optimal one we support this discussion by quantitative expansions of the functional cavity equations that describe inference problems on sparse graphs around their trivial solution these expansions shed light on the existence of hybrid hard phases for a large class of planted constraint satisfaction problems and on the question of the tightness of the kesten stigum ks bound for the associated tree reconstruction problem our results show that the instability of the trivial fixed point is not generic evidence for the bayes optimality of the message passing algorithms we clarify in particular the status of the symmetric sbm with four communities and of the tree reconstruction of the associated potts model in the assortative ferromagnetic case the ks bound is always tight whereas in the disassortative antiferromagnetic case we exhibit an explicit criterion involving the degree distribution that separates a large degree regime where the ks bound is tight and a low degree regime where it is not we also investigate the sbm with two communities of different sizes also known as the asymmetric ising model and describe quantitatively its computational gap as a function of its asymmetry as well as a version of the sbm with two groups of q 1 and q 2 communities we complement this study with numerical simulations of the belief propagation iterative algorithm confirming that its behavior on large samples is well described by the cavity method

when are prime formulae characteristic abstract in the setting of the modal logic that characterizes modal refinement over modal transition systems boudol and larsen showed that the formulae for which model checking can be reduced to preorder checking that is the characteristic formulae are exactly the consistent and prime ones this paper presents general sufficient conditions guaranteeing that characteristic formulae are exactly the consistent and prime ones it is shown that the given conditions apply to various behavioural relations in the literature in particular characteristic formulae are exactly the prime and consistent ones for all the semantics in van glabbeek s linear time branching time spectrum

storage capacity in symmetric binary perceptrons we study the problem of determining the capacity of the binary perceptron for two variants of the problem where the corresponding constraint is symmetric we call these variants the rectangle binary perceptron rpb and the u function binary perceptron ubp we show that unlike for the usual step function binary perceptron the critical capacity in these symmetric cases is given by the annealed computation in a large region of parameter space for all rectangular constraints and for narrow enough u function constraints k k we conclude that full step replica symmetry breaking would have to be evaluated in order to obtain the exact capacity in this case

reconstruction and estimation in the planted partition model the planted partition model also known as the stochastic blockmodel is a classical cluster exhibiting random graph model that has been extensively studied in statistics physics and computer science in its simplest form the planted partition model is a model for random graphs on n nodes with two equal sized clusters with an between class edge probability of q and a within class edge probability of p although most of the literature on this model has focused on the case of increasing degrees ie pn qn rightarrow infty as n rightarrow infty the sparse case p q o 1 n is interesting both from a mathematical and an applied point of view a striking conjecture of decelle krzkala moore and zdeborova based on deep non rigorous ideas from statistical physics gave a precise prediction for the algorithmic threshold of clustering in the sparse planted partition model in particular if p a n and q b n then decelle et al conjectured that it is possible to cluster in a way correlated with the true partition if a b 2 2 a b and impossible if a b 2 2 a b by comparison the best known rigorous result is that of coja oghlan who showed that clustering is possible if a b 2 c a b for some sufficiently large c we prove half of their prediction showing that it is indeed impossible to cluster if a b 2 2 a b furthermore we show that it is impossible even to estimate the model parameters from the graph when a b 2 2 a b on the other hand we provide a simple and efficient algorithm for estimating a and b when a b 2 2 a b following decelle et al our work establishes a rigorous connection between the clustering problem spin glass models on the bethe lattice and the so called reconstruction problem this connection points to fascinating applications and open problems

reduction of the hodgkin huxley equations to a single variable threshold model it is generally believed that a neuron is a threshold element that fires when some variable u reaches a threshold here we pursue the question of whether this picture can be justified and study the four dimensional neuron model of hodgkin and huxley as a concrete example the model is approximated by a response kernel expansion in terms of a single variable the membrane voltage the first order term is linear in the input and its kernel has the typical form of an elementary postsynaptic potential higher order kernels take care of nonlinear interactions between input spikes in contrast to the standard volterra expansion the kernels depend on the firing time of the most recent output spike in particular a zero order kernel that describes the shape of the spike and the typical after potential is included our model neuron fires if the membrane voltage given by the truncated response kernel expansion crosses a threshold the threshold model is tested on a spike train generated by the hodgkin huxley model with a stochastic input current we find that the threshold model predicts 90 percent of the spikes correctly our results show that to good approximation the description of a neuron as a threshold element can indeed be justified

hebbian learning and spiking neurons a correlation based hebbian learning rule at a spike level with millisecond resolution is formulated mathematically analyzed and compared with learning in a firing rate description the relative timing of presynaptic and postsynaptic spikes influences synaptic weights via an asymmetric learning window a differential equation for the learning dynamics is derived under the assumption that the time scales of learning and neuronal spike dynamics can be separated the differential equation is solved for a poissonian neuron model with stochastic spike arrival it is shown that correlations between input and output spikes tend to stabilize structure formation with an appropriate choice of parameters learning leads to an intrinsic normalization of the average weight and the output firing rate noise generates diffusion like spreading of synaptic weights s1063 651x 99 02804 4

pattern segmentation in associative memory the goal of this paper is to show how to modify associative memory such that it can discriminate several stored patterns in a composite input and represent them simultaneously segmention of patterns takes place in the temporal domain components of one pattern becoming temporally correlated with each other and anticorrelated with the components of all other patterns correlations are created naturally by the usual associative connections in our simulations temporal patterns take the form of oscillatory bursts of activity model oscillators consist of pairs of local cell populations connected appropriately transition of activity from one pattern to another is induced by delayed self inhibition or simply by noise

physiological gain leads to high isi variability in a simple model of a cortical regular spiking cell to understand the interspike interval isi variability displayed by visual cortical neurons softky koch 1993 it is critical to examine the dynamics of their neuronal integration as well as

spectral clustering and the high dimensional stochastic blockmodel networks or graphs can easily represent a diverse set of data sources that are characterized by interacting units or actors social ne tworks representing people who communicate with each other are one example communities or clusters of highly connected actors form an essential feature in the structure of several empirical networks spectral clustering is a popular and computationally feasi ble method to discover these communities the stochastic block model holland et al 1983 is a social network model with well defined communities each node is a member of one community for a network generated from the stochastic block model we bound the number of nodes misclus tered by spectral clustering the asymptotic results in th is paper are the first clustering results that allow the number of clusters in the model to grow with the number of nodes hence the name high dimensional in order to study spectral clustering under the stochastic block model we first show that under the more general latent space model the eigenvectors of the normalized graph laplacian asymptotically converge to the eigenvectors of a population normal ized graph laplacian aside from the implication for spectral clustering this provides insight into a graph visualization technique our method of studying the eigenvectors of random matrices is original ams 2000 subject classifications primary 62h30 62h25 secondary 60b20

a modal characterization of observational congruence on finite terms of ccs we propose a translation method of finite terms of ccs into formulas of a modal language representing their class of observational congruence for this purpose we define a modal language and a function associating with any finite term of ccs a formula of the language satisfied by the term furthermore this function is such that two terms are congruent if and only if the corresponding formulas are equivalent the translation method consists in associating with operations on terms action operations on the corresponding formulas this work is a first step towards the definition of a modal language with modalities expressing both possibility and inevitability and which is compatible with observational congruence

detection of hidden structures in nonstationary spike trains we propose an algorithm for simultaneously estimating state transitions among neural states and nonstationary firing rates using a switching state space model sssm this algorithm enables us to detect state transitions on the basis of not only discontinuous changes in mean firing rates but also discontinuous changes in the temporal profiles of firing rates e g temporal correlation we construct estimation and learning algorithms for a nongaussian sssm whose nongaussian property is caused by binary spike events local variational methods can transform the binary observation process into a quadratic form the transformed observation process enables us to construct a variational bayes algorithm that can determine the number of neural states based on automatic relevance determination additionally our algorithm can estimate model parameters from single trial data using a priori knowledge about state transitions and firing rates synthetic data analysis reveals that our algorithm has higher performance for estimating nonstationary firing rates than previous methods the analysis also confirms that our algorithm can detect state transitions on the basis of discontinuous changes in temporal correlation which are transitions that previous hidden markov models could not detect we also analyze neural data recorded from the medial temporal area the statistically detected neural states probably coincide with transient and sustained states that have been detected heuristically estimated parameters suggest that our algorithm detects the state transitions on the basis of discontinuous changes in the temporal correlation of firing rates these results suggest that our algorithm is advantageous in real data analysis

nearest neighbor walks with low predictability profile and percolation in 2 epsilon dimensions a few years ago grimmett kesten and zhang proved that for supercritical bond percolation on z 3 simple random walk on the infinite cluster is a s transient we generalize this result to a class of wedges in z 3 including for any e 0 1 the wedge w e x y z z 3 x 0 z x e which can be thought of as representing a 2 e dimensional lattice our proof builds on recent work of benjamini pemantle and peres and involves the construction of finite energy flows using nearest neighbor walks on z with low predictability profile along the way we obtain some new results on attainable decay rates for predictability profiles of nearest neighbor walks

achieving exact cluster recovery threshold via semidefinite programming extensions resolving a conjecture of abbe bandeira and hall the authors have recently shown that the semidefinite programming sdp relaxation of the maximum likelihood estimator achieves the sharp threshold for exactly recovering the community structure under the binary stochastic block model sbm of two equal sized clusters the same was shown for the case of a single cluster and outliers extending the proof techniques in this paper it is shown that sdp relaxations also achieve the sharp recovery threshold in the following cases 1 binary sbm with two clusters of sizes proportional to network size but not necessarily equal 2 sbm with a fixed number of equal sized clusters and 3 binary censored block model with the background graph being erd s renyi furthermore a sufficient condition is given for an sdp procedure to achieve exact recovery for the general case of a fixed number of clusters plus outliers these results demonstrate the versatility of sdp relaxation as a simple general purpose computationally feasible methodology for community detection

algebraic laws for nondeterminism and concurrency since a nondeterministic and concurrent program may in general communicate repeatedly with its environment its meaning cannot be presented naturally as an input output function as is often done in the denotational approach to semantics in this paper an alternative is put forth first a definition is given of what it is for two programs or program parts to be equivalent for all observers then two program parts are said to be observation congruent if they are in all program contexts equivalent the behavior of a program part that is its meaning is defined to be its observation congruence class the paper demonstrates for a sequence of simple languages expressing finite terminating behaviors that in each case observation congruence can be axiomatized algebraically moreover with the addition of recursion and another simple extension the algebraic language described here becomes a calculus for writing and specifying concurrent programs and for proving their properties

finite state automata resulting from temporal information maximization and a temporal learning rule we extend linkser s infomax principle for feedforward neural networks to a measure for stochastic interdependence that captures spatial and temporal signal properties in recurrent systems this measure stochastic interaction quantifies the kullback leibler divergence of a markov chain from a product of split chains for the single unit processes for unconstrained markov chains the maximization of stochastic interaction also called temporal infomax has been previously shown to result in almost deterministic dynamics this letter considers temporal infomax on constrained markov chains where some of the units are clamped to prescribed stochastic processes providing input to the system temporal infomax in that case leads to finite state automata either completely deterministic or weakly nondeterministic transitions between internal states of these systems are almost perfectly predictable given the complete current state and the input but the activity of each single unit alone is virtually random the results are demonstrated by means of computer simulations and confirmed analytically it is furthermore shown numerically that temporal infomax leads to a high information flow from the input to internal units and that a simple temporal learning rule can approximately achieve the optimization of temporal interaction we relate these results to experimental data concerning the correlation dynamics and functional connectivities observed in multiple electrode recordings

exact recovery in the hypergraph stochastic block model a spectral algorithm abstract we consider the exact recovery problem in the hypergraph stochastic block model hsbm with k blocks of equal size more precisely we consider a random d uniform hypergraph h with n vertices partitioned into k clusters of size s n k hyperedges e are added independently with probability p if e is contained within a single cluster and q otherwise where 0 q p 1 we present a spectral algorithm which recovers the clusters exactly with high probability given mild conditions on n k p q and d our algorithm is based on the adjacency matrix of h which is a symmetric n n matrix whose u v th entry is the number of hyperedges containing both u and v to the best of our knowledge our algorithm is the first to guarantee exact recovery when the number of clusters k n

computational barriers in minimax submatrix detection this paper studies the minimax detection of a small submatrix of elevated mean in a large matrix contaminated by additive gaussian noise to investigate the tradeoff between statistical performance and computational cost from a complexity theoretic perspective we consider a sequence of discretized models which are asymptotically equivalent to the gaussian model under the hypothesis that the planted clique detection problem cannot be solved in randomized polynomial time when the clique size is of smaller order than the square root of the graph size the following phase transition phenomenon is established when the size of the large matrix p to infty if the submatrix size k theta p alpha for any alpha in 0 2 3 computational complexity constraints can incur a severe penalty on the statistical performance in the sense that any randomized polynomial time test is minimax suboptimal by a polynomial factor in p if k theta p alpha for any alpha in 2 3 1 minimax optimal detection can be attained within constant factors in linear time using schatten norm loss as a representative example we show that the hardness of attaining the minimax estimation rate can crucially depend on the loss function implications on the hardness of support recovery are also obtained

achieving exact cluster recovery threshold via semidefinite programming the binary symmetric stochastic block model deals with a random graph of n vertices partitioned into two equal sized clusters such that each pair of vertices is independently connected with probability p within clusters and q across clusters in the asymptotic regime of p a log n n and q b log n n for fixed a b and n to infty we show that the semidefinite programming relaxation of the maximum likelihood estimator achieves the optimal threshold for exactly recovering the partition from the graph with probability tending to one resolving a conjecture of abbe et al furthermore we show that the semidefinite programming relaxation also achieves the optimal recovery threshold in the planted dense subgraph model containing a single cluster of size proportional to n

extracting oscillations neuronal coincidence detection with noisy periodic spike input how does a neuron vary its mean output firing rate if the input changes from random to oscillatory coherent but noisy activity what are the critical parameters of the neuronal dynamics and input statistics to answer these questions we investigate the coincidence detection properties of an integrate and fire neuron we derive an expression indicating how coincidence detection depends on neuronal parameters specifically we show how coincidence detection depends on the shape of the postsynaptic response function the number of synapses and the input statistics and we demonstrate that there is an optimal threshold our considerations can be used to predict from neuronal parameters whether and to what extent a neuron can act as a coincidence detector and thus can convert a temporal code into a rate code

information theoretic thresholds from the cavity method abstract vindicating a sophisticated but non rigorous physics approach called the cavity method we establish a formula for the mutual information in statistical inference problems induced by random graphs and we show that the mutual information holds the key to understanding certain important phase transitions in random graph models we work out several concrete applications of these general results for instance we pinpoint the exact condensation phase transition in the potts antiferromagnet on the random graph thereby improving prior approximate results contucci et al 2013 34 further we prove the conjecture from krzakala et al 2007 55 about the condensation phase transition in the random graph coloring problem for any number q 3 of colors moreover we prove the conjecture on the information theoretic threshold in the disassortative stochastic block model decelle et al 2011 35 additionally our general result implies the conjectured formula for the mutual information in low density generator matrix codes montanari 2005 73

termination deadlock and divergence in this paper a process algebra that incorporates explicit representations of successful termination deadlock and divergence is introduced and its semantic theory is analyzed both an operational and a denotational semantics for the language is given and it is shown that they agree the operational theory is based upon a suitable adaptation of the notion of bisimulation preorder the denotational semantics for the language is given in terms of the initial continuous algebra that satisfies a set of equations e ci e it is shown that ci e is fully abstract with respect to our choice of behavioral preorder several results of independent interest are obtained namely the finite approximability of the behavioral preorder and a partial completeness result for the set of equations e with respect to the preorder

markov analysis of stochastic resonance in a periodically driven integrate and fire neuron we model the dynamics of the leaky integrate and fire neuron under periodic stimulation as a markov process with respect to the stimulus phase this avoids the unrealistic assumption of a stimulus reset after each spike made in earlier papers and thus solves the long standing reset problem the neuron exhibits stochastic resonance both with respect to input noise intensity and stimulus frequency the latter resonance arises by matching the stimulus frequency to the refractory time of the neuron the markov approach can be generalized to other periodically driven stochastic processes containing a reset mechanism