the spike triggered average of the integrate and fire cell driven by gaussian white noise we compute the exact spike triggered average sta of the voltage for the nonleaky integrate and fire if cell in continuous time driven by gaussian white noise the computation is based on techniques from the theory of renewal processes and continuous time hidden markov processes e g the backward and forward fokker planck partial differential equations associated with first passage time densities from the sta voltage it is straightforward to derive the sta input current the theory also gives an explicit asymptotic approximation for the sta of the leaky if cell valid in the low noise regime 0 we consider both the sta and the conditional average voltage given an observed spike doublet event that is two spikes separated by some fixed period of silence in each case we find that the sta as a function of time preceding spike has a square root singularity as approaches zero from below and scales linearly with the scale of injected noise current we close by briefly examining the discrete time case where similar phenomena are observed

serial correlation in neural spike trains experimental evidence stochastic modeling and single neuron variability the activity of spiking neurons is frequently described by renewal point process models that assume the statistical independence and identical distribution of the intervals between action potentials however the assumption of independent intervals must be questioned for many different types of neurons we review experimental studies that reported the feature of a negative serial correlation of neighboring intervals commonly observed in neurons in the sensory periphery as well as in central neurons notably in the mammalian cortex in our experiments we observed the same short lived negative serial dependence of intervals in the spontaneous activity of mushroom body extrinsic neurons in the honeybee to model serial interval correlations of arbitrary lags we suggest a family of autoregressive point processes its marginal interval distribution is described by the generalized gamma model which includes as special cases the log normal and gamma distributions which have been widely used to characterize regular spiking neurons in numeric simulations we investigated how serial correlation affects the variance of the neural spike count we show that the experimentally confirmed negative correlation reduces single neuron variability as quantified by the fano factor by up to 50 which favors the transmission of a rate code we argue that the feature of a negative serial correlation is likely to be common to the class of spike frequency adapting neurons and that it might have been largely overlooked in extracellular single unit recordings due to spike sorting errors

regularity of spike trains and harmony perception in a model of the auditory system spike train regularity of the noisy neural auditory system model under the influence of two sinusoidal signals with different frequencies is investigated for the increasing ratio m n of the input signal frequencies m n are natural numbers the linear growth of the regularity is found at the fixed difference m

on the convergence of diffusion processes conditioned to remain in a bounded region for large time to limiting positive recurrent diffusion processes let x t be a diffusion process on r d with generator l 1 2 nabla cdot a nabla b nabla and let p x x in r d be the corresponding measures on paths pick 0 s which are always satisfied if a 1 b is a gradient function we show that y t s is an inhomogeneous diffusion process and that as t rightarrow infty y t s 0 leq s leq t converges to a limiting homogeneous positive recurrent diffusion y s 0 leq s leq t with state space g since t is arbitrary we actually obtain a limiting process y s 0 leq s infty the generator of the limiting process may be written in the form l g 1 2 nabla cdot a nabla b nabla a nabla g 0 g 0 nabla a nabla h g 0 nabla where g 0 is the square root of the density of a measure mu 0 which minimizes the i function for the process over all gamma in mathscr p bar g the set of probability measures on bar g the function h g 0 appears in the explicit calculation of i mu 0 and solves a certain variational equation the invariant measure for the process is mu 0

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

analytical model for the effects of learning on spike count distributions the spike count distribution observed when recording from a variety of neurons in many different conditions has a fairly stereotypical shape with a single mode at zero or close to a low average count and a long quasi exponential tail to high counts such a distribution has been suggested to be the direct result of three simple facts the firing frequency of a typical cortical neuron is close to linear in the summed input current entering the soma above a threshold the input current varies on several timescales both faster and slower than the window used to count spikes and the input distribution at any timescale can be taken to be approximately normal the third assumption is violated by associative learning which generates correlations between the synaptic weight vector on the dendritic tree of a neuron and the input activity vectors it is repeatedly subject to we show analytically that for a simple feed forward model the normal distribution of the slow components of the input current becomes the sum of two quasi normal terms the term important below threshold shifts with learning while the term important above threshold does not shift but grows in width these deviations from the standard distribution may be observable in appropriate recording experiments

a holonomic systems approach to special functions identities abstract we observe that many special functions are solutions of so called holonomic systems bernstein s deep theory of holonomic systems is then invoked to show that any identity involving sums and integrals of products of these special functions can be verified in a finite number of steps this is partially substantiated by an algorithm that proves terminating hypergeometric series identities and that is given both in english and in maple

multiscale codes in the nervous system the problem of noise correlations and the ambiguity of periodic scales encoding information about continuous variables using noisy computational units is a challenge nonetheless asymptotic theory shows that combining multiple periodic scales for coding can be highly precise despite the corrupting influence of noise mathis herz and stemmler phys rev lett 109 018103 2012 indeed the cortex seems to use periodic multiscale grid codes to represent position accurately here we show how such codes can be read out without taking the long term limit even on short time scales the precision of such codes scales exponentially in the number n of neurons does this finding also hold for neurons that are not firing in a statistically independent fashion to assess the extent to which biological grid codes are subject to statistical dependences we first analyze the noise correlations between pairs of grid code neurons in behaving rodents we find that if the grids of two neurons align and have the same length scale the noise correlations between the neurons can reach values as high as 0 8 for increasing mismatches between the grids of the two neurons the noise correlations fall rapidly incorporating such correlations into a population coding model reveals that the correlations lessen the resolution but the exponential scaling of resolution with n is unaffected

firing variability is higher than deduced from the empirical coefficient of variation a convenient and often used summary measure to quantify the firing variability in neurons is the coefficient of variation cv defined as the standard deviation divided by the mean it is therefore important to find an estimator that gives reliable results from experimental data that is the estimator should be unbiased and have low estimation variance when the cv is evaluated in the standard way empirical standard deviation of interspike intervals divided by their average then the estimator is biased underestimating the true cv especially if the distribution of the interspike intervals is positively skewed moreover the estimator has a large variance for commonly used distributions the aim of this letter is to quantify the bias and propose alternative estimation methods if the distribution is assumed known or can be determined from data parametric estimators are proposed which not only remove the bias but also decrease the estimation errors if no distribution is assumed and the data are very positively skewed we propose to correct the standard estimator when defining the corrected estimator we simply use that it is more stable to work on the log scale for positively skewed distributions the estimators are evaluated through simulations and applied to experimental data from olfactory receptor neurons in rats

representational accuracy of stochastic neural populations fisher information is used to analyze the accuracy with which a neural population encodes d stimulus features it turns out that the form of response variability has a major impact on the encoding capacity and therefore plays an important role in the selection of an appropriate neural model in particular in the presence of baseline firing the reconstruction error rapidly increases with d in the case of poissonian noise but not for additive noise the existence of limited range correlations of the type found in cortical tissue yields a saturation of the fisher information content as a function of the population size only for an additive noise model we also show that random variability in the correlation coefficient within a neural population as found empirically considerably improves the average encoding quality finally the representational accuracy of populations with inhomogeneous tuning properties either with variability in the tuning widths or fragmented into specialized subpopulations is superior to the case of identical and radially symmetric tuning curves usually considered in the literature

linearization of f 1 curves by adaptation we show that negative feedback to highly nonlinear frequency current f i curves results in an effective linearization by highly nonlinear we mean that the slope at threshold is infinite or very steep we then apply this to a specific model for spiking neurons and show that the details of the adaptation mechanism do not affect the results the crucial points are that the adaptation is slow compared to other processes and the unadapted f i curve is highly nonlinear

disappearance of spurious states in analog associative memories we show that symmetric n mixture states when they exist are almost never stable in autoassociative networks with threshold linear units only with a binary coding scheme we could find a limited region of the parameter space in which either 2 mixture or 3 mixture states are stable attractors of the dynamics

characterization and computation of optimal distributions for channel coding this paper concerns the structure of capacity achieving input distributions for stochastic channel models and a renewed look at their computational aspects the following conclusions are obtained under general assumptions on the channel statistics i the capacity achieving input distribution is binary for low signal to noise ratio snr the proof is obtained on comparing the optimization equations that determine channel capacity with a linear program over the space of probability measures ii simple discrete approximations can nearly reach capacity even in cases where the optimal distribution is known to be absolutely continuous with respect to lebesgue measure iii a new class of algorithms is introduced based on the cutting plane method to iteratively construct discrete distributions along with upper and lower bounds on channel capacity it is shown that the bounds converge to the true channel capacity and that the distributions converge weakly to a capacity achieving distribution

including long range dependence in integrate and fire models of the high interspike interval variability of cortical neurons many different types of integrate and fire models have been designed in order to explain how it is possible for a cortical neuron to integrate over many independent inputs while still producing highly variable spike trains within this context the variability of spike trains has been almost exclusively measured using the coefficient of variation of interspike intervals however another important statistical property that has been found in cortical spike trains and is closely associated with their high firing variability is long range dependence we investigate the conditions if any under which such models produce output spike trains with both interspike interval variability and long range dependence similar to those that have previously been measured from actual cortical neurons we first show analytically that a large class of high variability integrate and fire models is incapable of producing such outputs based on the fact that their output spike trains are always mathematically equivalent to renewal processes this class of models subsumes a majority of previously published models including those that use excitation inhibition balance correlated inputs partial reset or nonlinear leakage to produce outputs with high variability next we study integrate and fire models that have nonpoissonian renewal point process inputs instead of the poisson point process inputs used in the preceding class of models the confluence of our analytical and simulation results implies that the renewal input model is capable of producing high variability and long range dependence comparable to that seen in spike trains recorded from cortical neurons but only if the interspike intervals of the inputs have infinite variance a physiologically unrealistic condition finally we suggest a new integrate and fire model that does not suffer any of the previously mentioned shortcomings by analyzing simulation results for this model we show that it is capable of producing output spike trains with interspike interval variability and long range dependence that match empirical data from cortical spike trains this model is similar to the other models in this study except that its inputs are fractional gaussian noise driven poisson processes rather than renewal point processes in addition to this model s success in producing realistic output spike trains its inputs have long range dependence similar to that found in most subcortical neurons in sensory pathways including the inputs to cortex analysis of output spike trains from simulations of this model also shows that a tight balance between the amounts of excitation and inhibition at the inputs to cortical neurons is not necessary for high interspike interval variability at their outputs furthermore in our analysis of this model we show that the superposition of many fractional gaussian noise driven poisson processes does not approximate a poisson process which challenges the common assumption that the total effect of a large number of inputs on a neuron is well represented by a poisson process

onset of negative interspike interval correlations in adapting neurons negative serial correlations in single spike trains are an effective method to reduce the variability of spike counts one of the factors contributing to the development of negative correlations between successive interspike intervals is the presence of adaptation currents in this work based on a hidden markov model and a proper statistical description of conditional responses we obtain analytically these correlations in an adequate dynamical neuron model resembling adaptation we derive the serial correlation coefficients for arbitrary lags under a small adaptation scenario in this case the behavior of correlations is universal and depends on the first order statistical description of an exponentially driven time inhomogeneous stochastic process

how to modify a neural network gradually without changing its input output functionality it is generally unknown when distinct neural networks having different synaptic weights and thresholds implement identical input output transformations determining the exact conditions for structurally distinct yet functionally equivalent networks may shed light on the theoretical constraints on how diverse neural circuits might develop and be maintained to serve identical functions such consideration also imposes practical limits on our ability to uniquely infer the structure of underlying neural circuits from stimulus response measurements we introduce a biologically inspired mathematical method for determining when the structure of a neural network can be perturbed gradually while preserving functionality we show that for common three layer networks with convergent and nondegenerate connection weights this is possible only when the hidden unit gains are power functions exponentials or logarithmic functions which are known to approximate the gains seen in some biological neurons for practical applications our numerical simulations with finite and noisy data show that continuous confounding of parameters due to network functional equivalence tends to occur approximately even when the gain function is not one of the aforementioned three types suggesting that our analytical results are applicable to more general situations and may help identify a common source of parameter variability in neural network modeling

receiver operating characteristic curves for a simple stochastic process that carries a static signal the detection of a weak signal in the presence of noise is an important problem that is often studied in terms of the receiver operating characteristic roc curve in which the probability of correct detection is plotted against the probability for a false positive this kind of analysis is typically applied to the situation in which signal and noise are stochastic variables the detection problem emerges however also often in a context in which both signal and noise are stochastic processes and the correct or false detection is said to take place when the process crosses a threshold in a given time window here we consider the problem for a combination of a static signal which has to be detected against a dynamic noise process the well known ornstein uhlenbeck process we give exact but difficult to evaluate quadrature expressions for the detection rates for false positives and correct detections investigate systematically a simple sampling approximation suggested earlier compare to an approximation by stratonovich for the limit of high threshold and briefly explore the case of multiplicative signal all theoretical results are compared to extensive numerical simulations of the corresponding langevin equation our results demonstrate that the sampling approximation provides a reasonable description of the roc curve for this system and it clarifies limit cases for the roc curve

why two qubits are special we analyze some special properties of a system of two qubits and in particular of the so called bell basis for this system and discuss the possibility of extending these properties to higher dimensional systems we give a general construction for orthonormal bases of maximally entangled vectors which works in any dimension and is based on latin squares and complex hadamard matrices however for none of these bases the special properties of the operation of complex conjugation in bell basis hold namely that maximally entangled vectors have up to a phase real coefficients and that factorizable unitaries have real matrix elements

estimating input parameters from intracellular recordings in the feller neuronal model we study the estimation of the input parameters in a feller neuronal model from a trajectory of the membrane potential sampled at discrete times these input parameters are identified with the drift and the infinitesimal variance of the underlying stochastic diffusion process with multiplicative noise the state space of the process is restricted from below by an inaccessible boundary further the model is characterized by the presence of an absorbing threshold the first hitting of which determines the length of each trajectory and which constrains the state space from above we compare both in the presence and in the absence of the absorbing threshold the efficiency of different known estimators in addition we propose an estimator for the drift term which is proved to be more efficient than the others at least in the explored range of the parameters the presence of the threshold makes the estimates of the drift term biased and two methods to correct it are proposed

mean instantaneous firing frequency is always higher than the firing rate frequency coding is considered one of the most common coding strategies employed by neural systems this fact leads in experiments as well as in theoretical studies to construction of so called transfer functions where the output firing frequency is plotted against the input intensity the term firing frequency can be understood differently in different contexts basically it means that the number of spikes over an interval of preselected length is counted and then divided by the length of the interval but due to the obvious limitations the length of observation cannot be arbitrarily long then firing frequency is defined as reciprocal to the mean interspike interval in parallel an instantaneous firing frequency can be defined as reciprocal to the length of current interspike interval and by taking a mean of these the definition can be extended to introduce the mean instantaneous firing frequency all of these definitions of firing frequency are compared in an effort to contribute to a better understanding of the input output properties of a neuron

firing rate distributions and efficiency of information transmission of inferior temporal cortex neurons to natural visual stimuli the distribution of responses of sensory neurons to ecological stimulation has been proposed to be designed to maximize information transmission which according to a simple model would imply an exponential distribution of spike counts in a given time window we have used recordings from inferior temporal cortex neurons responding to quasi natural visual stimulation presented using a video of everyday lab scenes and a large number of static images of faces and natural scenes to assess the validity of this exponential model and to develop an alternative simple model of spike count distributions we find that the exponential model has to be rejected in 84 of cases at the p 0 01 level a new model which accounts for the firing rate distribution found in terms of slow and fast variability in the inputs that produce neuronal activation is rejected statistically in only 16 of cases finally we show that the neurons are moderately efficient at transmitting information but not optimally efficient

information capacity of stochastic pooling networks is achieved by discrete inputs stochastic pooling networks spn are sensor networks where multiple sensors make independently noisy and compressed measurements of the same information source which are combined via pooling examples of spns range from nanoelectronics to biological sensory neurons here it is shown that optimal information transmission in spns with nodes that quantize to a finite number of states requires the input signal distribution to be discrete this is illustrated numerically for a simple spn consisting of n binary quantizing sensors the resultant information capacity is shown to be independent of the noise distribution when the signal distribution can be freely chosen but to imply an optimal noise distribution if the signal distribution is fixed while larger than the best performance of previously studied continuously valued input signals the capacity does not scale faster than the previous best result of text log 2 sqrt n bits per channel use it is also shown that a plot of the optimal input distribution contains bifurcations as n increases and that suprathreshold stochastic resonance occurs when the mutual information is determined for a suboptimal noise distribution

quasistatic approximation of the interspike interval distribution of neurons driven by time dependent inputs variability in neural responses is a ubiquitous phenomenon in neurons usually modeled with stochastic differential equations in particular stochastic integrate and fire models are widely used to simplify theoretical studies the statistical properties of the generated spikes depend on the stimulating input current given that real sensory neurons are driven by time dependent signals here we study how the interspike interval distribution of integrate and fire neurons depends on the evolution of the stimulus in a quasistatic limit we obtain a closed form expression for this distribution and we compare it to the one obtained with numerical simulations for several time dependent currents for slow inputs the quasistatic distribution provides a very good description of the data the results obtained for the integrate and fire model can be extended to other nonautonomous stochastic systems where the first passage time problem has an explicit solution

models of stochastic biperiodic oscillations and extended serial correlations in electroreceptors of paddlefish two types of minimal models were used to study stochastic oscillations in sensory receptors composed of two coupled oscillators as in the electroreceptors of paddlefish they have populations of cells in sensory epithelia undergoing approximately 26 hz oscillations these are coupled unidirectionally via synaptic excitation to a few afferent neurons each of whose terminal contains a 30 70 hz oscillator expressed as a dominant peak in the power spectra of spontaneous afferent firing corresponding to the mean firing rate the two distinct types of internal noisy oscillators result in stochastic biperiodic firing patterns of the primary afferent sensory neurons however the functions of the oscillations have remained elusive motivating this study the models we used here are based on the circle map or on the ermentraut koppell canonical phase theta neuron model parameters were chosen according to experimental data we used the models to demonstrate that the presence of epithelial oscillations leads to extended negative correlations of afferent interspike intervals and to show that the correlation structure depends crucially on the ratio of the afferent to epithelial oscillation frequencies being most pronounced when this ratio is close to 2 as observed in experiments our studies of stochastic versions of these models are of general interest for a wide range of coupled excitable systems especially for understanding the functional roles of noisy oscillations in auditory and other types of hair cell primary afferent sensory receptors

maximum likelihood estimation of a stochastic integrate and fire neural encoding model we examine a cascade encoding model for neural response in which a linear filtering stage is followed by a noisy leaky integrate and fire spike generation mechanism this model provides a biophysically more realistic alternative to models based on poisson memoryless spike generation and can effectively reproduce a variety of spiking behaviors seen in vivo we describe the maximum likelihood estimator for the model parameters given only extracellular spike train responses not intracellular voltage data specifically we prove that the log likelihood function is concave and thus has an essentially unique global maximum that can be found using gradient ascent techniques we develop an efficient algorithm for computing the maximum likelihood solution demonstrate the effectiveness of the resulting estimator with numerical simulations and discuss a method of testing the model s validity using time rescaling and density evolution techniques

dynamics of deterministic and stochastic paired excitatory inhibitory delayed feedback we examine the effects of paired delayed excitatory and inhibitory feedback on a single integrate and fire neuron with reversal potentials embedded within a feedback network these effects are studied using bifurcation theory and numerical analysis the feedback occurs through modulation of the excitatory and inhibitory conductances by the previous firing history of the neuron as a consequence the feedback also modifies the membrane time constant such paired feedback is ubiquitous in the nervous system we assume that the feedback dynamics are slower than the membrane time constant which leads to a rate model formulation our article provides an extensive analysis of the possible dynamical behaviors of such simple yet realistic neural loops as a function of the balance between positive and negative feedback with and without noise and offers insight into the potential behaviors such loops can exhibit in response to time varying external inputs with excitatory feedback the system can be quiescent can be periodically firing or can exhibit bistability between these two states with inhibitory feedback quiescence oscillatory firing rates and bistability between constant and oscillatory firing rate solutions are possible the general case of paired feedback exhibits a blend of the behaviors seen in the extreme cases and can produce chaotic firing we further derive a condition for a dynamically balanced paired feedback in which there is neither bistability nor oscillations we also show how a biophysically plausible smoothing of the firing function by noise can modify the existence and stability of fixed points and oscillations of the system we take advantage in our mathematical analysis of the existence of an invariant manifold which reduces the dimensionality of the dynamics and prove the stability of this manifold the novel computational challenges involved in analyzing such dynamics with and without noise are also described our results demonstrate that a paired delayed feedback loop can act as a sophisticated computational unit capable of switching between a variety of behaviors depending on the input current the relative strengths and asymmetry of the two parallel feedback pathways and the delay distributions and noise level

statistical modelling of higher order correlations in pools of neural activity abstract simultaneous recordings from multiple neural units allow us to investigate the activity of very large neural ensembles to understand how large ensembles of neurons process sensory information it is necessary to develop suitable statistical models to describe the response variability of the recorded spike trains using the information geometry framework it is possible to estimate higher order correlations by assigning one interaction parameter to each degree of correlation leading to a 2 n 1 dimensional model for a population with n neurons however this model suffers greatly from a combinatorial explosion and the number of parameters to be estimated from the available sample size constitutes the main intractability reason of this approach to quantify the extent of higher than pairwise spike correlations in pools of multiunit activity we use an information geometric approach within the framework of the extended central limit theorem considering all possible contributions from higher order spike correlations the identification of a deformation parameter allows us to provide a statistical characterisation of the amount of higher order correlations in the case of a very large neural ensemble significantly reducing the number of parameters avoiding the sampling problem and inferring the underlying dynamical properties of the network within pools of multiunit neural activity

accuracy of rate coding when shorter time window and higher spontaneous activity help it is widely accepted that neuronal firing rates contain a significant amount of information about the stimulus intensity nevertheless theoretical studies on the coding accuracy inferred from the exact spike counting distributions are rare we present an analysis based on the number of observed spikes assuming the stochastic perfect integrate and fire model with a change point representing the stimulus onset for which we calculate the corresponding fisher information to investigate the accuracy of rate coding we analyze the effect of changing the duration of the time window and the influence of several parameters of the model in particular the level of the presynaptic spontaneous activity and the level of random fluctuation of the membrane potential which can be interpreted as noise of the system the results show that the fisher information is nonmonotonic with respect to the length of the observation period this counterintuitive result is caused by the discrete nature of the count of spikes we observe also that the signal can be enhanced by noise since the fisher information is nonmonotonic with respect to the level of spontaneous activity and in some cases also with respect to the level of fluctuation of the membrane potential

differences in spiking patterns among cortical neurons spike sequences recorded from four cortical areas of an awake behaving monkey were examined to explore characteristics that vary among neurons we found that a measure of the local variation of interspike intervals lv is nearly the same for every spike sequence for any given neuron while it varies significantly among neurons the distributions of lv values for neuron ensembles in three of the four areas were found to be distinctly bimodal two groups of neurons classified according to the spiking irregularity exhibit different responses to the same stimulus this suggests that neurons in each area can be classified into different groups possessing unique spiking statistics and corresponding functional properties

localized activity profiles and storage capacity of rate based autoassociative networks we study analytically the effect of metrically structured connectivity on the behavior of autoassociative networks the steady state equations are derived for a generic input output function and then we focus on their solutions in the case of networks composed of three alternative simple rate based model neurons threshold linear binary or smoothly saturating units for a connectivity which is short range enough the threshold linear network shows localized retrieval states the saturating and binary models also exhibit spatially modulated retrieval states if the highest activity level that they can achieve is above the maximum activity of the units in the stored patterns we show that this saturation level together with the linear gain of the transfer function are important parameters that determine the possibility of localized retrieval if the ratio of the number of stored patterns to the number of connections per unit goes to zero while the latter goes to infinity it is possible to derive an analytical formula for the critical value of the connectivity width below which one observes spatially nonuniform retrieval states the formula is also shown to offer a good first approximation for higher storage loads we show that even in the case of localized retrieval the storage capacity remains proportional to the number of connections per neurons with the proportionality constant lower by a factor of 3 char21 4 compared to uniform retrieval the approach that we present here is generic in the sense that there are no specific assumptions on the single unit input output function nor on the exact connectivity structure

adaptation reduces variability of the neuronal population code sequences of events in noise driven excitable systems with slow variables often show serial correlations among their intervals of events here we employ a master equation for generalized non renewal processes to calculate the interval and count statistics of superimposed processes governed by a slow adaptation variable for an ensemble of neurons with spike frequency adaptation this results in the regularization of the population activity and an enhanced postsynaptic signal decoding we confirm our theoretical results in a population of cortical neurons recorded in vivo

calculation of interspike intervals for integrate and fire neurons with poisson distribution of synaptic inputs we present a new technique for calculating the interspike intervals of integrate and fire neurons there are two new components to this technique first the probability density of the summed potential is calculated by integrating over the distribution of arrival times of the afferent post synaptic potentials psps rather than using conventional stochastic differential equation techniques a general formulation of this technique is given in terms of the probability distribution of the inputs and the time course of the postsynaptic response the expressions are evaluated in the gaussian approximation which gives results that become more accurate for large numbers of small amplitude psps second the probability density of output spikes which are generated when the potential reaches threshold is given in terms of an integral involving a conditional probability density this expression is a generalization of the renewal equation but it holds for both leaky neurons and situations in which there is no time translational invariance the conditional probability density of the potential is calculated using the same technique of integrating over the distribution of arrival times of the afferent psps for inputs with a poisson distribution the known analytic solutions for both the perfect integrator model and the stein model which incorporates membrane potential leakage in the diffusion limit are obtained the interspike interval distribution may also be calculated numerically for models that incorporate both membrane potential leakage and a finite rise time of the postsynaptic response plots of the relationship between input and output firing rates as well as the coefficient of variation are given and inputs with varying rates and amplitudes including inhibitory inputs are analyzed the results indicate that neurons functioning near their critical threshold where the inputs are just sufficient to cause firing display a large variability in their spike timings

capacity of a single spiking neuron channel information transfer through a single neuron is a fundamental component of information processing in the brain and computing the information channel capacity is important to understand this information processing the problem is difficult since the capacity depends on coding characteristics of the communication channel and optimization over input distributions among other issues in this letter we consider two models the temporal coding model of a neuron as a communication channel assumes the output is where is a gamma distributed random variable corresponding to the interspike interval that is the time it takes for the neuron to fire once the rate coding model is similar the output is the actual rate of firing over a fixed period of time theoretical studies prove that the distribution of inputs which achieves channel capacity is a discrete distribution with finite mass points for temporal and rate coding under a reasonable assumption this allows us to compute numerically the capacity of a neuron numerical results are in a plausible range based on biological evidence to date

the product of the eigenvalues of a symmetric tensor abstract we study e eigenvalues of a symmetric tensor f of degree d on a finite dimensional euclidean vector space v and their relation with the e characteristic polynomial of f we show that the leading coefficient of the e characteristic polynomial of f when it has maximum degree is the d 2 th power respectively the d 2 2 th power when d is odd respectively when d is even of the q discriminant where q is the d th veronese embedding of the isotropic quadric q p v this fact together with a known formula for the constant term of the e characteristic polynomial of f leads to a closed formula for the product of the e eigenvalues of f which generalizes the fact that the determinant of a symmetric matrix is equal to the product of its eigenvalues

efficient neural codes that minimize lp reconstruction error the efficient coding hypothesis assumes that biological sensory systems use neural codes that are optimized to best possibly represent the stimuli that occur in their environment most common models use information theoretic measures whereas alternative formulations propose incorporating downstream decoding performance here we provide a systematic evaluation of different optimality criteria using a parametric formulation of the efficient coding problem based on the reconstruction error of the maximum likelihood decoder this parametric family includes both the information maximization criterion and squared decoding error as special cases we analytically derived the optimal tuning curve of a single neuron encoding a one dimensional stimulus with an arbitrary input distribution we show how the result can be generalized to a class of neural populations by introducing the concept of a meta tuning curve the predictions of our framework are tested against previously measured characteristics of some early visual systems found in biology we find solutions that correspond to low values of suggesting that across different animal models neural representations in the early visual pathways optimize similar criteria about natural stimuli that are relatively close to the information maximization criterion

optimal signal estimation in neuronal models we study optimal estimation of a signal in parametric neuronal models on the basis of interspike interval data fisher information is the inverse asymptotic variance of the best estimator its dependence on the parameter value indicates accuracy of estimation our models assume that the input signal is estimated from neuronal output interspike interval data where the frequency transfer function is sigmoidal if the coefficient of variation of the interspike interval is constant with respect to the signal the fisher information is unimodal and its maximum for the most estimable signal can be found we obtain a general result and compare the signal producing maximal fisher information with the inflection point of the sigmoidal transfer function in several basic neuronal models

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

analysis of integrate and fire neurons synchronization of synaptic input and spike output a new technique for analyzing the probability distribution of output spikes for the integrate and fire model is presented this technique enables us to investigate models with arbitrary synaptic response functions that incorporate both leakage across the membrane and a rise time of the postsynaptic potential the results which are compared with numerical simulations are exact in the limit of a large number of small amplitude inputs this method is applied to the synchronization problem in which we examine the relationship between the spread in arrival times of the inputs the temporal jitter of the synaptic input and the resultant spread in the times at which the output spikes are generated output jitter the results of previous studies which indicated that the ratio of the output jitter to the input jitter is consistently less than one and that it decreases for increasing numbers of inputs are confirmed for three classes of the integrate and fire model in addition to the previously identified factors of axonal propagation times and synaptic jitter we identify the variation in the spike generating thresholds of the neurons and the variation in the number of active inputs as being important factors that determine the timing jitter in layered networks previously observed phase differences between optimally and suboptimally stimulated neurons may be understood in terms of the relative time taken to reach threshold

neural population coding is optimized by discrete tuning curves the sigmoidal tuning curve that maximizes the mutual information for a poisson neuron or population of poisson neurons is obtained the optimal tuning curve is found to have a discrete structure that results in a quantization of the input signal the number of quantization levels undergoes a hierarchy of phase transitions as the length of the coding window is varied we postulate using the mammalian auditory system as an example that the presence of a subpopulation structure within a neural population is consistent with an optimal neural code

a mapping approach to rate distortion computation and analysis in rate distortion theory results are often derived and stated in terms of the optimizing density over the reproduction space in this paper the problem is reformulated in terms of the optimal mapping from the unit interval with lebesgue measure that would induce the desired reproduction probability density this results in optimality conditions that are random relatives of the known lloyd 1982 optimality conditions for deterministic quantizers the validity of the mapping approach is assured by fundamental isomorphism theorems for measure spaces we show that for the squared error distortion the optimal reproduction random variable is purely discrete at supercritical distortion where the shannon 1948 lower bound is not tight the gaussian source is thus the only source that produces continuous reproduction variables for the entire range of positive rate to analyze the evolution of the optimal reproduction distribution we use the mapping formulation and establish an analogy to statistical mechanics the solutions are given by the distribution at isothermal statistical equilibrium and are parameterized by the temperature in direct correspondence to the parametric solution of the variational equations in rate distortion theory the analysis of an annealing process shows how the number of symbols grows as the system undergoes phase transitions thus an algorithm based on the mapping approach often needs but a few variables to find the exact solution while the blahut 1972 algorithm would only approach it at the limit of infinite resolution finally a quick deterministic annealing algorithm to generate the rate distortion curve is suggested the resulting curve is exact as long as continuous phase transitions in the process are accurately followed

parameters of spike trains observed in a short time window we study the estimation of statistical moments of interspike intervals based on observation of spike counts in many independent short time windows this scenario corresponds to the situation in which a target neuron occurs it receives information from many neurons and has to respond within a short time interval the precision of the estimation procedures is examined as the model for neuronal activity two examples of stationary point processes are considered renewal process and doubly stochastic poisson process both moment and maximum likelihood estimators are investigated not only the mean but also the coefficient of variation is estimated in accordance with our expectations numerical studies confirm that the estimation of mean interspike interval is more reliable than the estimation of coefficient of variation the error of estimation increases with increasing mean interspike interval which is equivalent to decreasing the size of window less events are observed in a window and with decreasing the number of neurons lower number of windows

noise adaptation in integrate and fire neurons the statistical spiking response of an ensemble of identically prepared stochastic integrate and fire neurons to a rectangular input current plus gaussian white noise is analyzed it is shown that on average integrate and fire neurons adapt to the root mean square noise level of their input this phenomenon is referred to as noise adaptation noise adaptation is characterized by a decrease in the average neural firing rate and an accompanying decrease in the average value of the generator potential both of which can be attributed to noise induced resets of the generator potential mediated by the integrate and fire mechanism a quantitative theory of noise adaptation in stochastic integrate and fire neurons is developed it is shown that integrate and fire neurons on average produce transient spiking activity whenever there is an increase in the level of their input noise this transient noise response is either reduced or eliminated over time depending on the parameters of the model neuron analytical methods are used to prove that nonleaky integrate and fire neurons totally adapt to any constant input noise level in the sense that their asymptotic spiking rates are independent of the magnitude of their input noise for leaky integrate and fire neurons the long run noise adaptation is not total but the response to noise is partially eliminated expressions for the probability density function of the generator potential and the first two moments of the potential distribution are derived for the particular case of a nonleaky neuron driven by gaussian white noise of mean zero and constant variance the functional significance of noise adaptation for the performance of networks comprising integrate and fire neurons is discussed

discrimination with spike times and isi distributions we study the discrimination capability of spike time sequences using the chernoff distance as a metric we assume that spike sequences are generated by renewal processes and study how the chernoff distance depends on the shape of interspike interval isi distribution first we consider a lower bound to the chernoff distance because it has a simple closed form then we consider specific models of isi distributions such as the gamma inverse gaussian ig exponential with refractory period er and that of the leaky integrate and fire lif neuron we found that the discrimination capability of spike times strongly depends on high order moments of isi and that it is higher when the spike time sequence has a larger skewness and a smaller kurtosis high variability in terms of coefficient of variation cv does not necessarily mean that the spike times have less discrimination capability spike sequences generated by the gamma distribution have the minimum discrimination capability for a given mean and variance of isi we used series expansions to calculate the mean and variance of isis for lif neurons as a function of the mean input level and the input noise variance spike sequences from an lif neuron are more capable of discrimination than those of ig and gamma distributions when the stationary voltage level is close to the neuron s threshold value of the neuron

how sample paths of leaky integrate and fire models are influenced by the presence of a firing threshold neural membrane potential data are necessarily conditional on observation being prior to a firing time in a stochastic leaky integrate and fire model this corresponds to conditioning the process on not crossing a boundary in the literature simulation and estimation have almost always been done using unconditioned processes in this letter we determine the stochastic differential equations of a diffusion process conditioned to stay below a level s up to a fixed time t1 and of a diffusion process conditioned to cross the boundary for the first time at t1 this allows simulation of sample paths and identification of the corresponding mean process differences between the mean of free and conditioned processes are illustrated as well as the role of noise in increasing these differences

conditions for uniqueness of product representations for separable quantum channels and separable quantum states we give a sufficient condition that an operator sum representation of a separable quantum channel in terms of product operators is the unique product representation for that channel and then provide examples of such channels for any number of parties this result has implications for efforts to determine whether or not a given separable channel can be exactly implemented by local operations and classical communication by the choi jamiolkowski isomorphism it also translates to a condition for the uniqueness of product state ensembles representing a given quantum state these ideas follow from considerations concerning whether or not a subspace spanned by a given set of product operators contains at least one additional product operator

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

mutual information expansion for studying the role of correlations in population codes how important are autocorrelations the role of correlations in the activity of neural populations responding to a set of stimuli can be studied within an information theory framework regardless of whether one approaches the problem from an encoding or decoding perspective the main measures used to study the role of correlations can be derived from a common source the expansion of the mutual information two main formalisms of mutual information expansion have been proposed the series expansion and the exact breakdown here we clarify that these two formalisms have a different representation of autocorrelations so that even when the total information estimated differs by less than 1 individual terms can diverge more precisely the series expansion explicitly evaluates the informational contribution of autocorrelations in the count of spikes that is count autocorrelations whereas the exact breakdown does not we propose a new formalism of mutual information expansion the poisson exact breakdown which introduces poisson equivalents in order to explicitly evaluate the informational contribution of count autocorrelations with no approximation involved because several widely employed manipulations of spike trains most notably binning and pooling alter the structure of count autocorrelations the new formalism can provide a useful general framework for studying the role of correlations in population codes

operator schmidt ranks of bipartite unitary matrices abstract the operator schmidt rank of an operator acting on the tensor product c n c m is the number of terms in a decomposition of the operator as a sum of simple tensors with factors forming orthogonal families in their respective matrix algebras it has been known that for unitary operators acting on two copies of c 2 the operator schmidt rank can only take the values 1 2 and 4 the value 3 being forbidden in this paper we settle an open question showing that the above obstruction is the only one occurring we do so by constructing explicit examples of bipartite unitary operators of all possible operator schmidt ranks for arbitrary dimensions n m 2 2

effects of synaptic depression and adaptation on spatiotemporal dynamics of an excitatory neuronal network we analyze the spatiotemporal dynamics of a system of integro differential equations that describes a one dimensional excitatory neuronal network with synaptic depression and spike frequency adaptation physiologically suggestive forms are used for both types of negative feedback we also consider the effects of employing two different types of firing rate function a heaviside step function and a piecewise linear function we first derive conditions for the existence of traveling fronts and pulses in the case of a heaviside step firing rate and show that adaptation plays a relatively minor role in determining the characteristics of traveling waves we then derive conditions for the existence and stability of stationary pulses or bumps and show that a purely excitatory network with synaptic depression cannot support stable bumps however bumps do not exist in the presence of adaptation finally in the case of a piecewise linear firing rate function we show numerically that the network also supports self sustained oscillations between an up state and a down state in which a spatially localized oscillating core periodically emits pulses at each cycle

neural dynamics bifurcations and firing rates in a quadratic integrate and fire model with a recovery variable i deterministic behavior we study the dynamics of a quadratic integrate and fire model of a single compartment neuron with a slow recovery variable as input current and parameters describing timescales recovery variable and postspike reset change analysis of a codimension 2 bifurcation reveals that the domain of attraction of a stable hyperpolarized rest state interacts subtly with reset parameters which reposition the system state after spiking we obtain explicit approximations of instantaneous firing rates for fixed values of the recovery variable and use the averaging theorem to obtain asymptotic firing rates as a function of current and reset parameters along with the different phase plane geometries these computations provide explicit tools for the interpretation of different spiking patterns and guide parameter selection in modeling different cortical cell types

time structure of the activity in neural network models several neural network models in continuous time are reconsidered in the framework of a general mean field theory which is exact in the limit of a large and fully connected network the theory assumes pointlike spikes which are generated by a renewal process the effect of spikes on a receiving neuron is described by a linear response kernel which is the dominant term in a weak coupling expansion it is shown that the resulting spike response model is the most general renewal model with linear inputs the standard integrate and fire model forms a special case in a network structure with several pools of identical spiking neurons the global states and the dynamic evolution are determined by a nonlinear integral equation which describes the effective interaction within and between different pools we derive explicit stability criteria for stationary incoherent and oscillatory coherent solutions it is shown that the stationary state of noiseless systems is almost always unstable noise suppresses fast oscillations and stabilizes the system furthermore collective oscillations are stable only if the firing occurs while the synaptic potential is increasing in particular collective oscillations in a network with delayless excitatory interaction are at most semistable inhibitory interactions with short delays or excitatory interactions with long delays lead to stable oscillations our general results allow a straightforward application to different network models with spiking neurons furthermore the theory allows an estimation of the errors introduced in firing rate or graded response models

eigenvalues and invariants of tensors abstract a tensor is represented by a supermatrix under a co ordinate system in this paper we define e eigenvalues and e eigenvectors for tensors and supermatrices by the resultant theory we define the e characteristic polynomial of a tensor an e eigenvalue of a tensor is a root of the e characteristic polynomial in the regular case a complex number is an e eigenvalue if and only if it is a root of the e characteristic polynomial we convert the e characteristic polynomial of a tensor to a monic polynomial and show that the coefficients of that monic polynomial are invariants of that tensor i e they are invariant under co ordinate system changes we call them principal invariants of that tensor the maximum number of principal invariants of mth order n dimensional tensors is a function of m and n we denote it by d m n and show that d 1 n 1 d 2 n n d m 2 m for m 3 and d m n m n 1 m for m n 3 we also define the rank of a tensor all real eigenvectors associated with nonzero e eigenvalues are in a subspace with dimension equal to its rank

suprathreshold stochastic resonance in neural processing tuned by correlation suprathreshold stochastic resonance ssr is examined in the context of integrate and fire neurons with an emphasis on the role of correlation in the neuronal firing we employed a model based on a network of spiking neurons which received synaptic inputs modeled by poisson processes stimulated by a stepped input signal the smoothed ensemble firing rate provided an output signal and the mutual information between this signal and the input was calculated for networks with different noise levels and different numbers of neurons it was found that an ssr effect was present in this context we then examined a more biophysically plausible scenario where the noise was not controlled directly but instead was tuned by the correlation between the inputs the ssr effect remained present in this scenario with nonzero noise providing improved information transmission and it was found that negative correlation between the inputs was optimal finally an examination of ssr in the context of this model revealed its connection with more traditional stochastic resonance and showed a trade off between supratheshold and subthreshold components we discuss these results in the context of existing empirical evidence concerning correlations in neuronal firing

estimating instantaneous irregularity of neuronal firing cortical neurons in vivo had been regarded as poisson spike generators that convey no information other than the rate of random firing recently using a metric for analyzing local variation of interspike intervals researchers have found that individual neurons express specific patterns in generating spikes which may symbolically be termed regular random or bursty rather invariantly in time in order to study the dynamics of firing patterns in greater detail we propose here a bayesian method for estimating firing irregularity and the firing rate simultaneously for a given spike sequence and we implement an algorithm that may render the empirical bayesian estimation practicable for data comprising a large number of spikes application of this method to electrophysiological data revealed a subtle correlation between the degree of firing irregularity and the firing rate for individual neurons irregularity of firing did not deviate greatly around the low degree of dependence on the firing rate and remained practically unchanged for individual neurons in the cortical areas v1 and mt whereas it fluctuated greatly in the lateral geniculate nucleus of the thalamus this indicates the presence and absence of autocontrolling mechanisms for maintaining patterns of firing in the cortex and thalamus respectively

analytic expressions for rate and cv of a type i neuron driven by white gaussian noise we study the one dimensional normal form of a saddle node system under the influence of additive gaussian white noise and a static bias current input parameter a model that can be looked upon as the simplest version of a type i neuron with stochastic input this is in contrast with the numerous studies devoted to the noise driven leaky integrate and fire neuron we focus on the firing rate and coefficient of variation cv of the interspike interval density for which scaling relations with respect to the input parameter and noise intensity are derived quadrature formulas for rate and cv are numerically evaluated and compared to numerical simulations of the system and to various approximation formulas obtained in different limiting cases of the model we also show that caution must be used to extend these results to the neuron model with multiplicative gaussian white noise the correspondence between the first passage time statistics for the saddle node model and the neuron model is obtained only in the stratonovich interpretation of the stochastic neuron model while previous results have focused only on the ito interpretation the correct stratonovich interpretation yields cvs that are still relatively high although smaller than in the ito interpretation it also produces certain qualitative differences especially at larger noise intensities our analysis provides useful relations for assessing the distance to threshold and the level of synaptic noise in real type i neurons from their firing statistics we also briefly discuss the effect of finite boundaries finite values of threshold and reset on the firing statistics

measures of statistical dispersion based on shannon and fisher information concepts we propose and discuss two information based measures of statistical dispersion of positive continuous random variables the entropy based dispersion and fisher information based dispersion although standard deviation is the most frequently employed dispersion measure we show that it is not well suited to quantify some aspects that are often expected intuitively such as the degree of randomness the proposed dispersion measures are not entirely independent though each describes the quality of probability distribution from a different point of view we discuss relationships between the measures describe their extremal values and illustrate their properties on the pareto the lognormal and the lognormal mixture distributions application possibilities are also mentioned

chaotic balanced state in a model of cortical circuits the nature and origin of the temporal irregularity in the electrical activity of cortical neurons in vivo are not well understood we consider the hypothesis that this irregularity is due to a balance of excitatory and inhibitory currents into the cortical cells we study a network model with excitatory and inhibitory populations of simple binary units the internal feedback is mediated by relatively large synaptic strengths so that the magnitude of the total excitatory and inhibitory feedback is much larger than the neuronal threshold the connectivity is random and sparse the mean number of connections per unit is large though small compared to the total number of cells in the network the network also receives a large temporally regular input from external sources we present an analytical solution of the mean field theory of this model which is exact in the limit of large network size this theory reveals a new cooperative stationary state of large networks which we term a balanced state in thi

estimating spiking irregularities under changing environments we considered a gammadistribution of interspike intervals as a statistical model for neuronal spike generation a gamma distribution is a natural extension of the poisson process taking the effect of a refractory period into account the model is specified by two parameters a time dependent firing rate and a shape parameter that characterizes spiking irregularities of individual neurons because the environment changes over time observed data are generated from a model with a time dependent firing rate which is an unknown function a statistical model with an unknown function is called a semiparametric model and is generally very difficult to solve we used a novel method of estimating functions in information geometry to estimate the shape parameter without estimating the unknown function we obtained an optimal estimating function analytically for the shape parameter independent of the functional form of the firing rate this estimation is efficient without fisher information loss and better than maximum likelihood estimation we suggest a measure of spiking irregularity based on the estimating function which may be useful for characterizing individual neurons in changing environments

parameters of stochastic diffusion processes estimated from observations of first hitting times application to the leaky integrate and fire neuronal model a theoretical model has to stand the test against the real world to be of any practical use the first step is to identify parameters in the model estimated from experimental data in many applications where renewal point data are available models of first hitting times of underlying diffusion processes arise despite the seemingly simplicity of the model the problem of how to estimate parameters of the underlying stochastic process has resisted solution the few attempts have either been unreliable difficult to implement or only valid in subsets of the relevant parameter space here we present an estimation method that overcomes these difficulties is computationally easy and fast to implement and also works surprisingly well on small data sets the method is illustrated on simulated and experimental data two common neuronal models the ornstein uhlenbeck and feller models are investigated

the number of eigenvalues of a tensor abstract eigenvectors of tensors as studied recently in numerical multilinear algebra correspond to fixed points of self maps of a projective space we determine the number of eigenvectors and eigenvalues of a generic tensor and we show that the number of normalized eigenvalues of a symmetric tensor is always finite we also examine the characteristic polynomial and how its coefficients are related to discriminants and resultants

spike train statistics and dynamics with synaptic input from any renewal process a population density approach in the probability density function pdf approach to neural network modeling a common simplifying assumption is that the arrival times of elementary postsynaptic events are governed by a poisson process this assumption ignores temporal correlations in the input that sometimes have important physiological consequences we extend pdf methods to models with synaptic event times governed by any modulated renewal process we focus on the integrate and fire neuron with instantaneous synaptic kinetics and a random elementary excitatory postsynaptic potential epsp a between presynaptic events the membrane voltage v decays exponentially toward rest while s the time since the last synaptic input event evolves with unit velocity when a synaptic event arrives v jumps by a and s is reset to zero if v crosses the threshold voltage an action potential occurs and v is reset to vreset the probability per unit time of a synaptic event at time t given the elapsed time s since the last event h s t depends on specifics of the renewal process we study how regularity of the train of synaptic input events affects output spike rate pdf and coefficient of variation cv of the interspike interval and the autocorrelation function of the output spike train in the limit of a deterministic clocklike train of input events the pdf of the interspike interval converges to a sum of delta functions with coefficients determined by the pdf for a the limiting autocorrelation function of the output spike train is a sum of delta functions whose coefficients fall under a damped oscillatory envelope when the epsp cv a a is equal to 0 45 a cv for the intersynaptic event interval t t 0 35 is functionally equivalent to a deterministic periodic train of synaptic input events cv 0 with respect to spike statistics we discuss the relevance to neural network simulations

an adjustment to the time rescaling method for application to short trial spike train data it is important to validate models of neural data using appropriate goodness of fit measures models summarizing some response features for example spike count distributions or peristimulus time histograms can be assessed using standard statistical tools measuring the fit of a full point process model of spike trains is more difficult recently barbieri quirk frank wilson and brown 2001 and brown barbieri ventura kass and frank 2002 presented a method for rescaling time so that if an underlying description correctly describes the conditional intensity function of a point process the rescaling will convert the process into a homogeneous poisson process the corresponding interevent intervals are exponential with mean 1 and can be transformed to be uniform tests of the uniformity of the transformed intervals are thus tests of how well the model fits the data when the lengths of interevent intervals are comparable to the length of the observation window as can happen in common neurophysiology paradigms using short trials the fact that long intervals cannot be observed are censored can cause the tests based on time rescaling to reject a correct model inappropriately this article presents a simple adjustment to the time rescaling method to account for interval censoring avoiding inappropriate rejection of acceptable models for short trial data we illustrate the adjustment s effect using both simulated data and short trial data from monkey primary visual cortex

on firing rate estimation for dependent interspike intervals if interspike intervals are dependent the instantaneous firing rate does not catch important features of spike trains in this case the conditional instantaneous rate plays the role of the instantaneous firing rate for the case of samples of independent interspike intervals if the conditional distribution of the interspikes intervals isis is unknown it becomes difficult to evaluate the conditional firing rate we propose a nonparametric estimator for the conditional instantaneous firing rate for markov stationary and ergodic isis an algorithm to check the reliability of the proposed estimator is introduced and its consistency properties are proved the method is applied to data obtained from a stochastic two compartment model and to in vitro experimental data

operator schmidt decompositions and the fourier transform with applications to the operator schmidt numbers of unitaries the operator schmidt decomposition is useful in quantum information theory for quantifying the nonlocality of bipartite unitary operations we construct a family of unitary operators on n n whose operator schmidt decompositions are computed using the discrete fourier transform as a corollary we produce unitaries on 3 3 with operator schmidt number s for every s 1 9 this corollary was unexpected since it contradicted reasonable conjectures of nielsen et al 2003 phys rev a 67 052301 based on intuition from a striking result in the two qubit case by the results of dur et al 2002 phys rev lett 89 057901 who also considered the two qubit case our result implies that there are nine equivalence classes of unitaries on 3 3 which are probabilistically interconvertible by stochastic local operations and classical communication as another corollary a prescription is produced for constructing maximally entangled unitaries from biunimodular functions reversing tact we state a generalized operator schmidt decomposition of the quantum fourier transform considered as an operator m1 m2 n1 n2 with m1m2 n1n2 this decomposition shows by nielsen s bound that the communication cost of the qft remains maximal when a net transfer of qudits is permitted in an appendix a canonical procedure is given for removing basis dependence for results and proofs depending on the magic basis introduced in s hill and w wootters 1997 entanglement of a pair of quantum bits phys rev lett 78 5022 5

neuronal tuning to sharpen or broaden sensory and motor variables are typically represented by a population of broadly tuned neurons a coarser representation with broader tuning can often improve coding accuracy but sometimes the accuracy may also improve with sharper tuning the theoretical analysis here shows that the relationship between tuning width and accuracy depends crucially on the dimension of the encoded variable a general rule is derived for how the fisher information scales with the tuning width regardless of the exact shape of the tuning function the probability distribution of spikes and allowing some correlated noise between neurons these results demonstrate a universal dimensionality effect in neural population coding

statistical analysis of stochastic resonance in a simple setting a subthreshold signal may be detected if noise is added to the data we study a simple model consisting of a constant signal to which at uniformly spaced times independent and identically distributed noise variables with known distribution are added a detector records the times at which the noisy signal exceeds a threshold there is an optimal noise level called stochastic resonance we explore the detectability of the signal in a system with one or more detectors with different thresholds we use a statistical detectability measure the asymptotic variance of the best estimator of the signal from the thresholded data or equivalently the fisher information in the data in particular we determine optimal configurations of detectors varying the distances between the thresholds and the signal as well as the noise level the approach generalizes to nonconstant signals

maximally informative stimuli and tuning curves for sigmoidal rate coding neurons and populations a general method for deriving maximally informative sigmoidal tuning curves for neural systems with small normalized variability is presented the optimal tuning curve is a nonlinear function of the cumulative distribution function of the stimulus and depends on the mean variance relationship of the neural system the derivation is based on a known relationship between shannon s mutual information and fisher information and the optimality of jeffrey s prior it relies on the existence of closed form solutions to the converse problem of optimizing the stimulus distribution for a given tuning curve it is shown that maximum mutual information corresponds to constant fisher information only if the stimulus is uniformly distributed as an example the case of sub poisson binomial firing statistics is analyzed in detail

dynamics of membrane excitability determine interspike interval variability a link between spike generation mechanisms and cortical spike train statistics we propose a biophysical mechanism for the high interspike interval variability observed in cortical spike trains the key lies in the nonlinear dynamics of cortical spike generation which are consistent with type i membranes where saddle node dynamics underlie excitability rinzel ermentrout 1989 we present a canonical model for type i membranes the neuron the neuron is a phase model whose dynamics reflect salient features of type i membranes this model generates spike trains with coefficient of variation cv above 0 6 when brought to firing by noisy inputs this happens because the timing of spikes for a type i excitable cell is exquisitely sensitive to the amplitude of the suprathreshold stimulus pulses a noisy input current giving random amplitude kicks to the cell evokes highly irregular firing across a wide range of firing rates an intrinsically oscillating cell gives regular spike trains we corroborate the results with simulations of the morris lecar m l neural model with random synaptic inputs type i m l yields high cvs when this model is modified to have type ii dynamics periodicity arises via a hopf bifurcation however it gives regular spike trains cv below 0 3 our results suggest that the high cv values such as those observed in cortical spike trains are an intrinsic characteristic of type i membranes driven to firing by random inputs in contrast neural oscillators or neurons exhibiting type ii excitability should produce regular spike trains

two computational regimes of a single compartment neuron separated by a planar boundary in conductance space recent in vitro data show that neurons respond to input variance with varying sensitivities here we demonstrate that hodgkin huxley hh neurons can operate in two computational regimes one that is more sensitive to input variance differentiating and one that is less sensitive integrating a boundary plane in the 3d conductance space separates these two regimes for a reduced hh model this plane can be derived analytically from the v nullcline thus suggesting a means of relating biophysical parameters to neural computation by analyzing the neuron s dynamical system

optimal short term population coding when fisher information fails efficient coding has been proposed as a first principle explaining neuronal response properties in the central nervous system the shape of optimal codes however strongly depends on the natural limitations of the particular physical system here we investigate how optimal neuronal encoding strategies are influenced by the finite number of neurons n place constraint the limited decoding time window length t time constraint the maximum neuronal firing rate fmax power constraint and the maximal average rate f max energy constraint while fisher information provides a general lower bound for the mean squared error of unbiased signal reconstruction its use to characterize the coding precision is limited analyzing simple examples we illustrate some typical pitfalls and thereby show that fisher information provides a valid measure for the precision of a code only if the dynamic range fmint fmaxt is sufficiently large in particular we demonstrate that the optimal width of gaussian tuning curves depends on the available decoding time t within the broader class of unimodal tuning functions it turns out that the shape of a fisher optimal coding scheme is not unique we solve this ambiguity by taking the minimum mean square error into account which leads to fiat tuning curves the tuning width however remains to be determined by energy constraints rather than by the principle of efficient coding

gaussian process approach to spiking neurons for inhomogeneous poisson inputs this article presents a new theoretical framework to consider the dynamics of a stochastic spiking neuron model with general membrane response to input spike we assume that the input spikes obey an inhomogeneous poisson process the stochastic process of the membrane potential then becomes a gaussian process when a general type of the membrane response is assumed the stochastic process becomes a markov gaussian process we present a calculation method for the membrane potential density and the firing probability density our new formulation is the extension of the existing formulation based on diffusion approximation although the single markov assumption of the diffusion approximation simplifies the stochastic process analysis the calculation is inaccurate when the stochastic process involves a multiple markov property we find that the variation of the shape of the membrane response which has often been ignored in existing stochastic process studies significantly affects the firing probability our approach can consider the reset effect which has been difficult to deal with by analysis based on the first passage time density

what causes a neuron to spike the computation performed by a neuron can be formulated as a combination of dimensional reduction in stimulus space and the nonlinearity inherent in a spiking output white noise stimulus and reverse correlation the spike triggered average and spike triggered covariance are often used in experimental neuroscience to ask neurons which dimensions in stimulus space they are sensitive to and to characterize the nonlinearity of the response in this article we apply reverse correlation to the simplest model neuron with temporal dynamics the leaky integrate and fire model and find that for even this simple case standard techniques do not recover the known neural computation to overcome this we develop novel reverse correlation techniques by selectively analyzing only isolated spikes and taking explicit account of the extended silences that precede these isolated spikes we discuss the implications of our methods to the characterization of neural adaptation although these methods are developed in the context of the leaky integrate and fire model our findings are relevant for the analysis of spike trains from real neurons

the degree of the e characteristic polynomial of an even order tensor the e characteristic polynomial of an even order supersymmetric tensor is a useful tool in determining the positive definiteness of an even degree multivariate form in this paper for an even order tensor we first establish the formula of its e characteristic polynomial by using the classical macaulay formula of resultants then give an upper bound for the degree of that e characteristic polynomial examples illustrate that this bound is attainable in some low order and dimensional cases