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

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

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

examples of factorization without bounded approximate units examples are given of banach algebras which do not have bounded approximate units but in which every element is a product another algebra is constructed in which there are elements which are not products but every element is the sum of two products most of the examples are commutative and separable these examples suggest that there may be a connection between factorization questions and the topology of the carrier space of a banach algebra

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

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

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

functional evolution equations abstract of concern is the functional evolution equation du dt bu f t u t t 0 u 0 s c 1 r 0 x where ut s u t s r 0 s r 0 and x is a real banach space it is shown by the method of lines combined with the crandall pazy theorem that for the initial data u0 in a generalized domain this equation has a limit solution which is lipschitz continuous in t and that this limit solution is a unique strong one if further assumptions on b are imposed and that the zero solution is asymptotically stable an application is given to a class of partial functional differential boundary value problems

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

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

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

nachbin spaces as cq algebras abstract we introduce a new class of commutative c q algebras whose topologies generalize the uniform and the strict topology we consider multipliers of these modules and describe their submodule structure a key role is played by nachbin spaces

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

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

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

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

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

on the existence of solutions for functional evolution equations soit x un espace de banach de norme soit pc l espace des fonctions continues par morceaux r 0 x pour un r fixe 0 pour pc on pose pc sup r 0 on considere l equation d evolution fonctionnelle non lineaire abstraite g t u t u t a t u t u 0 t 0 t

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

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

c segal algebras with order unit abstract we introduce the notion of a noncommutative c segal algebra as a banach algebra a a which is a dense ideal in a c algebra c c where a is strictly stronger than c on a several basic properties are investigated and with the aid of the theory of multiplier modules the structure of c segal algebras with order unit is determined

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

a method of lines for a nonlinear abstract functional evolution equation let x be a real banach space with x uniformly convex a method of lines is introduced and developed for the abstract functional problem lipschitzian like function in its two variables further conditions are given for the convergence of the method to a strong solution of e recent results for perturbed abstract ordinary equations are substantially improved the method applies also to large classes of functional parabolic problems as well as problems of integral perturbations the method is straightforward because it avoids the introduction of the operators a t and the corresponding use of nonlinear evolution operator theory 1 introduction preliminaries let x be a real banach space with norm 11 let