computation in a single neuron hodgkin and huxley revisited a spiking neuron computes by transforming a complex dynamical input into a train of action potentials or spikes the computation performed by the neuron can be formulated as dimensional reduction or feature detection followed by a nonlinear decision function over the low dimensional space generalizations of the reverse correlation technique with white noise input provide a numerical strategy for extracting the relevant low dimensional features from experimental data and information theory can be used to evaluate the quality of the low dimensional approximation we apply these methods to analyze the simplest biophysically realistic model neuron the hodgkin huxley hh model using this system to illustrate the general methodological issues we focus on the features in the stimulus that trigger a spike explicitly eliminating the effects of interactions between spikes one can approximate this triggering feature space as a two dimensional linear subspace in the high dimensional space of input histories capturing in this way a substantial fraction of the mutual information between inputs and spike time we find that an even better approximation however is to describe the relevant subspace as two dimensional but curved in this way we can capture 90 of the mutual information even at high time resolution our analysis provides a new understanding of the computational properties of the hh model while it is common to approximate neural behavior as integrate and fire the hh model is not an integrator nor is it well described by a single threshold

from spiking neurons to rate models a cascade model as an approximation to spiking neuron models with refractoriness a neuron that is stimulated repeatedly by the same time dependent stimulus exhibits slightly different spike timing at each trial we compared the exact solution of the time dependent firing rate for a stochastically spiking neuron model with refractoriness spike response model with that of an inhomogeneous poisson process subject to the same stimulus to arrive at a mapping between the two models we used alternatively i a systematic parameter free volterra expansion of the exact solution or ii a linear filter combined with nonlinear poisson rate model linear nonlinear poisson cascade model with a single free parameter both the cascade model and the second order volterra model showed excellent agreement with the exact rate dynamics of the spiking neuron model with refractoriness even for strong and rapidly changing input cascade rate models are widely used in systems neuroscience our method could help to connect experimental rate measurements to the theory of spiking neurons

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

estimating parameters of generalized integrate and fire neurons from the maximum likelihood of spike trains when a neuronal spike train is observed what can we deduce from it about the properties of the neuron that generated it a natural way to answer this question is to make an assumption about the type of neuron select an appropriate model for this type and then choose the model parameters as those that are most likely to generate the observed spike train this is the maximum likelihood method if the neuron obeys simple integrate and fire dynamics paninski pillow and simoncelli 2004 showed that its negative log likelihood function is convex and that at least in principle its unique global minimum can thus be found by gradient descent techniques many biological neurons are however known to generate a richer repertoire of spiking behaviors than can be explained in a simple integrate and fire model for instance such a model retains only an implicit through spike induced currents not an explicit memory of its input an example of a physiological situation that cannot be explained is the absence of firing if the input current is increased very slowly therefore we use an expanded model mihalas niebur 2009 which is capable of generating a large number of complex firing patterns while still being linear linearity is important because it maintains the distribution of the random variables and still allows maximum likelihood methods to be used in this study we show that although convexity of the negative log likelihood function is not guaranteed for this model the minimum of this function yields a good estimate for the model parameters in particular if the noise level is treated as a free parameter furthermore we show that a nonlinear function minimization method r algorithm with space dilation usually reaches the global minimum

analytical investigation of the effects of lateral connections on the accuracy of population coding we studied how lateral connections affect the accuracy of a population code by using a model of orientation selectivity in the primary visual cortex investigating the effects of lateral connections on population coding is a complex problem because these connections simultaneously change the shape of tuning curves and correlations between neurons both of these changes caused by lateral connections have to be taken into consideration to correctly evaluate their effects we propose a theoretical framework for analytically computing the fisher information which measures the accuracy of a population code in stochastic spiking neuron models with refractory periods within our framework we accurately evaluated both the changes in tuning curves and correlations caused by lateral connections and their effects on the fisher information we found that their effects conflicted with each other and the answer to whether or not the lateral connections increased the fisher information strongly depended on the intrinsic properties of the model neuron by systematically changing the coupling strengths of excitations and inhibitions we found the parameter regions of lateral connectivities where sharpening of tuning curves through mexican hat connectivities led to an increase in information which is in contrast to some previous findings

population dynamics of spiking neurons fast transients asynchronous states and locking an integral equation describing the time evolution of the population activity in a homogeneous pool of spiking neurons of the integrate and fire type is discussed it is analytically shown that transients from a state of incoherent firing can be immediate the stability of incoherent firing is analyzed in terms of the noise level and transmission delay and a bifurcation diagram is derived the response of a population of noisy integrate and fire neurons to an input current of small amplitude is calculated and characterized by a linear filter l the stability of perfectly synchronized locked solutions is analyzed

the solution of groundwater quality management problems with a nonconvex feasible region using a cutting plane optimization technique in groundwater quality management problems the concentration constraints have a nonlinear behavior which may be described either as a convex or a nonconvex function therefore the feasible region which is defined as the intersection of all of these constraints can be either a convex or a nonconvex set a review of existing optimization algorithms for the solution of the groundwater quality management problem indicates that the majority of them have an inability to determine a global optimum when nonconvexity occurs in an earlier paper that appeared in this journal karatzas and finder 1993 the outer approximation method a global optimization technique was presented for the solution of groundwater management problems with convex constraints the problem was formulated to minimize a concave objective function over a compact convex set of constraints in the present study the same concept is applied to problems with nonconvex constraints while the main concept of the current approach remains the same as that in our earlier study there is a significant difference in the determination of the cutting hyperplane the nonconvexity of the domain requires a special approach to insure that the introduction of the cutting hyperplane does not eliminate any part of the nonconvex feasible region in this work the theory of the developed algorithm is presented and subsequently applied to two groundwater quality problems in the first example a hypothetical aquifer is considered to illustrate the performance of the methodology in the second example a groundwater quality management problem in woburn massachusetts is solved results obtained are compared with those generated by minos 5 1

variational discretization of axisymmetric curvature flows we present natural axisymmetric variants of schemes for curvature flows introduced earlier by the present authors and analyze them in detail although numerical methods for geometric flows have been used frequently in axisymmetric settings numerical analysis results so far are rare in this paper we present stability equidistribution existence and uniqueness results for the introduced approximations numerical computations show that these schemes are very efficient in computing numerical solutions of geometric flows as only a spatially one dimensional problem has to be solved the good mesh properties of the schemes also allow them to compute in very complex axisymmetric geometries

enhancement of the outer approximation method for the solution of concentration constrained optimal design groundwater remediation problems 1 nonlinearity and nonconvexity are two major characteristics of groundwater quality management models the classical solutions of such problems require enormous computational effort without ensuring a global optimum to circumvent this problem the outer approximation method a global optimization technique was introduced to solve groundwater quality management problems characterized by a nonconvex objective function with minima at the boundary of the feasible region and constraints that have convex or nonconvex behavior in this paper a more sophisticated and computationally efficient approach for the case of the nonconvex constraints is presented to illustrate the efficacy and efficiency of this new approach a hypothetical and a field scale problem are considered

stable propagation of activity pulses in populations of spiking neurons we investigate the propagation of pulses of spike activity in a neuronal network with feedforward couplings the neurons are of the spike response type with a firing probability that depends linearly on the membrane potential after firing neurons enter a phase of refractoriness spike packets are described in terms of the moments of the firing time distribution so as to allow for an analytical treatment of the evolution of the spike packet as it propagates from one layer to the next analytical results and simulations show that depending on the synaptic coupling strength a stable propagation of the packet with constant waveform is possible crucial for this observation is neither the existence of a firing threshold nor a sigmoidal gain function both are absent in our model but the refractory behavior of the neurons

estimation of the input parameters in the feller neuronal model the stochastic feller neuronal model is studied and estimators of the model input parameters depending on the firing regime of the process are derived closed expressions for the first two moments of functionals of the first passage time ftp through a constant boundary in the suprathreshold regime are derived which are used to calculate moment estimators in the subthreshold regime the exponentiality of the ftp is utilized to characterize the input parameters the methods are illustrated on simulated data finally approximations of the first passage time moments are suggested and biological interpretations and comparisons of the parameters in the feller and the ornstein uhlenbeck models are discussed

a primal method for the solution of the groundwater quality management problem a new algorithm for the solution of the groundwater quality management problem is presented the method assumes that most of the computational effort in such problems involves evaluation of the concentrations and their derivatives with respect to the pumping rates at the control points the methodology proposed herein is a combination of the cutting plane method and the primal method in this approach each line search moves from a feasible point towards the solution of a subproblem with linear constraints and the same objective function as the original problem sensitivity analysis is used to compute the derivative of a single concentration with respect to all the pumping rates by solving only one linear system at each time step the method has been tested with analytical convex problems and a model example a comparison of the algorithm s performance was conducted using minos 5 1 for the case of a linear cost function the method shows a computational growth rate almost linearly proportional to the number of decision variables

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

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

switch of encoding characteristics in single neurons by subthreshold and suprathreshold stimuli spike triggered analysis is a statistical method used to elucidate encoding properties in neural systems by estimating the statistical structure of input stimulus preceding spikes a recent numerical study suggested that the profile of the spike triggered average sta changes depending on whether the mean input stimuli are subthreshold or suprathreshold here we analytically verify the difference between subthreshold sta and suprathreshold sta by using the spike response model srm we show by moment expansion that the suprathreshold sta is proportional to the first derivative of the response kernel and that the subthreshold sta is expressed by a linear combination of the response kernel and its first derivative we verify whether the analytical results obtained from the srm can be applied to a multicompartment model with hodgkin huxley type dynamics

optimal control of neuronal activity we investigate the optimal control of neuronal spiking activity for neurons receiving a class of random synaptic inputs characterized by a positive parameter alpha optimal control signals and optimal variances are found exactly for the diffusion process approximating an integrate and fire model when synaptic inputs are sub poisson alpha 0 5 inputs the optimal control is smooth and unique the optimal variance obtained in the current paper sets the lowest possible bound in controlling the stochasticity of neuronal activity we also discuss how to implement the optimal control signal for certain model neurons

piecewise linear differential equations and integrate and fire neurons insights from two dimensional membrane models we derive and study two dimensional generalizations of integrate and fire models which can be found from a piecewise linear idealization of the fitzhugh nagumo or morris lecar model these models give rise to new properties not present in one dimensional integrate and fire models a detailed analytical study of the models is presented in particular i for the piecewise linear fitzhugh nagumo model we determine analytically the bistability regime between stationary solutions and oscillations that is typical for class ii models ii in the piecewise morris lecar model we find a noncanonical class i transition from a stationary state to oscillations with logarithmic dependence similar to that found for leaky integrate and fire models iii furthermore we establish a relation to the recently proposed resonate and fire model and show that a short input current pulse can trigger several spikes

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

groundwater management using numerical simulation and the outer approximation method for global optimization groundwater quantity management problems with fixed charges have been formulated in the past as mixed integer and linear programming problems in this paper a new methodology is presented where the fixed charges are incorporated into the objective function in an exponential form and the problem is solved as a concave minimization problem the principal difficulty in the minimization of a concave function over a linear or nonlinear set of constraints is that the local minima which are determined by the classical minimization algorithms may not be global in an effort to circumvent this problem the outer approximation method is introduced this method is applicable to the global minimization of a concave function over a compact set of constraints in the present work the outer approximation is applied to concave minimization problems over a convex compact set of constraints two applications of the method to groundwater management problems are presented herein and the results are compared with an existing solution obtained using a different optimization approach

development of objective oriented groundwater models 2 robust experimental design 1 this paper continues the discussion in part 1 by considering the data collection strategy problem when the existing data are judged to be insufficient for constructing a reliable model designing an experiment for identifying a distributed parameter is very difficult because the identification of a more complex parameter structure requires more data moreover without knowing the sufficiency of a design finding an optimal design becomes meaningless these difficulties can be avoided if we turn to the construction of objective oriented models the identifiability of a distributed parameter as defined in this paper contains the reducibility of parameter structure sufficient conditions for this kind of identifiability are given when the structure error associated with a structure reduction is too large these conditions may not be satisfied no matter how much data are collected in this paper we formulate a new experimental design problem that consists of two objectives minimizing the cost and maximizing the information content with robustness and feasibility as constraints we develop an algorithm that can find a cost effective robust design for objective oriented parameter identification we also present a heuristic algorithm that can find a suboptimal design with less computational effort for real case studies the proposed methodology is used to design a pumping test for identifying a distributed hydraulic conductivity we verify the robustness of the obtained design by assuming that the true parameter may have continuous discrete random and fractured structures finally the presented procedure of constructing objective oriented models is described step by step

applying the multivariate time rescaling theorem to neural population models statistical models of neural activity are integral to modern neuroscience recently interest has grown in modeling the spiking activity of populations of simultaneously recorded neurons to study the effects of correlations and functional connectivity on neural information processing however any statistical model must be validated by an appropriate goodness of fit test kolmogorov smirnov tests based on the time rescaling theorem have proven to be useful for evaluating point process based statistical models of single neuron spike trains here we discuss the extension of the time rescaling theorem to the multivariate neural population case we show that even in the presence of strong correlations between spike trains models that neglect couplings between neurons can be erroneously passed by the univariate time rescaling test we present the multivariate version of the time rescaling theorem and provide a practical step by step procedure for applying it to testing the sufficiency of neural population models using several simple analytically tractable models and more complex simulated and real data sets we demonstrate that important features of the population activity can be detected only using the multivariate extension of the test

estimation of time dependent input from neuronal membrane potential the set of firing rates of the presynaptic excitatory and inhibitory neurons constitutes the input signal to the postsynaptic neuron estimation of the time varying input rates from intracellularly recorded membrane potential is investigated here for that purpose the membrane potential dynamics must be specified we consider the ornstein uhlenbeck stochastic process one of the most common single neuron models with time dependent mean and variance assuming the slow variation of these two moments it is possible to formulate the estimation problem by using a state space model we develop an algorithm that estimates the paths of the mean and variance of the input current by using the empirical bayes approach then the input firing rates are directly available from the moments the proposed method is applied to three simulated data examples constant signal sinusoidally modulated signal and constant signal with a jump for the constant signal the estimation performance of the method is comparable to that of the traditionally applied maximum likelihood method further the proposed method accurately estimates both continuous and discontinuous time variable signals in the case of the signal with a jump which does not satisfy the assumption of slow variability the robustness of the method is verified it can be concluded that the method provides reliable estimates of the total input firing rates which are not experimentally measurable

asynchronous states in networks of pulse coupled oscillators we use a mean field approach to analyze the stability of the asynchronous state in a population of all to all pulse coupled nonlinear oscillators we determine the conditions that must be satisfied by the time constants and phase dependence characterizing the coupling between the oscillators in order for the asynchronous state to be stable we also consider the effects of noise this work complements results on synchronous states in similar models and allows us to study the validity of firing rate models commonly used for neural networks

increase of coherence in excitable systems by delayed feedback the control of coherence and spectral properties of noise induced oscillations by time delayed feedback is studied in a fitzhugh nagumo system which serves as a paradigmatic model of excitable systems a semianalytical approach based on a discrete model with waiting time densities is developed which allows one to predict quantitatively the increase of coherence measured by the correlation time and the modulation of the main frequencies of the stochastic dynamics in dependence on the delay time the analytical mean field approximation is in good agreement with numerical results for the full nonlinear model

what matters in neuronal locking exploiting local stability we show what neuronal characteristics are essential to ensure that coherent oscillations are asymptotically stable in a spatially homogeneous network of spiking neurons under standard conditions a necessary and in the limit of a large number of interacting neighbors also sufficient condition is that the postsynaptic potential is increasing in time as the neurons fire if the postsynaptic potential is decreasing oscillations are bound to be unstable this is a kind of locking theorem and boils down to a subtle interplay of axonal delays postsynaptic potentials and refractory behavior the theorem also allows for mixtures of excitatory and inhibitory interactions on the basis of the locking theorem we present a simple geometric method to verify the existence and local stability of a coherent oscillation

phase locking in weakly heterogeneous neuronal networks abstract we examine analytically the existence and stability of phase locked states in a weakly heterogeneous neuronal network we consider a model of n neurons with all to all synaptic coupling where the heterogeneity is in the intrinsic firing frequency of the individual neurons we consider both inhibitory and excitatory coupling we derive the conditions under which stable phase locking is possible in homogeneous networks many different periodic phase locked states are possible their stability depends on the dynamics of the neuron and the coupling for weak heterogeneity the phase locked states are perturbed from the homogeneous states and can remain stable if their homogeneous counterparts are stable for enough heterogeneity phase locked solutions either lose stability or are destroyed completely we analyze the possible states the network can take when phase locking is broken

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

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

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