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 Theta neuron model with multiplicative gaussian white noise. The correspondence between the first passage time statistics for the saddle-node model and the Theta neuron model is obtained only in the Stratonovich interpretation of the stochastic Theta 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.     

Solving random diffusion models with nonlinear perturbations by the Wiener–Hermite expansion method

This paper deals with the construction of approximate series solutions of random nonlinear diffusion equations where nonlinearity is considered by means of a frank small parameter and uncertainty is introduced through white noise in the forcing term. For the simpler but important case in which the diffusion coefficient is time independent, we provide a Gaussian approximation of the solution stochastic process by taking advantage of the Wiener–Hermite expansion together with the perturbation method. In addition, approximations of the main statistical functions associated with a solution, such as the mean and variance, are computed. Numerical values of these functions are compared with respect to those obtained by applying the Runge–Kutta second-order stochastic scheme as an illustrative example.     

Optimal bang-bang control of linear stochastic systems with a small noise parameter

This study considers the problem of determining optimal feedback control laws for linear stochastic systems with amplitude-constrained control inputs. Two basic performance indices are considered, average time and average integral quadratic form. The optimization interval is random and defined as the first time a trajectory reaches the terminal regionR. The plant is modeled as a stochastic differential equation with an additive Wiener noise disturbance. The variance parameter of the Wiener noise process is assumed to be suitably small. A singular perturbation technique is presented for the solution of the stochastic optimization equations (second-order partial differential equation). A method for generating switching curves for the resulting optimal bang-bang control system is then developed. The results are applied to various problems associated with a second-order purely inertial system with additive noise at the control input. This problem is typical of satellite attitude control problems.     

Manoeuvring target tracking with coordinated-turn motion using stochastic non-linear filter

Allowing for perturbations in speed and turn rate of a target moving in a coordinated turn obeys a non-linear stochastic differential equation. Existing algorithms for coordinated turn tracking avoid this problem by ignoring perturbations in the continuous time model and adding process noise only after discretisation. The dynamic model used here adds small perturbations, modelled as independent Brownian motion processes, to the speed and turn rate. The target state is to be recursively estimated from noisy discrete-time measurements of the target's range and bearing. In particular, this paper examines the effect of the perturbations in speed and turn rate on the coordinated turn motion of the aircraft, and subsequently the stochastic algorithm is developed by deriving the evolutions of conditional means and variances for estimating the state of the aircraft. By linearizing the stochastic differential equations about the mean of the state vector using first-order approximation, the mean trajectory of the resulting first-order approximated stochastic differential model does not preserve the perturbation effect felt by the moving target; only the variance trajectory includes the perturbation effect. For this reason, the effectiveness of the perturbed model is examined on the basis of the second-order approximations of the system non-linearity. The theory of the non-linear filter of this paper is developed using the Kolmogorov forward equation ‘between the observation’ and a functional difference equation for the conditional probability density ‘at the observation’. The effectiveness of the second-order non-linear filter is examined on the basis of its ability to preserve perturbation effect felt by the aircraft. The Kolmogorov forward equation, however, is not appropriate for numerical simulations, since it is the equation for the evolution of the conditional probability density. Instead of the Kolmogorov equation, one derives the evolutions for the moments of the state vector, which in our case consists of positions, velocities and turn rate of the manoeuvring aircraft. Even these equations are not appropriate for the numerical simulations, since they are not closed in the sense that computing the evolution of a given moment involves the knowledge of higher-order moments. Hence we consider the approximations to these moment evolution equations. Simulation results are introduced to demonstrate the usefulness of an analytic theory developed in this paper.     

Generalized perturbation-based stochastic finite element method in elastostatics

Generalised nth order stochastic perturbation technique that can be applied to solve some boundary value or boundary initial problems in computational physics and/or engineering with random coefficients is presented here. This technique is implemented in conjunction with the finite element method (FEM) to model 1D linear elastostatics problem with a single random variable. Main motivation of this work is to improve essentially the accuracy of the stochastic perturbation technique, which in its second order realization was ineffective for large variations of the input random fields. The nth order approach makes it possible to specify the accuracy of the computations a priori for the expected values and variances separately. The symbolic computer program is employed to perform computational studies on convergence of the first two probabilistic moments for simple unidirectional tension of the bar. These numerical studies verify the influence of coefficient of variation of the random input and, in the same time, of the perturbation parameter on the first four probabilistic moments of the final solution vector.     

On stochastic finite element method for linear elastostatics by the Taylor expansion

The main aim of the paper is to present an application of the Taylor expansion in formulation and computational implementation of the perturbation-based stochastic finite element method. Random-input parameters as well as all-state functions included in static equilibrium equations are expanded in this approach around their expectations via Taylor series up the order given a priori. It further enables a dual computational approach for determination of probabilistic moments of the state functions—a formation and the solution of increasing order equilibrium equations and, on the other hand, polynomial approximation of deterministic state functions with respect to a given input random parameter. Theoretical and technical details of such methodology are explained also; some elementary engineering application with analytical solution is available to derive explicitly fundamental probabilistic moments of the resulting state function.     

A convergent approximation of the continuous-time optimal parameterestimator

Continuous-time linear stochastic systems that are bilinear in the state and parameters are considered. A specific approximation to the optimal nonlinear filter used as a recursive parameter estimator is derived by retaining third-order moments and using a Gaussian approximation for higher order moments. With probability one, the specific approximation is proved to converge to a minimum of the likelihood function. The proof uses the ordinary differential equation technique and requires that the trajectories of the slow system be bounded on finite time intervals and that the fixed parameter fast system by asymptotically stable. The fixed parameter fast system is proved to be asymptotically stable if the parameter update gain is small enough. Essentially, the specific approximation is asympotically equivalent to the recursive prediction error method, thus inheriting its asymptotic rate of convergence. A numerical simulation for a simple example indicates that the specific approximation has better transient response than other commonly used convergent parameter estimators     

A singular perturbation model of reliability in systems control

In many engineering systems, the intensity of the input noise is small and the state trajectory is mainly due to the deterministic part of the system structure. When this is the case for a white input noise, the mean exit-time from the reliability region may be represented as a solution to a partial differential equation with a small parameter multiplying the highest derivatives. Hence, the evaluation of the mean exit-time from the reliability region reduces to the solution of a singularly perturbed partial differential equation for which an asymptotic solution can be evaluated analytically in a fairly general case. The mean exit-time is then used to measure the reliability of some decentralized control policies for linear stochastic systems.     

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 t(1) and of a diffusion process conditioned to cross the boundary for the first time at t(1). 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.     

有界随机噪声参数激励下碰撞系统的矩稳定性

研究了单自由度线性单边碰撞系统在有界随机噪声参数激励下系统的矩稳定性问题. 用 Zhuravlev 变换将碰撞系统转化为连续的非碰撞系统,然后用随机平均法得到了关于慢变量的随机微分方程. 利用伊藤法则给出了系统一、二阶矩满足的常微分方程,根据微分方程的稳定性理论得到了系统一阶矩稳定充分必要条件的解析表达式和二阶矩稳定充分必要条件的数值算法,并对理论结果用数值方法进行了仿真计算.理论分析和数值仿真表明,无论是相对于一阶矩还是二阶矩的稳定性,随着随机激励振幅变大,系统的稳定性区域变小从而使得系统变得不稳定. 而当调谐参数趋于零系统达到参数主共振情形时,系统的稳定性区域变得最小. 当随机噪声强度逐渐变小趋于零时,由二种矩稳定性给出的稳定性区域变得一致. 在一定的参数区域内,随机噪声使得系统稳定化.     

Integrate-and-fire neurons driven by correlated stochastic input

Neurons are sensitive to correlations among synaptic inputs. However, analytical models that explicitly include correlations are hard to solve analytically, so their influence on a neuron's response has been difficult to ascertain. To gain some intuition on this problem, we studied the firing times of two simple integrate-and-fire model neurons driven by a correlated binary variable that represents the total input current. Analytic expressions were obtained for the average firing rate and coefficient of variation (a measure of spike-train variability) as functions of the mean, variance, and correlation time of the stochastic input. The results of computer simulations were in excellent agreement with these expressions. In these models, an increase in correlation time in general produces an increase in both the average firing rate and the variability of the output spike trains. However, the magnitude of the changes depends differentially on the relative values of the input mean and variance: the increase in firing rate is higher when the variance is large relative to the mean, whereas the increase in variability is higher when the variance is relatively small. In addition, the firing rate always tends to a finite limit value as the correlation time increases toward infinity, whereas the coefficient of variation typically diverges. These results suggest that temporal correlations may play a major role in determining the variability as well as the intensity of neuronal spike trains.     

Firing rate of the noisy quadratic integrate-and-fire neuron

We calculate the firing rate of the quadratic integrate-and-fire neuron in response to a colored noise input current. Such an input current is a good approximation to the noise due to the random bombardment of spikes, with the correlation time of the noise corresponding to the decay time of the synapses. The key parameter that determines the firing rate is the ratio of the correlation time of the colored noise, tau(s), to the neuronal time constant, tau(m). We calculate the firing rate exactly in two limits: when the ratio, tau(s)/tau(m), goes to zero (white noise) and when it goes to infinity. The correction to the short correlation time limit is O(tau(s)/tau(m)), which is qualita tively different from that of the leaky integrate-and-fire neuron, where the correction is O( radical tau(s)/tau(m)). The difference is due to the different boundary conditions of the probability density function of the membrane potential of the neuron at firing threshold. The correction to the long correlation time limit is O(tau(m)/tau(s)). By combining the short and long correlation time limits, we derive an expression that provides a good approximation to the firing rate over the whole range of tau(s)/tau(m) in the suprathreshold regime-that is, in a regime in which the average current is sufficient to make the cell fire. In the subthreshold regime, the expression breaks down somewhat when tau(s) becomes large compared to tau(m).     

A generalized linear integrate-and-fire neural model produces diverse spiking behaviors

For simulations of neural networks, there is a trade-off between the size of the network that can be simulated and the complexity of the model used for individual neurons. In this study, we describe a generalization of the leaky integrate-and-fire model that produces a wide variety of spiking behaviors while still being analytically solvable between firings. For different parameter values, the model produces spiking or bursting, tonic, phasic or adapting responses, depolarizing or hyperpolarizing after potentials and so forth. The model consists of a diagonalizable set of linear differential equations describing the time evolution of membrane potential, a variable threshold, and an arbitrary number of firing-induced currents. Each of these variables is modified by an update rule when the potential reaches threshold. The variables used are intuitive and have biological significance. The model's rich behavior does not come from the differential equations, which are linear, but rather from complex update rules. This single-neuron model can be implemented using algorithms similar to the standard integrate-and-fire model. It is a natural match with event-driven algorithms for which the firing times are obtained as a solution of a polynomial equation.     

The multilinear algebra of the moments of distributions in the analysis of nonlinear stochastic systems

The differential equations are obtained for multilinear moment forms of the phase vector of a stochastic system described by the nonlinear stochastic Ito equation. The equations are derived for mixed moment forms of the process whose phase vector takes values in the product space. Multilinear cumulant forms are defined, and the link is established between the moments and cumulants. The issues of approximate solution of an infinite set of equations for the moments are discussed. The exact solution is given to the equations for the moments of a specific two-dimensional bilinear system.     

Higher-order statistics of input ensembles and the response of simple model neurons

Pairwise correlations among spike trains recorded in vivo have been frequently reported. It has been argued that correlated activity could play an important role in the brain, because it efficiently modulates the response of a postsynaptic neuron. We show here that a neuron's output firing rate critically depends on the higher-order statistics of the input ensemble. We constructed two statistical models of populations of spiking neurons that fired with the same rates and had identical pairwise correlations, but differed with regard to the higher-order interactions within the population. The first ensemble was characterized by clusters of spikes synchronized over the whole population. In the second ensemble, the size of spike clusters was, on average, proportional to the pairwise correlation. For both input models, we assessed the role of the size of the population, the firing rate, and the pairwise correlation on the output rate of two simple model neurons: a continuous firing-rate model and a conductance-based leaky integrate-and-fire neuron. An approximation to the mean output rate of the firing-rate neuron could be derived analytically with the help of shot noise theory. Interestingly, the essential features of the mean response of the two neuron models were similar. For both neuron models, the three input parameters played radically different roles with respect to the postsynaptic firing rate, depending on the interaction structure of the input. For instance, in the case of an ensemble with small and distributed spike clusters, the output firing rate was efficiently controlled by the size of the input population. In addition to the interaction structure, the ratio of inhibition to excitation was found to strongly modulate the effect of correlation on the postsynaptic firing rate.     

Strong Convergence Analysis of the Stochastic Exponential Rosenbrock Scheme for the Finite Element Discretization of Semilinear SPDEs Driven by Multiplicative and Additive Noise

In this paper, we consider the numerical approximation of a general second order semilinear stochastic spartial differential equation (SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the nonlinear part is stronger than the linear part also called stochastic reactive dominated transport equations. Most numerical techniques, including current stochastic exponential integrators lose their good stability properties on such equations. Using finite element for space discretization, we propose a new scheme appropriated on such equations, called stochastic exponential Rosenbrock scheme based on local linearization at every time step of the semi-discrete equation obtained after space discretization. We consider noise with finite trace and give a strong convergence proof of the new scheme toward the exact solution in the root-mean-square \(L^2\) norm. Numerical experiments to sustain theoretical results are provided.     

The error in solving a wave equation based on a scheme with weights

The error in approximating a Cauchy problem for a two-dimensional wave equation based on a scheme with weights is studied. The dependence of the approximation error on the time step and weight parameter is considered. With this aim in view, the difference of second-order spatial derivatives in the wave equation is approximated, while the time derivative is preserved continuous; and an analytical solution of a Cauchy problem for a system of ordinary differential equations is obtained as the decomposition in the orthonormal basis consisting of the eigenvectors of the operator of the second difference derivative with respect to the spatial variables. Based on this solution, the errors of the approximation of the wave problem by three-layer difference schemes are studied and the conditions for the stability of a three-layer difference scheme are obtained. It is established that, when simulating the propagation of oscillation processes using difference methods, the oscillation frequency values differ from the real ones and depend on the weight parameter and time step. The optimal values of the weight parameter with which the deviation of the oscillation frequency for the difference scheme is minimal are obtained. The dependences of the approximation error on weight and spatial step are derived. The optimal values of the weight parameter with which the schemes are of the second and fourth order of accuracy with respect to the time step are found.     

Robust H∞ Filtering and Deconvolution for Continuous Time Delay Systems Based on Game‐Theoretic Approach

Many dynamical systems involve not only process and measurement noise signals but also parameter uncertainty and unknown input signals. This paper aims to estimate the state and unknown input for linear continuous time‐varying systems subject to time delay in state, norm‐bounded parameter uncertainty, and a known input. Such a problem is reformulated into a two‐player differential game whose saddle point solution gives rise to one sufficient solvable condition for the estimation problem. The possible optimal estimators are obtained by solving the two coupled Riccati differential equations. We demonstrate, through two examples, how the proposed estimator is valid for estimating state and unknown input.     

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.     

Theory of input spike auto- and cross-correlations and their effect on the response of spiking neurons

Spike correlations between neurons are ubiquitous in the cortex, but their role is not understood. Here we describe the firing response of a leaky integrate-and-fire neuron (LIF) when it receives a temporarily correlated input generated by presynaptic correlated neuronal populations. Input correlations are characterized in terms of the firing rates, Fano factors, correlation coefficients, and correlation timescale of the neurons driving the target neuron. We show that the sum of the presynaptic spike trains cannot be well described by a Poisson process. In fact, the total input current has a nontrivial two-point correlation function described by two main parameters: the correlation timescale (how precise the input correlations are in time) and the correlation magnitude (how strong they are). Therefore, the total current generated by the input spike trains is not well described by a white noise gaussian process. Instead, we model the total current as a colored gaussian process with the same mean and two-point correlation function, leading to the formulation of the problem in terms of a Fokker-Planck equation. Solutions of the output firing rate are found in the limit of short and long correlation timescales. The solutions described here expand and improve on our previous results (Moreno, de la Rocha, Renart, & Parga, 2002) by presenting new analytical expressions for the output firing rate for general IF neurons, extending the validity of the results for arbitrarily large correlation magnitude, and by describing the differential effect of correlations on the mean-driven or noise-dominated firing regimes. Also the details of this novel formalism are given here for the first time. We employ numerical simulations to confirm the analytical solutions and study the firing response to sudden changes in the input correlations. We expect this formalism to be useful for the study of correlations in neuronal networks and their role in neural processing and information transmission.