On the comparison of Feller and Ornstein-Uhlenbeck models for neural activity

Some Dissimilarity Measures of Branching Processes and Optimal Decision Making in the Presence of Potential Pandemics

We compute exact values respectively bounds of dissimilarity/distinguishability measures-in the sense of the Kullback-Leibler information distance (relative entropy) and some transforms of more general power divergences and Renyi divergences-between two competing discrete-time Galton-Watson branching processes with immigration GWI for which the offspring as well as the immigration (importation) is arbitrarily Poisson-distributed; especially, we allow for arbitrary type of extinction-concerning criticality and thus for non-stationarity. We apply this to optimal decision making in the context of the spread of potentially pandemic infectious diseases (such as e.g., the current COVID-19 pandemic), e.g., covering different levels of dangerousness and different kinds of intervention/mitigation strategies. Asymptotic distinguishability behaviour and diffusion limits are investigated, too.

Explicit asymptotics on first passage times of diffusion processes

We introduce a unified framework for solving first passage times of time-homogeneous diffusion processes. Using potential theory and perturbation theory, we are able to deduce closed-form truncated probability densities, as asymptotics or approximations to the original first passage time densities, for single-side level crossing problems. The framework is applicable to diffusion processes with continuous drift functions; in particular, for bounded drift functions, we show that the perturbation series converges. In the present paper, we demonstrate examples of applying our framework to the Ornstein–Uhlenbeck, Bessel, exponential-Shiryaev, and hypergeometric diffusion processes (the latter two being previously studied by Dassios and Li (2018) and Borodin (2009), respectively). The purpose of this paper is to provide a fast and accurate approach to estimating first passage time densities of various diffusion processes.

On two diffusion neuronal models with multiplicative noise: The mean first-passage time properties

Two diffusion processes with multiplicative noise, able to model the changes in the neuronal membrane depolarization between two consecutive spikes of a single neuron, are considered and compared. The processes have the same deterministic part but different stochastic components. The differences in the state-dependent variabilities, their asymptotic distributions, and the properties of the first-passage time across a constant threshold are investigated. Closed form expressions for the mean of the first-passage time of both processes are derived and applied to determine the role played by the parameters involved in the model. It is shown that for some values of the input parameters, the higher variability, given by the second moment, does not imply shorter mean first-passage time. The reason for that can be found in the complete shape of the stationary distribution of the two processes. Applications outside neuroscience are also mentioned.

A Recursion Formula for the Moments of the First Passage Time of the Ornstein-Uhlenbeck Process

In this paper we use the Siegert formula to derive alternative expressions for the moments of the first passage time of the Ornstein-Uhlenbeck process through a constant threshold. The expression for the nth moment is recursively linked to the lower-order moments and consists of only n terms. These compact expressions can substantially facilitate (numerical) applications also for higher-order moments.

Cut-off and hitting times of a sample of Ornstein-Uhlenbeck processes and its average

A cut-off phenomenon is shown to occur in a sample of n independent, identically distributed Ornstein-Uhlenbeck processes and its average. Their distributions stay far from equilibrium before a certain O(log(n)) time, and converge exponentially fast after. Precise estimates show that the total variation distance drops from almost 1 to almost 0 over an interval of time of length O(1) around log(n)/(2α), where α is the viscosity coefficient of the sampled process. The distribution of the hitting time of 0 by the average of the sample is computed. As n tends to infinity, the hitting time becomes concentrated around the cut-off instant, and its tails match the estimates given for the total variation distance.

Cut-off and hitting times of a sample of Ornstein-Uhlenbeck processes and its average

A cut-off phenomenon is shown to occur in a sample of n independent, identically distributed Ornstein-Uhlenbeck processes and its average. Their distributions stay far from equilibrium before a certain O(log(n)) time, and converge exponentially fast after. Precise estimates show that the total variation distance drops from almost 1 to almost 0 over an interval of time of length O(1) around log(n)/(2α), where α is the viscosity coefficient of the sampled process. The distribution of the hitting time of 0 by the average of the sample is computed. As n tends to infinity, the hitting time becomes concentrated around the cut-off instant, and its tails match the estimates given for the total variation distance.

Nonstationary Feller process with time-varying coefficients

We study the nonstationary Feller process with time varying coefficients. We obtain the exact probability distribution exemplified by its characteristic function and cumulants. In some particular cases we exactly invert the distribution and achieve the probability density function. We show that for sufficiently long times this density approaches a Γ distribution with time-varying shape and scale parameters. Not far from the origin the process obeys a power law with an exponent dependent of time, thereby concluding that accessibility to the origin is not static but dynamic. We finally discuss some possible applications of the process.

A Recursion Formula for the Moments of the First Passage Time of the Ornstein-Uhlenbeck Process

In this paper we use the Siegert formula to derive alternative expressions for the moments of the first passage time of the Ornstein-Uhlenbeck process through a constant threshold. The expression for the nth moment is recursively linked to the lower-order moments and consists of only n terms. These compact expressions can substantially facilitate (numerical) applications also for higher-order moments.

Diffusion approximation of neuronal models revisited

Leaky integrate-and-fire neuronal models with reversal potentials have a number of different diffusion approximations, each depending on the form of the amplitudes of the postsynaptic potentials. Probability distributions of the first-passage times of the membrane potential in the original model and its diffusion approximations are numerically compared in order to find which of the approximations is the most suitable one. The properties of the random amplitudes of postsynaptic potentials are discussed. It is shown on a simple example that the quality of the approximation depends directly on them.

First-passage and escape problems in the Feller process

The Feller process is an one-dimensional diffusion process with linear drift and state-dependent diffusion coefficient vanishing at the origin. The process is positive definite and it is this property along with its linear character that have made Feller process a convenient candidate for the modeling of a number of phenomena ranging from single-neuron firing to volatility of financial assets. While general properties of the process have long been well known, less known are properties related to level crossing such as the first-passage and the escape problems. In this work we thoroughly address these questions.

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.

Motoneuron membrane potentials follow a time inhomogeneous jump diffusion process

Stochastic leaky integrate-and-fire models are popular due to their simplicity and statistical tractability. They have been widely applied to gain understanding of the underlying mechanisms for spike timing in neurons, and have served as building blocks for more elaborate models. Especially the Ornstein-Uhlenbeck process is popular to describe the stochastic fluctuations in the membrane potential of a neuron, but also other models like the square-root model or models with a non-linear drift are sometimes applied. Data that can be described by such models have to be stationary and thus, the simple models can only be applied over short time windows. However, experimental data show varying time constants, state dependent noise, a graded firing threshold and time-inhomogeneous input. In the present study we build a jump diffusion model that incorporates these features, and introduce a firing mechanism with a state dependent intensity. In addition, we suggest statistical methods to estimate all unknown quantities and apply these to analyze turtle motoneuron membrane potentials. Finally, simulated and real data are compared and discussed. We find that a square-root diffusion describes the data much better than an Ornstein-Uhlenbeck process with constant diffusion coefficient. Further, the membrane time constant decreases with increasing depolarization, as expected from the increase in synaptic conductance. The network activity, which the neuron is exposed to, can be reasonably estimated to be a threshold version of the nerve output from the network. Moreover, the spiking characteristics are well described by a Poisson spike train with an intensity depending exponentially on the membrane potential.

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.

A review of the methods for signal estimation in stochastic diffusion leaky integrate-and-fire neuronal models

Parameters in diffusion neuronal models are divided into two groups; intrinsic and input parameters. Intrinsic parameters are related to the properties of the neuronal membrane and are assumed to be known throughout the paper. Input parameters characterize processes generated outside the neuron and methods for their estimation are reviewed here. Two examples of the diffusion neuronal model, which are based on the integrate-and-fire concept, are investigated--the Ornstein--Uhlenbeck model as the most common one and the Feller model as an illustration of state-dependent behavior in modeling the neuronal input. Two types of experimental data are assumed-intracellular describing the membrane trajectories and extracellular resulting in knowledge of the interspike intervals. The literature on estimation from the trajectories of the diffusion process is extensive and thus the stress in this review is set on the inference made from the interspike intervals.

The dynamic brain: from spiking neurons to neural masses and cortical fields

The cortex is a complex system, characterized by its dynamics and architecture, which underlie many functions such as action, perception, learning, language, and cognition. Its structural architecture has been studied for more than a hundred years; however, its dynamics have been addressed much less thoroughly. In this paper, we review and integrate, in a unifying framework, a variety of computational approaches that have been used to characterize the dynamics of the cortex, as evidenced at different levels of measurement. Computational models at different space-time scales help us understand the fundamental mechanisms that underpin neural processes and relate these processes to neuroscience data. Modeling at the single neuron level is necessary because this is the level at which information is exchanged between the computing elements of the brain; the neurons. Mesoscopic models tell us how neural elements interact to yield emergent behavior at the level of microcolumns and cortical columns. Macroscopic models can inform us about whole brain dynamics and interactions between large-scale neural systems such as cortical regions, the thalamus, and brain stem. Each level of description relates uniquely to neuroscience data, from single-unit recordings, through local field potentials to functional magnetic resonance imaging (fMRI), electroencephalogram (EEG), and magnetoencephalogram (MEG). Models of the cortex can establish which types of large-scale neuronal networks can perform computations and characterize their emergent properties. Mean-field and related formulations of dynamics also play an essential and complementary role as forward models that can be inverted given empirical data. This makes dynamic models critical in integrating theory and experiments. We argue that elaborating principled and informed models is a prerequisite for grounding empirical neuroscience in a cogent theoretical framework, commensurate with the achievements in the physical sciences.

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.

Parameter estimation for a leaky integrate-and-fire neuronal model from ISI data

The Ornstein-Uhlenbeck process has been proposed as a model for the spontaneous activity of a neuron. In this model, the firing of the neuron corresponds to the first passage of the process to a constant boundary, or threshold. While the Laplace transform of the first-passage time distribution is available, the probability distribution function has not been obtained in any tractable form. We address the problem of estimating the parameters of the process when the only available data from a neuron are the interspike intervals, or the times between firings. In particular, we give an algorithm for computing maximum likelihood estimates and their corresponding confidence regions for the three identifiable (of the five model) parameters by numerically inverting the Laplace transform. A comparison of the two-parameter algorithm (where the time constant tau is known a priori) to the three-parameter algorithm shows that significantly more data is required in the latter case to achieve comparable parameter resolution as measured by 95% confidence intervals widths. The computational methods described here are a efficient alternative to other well known estimation techniques for leaky integrate-and-fire models. Moreover, it could serve as a template for performing parameter inference on more complex integrate-and-fire neuronal models.

On a set of data for the membrane potential in a neuron

We consider a set of data where the membrane potential in a pyramidal neuron is measured almost continuously in time, under varying experimental conditions. We use nonparametric estimates for the diffusion coefficient and the drift in view to contribute to the discussion which type of diffusion process is suitable to model the membrane potential in a neuron (more exactly: in a particular type of neuron under particular experimental conditions).

A review of the integrate-and-fire neuron model: I. Homogeneous synaptic input

The integrate-and-fire neuron model is one of the most widely used models for analyzing the behavior of neural systems. It describes the membrane potential of a neuron in terms of the synaptic inputs and the injected current that it receives. An action potential (spike) is generated when the membrane potential reaches a threshold, but the actual changes associated with the membrane voltage and conductances driving the action potential do not form part of the model. The synaptic inputs to the neuron are considered to be stochastic and are described as a temporally homogeneous Poisson process. Methods and results for both current synapses and conductance synapses are examined in the diffusion approximation, where the individual contributions to the postsynaptic potential are small. The focus of this review is upon the mathematical techniques that give the time distribution of output spikes, namely stochastic differential equations and the Fokker-Planck equation. The integrate-and-fire neuron model has become established as a canonical model for the description of spiking neurons because it is capable of being analyzed mathematically while at the same time being sufficiently complex to capture many of the essential features of neural processing. A number of variations of the model are discussed, together with the relationship with the Hodgkin-Huxley neuron model and the comparison with electrophysiological data. A brief overview is given of two issues in neural information processing that the integrate-and-fire neuron model has contributed to - the irregular nature of spiking in cortical neurons and neural gain modulation.

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.

Sensitive dependence of the coefficient of variation of interspike intervals on the lower boundary of membrane potential for the leaky integrate-and-fire neuron model

After the report of Softky and Koch [Softky, W.R., Koch, C., 1993. The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs. J. Neurosci. 13, 334-350], leaky integrate-and-fire models have been investigated to explain high coefficient of variation (CV) of interspike intervals (ISIs) at high firing rates observed in the cortex. The purpose of this paper is to study the effect of the position of a lower boundary of membrane potential on the possible value of CV of ISIs based on the diffusional leaky integrate-and-fire models with and without reversal potentials. Our result shows that the irregularity of ISIs for the diffusional leaky integrate-and-fire neuron significantly changes by imposing a lower boundary of membrane potential, which suggests the importance of the position of the lower boundary as well as that of the firing threshold when we study the statistical properties of leaky integrate-and-fire neuron models. It is worth pointing out that the mean-CV plot of ISIs for the diffusional leaky integrate-and-fire neuron with reversal potentials shows a close similarity to the experimental result obtained in Softky and Koch [Softky, W.R., Koch, C., 1993. The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs. J. Neurosci. 13, 334-350].

On the Classification of Experimental Data Modeled Via a Stochastic Leaky Integrate and Fire Model Through Boundary Values

We present a computational algorithm aimed to classify single unit spike trains on the basis of observed interspikes intervals (ISI). The neuronal activity is modeled with a stochastic leaky integrate and fire model and the inverse first passage time method is extended to the Ornstein-Uhlenbeck (OU) process. Differences between spike trains are detected in terms of the boundary shape. The proposed classification method is applied to the analysis of multiple single units recorded simultaneously in the thalamus and in the cerebral cortex of unanesthetized rats during spontaneous activity. We show the existence of at least three different firing patterns that could not be classified using the usual statistical indices.

A stochastic model and a functional central limit theorem for information processing in large systems of neurons

The paper deals with information transmission in large systems of neurons. We model the membrane potential in a single neuron belonging to a cell tissue by a non time-homogeneous Cox-Ingersoll-Ross type diffusion; in terms of its time-varying expectation, this stochastic process can convey deterministic signals. We model the spike train emitted by this neuron as a Poisson point process compensated by the occupation time of the membrane potential process beyond the excitation threshold. In a large system of neurons 1 < or = i < or = N processing independently the same deterministic signal, we prove a functional central limit theorem for the pooled spike train collected from the N neurons. This pooled spike train allows to recover the deterministic signal, up to some shape transformation which is explicit.

Optimum signal in a simple neuronal model with signal-dependent noise

How does the information about a signal in neural threshold crossings depend on the noise acting upon it? Two models are explored, a binary McCulloch and Pitts (threshold exceedance) model and a model of waiting time to exceedance--a discrete-time version of interspike intervals. If noise grows linearly with the signal, we find the best identification of the signal in terms of the Fisher information is for signals that do not reach the threshold in the absence of noise. Identification attains the same precision under weak and strong signals, but the coding range decreases at both extremes of signal level. We compare the results obtained for Fisher information with those using related first and second moment measures. The maximum obtainable information is plotted as a function of the ratio of noise to signal.

Estimation of the input parameters in the Ornstein-Uhlenbeck neuronal model

The stochastic Ornstein-Uhlenbeck 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 Laplace transforms of the first two moments of the normalized first-passage time through a constant boundary in the suprathreshold regime are derived, which is used to define moment estimators. In the subthreshold regime, the exponentiality of the first-passage time is utilized to characterize the input parameters. In the threshold regime and for the Wiener process approximation, analytic expressions for the first-passage-time density are used to derive the maximum-likelihood estimators of the parameters. The methods are illustrated on simulated data under different conditions, including misspecification of the intrinsic parameters of the model. Finally, known approximations of the first-passage-time moments are improved.

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.

Characterization of Subthreshold Voltage Fluctuations in Neuronal Membranes

Synaptic noise due to intense network activity can have a significant impact on the electrophysiological properties of individual neurons. This is the case for the cerebral cortex, where ongoing activity leads to strong barrages of synaptic inputs, which act as the main source of synaptic noise affecting on neuronal dynamics. Here, we characterize the sub-threshold behavior of neuronal models in which synaptic noise is represented by either additive or multiplicative noise, described by Ornstein-Uhlenbeck processes. We derive and solve the Fokker-Planck equation for this system, which describes the time evolution of the probability density function for the membrane potential. We obtain an analytic expression for the membrane potential distribution at steady state and compare this expression with the subthreshold activity obtained in Hodgkin-Huxley-type models with stochastic synaptic inputs. The differences between multiplicative and additive noise models suggest that multiplicative noise is adequate to describe the high-conductance states similar to in vivo conditions. Because the steady-state membrane potential distribution is easily obtained experimentally, this approach provides a possible method to estimate the mean and variance of synaptic conduct ancesinreal neurons.

New parameter relationships determined via stochastic ordering for spike activity in a reversal potential neural model

Purpose of this work is to study the dependence of interspike interval distribution on the model parameters when use is made of the Feller diffusion process to describe the subthreshold membrane potential of a neuron. To this aim we make use of a new approach, namely the ordering of first passage times. The functional dependence among the model parameters (e.g, membrane time constant, reversal potential, etc.) resulting from the ordering criteria employed and from the study of mean trajectory plots is analyzed into detail for four different scenario.

Temporally correlated inputs to leaky integrate-and-fire models can reproduce spiking statistics of cortical neurons

There has been controversy over whether the standard neuro-spiking models are consistent with the irregular spiking of cortical neurons. In a previous study, we proposed examining this consistency on the basis of the high-order statistics of the inter-spike intervals (ISIs), as represented by the coefficient of variation and the skewness coefficient. In that study we found that a leaky integrate-and-fire model incorporating the assumption of temporally uncorrelated inputs is not able to account for the spiking data recorded from a monkey prefrontal cortex. In the present paper, we attempt to revise the neuro-spiking model so as to make it consistent with the biological data. Here we consider the correlation coefficient of consecutive ISIs, which was ignored in previous studies. Considering three statistical coefficients, we conclude that the leaky integrate-and-fire model with temporally correlated inputs does account for the biological data. The correlation time scale of the inputs needed to explain the biological statistics is found to be on the order of 100ms. We discuss possible origins of this input correlation.

Fast temporal encoding and decoding with spiking neurons

We propose a simple theoretical structure of interacting integrate-and-fire neurons that can handle fast information processing and may account for the fact that only a few neuronal spikes suffice to transmit information in the brain. Using integrate-and-fire neurons that are subjected to individual noise and to a common external input, we calculate their first passage time (FPT), or interspike interval. We suggest using a population average for evaluating the FPT that represents the desired information. Instantaneous lateral excitation among these neurons helps the analysis. By employing a second layer of neurons with variable connections to the first layer, we represent the strength of the input by the number of output neurons that fire, thus decoding the temporal information. Such a model can easily lead to a logarithmic relation as in Weber's law. The latter follows naturally from information maximization if the input strength is statistically distributed according to an approximate inverse law.

Input parameters in a one-dimensional neuronal model with reversal potentials

An equation for a stochastic neuronal model describing the initiation of action potentials is studied for the case in which the depolarization of the membrane potential is restricted by the reversal potentials. It is assumed that the values of the membrane potential can be continuously recorded between consecutive spikes. Under this assumption, the estimators of the model parameters are derived and the methods for testing the model are proposed. The objective of the methods presented in this contribution is to provide neuroscientists with quantitative means in order to estimate parameters of stochastic neuronal models.

Simulation methods in neuronal modelling

The interspike distribution can be modelled as the first-passage-time distribution of suitable diffusion processes with biologically meaningful boundaries. Since various mathematical difficulties arise when one attempts to obtain closed form solutions for first-passage-time problems, one can resort to simulation methods in order to study the problem. In this paper we pinpoint possible overestimations connected with simulations of first-passage-times for diffusion processes and propose a suitable simulation technique to determine the moments and the distribution of the firing times. After checking the validity of the proposed method in some instances where numerical evaluations for such quantities are available, we apply the simulation algorithm to model the spiking activity by means of a particular diffusion process constrained by a suitable time varying threshold.

Jump-diffusion processes as models for neuronal activity

Aiming at an improvement of the existing neuronal models, we consider a mixed process ensuing from the superposition of continuous diffusions and of Poisson time-distributed sequence of impulses and focus our attention on the moments of the firing time. In particular, we consider three different instances: the large jumps model in which each jump causes the neuron firing, the reset model characterized by jumps towards the resting potential and a more general model where constant amplitude excitatory and inhibitory jumps are superimposed on diffusion. By resorting to analytical arguments and to numerical computations, the main behavioral differences of the considered models are outlined.

Computation of first passage time moments for stochastic diffusion processes modelling nerve membrane depolarization

For further understanding of neural coding, stochastic variability of interspike intervals has been investigated by both experimental and theoretical neuroscientists. In stochastic neuronal models, the interspike interval corresponds to the time period during which the process imitating the membrane potential reaches a threshold for the first time from a reset depolarization. For neurons belonging to complex networks in the brain, stochastic diffusion processes are often used to approximate the time course of the membrane potential. The interspike interval is then viewed as the first passage time for the employed diffusion process. Due to a lack of analytical solution for the related first passage time problem for most diffusion neuronal models, a numerical integration method, which serves to compute first passage time moments on the basis of the Siegert recursive formula, is presented in this paper. For their neurobiological plausibility, the method here is associated with diffusion processes whose state spaces are restricted to finite intervals, but it can also be applied to other diffusion processes and in other (non-neuronal) contexts. The capability of the method is demonstrated in numerical examples and the relation between the integration step, accuracy of calculation and amount of computing time required is discussed.