On the parameter estimation for diffusion models of single neuron's activities. I. Application to spontaneous activities of mesencephalic reticular formation cells in sleep and waking states

The Jacobi diffusion process as a neuronal model

The Jacobi process is a stochastic diffusion characterized by a linear drift and a special form of multiplicative noise which keeps the process confined between two boundaries. One example of such a process can be obtained as the diffusion limit of the Stein's model of membrane depolarization which includes both excitatory and inhibitory reversal potentials. The reversal potentials create the two boundaries between which the process is confined. Solving the first-passage-time problem for the Jacobi process, we found closed-form expressions for mean, variance, and third moment that are easy to implement numerically. The first two moments are used here to determine the role played by the parameters of the neuronal model; namely, the effect of multiplicative noise on the output of the Jacobi neuronal model with input-dependent parameters is examined in detail and compared with the properties of the generic Jacobi diffusion. It appears that the dependence of the model parameters on the rate of inhibition turns out to be of primary importance to observe a change in the slope of the response curves. This dependence also affects the variability of the output as reflected by the coefficient of variation. It often takes values larger than one, and it is not always a monotonic function in dependency on the rate of excitation.

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.

Collective Activity of Many Bistable Assemblies Reproduces Characteristic Dynamics of Multistable Perception

The timing of perceptual decisions depends on both deterministic and stochastic factors, as the gradual accumulation of sensory evidence (deterministic) is contaminated by sensory and/or internal noise (stochastic). When human observers view multistable visual displays, successive episodes of stochastic accumulation culminate in repeated reversals of visual appearance. Treating reversal timing as a "first-passage time" problem, we ask how the observed timing densities constrain the underlying stochastic accumulation. Importantly, mean reversal times (i.e., deterministic factors) differ enormously between displays/observers/stimulation levels, whereas the variance and skewness of reversal times (i.e., stochastic factors) keep characteristic proportions of the mean. What sort of stochastic process could reproduce this highly consistent "scaling property?" Here we show that the collective activity of a finite population of bistable units (i.e., a generalized Ehrenfest process) quantitatively reproduces all aspects of the scaling property of multistable phenomena, in contrast to other processes under consideration (Poisson, Wiener, or Ornstein-Uhlenbeck process). The postulated units express the spontaneous dynamics of attractor assemblies transitioning between distinct activity states. Plausible candidates are cortical columns, or clusters of columns, as they are preferentially connected and spontaneously explore a restricted repertoire of activity states. Our findings suggests that perceptual representations are granular, probabilistic, and operate far from equilibrium, thereby offering a suitable substrate for statistical inference.

SIGNIFICANCE STATEMENT

Spontaneous reversals of high-level perception, so-called multistable perception, conform to highly consistent and characteristic statistics, constraining plausible neural representations. We show that the observed perceptual dynamics would be reproduced quantitatively by a finite population of distinct neural assemblies, each with locally bistable activity, operating far from the collective equilibrium (generalized Ehrenfest process). Such a representation would be consistent with the intrinsic stochastic dynamics of neocortical activity, which is dominated by preferentially connected assemblies, such as cortical columns or clusters of columns. We predict that local neuron assemblies will express bistable dynamics, with spontaneous active-inactive transitions, whenever they contribute to high-level perception.

The Gamma renewal process as an output of the diffusion leaky integrate-and-fire neuronal model

Statistical properties of spike trains as well as other neurophysiological data suggest a number of mathematical models of neurons. These models range from entirely descriptive ones to those deduced from the properties of the real neurons. One of them, the diffusion leaky integrate-and-fire neuronal model, which is based on the Ornstein-Uhlenbeck (OU) stochastic process that is restricted by an absorbing barrier, can describe a wide range of neuronal activity in terms of its parameters. These parameters are readily associated with known physiological mechanisms. The other model is descriptive, Gamma renewal process, and its parameters only reflect the observed experimental data or assumed theoretical properties. Both of these commonly used models are related here. We show under which conditions the Gamma model is an output from the diffusion OU model. In some cases, we can see that the Gamma distribution is unrealistic to be achieved for the employed parameters of the OU process.

Fluctuation scaling in neural spike trains

Fluctuation scaling has been observed universally in a wide variety of phenomena. In time series that describe sequences of events, fluctuation scaling is expressed as power function relationships between the mean and variance of either inter-event intervals or counting statistics, depending on measurement variables. In this article, fluctuation scaling has been formulated for a series of events in which scaling laws in the inter-event intervals and counting statistics were related. We have considered the first-passage time of an Ornstein-Uhlenbeck process and used a conductance-based neuron model with excitatory and inhibitory synaptic inputs to demonstrate the emergence of fluctuation scaling with various exponents, depending on the input regimes and the ratio between excitation and inhibition. Furthermore, we have discussed the possible implication of these results in the context of neural coding.

Reconstruction of the input signal of the leaky integrate-and-fire neuronal model from its interspike intervals

Extracting the input signal of a neuron by analyzing its spike output is an important step toward understanding how external information is coded into discrete events of action potentials and how this information is exchanged between different neurons in the nervous system. Most of the existing methods analyze this decoding problem in a stochastic framework and use probabilistic metrics such as maximum-likelihood method to determine the parameters of the input signal assuming a leaky and integrate-and-fire (LIF) model. In this article, the input signal of the LIF model is considered as a combination of orthogonal basis functions. The coefficients of the basis functions are found by minimizing the norm of the observed spikes and those generated by the estimated signal. This approach gives rise to the deterministic reconstruction of the input signal and results in a simple matrix identity through which the coefficients of the basis functions and therefore the neuronal stimulus can be identified. The inherent noise of the neuron is considered as an additional factor in the membrane potential and is treated as the disturbance in the reconstruction algorithm. The performance of the proposed scheme is evaluated by numerical simulations, and it is shown that input signals with different characteristics can be well recovered by this algorithm.

Modeling neural activity with cumulative damage distributions

Neurons transmit information as action potentials or spikes. Due to the inherent randomness of the inter-spike intervals (ISIs), probabilistic models are often used for their description. Cumulative damage (CD) distributions are a family of probabilistic models that has been widely considered for describing time-related cumulative processes. This family allows us to consider certain deterministic principles for modeling ISIs from a probabilistic viewpoint and to link its parameters to values with biological interpretation. The CD family includes the Birnbaum-Saunders and inverse Gaussian distributions, which possess distinctive properties and theoretical arguments useful for ISI description. We expand the use of CD distributions to the modeling of neural spiking behavior, mainly by testing the suitability of the Birnbaum-Saunders distribution, which has not been studied in the setting of neural activity. We validate this expansion with original experimental and simulated electrophysiological data.

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.

Stochastic optimal control of single neuron spike trains

OBJECTIVE

External control of spike times in single neurons can reveal important information about a neuron's sub-threshold dynamics that lead to spiking, and has the potential to improve brain-machine interfaces and neural prostheses. The goal of this paper is the design of optimal electrical stimulation of a neuron to achieve a target spike train under the physiological constraint to not damage tissue.

APPROACH

We pose a stochastic optimal control problem to precisely specify the spike times in a leaky integrate-and-fire (LIF) model of a neuron with noise assumed to be of intrinsic or synaptic origin. In particular, we allow for the noise to be of arbitrary intensity. The optimal control problem is solved using dynamic programming when the controller has access to the voltage (closed-loop control), and using a maximum principle for the transition density when the controller only has access to the spike times (open-loop control).

MAIN RESULTS

We have developed a stochastic optimal control algorithm to obtain precise spike times. It is applicable in both the supra-threshold and sub-threshold regimes, under open-loop and closed-loop conditions and with an arbitrary noise intensity; the accuracy of control degrades with increasing intensity of the noise. Simulations show that our algorithms produce the desired results for the LIF model, but also for the case where the neuron dynamics are given by more complex models than the LIF model. This is illustrated explicitly using the Morris-Lecar spiking neuron model, for which an LIF approximation is first obtained from a spike sequence using a previously published method. We further show that a related control strategy based on the assumption that there is no noise performs poorly in comparison to our noise-based strategies. The algorithms are numerically intensive and may require efficiency refinements to achieve real-time control; in particular, the open-loop context is more numerically demanding than the closed-loop one.

SIGNIFICANCE

Our main contribution is the online feedback control of a noisy neuron through modulation of the input, taking into account physiological constraints on the control. A precise and robust targeting of neural activity based on stochastic optimal control has great potential for regulating neural activity in e.g. prosthetic applications and to improve our understanding of the basic mechanisms by which neuronal firing patterns can be controlled in vivo.

Fokker-Planck and Fortet Equation-Based Parameter Estimation for a Leaky Integrate-and-Fire Model with Sinusoidal and Stochastic Forcing

Analysis of sinusoidal noisy leaky integrate-and-fire models and comparison with experimental data are important to understand the neural code and neural synchronization and rhythms. In this paper, we propose two methods to estimate input parameters using interspike interval data only. One is based on numerical solutions of the Fokker-Planck equation, and the other is based on an integral equation, which is fulfilled by the interspike interval probability density. This generalizes previous methods tailored to stationary data to the case of time-dependent input. The main contribution is a binning method to circumvent the problems of nonstationarity, and an easy-to-implement initializer for the numerical procedures. The methods are compared on simulated data.

LIST OF ABBREVIATIONS LIF

Leaky integrate-and-fireISI: Interspike intervalSDE: Stochastic differential equationPDE: Partial differential equation.

Estimating nonstationary inputs from a single spike train based on a neuron model with adaptation

Because every spike of a neuron is determined by input signals, a train of spikes may contain information about the dynamics of unobserved neurons. A state-space method based on the leaky integrate-and-fire model, describing neuronal transformation from input signals to a spike train has been proposed for tracking input parameters represented by their mean and fluctuation [11]. In the present paper, we propose to make the estimation more realistic by adopting an LIF model augmented with an adaptive moving threshold. Moreover, because the direct state-space method is computationally infeasible for a data set comprising thousands of spikes, we further develop a practical method for transforming instantaneous firing characteristics back to input parameters. The instantaneous firing characteristics, represented by the firing rate and non-Poisson irregularity, can be estimated using a computationally feasible algorithm. We applied our proposed methods to synthetic data to clarify that they perform well.

Estimating inputs and an internal neuronal parameter from a single spike train

Because neurons are integrating input signals and translating them into timed output spikes, examining spike timing may reveal information about inputs, such as population activities of excitatory and inhibitory presynaptic neurons. Here we construct a state-space method for estimating not only such extrinsic parameters, but also an intrinsic neuronal parameter such as the membrane time constant from a single spike train.

Estimating nonstationary input signals from a single neuronal spike train

Neurons temporally integrate input signals, translating them into timed output spikes. Because neurons nonperiodically emit spikes, examining spike timing can reveal information about input signals, which are determined by activities in the populations of excitatory and inhibitory presynaptic neurons. Although a number of mathematical methods have been developed to estimate such input parameters as the mean and fluctuation of the input current, these techniques are based on the unrealistic assumption that presynaptic activity is constant over time. Here, we propose tracking temporal variations in input parameters with a two-step analysis method. First, nonstationary firing characteristics comprising the firing rate and non-Poisson irregularity are estimated from a spike train using a computationally feasible state-space algorithm. Then, information about the firing characteristics is converted into likely input parameters over time using a transformation formula, which was constructed by inverting the neuronal forward transformation of the input current to output spikes. By analyzing spike trains recorded in vivo, we found that neuronal input parameters are similar in the primary visual cortex V1 and middle temporal area, whereas parameters in the lateral geniculate nucleus of the thalamus were markedly different.

Interspike interval distributions of spiking neurons driven by fluctuating inputs

Interspike interval (ISI) distributions of cortical neurons exhibit a range of different shapes. Wide ISI distributions are believed to stem from a balance of excitatory and inhibitory inputs that leads to a strongly fluctuating total drive. An important question is whether the full range of experimentally observed ISI distributions can be reproduced by modulating this balance. To address this issue, we investigate the shape of the ISI distributions of spiking neuron models receiving fluctuating inputs. Using analytical tools to describe the ISI distribution of a leaky integrate-and-fire (LIF) neuron, we identify three key features: 1) the ISI distribution displays an exponential decay at long ISIs independently of the strength of the fluctuating input; 2) as the amplitude of the input fluctuations is increased, the ISI distribution evolves progressively between three types, a narrow distribution (suprathreshold input), an exponential with an effective refractory period (subthreshold but suprareset input), and a bursting exponential (subreset input); 3) the shape of the ISI distribution is approximately independent of the mean ISI and determined only by the coefficient of variation. Numerical simulations show that these features are not specific to the LIF model but are also present in the ISI distributions of the exponential integrate-and-fire model and a Hodgkin-Huxley-like model. Moreover, we observe that for a fixed mean and coefficient of variation of ISIs, the full ISI distributions of the three models are nearly identical. We conclude that the ISI distributions of spiking neurons in the presence of fluctuating inputs are well described by gamma distributions.

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.

Stochastic modeling of the neuronal activity in the thalamus of Essential Tremor patient

Several stochastic models, with various degrees of complexity, have been proposed to model the neuronal activity from different parts of the human brain. In this paper, we use an Ornstein-Uhlenbeck Process (OUP) to model the spike activity recorded from the thalamus of a patient suffering from Essential Tremor at the time of implantation of the electrodes for Deep Brain Stimulation. From the recorded data, which contains information about the spike times of a single neuron, we identify the model parameters of the OUP.We then use these parameters to numerically simulate the inter-spike interval distribution. We show that the OUP provides excellent fits to the data recorded both without any external stimulation as well as with stimulation. We finally compare the fits with other stochastic models commonly used and we show the superiority of the OUP model in general.

On a Stochastic Leaky Integrate-and-Fire Neuronal Model

The leaky integrate-and-fire neuronal model proposed in Stevens and Zador ( 1998 ), in which time constant and resting potential are postulated to be time dependent, is revisited within a stochastic framework in which the membrane potential is mathematically described as a gauss-diffusion process. The first-passage-time probability density, miming in such a context the firing probability density, is evaluated by either the Volterra integral equation of Buonocore, Nobile, and Ricciardi ( 1987 ) or, when possible, by the asymptotics of Giorno, Nobile, and Ricciardi ( 1990 ). The model examined here represents an extension of the classic leaky integrate-and-fire one based on the Ornstein-Uhlenbeck process in that it is in principle compatible with the inclusion of some other physiological characteristics such as relative refractoriness. It also allows finer tuning possibilities in view of its accounting for certain qualitative as well as quantitative features, such as the behavior of the time course of the membrane potential prior to firings and the computation of experimentally measurable statistical descriptors of the firing time: mean, median, coefficient of variation, and skewness. Finally, implementations of this model are provided in connection with certain experimental evidence discussed in the literature.

Stochastic modeling of the neuronal activity in the subthalamic nucleus and model parameter identification from Parkinson patient data

Several stochastic models, with various degrees of complexity, have been proposed to model the neuronal activity from different parts of the human brain. In this article, we use a simple Ornstein-Uhlenbeck process (OUP) to model the spike activity recorded from the subthalamic nucleus of patients suffering from Parkinson's disease at the time of implantation of the electrodes for deep brain stimulation. From the recorded data, which contains information about the spike times of a single neuron, we identify and extract the model parameters of the OUP. We then use these parameters to numerically simulate the inter-spike intervals and the voltage across the neuron membrane. We finally assess how well the proposed mathematical model fits to the measured data and compare it with other commonly adopted stochastic models. We show an excellent agreement between the computer-generated data according to the OUP model and the measured one, as well as the superiority of the OUP model when compared to the Poisson process model and the random walk model; thus, establishing the validity of the OUP as a simple yet biologically plausible model of the neuronal activity recorded from the subthalamic nucleus of Parkinson's disease patients.

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.

Effect of stimulation on the input parameters of stochastic leaky integrate-and-fire neuronal model

The Ornstein-Uhlenbeck neuronal model is specified by two types of parameters. One type corresponds to the properties of the neuronal membrane, whereas the second type (local average rate of the membrane depolarization and its variability) corresponds to the input of the neuron. In this article, we estimate the parameters of the second type from an intracellular record during neuronal firing caused by stimulation (audio signal). We compare the obtained estimates with those from the spontaneous part of the record. As predicted from the model construction, the values of the input parameters are larger for the periods when neuron is stimulated than for the spontaneous ones. Finally, the firing regimen of the model is checked. It is confirmed that the neuron is in the suprathreshold regimen during the stimulation.

Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold

Information is transmitted in the brain through various kinds of neurons that respond differently to the same signal. Full characteristics including cognitive functions of the brain should ultimately be comprehended by building simulators capable of precisely mirroring spike responses of a variety of neurons. Neuronal modeling that had remained on a qualitative level has recently advanced to a quantitative level, but is still incapable of accurately predicting biological data and requires high computational cost. In this study, we devised a simple, fast computational model that can be tailored to any cortical neuron not only for reproducing but also for predicting a variety of spike responses to greatly fluctuating currents. The key features of this model are a multi-timescale adaptive threshold predictor and a nonresetting leaky integrator. This model is capable of reproducing a rich variety of neuronal spike responses, including regular spiking, intrinsic bursting, fast spiking, and chattering, by adjusting only three adaptive threshold parameters. This model can express a continuous variety of the firing characteristics in a three-dimensional parameter space rather than just those identified in the conventional discrete categorization. Both high flexibility and low computational cost would help to model the real brain function faithfully and examine how network properties may be influenced by the distributed characteristics of component neurons.

Are the input parameters of white noise driven integrate and fire neurons uniquely determined by rate and CV?

Integrate and fire (IF) neurons have found widespread applications in computational neuroscience. Particularly important are stochastic versions of these models where the driving consists of a synaptic input modeled as white Gaussian noise with mean mu and noise intensity D. Different IF models have been proposed, the firing statistics of which depends nontrivially on the input parameters mu and D. In order to compare these models among each other, one must first specify the correspondence between their parameters. This can be done by determining which set of parameters (mu,D) of each model is associated with a given set of basic firing statistics as, for instance, the firing rate and the coefficient of variation (CV) of the interspike interval (ISI). However, it is not clear a priori whether for a given firing rate and CV there is only one unique choice of input parameters for each model. Here we review the dependence of rate and CV on input parameters for the perfect, leaky, and quadratic IF neuron models and show analytically that indeed in these three models the firing rate and the CV uniquely determine the input parameters.

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.

Parameters of the Diffusion Leaky Integrate-and-Fire Neuronal Model for a Slowly Fluctuating Signal

Stochastic leaky integrate-and-fire (LIF) neuronal models are common theoretical tools for studying properties of real neuronal systems. Experimental data of frequently sampled membrane potential measurements between spikes show that the assumption of constant parameter values is not realistic and that some (random) fluctuations are occurring. In this article, we extend the stochastic LIF model, allowing a noise source determining slow fluctuations in the signal. This is achieved by adding a random variable to one of the parameters characterizing the neuronal input, considering each interspike interval (ISI) as an independent experimental unit with a different realization of this random variable. In this way, the variation of the neuronal input is split into fast (within-interval) and slow (between-intervals) components. A parameter estimation method is proposed, allowing the parameters to be estimated simultaneously over the entire data set. This increases the statistical power, and the average estimate over all ISIs will be improved in the sense of decreased variance of the estimator compared to previous approaches, where the estimation has been conducted separately on each individual ISI. The results obtained on real data show good agreement with classical regression methods.

Errors in estimation of the input signal for integrate-and-fire neuronal models

Estimation of the input parameters of stochastic (leaky) integrate-and-fire neuronal models is studied. It is shown that the presence of a firing threshold brings a systematic error to the estimation procedure. Analytical formulas for the bias are given for two models, the randomized random walk and the perfect integrator. For the third model considered, the leaky integrate-and-fire model, the study is performed by using Monte Carlo simulated trajectories. The bias is compared with other errors appearing during the estimation, and it is documented that the effect of the bias has to be taken into account in experimental studies.

Discrimination with Spike Times and ISI Distributions

We study the discrimination capability of spike time sequences using the Chernoff distance as a metric. We assume that spike sequences are generated by renewal processes and study how the Chernoff distance depends on the shape of interspike interval (ISI) distribution. First, we consider a lower bound to the Chernoff distance because it has a simple closed form. Then we consider specific models of ISI distributions such as the gamma, inverse gaussian (IG), exponential with refractory period (ER), and that of the leaky integrate-and-fire (LIF) neuron. We found that the discrimination capability of spike times strongly depends on high-order moments of ISI and that it is higher when the spike time sequence has a larger skewness and a smaller kurtosis. High variability in terms of coefficient of variation (CV) does not necessarily mean that the spike times have less discrimination capability. Spike sequences generated by the gamma distribution have the minimum discrimination capability for a given mean and variance of ISI. We used series expansions to calculate the mean and variance of ISIs for LIF neurons as a function of the mean input level and the input noise variance. Spike sequences from an LIF neuron are more capable of discrimination than those of IG and gamma distributions when the stationary voltage level is close to the neuron's threshold value of the neuron.

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.

Randomness and variability of the neuronal activity described by the Ornstein-Uhlenbeck model

Normalized entropy as a measure of randomness is explored. It is employed to characterize those properties of neuronal firing that cannot be described by the first two statistical moments. We analyze randomness of firing of the Ornstein-Uhlenbeck (OU) neuronal model with respect either to the variability of interspike intervals (coefficient of variation) or the model parameters. A new form of the Siegert's equation for first-passage time of the OU process is given. The parametric space of the model is divided into two parts (sub-and supra-threshold) depending upon the neuron activity in the absence of noise. In the supra-threshold regime there are many similarities of the model with the Wiener process model. The sub-threshold behavior differs qualitatively both from the Wiener model and from the supra-threshold regime. For very low input the firing regularity increases (due to increase of noise) cannot be observed by employing the entropy, while it is clearly observable by employing the coefficient of variation. Finally, we introduce and quantify the converse effect of firing regularity decrease by employing the normalized entropy.

The parameters of the stochastic leaky integrate-and-fire neuronal model

Five parameters of one of the most common neuronal models, the diffusion leaky integrate-and-fire model, also known as the Ornstein-Uhlenbeck neuronal model, were estimated on the basis of intracellular recording. These parameters can be classified into two categories. Three of them (the membrane time constant, the resting potential and the firing threshold) characterize the neuron itself. The remaining two characterize the neuronal input. The intracellular data were collected during spontaneous firing, which in this case is characterized by a Poisson process of interspike intervals. Two methods for the estimation were applied, the regression method and the maximum-likelihood method. Both methods permit to estimate the input parameters and the membrane time constant in a short time window (a single interspike interval). We found that, at least in our example, the regression method gave more consistent results than the maximum-likelihood method. The estimates of the input parameters show the asymptotical normality, which can be further used for statistical testing, under the condition that the data are collected in different experimental situations. The model neuron, as deduced from the determined parameters, works in a subthreshold regimen. This result was confirmed by both applied methods. The subthreshold regimen for this model is characterized by the Poissonian firing. This is in a complete agreement with the observed interspike interval data.

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.

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.

Effect of periodic stimulus on a neuronal diffusion model with signal-dependent noise

To relate the noise intensity with a periodically modulated input signal in a single neuron stochastic model we introduce a diffusion model with both time modulated drift and diffusion coefficient. Such a model is the continuous version of a Stein model with time oscillating frequencies for the Poisson processes describing the inputs impinging on the neuron. We focus here on some aspects of the resonance phenomenon for such a model. We compare the corresponding interspike interval distribution with the analogous distribution for a model sharing the same parameter values, but with constant noise intensity. Examples with two different levels for this noise intensity are discussed. The enhancement of the height of the peaks in the interspike interval distribution appearing at the modulation period, the improvement of the phase locking behavior and an enlargement of the noise ranges where a resonance like behavior arises are the main features observed in the considered cases.

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.

Neuronal integration mechanisms have little effect on spike auto-correlations of cortical neurons

Cortical neurons of behaving animals generate irregular spike sequences, but the sequences generally differ from an entirely random sequence (Poisson process), and they have temporal correlations (spike auto-correlations). Temporally correlated spike sequences can be brought about because of incoming synaptic inputs to the neuron, or because of the neuronal integration mechanism. In this paper, we attempt to determine which is the origin of spike auto-correlations observed in the spiking data recorded from neurons in the prefrontal cortex of a monkey preserving a cue information in the delay response task experiment. Each incoming input is assumed to be independent from its own spike events, and the temporal integration in the neuron is assumed to be reset by every spike event. So, the process to spike is assumed to be divided into two processes: the process independent from its own spikes, which drives the process reset by its own spikes. Under these assumptions, it is found that the spike-independent process needs to have temporal correlations, through examinations of two kinds of correlation coefficient of consecutive inter-spike intervals. It is also found that the spike-reset process has little effect on the spike auto-correlations and the interval distributions. This suggests that the spike auto-correlation does originate in the temporal correlation of incoming synaptic inputs and the neuronal integration mechanism has little effect on the spike auto-correlation.

Effect of spatial extension on noise-enhanced phase locking in a leaky integrate-and-fire model of a neuron

Signal transmission enhanced by noise has been recently investigated in detail on the single compartment, also referred to as single point, leaky integrate-and-fire model neuron under a subthreshold stimulation. In this paper we study how this phenomenon is influenced by taking into account the spatial characteristics of the neuron. A stochastic two-point leaky integrate-and-fire model, comprising a dendritic compartment and trigger zone, under periodic stimulation is studied. A method of how to measure synchronization between the signal and the output in both, experiments and models, is proposed. This method is based on a distance between the exact periodic spiking, as expected for sufficiently strong and noiseless stimulation, and neuronal activity evoked by a subthreshold signal corrupted by noise. It is shown that qualitatively the same phenomenon, phase-locking enhanced by the noise, as found in the spatially unstructured neuron is produced by the spatially complex neuron. However, quantitatively there are significant differences. Namely, the two-point model neuron is more robust against the noise and therefore its amplitude has to be higher to enhance the signal. Further, it is found that the range of the critical levels of noise is larger for the two-point model than for the single-point one. Finally, the enhancing effect at the optimal noise is more efficient in the single-point model and thus the firing patterns at their optimal noise levels are different in both models.

The Ornstein-Uhlenbeck process does not reproduce spiking statistics of neurons in prefrontal cortex

Cortical neurons of behaving animals generate irregular spike sequences. Recently, there has been a heated discussion about the origin of this irregularity. Softky and Koch (1993) pointed out the inability of standard single-neuron models to reproduce the irregularity of the observed spike sequences when the model parameters are chosen within a certain range that they consider to be plausible. Shadlen and Newsome (1994), on the other hand, demonstrated that a standard leaky integrate-and-fire model can reproduce the irregularity if the inhibition is balanced with the excitation. Motivated by this discussion, we attempted to determine whether the Ornstein-Uhlenbeck process, which is naturally derived from the leaky integration assumption, can in fact reproduce higher-order statistics of biological data. For this purpose, we consider actual neuronal spike sequences recorded from the monkey prefrontal cortex to calculate the higher-order statistics of the interspike intervals. Consistency of the data with the model is examined on the basis of the coefficient of variation and the skewness coefficient, which are, respectively, a measure of the spiking irregularity and a measure of the asymmetry of the interval distribution. It is found that the biological data are not consistent with the model if the model time constant assumes a value within a certain range believed to cover all reasonable values. This fact suggests that the leaky integrate-and-fire model with the assumption of uncorrelated inputs is not adequate to account for the spiking in at least some cortical neurons.

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.

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.

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

Diffusion processes have been extensively used to describe membrane potential behavior. In this approach the interspike interval has a theoretical counterpart in the first-passage-time of the diffusion model employed. Since the mathematical complexity of the first-passage-time problem increases with attempts to make the models more realistic it seems useful to compare the features of different models in order to highlight their relative performance. In this paper we compare the Feller and Ornstein-Uhlenbeck models under three different criteria derived from the level of information available about their parameters. We conclude that the Feller model is preferable when complete knowledge of the characterizing parameters is assumed. On the other hand, when only limited information about the parameters is available, such as the mean firing time and the histogram shape, no advantage arises from using this more complex model.