The effect of a random initial value in neural first-passage-time models

Large coefficient of variation of inter-spike intervals induced by noise current in the resonate-and-fire model neuron

Neuronal spike variability is a statistical property associated with the noise environment. Considering a linearised Hodgkin-Huxley model, we investigate how large spike variability can be induced in a typical stellate cell when submitted to constant and noise current amplitudes. For low noise current, we observe only periodic firing (active) or silence activities. For intermediate noise values, in addition to only active or inactive periods, we also identify a single transition from an initial spike-train (active) to silence dynamics over time, where the spike variability is low. However, for high noise current, we find intermittent active and silence periods with different values. The spike intervals during active and silent states follow the exponential distribution, which is similar to the Poisson process. For non-maximal noise current, we observe the highest values of inter-spike variability. Our results suggest sub-threshold oscillations as a possible mechanism for the appearance of high spike variability in a single neuron due to noise currents.

Some examples of solutions to an inverse problem for the first-passage place of a jump-diffusion process

We report some additional examples of explicit solutions to an inverse first-passage place problem for one-dimensional diffusions with jumps, introduced in a previous paper. If X(t) is a one-dimensional diffusion with jumps, starting from a random position η ∈ [a, b], let be τ a,b the time at which X(t) first exits the interval (a, b), and π a = P (X(τ a,b) ≤ a) the probability of exit from the left of (a, b). Given a probability q ∈ (0, 1), the problem consists in finding the density g of η (if it exists) such that π a = q; it can be seen as a problem of optimization.

A computational approach to first-passage-time problems for Gauss–Markov processes

A new computationally simple, speedy and accurate method is proposed to construct first-passage-time probability density functions for Gauss–Markov processes through time-dependent boundaries, both for fixed and for random initial states. Some applications to Brownian motion and to the Brownian bridge are then provided together with a comparison with some computational results by Durbin and by Daniels. Various closed-form results are also obtained for classes of boundaries that are intimately related to certain symmetries of the processes considered.

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.

Distinguishing the Causes of Firing with the Membrane Potential Slope

In this letter, we aim to measure the relative contribution of coincidence detection and temporal integration to the firing of spikes of a simple neuron model. To this end, we develop a method to infer the degree of synchrony in an ensemble of neurons whose firing drives a single postsynaptic cell. This is accomplished by studying the effects of synchronous inputs on the membrane potential slope of the neuron and estimating the degree of response-relevant input synchrony, which determines the neuron's operational mode. The measure is calculated using the normalized slope of the membrane potential prior to the spikes fired by a neuron, and we demonstrate that it is able to distinguish between the two operational modes. By applying this measure to the membrane potential time course of a leaky integrate-and-fire neuron with the partial somatic reset mechanism, which has been shown to be the most likely candidate to reflect the mechanism used in the brain for reproducing the highly irregular firing at high rates, we show that the partial reset model operates as a temporal integrator of incoming excitatory postsynaptic potentials and that coincidence detection is not necessary for producing such high irregular firing.

Learning optimisation by high firing irregularity

In a network of leaky integrate-and-fire (LIF) neurons, we investigate the functional role of irregular spiking at high rates. Irregular spiking is produced by either employing the partial somatic reset mechanism on every LIF neuron of the network or by using temporally correlated inputs. In both the benchmark problem of XOR (exclusive-OR) and in a general-sum game, it is shown that irrespective of the mechanism that is used to produce it, high firing irregularity enhances the learning capability of the spiking neural network trained with reward-modulated spike-timing-dependent plasticity. These results suggest that the brain may be utilising high firing irregularity for the purposes of learning optimisation.

A modeling study of the effects of membrane afterhyperpolarization on spike interval statistics and on ILD encoding in the lateral superior olive

The lateral superior olive (LSO) is the first nucleus in the ascending auditory pathway that encodes acoustic level information from both ears, the interaural level difference (ILD). This sensitivity is believed to result from the relative strengths of ipsilateral excitation and contralateral inhibition. The study reported here simulated sound-evoked responses of LSO chopper units with a focus on the role of the heterogeneity in membrane afterhyperpolarization (AHP) channels on spike interval statistics and on ILD encoding. A relatively simplified cell model was used so that the effects of interest could be isolated. Specifically, the amplitude and time constant of the AHP conductance within a leaky integrate-and-fire (LIF) cell model were studied. This extends the work of others who used a more physiologically detailed model. Results show that differences in these two parameters lead to both the distinctive chopper response patterns and to the level-dependent interval statistics as observed in vivo. In general, diverse AHP characteristics enable an enhanced contrast across population responses with respect to rate gain and temporal correlations. This membrane heterogeneity provides an internal, cell-specific dimension for the neural representation of stimulus information, allowing sensitivity to ILDs of dynamic stimuli.

Analysis of a stochastic neuronal model with excitatory inputs and state-dependent effects

We propose a stochastic model for the firing activity of a neuronal unit. It includes the decay effect of the membrane potential in absence of stimuli, and the occurrence of time-varying excitatory inputs governed by a Poisson process. The sample-paths of the membrane potential are piecewise exponentially decaying curves with jumps of random amplitudes occurring at the input times. An analysis of the probability distributions of the membrane potential and of the firing time is performed. In the special case of time-homogeneous stimuli the firing density is obtained in closed form, together with its mean and variance.

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.

Spike trains in a stochastic Hodgkin-Huxley system

We consider a standard Hodgkin-Huxley model neuron with a Gaussian white noise input current with drift parameter mu and variance parameter sigma(2). Partial differential equations of second order are obtained for the first two moments of the time taken to spike from (any) initial state, as functions of the initial values. The analytical theory for a 2-component (V,m) approximation is also considered. Let mu(c) (approximately 4.15) be the critical value of mu for firing when noise is absent. Large sample simulation results are obtained for mu<mu(c) and mu>mu(c), for many values of sigma between 0 and 25. For the time to spike, the 2-component approximation is accurate for all sigma when mu=10, for sigma>7 when mu=5 and only when sigma>15 when mu=2. When mu<mu(c), sigma must be large to induce firing so paths are always erratic. As the noise increases, the coefficient of variation (CV) has a well-defined minimum, and then climbs steadily over the range considered. If mu is just above mu(c), when the noise is small, paths are close to deterministic and the standard deviation and CV of the time to spike are small. As sigma increases, some very erratic paths (some almost oscillatory) appear, making the mean, standard deviation and CV of the spike time very large. These erratic paths start to have a large influence, so all three statistics have very pronounced maxima at intermediate sigma. When mu>>mu(c), most paths show similar behavior and the moments exhibit smoothly changing behavior as sigma increases. Thus there are a different number of regimes depending on the magnitude of mu relative to mu(c): one when mu is small and when mu is large; but three when mu is close to and above mu(c). Both for the Hodgkin-Huxley (HH) system and the 2-component approximation, and regardless of the value of mu, the CV tends to about 1.3 at the largest value (25) of sigma considered. We also discuss in detail the problem of determining the interspike interval and give an accurate method for estimating this random variable by decomposing the interval into stochastic and almost deterministic components.

Including Long-Range Dependence in Integrate-and-Fire Models of the High Interspike-Interval Variability of Cortical Neurons

Many different types of integrate-and-fire models have been designed in order to explain how it is possible for a cortical neuron to integrate over many independent inputs while still producing highly variable spike trains. Within this context, the variability of spike trains has been almost exclusively measured using the coefficient of variation of interspike intervals. However, another important statistical property that has been found in cortical spike trains and is closely associated with their high firing variability is long-range dependence. We investigate the conditions, if any, under which such models produce output spike trains with both interspike-interval variability and long-range dependence similar to those that have previously been measured from actual cortical neurons. We first show analytically that a large class of high-variability integrate-and-fire models is incapable of producing such outputs based on the fact that their output spike trains are always mathematically equivalent to renewal processes. This class of models subsumes a majority of previously published models, including those that use excitation-inhibition balance, correlated inputs, partial reset, or nonlinear leakage to produce outputs with high variability. Next, we study integrate-and-fire models that have (non-Poissonian) renewal point process inputs instead of the Poisson point process inputs used in the preceding class of models. The confluence of our analytical and simulation results implies that the renewal-input model is capable of producing high variability and long-range dependence comparable to that seen in spike trains recorded from cortical neurons, but only if the interspike intervals of the inputs have infinite variance, a physiologically unrealistic condition. Finally, we suggest a new integrate-and-fire model that does not suffer any of the previously mentioned shortcomings. By analyzing simulation results for this model, we show that it is capable of producing output spike trains with interspike-interval variability and long-range dependence that match empirical data from cortical spike trains. This model is similar to the other models in this study, except that its inputs are fractional-gaussian-noise-driven Poisson processes rather than renewal point processes. In addition to this model's success in producing realistic output spike trains, its inputs have longrange dependence similar to that found in most subcortical neurons in sensory pathways, including the inputs to cortex. Analysis of output spike trains from simulations of this model also shows that a tight balance between the amounts of excitation and inhibition at the inputs to cortical neurons is not necessary for high interspike-interval variability at their outputs. Furthermore, in our analysis of this model, we show that the superposition of many fractional-gaussian-noise-driven Poisson processes does not approximate a Poisson process, which challenges the common assumption that the total effect of a large number of inputs on a neuron is well represented by a Poisson process.

A neuronal modeling paradigm in the presence of refractoriness

A mathematical characterization of the membrane potential as an instantaneous return process in the presence of refractoriness is investigated for diffusion models of single neuron's activity, assuming that the firing threshold acts as an elastic barrier. Steady-state probability densities and asymptotic moments of the neuronal membrane potential are explicitly obtained in a form that is suitable for quantitative evaluations. For the Ornstein-Uhlenbeck (OU) and Feller neuronal models, closed form expression are obtained for asymptotic mean and variance of the neuronal membrane potential and an analysis of the different features exhibited by the above mentioned models is performed.

Near Poisson-type firing produced by concurrent excitation and inhibition

The effect of inhibition on the firing variability is examined in this paper using the biologically-inspired temporal noisy-leaky integrator (TNLI) neuron model. The TNLI incorporates hyperpolarising inhibition with negative current pulses of controlled shapes and it also separates dendritic from somatic integration. The firing variability is observed by looking at the coefficient of variation (C(V)) (standard deviation/mean interspike interval) as a function of the mean interspike interval of firing (delta tM) and by comparing the results with the theoretical curve for random spike trains, as well as looking at the interspike interval (ISI) histogram distributions. The results show that with 80% inhibition, firing at high rates (up to 200 Hz) is nearly consistent with a Poisson-type variability, which complies with the analysis of cortical neuron firing recordings by Softky and Koch [1993, J. Neurosci. 13(1) 334-530]. We also demonstrate that the mechanism by which inhibition increases the C(V) values is by introducing more short intervals in the firing pattern as indicated by a small initial hump at the beginning of the ISI histogram distribution. The use of stochastic inputs and the separation of the dendritic and somatic integration which we model in TNLI, also affect the high firing, near Poisson-type (explained in the paper) variability produced. We have also found that partial dendritic reset increases slightly the firing variability especially at short ISIs.

On a non-Markov neuronal model and its approximations

Single neuron's activity modeling is considered with reference to some earlier contributions in which a non-Markov Gaussian process is assumed to describe the time course of the neuron's membrane potential. After re-formulating the problem in a rigorous framework and pinpointing the limits of validity of such a model, the available results on the firing probability density are compared with those obtained by us by means of an ad hoc numerical algorithm implemented for the leaky integrator diffusion firing model and with some data constructed by a simulation procedure of non-Markov Gaussian processes with pre-assigned covariances. Throughout this paper, the notion of 'correlation time' plays a fundamental role for the neuronal coding process modeling.

A stochastic model for circulatory transport in pharmacokinetics

A new stochastic model for the residence time distribution of a drug injected instantaneously into the circulatory system is proposed and analyzed. The properties of the residence time are derived from the assumptions made about the cycle time distribution and the rule for elimination. This rule is given by the probability distribution of the number of cycles needed for elimination of a randomly selected molecule of the drug. Only the geometric distribution has been previously used for this purpose. Its transformation is applied here to get a boundary for the residence time. Other discrete distributions are applied with a view to describing different experimental situations. Suitable continuous probability distributions for the cycle time description are discussed.

First-passage-time problem for simulated stochastic diffusion processes

Solving the first-passage-time problem for one-dimensional stochastic diffusion processes is a task with many applications in biomedical research. It has been noted (Musila and Lánský, Int. J. Biomed. Comput. 31, 233-245, 1992) that the first-passage time is overestimated if computed as the time when the simulated trajectory of the process crosses the threshold. It is studied in this paper how the error depends on the simulation step and on the parameters of the process. We propose an adaptive algorithm to make the simulation faster. The presented examples are related to neuronal modelling, but application in other fields is straightforward.

A stochastic model for neuronal bursting

A new stochastic model for bursting of neuronal firing is proposed. It is based on stochastic diffusion and related to the first passage time problem. However, the model is not of renewal type. Its form and parameters are physiologically interpretable. Parametric and non-parametric inferential issues are discussed.

Stochastic model neuron without resetting of dendritic potential: application to the olfactory system

A two-dimensional neuronal model, in which the membrane potential of the dendrite evolves independently from that at the trigger zone of the axon, is proposed and studied. In classical one-dimensional neuronal models the dendritic and axonal potentials cannot be distinguished, and thus they are reset to resting level after firing of an action potential, whereas in the present model the dendritic potential is not reset. The trigger zone is modelled by a simplified leaky integrator (RC circuit) and the dendritic compartment can be described by any of the classical one-dimensional neuronal models. The new model simulates observed features of the firing dynamics which are not displayed by classical models, namely positive correlation between interspike intervals and endogenous bursting. It gives a more natural account of features already accounted for in previous models, such as the absence of an upper limit for the coefficient of variation of intervals (i.e. irregular firing). It allows the first- and second-order neurons of the olfactory system to be described with the same basic assumptions, which was not the case in one-point models. Nevertheless it keeps the main qualitative properties found previously, such as the existence of three regimens of firing with increasing stimulus concentration and the sigmoid shape of the firing frequency of first-order neurons as a function of the logarithm of stimulus concentration.

Coding of odor intensity

A model for coding of odor intensity in the first two neuronal layers of olfactory systems is proposed. First, the occupation and activation by odorant molecules of receptor proteins of different types borne by the first order neurons are described as birth and death processes. The occupation (birth) rate depends on the concentration of the odorant, whereas the probability of activation of an occupied receptor depends on the type of the odorant. Second, the spike generation mechanism proposed for the first order neuron depends on the level of the generator potential evoked by the activated receptors and on a time-decaying threshold which is reset to infinity after each spike. The various resulting stochastic regimes of firing activity at different concentrations are described. Third, each second order neuron is influenced by excitation coming from numerous first order neurons, lateral inhibition from other second order neurons, and self-inhibition. All these incoming signals are integrated at the second order neuron. The firing activity of the first and second order neurons is modeled by a first passage time scheme. For both types of neuron the shapes of the curves predicted by the model for the mean firing frequency as a function of stimulus concentration are shown to be in accordance with available experimental results.

Effects of afterhyperpolarization on neuronal firing

Stein's model for a neuron is studied. This model is modified to take into account the effects of afterhyperpolarization on the neuronal firing. The relative refractory phase, following the absolute one, is modelled by a time-increasing amplitude of postsynaptic potentials and it is also incorporated into the model. Besides the simulation of the model, some theoretical results and approximation methods are derived. Afterhyperpolarization tends to preserve the linearity of the frequency transfer characteristic and it has a limited effect on the moments of the interspike intervals in general. The main effects are seen at high firing rates and in the removal of short intervals in the interspike interval histogram.

Generalized Stein's model for anatomically complex neurons

A neuron with a large dendritic structure is considered. The number of synapses located on the dendrites is substantially higher than on the soma. The synaptic input effect on the neuronal excitability decreases with distance between a synapse ending and the trigger zone. Two areas are distinguished in accordance with the effect of synaptic input--dendritic and somatic. The dendritic area, when compared to the soma, is characterized by much higher intensity of its activation but the amplitudes of synaptically evoked changes of the membrane potential at the trigger zone are in general small. This situation is suitable for a diffusion approximation. However, on the soma, especially in the proximity of the trigger zone, the membrane potential changes are a large fraction of the threshold depolarization. The membrane potential at the trigger zone is modelled by a one-dimensional stochastic process. The diffusion Ornstein-Uhlenbeck process serves as a basis of the model; however, at the moments of somatic synapses activation its voltage changes in jumps. Their sizes represent the amplitudes of the evoked postsynaptic potentials. The unimodal histograms of interspike intervals can be explained by the model. The values of the coefficient of variation greater than one are connected with substantial inhibition.