Statistical analysis and stochastic modeling of neuronal spike-train activity

A Markov model for interspike interval distributions of auditory cortical neurons that do not show periodic firings

Spontaneous firing properties of individual auditory cortical neurons are interpreted in terms of local and global order present in functioning brain networks, such as alternating "up" and "down" states. A four-state modulated Markov process is used to model neuronal firings. The system alternates between a bound and an unbound state, both with Poisson-distributed lifetimes. During the unbound state, active and closed states alternate with Poisson-distributed lifetimes. Inside the active state, spikes are generated as a realization of a Poisson process. This combination of processes constitutes a four-state modulated Markov process, determined by five independent parameters. Analytical expressions for the probability density functions (pdfs) that describe the interspike interval (ISI) distribution and autocorrelation function are derived. The pdf for the ISI distribution is shown to be a linear combination of three exponential functions and is expressed through the five system parameters. Through fitting experimental ISI histograms by the theoretical ones, numerical values of the system parameters are obtained for the individual neurons. Both Monte Carlo simulations and goodness-of-fit tests are used to validate the fitting procedure. The values of the estimated system parameters related to the active-closed and bound-unbound processes and their independence on the neurons' mean firing rate suggest that the underlying quasi-periodic processes reflect properties of the network in which the neurons are embedded. The characteristic times of autocorrelations, determined by the bound-unbound and active-closed processes, are also independent of the neuron's firing rate. The agreement between experimental and theoretical ISI histograms and autocorrelation functions allows interpretation of the system parameters of the individual neurons in terms of slow and delta waves, and high-frequency oscillations observed in cortical networks. This procedure can identify and track the influence of changing brain states on the single-unit firing patterns in experimental animals.

Distribution-free graphical and statistical analysis of serial dependence in neuronal spike trains

Two-dimensional 'joint' interval distributions of sequential interspike (ISIs) are a commonly used tool in neuronal spike train analysis. We present and evaluate here a modification of the joint interval plot using ranked ISIs. This modification provides clearer graphical evaluation of serial dependence in ISI sequences, a distribution-free basis for isolating changes in serial dependence across experimental treatments from changes in ISI distributions, and a basis for unambiguous statistical tests of serial dependence and stationarity. To validate this method and illustrate the advantages of its use we have applied it to both single-neuron spike trains recorded from cultured mammalian spinal cord neurons and artificial spike trains generated by stochastic models with defined burst envelopes and serial dependencies.

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.

Electrical activity of neurons in the ventrolateral septum and bed nuclei of the stria terminalis in suckled rats: statistical analysis gives evidence for sensitivity to oxytocin and for relation to the milk-ejection reflex

Our previous results obtained by lesioning or stimulating the ventrolateral part of the lateral septum and the bed nuclei of the stria terminalis suggested that this area is involved in the control of milk ejection pattern in rats. The present study was undertaken with the aim of testing ventrolateral part of the lateral septum-bed nuclei of the stria terminalis neurons as a putative link of the neuronal network controlling the bursting activity of oxytocin neurons in suckled lactating rats (anaesthetized with urethane). Ventrolateral part of the lateral septum-bed nuclei of the stria terminalis neurons were recorded simultaneously with hypothalamic oxytocin neurons in either the paraventricular or supraoptic nucleus in rats with (n = 26) or without (n = 29) periodic milk ejections. Analysis of their firing pattern enabled differentiation of two subgroups: type I, characterized by numerous high frequency spikes, often grouped in clusters; and type II with very few or no high frequency clusters of spikes. The probability density function of the interspike intervals of both patterns could be modelled using a mixture of two log-normal distributions, the parameters of which differed significantly. The presence of absence of milk ejections did not influence the overall mean level of activity (2.0 +/- 0.5 and 1.9 +/- 0.4 spikes/s, respectively). However, the characteristics of the type I firing pattern were affected by the presence of the milk-ejection reflex. The average level of activity was not always constant and 16/55 ventrolateral part of the lateral septum-bed nuclei of the stria terminalis neurons displayed cyclical activity (from 0.6 +/- 0.2 to 4.0 +/- 0.5 spikes/s) both in the presence (n = 8) and absence (n = 8) of the milk-ejection reflex. In five of eight neurons recorded during milk-ejection reflex, the cycles in firing were clearly correlated with the bursting of oxytocin neurons. These five neurons exhibited the type I firing pattern. The three remaining neurons and the eight neurons recorded in the absence of milk-ejection reflex displayed the type II firing pattern. Oxytocin (1-2 ng = 0.45-0.9 mU) was injected into the third ventricle (i.c.v.) in order to examine the possible involvement of ventrolateral part of the lateral septum-bed nuclei of the stria terminalis neurons in the facilitatory effect of oxytocin on the reflex.(ABSTRACT TRUNCATED AT 400 WORDS)

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.

Steady discharges of X and Y retinal ganglion cells of cat under photopic illuminance

The discharges of ON- and OFF-center X and Y retinal ganglion cells in the presence of stationary patterns or of a uniform field of photopic luminance were recorded from urethane-anesthetized adult cats. The interval statistics and power spectra of these discharges were determined from these discharge records. The patterned stimuli were selected and positioned with respect to a cell's receptive field so as to generate steady discharges that were different in mean discharge rate from that cell's discharge for the diffuse field. The interval statistics of discharges recorded for diffuse or patterned illumination for all cell types can be modeled, approximately, as coming from renewal processes with gamma-distributed intervals. The gamma order of the interval distributions was found to be nearly proportional to the mean discharge rate for X cells, but not for Y cells. Typical values for the gamma orders and their dependence on mean rate for different cell types are given. The same model of a renewal process with gamma-distributed intervals is used to model the measured power spectra and performs well. When the gamma order is proportional to mean rate, the power spectral density at low temporal frequencies is independent of discharge rate. Gamma order was proportional to mean rate for X cells but not for Y cells. Nonetheless, the power spectral densities of both cell types at low frequencies were approximately independent of discharge rate. Hence, noise in this band of frequencies can be considered additive. The consequences of departures from the renewal process and of the gamma order not being proportional to mean rate are considered. The significance of different rates of discharge for signaling is discussed.

Identification of bursts in spike trains

A computer algorithm to identify 'bursts' in trains of spikes is described. The algorithm works by constructing a histogram of interspike intervals, then analyzing the histogram to detect the critical interval value in the distribution that represents the break between short intervals within a burst and the longer intervals between bursts. When such a value is found, it is used as the 'threshold' to determine those intervals in the spike train that lie within a burst and those that lie between bursts and, thereby, to identify the beginning and end of each burst in the train. The validity of the bursts is evaluated with a chi-square test. The performance of the algorithm and how it can be assessed is discussed.

Synaptic noise in spike trains of normal and denervated electroreceptor organs

The sequence of interspike intervals of the spontaneous activity in denervated electroreceptor organs of the catfish is analysed with several statistical techniques: visual inspection of dot displays, interval histograms, serial correlograms, tests for stationary and trends, and tests for renewal properties, based on the spectrum of intervals. The interspike interval train of primary afferents can usually be treated as a renewal process. Following denervation, a number of non-renewal properties emerge. The interval histogram of the renewal spike trains can be fitted well with a gamma probability density function; non-renewal cases need a more complex approach. We propose that the stochastic fluctuations in interval duration arise from randomly occurring quantal depolarizations (Stein's model of stochastic neuronal excitations. Two important properties with regard to synaptic transduction are: afferent firing results from capture of a limited number of depolarizing quanta; and the quantal input rate and the threshold for firing appear to be correlated.

Quantitative aspects of transduction in an electroreceptor organ studied by means of experimental manipulation of the interspike interval

The interspike interval histogram of spontaneously firing electroreceptor organs of freshwater catfish (Ictalurus nebulosus LeS.) fits well with the probability density function of a gamma distribution. The shape parameter r of this probability density function can be used as a measure for the firing threshold of the spike generator, and the scale parameter lambda provides a measure for the input rate of synaptic quanta. Here, we have studied the physiological meaning of the two parameters of the fitted gamma probability density function by manipulating the mean firing rate. This was done in two ways: "frequency clamp" stimulation and a change in temperature. Frequency clamp stimulation, compensating for adaptation of the response, keeps the afferent firing rate at a fixed value for a limited period. We show that within such a period not only the mean spike rate remains constant, but these spike trains can be regarded as renewal series: the spike generator is functionally uncoupled from the frequency-dependent parts of the transduction path. Concerning the gamma parameters, two types of responses are found: at strong excitation lambda increases, r does not change; with small stimuli r is inversely proportional to the stimulus strength, lambda remains unchanged. This is difficult to explain when stimulus transduction acts via the synaptic input rate only. Stimulation seems to influence the firing threshold of the spike generator directly. At strong inhibition a regular firing pattern suddenly takes over; the major part of the variability suddenly disappears from the spike train. A change in temperature causes a shift in lambda (Q10 approximately 2.3), but not in r. This points to a temperature effect on the input rate only.

Spontaneous neuronal firing patterns in the occipital cortex of developing rats

Spontaneous action potentials have been recorded in the cerebral cortex of developing rats under chloral hydrate anaesthesia. These "spike-trains" were subjected to a multivariate analysis of parameters describing different aspects of temporal patterning within each train. Three parameters which reflected clustering of action potentials in, respectively, the range of: (i) milliseconds (the modal interval, derived from the interval histograms); (ii) seconds (the mean "burst period", i.e. the time elapsing between the onset of successive clusters of relatively short intervals); and (iii) minutes (the coefficient of variation for the number of spikes in consecutive 2 min time bins) showed distinctive developmental patterns.

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

The effect of a random initial value is examined in several stochastic integrate-and-fire neural models with a constant threshold and a constant input. The three models considered are approximations of Stein's model, namely: (1) a leaky integrator with deterministic trajectories, (2) a Wiener process with drift, and (3) an Ornstein-Uhlenbeck process. For model 1, different distributions for the initial value lead to commonly observed interspike interval distributions. For model 2, a discrete and a uniform distribution for the initial value are examined along with some parameter estimation procedures. For model 3, with a truncated normal distribution for the initial value, the coefficient of variation is shown to be greater than 1, and as the threshold becomes large the first-passage-time distribution approaches an exponential distribution. The relationships among the models and between them and previous models are also discussed, along with the robustness of the model assumptions and methods of their verification. The effects of a random initial value are found to be most pronounced at high firing rates.

The analysis of neuronal discharge sequences: change-point estimation and comparison of variances

A key problem in neurophysiology is to determine whether, after presentation of a stimulus, there has been a modification in the discharge of a recorded neuron and if so, an attempt is made to estimate the latency of the response. The estimation problem can be considered as that of the estimation of a change-point in a sequence of random variables. The gamma distribution is adequate to model the distribution of intervals between action potentials for different types of neurons. Simulations show that the maximum likelihood estimator based on this model is efficient and robust. An additional problem, in the case of experiments in which a movement follows the stimulus, is to determine whether a response is related to the stimulus or to the movement. A test based on the comparison of marginal scales of a bivariate distribution is proposed. The whole procedure has been tested in simulation and with real examples.