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PubMed1967

Neuronal spike trains and stochastic point processes. I. The single spike train.

In a growing class of neurophysiological experiments, the train of impulses ("spikes") produced by a nerve cell is subjected to statistical treatment involving the time intervals between spikes. The statistical techniques available for the analysis of single spike trains are described and related to the underlying mathematical theory, that of stochastic point processes, i.e., of stochastic processes whose realizations may be described as series of point events occurring in time, separated by random intervals. For single stationary spike trains, several orders of complexity of statistical treatment are described; the major distinction is that between statistical measures that depend in an essential way on the serial order of interspike intervals and those that are order-independent. The interrelations among the several types of calculations are shown, and an attempt is made to ameliorate the current nomenclatural confusion in this field. Applications, interpretations, and potential difficulties of the statistical techniques are discussed, with special reference to types of spike trains encountered experimentally. Next, the related types of analysis are described for experiments which involve repeated presentations of a brief, isolated stimulus. Finally, the effects of nonstationarity, e.g. long-term changes in firing rate, on the various statistical measures are discussed. Several commonly observed patterns of spike activity are shown to be differentially sensitive to such changes. A companion paper covers the analysis of simultaneously observed spike trains.

Action Potentials↗
PubMed1999

Stochastic multiplicative processes with reset events.

We study a stochastic multiplicative process with reset events. It is shown that the model develops a stationary power-law probability distribution for the relevant variable, whose exponent depends on the model parameters. Two qualitatively different regimes are observed, corresponding to intermittent and regular behavior. In the boundary between them, the mean value of the relevant variable is time independent, and the exponent of the stationary distribution equals -2. The addition of diffusion to the system modifies in a nontrivial way the profile of the stationary distribution. Numerical and analytical results are presented.

Journal Article↗
PubMed1994

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.

Algorithms↗
PubMed1984

Identification and estimation algorithm for stochastic neural system.

An algorithm for the estimation of stochastic processes in a neural system is presented. This process is defined here as the continuous stochastic process reflecting the dynamics of the neural system which has some inputs and generates output spike trains. The algorithm proposed here is to identify the system parameters and then estimate the stochastic process called neural system process here. These procedures carried out on the basis of the output spike trains which are supposed to be the data observed in the randomly missing way by the threshold time function in the neural system. The algorithm is constructed with the well-known Kalman filters and realizes the estimation of the neural system process by cooperating with the algorithm for the parameter estimation of the threshold time function presented previously (Nakao et al., 1983). The performance of the algorithm is examined by applying it to the various spike trains simulated by some artificial models and also to the neural spike trains recorded in cat's optic tract fibers. The results in these applications are thought to prove the effectiveness of the algorithm proposed here to some extent. Such attempts, we think, will serve to improve the characterizing and modelling techniques of the stochastic neural systems.

Animals↗
PubMed1999

Locations of radiation-produced DNA double strand breaks along chromosomes: a stochastic cluster process formalism.

Ionizing radiation produces DNA double strand breaks (DSBs) in chromosomes. For densely ionizing radiation, the DSBs are not spaced randomly along a chromosome: recent data for size distributions of DNA fragments indicate break clustering on kbp-Mbp scales. Different DSB clusters on a chromosome are typically made by different, statistically independent, stochastically structured radiation tracks, and the average number of tracks involved can be small. We therefore model DSB positions along a chromosome as a stationary Poisson cluster process, i.e. a stochastic process consisting of secondary point processes whose locations are determined by a primary point process that is Poisson. Each secondary process represents a break cluster, typically consisting of 1-10 DSBs in a comparatively localized stochastic pattern determined by chromatin geometry and radiation track structure. Using this Poisson cluster process model, which we call the randomly located clusters (RLC) formalism, theorems are derived for how the DNA fragment-size distribution depends on radiation dose. The RLC dose-response relations become non-linear when the dose becomes so high that DSB clusters from different tracks overlap or adjoin closely. The RLC formalism generalizes previous models, fits current data adequately and facilitates mechanistically based extrapolations from high-dose experiments to the much lower doses of interest for most applications.

Chromosome Aberrations↗
PubMed2003

Queuing transitions in the asymmetric simple exclusion process.

Stochastic driven flow along a channel can be modeled by the asymmetric simple exclusion process. We confirm numerically the presence of a dynamic queuing phase transition at a nonzero obstruction strength, and establish its scaling properties. Below the transition, the traffic jam is macroscopic in the sense that the length of the queue scales linearly with system size. Above the transition, only a power-law shaped queue remains. Its density profile scales as deltarho approximately x(-nu) with nu=1/3, and x is the distance from the obstacle. We construct a heuristic argument, indicating that the exponent nu=1/3 is universal and independent of the dynamic exponent of the underlying dynamic process. Fast bonds create only power-law shaped depletion queues, and with an exponent that could be equal to nu=2/3, but the numerical results yield consistently somewhat smaller values nu approximately 0.63(3). The implications of these results to faceting of growing interfaces and localization of directed polymers in random media, both in the presence of a columnar defect are pointed out as well.

Journal Article↗
PubMed1991

Development of skin tumors in hairless mice after discontinuation of ultraviolet irradiation.

The development of skin tumors (mainly squamous cell carcinomas) in hairless Skh-HR1 mice after discontinuation of a course of daily UV irradiations (wavelengths, 280-370 nm) is compared to that when the daily irradiations are continued. Under conditions of continued daily exposures 50% of 22 animals contracted tumors with diameters of at least 4 mm in 135 days. With exposures stopped after 35 or 19 days (2 groups with 24 and 23 mice) this time interval increased to 280 and 645 days, respectively; the rate at which multiple tumors developed on the mice was correspondingly lower. A mathematical model, derived from a larger experiment (223 mice) with different levels of chronic UV exposure, successfully predicts the tumor development after discontinuation of UV exposure. This model is similar to those used in risk assessments for skin cancers in human populations, e.g., in relation to stratospheric ozone depletion, sunbeds, etc. The model separates UV-driven processes from purely time-dependent processes. These stochastic processes, described by Weibull statistics, form stages in the tumorigenesis. This interpretation of the data indicates that a late, UV-independent stage occurs between the smallest observable tumors and larger ones with diameters of over 4 mm. This could be a simple growth stage, but histopathology suggests that it may also entail a transition from actinic keratosis to squamous cell carcinoma.

Animals↗
PubMed1996

The dynamical theory of coevolution: a derivation from stochastic ecological processes.

In this paper we develop a dynamical theory of coevolution in ecological communities. The derivation explicitly accounts for the stochastic components of evolutionary change and is based on ecological processes at the level of the individual. We show that the coevolutionary dynamic can be envisaged as a directed random walk in the community's trait space. A quantitative description of this stochastic process in terms of a master equation is derived. By determining the first jump moment of this process we abstract the dynamic of the mean evolutionary path. To first order the resulting equation coincides with a dynamic that has frequently been assumed in evolutionary game theory. Apart from recovering this canonical equation we systematically establish the underlying assumptions. We provide higher order corrections and show that these can give rise to new, unexpected evolutionary effects including shifting evolutionary isoclines and evolutionary slowing down of mean paths as they approach evolutionary equilibria. Extensions of the derivation to more general ecological settings are discussed. In particular we allow for multi-trait coevolution and analyze coevolution under nonequilibrium population dynamics.

Animals↗
PubMed1996

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.

Action Potentials↗
PubMed1995

Hominoid phylogeny estimated by model selection using goodness of fit significance tests.

Phylogeny estimation from nucleotide sequence data may be thought of as a problem of choosing between different evolutionary models that vary with the branching pattern of the phylogeny and with the stochastic process of nucleotide sequence change occurring on the branches of the phylogenetic tree. Thus, each evolutionary model consists of both a particular stochastic process and a particular phylogeny. Such models produce multinomial distributions of nucleotide character patterns. As first suggested by Cavalli-Sforza and Edwards [Evolution 21: 550-570 (1967)] the distribution of patterns expected under each model can be compared to the actual observed distribution of patterns by a goodness of fit statistic such as the loglikelihood ratio G2 or Pearson's X2 after the numerical parameters for the model have been chosen to minimize the respective statistic. For each evolutionary model, the probability P of getting a value of the goodness of fit statistic greater than the observed value is computed. A very small P value means that either a rare event has occurred or that the model is false. Employing for each of 16 models a stochastic process which has 12 parameters to describe the mode of nucleotide change on each branch of each putative phylogenetic tree, we examined all 15 unrooted dichotomously branching arrangements of orthologous noncoding sequences from the gamma hemoglobin genomic region of the five hominoids (gibbon, orangutan, gorilla, chimpanzee, and human) plus the branching arrangement with a trichotomous separation of gorilla, chimpanzee, and human. Of these 16 models, all had P values less than 0.01, except for the arrangement of human joined by chimpanzee, in turn joined by gorilla, and then orangutan and gibbon. This analysis allows convincing claims to be made about hominoid phylogenetic relationships by testing the applicability of the assumed stochastic process for nucleotide sequence evolution at the same time as testing the inferred phylogenetic branching arrangement.

Animals↗
PubMed2002

Additive noise, Weibull functions and the approximation of psychometric functions.

The Weibull function is frequently chosen to define psychometric functions. Tyler and Chen (Vis. Res. 40 (2000) 3121) criticised the high-threshold postulate implied by the Weibull function and argued that this function implies the assumption of multiplicative noise. It will be shown in this paper that in fact the Weibull function is compatible with the assumption of additive noise, and that the Weibull function may be generalised to the case of detection not being high threshold. The derivations rest, however, on a representation of sensory activity lacking a satisfying degree of generality. Therefore, a more general representation of sensory activity in terms of stochastic processes will be suggested, with detection being defined as a level-crossing process, containing the original representation as a special case. Two classes of stochastic processes will be considered: one where the noise is assumed to be additive, stationary Gaussian, and another resulting from cascaded Poisson processes, representing a form of multiplicative noise. While Weibull functions turn out to approximate well psychometric functions generated by both types of stochastic processes, it also becomes obvious that there is no simple interpretation of the parameters of the fitted Weibull functions. Moreover, corresponding to Tyler and Chen's discussion of the role of multiplicative noise particular sources of this type of noise will be considered and shown to be compatible with the Weibull. It is indicated how multiplicative noise may be defined in general; however, it will be argued that in the light of certain empirical data the role of this type of noise may be negligible in most detection tasks.

Contrast Sensitivity↗
PubMed2004

Extinction times and moment closure in the stochastic logistic process.

We investigate the statistics of extinction times for an isolated population, with an initially modest number M of individuals, whose dynamics are controlled by a stochastic logistic process (SLP). The coefficient of variation in the extinction time V is found to have a maximum value when the death and birth rates are close in value. For large habitat size K we find that Vmax is of order K1/4 / M1/2, which is much larger than unity so long as M is small compared to K1/2. We also present a study of the SLP using the moment closure approximation (MCA), and discuss the successes and failures of this method. Regarding the former, the MCA yields a steady-state distribution for the population when the death rate is low. Although not correct for the SLP model, the first three moments of this distribution coincide with those calculated exactly for an adjusted SLP in which extinction is forbidden. These exact calculations also pinpoint the breakdown of the MCA as the death rate is increased.

Animals↗
PubMed1998

Transit compartments versus gamma distribution function to model signal transduction processes in pharmacodynamics.

Delayed effects for pharmacodynamic responses can be observed for many signal transduction processes. Three approaches are summarized in this report to describe such effects caused by cascading steps: stochastic process model, gamma distribution function, and transit compartment model. The gamma distribution function, a probability density function of the waiting time for the final step in a stochastic process model, is a function of time with two variables: number of compartments N, and the expected number of compartments occurring per unit time k. The parameter k is equal to 1/tau, where tau is the mean transit time in the stochastic process model. Effects of N and k on the gamma distribution function were examined. The transit compartment model can link the pharmacokinetic profile of the tested compound, receptor occupancy, and cascade steps for the signal transduction process. Time delays are described by numbers of steps, the mean transit time tau, and the amplification or suppression of the process as characterized by a power coefficient gamma. The effects of N, tau, and gamma on signal transduction profiles are shown. The gamma distribution function can be utilized to estimate N and k values when the final response profile is available, but it is less flexible than transit compartments when dose-response relationships, receptor dynamics, and efficiency of the transduction process are of concern. The transit compartment model is useful in pharmacokinetic/pharmacodynamic modeling to describe precursor/product relationships in signal transduction process.

Animals↗
PubMed1995

Trivial influences: a doubly stochastic Poisson process model permits the detection of arbitrarily small electromagnetic signals.

If a weak, exogenous, extremely low-frequency (ELF) electric or magnetic field is to produce biological sequelae, then there must exist averaging sufficient to lift some primary effect of that field above the endogenous stochastic variations of the biological system. One way in which a field could accomplish this is by changing the intensity of some stochastic operation that controls an important and not trivially reversible biological transformation. In this paper, this operation is modeled as a doubly stochastic Poisson process. It is then shown, first, that (in theory) even a minuscule exogenous influence might appreciably shift the incidence of a sufficiently rare transformation and, second, that this shift might be observable if a trial were allowed to run long enough over a sufficiently large population of exposed entities.

Algorithms↗
PubMed2002

Testing for non-linearity in EEG signal of healthy subjects.

Spontaneous EEG of 21 healthy human subjects obtained by standard procedure of recording is analysed using non-linear prediction methods to check whether the signals were generated by a non-linear dynamics process or by a linear stochastic process. The test for non-linearity is performed by surrogate data method with non-linear prediction error as the test statistic. The null hypothesis that EEG signal (in rest, with eyes closed) is generated by linear stochastic process can be rejected in 17 cases (5%) out of the 336 (21 subjects, 16 channels) studied epochs. However, most of these rejections concern 3 subjects. The 88% of rejections of the null hypothesis concern frontal channels. The null hypothesis is not rejected for epochs recorded with eyes open and during photostimulation.

Algorithms↗
PubMedDate not supplied

[Dynamic mobility of the histidine-containing domain of spin-labeled lysozyme].

The hen egg-white lysozyme was modified by the spin label (2,2,6,6-tetramethylpiperidine-N1-oxyl-4-iodacetamide) at the single histidine residue His-15. The rotational correlation time of the molecular carrier was found to be defined by the mobility of the histidine-bearing domain and not influenced by the protein monomer shape at pH 4.7 and dimer shape at pH 7.1. The dependence of viscosity at 1 degree C on the distance between outer wide peaks in the immobilized EPR spectra enabled us to evaluate rotational correlation time of the domain. The molecular mass of the latter was close to the data obtained by X-ray analysis. The spin label was highly mobile at room temperature, as the EPR spectrum displayed the triple shape; at 1 degree C it was immobilized. The new general approach to the EPR spectra simulation was applied to all experimental EPR spectra. This approach is based on a substitution of an undefined stochastic process of the spin label reorientation relative to the lysozyme domain by the defined modelled stochastic processes: axial rotation of the nitroxide relative to the preferable axis and angular oscillations of the nitroxide relative to axes of the molecular coordinate system. Each of the modelled stochastic processes leads to a relative partial averaging of the magnetic tensor components. A set of discrete partially averaged states is introduced with the relative cluster of the spin-labelled molecules. The resulting EPR spectrum is assumed to be the sum of EPR spectra from all the clusters. A good fitting of all simulated EPR spectra is obtained.

Animals↗
PubMed1993

Analyzing and modeling fractal intensity point processes.

Fractal intensity point processes--doubly stochastic point processes with a fractal waveform intensity process--are required to describe the discharge patterns recorded from the auditory and visual systems. The Fano factor--the ratio of the variance of the number of events in an interval to the mean of this number--captures the self-similar characteristics of the intensity via two quantities: fractal dimension and fractal time. The fractal dimension is the exponent of the asymptotic power law behavior of the Fano factor with interval duration. The fractal time delineates long-term fractal behavior from short-term characteristics of the data. The average rate and self-similarity parameter of the intensity process, absolute and relative refractory effects, and serial dependence all modify the fractal time. To generate fractal intensity point processes, stochastic fractal processes are derived by applying memoryless, nonlinear transformations to fractional Gaussian noise. The intensity's amplitude distribution in combination with the Fano factor form criteria to choose the transformation that best describes data.

Auditory Perception↗
PubMed2001

Stochastic exclusion processes with extended hopping.

We study the effects of extended hopping in the stochastic asymmetric simple exclusion process (ASEP) of one dimension. A modified ASEP with four parameters is proposed. The current and bulk density are calculated. Two different phases with a curved boundary are observed. The phase of maximum current is absent. We also present the unusual density fluctuations near both boundaries. The characteristic of the stochastic dynamics with extended hopping is pointed out.

Journal Article↗