Poisson-like Process Research Articles

How variable progenitor clones construct a largely invariant neocortex.

The neocortex contains a vast collection of diverse neurons organized into distinct layers. While nearly all neocortical neurons are generated by radial glial progenitors (RGPs), it remains largely unclear how a complex yet organized neocortex is constructed reliably and robustly. Here, we show that the division behavior and neuronal output of RGPs are highly constrained with patterned variabilities to support the reliable and robust construction of the mouse neocortex. The neurogenic process of RGPs can be well-approximated by a consistent Poisson-like process unfolding over time, producing deep to superficial layer neurons progressively. The exact neuronal outputs regarding layer occupation are variable; yet, this variability is constrained systematically to support all layer formation, largely reflecting the variable intermediate progenitor generation and RGP neurogenic entry and exit timing differences. Together, these results define the fundamental features of neocortical neurogenesis with a balanced reliability and variability for the construction of the complex neocortex.

Temporal Patterns of Spontaneous Fixational Eye Movements: The Influence of Basal Ganglia.

Spontaneity is a unique feature of the nervous system. One of the fundamentally critical and recognized forms of spontaneous motor activity is witnessed in the visuomotor system. Microsaccades, the miniature spontaneous eye movements, are critical for the visual perception. We hypothesized that microsaccades follow specific temporal patterns that are modulated by the basal ganglia output. We used high-resolution video-oculography to capture microsaccades in 48 subjects (31 healthy and 17 with Parkinson's disease) when subjects were asked to hold their gaze on a straight-ahead target projected on white background. We analyzed spontaneous discharge patterns of microsaccades. The first analysis considering coefficient of variation in intersaccadic interval distribution demonstrated that microsaccades in Parkinson's disease are more dispersed than the control group. The second analysis scrutinized microsaccades' temporal variability and revealed 3 distinct occurrence patterns: regular rhythmic, clustered, and randomly occurring following a Poisson-like process. The regular pattern was relatively more common in Parkinson's disease. Subthalamic DBS modulated this temporal pattern. The amount of change in the temporal variability depended on the DBS-induced volume of tissue activation and its overlap with the subthalamic nucleus. The third analysis determined the autocorrelations of microsaccades within 2-second time windows. We found that Parkinson's disease altered local temporal organization in microsaccade generation, and DBS had a modulatory effect. The microsaccades occur in 3 temporal patterns. The basal ganglia are one of the modulators of the microsaccade spontaneity.

Dark counts double switching rates in NbTiN Superconducting Nanowire Single Photon Detectors

In this work we present a study of the dark counts rate in a NbTiN Superconducting Nanowire Single Photon Detectors (SNSPD). We measure the distribution of the time intervals elapsed between two consecutive dark pulses at the fixed temperature of 4,2K. Due to the stochastic nature of the dark counts, the distribution is expected to have a Poisson shape but what we observe is a combination of two Poisson-like processes. A further analysis of the distributions dependence on the bias current highlights that the weight of the two process is not constant. In the scenario presented by Ejrnaes et al. [1], this result can confirm that, in this temperature regime, dark counts are generated mainly by multiple consecutive fluctuation events and the contribution to the dark counts rate coming from single fluctuations increases in the high bias current region.

Shear-driven flow of athermal, frictionless, spherocylinder suspensions in two dimensions: Particle rotations and orientational ordering.

We use numerical simulations to study the flow of a bidisperse mixture of athermal, frictionless, soft-core two-dimensional spherocylinders driven by a uniform steady-state simple shear applied at a fixed volume and a fixed finite strain rate γ[over ̇]. Energy dissipation is via a viscous drag with respect to a uniformly sheared host fluid, giving a simple model for flow in a non-Brownian suspension with Newtonian rheology. Considering a range of packing fractions ϕ and particle asphericities α at small γ[over ̇], we study the angular rotation θ[over ̇]_{i} and the nematic orientational ordering S_{2} of the particles induced by the shear flow, finding a nonmonotonic behavior as the packing ϕ is varied. We interpret this nonmonotonic behavior as a crossover from dilute systems at small ϕ, where single-particle-like behavior occurs, to dense systems at large ϕ, where the geometry of the dense packing dominates and a random Poisson-like process for particle rotations results. We also argue that the finite nematic ordering S_{2} is a consequence of the shearing serving as an ordering field, rather than a result of long-range cooperative behavior among the particles. We arrive at these conclusions by consideration of (i) the distribution of waiting times for a particle to rotate by π, (ii) the behavior of the system under pure, as compared to simple, shearing, (iii) the relaxation of the nematic order parameter S_{2} when perturbed away from the steady state, and (iv) by construction, a numerical mean-field model for the rotational motion of a particle. Our results also help to explain the singular behavior observed when taking the α→0 limit approaching circular disks.

Effect of Length on the Dynamics of Wall Tethered Polymers in Shear Flow

We study the configuration and wall collision process of a wall tethered polymer in shear flow using a Brownian dynamics technique. We develop scalings based on the flow strength and polymer length to explain the nonequilibrium polymer dimensions and wall collision rates along the polymer backbone. Our simulation results corroborate this theory and collapse under these scalings. For the first time, previous experiments conducted with DNA molecules showing the strong dependence of polymer dimensions on length are successfully compared with simulations. Our results highlight the connection between the tumbling of a free polymer and the collision process of a tethered polymer. We successfully describe the chain collisions as a Poisson-like process, where we observe that a minimum number of wall collisions is experienced by the segment located at approximately 3/4 the distance along the polymer backbone from the tethering point. Our bead–spring model captures all the essential physics. Hydrodynamic and exclud...

Poisson pre-processing of nonstationary photonic signals: Signals with equality between mean and variance.

Photonic signals are broadly exploited in communication and sensing and they typically exhibit Poisson-like statistics. In a common scenario where the intensity of the photonic signals is low and one needs to remove a nonstationary trend of the signals for any further analysis, one faces an obstacle: due to the dependence between the mean and variance typical for a Poisson-like process, information about the trend remains in the variance even after the trend has been subtracted, possibly yielding artifactual results in further analyses. Commonly available detrending or normalizing methods cannot cope with this issue. To alleviate this issue we developed a suitable pre-processing method for the signals that originate from a Poisson-like process. In this paper, a Poisson pre-processing method for nonstationary time series with Poisson distribution is developed and tested on computer-generated model data and experimental data of chemiluminescence from human neutrophils and mung seeds. The presented method transforms a nonstationary Poisson signal into a stationary signal with a Poisson distribution while preserving the type of photocount distribution and phase-space structure of the signal. The importance of the suggested pre-processing method is shown in Fano factor and Hurst exponent analysis of both computer-generated model signals and experimental photonic signals. It is demonstrated that our pre-processing method is superior to standard detrending-based methods whenever further signal analysis is sensitive to variance of the signal.

Forecasting a point process with an ARIMA model

ABSTRACTIn fitting a power-law process, we show that the construction of the empirical recurrence rate time series either simplifies the modeling task, or liberates a point process restrained by a key parametric model assumption such as the monotonicity requirement of the intensity function. The technique can be applied to seasonal events occurring in spurts or clusters, because the autoregressive integrated moving average (ARIMA) procedure provides a comprehensive set of tools with great flexibility. Essentially, we consolidate two of the most powerful modeling tools for the stochastic process and time series in the statistical literature to handle counts of events in a Poisson or Poisson-like process.

Analytically Solvable Model of Spreading Dynamics with Non-Poissonian Processes

Non-Poissonian bursty processes are ubiquitous in natural and social phenomena, yet little is known about their effects on the large-scale spreading dynamics. In order to characterize these effects we devise an analytically solvable model of Susceptible-Infected (SI) spreading dynamics in infinite systems for arbitrary inter-event time distributions and for the whole time range. Our model is stationary from the beginning, and the role of lower bound of inter-event times is explicitly considered. The exact solution shows that for early and intermediate times the burstiness accelerates the spreading as compared to a Poisson-like process with the same mean and same lower bound of inter-event times. Such behavior is opposite for late time dynamics in finite systems, where the power-law distribution of inter-event times results in a slower and algebraic convergence to fully infected state in contrast to the exponential decay of the Poisson-like process. We also provide an intuitive argument for the exponent characterizing algebraic convergence.

Cell-to-cell diversity in protein levels of a gene driven by a tetracycline inducible promoter

BackgroundGene expression in Escherichia coli is regulated by several mechanisms. We measured in single cells the expression level of a single copy gene coding for green fluorescent protein (GFP), integrated into the genome and driven by a tetracycline inducible promoter, for varying induction strengths. Also, we measured the transcriptional activity of a tetracycline inducible promoter controlling the transcription of a RNA with 96 binding sites for MS2-GFP.ResultsThe distribution of GFP levels in single cells is found to change significantly as induction reaches high levels, causing the Fano factor of the cells' protein levels to increase with mean level, beyond what would be expected from a Poisson-like process of RNA transcription. In agreement, the Fano factor of the cells' number of RNA molecules target for MS2-GFP follows a similar trend. The results provide evidence that the dynamics of the promoter complex formation, namely, the variability in its duration from one transcription event to the next, explains the change in the distribution of expression levels in the cell population with induction strength.ConclusionsThe results suggest that the open complex formation of the tetracycline inducible promoter, in the regime of strong induction, affects significantly the dynamics of RNA production due to the variability of its duration from one event to the next.

SUPERPOSITION OF STOCHASTIC PROCESSES AND THE RESULTING PARTICLE DISTRIBUTIONS

Many observations of suprathermal and energetic particles in the solar wind and the inner heliosheath show that distribution functions scale approximately with the inverse of particle speed (v) to the fifth power. Although there are exceptions to this behavior, there is a growing need to understand why this type of distribution function appears so frequently. This paper develops the concept that a superposition of exponential and Gaussian distributions with different characteristic speeds and temperatures show power-law tails. The particular type of distribution function, f ∝ v –5, appears in a number of different ways: (1) a series of Poisson-like processes where entropy is maximized with the rates of individual processes inversely proportional to the characteristic exponential speed, (2) a series of Gaussian distributions where the entropy is maximized with the rates of individual processes inversely proportional to temperature and the density of individual Gaussian distributions proportional to temperature, and (3) a series of different diffusively accelerated energetic particle spectra with individual spectra derived from observations (1997-2002) of a multiplicity of different shocks. Thus, we develop a proof-of-concept for the superposition of stochastic processes that give rise to power-law distribution functions.

Application of the generalised ballot theorem for evaluation of performance in packet buffers with non-first in first out scheduling

Packet scheduling is a vital component to support different classes of service in all-packet networks. In classical queuing systems, the waiting-time performance of non-first in first out buffer scheduling systems could be predicted through the use of analysis. However, all-packet networks feature traffic patterns that do not conform to classical Poisson-like processes, and this greatly complicates the evaluation of their performance. Our novel approach to this problem is through a hybrid combination of analysis and simulation. The authors derive a combinatorial algorithm, using the generalised ballot theorem, which predicts waiting times for low-priority traffic. When this algorithm is combined with prior work on traffic aggregation, the authors achieve a significant reduction in the state space associated with the buffer under study. To numerically test this algorithm, the authors demonstrate its use in simulation, where state space and event count reduction is a fundamental requirement to ensure experiments complete in a timely fashion. Numerical results from these simulations show a very significant reduction in the number of events processed combined with improved state coverage. This is achieved while maintaining a highly accurate representation of packet delays compared with a conventional approach.

Statistics of Midbrain Dopamine Neuron Spike Trains in the Awake Primate

Work in behaving primates indicates that midbrain dopamine neurons encode a prediction error, the difference between an obtained reward and the reward expected. Studies of dopamine action potential timing in the alert and anesthetized rat indicate that dopamine neurons respond in tonic and phasic modes, a distinction that has been less well characterized in the primates. We used spike train models to examine the relationship between the tonic and burst modes of activity in dopamine neurons while monkeys were performing a reinforced visuo-saccadic movement task. We studied spiking activity during four task-related intervals; two of these were intervals during which no task-related events occurred, whereas two were periods marked by task-related phasic activity. We found that dopamine neuron spike trains during the intervals when no events occurred were well described as tonic. Action potentials appeared to be independent, to occur at low frequency, and to be almost equally well described by Gaussian and Poisson-like (gamma) processes. Unlike in the rat, interspike intervals as low as 20 ms were often observed during these presumptively tonic epochs. Having identified these periods of presumptively tonic activity, we were able to quantitatively define phasic modulations (both increases and decreases in activity) during the intervals in which task-related events occurred. This analysis revealed that the phasic modulations of these neurons include both bursting, as has been described previously, and pausing. Together bursts and pauses seemed to provide a continuous, although nonlinear, representation of the theoretically defined reward prediction error of reinforcement learning.

On the analysis of count data of birth‐and‐death process type: with application to molecularly targeted cancer therapy

For molecularly targeted cancer therapy, potency of a novel monoclonal antibody is typically studied in vitro in order to observe directly the effect of the antibody on a malignant tumour cell population. Numbers of viable cells of the population are counted over time to evaluate how the antibody may change the population growth pattern. Usually, such count data is analysed by a log-linear model to estimate the average antibody effects, but it is based on a Poisson-like process and is not appropriate for modelling the growth pattern of the tumour cells. In this paper, we propose to analyse the count data from the point of view of a birth-and-death process, which is a more natural description of the biological process in these experiments. Assuming a simple birth-and-death process and a naïve regression model for the birth rate and death rate, we show that the log-linear relationship still holds for the average count and the covariates, although the linear predictor needs to satisfy a specific functional form. The estimation can be based on the quasi-likelihood method. The variance is not a simple function of the mean, but can be adequately approximated by a function of mean of either quadratic form or linear form, available in most standard statistical packages. We perform a simulation study to show that the proposed method provides consistent and robust estimations. The utility of the method is demonstrated by the analysis of a data set for experiment for anti-lymphoma monoclonal antibodies.

POISSON REGRESSION WITH A PERIODIC FUNCTION

ABSTRACT Let {yt } be a Poisson-like process with the mean μ t which is a periodic function of time t. We discuss how to fit this type of data set using quasi-likelihood method. Our method provides a new avenue to fit a time series data when the usual assumption of stationarity and homogeneous residual variances are invalid. We show that the estimators obtained are strongly consistent and also asymptotically normal.

The significance of precisely replicating patterns in mammalian CNS spike trains

Neuronal spike trains from both single and multi-unit recordings often contain patterns such as doublets and triplets of spikes that precisely replicate themselves at a later time. The presence of such precisely replicating patterns can still be detected when the tolerance on interval replication is shortened to a fraction of a millisecond. In this context we examine here data taken from various parts of the central nervous systems of anesthetized rats, cats and monkeys. The relative abundance of replicating triplets varies from centre to centre, and is nearly always significantly greater than obtained in Monte-Carlo simulations of either a Poisson-like process or a renewal process having the same interspike interval distribution as the neuronal data. However, a remarkable exception is found in the activity of retinal ganglion cells. Significant deviations were found in the primary visual cortex and, even more so, in the lateral geniculate body and the mitral cells of the olfactory bulb. Using a fixed tolerance for the replication of intervals (0.5 ms) it is usually observed that replicating patterns are produced in excess (with respect to renewal process models) mostly in low firing rate episodes (≤100 Hz). However, using a tolerance that varies in direct proportion to the mean interval (i.e. as the reciprocal of the firing rate), one generally observes that replicating triplets occur with higher than expected frequency in comparable proportions at all firing rates. This observation suggests the existence of a scale invariance principle in these phenomena with respect to certain neuronal codes. In order to decrease the influence of the estimated neuronal firing rate on the results of the comparisons, we computed also the ratio NT2/ND3, of the number of replicating triplets to the number of doublets replicating three times, [Lestienne R. (1994) Proc. Soc. Neurosci. 20, 22; Lestienne R. (1996) Bio. Cybern. 74, 55–61] using both a fixed or a variable tolerance. In spike trains obeying a Poisson process, NT2/ND3 ratios should be nearly independent of the frequency, especially when using a variable tolerance. These studies supported previous results : significant deviations from the models are found in all the spike trains examined, except in the case of retinal ganglion cells, and the most significant deviations are found in recordings from the lateral geniculate nucleus and the mitral cells of the olfactory bulb. Removing spikes that belong to bursts having large “Poisson surprise” values [Legéndy C. R. and Salcman M. (1985) J. Neurophysiol. 53, 926-939] (except the very first spike of the burst) significantly decreases NT2/ND3 ratios in the record from the lateral geniculate nucleus, suggesting that in this case bursty episodes greatly contribute to the production of replicating patterns, but such a removal does not affect results from the piriform record. Finally, in both the lateral geniculate nucleus and in the mitral cells of the olfactory bulb records, perturbing the timing of spikes by applying to interspike intervals small jitters of uniform probability density with amplitude up to 3 ms, very significantly decrease NT2/ND3 ratios in these centres, but does not change much the NT2/ND3 ratios in other neuronal recordings. Implications of these findings for a possible role of precisely replicating patterns in temporal coding of neuronal information is discussed, as well as possible mechanisms for their production.

Analysis of truck accident rates using loglinear models

Alternative approaches for the analysis of factors affecting truck accident rates are assessed. The scope of the discussion is methodological in nature, with emphasis on exploring the limitations of each approach. Several important statistical concerns associated with the weighted least squares algorithm for calibrating loglinear models are addressed, including reduced cell membership in the contingency tables of accidents, scaling factors for measuring truck exposure, incompatibility between continuous exposure variables and categorical accident data, and sensitivity of calibrated coefficients to changes in accident characteristics for different data sets. An alternative approach for fitting loglinear models is proposed, which makes use of the generalized linear interactive models (GLIM) package. This approach assumes that the dependent variables in the contingency table of accidents behave in a Poisson-like process with values ranging from 0 to infinity. The calibration of the beta parameters in the loglinear expressions uses maximum likelihood techniques. Continuous exposure variates are incorporated into these models as offsets. As in the classical weighted least squares algorithm, this approach permits a stepwise statistical analysis of higher-order interactions for categorical accident frequency data, while adjusting directly for continuous measures of exposure. The results of a calibration of loglinear expressions for Ontario truck accident data are presented. Key words: truck accident rate, exposure, loglinear calibration, causal interaction.

Poisson process stimulation of an excitable membrane cable model

The convergence of multiple inputs within a single-neuronal substrate is a common design feature of both peripheral and central nervous systems. Typically, the result of such convergence impinges upon an intracellularly contiguous axon, where it is encoded into a train of action potentials. The simplest representation of the result of convergence of multiple inputs is a Poisson process; a general representation of axonal excitability is the Hodgkin-Huxley/cable theory formalism. The present work addressed multiple input convergence upon an axon by applying Poisson process stimulation to the Hodgkin-Huxley axonal cable. The results showed that both absolute and relative refractory periods yielded in the axonal output a random but non-Poisson process. While smaller amplitude stimuli elicited a type of short-interval conditioning, larger amplitude stimuli elicited impulse trains approaching Poisson criteria except for the effects of refractoriness. These results were obtained for stimulus trains consisting of pulses of constant amplitude and constant or variable durations. By contrast, with or without stimulus pulse shape variability, the post-impulse conditional probability for impulse initiation in the steady-state was a Poisson-like process. For stimulus variability consisting of randomly smaller amplitudes or randomly longer durations, mean impulse frequency was attenuated or potentiated, respectively. Limitations and implications of these computations are discussed.

Spontaneous Activity of Single Auditory Nerve Fibers in Cats

In a study of over 100 auditory nerve fibers, every fiber exhibited spontaneous spike discharges although the rates ranged between <1/sec to 100/sec. Histograms of the interspike interval of this activity all show a mode at about 4 msec and a tail that decays exponentially. This pattern is consistent with that generated by a Poisson-like process with a dead time. A histogram of the number of fibers encountered vs spontaneous rate shows that certain ranges of rates occur more frequently than others. Distinct peaks occur at <5/sec and 40/sec with less distinct peaks at 60/sec and 70/sec. This distribution of spontaneous activity can be seen in individual cats and is apparently unrelated to either physiological condition or level of anesthetization of the animal. There is no correlation of rate of spontaneous discharge with the frequency of tone to which the unit is most sensitive. However, there is some indication that units with low rates of spontaneous discharge have higher thresholds to acoustic stimuli than units with high spontaneous rates. [This work was supported in part by the U. S. Army, U. S. Air Force Office of Scientific Research, and the U. S. Office of Naval Research; in part by the National Science Foundation (Grant G-16526): in part by the National Institutes of Health (Grant MH-04737-02); and in part by research grant B-1344, National Institute of Neurological Diseases and Blindness, National Institutes of Health, Public Health Service, U. S. Department of Health, Education, and Welfare.]