Renewal Point Process Research Articles

Deviation of spontaneous firing in frog medullar auditory units from the renewal point process

Statistical characteristics of spontaneous activity (distribution of interspike intervals, hazard function, autocorrelation function, autocorrelation function for the process with mixed intervals, and interdependence between adjoining intervals) were analyzed for 123 neurons of the frog medullar dorsal nucleus (homolog of the mammalian cochlear nucleus). In the majority of cells, this activity was distinct from the Poisson process, and firing periodicity was noticed in some cases. In addition, deviations of the spontaneous activity from the renewal process were usually observed. Weak yet reliable positive interspike-interval correlation was typical of most neurons; however, a negative correlation between short adjoining intervals was recorded for some units. These data suggest the effects of memory in the activity of single neurons of the auditory system.

On the correlation structure of some random point processes on the line

The correlation structure of some remarkable point processes on the one-dimensional real line is investigated. More specifically, focus is on translation invariant determinantal, permanental and/or renewal point processes. In some cases, anomalous (non-Poissonian) fluctuations for the number of points in a large window can be observed. This may be read from the total correlation function of the point process. We try to understand when and why this occurs and what are the anomalous behaviors to be expected. From examples, it is shown that determinantal (fermion) point processes can be super-homogeneous (the number variance grows slower than the number mean) and even hyper-uniform (when variance growth saturates). Renewal point processes with bounded spacings variance are essentially Poissonian (the number variance grows like the number mean as in Poisson models). Under certain conditions, permanental (boson) point processes can be sub-homogeneous or critical (in the sense that the number variance grows faster than the number mean). We give several detailed examples illustrating these properties of interest together with unexpected behaviors.

Dependent multivariate diffusion models and related point process models of ensemble spiking neurons

The first-passage time of a diffusion process through a constant or variable boundary has been the focus of many stochastic models of neuronal membrane potential dynamics. Diffusion processes have been used extensively to model a latent process that may only be observable through consequent renewal point process events. The mathematical relationship between inter-spike intervals and the first-passage time of simple diffusion models is well-known, however this relationship becomes increasingly more complex as the diffusion models become more physiologically realistic, and when multivariate diffusion processes are no longer considered to be independent. The probability density of a diffusing particle position at a particular point in time P(x, t) as defined by the Fokker-Planck equation can be solved, under suitable conditions, using the method of images. We show how the method of images can be extended to a multivariate probability density constructed from marginal densities modeling simple individual spiking neurons using a copula construction that factors out the correlated (dependent) noise structure. This in turn provides a straightforward method for estimating multivariate spike survival and hazard functions from simultaneously recorded single unit activity and, indirectly, the ensemble neuron diffusion noise dependence. The analytical approach is supported by simulated Wiener processes with drift and applied to simultaneous single unit recordings from Heschl's gyrus in awake human subjects. Extensions of this approach to more physiologically realistic diffusion models such as the Ornstein-Uhlenbeck and Feller processes will be discussed.

Serial interval statistics of spontaneous activity in cortical neurons in vivo and in vitro

Stationary spiking of single neurons is often modelled by a renewal point process. Here, we tested the underlying model assumption that the inter-spike intervals are mutually independent by analyzing stationary spike train recordings from individual rat neocortical neurons in vivo and in vitro. All neurons exhibited moderate ( in vivo) or weak ( in vitro) negative first order serial correlation of neighboring intervals which was found to be significant in most cases. No significant higher order serial correlations were detected. The observed negative correlation lead to a strong reduction of the spike count variability by about 30% in vivo.

State-dependent fire models and related renewal processes

We introduce a general class of stochastic processes forced by instantaneous random fires (i.e., jumps) that reset the state variable x to a given value. Since in many physical systems the fire activity is often dependent on the actual value of the state variable, as in the case of natural fires in ecosystems and firing dynamics in neuronal activity, the frequency of fire occurrence is assumed to be state dependent. Such dynamics leads to independent interfire statistics--i.e., to renewal point processes. Various functions relating the frequency of fire occurrence to x(t) are analyzed and compared. The relation between the probabilistic dynamics of x(t) and the interfire statistics is derived and some exact probability distribution of both x(t) and the interfire times are obtained for systems with different degrees of complexity. After studying processes in which the fire activity is coupled only to a deterministic drift, we also analyze processes forced by either additive or multiplicative Gaussian white noise.

Topological Analysis of Linear Polymer Melts: A Statistical Approach

We introduce an algorithm for the reduction of a computer generated atomistic polymer sample to an entanglement network of primitive paths. These networks are structural representations of the topology underlying a polymer melt. By examining network ensembles of polyethylene and cis-1,4-polybutadiene melts, we provide topological measures and statistical properties of primitive paths. We present the radial distribution function of entanglements and the distribution of the number of monomers between entanglements. A renewal point process that generates entanglement events along the monomer sequence of a chain is found to describe the statistics of detected topological constraints. We discuss chain thickness effects on topological measures and provide a method for detecting persistent chain contacts in melt configurations. A suitable scaling of acquired data leads to a unifying microscopic topological description of the melts studied.

Strong memoryless times and rare events in Markov renewal point processes

Let W be the number of points in (0,t] of a stationary finite-state Markov renewal point process. We derive a bound for the total variation distance between the distribution of W and a compound Poisson distribution. For any nonnegative random variable ζ, we construct a “strong memoryless time” $\hat{\zeta}$ such that ζ−t is exponentially distributed conditional on $\{\hat{\zeta}\leq t,\,\zeta>t\}$, for each t. This is used to embed the Markov renewal point process into another such process whose state space contains a frequently observed state which represents loss of memory in the original process. We then write W as the accumulated reward of an embedded renewal reward process, and use a compound Poisson approximation error bound for this quantity by Erhardsson. For a renewal process, the bound depends in a simple way on the first two moments of the interrenewal time distribution, and on two constants obtained from the Radon–Nikodym derivative of the interrenewal time distribution with respect to an exponential distribution. For a Poisson process, the bound is 0.

A general shock model for a reliability system

We study a reliability system subject to shocks generated by a renewal point process. When a shock occurs, components fail independently of each other with equal probabilities that are random numbers drawn from a distribution that may differ from shock to shock. We first consider the case of a parallel system and derive closed expressions for the Laplace-Stieltjes transform and the expectation of the time to system failure and for its density in the case that the distribution function of the renewal process possesses a density. We then treat a more general system structure, which has some very important special cases, such as k-out-of-n:F systems, and derive analogous formulae.

A general shock model for a reliability system

We study a reliability system subject to shocks generated by a renewal point process. When a shock occurs, components fail independently of each other with equal probabilities that are random numbers drawn from a distribution that may differ from shock to shock. We first consider the case of a parallel system and derive closed expressions for the Laplace-Stieltjes transform and the expectation of the time to system failure and for its density in the case that the distribution function of the renewal process possesses a density. We then treat a more general system structure, which has some very important special cases, such as k-out-of-n:F systems, and derive analogous formulae.

A unified approach to fast teller queues and ATM

This paper examines a problem of importance to the telecommunications industry. In the design of modern ATM switches, it is necessary to use simulation to estimate the probability that a queue within the switch exceeds a given large value. Since these are extremely small probabilities, importance sampling methods must be used. Here we obtain a change of measure for a broad class of models with direct applicability to ATM switches.We consider a model with A independent sources of cells where each source is modeled by a Markov renewal point process with batch arrivals. We do not assume the sources are necessarily identically distributed, nor that batch sizes are independent of the state of the Markov process. These arrivals join a queue served by multiple independent servers, each with service times also modeled as a Markov renewal process. We only discuss a time-slotted system. The queue is viewed as the additive component of a Markov additive chain subject to the constraint that the additive component remains non-negative. We apply the theory in McDonald (1999) to obtain the asymptotics of the tail of the distribution of the queue size in steady state plus the asymptotics of the mean time between large deviations of the queue size.

A unified approach to fast teller queues and ATM

This paper examines a problem of importance to the telecommunications industry. In the design of modern ATM switches, it is necessary to use simulation to estimate the probability that a queue within the switch exceeds a given large value. Since these are extremely small probabilities, importance sampling methods must be used. Here we obtain a change of measure for a broad class of models with direct applicability to ATM switches. We consider a model with A independent sources of cells where each source is modeled by a Markov renewal point process with batch arrivals. We do not assume the sources are necessarily identically distributed, nor that batch sizes are independent of the state of the Markov process. These arrivals join a queue served by multiple independent servers, each with service times also modeled as a Markov renewal process. We only discuss a time-slotted system. The queue is viewed as the additive component of a Markov additive chain subject to the constraint that the additive component remains non-negative. We apply the theory in McDonald (1999) to obtain the asymptotics of the tail of the distribution of the queue size in steady state plus the asymptotics of the mean time between large deviations of the queue size.

Multivariate regression estimation of continuous-time processes from sampled data: local polynomial fitting approach

Let (Y,X)={Y(t),X(t),-/spl infin/<t</spl infin/} be real-valued continuous-time jointly stationary processes and let (t/sub j/) be a renewal point processes on (0,/spl infin/), with a finite mean rate, independent of (Y,X). We consider the estimation of regression function r(x/sub 0/, x/sub 1/,...,x/sub m-1/; /spl tau//sub 1/,...,/spl tau//sub m/) of /spl psi/(Y(/spl tau//sub m/)) given (X(0)=x/sub 0/, X(/spl tau//sub 1/)=x/sub 1/,...,X(/spl tau//sub m-1/)=/sub x-1/) for arbitrary lags 0</spl tau//sub 1/<...< /spl tau//sub m/ on the basis of the discrete-time observations {Y(t/sub j/),X(t/sub j/),t/sub j/)/sub j=1//sup n/. We estimate the regression function and all its partial derivatives up to a total order p/spl ges/1 using high-order local polynomial fitting. We establish the weak consistency of such estimates along with rates of convergence. We also establish the joint asymptotic normality of the estimates for the regression function and all its partial derivatives up to a total order p/spl ges/1 and provide explicit expressions for the bias and covariance matrix (of the asymptotically normal distribution).

Molecular evolution modeled as a fractal renewal point process in agreement with the dispersion of substitutions in mammalian genes.

A fractal renewal point process (FRPP) is used to model molecular evolution in agreement with the relationship between the variance and the mean numbers of nonsynonymous and synonymous substitutions in mammals. Like other episodic models such as the doubly stochastic Poisson process, this model accounts for the large variances observed in amino acid substitution rates, but unlike certain other episodic models, it also accounts for the increase in the index of dispersion with the mean number of substitutions in Ohta's (1995) data. We find that this correlation is significant for nonsynonymous substitutions at the 1% level and for synonymous substitutions at the 10% level, even after removing lineage effects and when using Bulmer's (1989) unbiased estimator of the index of dispersion. This model is simpler than most other overdispersed models of evolution in the sense that it is fully specified by a single interevent probability distribution. Interpretations in terms of chaotic dynamics and in terms of chance and selection are discussed.

DYNAMIC RESPONSE OF NON-LINEAR SYSTEMS TO RENEWAL IMPULSES BY PATH INTEGRATION

The cell-to-cell mapping (path integration) technique has been devised for non-linear and non-hysteretic systems subjected to random trains of impulses driven by an ordinary renewal point process with gamma-distributed integer parameter inter-arrival times (an Erlang process). Since the renewal point process does not have independent increments the state vector of the system, consisting of the generalized displacements and velocities, is not a Markov process. Initially, it is shown how the indicated system can be converted to an equivalent Poisson driven system at the expense of introducing additional discrete-valued state variables for which the stochastic equations are also formulated. Thereby the original non-Markov system is converted into an equivalent system which does possess the Markov property. The continuous non-diffusive Markov process is converted, by discretization in time and in phase space, to a Markov chain. The transition probability matrix is next constructed for a transition time interval short enough to neglect the probability of occurrence of more than one impulse in this interval. The technique of convection and lumping of the probability mass is essentially based on the technique devised by Köylüoğlu and the authors [1] for Poisson driven systems. In the present problem this technique has been suitably modified with due account taken of the artificial, auxiliary, state variables introduced in the description. The method has been applied to the Duffing oscillator, and the results for the stationary probability density function have been compared with those obtained from extensive Monte Carlo simulations.

Mean time to finish a randomly disturbed 2-phase mission

This paper shows the average time to finish a mission consisting of a sequence of two noninterruptible phases of deterministic, given lengths, when after a short disturbance the interrupted phase has to be restarted. Disturbances, e.g., intermittent faults, are modeled according to a renewal point-process. In order to overcome limitations of Markov modeling, this point process need not be Poisson. Fairly simple closed-form solutions are derived. The Poisson case treated, is a practical example. Upper bounds are derived for easy conservative approximations. The approach can be easily extended to cover the determination of probability distributions instead of only mean values.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

ON THE SELECTION OF RANDOM SAMPLING SCHEMES FOR THE SPECTRAL ESTIMATION OF CONTINUOUS TIME PROCESSES

Abstract. Let X={X(t), ‐∝&lt;t&lt;∝} be a continuous time stationary process with spectral density φx(Λ) and {φk} be a stationary point process independent of X. The problem of selecting the sampling point process {τk} for the estimation of φx(Λ) based on the discrete time observations {X(τk), τk} is considered. Sufficient conditions are established for the admittability of a broad class of delayed renewal point processes. Examples are given for delayed renewal point processes {τk} which have a mixture of Gamma densities for their inter‐arrival times. Guidelines are given for the selection of specific point processes in the estimation of broadband and narrowband spectral density functions.

A queueing model in an alternating random environment

In this paper, we consider a single server queueing model with alternately varying Poisson arrival rates. Such arrival processes are usually modelled using the Doubly Stochastic Poisson (DSP) processes whose Poisson rates vary alternately and remains the same during an exponential time duration called a switching time. Queues with the DSP arrival processes are often employed to describe production, computer and communication systems. A DSP process includes both renewal and nonrenewal point processes. Since performance measures of a single server queue with a DSP input are difficult to evaluate, we approximate a DSP arrival process by a renewal process. We then use parameters so developed in an expression for the expected waiting time in a GI/ G/1. To demonstrate the quality of our approximations, we also present exact results. Our approximations have applications in production, inventory, insurance and maintenance systems.

Fractal patterns in auditory nerve-spike trains

The authors discuss the following topics: traditional renewal point process models; long-term correlations; self-similarity of neuronal firing rates; power-law behavior of the Fano-factor time curve; change in the firing pattern induced by the presence of a stimulus; neural information processing with fractal nerve spikes; biophysical origin of the fractal behavior; fractal point-process model; short-term correlations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Convergence of auditory nerve fibers onto bushy cells in the ventral cochlear nucleus: implications of a computational model.

1. Convergence of auditory nerve (AN) fibers onto bushy cells of the ventral cochlear nucleus (VCN) was investigated with a model that describes the electrical membrane properties of these cells. The model consists of a single compartment, representing the soma, and includes three voltage-sensitive ion channels (fast sodium, delayed-rectifier-like potassium, and low-threshold potassium). These three channels have characteristics derived from voltage clamp data of VCN bushy cells. The model also contains a leakage channel, membrane capacitance, and synaptic inputs. The model accurately reproduces the membrane rectification seen in current clamp studies of bushy cells, as well as their unique current clamp responses. 2. In this study, the number and synaptic strength of excitatory AN inputs to the model were varied to investigate the relationship between input convergence parameters and response characteristics. From 1 to 20 excitatory synaptic inputs were applied through channels in parallel with the voltage-gated channels. Each synapse was driven by an independent AN spike train; spike arrivals produced brief (approximately 0.5 ms) conductance increases. The number (NS) and conductance (AE) of these inputs were systematically varied. The input spike trains were generated as a renewal point process that accurately models characteristics of AN fibers (refractoriness, adaptation, onset latency, irregularity of discharge, and phase locking). Adaptation characteristics of both high and low spontaneous rate (SR) AN fibers were simulated. 3. As NS and AE vary over the ranges 1-20 and 3-80 nS, respectively, the full range of response types seen in VCN bushy cells are produced by the model, with AN inputs typical of high-SR AN fibers. These include primarylike (PL), primarylike-with-notch (Pri-N), and onset-L (On-L). In addition, Onset responses, whose association with bushy cells in uncertain, and "dip" responses, which are not seen in the VCN, are produced. Dip responses occur with large NS and/or AE, and are due to depolarization block. When the AN inputs have the adaptation characteristics of low-SR AN fibers, PL--but not Pri-N or On-L responses--are produced. This suggests that neurons showing Pri-N and On-L responses must receive high-SR AN inputs. 4. The regularity of discharge of the model is compared with that of VCN bushy cells, using a measure derived from the mean and standard deviation of interspike intervals. Regularity is an important constraint on the model because the regularity of VCN bushy cells is the same as that of their AN inputs.(ABSTRACT TRUNCATED AT 400 WORDS)

Fractal renewal processes generate 1/f noise.

1/${\mathit{f}}^{\mathit{D}}$ noise occurs in an impressive variety of physical systems, and numerous complex theories have been proposed to explain it. We construct two relatively simple renewal processes whose power spectral densities vary as 1/${\mathit{f}}^{\mathit{D}}$: (i) a standard renewal point process, with 0D1, and (ii) a finite-valued alternating renewal process, with 0D2. The resulting event number statistics, coincidence rates, minimal coverings, and autocorrelation functions are shown also to follow power-law forms. These fractal characteristics derive from interevent-time probability density functions which themselves decay in a power-law fashion. A number of applications are considered: trapping in amorphous semiconductors, electronic burst noise, movement in systems with fractal boundaries, the digital generation of 1/${\mathit{f}}^{\mathit{D}}$ noise, and ionic currents in cell membranes.