Autocorrelation and spectrum of the modulated retinal discharge: A comparison of theoretical and experimental results

The theoretical autocorrelation of the cat ganglion cell discharge under stationary and dynamic conditions of light stimulus is compared with the autocorrelation determined experimentally. To obtain the theoretical autocorrelation, the stationary discharge is described by a stationary random point process of independent intervals equally distributed according to a gamma function, and the dynamic behaviour is described by a model defined in a previous paper. Comparison shows that the model predicts the experimental results. Finally, the power density spectrum is analysed and the relevance of the results to signal transmission by the retinal system is discussed.     

On estimating critical population size for an endangered species in the presence of environmental stochasticity

A stationary second order autoregressive process with Gaussian noise, which was linked to survivorship and reproductive success by logistic transformations, was used as a model for an environmental process. Computer experiments in Monte Carlo integration, with the objective of exploring the sensitivity of estimates of mean critical population size to variations in the parameters of the environmental process, were then conducted. These experiments suggest that estimates of mean critical population size are very sensitive to the form of the autocorrelation function of the stationary environmental process. For the most part, those experiments in which the autocorrelation function was strictly positive not only resulted in the largest estimates of mean critical population size but also led to the highest levels of environmental stochasticity as measured by its coefficient of variation. As in previous work, these experiments suggest that concerted efforts should be made to model those environmental factors that are critical to the survivability of an endangered species in assessing its chances for continued existence.     

A note on the sample serial autocorrelation function of stationary spike trains

The computation of the sample serial autocorrelation of a spike train is reasonable under the condition that the underlying process is stationary and linear. If these conditions are not fulfilled, the sample serial autocorrelation is random and any interpretation of it is erroneous: On the other hand, if the conditions are proved to hold, there is little need for the computation of the sample serial correlation, since in nearly all practical cases the stationarity is approximated by stationarity in the wide sense, which means that the deterministic autocorrelation of the process is already known.     

Low-level and high-level processes in apparent motion

When a group of dots within a random-dot array is discontinuously displaced, it appears as a moving region perceptually segregated from its stationary surround. The spastial, temporal and other constraints governing this effect are markedly different from those classically found for the apparent motion of isolated stimulus elements. The random-dot display appears to tap a low-level motion-detecting process, distinct from the more interpretive process elicited by the classical displays. The distinct contributions of these processes can be identified in 'multi-stable' displays which yield alternative percepts of apparent motion depending on which one or both of the processes is activated. Such experiments illustrate the interaction of relatively stimulus-constrained and relatively autonomous processes invisual perception.     

Accounting for spatial pattern when modeling organism-environment interactions

Statistical models of environment-abundance relationships may be influenced by spatial autocorrelation in abundance, environmental variables, or both. Failure to account for spatial autocorrelation can lead to incorrect conclusions regarding both the absolute and relative importance of environmental variables as determinants of abundance. We consider several classes of statistical models that are appropriate for modeling environment-abundance relationships in the presence of spatial autocorrelation, and apply these to three case studies: 1) abundance of voles in relation to habitat characteristics; 2) a plant competition experiment; and 3) abundance of Orbatid mites along environmental gradients. We find that when spatial pattern is accounted for in the modeling process, conclusions about environmental control over abundance can change dramatically. We conclude with five lessons: 1) spatial models are easy to calculate with several of the most common statistical packages; 2) results from spatially-structured models may point to conclusions radically different from those suggested by a spatially independent model; 3) not all spatial autocorrelation in abundances results from spatial population dynamics; it may also result from abundance associations with environmental variables not included in the model; 4) the different spatial models do have different mechanistic interpretations in terms of ecological processes – thus ecological model selection should take primacy over statistical model selection; 5) the conclusions of the different spatial models are typically fairly similar – making any correction is more important than quibbling about which correction to make.     

A Test of the Equivalence of Four Random Number Generators – A Simulation Study

Four pseudo random number generators were tested for equivalence. Using two different seed values for each generator, the robustness of a one-sample t-statistic was assessed under a stationary auto-regressive process. The degree of autocorrelation, sample size and significance level were varied for the eight sets of random numbers generated. Results showed an interaction effect between random number generator and alpha level of the t-distribution, suggesting that simulation results could depend on the random number generator selected.     

The spatial structure of the gene pool of a viviparous population of Poa alpina - environmental controls and spatial constraints

Bjømstad, O. N., Iversen, A. & Hansen, M. 1995. The spatial structure of the gene pool of a viviparous population of Poa alpina — environmental controls and spatial constraints. — Nord. J. Bot. 15: 347–354. Copenhagen. ISSN 0107–055X.
Because both the genetic make-up and the environmental conditions of a population are spatially autocorrelated, it is difficult to infer processes of selection or drift for population genetic mappings. We propose a methodology based on partial Mantel techniques and partial autocorrelation techniques to separate the action of these processes. The method is applied to data on Poa alpina to indicate that isolation-by-distance (drift) is the main process inducing positive autocorrelation at the scale of diaspore dispersal (< 100m). The pattern at larger distances is more consistent with selection.     

A Test of Spatial Autocorrelation Analysis Using an Isolation-by-Distance Model

Using the isolation-by-distance model as an example, we have examined several assumptions of spatial autocorrelation analysis applied to gene frequency surfaces. Gene frequency surfaces generated by a simulation of Wright's isolation-by-distance model were shown to exhibit spatial autocorrelation, except in the panmictic case. Identical stochastic generating processes result in surfaces with characteristics that are functions of the process parameters, such as parental vagility and neighborhood size. Differences in these parameters are detectable as differences in spatial autocorrelations after only a few generations of the simulations. Separate realizations of processes with identical parameters yield similar spatial correlograms. We have examined the inferences about population structure that could have been made from these observations if they had been real, rather than simulated, populations. From such inferences, we could have drawn conclusions about the presence of selection, migration and drift in given natural systems.     

Properties of membrane stationary states

Summary The flux of permeant species through a membrane is examined using discrete state stochastic models for the transport process within the membrane. While a membrane flux is maintained due to a concentration gradient between bathing solutions, the distribution of species within the membrane evolves to a time invariant configuration which can differ significantly from the equilibrium configuration. Some special properties of these stationary states are examined using linear, microcanonical models for the membrane transport process. Analysis of these models reveals properties which are masked by the phenomenological analysis of irreversible thermodynamics. For example, the models can be used to study the nature of multi-state relaxation within the membrane by observation of the time dependence of the net membrane flux when the membrane is perturbed from its stationary state distribution. Under some conditions, multi-state models will produce relaxation similar to that observed for single-state processes. The symmetry within the membrane is a critical factor for monitoring relaxation processes within the membrane. Because of the stationary nature of the membrane configuration, statistical thermodynamic variables can be defined for the membrane configuration. The total system is not in equilibrium since the baths must still be described by dissipation functions. In the stationary state, the configurational entropy of the membrane is lowered relative to equilibrium and is shown to depend quadratically on the time independent parameter (j/p) wherej is the membrane flux andp is a characteristic transition probability for intra-membrane transitions. The basic membrane system serves as a quantitative example of the entropy reduction possible in a stationary state system. An allosteric transition mediated by the stationary state configuration is examined as a means of utilizing this negentropy production.     

Detecting clinal and balanced selection using spatial autocorrelation analysis under kin-structured migration

Recently spatial autocorrelation has been employed to infer microevolutionary processes from patterns of genetic variation. In theory, different processes should show characteristic signature correlograms; e. g., clinal selection should produce correlograms decreasing from positive to negative autocorrelation, whereas uniform balanced selection should lead to no spatial autocorrelation. The ability of a statistical method such as spatial autocorrelation analysis to distinguish between these selective regimes or even to detect departures from neutrality is dependent on the strength of the evolutionary force and the population structure. Weak selection or migration will not be apparent against the expected background of stochastic noise. Moreover, the population structure may generate sufficient stochastic variation such that even strong evolutionary forces may fail to be detected. This study uses computer simulation to examine the effects of kin-structured migration and three different selective regimes on the shape of spatial correlograms to assess the ability of this technique to detect different microevolutionary processes. Genetic variation among 8 loci is simulated in a linear set of 25 artificial populations. Kin-structured stepping-stone migration among adjacent populations is modeled; directional, balanced, and clinal selection, as well as neutral loci are considered. These experiments show that strong selection produces correlograms of the predicted shape. However, with an anthropologically reasonable population structure, considerable stochastic variation among correlograms for different alleles may still exist. This suggests the need for caution in inferring genetic process from spatial patterns. © 1994 Wiley-Liss, Inc.     

A latent autoregressive model for longitudinal binary data subject to informative missingness

Longitudinal clinical trials often collect long sequences of binary data. Our application is a recent clinical trial in opiate addicts that examined the effect of a new treatment on repeated binary urine tests to assess opiate use over an extended follow-up. The dataset had two sources of missingness: dropout and intermittent missing observations. The primary endpoint of the study was comparing the marginal probability of a positive urine test over follow-up across treatment arms. We present a latent autoregressive model for longitudinal binary data subject to informative missingness. In this model, a Gaussian autoregressive process is shared between the binary response and missing-data processes, thereby inducing informative missingness. Our approach extends the work of others who have developed models that link the various processes through a shared random effect but do not allow for autocorrelation. We discuss parameter estimation using Monte Carlo EM and demonstrate through simulations that incorporating within-subject autocorrelation through a latent autoregressive process can be very important when longitudinal binary data is subject to informative missingness. We illustrate our new methodology using the opiate clinical trial data.     

Laser light-scattering analysis of protoplasmic streaming in the slime mold Physarum polycephalum

Laser light scattering is shown to be an effective means of obtaining a rapid, objective assessment of dynamic changes in the intact plasmodium of the myxomycete Physarum polycephalum during bidirectional (shuttle) streaming. The motion of material in a 100 mum diameter region of a plasmodial vein was studied by following changes in the autocorrelation function of the fluctuations in the scattered light intensity. The autocorrelation function was recorded at 10 s intervals and analyzed to follow changes in the flow velocity of protoplasm associated with shuttle streaming. Rhythmic velocity changes and a "beating" pattern of velocity maxima were readily observed. In an attempt to locate the site of underlying structural changes in the vein responsible for the changing pattern of flow, the average scattered intensity was separated into components derived from moving and stationary scatterers. Periodic variations in the light intensity due to stationary scatterers are related to the streaming cycle and indicate the occurrence of important structural changes in the vein walls. Two possible interpretations of the data are offered; one involving gross dynamic changes in vein structure, the other involving the formation, contraction, or breakdown of fibrillar material in the vein wall during the streaming cycle.     

Phylogenetic autocorrelation and heritability of geographic range size,shape and position of fiddler crabs,genus Uca (Crustacea,Decapoda)

The aim of this study was to evaluate the levels of phylogenetic heritability of the geographical range size, shape and position for 88 species of fiddler crabs of the world, using phylogenetic comparative methods and simulation procedures to evaluate their fit to the neutral model of Brownian motion. The geographical range maps were compiled from literature, and range size was based on the entire length of coastline occupied by each species, and the position of each range was calculated as its latitudinal and longitudinal midpoint. The range shape of each species was based in fractal dimension (box‐counting technique). The evolutionary patterns in the geographical range metrics were explored by phylogenetic correlograms using Moran’s I autocorrelation coefficients, autoregressive method (ARM) and phylogenetic eigenvector regression (PVR). The correlograms were compared with those obtained by simulations of Brownian motion processes across phylogenies. The distribution of geographical range size of fiddler crabs is right‐skewed and weak phylogenetic autocorrelation was observed. On the other hand, there was a strong phylogenetic pattern in the position of the range (mainly along longitudinal axis). Indeed, the ARM and PVR evidenced, respectively, that ca. 86% and 91% of the longitudinal midpoint could be explained by phylogenetic relationships among the species. The strong longitudinal phylogenetic pattern may be due to vicariant allopatric speciation and geographically structured cladogenesis in the group. The traits analysed (geographical range size and position) did not follow a Brownian motion process, thus suggesting that both adaptive ecological and evolutionary processes must be invoked to explain their dynamics, not following a simple neutral inheritance in the fiddler‐crab evolution.     

Stochastic Models of Gene Expression with Delayed Degradation

In many biochemical reactions occurring in living cells, number of various molecules might be low which results in significant stochastic fluctuations. In addition, most reactions are not instantaneous, there exist natural time delays in the evolution of cell states. It is a challenge to develop a systematic and rigorous treatment of stochastic dynamics with time delays and to investigate combined effects of stochasticity and delays in concrete models.We propose a new methodology to deal with time delays in biological systems and apply it to simple models of gene expression with delayed degradation. We show that time delay of protein degradation does not cause oscillations as it was recently argued. It follows from our rigorous analysis that one should look for different mechanisms responsible for oscillations observed in biological experiments.We develop a systematic analytical treatment of stochastic models of time delays. Specifically we take into account that some reactions, for example degradation, are consuming, that is: once molecules start to degrade they cannot be part in other degradation processes.We introduce an auxiliary stochastic process and calculate analytically the variance and the autocorrelation function of the number of protein molecules in stationary states in basic models of delayed protein degradation.     

A review of techniques for spatial modeling in geographical, conservation and landscape genetics

Most evolutionary processes occur in a spatial context and several spatial analysis techniques have been employed in an exploratory context. However, the existence of autocorrelation can also perturb significance tests when data is analyzed using standard correlation and regression techniques on modeling genetic data as a function of explanatory variables. In this case, more complex models incorporating the effects of autocorrelation must be used. Here we review those models and compared their relative performances in a simple simulation, in which spatial patterns in allele frequencies were generated by a balance between random variation within populations and spatially-structured gene flow. Notwithstanding the somewhat idiosyncratic behavior of the techniques evaluated, it is clear that spatial autocorrelation affects Type I errors and that standard linear regression does not provide minimum variance estimators. Due to its flexibility, we stress that principal coordinate of neighbor matrices (PCNM) and related eigenvector mapping techniques seem to be the best approaches to spatial regression. In general, we hope that our review of commonly used spatial regression techniques in biology and ecology may aid population geneticists towards providing better explanations for population structures dealing with more complex regression problems throughout geographic space.     

An updated perspective on the role of environmental autocorrelation in animal populations

Ecological theory predicts that the presence of temporal autocorrelation in environments can considerably affect population extinction risk. However, empirical estimates of autocorrelation values in animal populations have not decoupled intrinsic growth and density feedback processes from environmental autocorrelation. In this study, we first discuss how the autocorrelation present in environmental covariates can be reduced through nonlinear interactions or by interactions with multiple limiting resources. We then estimated the degree of environmental autocorrelation present in the Global Population Dynamics Database using a robust, model-based approach. Our empirical results indicate that time series of animal populations are affected by low levels of environmental autocorrelation, a result consistent with predictions from our theoretical models. Claims supporting the importance of autocorrelated environments have been largely based on indirect empirical measures and theoretical models seldom anchored in realistic assumptions. It is likely that a more nuanced understanding of the effects of autocorrelated environments is necessary to reconcile our conclusions with previous theory. We anticipate that our findings and other recent results will lead to improvements in understanding how to incorporate fluctuating environments into population risk assessments.     

Weak convergence of a sequence of stochastic difference equations to a stochastic ordinary differential equation

We consider a sequence of discrete parameter stochastic processes defined by solutions to stochastic difference equations. A condition is given that this sequence converges weakly to a continuous parameter process defined by solutions to a stochastic ordinary differential equation. Applying this result, two limit theorems related to population biology are proved. Random parameters in stochastic difference equations are autocorrelated stationary Gaussian processes in the first case. They are jump-type Markov processes in the second case. We discuss a problem of continuous time approximations for discrete time models in random environments.     

Laser light-scattering analysis of protoplasmic streaming in the slime mold physarum polycephalum

Laser light scattering is shown to be an effective means of obtaining a rapid, objective assessment of dynamic changes in the intact plasmodium of the myxomycete Physarum polycephalum during bidirectional (shuttle) streaming. The motion of material in a 100 μm diameter region of a plasmodial vein was studied by following changes in the autocorrelation function of the fluctuations in the scattered light intensity. The autocorrelation function was recorded at 10 s intervals and analyzed to follow changes in the flow velocity of protoplasm associated with shuttle streaming. Rhythmic velocity changes and a “beating” pattern of velocity maxima were readily observed. In an attempt to locate the site of underlying structural changes in the vein responsible for the changing pattern of flow, the average scattered intensity was separated into components derived from moving and stationary scatterers. Periodic variations in the light intensity due to stationary scatterers are related to the streaming cycle and indicate the occurrence of important structural changes in the vein walls. Two possible interpretations of the data are offered; one involving gross dynamic changes in vein structure, the other involving the formation, contraction, or breakdown of fibrillar material in the vein wall during the streaming cycle.     

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.     

Autocorrelation function and power spectrum of two-state random processes used in neurite guidance.

During development neurons extend and retract cytoskeletal structures, chiefly microtubules and filopodia, to process informational cues from the extracellular environment and thereby guide growth cone migration toward an appropriate synaptic partner. This cytoskeleton-based exploration is achieved by stochastic switching, with microtubules and filopodia alternating between growing and shortening phases apparently at random. If stabilizing signals are not detected during the growth phase, then the structures switch to a shortening state, from which they can again return to a growth phase, and so forth. A useful means of characterizing these stochastic processes in a model-independent way is by autocorrelation and spectral analysis. Previously, we compared experiment to theory by performing Monte Carlo simulations and computing the autocorrelation function and power spectrum from the simulated dynamics, an approach that is computationally intensive and requires recalculation whenever model parameters are changed. Here we present analytical expressions for the autocorrelation function and power spectrum, which compactly characterize microtubule and filopodial dynamics based on the stochastic, two-state model. The model assumes that the phase times are of variable duration and gamma-distributed, consistent with experimental evidence for microtubules assembled in vitro from purified tubulin. The analytical expressions permit the precise quantitative characterization of changes in microtubule and filopodial searching behavior corresponding to changes in the shape of the gamma distribution.