Can Power-Law Scaling and Neuronal Avalanches Arise from Stochastic Dynamics?

The presence of self-organized criticality in biology is often evidenced by a power-law scaling of event size distributions, which can be measured by linear regression on logarithmic axes. We show here that such a procedure does not necessarily mean that the system exhibits self-organized criticality. We first provide an analysis of multisite local field potential (LFP) recordings of brain activity and show that event size distributions defined as negative LFP peaks can be close to power-law distributions. However, this result is not robust to change in detection threshold, or when tested using more rigorous statistical analyses such as the Kolmogorov–Smirnov test. Similar power-law scaling is observed for surrogate signals, suggesting that power-law scaling may be a generic property of thresholded stochastic processes. We next investigate this problem analytically, and show that, indeed, stochastic processes can produce spurious power-law scaling without the presence of underlying self-organized criticality. However, this power-law is only apparent in logarithmic representations, and does not survive more rigorous analysis such as the Kolmogorov–Smirnov test. The same analysis was also performed on an artificial network known to display self-organized criticality. In this case, both the graphical representations and the rigorous statistical analysis reveal with no ambiguity that the avalanche size is distributed as a power-law. We conclude that logarithmic representations can lead to spurious power-law scaling induced by the stochastic nature of the phenomenon. This apparent power-law scaling does not constitute a proof of self-organized criticality, which should be demonstrated by more stringent statistical tests.     

An assessment of preferential attachment as a mechanism for human sexual network formation

Recent research into the properties of human sexual-contact networks has suggested that the degree distribution of the contact graph exhibits power-law scaling. One notable property of this power-law scaling is that the epidemic threshold for the population disappears when the scaling exponent rho is in the range 2 < rho < or = 3. This property is of fundamental significance for the control of sexually transmitted diseases (STDs) such as HIV/AIDS since it implies that an STD can persist regardless of its transmissibility. A stochastic process, known as preferential attachment, that yields one form of power-law scaling has been suggested to underlie the scaling of sexual degree distributions. The limiting distribution of this preferential attachment process is the Yule distribution, which we fit using maximum likelihood to local network data from samples of three populations: (i) the Rakai district, Uganda; (ii) Sweden; and (iii) the USA. For all local networks but one, our interval estimates of the scaling parameters are in the range where epidemic thresholds exist. The estimate of the exponent for male networks in the USA is close to 3, but the preferential attachment model is a very poor fit to these data. We conclude that the epidemic thresholds implied by this model exist in both single-sex and two-sex epidemic model formulations. A strong conclusion that we derive from these results is that public health interventions aimed at reducing the transmissibility of STD pathogens, such as implementing condom use or high-activity anti-retroviral therapy, have the potential to bring a population below the epidemic transition, even in populations exhibiting large degrees of behavioural heterogeneity.     

Network dynamics of the nonlinear power-law relaxation of cell cortex

Living cells are known to exhibit universal power-law rheological behaviors, but their underlying biomechanical principles are still not fully understood. Here, we present a network dynamics picture to decipher the nonlinear power-law relaxation of cortical cytoskeleton. Under step strains, we present a scaling relation between instantaneous differential stiffness and external stress as a result of chain reorientation. Then, during the relaxation, we show how the scaling law theoretically originates from an exponential form of cortical disorder, with the scaling exponent decreased by the imposed strain or crosslinker density in the nonlinear regime. We attribute this exponent variation to the molecular realignment along the stretch direction or the transition of network structure from in-series to in-parallel modes, both solidifying the network toward our one-dimensional theoretical limit. In addition, the rebinding of crosslinkers is found to be crucial for moderating the relaxation speed under small strains. Together with the disorder nature, we demonstrate that the structural effects of networks provide a unified interpretation for the nonlinear power-law relaxation of cell cortex, and may help to understand cell mechanics from the molecular scale.     

Simplified method for the computation of parameters of power-law rate equations from time-series

Modeling biological processes from time-series data is a resourceful procedure which has received much attention in the literature. For models established in the context of non-linear differential equations, parameter-dependent phenomenological tentative response functions are tested by comparing would-be solutions of those models to the experimental time-series. Those values of the parameters for which a tested solution is a best fit are then retained. It is done with the help of some appropriate optimization algorithm which simplifies the searching procedure within the range of variability of the parameters that are to be estimated. The procedure works well in problems with a small number of adjustable parameters or/and with narrow searching ranges. However, it may start to be problematic for models with a large number of problem parameters inasmuch as convergence to the best fit is not necessarily ensured. In this case, a reduction in size of the parameter estimation problem must be undertaken. We presently address this issue by proposing a systematic procedure that does so in problems in which the system's response to a sufficiently small pulse perturbation of steady-state can be obtained. The response is then assumed to be a solution of the linearized equations, the Jacobian of which can be retrieved by a simple multilinear regression. The calculated n(2) Jacobian entries provide as many relationships among problem parameters, thus cutting substantially the size of the starting problem. After this preliminary treatment is applied, only (kappa-n(2)) of the initial kappa adjustable parameters are left for evaluation by means of a non-linear optimization procedure. The benefits of the present variant are both in economy of computation and in accuracy in determining the parameter values. The performance of the method is established under different circumstances. It is illustrated in the context of power-law rates, although this does not preclude its applicability to more general functional responses.     

A theoretical interpretation of length-biomass allometry of predominantly bidimensional seaweeds

Experimental studies on bidimensional seaweeds revealed a scaling exponent of 0.472 for their length-biomass allometry. This was significantly higher than the value 0.25, which was proposed earlier as universal for all primary producers, based on the data for unicellular microalgae and vascular plants. Later, an exponent of 0.5 was theoretically derived, which agreed, to some extent, with experimental findings. Here, it is shown that there exists a power-law relation between the two perpendicular length parameters along the directions of growth of a bidimensional organism. The length-biomass allometric parameters can be expressed in terms of this power index. A relation between the allometric scaling exponent and allometric constant, involving the mass per unit area, has been obtained analytically. A method is proposed to determine the power index experimentally. Some mathematical expressions, relating mass, length and other parameters, have been formulated and these would be useful for experimental purposes in allometric studies. Analyzing images from an experimental study, a lot of parameters, regarding flat seaweeds, have been determined by analytical and numerical techniques.     

Dynamics of diluted attractor neural networks with delays

We present an analysis of the attractors of a deterministic dynamics in formal neural networks characterized by binary threshold units and a nonsymmetric connectivity. It is shown that in these networks a stored pattern or a pattern sequence is represented by a cloud of attractors rather than by a single attractor. Dilution, which we describe by a power-law scaling, and delayed couplings are shown to equip this type of network with a dynamic behaviour that is interesting enough for simplified models of biological motor systems. Received: 27 November 1992/Accepted in revised form: 22 September 1993     

Semi-empirical power-law scaling of new infection rate to model epidemic dynamics with inhomogeneous mixing

The expected number of new infections per day per infectious person during an epidemic has been found to exhibit power-law scaling with respect to the susceptible fraction of the population. This is in contrast to the linear scaling assumed in traditional epidemiologic modeling. Based on simulated epidemic dynamics in synthetic populations representing Los Angeles, Chicago, and Portland, we find city-dependent scaling exponents in the range of 1.7-2.06. This scaling arises from variations in the strength, duration, and number of contacts per person. Implementation of power-law scaling of the new infection rate is quite simple for SIR, SEIR, and histogram-based epidemic models. Treatment of the effects of the social contact structure through this power-law formulation leads to significantly lower predictions of final epidemic size than the traditional linear formulation.     

On the algorithms for determining the primary structure of biopolymers

The algorithm for determining the primary structure of biopolymers from complete and partial digests are analyzed. The problem of determining the primary structure is formulated in the form of the problem of word reconstruction in the limits of which the corresponding algorithms are analyzed. Difficulties arising in constructing the algorithms for determining the primary structure of nucleic acids from a partial digest are discussed. They seem to be due to the extensive testing of variants. When there is a certain scheme of the initial data from a partial digest we propose an economical testing (searching) algorithm. The scheme of an effective algorithm for reconstruction of the primary structure fromN complete digests is given.     

Deriving reaction–diffusion models in ecology from interacting particle systems

We use a scaling procedure based on averaging Poisson distributed random variables to derive population level models from local models of interactions between individuals. The procedure is suggested by using the idea of hydrodynamic limits to derive reaction-diffusion models for population interactions from interacting particle systems. The scaling procedure is formal in the sense that we do not address the issue of proving that it converges; instead we focus on methods for computing the results of the scaling or deriving properties of rescaled systems. To that end we treat the scaling procedure as a transform, in analogy with the Laplace or Fourier transform, and derive operational formulas to aid in the computation of rescaled systems or the derivation of their properties. Since the limiting procedure is adapted from work by Durrett and Levin, we refer to the transform as the Durrett-Levin transform. We examine the effects of rescaling in various standard models, including Lotka-Volterra models, Holling type predator-prey models, and ratio-dependent models. The effects of scaling are mostly quantitative in models with smooth interaction terms, but ratio-dependent models are profoundly affected by the scaling. The scaling transforms ratio-dependent terms that are singular at the origin into smooth terms. Removing the singularity at the origin eliminates some of the unique dynamics that can arise in ratio-dependent models.Research partially supported by NSF grants DMS 99-73017 and DMS 02-11367     

Physical meaning of one-machine and multimachine tokamak scalings

Specific features of energy confinement scalings constructed using different experimental databases for tokamak plasmas are considered. In the multimachine database, some pairs of engineering variables are collinear; e.g., the current I and the input power P both increase with increasing minor radius a. As a result, scalings derived from this database are reliable only for discharges in which such ratios as I/a ² or P/a ² are close to their values averaged over the database. The collinearity of variables allows one to exclude the normalized Debye radius d* from the scaling expressed in a nondimensional form. In one-machine databases, the dimensionless variables are functionally dependent, which allow one to cast a scaling without d*. In a database combined from two devices, the collinearity may be absent, so the Debye radius cannot generally be excluded from the scaling. It is shown that the experiments performed in support of the absence of d* in the two-machine scaling are unconvincing. Transformation expressions are given that allow one to compare experiments for the determination of scaling in any set of independent variables.     

Testing the Equality of Dependent Variances

The likelihood ratio test for testing equality of vgE;2 correlated variables is developed. In general, evaluation of the test statistic involves the iterative optimization of a likelihood function with 1 + v(v – 1)/2 parameters. The explicit form of the test statistic is derived in the bivariate case, and an iterative algorithm for determining the maximum likelihood estimates is suggested. A limited Monte Carlo study determines the behavior of the proposed procedure under the null hypothesis and variety of parameter values.     

Altitudinal Variations of Ground Tissue and Xylem Tissue in Terminal Shoot of Woody Species: Implications for Treeline Formation

1. The terminal shoot (or current-year shoot), as one of the most active parts on a woody plant, is a basic unit determining plant height and is potentially influenced by a variety of environmental factors. It has been predicted that tissues amount and their allocation in plant stems may play a critical role in determining plant size in alpine regions. The primary structure in terminal shoots is a key to our understanding treeline formation. The existing theories on treeline formation, however, are still largely lacking of evidence at the species level, much less from anatomy for the terminal shoot.2. The primary structures within terminal shoot were measured quantitatively for 100 species from four elevation zones along the eastern slope of Gongga Mountain, southwestern China; one group was sampled from above the treeline. An allometric approach was employed to examine scaling relationships interspecifically, and a principal components analysis (PCA) was performed to test the relation among primary xylem, ground tissue, species growth form and altitude.3. The results showed that xylem tissue size was closely correlated with ground tissue size isometrically across species, while undergoing significant y- or/and x-intercept shift in response to altitudinal belts. Further, a conspicuous characteristic of terminal shoot was its allocation of contrasting tissues between primary xylem and ground tissues with increasing elevation. The result of the PCA showed correlations between anatomical variation, species growth form/height classes and environment.4. The current study presents a comparative assessment of the allocation of tissue in terminal shoot across phylogenically and ecologically diverse species, and analyzes tissue, function and climate associations with plant growth forms and height classes among species. The interspecific connection between primary xylem ratio and plant size along an elevation gradient suggests the importance of primary xylem in explaining the treeline formation.     

Using two palpable measurements improves the subject-specific femoral modeling

Subject-specific musculoskeletal models are essential to biomedical research and clinical applications, such as customized joint replacement, computer-aided surgical planning, gait analysis and automated segmentation. Generating these models from CT or magnetic resonance imaging (MRI) is time and resource intensive, requiring special skills. Therefore, in many studies individual bone models are approximated by scaling a generic template. Thus, the primary goal of this study was to determine a set of clinically available parameters (palpable measures and demographic data) that could improve the prediction of femoral dimensions, as compared to predicting these variables using uniform scaling based on palpable length. Similar to previous non-homogenous anthropometric scaling methods, the non-homogenous scaling method proposed in this study improved the prediction over uniform scaling of five key femoral measures. Homogenous scaling forces all dimensions of an object to be scaled equally, whereas non-homogenous scaling allows the dimensions to be scaled independently. The largest improvement was in femoral depth, where the coefficient of determination (r²) improved from 0.22 (homogenous) to 0.60 (non-homogeneous). In general, the major advantage of this non-homogenous scaling method is its ability to support the accurate and rapid generation of subject-specific femoral models since all parameters can be collected clinically, without imaging or invasive methods.     

MANOVA multiple comparisons using the generalized step-down procedure

If the variables in MANOVA problem can be arranged according to the order of their importance, then J. ROY'S (1958) step-down procedure may be more appropriate than the conventional invariant inference techniques. However, it may often be possible only to identify subsets such that variables within subsets are equally important and subsets are of unequal importance. In experimental situations, it is common to have a set of variables of primary interest and another of “addon” variables. The step-down reasoning is extended to such cases and a set of simultaneous confidence bounds based upon the procedure which uses the largest root criterion at each stage are derived. The confidence bounds are on all linear functions of means only that do not involve nuisance parameters, and are therefore suitable for studying the configuration of means. This method yields shorter intervals for contrasts among the means of the variables of primary interest compared with the conventional intervals based upon the largest root. The method is illustrated using BARNARD'S data (1935) on skull characters.     

A stability analysis of the power-law steady state of marine size spectra

This paper investigates the stability of the power-law steady state often observed in marine ecosystems. Three dynamical systems are considered, describing the abundance of organisms as a function of body mass and time: a “jump-growth” equation, a first order approximation which is the widely used McKendrick–von Foerster equation, and a second order approximation which is the McKendrick–von Foerster equation with a diffusion term. All of these yield a power-law steady state. We derive, for the first time, the eigenvalue spectrum for the linearised evolution operator, under certain constraints on the parameters. This provides new knowledge of the stability properties of the power-law steady state. It is shown analytically that the steady state of the McKendrick–von Foerster equation without the diffusion term is always unstable. Furthermore, numerical plots show that eigenvalue spectra of the McKendrick–von Foerster equation with diffusion give a good approximation to those of the jump-growth equation. The steady state is more likely to be stable with a low preferred predator:prey mass ratio, a large diet breadth and a high feeding efficiency. The effects of demographic stochasticity are also investigated and it is concluded that these are likely to be small in real systems.     

Inbreeding reduces power-law scaling in the distribution of fluctuating asymmetry: an explanation of the basis of developmental instability

The study of fluctuating asymmetry has been controversial because of conflicting results found in much of the primary literature. It has been suggested that the source of this conflict is the fact that the basis of fluctuating asymmetry is poorly understood and that, as a consequence, methodology of fluctuating asymmetry studies may be flawed. A new model for the phenomenological basis of fluctuating asymmetry, that variation in fluctuating asymmetry is in large part due to the random exponential growth of cell populations (geometric Brownian motion) that are terminated randomly around a genetically programmed development time, is presented here. If termination of development has a genetic component, then scaling effects and kurtosis in the distribution of fluctuating asymmetry should increase with genetic redundancy of the population. This model prediction was tested by comparing the distribution of multivariate size and shape fluctuating asymmetry in large samples collected from both wild populations and four moderately inbred lines of Drosophila simulans. It was found that while wild populations were best described by a lognormal distribution with power-law scaled tails, the inbred lines derived from the wild stock were dramatically normalized (half-normal) in three of four cases. As predicted, the scaling exponent of the upper tail of the distribution of fluctuating asymmetry increased with inbreeding while the kurtosis and mean fluctuating asymmetry decreased with inbreeding. The model suggests an additional explanation of leptokurtosis in fluctuating asymmetry. Kurtosis and scaling of the statistical distribution of fluctuating asymmetry in a population is related directly to genetic differences between individuals and these differences affect their ability to buffer the process of development against random perturbations.     

Phylogenetic grouping, curvature and metabolic scaling in terrestrial invertebrates

For more than a century, the scaling of animal metabolic rates with individual body masses and environmental temperature has predominantly been described by power-law and exponential relationships respectively. Many theories have been proposed to explain these scaling relationships, but were challenged by empirically documented curvatures on double-logarithmic scales. In the present study, we present a novel data set comprising 3661 terrestrial (mainly soil) invertebrate respiration rates from 192 independent sources across a wide range in body masses, environmental temperatures and phylogenetic groups. Although our analyses documented power-law and exponential scaling with body masses and temperature, respectively, polynomial models identified curved deviations. Interestingly, complex scaling models accounting for phylogenetic groups were able to remove curvatures except for a negative curvature at the highest temperatures (>30 °C) indicating metabolic down regulation. This might indicate that the tremendous differences in invertebrate body architectures, ecology and physiology may cause severely different metabolic scaling processes.     

Rural to Urban Population Density Scaling of Crime and Property Transactions in English and Welsh Parliamentary Constituencies

Urban population scaling of resource use, creativity metrics, and human behaviors has been widely studied. These studies have not looked in detail at the full range of human environments which represent a continuum from the most rural to heavily urban. We examined monthly police crime reports and property transaction values across all 573 Parliamentary Constituencies in England and Wales, finding that scaling models based on population density provided a far superior framework to traditional population scaling. We found four types of scaling: i) non-urban scaling in which a single power law explained the relationship between the metrics and population density from the most rural to heavily urban environments, ii) accelerated scaling in which high population density was associated with an increase in the power-law exponent, iii) inhibited scaling where the urban environment resulted in a reduction in the power-law exponent but remained positive, and iv) collapsed scaling where transition to the high density environment resulted in a negative scaling exponent. Urban scaling transitions, when observed, took place universally between 10 and 70 people per hectare. This study significantly refines our understanding of urban scaling, making clear that some of what has been previously ascribed to urban environments may simply be the high density portion of non-urban scaling. It also makes clear that some metrics undergo specific transitions in urban environments and these transitions can include negative scaling exponents indicative of collapse. This study gives promise of far more sophisticated scale adjusted metrics and indicates that studies of urban scaling represent a high density subsection of overall scaling relationships which continue into rural environments.     

Joint scaling tests.

It is shown that a joint scaling test developed by Tan (1974) is closely related to the widely used standard Cavalli joint scaling test (described by Mather and Jinks, 1971) which was not referred to in Tan's paper. With the numbers of individuals per generation observed in practice, the two tests give essentially similar results. The Cavalli procedure also provides estimates of genetical parameters and is more readily extended to a wider range of situations.