A nonstationary Poisson point process describes the sequence of action potentials over long time scales in lateral-superior-olive auditory neurons

The behavior of lateral-superior-olive (LSO) auditory neurons over large time scales was investigated. Of particular interest was the determination as to whether LSO neurons exhibit the same type of fractal behavior as that observed in primary VIII-nerve auditory neurons. It has been suggested that this fractal behavior, apparent on long time scales, may play a role in optimally coding natural sounds. We found that a nonfractal model, the nonstationary dead-time-modified Poisson point process (DTMP), describes the LSO firing patterns well for time scales greater than a few tens of milliseconds, a region where the specific details of refractoriness are unimportant. The rate is given by the sum of two decaying exponential functions. The process is completely specified by the initial values and time constants of the two exponentials and by the dead-time relation. Specific measures of the firing patterns investigated were the interspike-interval (ISI) histogram, the Fano-factor time curve (FFC), and the serial count correlation coefficient (SCC) with the number of action potentials in successive counting times serving as the random variable. For all the data sets we examined, the latter portion of the recording was well approximated by a single exponential rate function since the initial exponential portion rapidly decreases to a negligible value. Analytical expressions available for the statistics of a DTMP with a single exponential rate function can therefore be used for this portion of the data. Good agreement was obtained among the analytical results, the computer simulation, and the experimental data on time scales where the details of refractoriness are insignificant. For counting times that are sufficiently large, yet much smaller than the largest time constant in the rate function, the Fano factor is directly proportional to the counting time. The nonstationarity may thus mask fractal fluctuations, for which the Fano factor increases as a fractional power (less than unity) of the counting time.     

Diameters, generations, and orders of branches in the bronchial tree

Studies of bronchial tree data by West et al. (J. Appl. Physiol. 60: 1089-1097, 1986) have shown that plots of mean diameter against generation, using log-log scales, can be represented by a power function with harmonic modulations. Other studies have shown that the mean diameter of the airways is exponentially related to order of branching. This paper demonstrates that both observations are compatible with a fractal model of branching, and because airway branching is fractal, this may explain why both are also true of the bronchial tree. Furthermore, the exponential relationship of mean diameter with generation in the larger airways, demonstrated by Weibel, is shown to result from the exponential relation of diameter with order in the fractal model.     

Fractional Poisson–Nernst–Planck Model for Ion Channels I: Basic Formulations and Algorithms

In this work, we propose a fractional Poisson–Nernst–Planck model to describe ion permeation in gated ion channels. Due to the intrinsic conformational changes, crowdedness in narrow channel pores, binding and trapping introduced by functioning units of channel proteins, ionic transport in the channel exhibits a power-law-like anomalous diffusion dynamics. We start from continuous-time random walk model for a single ion and use a long-tailed density distribution function for the particle jump waiting time, to derive the fractional Fokker–Planck equation. Then, it is generalized to the macroscopic fractional Poisson–Nernst–Planck model for ionic concentrations. Necessary computational algorithms are designed to implement numerical simulations for the proposed model, and the dynamics of gating current is investigated. Numerical simulations show that the fractional PNP model provides a more qualitatively reasonable match to the profile of gating currents from experimental observations. Meanwhile, the proposed model motivates new challenges in terms of mathematical modeling and computations.     

A flexible sigmoid function of determinate growth

A new empirical equation for the sigmoid pattern of determinate growth, 'the beta growth function', is presented. It calculates weight (w) in dependence of time, using the following three parameters: t(m), the time at which the maximum growth rate is obtained; t(e), the time at the end of growth; and w(max), the maximal value for w, which is achieved at t(e). The beta growth function was compared with four classical (logistic, Richards, Gompertz and Weibull) growth equations, and two expolinear equations. All equations described successfully the sigmoid dynamics of seed filling, plant growth and crop biomass production. However, differences were found in estimating w(max). Features of the beta function are: (1) like the Richards equation it is flexible in describing various asymmetrical sigmoid patterns (its symmetrical form is a cubic polynomial); (2) like the logistic and the Gompertz equations its parameters are numerically stable in statistical estimation; (3) like the Weibull function it predicts zero mass at time zero, but its extension to deal with various initial conditions can be easily obtained; (4) relative to the truncated expolinear equation it provides more reasonable estimates of final quantity and duration of a growth process. In addition, the new function predicts a zero growth rate at both the start and end of a precisely defined growth period. Therefore, it is unique for dealing with determinate growth, and is more suitable than other functions for embedding in process-based crop simulation models to describe the dynamics of organs as sinks to absorb assimilates. Because its parameters correspond to growth traits of interest to crop scientists, the beta growth function is suitable for characterization of environmental and genotypic influences on growth processes. However, it is not suitable for estimating maximum relative growth rate to characterize early growth that is expected to be close to exponential.     

Estimation of growth regulation in natural populations by extended family of growth curve models with fractional order derivative: Case studies from the global population dynamics database

Estimating the trend in population time series data using growth curve models is a central idea in population ecology. Several models, mainly governed by differential or difference equations, have been applied to real data sets to identify general growth pattern and make predictions. In this article, we analyze ecological time series data by fitting mathematical models governed by fractional differential equations (FDE). The order of the FDE (α) is used to quantify the evidence of memory in the population processes. The application of FDE is exemplified by analyzing time series data on two bird species Phalacrocorax carbo (Great cormorant) and Parus bicolor (Tufted titmouse) and two mammal species Castor canadensis (Beaver) and Ursus americanus (American black bear) extracted from the global population dynamics database. Five different population growth models were fitted to these data; density-independent exponential, negative density-dependent logistic and θ-logistic model, positive density-dependent exponential Allee and strong Allee model. Both ordinary and fractional derivative representations of these models were fitted to the time series data. Markov chain Monte Carlo (MCMC) method was used to estimate the model parameters and Akaike information criterion was used to select the best model. By estimating the return rate for each of the time series, we have shown that populations governed by FDE with a small value of α (high level of memory) return to the stable equilibrium faster. This demonstrates a synergistic interplay between memory and stability in natural populations.     

Empirical and theoretical analysis of the extremely low frequency arterial blood pressure power spectrum in unanesthetized rat

The slope of the log of power versus the log of frequency in the arterial blood pressure (BP) power spectrum is classically considered constant over the low-frequency range (i.e., "fractal" behavior), and is quantified by beta in the relationship "1/f(beta)." In practice, the fractal range cannot extend to indefinitely low frequencies, but factor(s) that terminate this behavior, and determine beta, are unclear. We present 1) data in rats (n = 8) that reveal an extremely low frequency spectral region (0.083-1 cycle/h), where beta approaches 0 (i.e., the "shoulder"); and 2) a model that 1) predicts realistic values of beta within that range of the spectrum that conforms to fractal dynamics (approximately 1-60 cycles/h), 2) offers an explanation for the shoulder, and 3) predicts that the "successive difference" in mean BP (mBP) is an important parameter of circulatory function. We recorded BP for up to 16 days. The absolute difference between successive mBP samples at 0.1 Hz (the successive difference, or Delta) was 1.87 +/- 0.21 mmHg (means +/- SD). We calculated beta for three frequency ranges: 1) 0.083-1; 2) 1-6; and 3) 6-60 cycles/h. The beta for all three regions differed (P < 0.01). For the two higher frequency ranges, beta indicated a fractal relationship (beta(6-60/h) = 1.27 +/- 0.01; beta(1-6/h) = 1.80 +/- 0.16). Conversely, the slope of the lowest frequency region (i.e., the shoulder) was nearly flat (beta(0.083-1 /h) = 0.32 +/- 0.28). We simulated the BP time series as a random walk about 100 mmHg with ranges above and below of 10, 30, and 50 mmHg and with Delta from 0.5 to 2.5. The spectrum for the conditions mimicking actual BP time series (i.e., range, 85-115 mmHg; Delta, 2.00) resembled the observed spectra, with beta in the lowest frequency range = 0.207 and fractal-like behavior in the two higher frequency ranges (beta = 1.707 and 2.057). We suggest that the combined actions of mechanisms limiting the excursion of arterial BP produce the shoulder in the spectrum and that Delta contributes to determining beta.     

Dynamic Modeling of a Non-Uniform Flexible Tail for a Robotic Fish

In this paper, a non-uniform flexible tail of a fish robot was presented and the dynamic model was developed. In this model, the non-uniform flexible tail was modeled by a rotary slender beam. The hydrodynamics forces, including the reactive force and resistive force, were analyzed in order to derive the governing equation. This equation is a fourth-order in space and second-order in time Partial Differential Equation (PDE) of the lateral movement function. The coefficients of this PDE were not constants because of the non-uniform beams, so they were approximated by exponential functions in order to obtain an analytical solution. This solution describes the lateral movement of the flexible tail as a function of material, geometrical and actuator properties. Experiments were then carried out and compared to simulations. It was proved that the proposed model is suitable for predicting the real behavior of fish robots.     

Epidemiological Models with Non-Exponentially Distributed Disease Stages and Applications to Disease Control

SEIR epidemiological models with the inclusion of quarantine and isolation are used to study the control and intervention of infectious diseases. A simple ordinary differential equation (ODE) model that assumes exponential distribution for the latent and infectious stages is shown to be inadequate for assessing disease control strategies. By assuming arbitrarily distributed disease stages, a general integral equation model is developed, of which the simple ODE model is a special case. Analysis of the general model shows that the qualitative disease dynamics are determined by the reproductive number , which is a function of control measures. The integral equation model is shown to reduce to an ODE model when the disease stages are assumed to have a gamma distribution, which is more realistic than the exponential distribution. Outcomes of these models are compared regarding the effectiveness of various intervention policies. Numerical simulations suggest that models that assume exponential and non-exponential stage distribution assumptions can produce inconsistent predictions.     

Relating individual behaviour to population dynamics

How do the behavioural interactions between individuals in an ecological system produce the global population dynamics of that system? We present a stochastic individual-based model of the reproductive cycle of the mite Varroa jacobsoni, a parasite of honeybees. The model has the interesting property in that its population level behaviour is approximated extremely accurately by the exponential logistic equation or Ricker map. We demonstrated how this approximation is obtained mathematically and how the parameters of the exponential logistic equation can be written in terms of the parameters of the individual-based model. Our procedure demonstrates, in at least one case, how study of animal ecology at an individual level can be used to derive global models which predict population change over time.     

On time-space of nonlinear phenomena with Gompertzian dynamics

This paper describes a universal relationship between time and space for a nonlinear process with Gompertzian dynamics, such as growth. Gompertzian dynamics implicates a coupling between time and space. Those two categories are related to each other through a linear function of their logarithms. Moreover, we demonstrate that the spatial fractal dimension is a function of both scalar time and the temporal fractal dimension. The Gompertz function reflects the equilibrium of regular states, that is, states with dynamics that are predictable for any time-point (e.g., sinusoidal glycolytic oscillations) and chaotic states, that is, states with dynamics that are unpredictable in time, but are characterized by certain regularities (e.g., the existence of strange attractor for any biochemical reaction). We conclude that both this equilibrium and volume of the available complementary Euclidean space determine temporal and spatial expansion of a process with Gompertzian dynamics.     

Population dynamics with a refuge: fractal basins and the suppression of chaos

We consider the effect of coupling an otherwise chaotic population to a refuge. A rich set of dynamical phenomena is uncovered. We consider two forms of density dependence in the active population: logistic and exponential. In the former case, the basin of attraction for stable population growth becomes fractal, and the bifurcation diagrams for the active and refuge populations are chaotic over a wide range of parameter space. In the case of exponential density dependence, the dynamics are unconditionally stable (in that the population size is always positive and finite), and chaotic behavior is completely eradicated for modest amounts of dispersal. We argue that the use of exponential density dependence is more appropriate, theoretically as well as empirically, in a model of refuge dynamics.     

Time domain analysis of oxygen uptake during pseudorandom binary sequence exercise tests.

Pseudorandom binary sequence (PRBS) exercise tests involve repeated switching between two work rates (WR) according to a computer-generated pattern. This paper presents an approach to analysis of O2 uptake (VO2) in the time domain. First, the autocorrelation function (ACF) of the input WR was recognized to be a triangular-shaped pulse that can be taken to be equivalent to a ramp increase followed by a ramp decrease in WR. Then the cross-correlation function of the input (WR) and the output (VO2) was treated as if it were the response to a triangular-shaped pulse. The cross-correlation function was analyzed by fitting a linear summation of the ramp form of a two-component exponential function to this triangular pulse. VO2 responses of eight subjects were obtained from two different PRBS tests, as well as step changes in WR. The first PRBS test consisted of 15 units, each 30 s in duration. Its ACF had a base width of 60 s. The ramp increase-ramp decrease model fit the data throughout the range of response. The second PRBS test had 63 units, each 5 s in duration; thus its ACF base width was 10 s. Again, the ramp model fit adequately. The data from the second PRBS test could be fit by the impulse form of the two-component exponential equation, although the fit in the first 30 s tended to be poorer. The time constants of VO2 dynamics estimated from step and PRBS tests were not significantly different. PRBS tests can be analyzed in the time domain, and the indicators of system dynamics reflect physiological properties similar to those investigated during step changes in WR.     

Curvature-Driven Pore Growth in Charged Membranes during Charge-Pulse and Voltage-Clamp Experiments

We find that curvature-driven growth of pores in electrically charged membranes correctly reproduces charge-pulse experiments. Our model, consisting of a Langevin equation for the time dependence of the pore radius coupled to an ordinary differential equation for the number of pores, captures the statistics of the pore population and its effect on the membrane conductance. The calculated pore radius is a linear, and not an exponential, function of time, as observed experimentally. Two other important features of charge-pulse experiments are recovered: pores reseal for low and high voltages but grow irreversibly for intermediate values of the voltage. Our set of coupled ordinary differential equations is equivalent to the partial differential equation used previously to study pore dynamics, but permits the study of longer timescales necessary for the simulations of voltage-clamp experiments. An effective phase diagram for such experiments is obtained.     

Sorption rates for the uptake of Cr6+ by a consortia of denitrifying bacteria

Summary A fractional factorial statistical experimental design was used to ascertain which experimental parameters affect the rate at which Cr⁶⁺ is sorbed by a consortium of denitrifying bacteria. Data from this set of experiments indicates that the amount of chromium sorbed as a function of time could be described by an exponential rise function. Additionally, the data indicate that at least 15 hours were required to establish equilibrium. Finally, observed variations in the parameters in this rate equation were found to be due to the imposed changes in the experimental variables. These results indicate that the processes by which the chromate ion was removed from solution may be associated with the cells' metabolic processes.     

A quantitative method for the analysis of mammalian cell proliferation in culture in terms of dividing and non-dividing cells

The application of the exponential growth equation is the standard method employed in the quantitative analyses of mammalian cell proliferation in culture. This method is based on the implicit assumption that, within a cell population under study, all division events give rise to daughter cells that always divide. When a cell population does not adhere to this assumption, use of the exponential growth equation leads to errors in the determination of both population doubling time and cell generation time. We have derived a more general growth equation that defines cell growth in terms of the dividing fraction of daughter cells. This equation can account for population growth kinetics that derive from the generation of both dividing and non-dividing cells. As such, it provides a sensitive method for detecting non-exponential division dynamics. In addition, this equation can be used to determine when it is appropriate to use the standard exponential growth equation for the estimation of doubling time and generation time.     

The study of bimolecular reactions under non-pseudo-first order conditions

In this work a new equation which describes the time evolution of bimolecular reactions is derived and tested by experiment. The equation is general and the results show that second-order reactions of any simple type may be accurately described by a quotient of exponential functions. The model and reagent concentration dependent observed rate constants show a complex non-linear behaviour when experimental conditions deviate from pseudo-first order nevertheless reducing to the well-known linear dependence when pseudo-first order conditions are met.     

Kinetics of gelatin transitions with phase separation: T-jump and step-wise DSC study

The isothermal gelation (or melting) of gelatin after fast cooling (or heating) steps is studied by using high sensitivity differential scanning micro-calorimetry, in order to determine the dependence of the kinetic and thermodynamic parameters upon changes in composition and in temperature. The calorimetric heat flow curves, obtained according to defined temperature profiles, have been fitted with exponential functions (simple exponentials or stretched exponentials for the step-wise and for T-jump experiments, respectively). The gelation process of gelatin alone for t<300 min shows that the characteristic time tau and the fractional exponent are beta very sensitive to the concentration of gelatin chains and to the microscopic phase segregation due to the presence of another polymeric component.