Dynamics extraction in multivariate biomedical time series

A nonlinear analysis of the underlying dynamics of a biomedical time series is proposed by means of a multi-dimensional testing of nonlinear Markovian hypotheses in the observed time series. The observed dynamics of the original N-dimensional biomedical time series is tested against a hierarchy of null hypotheses corresponding to N-dimensional nonlinear Markov processes of increasing order, whose conditional probability densities are estimated using neural networks. For each of the N time series, a measure based on higher order cumulants quantifies the independence between the past of the N-dimensional time series, and its value r steps ahead. This cumulant-based measure is used as a discriminating statistic for testing the null hypotheses. Experiments performed on artificial and real world examples, including autoregressive models, noisy chaos, and nonchaotic nonlinear processes, show the effectiveness of the proposed approach in modeling multivariate systems, predicting multidimensional time series, and characterizing the structure of biological systems. Electroencephalogram (EEG) time series and heart rate variability trends are tested as biomedical signal examples. Received: 2 July 1997 / Accepted in revised form: 26 March 1998     

Development and application of white-noise modeling techniques for studies of insect visual nervous system

The nonlinear system identification technique through white-noise stimulation is extended to multi-input, -output systems with consideration given to applications in the functional study of the nervous system. The applicability of the method is discussed in general and in particular for the motion detection neuronal system of the fly. Two series of experiments are performed; one with moving striped-pattern stimuli and the other with spot stimuli of fluctuating intensity. In both cases nonlinear dynamic models are derived which describe the system with considerable accuracy over the frequency range of 0.2–50 Hz and a dynamic amplitude range of about 40-1. These models are able to predict accurately all the discrete experiments so far performed on this system for which the models are applicable. The differences in dynamic characteristics between the corresponding system of the Musca and Phoenicia families of flies are minor except for a difference in latencies and if the difference in geometry of their faceted eyes is taken into account. The large field response of the motion detection unit is a linear weighted summation of all the smaller field highly nonlinear subsystems of which the large field is comprised.     

Crossover inhibition in the retina: circuitry that compensates for nonlinear rectifying synaptic transmission

In the mammalian retina, complementary ON and OFF visual streams are formed at the bipolar cell dendrites, then carried to amacrine and ganglion cells via nonlinear excitatory synapses from bipolar cells. Bipolar, amacrine and ganglion cells also receive a nonlinear inhibitory input from amacrine cells. The most common form of such inhibition crosses over from the opposite visual stream: Amacrine cells carry ON inhibition to the OFF cells and carry OFF inhibition to the ON cells (”crossover inhibition”). Although these synapses are predominantly nonlinear, linear signal processing is required for computing many properties of the visual world such as average intensity across a receptive field. Linear signaling is also necessary for maintaining the distinction between brightness and contrast. It has long been known that a subset of retinal outputs provide exactly this sort of linear representation of the world; we show here that rectifying (nonlinear) synaptic currents, when combined thorough crossover inhibition can generate this linear signaling. Using simple mathematical models we show that for a large set of cases, repeated rounds of synaptic rectification without crossover inhibition can destroy information carried by those synapses. A similar circuit motif is employed in the electronics industry to compensate for transistor nonlinearities in analog circuits.     

Two Bayesian methods for estimating parameters of the von Bertalanffy growth equation

The von Bertalanffy growth equation (VBGE) is commonly used in ecology and fisheries management to model individual growth of an organism. Generally, a nonlinear regression is used with length-at-age data to recover key life history parameters: L _∞ (asymptotic size), k (the growth coefficient), and t ₀ (a time used to calculate size at age 0). However, age data are often unavailable for many species of interest, which makes the regression impossible. To confront this problem, we have developed a Bayesian model to find L _∞ using only length data. We use length-at-age data for female blue shark, Prionace glauca, to test our hypothesis. Preliminary comparisons of the model output and the results of a nonlinear regression using the VBGE show similar estimates of L _∞ . We also developed a full Bayesian model that fits the VBGE to the same data used in the classical regression and the length-based Bayesian model. Classical regression methods are highly sensitive to missing data points, and our analysis shows that fitting the VBGE in a Bayesian framework is more robust. We investigate the assumptions made with the traditional curve fitting methods, and argue that either the full Bayesian or the length-based Bayesian models are preferable to classical nonlinear regressions. These methods clarify and address assumptions␣made in classical regressions using von Bertalanffy growth and facilitate more detailed stock assessments of species for which data are sparse.     

Accounting for wildlife life-history strategies when modeling stochastic density-dependent populations: A review

This article briefly reviews and provides discussion on the evidence for, and nature of, density-dependence patterns in r and K-selected species. In this review, I discuss how life-history strategies cause different nonlinear density-dependence patterns and I provide a simple modeling recommendation to incorporate nonlinear density dependence in population growth equations. Second, I discuss the importance of incorporation of environmental stochasticity and local extinction associated with nonlinear density dependence associated with life-history patterns through a novel modeling exercise. Last, I discuss the importance of considering how life-history nonlinear density dependence could affect optimal harvest yields. Though these topics are extensive, this review should spur wildlife biologists and managers to consider more inclusive population models that incorporate life-history strategies and stochasticity in their decision-making processes. © 2012 The Wildlife Society.     

Otoacoustic emissions from insect ears having just one auditory neuron

Sensitive hearing organs often employ nonlinear mechanical sound processing which produces distortion-product otoacoustic emissions. Such emissions are also recorded from insect tympanal organs. Here we report high frequency distortion-product emissions, evoked by stimulus frequencies up to 95 kHz, from the tympanal organ of a notodontid moth, Ptilodon cucullina, which contains only a single auditory receptor neuron. The 2f1–f2 distortion-product emission reaches sound levels above 40 dB SPL. Most emission growth functions show a prominent notch of 20 dB depth (n = 20 trials), accompanied by an average phase shift of 119°, at stimulus levels between 60 and 70 dB SPL, which separates a low- and a high-level component. The emissions are vulnerable to topical application of ethyl ether which shifts growth functions by about 20 dB towards higher stimulus levels. For the mammalian cochlea, Lukashkin and colleagues have proposed that distinct level-dependent components of nonlinear amplification do not necessarily require interaction of several cellular sources but could be due to a single nonlinear source. In notodontids, such a physiologically vulnerable source could be the single receptor cell. Potential contributions from accessory cells to the nonlinear properties of the scolopidial hearing organ are still unclear.     

An age-and-cyclin-structured cell population model for healthy and tumoral tissues

We present a nonlinear model of the dynamics of a cell population divided into proliferative and quiescent compartments. The proliferative phase represents the complete cell cycle (G ₁−S−G ₂−M) of a population committed to divide at its end. The model is structured by the time spent by a cell in the proliferative phase, and by the amount of Cyclin D/(CDK4 or 6) complexes. Cells can transit from one compartment to the other, following transition rules which differ according to the tissue state: healthy or tumoral. The asymptotic behaviour of solutions of the nonlinear model is analysed in two cases, exhibiting tissue homeostasis or tumour exponential growth. The model is simulated and its analytic predictions are confirmed numerically.     

Metabolic engineering in silico

This review briefs on the main directions in the field of mathematical modeling of metabolic processes aimed at a rational design of genetically modified organisms. The class of generalized Hill functions is described, and their application to modeling of nonlinear processes in Escherichia coli metabolic systems is illustrated by several examples. A model for the pyrimidine biosynthesis in E. coli, taking into account the nonlinear effects of a negative allosteric regulation of enzyme activities involved in the control of the subsequent stages by the end products of synthesis, is considered. It has been shown that the model displays its own continuous oscillation mode of functioning with a period of approximately 50 min, which is close to the duration of E. coli cell cycle. The need in considering the nonlinear effects in the models as essential elements in the function of metabolic systems far from equilibrium is discussed.     

Backward bifurcations and strong Allee effects in matrix models for the dynamics of structured populations

In nonlinear matrix models, strong Allee effects typically arise when the fundamental bifurcation of positive equilibria from the extinction equilibrium at r=1 (or R₀=1) is backward. This occurs when positive feedback (component Allee) effects are dominant at low densities and negative feedback effects are dominant at high densities. This scenario allows population survival when r (or equivalently R₀) is less than 1, provided population densities are sufficiently high. For r>1 (or equivalently R₀>1) the extinction equilibrium is unstable and a strong Allee effect cannot occur. We give criteria sufficient for a strong Allee effect to occur in a general nonlinear matrix model. A juvenile–adult example model illustrates the criteria as well as some other possible phenomena concerning strong Allee effects (such as positive cycles instead of equilibria).     

Nonlinear electric properties of DNA solutions.

DNA solutions are shown to present a nonlinear electric behavior. This property is measured through the third harmonic current intensity, which appears when the solution is placed in a sinusoidal electric field of moderately high strength (about 100 V·cm^?1). The influence of different parameters has been examined: fundamental frequency, field strength, concentration, molecular weight, and conformation. By progressive sonication, it is shown that the harmonic current is linearly proportional to the DNA molecular weight, but under an M_r of approximately 10⁶, the nonlinear electrical property decreases sharply and its spectrum shows a drastic change. It is thought that the harmonic current is not related to an orientational phenomenon; an explanation based on the electrical deformation of the molecule is suggested.     

Cation effects on the nonlinear electrical properties of DNA solutions

The nonlinear electrical properties of DNA solutions were measured when different monovalent cations were added to DNA. The influence of different parameters has been examined: fundamental frequency, field strength, and concentration. A linear relationship between the harmonic current I_h and the DNA concentration is shown, even for higher concentration values (400 mg/l.). The frequency dispersion of I_h has the same shape for all the cations and the low-frequency amplitude of I_h increases in the following order: Li⁺ < Na⁺ < K⁺ < NH < Cs⁺. The nonlinear polarizability values are compared with the linear ones determined using the very low field electric birefringence technique. Both linear and nonlinear values are of the same order of magnitude. It is thought that the nonlinear electrical property of high-molecular-weight DNA mainly results from the deformation of the DNA coils by the electric field.     

Adjustment of Prediction Intervals in Nonlinear Regression

By treating the nonlinear model as if it were linear in the parameterization θ in the neighbourhood of the least squares estimate θC, two-sided nominally-q-prediction intervals can be constructed by applying the usual linear model theory. The quadratic approximation of the expected coverage of the prediction intervals is derived for a p-parameter nonlinear model. An adjustment of the nominally-q-prediction intervals is proposed. It is shown that, to the extent that quadratic approximation is adequate, the actual expected coverage of the adjusted prediction intervals is q.     

Spatio-temporal cross-correlation analysis of catfish retinal neurons

1. The receptive field properties of visual neurons in the retina of the catfish are studied by a white noise spatio-temporal stimulus. The spatial and temporal inputs of the stimulus are independent and lead to complete linear characterizations and local nonlinear characterizations of the neural response. 2. Horizontal cells, bipolar cells, and sustained or Type N amacrine cells all yield spatially coherent linear correlations. The horizontal cells have the shortest latency by these methods and exhibit a late depolarizing component that is wider in spatial extent than the initial hyperpolarizing component. Depolarizing Type N neurons have center-hyperpolarizing local nonlinearity. 3. Transient or Type C amacrine cells do not correlate well with the intensity of the stimulus, even though the Fast variety responds vigorously to the stimulus. 4. Ganglion cells are classified into Excitatory, Inhibitory and Biphasic classes based upon their linear correlations. Some ganglions exhibit responses dependent upon the orientation of stimulus. Although linear correlation of the Excitatory class is similar to that of the depolarizing Type N cell, the locally nonlinear character of these cell types is distinct. The receptive field of the Inhibitory ganglion cells has strong locally excitatory nonlinearity.     

Linear and nonlinear models of the basilar membrane motion

A linear and a nonlinear transmission line model of the basilar membrane is described. The motion of the basilar membrane model has been simulated by numerical methods and compared with physiological data for several types of sound stimuli. It is shown that a linear model exhibits a frequency modulation in its impulse response that is in accordance with physiological data. The nonlinear model displays a sharpened frequency response for lower sound intensities. Futhermore, a nonlinear model explains why hearing damage imposed by short, high-intensity, sounds is extended to the low-frequency regions of the cochlea.     

A Derivative Matching Approach to Moment Closure for the Stochastic Logistic Model

Continuous-time birth-death Markov processes serve as useful models in population biology. When the birth-death rates are nonlinear, the time evolution of the first n order moments of the population is not closed, in the sense that it depends on moments of order higher than n. For analysis purposes, the time evolution of the first n order moments is often made to be closed by approximating these higher order moments as a nonlinear function of moments up to order n, which we refer to as the moment closure function. In this paper, a systematic procedure for constructing moment closure functions of arbitrary order is presented for the stochastic logistic model. We obtain the moment closure function by first assuming a certain separable form for it, and then matching time derivatives of the exact (not closed) moment equations with that of the approximate (closed) equations for some initial time and set of initial conditions. The separable structure ensures that the steady-state solutions for the approximate equations are unique, real and positive, while the derivative matching guarantees a good approximation, at least locally in time. Explicit formulas to construct these moment closure functions for arbitrary order of truncation n are provided with higher values of n leading to better approximations of the actual moment dynamics. A host of other moment closure functions previously proposed in the literature are also investigated. Among these we show that only the ones that achieve derivative matching provide a close approximation to the exact solution. Moreover, we improve the accuracy of several previously proposed moment closure functions by forcing derivative matching.     

Chaotic behavior in the locomotion ofamoeba proteus

Summary The locomotion ofAmoeba proteus has been investigated by algorithms evaluating correlation dimension and Lyapunov spectrum developed in the field of nonlinear science. It is presumed by these parameters whether the random behavior of the system is stochastic or deterministic. For the analysis of the nonlinear parameters, n-dimensional time-delayed vectors have been reconstructed from a time series of periphery and area ofA. proteus images captured with a charge-coupled-device camera, which characterize its random motion. The correlation dimension analyzed has shown the random motion ofA. proteus is subjected only to 3–4 macrovariables, though the system is a complex system composed of many degrees of freedom. Furthermore, the analysis of the Lyapunov spectrum has shown its largest exponent takes positive values. These results indicate the random behavior ofA. proteus is chaotic and deterministic motion on an attractor with low dimension. It may be important for the elucidation of the cell locomotion to take account of nonlinear interactions among a small number of dynamics such as the sol-gel transformation, the cytoplasmic streaming, and the relating chemical reaction occurring in the cell.     

Population extinction and quasi-stationary behavior in stochastic density-dependent structured models

Density-independent and density-dependent, stochastic and deterministic, discrete-time, structured models are formulated, analysed and numerically simulated. A special case of the deterministic, density-independent, structured model is the well-known Leslie age-structured model. The stochastic, density-independent model is a multitype branching process. A review of linear, density-independent models is given first, then nonlinear, density-dependent models are discussed. In the linear, density-independent structured models, transitions between states are independent of time and state. Population extinction is determined by the dominant eigenvalue λ of the transition matrix. If λ ≤ 1, then extinction occurs with probability one in the stochastic and deterministic models. However, if λ > 1, then the deterministic model has exponential growth, but in the stochastic model there is a positive probability of extinction which depends on the fixed point of the system of probability generating functions. The linear, density-independent, stochastic model is generalized to a nonlinear, density-dependent one. The dependence on state is in terms of a weighted total population size. It is shown for small initial population sizes that the density-dependent, stochastic model can be approximated by the density-independent, stochastic model and thus, the extinction behavior exhibited by the linear model occurs in the nonlinear model. In the deterministic models there is a unique stable equilibrium. Given the population does not go extinct, it is shown that the stochastic model has a quasi-stationary distribution with mean close to the stable equilibrium, provided the population size is sufficiently large. For small values of the population size, complete extinction can be observed in the simulations. However, the persistence time increases rapidly with the population size. This author received partial support by the National Science Foundation grant # DMS-9626417.     

Evidence for different nonlinear summation schemes for lines and gratings at threshold

Within a wide class of multichannel models of the visual system it is suggested that spatial distributions of luminance are processed by the independent activation of grating detectors, or spatial frequency channels. Probability summation is often described in terms of Quick's nonlinear pooling model [Quick RF (1974) Kybernetik 16:65–67]. Using this model, we find evidence for the existence of different kinds of nonlinear summation at threshold; for compound gratings with well-separated spatial frequency components, the threshold functions indicate nonlinear summation which is not compatible with probability summation, while for line patterns well separated in the spatial domain the probability summation rule proves compatible with the data. Received: 24 June 1998 / Accepted in revised form: 16 March 1999     

A new asymptotic analysis of the nth order reaction-diffusion problem: analytical and numerical studies

We present a new formulation of the steady state, isothermal, nonlinear reaction-diffusion problem involving nth order reaction kinetics for slab geometry. This results in tractable expressions for the effectiveness factor as a function of the Thiele modulus, the Thiele modulus as a function of the centerline concentration, and the concentration profiles in the slab. The expressions are valid asymptotically in the limit of large orders n. We compare these results with the exact numerical solutions obtained by transforming the nonlinear differential equation into an integral form, using Green's function methods, and solving by successive approximations. The formulation for a membrane is also given, and the nature of the asymmetrical solution discussed. The analysis is facilitated through the introduction of pseudo-reaction orders. A comparison of the asymptotic Thiele modulus obtained herein with a previously given expression shows the present theory to be an improvement.