A note on nonperiodic undamped oscillations,with special reference to brain waves

Nonperiodical sustained fluctuations of electrical potential are characteristic of the rhythms of cortical electric activities. A nonperiodical negatively or positively damped oscillation would be a general solution of any first order linear system of differential equations. Attention is called to a somewhat specialized type of a system of ordinary linear differential equations, which are used by Vito Volterra (Leçons Sur la Théorie Mathématique de la Lutte Pour la Vie, Paris, Gauthier-Villars, 1931) in his theory of interdevouring species. This system of linear equations leads to nonperiodicalundamped oscillations and admits several plausible interpretations in terms of possible biophysical reactions in the brain. A number of new problems is posed and discussed by this study.     

The Spectro-Temporal Receptive Field

The Spectro-Temporal Receptive Field (STRF) of an auditory neuron has been introduced experimentally on the base of the average spectrotemporal structure of the acoustic stimuli which precede the occurrence of action potentials (Aertsen et al., 1980, 1981). In the present paper the STRF is considered in the general framework of nonlinear system theory, especially in the form of the Volterra integral representation. The STRF is proposed to be formally identified with a linear functional of the second order Volterra kernel. The experimental determination of the STRF leads to a formulation in terms of the Wiener expansion where the kernels can be identified by evaluation of the system's input-output correlations. For a Gaussian stimulus ensemble and a nonlinear system with no even order contributions of order higher than two, it is shown that the second order cross correlation of stimulus and response, normalized with respect to the spectral contents of the stimulus ensemble, leads to the stimulus-invariant spectrotemporal receptive field. The investigation of stimulus-invariance of the STRF for more general nonlinear systems and for stimulus ensembles which can be generated by nonlinear transformations of Gaussian noise involve the evaluation of higher order stimulus-response correlation functions.     

On representation and approximation of nonlinear systems

Many methods for the analysis of nonlinear systems rely on a Volterra system-representation in terms of integral kernels. This paper considers two questions: 1) whether Volterra-like representations are possible for all smooth systems (i.e. analytic operators),-2) which classes of systems can be approximated by some interesting subclasses of the smooth systems, e.g. Volterra systems with separable kernels and sandwich systems (cascades of linear systems and nomemory nonlinearities). Whereas the answer to question 1 is positive, the answer to question 2 depends on the type of approximation, i.e. topology, that is used.     

Modeling the batch bacteriocin production system by lactic acid bacteria by using modified three-dimensional Lotka–Volterra equations

Different batch cultures of Lactococcus lactis CECT 539, a nisin-producing strain, were carried out in culture media prepared with whey and mussel processing wastes. From these cultures, a reasonable system of differential equations, similar to the three-dimensional Lotka–Volterra two predators-one prey model, was set up to describe, for the first time, the relationship between the absolute rates of growth, pH drop and nisin production.Thus, the nisin production system was described as a three-species (pH, biomass and nisin) ecosystem. In this case, both nisin and biomass production were considered as two pH-dependent species that compete for the nitrogen source. Excellent agreement (R² values ≥0.9885) resulted between model predictions and the experimental data, and significant values for all the model parameters were obtained. The developed model was demonstrated (R² values ≥0.9874) for five batch cultivations of the strains L. lactis CECT 539 in MRS broth and Lactobacillus sakei LB 706 (sakacin A producer), Pediococcus acidilactici LB42-923 (pediocin AcH producer), L. lactis ATCC 11454 (nisin producer) and Leuconostoc carnosum Lm1 (leuconocin Lcm1 producer) in TGE broth. These results suggest that the batch bacteriocin production system in these culture media can be successfully described by using the Lotka–Volterra approach.     

Adaptive dynamics for physiologically structured population models

We develop a systematic toolbox for analyzing the adaptive dynamics of multidimensional traits in physiologically structured population models with point equilibria (sensu Dieckmann et al. in Theor. Popul. Biol. 63:309–338, 2003). Firstly, we show how the canonical equation of adaptive dynamics (Dieckmann and Law in J. Math. Biol. 34:579–612, 1996), an approximation for the rate of evolutionary change in characters under directional selection, can be extended so as to apply to general physiologically structured population models with multiple birth states. Secondly, we show that the invasion fitness function (up to and including second order terms, in the distances of the trait vectors to the singularity) for a community of N coexisting types near an evolutionarily singular point has a rational form, which is model-independent in the following sense: the form depends on the strategies of the residents and the invader, and on the second order partial derivatives of the one-resident fitness function at the singular point. This normal form holds for Lotka–Volterra models as well as for physiologically structured population models with multiple birth states, in discrete as well as continuous time and can thus be considered universal for the evolutionary dynamics in the neighbourhood of singular points. Only in the case of one-dimensional trait spaces or when N = 1 can the normal form be reduced to a Taylor polynomial. Lastly we show, in the form of a stylized recipe, how these results can be combined into a systematic approach for the analysis of the (large) class of evolutionary models that satisfy the above restrictions.      

Calculation of the Volterra kernels of non-linear dynamic systems using an artificial neural network

The Volterra series is a well-known method of describing non-linear dynamic systems. A major limitation of this technique is the difficulty involved in the calculation of the kernels. More recently, artificial neural networks have been used to produce black box models of non-linear dynamic systems. In this paper we show how a certain class of artificial neural networks are equivalent to Volterra series and give the equation for the nth order Volterra kernel in terms of the internal parameters of the network. The technique is then illustrated using a specific non-linear system. The kernels obtained by the method described in the paper are compared with those obtained by a Toeplitz matrix inversion technique. Received: 4 June 1993/Accepted in revised form: 2 March 1994     

Fast Second Degree Total Variation Method for Image Compressive Sensing

This paper presents a computationally efficient algorithm for image compressive sensing reconstruction using a second degree total variation (HDTV2) regularization. Firstly, a preferably equivalent formulation of the HDTV2 functional is derived, which can be formulated as a weighted L ₁-L ₂ mixed norm of second degree image derivatives under the spectral decomposition framework. Secondly, using the equivalent formulation of HDTV2, we introduce an efficient forward-backward splitting (FBS) scheme to solve the HDTV2-based image reconstruction model. Furthermore, from the averaged non-expansive operator point of view, we make a detailed analysis on the convergence of the proposed FBS algorithm. Experiments on medical images demonstrate that the proposed method outperforms several fast algorithms of the TV and HDTV2 reconstruction models in terms of peak signal to noise ratio (PSNR), structural similarity index (SSIM) and convergence speed.     

Nonlinear mechanisms for gain adaptation in locust photoreceptors.

Intracellular membrane potential responses were recorded from locust photoreceptors under two stimulus conditions: pairs of flashes to dark-adapted receptors, and white-noise modulated light at a range of background intensities from 500 to 15,000 effective photons per second. Nonlinear analysis of the input-output relationships were performed by estimating the Volterra and Wiener kernels of the system. The Volterra kernels obtained from the double-flash experiments were similar to the Wiener kernels obtained from the white-noise experiments, except for a change of time scale. The structure of the second-order kernels obtained with either method gave evidence for a gain control mechanism acting at an early stage of the cascade. Both feedforward and feedback nonlinearities could account for the observed system behavior at any one background level. The differences in amplitude between the kernels obtained at different background levels could be accounted for by an adaptation process which further decreased the gain of the system, acting on a slower time scale, also at some early stage of the cascade.     

Parametric and non-parametric modeling of short-term synaptic plasticity. Part I: Computational study

Parametric and non-parametric modeling methods are combined to study the short-term plasticity (STP) of synapses in the central nervous system (CNS). The nonlinear dynamics of STP are modeled by means: (1) previously proposed parametric models based on mechanistic hypotheses and/or specific dynamical processes, and (2) non-parametric models (in the form of Volterra kernels) that transforms the presynaptic signals into postsynaptic signals. In order to synergistically use the two approaches, we estimate the Volterra kernels of the parametric models of STP for four types of synapses using synthetic broadband input–output data. Results show that the non-parametric models accurately and efficiently replicate the input–output transformations of the parametric models. Volterra kernels provide a general and quantitative representation of the STP.     

The Evolution of One- and Two-Locus Systems

Assuming age-independent fertilities and mortalities and random mating, continuous-time models for a monoecious population are investigated for weak selection. A single locus with multiple alleles and two alleles at each of two loci are considered. A slow-selection analysis of diallelic and multiallelic two-locus models with discrete nonoverlapping generations is also presented. The selective differences may be functions of genotypic frequencies, but their rate of change due to their explicit dependence on time (if any) must be at most of the second order in s, (i.e., O( s²)), where s is the intensity of natural selection. Then, after several generations have elapsed, in the continuous time models the time-derivative of the deviations from Hardy-Weinberg proportions is of O(s²), and in the two-locus models the rate of change of the linkage disequilibrium is of O(s²). It follows that, if the rate of change of the genotypic fitnesses is smaller than second order in s (i.e., o(s²)), then to O(s²) the rate of change of the mean fitness of the population is equal to the genic variance. For a fixed value of s, however, no matter how small, the genic variance may occasionally be smaller in absolute value than the (possibly negative) lower order terms in the change in fitness, and hence the mean fitness may decrease. This happens if the allelic frequencies are changing extremely slowly, and hence occurs often very close to equilibrium. Some new expressions are derived for the change in mean fitness. It is shown that, with an error of O( s), the genotypic frequencies evolve as if the population were in Hardy-Weinberg proportions and linkage equilibrium. Thus, at least for the deterministic behavior of one and two loci, deviations from random combination appear to have very little evolutionary significance.     

Plasma equilibrium in axisymmetric open divertor configurations

The equilibrium of a plasma with isotropic pressure in a periodic divertor configuration with a poloidal magnetic field is calculated. The issue of how the plasma equilibrium changes as the parameter β≡8πp/B ² increases is considered for a fairly representative class of pressure profiles p(ψ) (where ψ is the flux coordinate). It is shown that the plasma can be in equilibrium up to β values (in terms of the vacuum magnetic field at the divertor axis) on the order of unity.     

The Evolution of One- and Two-Locus Systems. II

Weak selection in a monoecious population is studied in two multiallelic panmictic models. In the first, a single locus is considered with continuous time and age-independent fertilities and mortalities. If the fertilities of the various matings and the genotypic mortalities may be expressed with an error at most of the second order in s (i.e., O(s ²)), where s is the intensity of selection, as sums of terms corresponding to the different genotypes and alleles, respectively, then after several generations the deviations from Hardy-Weinberg proportions are of O(s²). In the second model, two loci are treated with discrete nonoverlapping generations. It is shown that if the epistatic parameters are of O(s²), then after several generations the linkage disequilibria are reduced to O(s²). Assuming only weak selection, it is proved that in both models, after several generations, the total change in mean fitness is generally positive. It is likely that the exclusion of the initial period is usually unnecessary in natural populations. Exceptions are discussed.     

Starting structure dependence of NMR order parameters derived from MD simulations: implications for judging force-field quality

Comparing experimental generalized N-H S² order parameters to those calculated from molecular dynamics trajectories is increasingly used to judge force-field quality and completeness of sampling. Herein we demonstrate for the well-investigated system hen egg white lysozyme that different experimental starting structures can lead to significant differences in molecular-dynamics-derived S² parameters that can be even larger than S² parameter deviations due to different force fields. Caution should thus be taken in general when simulated S² parameters are compared to experimental data with the aim of judging force-field quality. We show that adequately sampling flexible regions (∼100 ns) and only calculating S² parameters averaged over short time windows proved necessary to obtain consistent results irrespective of the starting structure.     

Analysis of all possible combinations of four measurements determining true propagation in arteries

All possible combinations of four measurements of blood pressure, blood flow and vascular diameter are examined by transmission-line theory. It is found that only nine measurement combinations can give the attenuation coefficient γ, reflection coefficient R and characteristic impedance Z₀ simultaneously. At least one pressure measurement must be included. Their general expressions with arbitrary measuring locations are presented, together with some simplified forms which cover all the previous methods. A greater choice of method is thereby made available for use in practice. The results show that, regardless of the measurement locations, all combinations can be solved in the order γ first, R second and Z₀ last.     

Persistent distribution functions for the dispersion of structured populations

This paper primarily expounds upon the problem of persistent age-state distribution functions for the dispersion of structured populations. A general model is introduced, based on the following assumptions: 1) the state of an individual of age a is characterized by a set of random variables X₁, X₂,…, X_Q (weight, size, etc.) obeying a phenomenological master equation; 2) the birth function λ depends on the age a' of parents and on the state variables X₁,…, X_Q of the newborns; 3) the mortality function is composed of two additive terms—the first contribution depends only on age while the second contribution depends on the total population density; 4) the population diffuses to avoid crowding. These hypotheses define a nonlinear population model for which time- and space-persistent age-state distribution functions eventually may occur even if the total population density is time- and space-dependent. A biological interpretation of the main results is given in terms of the distribution function of the state vector at birth. In the last part of the paper a generalized model is presented, assuming that the behavior of an individual is described by a system of age-dependent master equations [29].     

Mixed Moment Estimation in Hermite Distribution and its Comparison with Other Estimation Procedures from Sample Size Considerations

In this paper (1) expressions (correct to n^?2 terms) for biases, variances, and covariances of the estimators a and b of Hermite distribution with probability generating function Exp[a(t–1) + b(t–1)] are obtained for two mixed moment estimates; (2) for the biases and variance-covariances, approximate regions of the parameter space (a>0, b>0) have been outlined where a sample of size 100 can be considered as “safe” in the sense that contribution of second order terms in them is 5% of that from the first order term; (3) comparison of the biases and variance-covariances of these two sets of estimators are made with those for the moment estimators, maximum likelihood estimates and the even point estimators for a sample of size 100 using the terms up to order n^?2; (4) the comparisons based on n^?2 terms in (3) have not only provided information on the estimation procedures included in the Hermite distribution, but also demonstrated the importance of higher order terms in the sampling properties of the various alternative techniques for the Hermite distribution.