Seidel–Herzel model of human baroreflex in cardiorespiratory system with stochastic delays

The stochastic versus deterministic solution of the Seidel–Herzel model describing the baroreceptor control loop (which regulates the short-time heart rate) are compared with the aim of exploring the heart rate variability. The deterministic model solutions are known to bifurcate from the stable to sustained oscillatory solutions if time delays in transfer of signals by sympathetic nervous system to the heart and vasculature are changed. Oscillations in the heart rate and blood pressure are physiologically crucial since they are recognized as Mayer waves. We test the role of delays of the sympathetic stimulation in reconstruction of the known features of the heart rate. It appears that realistic histograms and return plots are attainable if sympathetic time delays are stochastically perturbed, namely, we consider a perturbation by a white noise. Moreover, in the case of stochastic model the bifurcation points vanish and Mayer oscillations in heart period and blood pressure are observed for whole considered space of sympathetic time delays.      

The effects of extra-low-frequency atmospheric pressure oscillations on human mental activity

 Slight atmospheric pressure oscillations (APO) in the extra-low-frequency range below 0.1 Hz, which frequently occur naturally, can influence human mental activity. This phenomenon has been observed in experiments with a group of 12 healthy volunteers exposed to experimentally created APO with amplitudes 30–50 Pa in the frequency band 0.011–0.17 Hz. Exposure of the subjects to APO for 15–30 min caused significant changes in attention and short-term memory functions, performance rate, and mental processing flexibility. The character of the response depended on the APO frequency and coherence. Periodic APO promoted purposeful mental activity, accompanied by an increase in breath-holding duration and a slower heart rate. On the other hand, quasi-chaotic APO, similar to the natural perturbations of atmospheric pressure, disrupted mental activity. These observations suggest that APO could be partly responsible for meteorosensitivity in humans. Received: 22 August 1997/Revised: 3 December 1998/Accepted: 25 February 1999     

Spatial instabilities and local oscillations in a lattice gas Lotka–Volterra model

 We analyze the evolution of spatially inhomogeneous perturbations in a lattice gas model for a prey-predator population. Starting with the master equation of the model, decoupled by means of a mean field approximation, spatial instabilities are seen to take place in a region of the phase diagram. This is in qualitative agreement with local oscillations already observed in computer simulations. We determine the transition line that separates the homogeneous region from the inhomogeneous region and we study the spatio-temporal self-organized structures that appear inside the inhomogeneous region. Received 3 November 1995; received in revised form 26 January 1996     

Spontaneous oscillations in two 2-component cells coupled by diffusion

Mathematical examples are presented of oscillators with two variables which do not oscillate in isolation, but which do oscillate stably when coupled with a twin via difiusion. Two examples are presented, the LefeverPrigogine Brusselator and a system used to model glycolytic oscillations. The mathematical method is not the usual bifurcation theory, but rather a type of singular perturbation theory combined with bifurcation theory. For both examples, it is shown that all stationary solutions are unstable for appropriate parameter settings. In the case of the Brusselator, it is further shown that there exist limit cycles; i.e. stable oscillations, in this parameter range. A numerical example is presented.Partially supported by NSF     

Transient oscillations induced by delayed growth response in the chemostat

In this paper, in order to try to account for the transient oscillations observed in chemostat experiments, we consider a model of single species growth in a chemostat that involves delayed growth response. The time delay models the lag involved in the nutrient conversion process. Both monotone response functions and nonmonotone response functions are considered. The nonmonotone response function models the inhibitory effects of growth response of certain nutrients when concentrations are too high. By applying local and global Hopf bifurcation theorems, we prove that the model has unstable periodic solutions that bifurcate from unstable nonnegative equilibria as the parameter measuring the delay passes through certain critical values and that these local periodic solutions can persist, even if the delay parameter moves far from the critical (local) bifurcation values.When there are two positive equilibria, then positive periodic solutions can exist. When there is a unique positive equilibrium, the model does not have positive periodic oscillations and the unique positive equilibrium is globally asymptotically stable. However, the model can have periodic solutions that change sign. Although these solutions are not biologically meaningful, provided the initial data starts close enough to the unstable manifold of one of these periodic solutions they may still help to account for the transient oscillations that have been frequently observed in chemostat experiments. Numerical simulations are provided to illustrate that the model has varying degrees of transient oscillatory behaviour that can be controlled by the choice of the initial data.Mathematics Subject Classification: 34D20, 34K20, 92D25Research was partially supported by NSERC of Canada.This work was partly done while this author was a postdoc at McMaster.     

Structured population on two patches: modeling dispersal and delay

We derive from the age-structured model a system of delay differential equations to describe the interaction of spatial dispersal (over two patches) and time delay (arising from the maturation period). Our model analysis shows that varying the immature death rate can alter the behavior of the homogeneous equilibria, leading to transient oscillations around an intermediate equilibrium and complicated dynamics (in the form of the coexistence of possibly stable synchronized periodic oscillations and unstable phase-locked oscillations) near the largest equilibrium.     

A delay recruitment model of the cardiovascular control system

We develop a nonlinear delay-differential equation for the human cardiovascular control system, and use it to explore blood pressure and heart rate variability under short-term baroreflex control. The model incorporates an intrinsically stable heart rate in the absence of nervous control, and allows us to compare the baroreflex influence on heart rate and peripheral resistance. Analytical simplifications of the model allow a general investigation of the rôles played by gain and delay, and the effects of ageing.     

Two SIS epidemiologic models with delays

 The SIS epidemiologic models have a delay corresponding to the infectious period, and disease-related deaths, so that the population size is variable. The population dynamics structures are either logistic or recruitment with natural deaths. Here the thresholds and equilibria are determined, and stabilities are examined. In a similar SIS model with exponential population dynamics, the delay destabilized the endemic equilibrium and led to periodic solutions. In the model with logistic dynamics, periodic solutions in the infectious fraction can occur as the population approaches extinction for a small set of parameter values. Received: 10 January 1997 / 18 November 1997     

Periodic orbits arising from Hopf bifurcations in a Volterra prey-predator model

 In this paper we derive a formula which enables the stability of periodic solutions to a Volterra integro-differential system to be determined. This system which has been studied by Cushing [1], models a predator-prey interaction with distributed delays. The results are obtained by using the algorithm developed by Kazarinoff, Wan and van den Driessche [2] based on the centre manifold formulas of Hassard and Wan [3]. We discuss an example of the formula for the case of weak kernels and show that under certain conditions stable periodic solutions arising from Hopf bifurcations at different critical values of the parameters can exist together. Received 30 December 1994; received in revised form 12 December 1995     

Oscillations in a refractory neural net

A functional differential equation that arises from the classic theory of neural networks is considered. As the length of the absolute refractory period is varied, there is, as shown here, a super-critical Hopf bifurcation. As the ratio of the refractory period to the time constant of the network increases, a novel relaxation oscillation occurs. Some approximations are made and the period of this oscillation is computed.     

Traveling wave solutions of a nonlinear reaction–advection equation

 We establish the existence of traveling wave solutions for a nonlinear partial differential equation that models a logistically growing population whose movement is governed by an advective process. Conditions are presented for which traveling wave solutions exist and for which they are stable to small semi-finite domain perturbations. The wave is of mathematical interest because its behavior is determined by a singular differential equation and those with small speed of propagation steepen into a shock-like solutions. Finally, we indicate that the smoothing presence of diffusion allows wave persistence when an advective slow moving wave may collapse. Received: 24 November 1997 / Revised version: 13 July 1998     

Analysis of an excitable dendritic spine with an activity-dependent stem conductance

 Dendritic spines are the major target for excitatory synaptic inputs in the vertebrate brain. They are tiny evaginations of the dendritic surface consisting of a bulbous head and a tenuous stem. Spines are considered to be an important locus for plastic changes underlying memory and learning processes. The findings that synaptic morphology may be activity-dependent and that spine head membrane may be endowed with voltage-dependent (excitable) channels is the motivation for this study. We first explore the dynamics, when an excitable, yet morphologically fixed spine receives a constant current input. Two parameter Andronov–Hopf bifurcation diagrams are constructed showing stability boundaries between oscillations and steady-states. We show how these boundaries can change as a function of both the spine stem conductance and the conductance load of the attached dendrite. Building on this reference case an idealized model for an activity-dependent spine is formulated and analyzed. Specifically we examine the possibility that the spine stem resistance, the tunable “synaptic weight” parameter identified by Rall and Rinzel, is activity-dependent. In the model the spine stem conductance depends (slowly) on the local electrical interactions between the spine head and the dendritic cable; parameter regimes are found for bursting, steady states, continuous spiking, and more complex oscillatory behavior. We find that conductance load of the dendrite strongly influences the burst pattern as well as other dynamics. When the spine head membrane potential exhibits relaxation oscillations a simple model is formulated that captures the dynamical features of the full model. Received: 10 January 1997/Revised version: 25 March 1997     

Uncertainty and predictability in population dynamics of a bitrophic ecological model: Mixed-mode oscillations,bistability and sensitivity to parameters

We consider a two-trophic ecological model comprising of two predators competing for their common prey. We cast the model into the framework of a singular perturbed system of equations in one fast variable (prey population density) and two slow variables (predator population densities), mimicking the common observation that the per-capita productivity rate decreases from bottom to top along the trophic levels in Nature. We assume that both predators exhibit Holling II functional response with one of the predators (territorial) having a density dependent mortality rate. Depending on the system parameters, the model exhibits small, intermediate and/or large fluctuations in the population densities. The large fluctuations correspond to periodic population outbreaks followed by collapses (commonly known as cycles of “boom and bust”). The small fluctuations arise due to a singular Hopf bifurcation in the system, and are ecologically more desirable. However, more interestingly, the system exhibits mixed-mode oscillations (which are concatenations of the large amplitude oscillations and the small amplitude oscillations) that indicate the adaptability of the species to prolong the time gap between successive cycles of boom and bust. Numerical simulations are carried out to demonstrate the extreme sensitivity of the system to initial conditions (chaos and bistability of limit cycles are observed) as well as to the system parameters (here we only show the sensitivity to the density dependent mortality rate of the territorial predator). This model throws light at the uncertainties in long term behaviors that are associated with a real ecological system. We show that even very small changes in the system parameters due to natural or human-induced causes can lead to a complete different ecological phenomenon, thus affecting the predictability of the density of the prey population. In this paper, we explain the mechanisms behind the irregular fluctuations in the population sizes in an attempt to understand the dynamics occurring in a natural population and also comment on the inherent uncertainties associated with the system.     

Bifurcations in haploid and diploid sequence space models

 Deterministic models of mutation and selection in the space of (binary) nucleotide-type sequences have been investigated for haploid populations during the past 25 years, and, recently, for diploid populations as well. These models, in particular their ‘error thresholds’, have mainly been analyzed by numerical methods and perturbation techniques. We consider them here by means of bifurcation theory, which improves our understanding of both equilibrium and dynamical properties. In a caricature obtained from the original model by neglecting back mutation to the favourable allele, the familiar error threshold of the haploid two-class model turns out to be a simple transcritical bifurcation, whereas its diploid counterpart exhibits an additional saddle node. This corresponds to a second error threshold. Three-class models with neutral spaces of unequal size introduce further features. Such are a global bifurcation in haploid populations, and simple examples of Hopf bifurcations (as predicted by Akin’s theorem) in the diploid case. Received 13 June 1995; received in revised form 26 July 1996     

A mathematical model for drug administration by using the phagocytosis of red blood cells

 A mathematical model for the delivery of drug directly to the macrophages by using the phagocytosis of senescent red blood cells is proposed. The model is based on the following assumption: At time t=0 a preassigned red blood cell population n(0, a)=φ(a), a>0, loaded by the drug, is injected in the blood circulation. Among the cells of that population only those with an age a≧ā (ā=120 days) will be phagocytosed by macrophages. Of course, the lifetime of the drug must be higher than ā. Within the red blood cells it cannot be metabolized, neither can it diffuse through their membranes. The emphasis of the paper is on the mathematical properties and on the formulation of the control problem. Received 15 December 1994; received in revised form 20 July 1995     

Equality of average and steady-state levels in some nonlinear models of biological oscillations

Nonlinear oscillatory systems, playing a major role in biology, do not exhibit harmonic oscillations. Therefore, one might assume that the average value of any of their oscillating variables is unequal to the steady-state value. For a number of mathematical models of calcium oscillations (e.g. the Somogyi–Stucki model and several models developed by Goldbeter and co-workers), the average value of the cytosolic calcium concentration (not, however, of the concentration in the intracellular store) does equal its value at the corresponding unstable steady state at the same parameter values. The average value for parameter values in the unstable region is even equal to the level at the stable steady state for other parameter values, which allow stability. This holds for all parameters except those involved in the net flux across the cell membrane. We compare these properties with a similar property of the Higgins–Selkov model of glycolytic oscillations and two-dimensional Lotka–Volterra equations. Here, we show that this equality property is critically dependent on the following conditions: There must exist a net flux across the model boundaries that is linearly dependent on the concentration variable for which the equality property holds plus an additive constant, while being independent of all others. A number of models satisfy these conditions or can be transformed such that they do so. We discuss our results in view of the question which advantages oscillations may have in biology. For example, the implications of the findings for the decoding of calcium oscillations are outlined. Moreover, we elucidate interrelations with metabolic control analysis. This paper is dedicated to the memory of the late Reinhart Heinrich, who was the academic teacher of S.S. and, to a great extent, also of M.M.     

Specifying the introgressed regions from H. argophyllus in cultivated sunflower (Helianthus annuus L.) to mark Phomopsis resistance genes

A method based upon targetting of intro-gressed markers in a Phomopsis-resistant line (R) of cultivated sunflower, issuing from a H. argophyllus cross was used to mark the Phomopsis resistance regions. Our study was based upon 203 families derived from a cross between an inbred line susceptible to Phomopsis (S1) and the introgressed resistant line (R). Families were checked for Phomopsis resistance level in a design with replicated plots and natural infection was re-inforced by pieces of contaminated stems. Thirty four primers were employed for RAPD analysis. Out of 102 polymorphic fragments between (S1) and H. argophyllus, seven were still present in (R) suggesting that they marked introgressions of H. argophyllus into (R). The plants were scored for the presence or absence of 19 fragments obtained from five primers, and the relationships between the presence/absence of fragments in plants and Phomopsis resistance/susceptiblity in the progenies was determined by using an analysis of variance. We found that at least two introgressed regions, as well as favourable factors from sunflower, contributed to the level of Phomopsis resistance in cultivated sunflower. Received: 28 June 1996 / Accepted: 5 July 1996     

Characteristics of genetic variation in the progenies of protoplast-derived plants of rice, Oryza sativa cv Nipponbare

Genetic variation in protoplast-derived rice (Oryza sativa L.) plants was characterized using first and second generation selfed progenies. A total of 133 regenerated plants were obtained from ten protoplasts of the japonica rice cultivar Nipponbare. Sixty two regenerated plants which set enough seeds for the subsequent field tests at the next generation and were derived from five protoplasts were selected, and their selfed seeds were used as the first selfed-seed progeny generation). Fifteen plants were selected from each of the 15 lines, and their selfed seeds were used for tests at the generation. Thirty seven lines (60%) segregated plants with detrimental mutant characters of yellow-green phenotype, dwarf stature, dense and short panicle, or low seed fertility. According to the segregation patterns in the lines having mutated plants among those originated from the same protoplasts, the stages of mutation induction were estimated. Additionally, five quantitative traits were changed in almost all and lines. Varied quantitative traits of heading date, number of spikelets per panicle, and seed fertility, were in a heterozygous state. However, culm and panicle lengths showed high uniformity, whereas reduced culm and panicle lengths were caused by mutational changes in polygenes and/or multiple genes. Received: 20 March 1996 / Accepted: 21 June 1996     

Modelling of the baroreflex-feedback mechanism with time-delay

 The cardiovascular system is considered. A direct modelling of the non-linear baroreflex-feedback mechanism, including time-delay, is developed based on physiological theory and empirical facts. The feedback model is then evaluated on an expanded, but simple, non-pulsatile Windkessel model of the cardiovascular system. The stability of the entire model is analyzed and the effect of the value of the time-delay is investigated and discussed. The time-delay may cause oscillations. A finite number of stability switches may occur dependent on the value of the time-delay. The location of these stability switches turns out to be sensitive to the value of the parameters in the model. We suggest a simple experiment to determine whether or not the time-delay is responsible for the 10 second Mayer waves. Data from an ergometer bicycle test is used for evaluation of the model. Received 1 June 1996; received in revised form 20 November 1996     

Periodic oscillations in leukopoiesis models with two delays

The term leukopoiesis describes processes leading to the production and regulation of white blood cells. It is based on stem cells differentiation and may exhibit abnormalities resulting in severe diseases, such as cyclical neutropenia and leukemias. We consider a nonlinear system of two equations, describing the evolution of a stem cell population and the resulting white blood cell population. Two delays appear in this model to describe the cell cycle duration of the stem cell population and the time required to produce white blood cells. We establish sufficient conditions for the asymptotic stability of the unique nontrivial positive steady state of the model by analysing roots of a second degree exponential polynomial characteristic equation with delay-dependent coefficients. We also prove the existence of a Hopf bifurcation which leads to periodic solutions. Numerical simulations of the model with parameter values reported in the literature demonstrate that periodic oscillations (with short and long periods) agree with observations of cyclical neutropenia in patients.