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.      

Gain-induced oscillations in blood pressure

 “Mayer waves” are long-period (6 to 12 seconds) oscillations in arterial blood pressure, which have been observed and studied for more than 100 years in the cardiovascular system of humans and other mammals. A mathematical model of the human cardiovascular system is presented, incorporating parameters relevant to the onset of Mayer waves. The model is analyzed using methods of Liapunov stability and Hopf bifurcation theory. The analysis shows that increase in the gain of the baroreflex feedback loop controlling venous volume may lead to the onset of oscillations, while changes in the other parameters considered do not affect stability of the equilibrium state. The results agree with clinical observations of Mayer waves in human subjects, both in the period of the oscillations and in the observed age-dependence of Mayer waves. This leads to a proposed explanation of their occurrence, namely that Mayer waves are a gain-induced oscillation. Received: 15 September 1997/Revised version: 15 March 1998     

A bidomain threshold model of propagating calcium waves

We present a bidomain fire-diffuse-fire model that facilitates mathematical analysis of propagating waves of elevated intracellular calcium (Ca²⁺) in living cells. Modeling Ca²⁺ release as a threshold process allows the explicit construction of traveling wave solutions to probe the dependence of Ca²⁺ wave speed on physiologically important parameters such as the threshold for Ca²⁺ release from the endoplasmic reticulum (ER) to the cytosol, the rate of Ca²⁺ resequestration from the cytosol to the ER, and the total [Ca²⁺] (cytosolic plus ER). Interestingly, linear stability analysis of the bidomain fire-diffuse-fire model predicts the onset of dynamic wave instabilities leading to the emergence of Ca²⁺ waves that propagate in a back-and-forth manner. Numerical simulations are used to confirm the presence of these so-called ‘tango waves’ and the dependence of Ca²⁺ wave speed on the total [Ca²⁺].      

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.     

Neural control of heart rate: the role of neuronal networking

Neural control of heart rate, particularly its sympathetic component, is generally thought to reside primarily in the central nervous system, though accumulating evidence suggests that intrathoracic extracardiac and intrinsic cardiac ganglia are also involved. We propose an integrated model in which the control of heart rate is achieved via three neuronal “levels” representing three control centers instead of the conventional one. Most importantly, in this model control is effected through networking between neuronal populations within and among these layers. The results obtained indicate that networking serves to process demands for systemic blood flow before transducing them to cardiac motor neurons. This provides the heart with a measure of protection against the possibility of “overdrive” implied by the currently held centrally driven system. The results also show that localized networking instabilities can lead to sporadic low frequency oscillations that have the characteristics of the well-known Mayer waves. The sporadic nature of Mayer waves has been unexplained so far and is of particular interest in clinical diagnosis.     

A condition for setting off ectopic waves in computational models of excitable cells

The purpose of this paper is to study the stability of steady state solutions of the Monodomain model equipped with Luo-Rudy I kinetics. It is well established that re-entrant arrhythmias can be created in computational models of excitable cells. Such arrhythmias can be initiated by applying an external stimulus that interacts with a partially refractory region, and spawn breaking waves that can eventually generate extremely complex wave patterns commonly referred to as fibrillation. An ectopic wave is one possible stimulus that may initiate fibrillation. Physiologically, it is well known that ectopic waves exist, but the mechanism for initiating ectopic waves in a large collection of cells is poorly understood. In the present paper we consider computational models of collections of excitable cells in one and two spatial dimensions. The cells are modeled by Luo-Rudy I kinetics, and we assume that the spatial dynamics is governed by the Monodomain model. The mathematical analysis is carried out for a reduced model that is known to provide good approximations of the initial phase of solutions of the Luo-Rudy I model. A further simplification is also introduced to motivate and explain the results for the more complicated models. In the analysis the cells are divided into two regions; one region (N) consists of normal cells as model by the standard Luo-Rudy I model, and another region (A) where the cells are automatic in the sense that they would act as pacemaker cells if they where isolated from their surroundings. We let delta denote the spatial diffusion and a denote a characteristic length of the automatic region. It has previously been shown that reducing diffusion or increasing the automatic region enhances ectopic activity. Here we derive a condition for the transition from stable resting state to ectopic wave spread. Under suitable assumptions on the model we provide mathematical and computational arguments indicating that there is a constant eta such that a steady state solution of this system is stable whenever delta approximately > etaa(2), and unstable whenever delta approximately < etaa(2).     

Mathematical analysis of a basic model for epidermal wound healing

The stimuli for the increase in epidermal mitosis during wound healing are not fully known. We construct a mathematical model which suggests that biochemical regulation of mitosis is fundamental to the process, and that a single chemical with a simple regulatory effect can account for the healing of circular epidermal wounds. The numerical results of the model compare well with experimental data. We investigate the model analytically by making biologically relevant approximations. We then obtain travelling wave solutions which provide information about the accuracy of these approximations and clarify the roles of the various model parameters.     

Three-dimensional waves in excitable reaction-diffusion systems

The eikonal equation [5] for excitable media is generalised to three dimensional systems. The main result of the investigation is the demonstration of the existence of toroidal and twisted toroidal scroll waves in the limit of large values of the major radius of the torus. The existence of a helical wave near the z-axis follows from the eidonal equation but its connection with the twisted toroidal scroll remains to be demonstrated. The eikonal equation also predicts a non-uniform rate of rotation of the cross-sectional spiral wave near the toroidal axis. The notion of geometrical stability is introduced for the case of an expanding sphere; in particular it is shown that a discussion of stability of solutions of the eikonal equation must take into account the possible shift in the origin of the coordinate systems with respect to which patterns are defined. On leave of absence from: The Department of Mathematics, Glasgow College of Technology, Cowcaddens Road, Glasgow G4 0BA, UK     

From periodic travelling waves to travelling fronts in the spike-diffuse-spike model of dendritic waves

In the vertebrate brain excitatory synaptic contacts typically occur on the tiny evaginations of neuron dendritic surface known as dendritic spines. There is clear evidence that spine heads are endowed with voltage-dependent excitable channels and that action potentials invade spines. Computational models are being increasingly used to gain insight into the functional significance of a spine with an excitable membrane. The spike-diffuse-spike (SDS) model is one such model that admits to a relatively straightforward mathematical analysis. In this paper we demonstrate that not only can the SDS model support solitary travelling pulses, already observed numerically in more detailed biophysical models, but that it has periodic travelling wave solutions. The exact mathematical treatment of periodic travelling waves in the SDS model is used, within a kinematic framework, to predict the existence of connections between two periodic spike trains of different interspike interval. The associated wave front in the sequence of interspike intervals travels with a constant velocity without degradation of shape, and might therefore be used for the robust encoding of information.     

Mathematical analysis of delay differential equation models of HIV-1 infection

Models of HIV-1 infection that include intracellular delays are more accurate representations of the biology and change the estimated values of kinetic parameters when compared to models without delays. We develop and analyze a set of models that include intracellular delays, combination antiretroviral therapy, and the dynamics of both infected and uninfected T cells. We show that when the drug efficacy is less than perfect the estimated value of the loss rate of productively infected T cells, delta, is increased when data is fit with delay models compared to the values estimated with a non-delay model. We provide a mathematical justification for this increased value of delta. We also provide some general results on the stability of non-linear delay differential equation infection models.     

Calcium waves

Waves through living systems are best characterized by their speeds at 20 degrees C. These speeds vary from those of calcium action potentials to those of ultraslow ones which move at 1-10 and/or 10-20 nm s(-1). All such waves are known or inferred to be calcium waves. The two classes of calcium waves which include ones with important morphogenetic effects are slow waves that move at 0.2-2 microm s(-1) and ultraslow ones. Both may be propagated by cycles in which the entry of calcium through the plasma membrane induces subsurface contraction. This contraction opens nearby stretch-sensitive calcium channels. Calcium entry through these channels propagates the calcium wave. Many slow waves are seen as waves of indentation. Some are considered to act via cellular peristalsis; for example, those which seem to drive the germ plasm to the vegetal pole of the Xenopus egg. Other good examples of morphogenetic slow waves are ones through fertilizing maize eggs, through developing barnacle eggs and through axolotl embryos during neural induction. Good examples of ultraslow morphogenetic waves are ones during inversion in developing Volvox embryos and across developing Drosophila eye discs. Morphogenetic waves may be best pursued by imaging their calcium with aequorins.     

Traveling waves in a chemotactic model

A model for chemotaxis in a bacteria-substrate mixture introduced by Keller and Segel, which is described by nonlinear partial differential equations, is studied analytically. The existence of traveling waves is shown for the system in which the substrate diffusion is taken into account and the chemotactic coefficient is greater than the motility one, and the instability of traveling waves is discussed.     

Gene-culture waves of advance

Major genetic and cultural changes may have been coupled during hominid evolution. Since hominids have had a wide geographical distribution for about one million years, any mutant gene or cultural innovation that became established had to spread from its origin. A pair of nonlinear diffusion equations is derived which models the propagation of a mutant gene and a cultural innovation. Both are assumed to originate in the same locality along a linear habitat. The mutant gene and its allele are semidominant, and the two cultural choices are transmitted according to what I call the logistic attraction-repulsion model. The genes influence cultural choice, and the two interact to determine fitness. Of particular interest is the case in which mutant gene and cultural innovation are mutually dependent, neither being able to spread without the other. Each equation of the pair is similar in form to Fisher's equation, with a linear function of the other dependent variable replacing the constant coefficient in the reaction term. The partial differential equations are solved numerically to obtain the asymptotic speeds. Their form also suggests an heuristic argument which has proved useful, but I have been unable to obtain any analytic results. The waves of the system are shown to be of two types, synchronous and asynchronous. When genes and culture are mutually dependent, synchronous travelling waves can exist. However, their existence is dependent on initial conditions, and the speed of propagation is slow.     

Solitonlike and nonsoliton modes of interaction of taxis waves (illustrated with an example of bacterial population waves)

It was shown earlier that during collisions bacterial population waves may either penetrate one another or stop. In this communication, the mechanism of these two interaction modes is considered in detail. It is shown on the basis of theoretical and experimental results that this interaction is a graphic example confirming one of the characteristic properties of waves in cross-diffusion systems.     

Discrete-time travelling waves: Ecological examples

Integrodifference equations are discrete-time models that possess many of the attributes of continuous-time reaction-diffusion equations. They arise naturally in population biology as models for organisms with discrete nonoverlapping generations and well-defined growth and dispersal stages. I examined the varied travelling waves that arise in some simple ecologically-interesting integrodifference equations. For a scalar equation with compensatory growth, I observed only simple travelling waves. For carefully chosen redistribution kernels, one may derive the speed and approximate the shape of the observed waveforms. A model with overcompensation exhibited flip bifurcations and travelling cycles in addition to simple travelling waves. Finally, a simple predator-prey system possessed periodic wave trains and a variety of travelling waves.     

Small amplitude periodic waves for the FitzHugh-Nagumo equations

Periodic waves of small amplitude are shown to exist for a simplified version of the FitzHugh-Nagumo equations and their stability is analyzed.UDDM Report DE 81:2     

The hypercycle,traveling waves,and Wright's equation

A formal relation between the hypercycle equation and the delay differential equation of E. M. Wright is exhibited using a traveling waves approach. Several unsolved questions in either problem can be related and interpreted, in particular new motivation for the study of Wright's equation is obtained.     

Formation of demarcation zones when bacterial population waves are drawn together

Many motile bacteria (for instance, Escherichia coli) inoculated at some point in a semisolid nutrient medium can form population waves: bands or rings. The formation of these motile structures is due to chemotaxis. The population waves when they are drawn together can form two types of non-motile structures. Firstly, the population waves can collide. Secondly, in certain conditions, the waves can slow down and stop without coming into contact directly with each other. In this way demarcation zones are formed. The mechanism of the occurrence of the demarcation zones has been unknown. In this paper we show that formation of these zones is due to lack of nutrients (which at the same time act as attractants) within the narrow gap between individual bacterial populations.     

Causal transfer function analysis to describe closed loop interactions between cardiovascular and cardiorespiratory variability signals

Although the concept of transfer function is intrinsically related to an input–output relationship, the traditional and widely used estimation method merges both feedback and feedforward interactions between the two analyzed signals. This limitation may endanger the reliability of transfer function analysis in biological systems characterized by closed loop interactions. In this study, a method for estimating the transfer function between closed loop interacting signals was proposed and validated in the field of cardiovascular and cardiorespiratory variability. The two analyzed signals x and y were described by a bivariate autoregressive model, and the causal transfer function from x to y was estimated after imposing causality by setting to zero the model coefficients representative of the reverse effects from y to x. The method was tested in simulations reproducing linear open and closed loop interactions, showing a better adherence of the causal transfer function to the theoretical curves with respect to the traditional approach in presence of non-negligible reverse effects. It was then applied in ten healthy young subjects to characterize the transfer functions from respiration to heart period (RR interval) and to systolic arterial pressure (SAP), and from SAP to RR interval. In the first two cases, the causal and non-causal transfer function estimates were comparable, indicating that respiration, acting as exogenous signal, sets an open loop relationship upon SAP and RR interval. On the contrary, causal and traditional transfer functions from SAP to RR were significantly different, suggesting the presence of a considerable influence on the opposite causal direction. Thus, the proposed causal approach seems to be appropriate for the estimation of parameters, like the gain and the phase lag from SAP to RR interval, which have a large clinical and physiological relevance.