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 model for slow axonal transport and its application to neurofilamentous neuropathies

A model for slow axonal transport is developed in which the essential features are reversible binding of cytoskeletal elements and of soluble cytosolic proteins to each other and to motile elements such as actin microfilaments. Computer simulation of the equations of the model demonstrate that the model can account for many of the features of the SCa and SCb waves observed in pulse experiments. The model also provides a unified explanation for the increase and decrease of neurofilament transport rates observed in various toxicant-induced neuropathies.     

Mechanisms of Seizure Propagation in 2-Dimensional Centre-Surround Recurrent Networks

Understanding how seizures spread throughout the brain is an important problem in the treatment of epilepsy, especially for implantable devices that aim to avert focal seizures before they spread to, and overwhelm, the rest of the brain. This paper presents an analysis of the speed of propagation in a computational model of seizure-like activity in a 2-dimensional recurrent network of integrate-and-fire neurons containing both excitatory and inhibitory populations and having a difference of Gaussians connectivity structure, an approximation to that observed in cerebral cortex. In the same computational model network, alternative mechanisms are explored in order to simulate the range of seizure-like activity propagation speeds (0.1–100 mm/s) observed in two animal-slice-based models of epilepsy: (1) low extracellular , which creates excess excitation and (2) introduction of gamma-aminobutyric acid (GABA) antagonists, which reduce inhibition. Moreover, two alternative connection topologies are considered: excitation broader than inhibition, and inhibition broader than excitation. It was found that the empirically observed range of propagation velocities can be obtained for both connection topologies. For the case of the GABA antagonist model simulation, consistent with other studies, it was found that there is an effective threshold in the degree of inhibition below which waves begin to propagate. For the case of the low extracellular model simulation, it was found that activity-dependent reductions in inhibition provide a potential explanation for the emergence of slowly propagating waves. This was simulated as a depression of inhibitory synapses, but it may also be achieved by other mechanisms. This work provides a localised network understanding of the propagation of seizures in 2-dimensional centre-surround networks that can be tested empirically.     

The effects of the size and shape of landscape features on the formation of traveling waves in cyclic populations

Recent field data indicate that in a number of cyclic populations, the cycles are organized spatially with the form of a periodic traveling wave. One way in which this type of wave is generated is when dispersing individuals encounter landscape features that impede movement in certain directions. In this article, we investigate the dependence of such periodic waves on ecological parameters and on the form of the landscape feature. Using a standard predator-prey model as a prototype for a cyclic population, we calculate the speed and amplitude of waves generated by a large landscape feature. This enables us to determine parameters for which the waves are stable; in other cases, they evolve into irregular oscillations. We then undertake for the first time a detailed study of the effects of the size and shape of a landscape feature on the waves that it generates. We show that size rather than shape is the key wave-forming property, with smaller obstacles generating waves with longer wavelength and waves from larger landscape features dominating those from smaller ones. Our results suggest that periodic traveling waves may be much more common than has previously been assumed in real ecological systems, and they enable quantitative predictions on the properties of these waves for particular cases.     

Transition from an excitable to an oscillatory state in dictyostelium discoideum

Under conditions of starvation, populations of the amoebae Dictyostelium discoideum aggregate are mediated by chemical excitation waves of cAMP. Two types of waves can be observed, either spiral or circular-shaped ones. We investigate transitions from rotating spirals to circular shaped waves (target patterns). Two different experiments demonstrating this phenomenon are presented. In the first case a continuous transition from the spiral type pattern to target waves was observed at the later stages of aggregation. In the second case the transition was induced by annihilation of waves by a spatially homogeneous cAMP pulse. Instead of the originally present spiral waves, oscillating spots bearing target patterns emerged. On the basis of a model for Dictyostelium aggregation, we provide a theoretical explanation for such transitions. It is shown that cell density can be an effective bifurcation parameter. Under certain conditions, the system is shifted from the excitable to the oscillatory state while the frequency of oscillations is proportional to the square root of the cell density. Thus, the regions with the highest cell density during the early stages of the spatial rearrangement of the cells become pacemakers and produce target patterns. The analytic results were confirmed in numerical simulations of the model.     

Mathematical approach to the stimulation of the competence development in Bacillus subtilis.

The stimulation of competence development by culture fluids of B. subtilis has been studied at very short intervals. In these conditions, successive competence waves as well as changes in the level of competence stimulating activity are observed. A mathematical model for the explanation of the phenomenon is developed. Two methods have been followed to study the model: construction of a simulation by using a computer system and analytic study of the equations.     

Generation of periodic waves by landscape features in cyclic predator-prey systems

The vast majority of models for spatial dynamics of natural populations assume a homogeneous physical environment. However, in practice, dispersing organisms may encounter landscape features that significantly inhibit their movement. We use mathematical modelling to investigate the effect of such landscape features on cyclic predator-prey populations. We show that when appropriate boundary conditions are applied at the edge of the obstacle, a pattern of periodic travelling waves develops, moving out and away from the obstacle. Depending on the assumptions of the model, these waves can take the form of roughly circular 'target patterns' or spirals. This is, to our knowledge, a new mechanism for periodic-wave generation in ecological systems and our results suggest that it may apply quite generally not only to cyclic predator-prey interactions, but also to populations that oscillate for other reasons. In particular, we suggest that it may provide an explanation for the observed pattern of travelling waves in the densities of field voles (Microtus agrestis) in Kielder Forest (Scotland-England border) and of red grouse (Lagopus lagopus scoticus) on Kerloch Moor (northeast Scotland), which in both cases move orthogonally to any large-scale obstacles to movement. Moreover, given that such obstacles to movement are the rule rather than the exception in real-world environments, our results suggest that complex spatio-temporal patterns such as periodic travelling waves are likely to be much more common in the natural world than has previously been assumed.     

Determination of the average shape of flagellar bends: a gradient curvature model

Data obtained by manual digitization of photographs of flagellar bending waves have been analyzed by determining size parameters for the bends by least-squares fitting of a model waveform. These parameters were then used to normalize the data so that the average shape of the bends could be determined. Best fits were obtained with a model waveform derived from the constant curvature waveforms used previously but with provision for a linear change in curvature across the central region of the bend-the gradient curvature model (GCM). The central regions of the GCM bending waves are separated by transition regions with length determined by a parameter called the truncation factor (FT). Fitting the GCM to sine-generated bending waves give optimal fit when FT = 0.34. Fitting the GCM to four different samples of flagellar bending waves gave best fits with values of FT ranging from 0.17 for ATP-reactivated Lytechinus spermatozoa beating at approximately 10 Hz to 0.32 for live spermatozoa of Arbacia. The difference between the Arbacia waveforms and a sine-generated waveform is therefore very small, but a sine-generated waveform lacks the degree of freedom represented by FT that is required to fit other waveforms optimally. The residual differences between the waveform data and optimal GCM waveforms were averaged and found to be small. In most cases, the curvature in the central region of the optimal GCM decreased in magnitude towards the tip of the flagellum; however, this slope was highly variable and sometimes positive. Significant variations in both this slope and FT were found in individual bends as they propagated along a flagellum.     

Lamellipodial contractions during crawling and spreading

Most eukaryotic cells can crawl over surfaces. In general, this motility requires three distinct actions: polymerization at the leading edge, adhesion to the substrate, and retraction at the rear. Recent experiments with mouse embryonic fibroblasts showed that during spreading and crawling the lamellipodium undergoes periodic contractions that are substrate-dependent. Here I show that a simple model incorporating stick-slip adhesion and a simplified mechanism for the generation of contractile forces is sufficient to explain periodic lamellipodial contractions. This model also explains why treatment of cells with latrunculin modifies the period of these contractions. In addition, by coupling a diffusing chemical species that can bind actin, such as myosin light-chain kinase, with the contractile model leads to periodic rows and waves in the chemical species, similar to what is observed in experiments. This model provides a novel and simple explanation for the generation of contractile waves during cell spreading and crawling that is only dependent on stick-slip adhesion and the generation of contractile force and suggests new experiments to test this mechanism.     

The logic of Ménage à Trois.

Recent studies of socially monogamous species have shown that in many cases females do not copulate exclusively with their pair mates, but are also receptive to other males. The explanation usually given for unfaithful female behavior is that most females are unable to bond with a male they would prefer as genetic father to their offspring. To secure male assistance the female therefore pairs with an available male but also copulates with males of supposedly higher genetic quality. Here we offer an alternative evolutionary explanation for female infidelity, which does not rely upon this ''Good Genes hypothesis of female choice. We show with a simple model that in an evolutionary game between three players, a male, a female and a male lover, solutions exist in which the female can secure more assistance from her mate by being receptive to other males. We conclude that female sexuality can have a decisive role in regulating social behaviour, in which the fertile female is the driving force.     

Dynamics of intramural scroll waves in three-dimensional continuous myocardium with rotational anisotropy.

It has been suggested that reentrant activity in three-dimensional cardiac muscle may be organized as a scroll wave rotating around a singularity line called the filament. Experimental studies indicate that filaments are often concealed inside the ventricular wall and consequently, scroll waves do not manifest reentrant activity on the surface. Here we analyse how such concealed scroll waves are affected by a twisted anisotropy resulting from rotation of layers of muscle fibers inside the ventricular wall. We used a computer model of a ventricular slab (15x15x15 mm(3)) with a fiber twist of 120 degrees from endocardium to epicardium. The action potential was simulated using FitzHugh-Nagumo equations. Scroll waves with rectilinear filaments were initiated at various depths of the slab and at different angles with respect to fiber orientation. The analysis shows that independent of initial conditions, after a certain transitional period, the filament aligns with the local fiber orientation. The alignment of the filament is determined by the directional variations in cell coupling due to fiber rotation and by boundary conditions. Our findings provide a mechanistic explanation for the prevalence of intramural reentry over transmural reentry during polymorphic ventricular tachycardia and fibrillation.     

A proton-led model of fast calcium waves

Fast (10-30 microm/s) calcium waves can be propagated through all nucleated eukaryotic cells that have been tested as well as certain cell-free extracts. In a widely used model, they are propagated by a reaction-diffusion cycle in which calcium ions diffuse along the outside of endoplasmic reticula and induce their own release from calsequestrin or calreticulin molecules stored within the reticulum's lumen. Here we propose a new tandem wave model in which they are also propagated by a reaction-diffusion cycle within a reticulum's lumen. In this cycle, increases in luminal [H(+)] induce proton release from luminal calsequestrin or calreticulin. The released protons diffuse ahead to where they release more protons from these luminal storage proteins. What might be called proton induced proton release. They also raise luminal electropositivity. The resultant luminal waves are coordinated with extrareticular ones by movements of calcium and hydrogen ions through the reticular membrane. This model makes five testable predictions which include the autorelease of protons in solutions of calsequestrins or calreticulins as well as waves of increased [H(+)], of increased [Ca(2+)] and of more positive voltage within the reticula of whole cells. Moreover, under some conditions, such luminal waves should cross regions without cytosolic ones.     

One-dimensional model for propagation of a pressure wave in a model of the human arterial network: comparison of theoretical and experimental results

Pulse wave evaluation is an effective method for arteriosclerosis screening. In a previous study, we verified that pulse waveforms change markedly due to arterial stiffness. However, a pulse wave consists of two components, the incident wave and multireflected waves. Clarification of the complicated propagation of these waves is necessary to gain an understanding of the nature of pulse waves in vivo. In this study, we built a one-dimensional theoretical model of a pressure wave propagating in a flexible tube. To evaluate the applicability of the model, we compared theoretical estimations with measured data obtained from basic tube models and a simple arterial model. We constructed different viscoelastic tube set-ups: two straight tubes; one tube connected to two tubes of different elasticity; a single bifurcation tube; and a simple arterial network with four bifurcations. Soft polyurethane tubes were used and the configuration was based on a realistic human arterial network. The tensile modulus of the material was similar to the elasticity of arteries. A pulsatile flow with ejection time 0.3 s was applied using a controlled pump. Inner pressure waves and flow velocity were then measured using a pressure sensor and an ultrasonic diagnostic system. We formulated a 1D model derived from the Navier-Stokes equations and a continuity equation to characterize pressure propagation in flexible tubes. The theoretical model includes nonlinearity and attenuation terms due to the tube wall, and flow viscosity derived from a steady Hagen-Poiseuille profile. Under the same configuration as for experiments, the governing equations were computed using the MacCormack scheme. The theoretical pressure waves for each case showed a good fit to the experimental waves. The square sum of residuals (difference between theoretical and experimental wave-forms) for each case was <10.0%. A possible explanation for the increase in the square sum of residuals is the approximation error for flow viscosity. However, the comparatively small values prove the validity of the approach and indicate the usefulness of the model for understanding pressure propagation in the human arterial network.     

On the mysterious propulsion of Synechococcus

We propose a model for the self-propulsion of the marine bacterium Synechococcus utilizing a continuous looped helical track analogous to that found in Myxobacteria [1]. In our model cargo-carrying protein motors, driven by proton-motive force, move along a continuous looped helical track. The movement of the cargo creates surface distortions in the form of small amplitude traveling ridges along the S-layer above the helical track. The resulting fluid motion adjacent to the helical ribbon provides the propulsive thrust. A variation on the helical rotor model of [1] allows the motors to be anchored to the peptidoglycan layer, where they drive rotation of the track creating traveling helical waves along the S-layer. We derive expressions relating the swimming speed to the amplitude, wavelength, and velocity of the surface waves induced by the helical rotor, and show that they fall in reasonable ranges to explain the velocity and rotation rate of swimming Synechococcus.     

A Comparative Study of Early Afterdepolarization-Mediated Fibrillation in Two Mathematical Models for Human Ventricular Cells

Early afterdepolarizations (EADs), which are abnormal oscillations of the membrane potential at the plateau phase of an action potential, are implicated in the development of cardiac arrhythmias like Torsade de Pointes. We carry out extensive numerical simulations of the TP06 and ORd mathematical models for human ventricular cells with EADs. We investigate the different regimes in both these models, namely, the parameter regimes where they exhibit (1) a normal action potential (AP) with no EADs, (2) an AP with EADs, and (3) an AP with EADs that does not go back to the resting potential. We also study the dependence of EADs on the rate of at which we pace a cell, with the specific goal of elucidating EADs that are induced by slow or fast rate pacing. In our simulations in two- and three-dimensional domains, in the presence of EADs, we find the following wave types: (A) waves driven by the fast sodium current and the L-type calcium current (Na-Ca-mediated waves); (B) waves driven only by the L-type calcium current (Ca-mediated waves); (C) phase waves, which are pseudo-travelling waves. Furthermore, we compare the wave patterns of the various wave-types (Na-Ca-mediated, Ca-mediated, and phase waves) in both these models. We find that the two models produce qualitatively similar results in terms of exhibiting Na-Ca-mediated wave patterns that are more chaotic than those for the Ca-mediated and phase waves. However, there are quantitative differences in the wave patterns of each wave type. The Na-Ca-mediated waves in the ORd model show short-lived spirals but the TP06 model does not. The TP06 model supports more Ca-mediated spirals than those in the ORd model, and the TP06 model exhibits more phase-wave patterns than does the ORd model.     

Evolution of spiral and scroll waves of excitation in a mathematical model of ischaemic border zone

Abnormal electrical activity from the boundaries of ischemic cardiac tissue is recognized as one of the major causes in generation of ischemia-reperfusion arrhythmias. Here we present theoretical analysis of the waves of electrical activity that can rise on the boundary of cardiac cell network upon its recovery from ischaemia-like conditions. The main factors included in our analysis are macroscopic gradients of the cell-to-cell coupling and cell excitability and microscopic heterogeneity of individual cells. The interplay between these factors allows one to explain how spirals form, drift together with the moving boundary, get transiently pinned to local inhomogeneities, and finally penetrate into the bulk of the well-coupled tissue where they reach macroscopic scale. The asymptotic theory of the drift of spiral and scroll waves based on response functions provides explanation of the drifts involved in this mechanism, with the exception of effects due to the discreteness of cardiac tissue. In particular, this asymptotic theory allows an extrapolation of 2D events into 3D, which has shown that cells within the border zone can give rise to 3D analogues of spirals, the scroll waves. When and if such scroll waves escape into a better coupled tissue, they are likely to collapse due to the positive filament tension. However, our simulations have shown that such collapse of newly generated scrolls is not inevitable and that under certain conditions filament tension becomes negative, leading to scroll filaments to expand and multiply leading to a fibrillation-like state within small areas of cardiac tissue.     

Spatial pattern switching enables cyclic evolution in spatial epidemics

Infectious diseases often spread as spatial epidemic outbreak waves. A number of model studies have shown that such spatial pattern formation can have important consequences for the evolution of pathogens. Here, we show that such spatial patterns can cause cyclic evolutionary dynamics in selection for the length of the infectious period. The necessary reversal in the direction of selection is enabled by a qualitative change in the spatial pattern from epidemic waves to irregular local outbreaks. The spatial patterns are an emergent property of the epidemic system, and they are robust against changes in specific model assumptions. Our results indicate that emergent spatial patterns can act as a rich source for complexity in pathogen evolution.     

Interpreting the equatorial diffraction pattern of collagenous tissues in the light of molecular motion.

The equatorial diffraction pattern associated with collagenous tissues, particularly type I collagen, is diffuse and clearly unlike that from crystals. Hukins and Woodhead-Galloway proposed a statistical model that they termed a "liquid crystal" for collagen fibers in tendons. Fratzl et al. applied this model to both unmineralized and mineralized turkey leg tendon, a model that ignores the organization imposed by the well-known cross-linking. The justification for adopting this model is that the curve fits the data. It is shown that the data can be equally well matched by fitting a least-squares curve consisting of a second-order polynomial plus a Gaussian. The peak of the Gaussian is taken as the equatorial spacing of the collagen. A physical explanation for this model is given, as is a reason for the changes in the spacing with changes in water content of the tissue. The diffusion is attributed to thermally driven agitation of the molecules, in accordance with the Debye-Waller theory including the Gaussian distribution. The remainder of the diffusion is attributed to other scattering sources like the mineral crystallites.     

Nearly circular,oval and irregular holes in Cretaceous ammonoids from Nigeria

Nearly circular‐, oval‐ and irregular‐shaped holes are present in a collection of Late Cretaceous ammonoid cephalopods from southern Nigeria. Two competing hypotheses have been advanced to explain these holes: one is they were produced by diagenetic crushing of limpet home scars and the other is that they are predator's teeth marks. The latter explanation appears to be the best explanation for some of the damage seen in the Nigerian specimens. The suspected predator for some of the specimens was probably an unidentified reptile based on the diameters of the holes. Insofar as we are aware, this is the first recorded predatory damage reported on Cretaceous ammonoids from West‐Central Africa.