A mathematical model for the dynamics of clustering

The formation of several clusters, arising from attracting forces between nonidentical entities or agents, is a phenomenon observed in diverse fields. Think of people gathered through a mutual interest, swarm behaviour of animals or clustering of oscillators in brain cells.We introduce a dynamic model of mutually attracting agents for which we prove that the long-term behaviour consists of agents organized into several groups or clusters. We have completely characterized the cluster structure (i.e. the number of clusters and their composition) by means of a set of inequalities in the parameters of the model and have identified the intensity of the attraction as a key parameter governing the transition between different cluster structures.The versatility of the model will be illustrated by discussing its relation to the Kuramoto model and by describing how it applies to a system of interconnected water basins.     

A mathematical model for the dynamics and synchronization of cows

We formulate a mathematical model for the daily activities of a cow (eating, lying down, and standing) in terms of a piecewise linear dynamical system. We analyze the properties of this bovine dynamical system representing the single animal and develop an exact integrative form as a discrete-time mapping. We then couple multiple cow “oscillators” together to study synchrony and cooperation in cattle herds. We comment on the relevant biology and discuss extensions of our model. With this abstract approach, we not only investigate equations with interesting dynamics but also develop biological predictions. In particular, our model illustrates that it is possible for cows to synchronize less when the coupling is increased.     

Nonlinear oscillator model reproducing various phenomena in the dynamics of the conduction system of the heart

A dedicated nonlinear oscillator model able to reproduce the pulse shape, refractory time, and phase sensitivity of the action potential of a natural pacemaker of the heart is developed. The phase space of the oscillator contains a stable node, a hyperbolic saddle, and an unstable focus. The model reproduces several phenomena well known in cardiology, such as certain properties of the sinus rhythm and heart block. In particular, the model reproduces the decrease of heart rate variability with an increase in sympathetic activity. A sinus pause occurs in the model due to a single, well-timed, external pulse just as it occurs in the heart, for example due to a single supraventricular ectopy. Several ways by which the oscillations cease in the system are obtained (models of the asystole). The model simulates properly the way vagal activity modulates the heart rate and reproduces the vagal paradox. Two such oscillators, coupled unidirectionally and asymmetrically, allow us to reproduce the properties of heart rate variability obtained from patients with different kinds of heart block including sino-atrial blocks of different degree and a complete AV block (third degree). Finally, we demonstrate the possibility of introducing into the model a spatial dimension that creates exciting possibilities of simulating in the future the SA the AV nodes and the atrium including their true anatomical structure.     

A macro-mechanical model of the guinea pig cochlea with realistic parameters

The post-mortem transfer function of the cochlea of the guinea pig was compared to the transfer function generated by a model with parameters derived from physical measurements of the guinea pig cochlea. Both the formulation and parameters of the model were carefully chosen to be realistic using evidence from published measurements. The fit between the transfer function of the model and recent mechanical measurements of the passive guinea pig cochlear response was good, with a root mean square ratio of 6.3 dB in amplitude and 0.33 pi rad in phase. The model was used to explore the effect of cochlear partition mode factor and duct geometry upon the mechanical response of the cochlea. Possible inadequacies of the model which could explain the remaining differences between the output of the model and measurements are discussed.     

Astrocyte signaling in the presence of spatial inhomogeneities

Astrocytes, a special type of glial cells, were considered to have just a supporting role in information processing in the brain. However, several recent studies have shown that they can be chemically stimulated by various neurotransmitters, such as ATP, and can generate Ca2+ and ATP waves, which can propagate over many cell lengths before being blocked. Although pathological conditions, such as spreading depression and epilepsy, have been linked to abnormal wave propagation in astrocytic cellular networks, a quantitative understanding of the underlying characteristics is still lacking. Astrocytic cellular networks are inhomogeneous, in the sense that the domain they occupy contains passive regions or gaps, which are unable to support wave propagation. Thus, this work focuses on understanding the complex interplay between single-cell signal transduction, domain inhomogeneity, and the characteristics of wave propagation and blocking in astrocytic cellular networks. The single-cell signal transduction model that was employed accounts for ATP-mediated IP3 production, the subsequent Ca2+ release from the ER, and ATP release into the extracellular space. The model is excitable and thus an infinite range of wave propagation is observed if the domain of propagation is homogeneous. This is not always the case for inhomogeneous domains. To model wave propagation in inhomogeneous astrocytic networks, a reaction-diffusion framework was developed and one-gap as well as multiple-gap cases were simulated using an efficient finite-element algorithm. The minimum gap length that blocks the wave was computed as a function of excitability levels and geometric characteristics of the inhomogeneous network, such as the length of the active regions (cells). Complex transient patterns, such as wave reflection, wave trapping, and generation of echo waves, were also predicted by the model, and their relationship to the geometric characteristics of the network was evaluated. Therefore, the proposed model can help in the formulation of testable hypotheses to explain the observed abnormal wave propagation in pathological situations.     

Strong-coupling electrostatics in the presence of dielectric inhomogeneities

We study the strong-coupling (SC) interaction between two like-charged membranes of finite thickness embedded in a medium of higher dielectric constant. A generalized SC theory is applied along with extensive Monte Carlo simulations to study the image charge effects induced by multiple dielectric discontinuities in this system. These effects lead to strong counterion crowding in the central region of the intersurface space upon increasing the solvent-membrane dielectric mismatch and change the membrane interactions from attractive to repulsive at small separations. These features agree quantitatively with the SC theory at elevated couplings or dielectric mismatch where the correlation hole around counterions is larger than the thickness of the central counterion layer.     

A mathematical model for wave propagation in elastic tubes with inhomogeneities: Application to blood waves propagation

We study the propagation of waves in an elastic tube filled with an inviscid fluid. We consider the case of inhomogeneity whose mechanical and geometrical properties vary in space. We deduce a system of equations of the Boussinesq type as describing the wave propagation in the tube. Numerical simulations of these equations show that inhomogeneities prevent separation of right-going from left-going waves.Then reflected and transmitted coefficients are obtained in the case of localized constriction and localized rigidity. Next we focus on wavetrains incident on various types of anomalous regions. We show that the existence of anomalous regions modifies the wavetrain patterns.     

Electron-level dynamics in the presence of impurities and the Ruijsenaars-Schneider model

The dynamic equations for the energy level of a finite system with impurities are shown to be equivalent to the rational Ruijsenaars-Schneider system. The action, which is simultaneously the generating function of the Bäcklund canonical transformation for this system, is calculated. Various variants of statistical averaging of the energy-level distribution are discussed.     

Membrane stress tensor in the presence of lipid density and composition inhomogeneities

We derive the expression of the stress tensor for one- and two-component lipid membranes with density and composition inhomogeneities. We first express the membrane stress tensor as a function of the free-energy density by means of the principle of virtual work. We then apply this general result to a monolayer model which is shown to be a local version of the area-difference elasticity (ADE) model. The resulting stress tensor expression generalizes the one associated with the Helfrich model, and can be specialized to obtain the one associated with the ADE model. Our stress tensor directly gives the force exchanged through a boundary in a monolayer with density and composition inhomogeneities. Besides, it yields the force density, which is also directly obtained in covariant formalism. We apply our results to study the forces induced in a membrane by a local perturbation.     

Characterization and control of chaotic dynamics in a nerve conduction model equation

In this paper we consider the Bonhoeffer-van der Pol (BVP) equation which describes propagation of nerve pulses in a neural membrane, and characterize the chaotic attractor at various bifurcations, and the probability distribution associated with weak and strong chaos. We illustrate control of chaos in the BVP equation by the Ott-Grebogi-Yorke method as well as through a periodic instantaneous burst.     

Scroll wave dynamics in a model of the heterogeneous heart

Scroll waves are found in physical, chemical and biological systems and underlie many significant processes including life-threatening cardiac arrhythmias. The theory of scroll waves predicts scroll wave dynamics should be substantially affected by heterogeneity of cardiac tissue together with other factors including shape and anisotropy. In this study, we used our recently developed analytical model of the human ventricle to identify effects of shape, anisotropy, and regional heterogeneity of myocardium on scroll wave dynamics. We found that the main effects of apical-base heterogeneity were an increased scroll wave drift velocity and a shift towards the region of maximum action potential duration. We also found that transmural heterogeneity does not substantially affect scroll wave dynamics and only in extreme cases changes the attractor position.     

Simulation of the nonlinear dynamics of magnetic inhomogeneities in real magnets

The dynamics of domain walls (DW) in an infinite ferromagnet containing a flat layer with parameters of magnetic anisotropy and exchange interaction differing from the bulk values was investigated. The dependences of the minimum velocity of DW transmission through the defect region and the translational and pulsation modes of DW fluctuations on the parameters describing the inhomogeneity of magnetic anisotropy and exchange interaction were found.     

Improving MRI's slice selectivity in the presence of strong,metal-derived inhomogeneities

PurposeTo develop schemes that deliver faithful 2D slices near field heterogeneities of the kind arising from non-ferromagnetic metal implants, with reduced artifacts and shorter scan times.MethodsAn excitation scheme relying on cross-term spatio-temporal encoding (xSPEN) was used as basis for developing the new inhomogeneity-insensitive, slice-selective pulse scheme. The resulting Fully refOCUSED cross-term SPatiotemporal ENcoding (FOCUSED-xSPEN) approach involved four adiabatic sweeps. The method was evaluated in silico, in vitro and in vivo using mice models, and compared against a number of existing and of novel alternatives based on both conventional and swept RF pulses, including an analogous method based on LASER's selectivity spatial selectivity.ResultsCalculations and experiments confirmed that multi-sweep derivatives of xSPEN and LASER can deliver localized excitation profiles, centered at the intended positions and endowed with enhanced immunity to B₀ and B₁ distortions. This, however, is achieved at the expense of higher SAR than non-swept counterparts. Furthermore, single-shot FOCUSED-xSPEN and LASER profiles covered limited off-resonance ranges. This could be extended to bands covering arbitrary off-resonance values with uniform slice widths, by looping the experiments over a number of scans possessing suitable transmission and reception offsets.ConclusionsA series of novel approaches were introduced to select slices near metals, delivering robustness against B_o and B₁⁺ field inhomogeneities.     

Positron annihilation in ionic media and an optical model of the positron

It is proposed that positron motion in quasiatomic positron + anion systems formed in anionic media can be described by a potential of the form V_eff(r) = Z_eff/r²-/r, where Z_eff is the effective charge of the nucleus, and n is the effective charge of the anion. It is shown that the positron wave function of the ground state of the quasiatomic positron + anion system in the field of such a potential is _X(r) = l/4·A_nx·r^X·e^–ar. Thus the validity of selecting a test variation positron wave function (r) = l/4·A·r·e^–ar is demonstrated for the potential V_eff = at r = 0 and V_eff = –/r for r > 0 (Gol'danskii-Prokop'ev optical positron model, Fiz. Tverd. Tela,8, 515 (1966)), belonging to the class of functions _X(r). Having the wave function _X(r) and Slater wave functions _ns,p(r) of the electrons, annihilation photon angular distribution (APAD) curves are calculated, together with halfwidths of the APAD curves and positron lifetimes _ns,p.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 52–56, May, 1990.     

Angular scattering pattern of waves in the ocean in the presence of anisotropic inhomogeneities

Closed integrodifferential equations are derived for the coherence function and the ray intensity in the ocean. Random inhomogeneities of the medium are assumed to be large-scale and statistically anisotropic. The influence of anisotropy on the angular scattering pattern is analyzed.     

A mathematical model of hotspot condensed phase ignition in the presence of a competitive endothermic reaction

We consider the propagation of a combustion front resulting from the gasless combustion of a condensed state fuel. The propagation of the front, essentially a premixed laminar flame, is supported by an exothermic reaction subject to possible heat loss through a competitive endothermic reaction. The dynamics of the endothermic process inducing the heat loss strongly depend on the temperature and the local fuel concentration. Through an analysis based on high activation energy, the steady-state values of the final burnt temperature as well as the burning velocity are obtained, and the control parameters are identified. Using a linear perturbation method, we assess the stability of the propagating front and obtain a condition for oscillatory behaviour. The critical parameter values for the transition from steady to oscillatory burning speeds are identified. The results represent a generalization of those obtained by Matkowsky and Sivashinsky to include the effects of heat loss induced by a competitive endothermic reaction.