Volume 31, Issue 5

We simulated the electric current distribution in the brain during transcranial direct current stimulation (tDCS) using an anatomically accurate human head model. We estimated an effect of common electrode montages on spatial distribution of the electric field during tDCS procedure and analyzed a sensitivity of the technique to variations of electrode size and orientation. We concluded that the used electrode montages are stable with respect to minor changes in electrode size and position, while an assumption of homogeneity and isotropy of the head model results in crucial changes of the electric field distribution. We determined the electrode montages suited to deliver strong effect on hippocampus and cerebellum.

In this paper we consider properties of the difference equations (discrete mappings) obtained in the study of the population dynamics of lemmings. A bifurcation scenario is proposed for obtained equations. Certain stability zones appear under this scenario with periods varying in order of natural series and also zones with more complicated modes. The study of transitional zones (‘ordering of the chaos’) is performed with the use of analytic calculations and computational experiments. Numerical analysis of mappings uses the methods of approximation of implicitly specified sets allowing us to construct and visualize sets of ‘resonance’ parameters including the front of the so-called singularity of ‘blue sky’.

In this paper we present a computational algorithm aimed at fitting a SEIR populational model to the influenza outbreaks incidence in Russian cities. The input data are derived from the long-term records on the incidence of acute respiratory diseases in Moscow, St. Petersburg, and Novosibirsk. It is shown that the classical SEIR model could provide a satisfactory fit for the majority of employed influenza outbreak incidence data sets ( R 2 > 0.91 for the 64 curves out of 67). Nevertheless, the model fitting algorithm in its current implementation has a number of drawbacks, which are discussed in the paper along with the ways of overcoming them.

Equations of a continuous-discrete mathematical model describing the propagation of HIV infection among the population of several regions are presented. The model equations take into account the reproduction and migration of the population, the risk of infection of individuals from different social groups, an impulse change in the number of individuals at discrete time moments under the action of various factors. The results of the study of the model solutions are also presented. We obtain conditions for the model parameters and initial data that provide the existence of solutions interpreted as full eradication of HIV infection in all considered regions or maintenance of sizes of groups of infected individuals at some acceptable level. The solutions analysis uses the monotone operators method and properties of nonsingular M-matrices.

The dynamics of scroll waves in a symmetric isotropic model of the human cardiac left ventricle is considered. The position of the attractor and the wave rotation velocity over the attractor were determined depending on the wall thickness, parameters of the cell model, chirality of the wave, and the initial position. Mechanisms of observed phenomena are discussed.

This paper presents first results on the dynamics of filaments of scroll waves of myocardium excitation obtained for personalized models of the left ventricle of the human heart. The paper describes a mathematical model of the left ventricle of the human heart and its electrical activity, numerical methods for the model calculation within the framework of computer implementation, and also the method of personalization of the model according to data of ultrasound examination. We found that regardless of the starting point of wave the filaments of wave drift along a spiral to the attractor located near the apex of the ventricle. The attractor position was essentially different in models constructed from the data of patients without identified pathology and those for patients with an increased left ventricular cavity.

Correct diagnostics of vascular pathologies underlies treatment success for patients with cerebrovascular diseases. Transcranial ultrasound is the well-known method for diagnostic of cerebrovascular diseases. Despite high sensitivity and specificity of the method, transcranial ultrasound has some limitations related to the B-mode image quality and accurate insonation of vessels of interest. Overcoming these limitations enables to enhance the quality of the diagnostic procedure. The present work addresses the numerical simulation of ultrasound propagation in a human head by a grid-characteristic method. We used a human tissue-mimicking phantom to verify our numerical model in terms of the accuracy of distance estimation. We obtained pressure distributions within a 3D segmented model of a human head. Our pilot study has some limitations, nevertheless the simulation results demonstrate that mathematical modelling of the transcranial ultrasound can be an effective tool to enhance the ultrasound examination.