Scientific Interests

Science supervisor: Prof. Dr. Alexander Loskutov

Current Position

         I  have been working in the Scientific Group "Non-linear Dynamics and Chaos" at the Physics Faculty of the Moscow State University (MSU) since 1999. At the moment I am a PhD-student of the MSU, and will complete my PhD in April 2006. My research topic is application of the theory of dynamical systems to models of excitable media.

I  have received an INTAS PhD fellowship (March 2004 - March 2006), Ref.Nr. 03-55-1920, to investigate filament dynamics in computational models of re-entry and fibrillation in the heart. This project is carried out in collaboration with Dr. Richard Clayton from the Department of Computer Science, University of Sheffield , and involves two visits, each of 4 months, to Sheffield.  

          Excitable media are spatially distributed systems, which have the ability to propagate signals without damping. For instance, a forest fire travels as a wave from its initiation point, and regenerates with every tree it ignites. One of the remarkable examples of an excitable medium is heart tissue. Under some conditions it can be considered as a system, formed of discrete elements, locally interacting with each other.

At first we proposed a model of the cardiac tissue as a conductive system with two interacting pacemakers (sources of excitation) and a refractory time, which plays an important role in normal cardiac functioning. This model can be used to describe certain types of cardiac arrhythmias, caused by disturbances in excitation initiation: AV-blocks and parasystoles. The interaction of the spontaneously oscillating nonlinear sources was analyzed on the base of the circle map derived for bidirectional influence of the pacemakers. In the parametric space of the model the phase locking areas were investigated in detail. We observed splitting of the resonance tongues and the superposition of the synchronization areas. The obtained results suggest that the model can be a useful tool for investigating the dynamical interaction of the cardiac nodes: can be applied to describe their entrainment and synchronization, and in the more general sence the results make possible to predict the behaviour of excitable systems with two pacemakers, depending on the type and intensity of their interaction and the initial phase. The model occurs to be a universal in the sense that its predictions are not sensitive to the specific form of interactions, i.e. on the phase response curve (PRC), which determines a change in phase after the action of stimulus.

Our study clearly indicates that this PRC based model can be applied to understand the response to an external stimulus of variable intensity and duration, as was previously observed in experimental investigations. Using the above results and the fact that excitation generated by the discharge of one cell (the action potential) induces a subthreshold depolarization in the adjacent cell, we have extrapolated our approach to study the bidirectional interaction among an arbitrary large ensemble of the pacemaker cells. Investigations on the base of the unified model lead to the development of the theory of oscillatory media with a set of interacting pacemakers, coupled by their PRC. This can be of a great practical importance due to possible application in controlling cardiac rhythms by external stimuli.

In the second approach the cardiac tissue is considered as a spatio-extended system, where action potential propagation is described by nonlinear differential equations. As is known, one of the more intriguing properties of excitable media driven by reaction-diffusion equations is the ability to support vortices. Therefore, this approach is highly useful for description of re-entrant cardiac arrhythmias, the most life-threatening of which is the ventricular fibrillation (VF).

During re-entry electrical activity propagates repeatedly along a closed path, forming a spiral wave of activation in two dimensions (2D) and a scroll wave in three dimensions (3D). A characteristic feature of a spiral wave is presence of a wavebreak at the core of the spiral (spiral wave tip). A spiral wave tip is sometimes referred as a point of phase singularity (PS), as at this point the phase of the wave is undefined. 3D scroll waves are characterised by lines of singularities called filaments. When more than one spiral wave or scroll wave is present, the pattern of spatio-temporal activity can be very complex. Using filaments enables us to understand this turbulent pattern of activity.

We have compared all existing methods for PSs detection, since although choosing an appropriate approach to identify the location of PSs is extremely important, investigators use different algorithms, and little is known about the distinction between these methods. We have found that number and location of PSs detected using different algorithms differ markedly on each time step, however some methods locate PSs earlier than others and if time scale is shifted properly (2-3 ms), then discrepancies between algorithms are caused by transient PSs.

Neither method is ideal since some of them require some parameter values to be chosen, while for other techniques transmembrane potential has to be processed before detecting PSs. However, there is clear advantage of using techniques that are based on topological charge. They immediately give us knowledge about the sence of spiral wave rotation (chirality), and this is an additional feature that is extremely useful for PS tracking.

We have observed dynamics of phase singularities using some of the techniques for PS detection on the base of just one framework three-variable Fenton-Karma model with different parameters sets to reproduce various mechanisms of spiral wave breakup which in principle can occur in cardiac tissue. A difference between constructed PS trajectories and fluctuation of their number has shown that a type of a breakup mechanism has a strong influence on the dynamics of singularities. After comparison of the results for several breakups, we have got some preliminary ideas concerning which parameters are responsible for the balance of PS creation and destruction due to positive-negative collision or collision with boundaries and how the number of singularities of both chiralities can decrease which leads to VF termination.