arXiv subjects
Publications and source records attributed to G. Karev.
A class of one-dimensional Fokker-Plank equations having a common stationary solution, which is a power function of the state of the process, was found. We prove that these equations also have generalized self-similar solutions which describe the temporary transition from one stationary state to another. The study was motivated by problems arising in mathematical modeling of genome size evolution.
The Fitzhugh-Nagumo equations have been used as a caricature of the Hodgkin-Huxley equations of neuron firing to better understand the essential dynamics of the interaction of the membrane potential and the restoring force and to capture, qualitatively, the general properties of an excitable membrane. Even though its simplicity allows very valuable insight to be gained, the accuracy of reproducing real experimental results is limited. In this paper, we utilize a modified version of the Fitzhugh-Nagumo equations to model the spatial propagation of neuron firing; we assume that this propagation is (at least, partially) caused by the cross-diffusion connection between the potential and recovery variables. We show that the cross-diffusion version of the model, besides giving rise to the typical fast traveling wave solution exhibited in the original diffusion Fitzhugh-Nagumo equations, also gives rise to a slow traveling wave solution. We analyze all possible traveling wave solutions of the Fitzhugh-Nagumo equations with this cross-diffusion term and show that there exists a threshold of the cross-diffusion coefficient (the maximum value for a given speed of propagation), which bounds the area where normal impulse propagation is possible.