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Computer Research and Modeling, 2020, Volume 12, Issue 4, Pages 845–864
DOI: https://doi.org/10.20537/2076-7633-2020-12-4-845-864
(Mi crm821)
 

ANALYSIS AND MODELING OF COMPLEX LIVING SYSTEMS

Application of simplified implicit Euler method for electrophysiological models

A. A. Karpaeva, R. R. Alievab

a Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, 141700, Russia
b Institute of Theoretical and Experimental Biophysics RAS, 3 Institutskaya st., Puschino, Moscow Region, 142290, Russia
References:
Abstract: A simplified implicit Euler method was analyzed as an alternative to the explicit Euler method, which is a commonly used method in numerical modeling in electrophysiology. The majority of electrophysiological models are quite stiff, since the dynamics they describe includes a wide spectrum of time scales: a fast depolarization, that lasts milliseconds, precedes a considerably slow repolarization, with both being the fractions of the action potential observed in excitable cells. In this work we estimate stiffness by a formula that does not require calculation of eigenvalues of the Jacobian matrix of the studied ODEs. The efficiency of the numerical methods was compared on the case of typical representatives of detailed and conceptual type models of excitable cells: Hodgkin–Huxley model of a neuron and Aliev–Panfilov model of a cardiomyocyte. The comparison of the efficiency of the numerical methods was carried out via norms that were widely used in biomedical applications. The stiffness ratio's impact on the speedup of simplified implicit method was studied: a real gain in speed was obtained for the Hodgkin–Huxley model. The benefits of the usage of simple and high-order methods for electrophysiological models are discussed along with the discussion of one method's stability issues. The reasons for using simplified instead of high-order methods during practical simulations were discussed in the corresponding section. We calculated higher order derivatives of the solutions of Hodgkin–Huxley model with various stiffness ratios; their maximum absolute values appeared to be quite large. A numerical method's approximation constant's formula contains the latter and hence ruins the effect of the other term (a small factor which depends on the order of approximation). This leads to the large value of global error. We committed a qualitative stability analysis of the explicit Euler method and were able to estimate the model's parameters influence on the border of the region of absolute stability. The latter is used when setting the value of the timestep for simulations a priori.
Keywords: electrophysiology, ionic models, conceptual models, stiff systems, numerical methods.
Funding agency Grant number
National Research University Higher School of Economics RAEP 5-100
The work was partially supported by programme RAEP 5-100 at Human Physiology Lab at Moscow Institute of Physics and Technology (state university).
Received: 06.01.2020
Revised: 18.04.2020
Accepted: 01.06.2020
Document Type: Article
UDC: 519.622.2
Language: Russian
Citation: A. A. Karpaev, R. R. Aliev, “Application of simplified implicit Euler method for electrophysiological models”, Computer Research and Modeling, 12:4 (2020), 845–864
Citation in format AMSBIB
\Bibitem{KarAli20}
\by A.~A.~Karpaev, R.~R.~Aliev
\paper Application of simplified implicit Euler method for electrophysiological models
\jour Computer Research and Modeling
\yr 2020
\vol 12
\issue 4
\pages 845--864
\mathnet{http://mi.mathnet.ru/crm821}
\crossref{https://doi.org/10.20537/2076-7633-2020-12-4-845-864}
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