== Analysis of Numerical Methods for Markov Chain Models of Ionic Channels ==
Tomas Stary (University of Exeter), Vadim N. Biktashev (University of Exeter)
Stochastic transitions between states of a system can be described by
a class of models known as Markov chain (MC). Important applications
of the MC include models of ionic channels in cell membranes.
The master equation of a MC model is a system of coupled ordinary
differential equations (ODEs). Numerical solution of these ODEs
involves discretising time into short intervals called time steps and
iteratively finding the solution after each consecutive time step. The
longer the time steps, the less demanding is the computation. However,
the time step size in the simplest (explicit) solvers is limited by
numerical instabilities, and implicit solvers for nonlinear models are
complicated.
A difficulty associated with of many popular MC models of ionic
channels are their fast dynamics, causing numerical instabilities at
relatively small time steps. To address this issue we exploit specific
properties of MC models in an exponential integrator (IEEE Trans. BME
62(4): 1070-1076, 2015), generalising a well-known Rush-Larsen
technique. Assuming only small variation of the transitions rates
within one time step, we approximate the MC ODEs by a system with
constant coefficients which is solved exactly.
In this presentation, we evaluate the accuracy and efficiency of
exponential integration method on two MC models, namely L-type calcium
current $I_{\mathrm{Ca}(L)}$ and calcium release $I_\mathrm{rel}$.
The exponential integration of $I_{\mathrm{Ca}(L)}$ is
straightforward. In $I_\mathrm{rel}$ we have to deal with the
dependence of the transition rates on two dynamic variables. Using
asymptotic properties of the $I_\mathrm{rel}$ MC, we divide the system
into fast and slow subsystems and obtain solution using operator
splitting: the fast subsystem uses exponential integrators; the slow
subsystem allows large time steps in explicit solvers.
The exponential integration allows up to 30-fold increase in the time
step size while maintaining numerical stability and accuracy of the
solution.