  In this work we analyze the resort to high order exponential solvers for
stiff ODEs in the context of cardiac electrophysiology modeling. The
exponential Adams-Bashforth and the Rush-Larsen schemes will be considered up
to order 4. These methods are explicit multistep schemes.The accuracy and the
cost of these methods are numerically analyzed in this paper and benchmarked
with several classical explicit and implicit schemes at various orders. This
analysis has been led considering data of high particular interest in cardiac
electrophysiology : the activation time ($t\_a$ ), the recovery time ($t\_r $)
and the action potential duration ($APD$). The Beeler Reuter ionic model,
especially designed for cardiac ventricular cells, has been used for this
study. It is shown that, in spite of the stiffness of the considered model,
exponential solvers allow computation at large time steps, as large as for
implicit methods. Moreover, in terms of cost for a given accuracy, a
significant gain is achieved with exponential solvers. We conclude that
accurate computations at large time step are possible with explicit high order
methods. This is a quite important feature when considering stiff non linear
ODEs.
