Systems with a partial differential equation (PDE) coupled to ordinary differential equations (ODEs) often appear in applications where the ODEs model microscopic chemical reactions, and the PDE models a macroscopic behaviour. Such applications include simulation of combustion, pollution and semiconductors. In this talk, we consider a PDE–ODE system for the electrical activity of the heart consisting of a semilinear parabolic PDE coupled to a set of nonlinear ODEs. Motivated by the fact that the dynamics of the ODEs may be much faster than that of the PDE, we present a multirate finite element method for the discretization. To control the error, a posteriori error estimates are derived using the method of dual-weighted residuals, giving error indicators useful for constructing adaptive algorithms. The indicators distinguish the errors in time and space for the PDE and the ODEs separately and include errors due to the transfer of data between the PDE and the ODEs. The method is computationally expensive, why a novel adaptive algorithm using space-time blocks is used.