Approximate matrix exponential: a fast and accurate solver for Markov kinetic models of ion channels

A new method is presented for simulating Markov kinetic models of ion channels, which are biologically realistic but difficult to simulate. Computing the analytic solution is prohibitively expensive, necessitating the use of numerical methods. The new method creates a piecewise polynomial approximation of the propagator matrix, which is the analytic solution operator. The new method is an order of magnitude faster and more accurate than previous numerical methods, but it is limited to models with one or two input variables and its time complexity scales quadratically with the number of model states. The accompanying program, ame, applies the new method to the NEURON simulator. For comparison, the analytic solution was also added to NEURON. The new method achieves real-time performance on a state-of-the-art GPU for tens to hundreds of thousands of model instances. This method will allow large-scale neuronal simulations to use more detailed models of ion channels.