Time-stepping techniques to enable the simulation of bursting behavior in a physiologically realistic computational islet

Physiologically realistic simulations of computational islets of beta cells require the long-time solution of several thousands of coupled ordinary differential equations (ODEs), resulting from the combination of several ODEs in each cell and realistic numbers of several hundreds of cells in an islet. For a reliable and accurate solution of complex nonlinear models up to the desired final times on the scale of several bursting periods, an appropriate ODE solver designed for stiff problems is eventually a necessity, since other solvers may not be able to handle the problem or are exceedingly inefficient. But stiff solvers are potentially significantly harder to use, since their algorithms require at least an approximation of the Jacobian matrix. For sophisticated models, systems of several complex ODEs in each cell, it is practically unworkable to differentiate these intricate nonlinear systems analytically and to manually program the resulting Jacobian matrix in computer code. This paper demonstrates that automatic differentiation can be used to obtain code for the Jacobian directly from code for the ODE system, which allows a full accounting for the sophisticated model equations. This technique is also feasible in source-code languages Fortran and C, and the conclusions apply to a wide range of systems of coupled, nonlinear reaction equations. However, when we combine an appropriately supplied Jacobian with slightly modified memory management in the ODE solver, simulations on the realistic scale of one thousand cells in the islet become possible that are several orders of magnitude faster than the original solver in the software Matlab, a language that is particularly user friendly for programming complicated model equations. We use the efficient simulator to analyze electrical bursting and show non-monotonic average burst period between fast and slow cells for increasing coupling strengths. We also find that interestingly, the arrangement of the connected fast and slow heterogeneous cells impacts the peak bursting period monotonically.