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

- Article in a journal -
 

Area
Biomedicine

Author(s)
Samuel Khuvis , Matthias K. Gobbert , Bradford E. Peercy

Published in
Mathematical Biosciences

Year
2015

Abstract
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.

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BibTeX
@ARTICLE{
         Khuvis2015Tst,
       author = "Samuel Khuvis and Matthias K. Gobbert and Bradford E. Peercy",
       title = "Time-stepping techniques to enable the simulation of bursting behavior in a
         physiologically realistic computational islet",
       journal = "Mathematical Biosciences",
       volume = "263",
       pages = "1--17",
       year = "2015",
       issn = "0025-5564",
       doi = "10.1016/j.mbs.2015.02.001",
       url = "http://www.sciencedirect.com/science/article/pii/S0025556415000334",
       keywords = "Computational islet, Beta cells, Stiff ordinary differential equations, Numerical
         differentiation formulas, Automatic differentiation",
       abstract = "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.",
       ad_area = "Biomedicine",
       ad_tools = "ADiMat"
}


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