## Mathematical model order reduction in computational neuroscience

Research output: Other conference contribution › Paper, poster or abstract › Scientific

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**Mathematical model order reduction in computational neuroscience.** / Lehtimäki, Mikko; Paunonen, Lassi; Linne, Marja-Leena.

Research output: Other conference contribution › Paper, poster or abstract › Scientific

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*Mathematical model order reduction in computational neuroscience*. Paper presented at 3rd HBP Student Conference on Interdisciplinary Brain Research, Ghent, Belgium.

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TY - CONF

T1 - Mathematical model order reduction in computational neuroscience

AU - Lehtimäki, Mikko

AU - Paunonen, Lassi

AU - Linne, Marja-Leena

PY - 2019/2/6

Y1 - 2019/2/6

N2 - Multi-scale models in neuroscience typically integrate detailed biophysicalneurobiological phenomena from molecular level up to network and system levels. Such models are very challenging to simulate despite the availability ofmassively parallel computing systems. Model Order Reduction (MOR) is anestablished method in engineering sciences, such as control theory. MOR is used in improving computational efficiency of simulations of large-scale and complexnonlinear mathematical models. In this study the dimension of a nonlinearmathematical model of plasticity in the brain is reduced using mathematical MOR methods. Traditionally, models are simplified by eliminating variables, such asmolecular entities and ionic currents, from the system. Additionally,assumptions of the system behavior can be made, for example regarding thesteady state of the chemical reactions. However, the current trend inneuroscience is incorporating multiple physical scales of the brain insimulations. Comprehensive models with full system dynamics are needed in order to increase understanding of different mechanisms in one brain area. Thus the elimination approach is not suitable for the consequent analysis of neuralphenomena.The loss of information typically induced by eliminating variables of the systemcan be avoided by mathematical MOR methods that strive to approximate theentire system with a smaller number of dimensions compared to the originalsystem. Here, the effectiveness of MOR in approximating the behavior of allthe variables in the original system by simulating a model with a radicallyreduced dimension, is demonstrated.In the present work, mathematical MOR is applied in the context of an experimentally verified signaling pathway model of plasticity [kim2013]. This nonlinear chemical equation based model describes the biochemical calcium signaling steps required for plasticity and learning in the subcortical area of the brain. In addition to nonlinear characteristics, the model includes time-dependent terms which pose an additional challenge both computational efficiency and reduction wise. The MOR method employed in this study is Proper Orthogonal Decomposition with Discrete Empirical Interpolation Method (POD+DEIM), a subspace projectionmethod for reducing the dimensionality of nonlinear systems [chaturantabut2010]. By applying these methods, the simulation time of themodel is radically shortened. However, our preliminary studies showapproximation error if the model is simulated for a very long time. Thetolerated amount of approximation error depends on the final application of themodel. Based on these promising results, POD+DEIM is recommended fordimensionality reduction in computational neuroscience.In summary, the reduced order model consumes a considerably smaller amount ofcomputational resources than the original model, while maintaining a low rootmean square error between the variables in the original and reduced models.This was achieved by simulating the system dynamics in a lower dimensionalsubspace without losing any of the variables from the model. The results presented here are novel as mathematical MOR has not been studied in neuroscience without linearisation of the mathematical model and never in the context of the model presented here.

AB - Multi-scale models in neuroscience typically integrate detailed biophysicalneurobiological phenomena from molecular level up to network and system levels. Such models are very challenging to simulate despite the availability ofmassively parallel computing systems. Model Order Reduction (MOR) is anestablished method in engineering sciences, such as control theory. MOR is used in improving computational efficiency of simulations of large-scale and complexnonlinear mathematical models. In this study the dimension of a nonlinearmathematical model of plasticity in the brain is reduced using mathematical MOR methods. Traditionally, models are simplified by eliminating variables, such asmolecular entities and ionic currents, from the system. Additionally,assumptions of the system behavior can be made, for example regarding thesteady state of the chemical reactions. However, the current trend inneuroscience is incorporating multiple physical scales of the brain insimulations. Comprehensive models with full system dynamics are needed in order to increase understanding of different mechanisms in one brain area. Thus the elimination approach is not suitable for the consequent analysis of neuralphenomena.The loss of information typically induced by eliminating variables of the systemcan be avoided by mathematical MOR methods that strive to approximate theentire system with a smaller number of dimensions compared to the originalsystem. Here, the effectiveness of MOR in approximating the behavior of allthe variables in the original system by simulating a model with a radicallyreduced dimension, is demonstrated.In the present work, mathematical MOR is applied in the context of an experimentally verified signaling pathway model of plasticity [kim2013]. This nonlinear chemical equation based model describes the biochemical calcium signaling steps required for plasticity and learning in the subcortical area of the brain. In addition to nonlinear characteristics, the model includes time-dependent terms which pose an additional challenge both computational efficiency and reduction wise. The MOR method employed in this study is Proper Orthogonal Decomposition with Discrete Empirical Interpolation Method (POD+DEIM), a subspace projectionmethod for reducing the dimensionality of nonlinear systems [chaturantabut2010]. By applying these methods, the simulation time of themodel is radically shortened. However, our preliminary studies showapproximation error if the model is simulated for a very long time. Thetolerated amount of approximation error depends on the final application of themodel. Based on these promising results, POD+DEIM is recommended fordimensionality reduction in computational neuroscience.In summary, the reduced order model consumes a considerably smaller amount ofcomputational resources than the original model, while maintaining a low rootmean square error between the variables in the original and reduced models.This was achieved by simulating the system dynamics in a lower dimensionalsubspace without losing any of the variables from the model. The results presented here are novel as mathematical MOR has not been studied in neuroscience without linearisation of the mathematical model and never in the context of the model presented here.

KW - Computational Neuroscience

KW - Control theory

KW - Mathematics

M3 - Paper, poster or abstract

ER -