# Dimensionality reduction for mathematical models in neuroscience

##### Lehtimäki, Mikko (2016)

Lehtimäki, Mikko

2016

Biotekniikan koulutusohjelma

Luonnontieteiden tiedekunta - Faculty of Natural Sciences

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##### Hyväksymispäivämäärä

2016-08-17**Julkaisun pysyvä osoite on**

https://urn.fi/URN:NBN:fi:tty-201608034370

##### Tiivistelmä

Dimensionality reduction is a commonly used method in engineering sciences, such as control theory, for improving computational efficiency of simulations of complex nonlinear mathematical models. Additionally, it is a way of surfacing the most important factors that drive the dynamics of the system. In the field of neuroscience, there is a great demand to incorporate molecular and cellular level detail in large-scale models of the brain in order to produce phenomena such as learning and behavior. This cannot be achieved with the computing power available today, since the detailed models are unsuitable for large-scale network or system level simulations.

In this thesis, methods for mathematical model reduction are reviewed. In the field of systems biology, models are typically simplified by completely eliminating variables, such as molecules, from the system, and making assumptions of the system behavior, for example regarding the steady state of the chemical reactions. However, this approach is not meaningful in neuroscience since comprehensive models are needed in order to increase understanding of the target systems. This information loss problem is solved by mathematical reduction methods that strive to approximate the entire system with a smaller number of dimensions compared to the original system.

In this study, mathematical model reduction is applied in the context of an experimentally verified signaling pathway model of plasticity. The chosen biophysical model is one of the most comprehensive models out of those that are currently able to explain aspects of plasticity on the molecular level with chemical interactions and the law of mass action. The employed reduction method is Proper Orthogonal Decomposition with Discrete Empirical Interpolation Method (POD+DEIM), a subspace projection method for reducing the dimensionality of nonlinear systems. By applying these methods, the simulation time of the plasticity model was radically shortened although approximation errors are present if the model is reviewed on large time scales. It is up to the final application of the model whether some error or none at all is tolerated. Based on these promising results, subspace projection methods are recommended for dimensionality reduction in computational neuroscience.

In this thesis, methods for mathematical model reduction are reviewed. In the field of systems biology, models are typically simplified by completely eliminating variables, such as molecules, from the system, and making assumptions of the system behavior, for example regarding the steady state of the chemical reactions. However, this approach is not meaningful in neuroscience since comprehensive models are needed in order to increase understanding of the target systems. This information loss problem is solved by mathematical reduction methods that strive to approximate the entire system with a smaller number of dimensions compared to the original system.

In this study, mathematical model reduction is applied in the context of an experimentally verified signaling pathway model of plasticity. The chosen biophysical model is one of the most comprehensive models out of those that are currently able to explain aspects of plasticity on the molecular level with chemical interactions and the law of mass action. The employed reduction method is Proper Orthogonal Decomposition with Discrete Empirical Interpolation Method (POD+DEIM), a subspace projection method for reducing the dimensionality of nonlinear systems. By applying these methods, the simulation time of the plasticity model was radically shortened although approximation errors are present if the model is reviewed on large time scales. It is up to the final application of the model whether some error or none at all is tolerated. Based on these promising results, subspace projection methods are recommended for dimensionality reduction in computational neuroscience.