Mathematical model order reduction in computational neuroscience

Multi-scale models in neuroscience typically integrate detailed biophysical
neurobiological phenomena from molecular level up to network and system levels. Such models are very challenging to simulate despite the availability of
massively parallel computing systems. Model Order Reduction (MOR) is an
established method in engineering sciences, such as control theory. MOR is used in improving computational efficiency of simulations of large-scale and complex
nonlinear mathematical models. In this study the dimension of a nonlinear
mathematical model of plasticity in the brain is reduced using mathematical MOR methods.

Traditionally, models are simplified by eliminating variables, such as
molecular entities and ionic currents, from the system. Additionally,
assumptions of the system behavior can be made, for example regarding the
steady state of the chemical reactions. However, the current trend in
neuroscience is incorporating multiple physical scales of the brain in
simulations. 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 neural
phenomena.

The loss of information typically induced by eliminating variables of the system
can be avoided by mathematical MOR methods that strive to approximate the
entire system with a smaller number of dimensions compared to the original
system. Here, the effectiveness of MOR in approximating the behavior of all
the variables in the original system by simulating a model with a radically
reduced 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 projection
method for reducing the dimensionality of nonlinear systems [chaturantabut2010]. By applying these methods, the simulation time of the
model is radically shortened. However, our preliminary studies show
approximation error if the model is simulated for a very long time. The
tolerated amount of approximation error depends on the final application of the
model. Based on these promising results, POD+DEIM is recommended for
dimensionality reduction in computational neuroscience.

In summary, the reduced order model consumes a considerably smaller amount of
computational resources than the original model, while maintaining a low root
mean square error between the variables in the original and reduced models.
This was achieved by simulating the system dynamics in a lower dimensional
subspace 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.