MATHEMATICAL MODEL ORDER REDUCTION OF A FOKKER-PLANCK MEAN-FIELD MODEL

Development of large-scale models of neuronal network dynamics is important in order to increase understanding of the whole brain. Although mathematical analysis of these models is intractable and their numerical simulation is very resource intensive, mean-field approximation has been shown to be an effective tool for studying populations of biophysically detailed neurons [1]. Here we compare state-of-the-art methods for improving the simulation time of mathematical models and show that a nonlinear Fokker-Planck-McKean-Vlasov-type neuronal mean-field model can be accurately approximated in low-dimensional subspaces with these methods.

Using mean-field approximation, cells are grouped together into populations
based on their statistical similarities, in order to represent the dynamics of
the system in terms of the ensemble behaviour. These populations can then be
described by a probability density function expressing the distribution of
neuronal states at a given time. In this study we focus on a mean-field model of
a network of FitzHugh-Nagumo neurons with chemical synapses using the
Fokker-Planck formalism, which results in a nonlinear McKean-Vlasov partial
differential equation (PDE) [1]. For numerical simulations the PDE is discretized in
space over three variables and a high-dimensional system, whose domain is a cube, is obtained.

The dimensionality, and hence simulation time, of discretized PDE systems can be reduced using mathematical model order reduction (MOR) methods. MOR methods are well established in engineering sciences, such as control theory. However, in computational neuroscience MOR is underutilised, although the potential benefits in enabling large-scale simulations are obvious [3].

Here we use recently developed advanced variants of the Discrete Empirical
Interpolation Method (DEIM) [2] to reduce a nonlinear mean-field model. The
system can be reduced with minimal information loss by deriving subspaces where the entire system is approximated with a small number of dimensions during the simulation phase, and after simulation the original model can be fully
reconstructed (see Fig. 1). By applying these methods, the simulation time of
the model is radically shortened, albeit not without dimension-dependent
approximation error. This can be particularly useful when attempting to model
whole-brain activity, for which there is an immediate demand in clinical and
robotic applications.

1. Baladron J, Fasoli D, Faugeras O, Touboul J. Mean-field description and
propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo
neurons. The Journal of Mathematical Neuroscience. 2012 Dec 1;2(1):10.
2. Chaturantabut, S. and Sorensen, D.C., Nonlinear model reduction via discrete
empirical interpolation. SIAM Journal on Scientific Computing. 2010, 32(5),
pp.2737-2764.
3. Lehtimäki, M. , Paunonen, L., Pohjolainen, S. and Linne, M.-L., Order
reduction for a signaling pathway model of neuronal synaptic plasticity.
IFAC Papers OnLine. 2017, 50-1:7687–7692.