## Abstract

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.

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.

Original language | English |
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Publication status | Published - 18 Sept 2019 |

Publication type | Not Eligible |

Event | Bernstein Conference 2019 - Technische Universität Berlin, Berlin, Germany Duration: 18 Sept 2019 → 20 Dec 2019 http://bernstein-conference.de/ |

### Conference

Conference | Bernstein Conference 2019 |
---|---|

Country/Territory | Germany |

City | Berlin |

Period | 18/09/19 → 20/12/19 |

Internet address |

## Keywords

- Neuroscience
- Control theory
- Model reduction