## Abstract

The current trend in computational neuroscience is to incorporate multiple physical levels of the brain into mathematical models, which often results in large networks of interconnected neural cells. Comprehensive models with accurate system dynamics are necessary in order to increase understanding of different mechanisms in the whole brain, but these models are analytically intractable. Additionally, their numerical simulation is very resource intensive. Useful ways of mitigating the computational burden include using a mean-field approach, as well as mathematical model order reduction (MOR).

Using mean-field approximation, one can account for the random fluctuations of variables by replacing them by their mean averages. The cells are grouped together into populations based on their statistical similarities, in order to represent the dynamics of the system in terms of the averaged out ensemble behaviour. These populations can then be described by a probability density function expressing the distribution of neuronal states at a given time. This approach ensures that the essential system dynamics converge to a stationary attractor consistent with the steady-state dynamics of the original system.Here we use the Fokker-Planck formalism, which results in a nonlinear system of partial differential equations (PDEs).

PDE systems can be difficult to solve analytically, and thus discretisation for numerical analysis is necessary. This discretisation often leads to very high-dimensional numerical models that correspond to equally high computational demand. Discretised PDE systems can be reduced using mathematical model order reduction methods [1]. MOR methods are well established in engineering sciences, such as control theory, as they improve computational efficiency of simulations of large-scale nonlinear mathematical models. In computational neuroscience MOR is underutilised, although the potential benefits in enabling multilevel simulations are obvious [2].

In this study we use mathematical MOR methods to reduce the dimensions of a PDE model derived using the mean-field approach. The system can be reduced with minimal information loss, by deriving a subspace that approximates the entire system and its dynamics with a smaller number of dimensions compared to the original model. Here we use Proper Orthogonal Decomposition with Discrete Empirical Interpolation Method (POD+DEIM), a subspace projection method for reducing the dimensionality of general nonlinear systems [1]. By applying these methods, the simulation time of the model is radically shortened, albeit not without dimension-dependent approximation error. The tolerated amount of inaccuracy depends on the final application of the model.

Due to being well-suited for depicting mesoscopic behaviour, the mean-field approach in combination with the POD+DEIM method allows us to describe the behaviour of any large multiscale brain model with a relatively low computational burden. This can be particularly useful when attempting to model whole-brain connectivity, for which there is an immediate demand in clinical and robotic applications.

1.Chaturantabut, S. and Sorensen, D.C., Nonlinear model reduction via discrete empirical interpolation. SIAM Journal on Scientific Computing. 2010, 32(5),pp.2737-2764.

2. 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, one can account for the random fluctuations of variables by replacing them by their mean averages. The cells are grouped together into populations based on their statistical similarities, in order to represent the dynamics of the system in terms of the averaged out ensemble behaviour. These populations can then be described by a probability density function expressing the distribution of neuronal states at a given time. This approach ensures that the essential system dynamics converge to a stationary attractor consistent with the steady-state dynamics of the original system.Here we use the Fokker-Planck formalism, which results in a nonlinear system of partial differential equations (PDEs).

PDE systems can be difficult to solve analytically, and thus discretisation for numerical analysis is necessary. This discretisation often leads to very high-dimensional numerical models that correspond to equally high computational demand. Discretised PDE systems can be reduced using mathematical model order reduction methods [1]. MOR methods are well established in engineering sciences, such as control theory, as they improve computational efficiency of simulations of large-scale nonlinear mathematical models. In computational neuroscience MOR is underutilised, although the potential benefits in enabling multilevel simulations are obvious [2].

In this study we use mathematical MOR methods to reduce the dimensions of a PDE model derived using the mean-field approach. The system can be reduced with minimal information loss, by deriving a subspace that approximates the entire system and its dynamics with a smaller number of dimensions compared to the original model. Here we use Proper Orthogonal Decomposition with Discrete Empirical Interpolation Method (POD+DEIM), a subspace projection method for reducing the dimensionality of general nonlinear systems [1]. By applying these methods, the simulation time of the model is radically shortened, albeit not without dimension-dependent approximation error. The tolerated amount of inaccuracy depends on the final application of the model.

Due to being well-suited for depicting mesoscopic behaviour, the mean-field approach in combination with the POD+DEIM method allows us to describe the behaviour of any large multiscale brain model with a relatively low computational burden. This can be particularly useful when attempting to model whole-brain connectivity, for which there is an immediate demand in clinical and robotic applications.

1.Chaturantabut, S. and Sorensen, D.C., Nonlinear model reduction via discrete empirical interpolation. SIAM Journal on Scientific Computing. 2010, 32(5),pp.2737-2764.

2. 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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DOIs | |

Publication status | Published - 14 Nov 2019 |

Publication type | Not Eligible |

Event | 28th Annual Computational Neuroscience Meeting (CNS*2019) - Barcelona, Spain Duration: 13 Jul 2019 → 17 Jul 2019 https://www.cnsorg.org/cns-2019 |

### Conference

Conference | 28th Annual Computational Neuroscience Meeting (CNS*2019) |
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Country/Territory | Spain |

City | Barcelona |

Period | 13/07/19 → 17/07/19 |

Internet address |

## Keywords

- Neuroscience
- Control theory
- model reduction