Mean-field methods for multiscale models in neuroscience

Multiscale modelling of the brain is necessary in order to understand how
interactions on the molecular and cellular levels can give rise to higher-level
brain functions. As microscale processes tie into mesoscopic populations that
facilitate whole-brain behaviour, being able to describe the full-scale
interconnectivity of the brain is clearly imperative. In order to interpret all
of the different mechanisms, we need comprehensive models with accurate system dynamics. However, incorporating multiple levels into mathematical models often results in large networks of interlinked neural cells that are analytically intractable. Additionally, their numerical simulation is 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, random fluctuations of variables can be
accounted for by replacing them by their averages. Cells are grouped together
into populations based on their statistical similarities, in order to represent
the dynamics of the system in terms of the mean ensemble behaviour. These
populations can then be described by a probability density function expressing
the distribution of neuronal states at a given time. We use the Fokker-Planck
formalism, which results in a nonlinear system of partial differential
equations (PDEs).

With mathematical MOR methods the dimensions of a PDE model can be reduced with minimal information loss. 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 MOR methods 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.