Andrea Clini, University of Oxford
Grid cells are a particular type of neuron in the brain of mammals constituting a positioning system.
Grid cells fire at regular intervals as the animal navigates an open area, creating specific patterns which serve to store information about location, distance, and direction.
In this talk we analyze a generalized SDE system modeling grid cells, which includes several models commonly proposed in neuroscience.
The main difficulty in the analysis is the spatial interaction.
Grid cells are indeed observed to organize in columns of closeby neurons distributed along the cortex, interacting within and among columns, allowing for the creation of the aforementioned patterns.
Mathematically, this corresponds to including space dependence into the equations.
We prove the existence of a mean-field description of the system in the thermodynamic limit, given by a family of space-dependent McKean–Vlasov SDEs.
We then characterize the fluctuations of the system in terms of a suitable Langevin SPDE and discuss ongoing work on its large deviations.