  We study the statistics of spike trains of simultaneously recorded grid cells
in freely behaving rats. We evaluate pairwise correlations between these cells
and, using a generalized linear model (kinetic Ising model), study their
functional connectivity. Even when we account for the covariations in firing
rates due to overlapping fields, both the pairwise correlations and functional
connections decay as a function of the shortest distance between the vertices
of the spatial firing pattern of pairs of grid cells, i.e. their phase
difference. The functional connectivity takes positive values between cells
with nearby phases and approaches zero or negative values for larger phase
differences. We also find similar results when, in addition to correlations due
to overlapping fields, we account for correlations due to theta oscillations
and head directional inputs. The inferred connections between neurons can be
both negative and positive regardless of whether the cells share common spatial
firing characteristics, that is, whether they belong to the same modules, or
not. The mean strength of these inferred connections is close to zero, but the
strongest inferred connections are found between cells of the same module.
Taken together, our results suggest that grid cells in the same module do
indeed form a local network of interconnected neurons with a functional
connectivity that supports a role for attractor dynamics in the generation of
the grid pattern.
