  Animals must constantly make decisions on the move, such as when choosing
among multiple options, or "targets", in space. Recent evidence suggests that
this results from a recursive feedback between the (vectorial) neural
representation of the targets and the resulting motion defined by this
consensus, which then changes the egocentric neural representation of the the
options, and so on. Here we employ a simple model of this process to both
explore how its dynamics account for the experimentally-observed
abruptly-branching trajectories exhibited by animals during spatial
decision-making, and to provide new insights into spatiotemporal computation.
Essential neural dynamics, notably local excitation and long-range inhibition,
are captured in our model via spin-system dynamics, with groups of Ising-spins
representing neural "activity bumps" corresponding to target directions.
Analysis, employing a novel "mean-field trajectory" approach, reveals the
nature of the spontaneous symmetry breaking - bifurcations in the model that
result in literal bifurcations in trajectory space and how it results in new
geometric principles for spatiotemporal decision-making. We find that all
bifurcation points, beyond the very first, fall on a small number of
"bifurcation curves". It is the spatial organization of these curves that is
shown to be key to determining the shape of the trajectories, such as
self-similar or space filling, exhibited during decision-making, irrespective
of the trajectory's starting point. Furthermore, we find that a non-Euclidean
representation of space considerably reduces the number of bifurcation points
in many geometrical configurations, preventing endless indecision and promoting
effective spatial decision-making. This suggests that a non-Euclidean neural
representation of space may be expected to have evolved across species in order
to facilitate spatial decision-making.
