Geometrical Structure of Bifurcations during Spatial Decision-Making

Animals are constantly making decisions while on the move. Despite this, most studies have considered only the outcome of, and time taken to make, decisions. Motion is, however, crucial in terms of how space is represented by organisms during spatial decision-making. Time-varying representation results in the emergence of new and fundamental geometric principles that considerably impact decision-making.

Here, we employ a simple spin-based model of this time dependent process to 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. In this model, each potential target is represented by a group of Ising spins with all-to-all connectivity and ferromagnetic intra-group interactions. In this way, we extend the Ising model to describe spatial phase transitions, which are trajectory bifurcation points, bridging statistical physics with dynamical systems.

Using this model, we find how the brain spontaneously breaks the symmetry in multi-choice decisions into a series of binary decisions, a process that repeats until one option remains. Among other methods, we employ a novel “mean-field trajectory” approach to reveal the new geometric principles for spatiotemporal decision-making. We find that all bifurcation points, beyond the very first, fall on a small number of bifurcation circles which act as attractors for the mean field trajectories, and their spatial organization determines the shape of the trajectories.

Furthermore, we find that a non-Euclidean (neural) representation of space (effectively an elliptic geometry) considerably reduces the number of bifurcation points in many geometrical configurations (including from an infinite number to only three), preventing endless indecision and promoting effective spatial decision-making. Further insights from the connection of this model to dynamical systems will be discussed as well.