Mean Field trajectories in a spin model for decision making on the move

How animals navigate and perform directional decision making while migrating and foraging, is an open puzzle. We have recently proposed a spin-based model for this process, where each optional target that is presented to the animal is represented by a group of Ising spins, that have all-to-all connectivity, with ferromagnetic intra-group interactions. The inter-group interactions are in the form of a vector dot product, depending on the instantaneous relative, and deformed, angle between the targets. The deformation of the angle in these interactions enhances the effective angular differences for small angles, as was found by fitting data from several animal species. We expose here the rich variety of trajectories predicted by the mean-field solutions of the model, for systems of three and four targets. We find that depending on the arrangement of the targets the trajectories may have an infinite series of self-similar bifurcations, or have a space-filling property. The bifurcations along the trajectories occur on "bifurcation curves'', that determine the overall nature of the trajectories. The angular deformation that was found to fit experimental data, is shown to greatly simplify the trajectories. This work demonstrates the rich space of trajectories that emerge from the model.