Fig. 1: Ring-attractor networks with an allocentric and an egocentric representation of space.

A Individuals are equipped with a ring-attractor network in which neurons are arranged on a ring. Each neuron receives sensory input from the external world through a Gaussian receptive field centered on an angle α_i (with respect to the individual’s allocentric or egocentric reference frame) and encodes for movement along the same direction, α_i. Besides, neurons interact with other neurons via excitatory or inhibitory synapses, depending on their distance along the ring. B With an egocentric representation of space, the animal encodes directions with respect to a self-body coordinate (head direction), α_ego. Whereas, with an allocentric representation, directions towards targets are encoded via an allocentric frame of reference, α_allo = α_r + α_ego, where α_r is the direction of the individual’s body axis. Thus, with an allocentric representation, directions are independent of the agent’s body coordinate. C To model an allocentric representation of space, we assume the neural network (represented by only four circles for better visibility) encodes for directions in a world-centric reference frame, which does not rotate with the individual’s movement in space (as if it is anchored in the external world, using one or more external cues), such that, neuron i, encodes for a direction, 2π(i−1)/N_s with respect to a world-centric axis. In the egocentric case, the network’s reference frame is attached to the individual and rotates with the individual as the individual moves in space, such that neuron i encodes for a direction, 2π(i−1)/N_s, with respect to the animal’s body axis.