Continuous quasi-attractors thrive on irregularity, up to a limit

Recent research involving bats navigating lengthy tunnels has confirmed that hippocampal place cells can be active at multiple locations, with considerable variability in place field size and peak rate. In a recurrent networks, such variability implies quenched disorder in the synaptic weights, impeding the establishment of a continuous manifold of fixed points. If most positions in space are not any longer fixed points, is the popular notion of continuous attractors still relevant? What would be the neural dynamics underlying retrieval of spatial representations?We study the dynamics of recurrent networks with similar field statistics as those recorded in bats. We show that spatial memory can be effective even if, eventually, neural dynamics converge onto few real fixed points: a quasi-attractive continuous manifold supports dynamically localized activity. Within limits, though: when the variability is too limited or too large, either activity delocalizes or its continuity is disrupted, beyond two boundaries that we can estimate analytically. One depends only on the number of fields per cell, and as this number grows, the viable region shrinks; the other boundary reduces to a pure number, quantifying field size variability very close to that experimentally observed in bats. In longer tunnels the two boundaries would eventually collide and the quasi-attractive region vanish, implying a maximal size of the environment that can be stored in memory.