Recent research involving bats flying in long tunnels has confirmed that hippocampal place cells can be active at multiple locations, with considerable variability in place field size and peak rate. With self-organizing recurrent networks, variability implies inhomogeneity in the synaptic weights, impeding the establishment of a continuous manifold of fixed points. Are continuous attractor neural networks still valid models for understanding spatial memory in the hippocampus, given such variability? Here, we ask what are the noise limits, in terms of an experimentally inspired parametrization of the irregularity of a single map, beyond which the notion of continuous attractor is no longer relevant. Through numerical simulations we show that (i) a continuous attractor can be approximated even when neural dynamics ultimately converge onto very few fixed points, since a quasi-attractive continuous manifold supports dynamically localized activity; (ii) excess irregularity in field size however disrupts the continuity of the manifold, while too little irregularity, with multiple fields, surprisingly prevents localized activity; and (iii) the boundaries in parameter space among these three regimes, extracted from simulations, are well matched by analytical estimates. These results lead to predict that there will be a maximum size of a 1D environment which can be retained in memory, and that the replay of spatial activity during sleep or quiet wakefulness will be for short segments of the environment.
