(a) The population pattern period in an aperiodic network expands continuously with increasing inhibition strength () over a range. The ordinate shows the stretch-factor , which quantifies the deviation of the period post-perturbation from that pre-perturbation, normalized by the pre-perturbation period (see Materials and methods). Altering the network’s connectivity to even slightly take into account the periodic activity pattern by adding weak connections between neurons in adjacent activity bumps (as in [b]) transforms the network into one that will not stretch at all (cyan curve). This network with coupled activity bumps, despite the weakness of the connectivity, is in principle mathematically analogous to the fully periodic network. Indeed, the population period in the network with cyan connectivity can no longer gradually vary with inhibition strength (cyan curve, (a)). Simulation details: The network connectivity is a hybrid of the aperiodic network in Burak and Fiete, 2009 with the fully periodic network of Fuhs and Touretzky, 2006 (note that, while the model of Fuhs and Touretzky, 2006 does not have explicit periodic boundary conditions, the multimodality of the synaptic weights couples adjacent activity bumps so that the network acts as a single-bump, fully periodic network). The dynamics are LNP-based (see Materials and methods) and driven with inputs simulating animal motion at constant speed (v = 0.3 m/s) for 10 s. There are only two populations (call them R and L), distinct in their directional preferences ( (0,1), (0,–1) for the R and L populations, respectively) and output synaptic asymmetries (see below). The shifted output weight profiles are sinusoids with gaussian envelopes, the latter which constrain the non-locality of the projections. For a narrow gaussian envelope, the weights resemble the purely local, center-surround profiles of Burak and Fiete, 2009, whereas for wide gaussian envelopes, the weights resemble the non-local, multimodal projections of Fuhs and Touretzky, 2006. The weights going from population to and from cells i and j, are given by , where ( for and ), is a scaling factor that modulates the amplitude of the weights, is a normalization factor, determines the width of the gaussian envelope, and determines the period of the underlying sinusoid. Parameters. 200 neurons; CV = 0.5; 0.5 ms; 30 ms; 50; 0; 1; ; ; = 200; 412.