(A,B) Data from simulations with fixed inhibitory drive and constant animal velocity. (A) Snapshots of neural activity. Scale bar, 50 neurons. (B) The diameter of bumps on the neural sheet decreases linearly with inhibitory drive (linear regression , ANOVA ). (C) Phase precession in a conceptual model with bump size oscillations. We imagine an attractor bump, with size oscillations described by B, passing through a recorded grid cell. Top, a single lap. The recorded neuron is at location 0 and fires a spike (black dot) whenever contained within the bump (gray area), subject to a 40 ms refractory period. Bottom, relationship between theta phase and time across multiple laps with different initial phases. Spikes occur around 360° (equivalently, 0°) at the start of the field, and their phase generally decreases with time within in the grid field. (D) Attractor bump shape at different theta phases averaged over theta cycles and individual bumps (Appendix 1). Grays scaled separately for each theta phase. Scale bar, three neurons. (E,F) Phase behavior in a simplified model using average bump dynamics. (E) We imagine the average attractor bump passing through a recorded grid cell. Top, a single lap. The recorded neuron is at location 0 and stochastically fires a spike (black dot) with rate proportional to bump activity, subject to a 40 ms refractory period. Bottom, relationship between time and theta phase across multiple laps with different initial phases. (F) Relationship between time and theta phase using average bumps whose activity has been rescaled to different maximum values: 40, 50, and 100 spikes/s. Dots represent spikes generated according to E. Lines indicate fit by circular-linear regression. Numbers in each panel from top left to top right indicate magnitude of correlation coefficient, regression fit score, and regression slope.