A) Each red dot depicts the amplitude of slow gamma in a window of 250 ms around each LFP theta peak as a function of running speed. The value of the slow gamma amplitude was averaged within a given speed bin (∼7 cm/s wide, with 80% overlap between neighboring bins) (red squares). Black line shows the best linear fit. B) Same as A for fast gamma (blue dots and squares) with logarithmic fit. (See for methods and , for details). C) Ensemble averaged data showing linear increases in slow gamma amplitude with speed. D) Same as C with logarithmic speed-dependence for fast gamma. E) Each vertical panel shows the cross-frequency coupling between the amplitude of a fast (15–300 Hz) signal (y-axis) and the phase of a slow (2–20 Hz) signal, whose frequency is shown on the x-axis). Separate panels show coupling at different running speeds (top) for the example data in figures 2A,B. Colorbar to the right indicates modulation index (see ). Significant cross-frequency coupling is found only between the phase of the theta (6–12 Hz) oscillation and the gamma amplitude (20–120 Hz). Fast-gamma-theta coupling is greater than slow-gamma-theta coupling (bottom panel) at all speeds. The coupling increases logarithmically and linearly with speed for fast and slow gamma respectively (see ). F) Slow gamma amplitude changes with running speed and theta phase for the example data set in figure 2A,B. G) Similarly for fast gamma. H) Slow gamma amplitude at the preferred phase (at 236±2.2°) of theta, averaged across all data, is linearly correlated with running speed (solid line, R2 = 0.90±0.018, median±s.e.m.), but slow gamma amplitude around the theta trough changed minimally (dotted line). I) Similarly, fast gamma amplitude around the peak (260±1.8°) of theta increased logarithmically with speed (solid line, R2 = 0.94±0.016), but fast gamma amplitude around the theta trough changed minimally (dotted line). J) Distribution of the slope of slow gamma amplitude around the theta peak as a function of running speed (solid line) across the ensemble of data, showing that it was significantly positive (0.017±0.0012, p = 1.9e-40) and far greater than the slope around the theta-trough (dotted line, 0.00067±0.00032, p = 4.3e-4). K) Similar results were true for the slope of fast gamma amplitude as a function of the logarithm of running speed around the theta peak (solid line, 0.17±0.0057, p = 4.1e-68) and the theta trough (0.013±0.0040, p = 0.0054).