**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, R^{2} = 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, R^{2} = 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).

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