Figure 1.
Statistical properties of runs through single grid fields.
(A) Distribution of path length within grid fields for all runs and for all runs with more than four spikes. In many runs, the animal's paths graze the field. Most of these very short runs show only little spiking activity. (B) Distribution of path tortuosity, as measured by the ratio of the actual path length to the length of the straight line connecting field entry to field exit. Short runs from (A) are usually straight. (C) Distribution of path eccentricity, the shortest distance between the path and the location of the maximum firing rate within the field (the “firing-field peak”). Note that many runs with large eccentricities have only few spikes. (D) Distribution of average speed in single runs. The constraint on spiking activity only mildly affects this distribution.
Figure 2.
Grid cells exhibit phase precession in two-dimensional environments.
(A) Trajectory (white line) of a rat over 10 minutes in a 1 m2 square enclosure together with the firing pattern (black dots) and color-coded firing-rate map of a single grid cell. Data from Sargolini et al. [26]. Note that many different paths traverse each grid field. (B) Spike phase relative to the local field potential for all passages through three example grid fields. Runs with varying directions and originating from various points in the two-dimensional environment are pooled. The position along each path within the firing field is normalized by the path's total length. (C) Three examples of single runs with different phase-precession slopes m and circular-linear correlation values r. Circular-linear regression lines are indicated. (D) Running direction has no consistent influence on the phase-precession slope. Histogram of p-values of the correlation between entry direction of the animal into a firing field and single-run phase-precession slope. The analysis is restricted to straight runs. Red dashed line indicates significance level p = 0.05. (E) Comparison of single-run phase precession and phase precession assessed by pooling all runs through a particular grid field. Each dot represents a single run; the left panel shows the place-phase correlation, the right panel depicts the slope of phase versus location. A negative slope implies phase precession; note the large variability across different runs. Red crosses denote the average correlation and the average slope. The diagonal line marks the identity. (F) Single-run phase precession in one and two-dimensional environments. (left) Distribution of circular-linear correlation values for runs on a linear track (dashed lines) and in the square arena (full lines). (right) Distribution of phase-precession slopes for the same two conditions. Despite the large speed and movement differences between the linear track and the open field, the phase-precession statistics are similar.
Figure 3.
Salient features of the animal's path through a grid field affect phase precession.
(A) The shorter the path is, the steeper the phase precession becomes. (B) The path length and phase precession correlate on a grid field by grid field basis, not just on average across grid fields. (C) First-half slopes are steeper than second-half slopes. The histogram in the inset shows the distance of data points from the diagonal, which is skewed towards smaller slopes in the second half of runs. (D) The phase range increases with path length and saturates at about 210°. (E) More meandering runs (increasing tortuosity) exhibit a less pronounced phase precession. As tortuosity correlates with the path length in a firing field, this finding is consistent with (A). (F) The animal's speed affects the phase-precession slope only weakly, and this effect primarily reflects a correlation between speed and tortuosity. For straight runs through the field, a statistically significant effect of speed on phase precession was not found. (G) Tangential paths lead to steeper phase precession than paths through the center of the field. The eccentricity measures the shortest distance between the path and the center of the firing field. For straight runs, the effect is not statistically significant. (H) Summary of the observed phenomena, with asterisks indicating statistical significance (p<0.05). For all investigated measures, restricting the analysis to straight runs weakens the effects. Error bars indicate one s.e.m. and are slightly offset for clarity in (A), (F), and (G).
Figure 4.
Negative values of time-phase correlation (A) and time-phase slope (B) of single runs indicate phase precession. (C) Time-phase correlation and position-phase correlation are statistically indistinguishable in single runs. (D) Time-phase slope and position-phase slope are highly correlated. The slope of the solid red line indicates the median of average speeds (19.5 cm/s) in single runs. Red and blue dashed lines mark 5-percentiles (6.5 cm/s and 44.1 cm/s, region shown in magenta) and 1-percentiles (2.7 cm/s and 59.9 cm/s) of the speed distribution, respectively. These data show that the variability between the phase-time and phase-position slope is mainly due to variations of the animals' running speed.
Figure 5.
Phase precession in different cortical layers.
Phase-precession slope generally does not depend on the cell's cortical layer (A and C). However, phase precession is decreased in layer III, as measured by the correlation (B and D). The single-run correlation of phase precession is lowest in layer III, and in layers II, V and VI the place-phase correlation is similar. Single-run effects are reproduced when the analysis is restricted to significantly correlated runs (cross-hatched bars). All bars show mean values, error bars depict one s.e.m. and asterisks indicate statistical significance (p<0.05). (E) Spikes show a preferred theta phase. The theta-phase preference is mild, and the weakest phase locking is encountered in layer III. (F) The first spike in a grid-field traversal generally occurs late in the theta cycle for layer II and VI, while it occurs rather early in layers III and V. In (E) and (F), the spike count histogram is normalized so that the sum of all ten bins equals 1. Colors label the cortical layer. Black arrows indicate the vector strength of the spike-phase theta modulation. All spikes were included in the analysis; no prior selection was made. The analysis is based on a total of 95 cells: 20 cells for layer II, 36 from layer III, 10 from layer V and 29 from layer VI.
Figure 6.
Testing predictions of oscillatory interference models.
(A) Spikes (dots) of an example grid cell in a two-dimensional environment. Colors indicate the theta phase of spiking. (B) Oscillatory interference model with three “dendritic” oscillations; preferred directions are separated by 60°, as indicated in the inset at the top left, yields direction-dependent phase coding. The top right inset shows spike phases along a linear run through the central firing field as indicated by the arrow in the main panel. (C) Preferred directions separated by 120° lead to nonmonotonic phase coding so that spike phases first precess, then recess. Insets show phases for linear runs through the center, as in (B). (D) Model with six “dendritic” oscillators whose frequency modulation with speed is half-wave rectified such that the frequency never falls below theta frequency. This model leads to saturating phase precession for any run through a grid field, but the phase-precession slope is independent of path length (E), tortuosity (F), and eccentricity (G), which is in contrast to the data analysis in Figure 3. (H) The rate of phase precession does not depend on speed, which is consistent with the experimental data for straight runs.