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Figure C.1: A jitter model leading to the Beta-Poisson spiking model. A: Simulated 1D trial displacement, drawn from the jitter distribution in the inset (P (jitter) ∝ g(x)^p−1). The trial-averaged firing field (solid black line) is broader and has a lower peak rate as compared to the underlying tuning curve without jitter (solid blue line, parameters used: λmax = 10, p = .6, q = .5.

B: The resulting distribution of λ at x₀ across N=100 trials and the analytical pdf of the Beta distribution. C: Simulated spike count distribution from a Poisson process with parameters λ_i where i denote the indices of the trials and the resulting Beta-Poisson distribution as derived in the Appendix D. D: Analogous situation in 2D with random shift angles leading to the identical statistics.

or rather flat distributions and we observed no stereotypical shape of the jitter.

The BP model connects the spike count statistics from the open field to the field shape and the spatial jitter distribution. The measures estimated for two-dimensional arenas can then be corroborated against the same measures on the linear track, as shown in Fig. C.3;

on the linear track, we have direct access to the jitter in field positions from trial to trial.

While fitting multiple models to the spike count data, we observed that the skewness in the BP model’s firing rate distribution is highly correlated with the zero inflation pa-rameters of the ZIP model as well as the ZINB model, both for the 1D and the 2D data (r = .68, p ≪ 1e − 10 and r = .63, p ≪ 1e − 10, see Table C.1). That is, because for a positive skew in the rate distribution a highly zero-inflated regime regime is approached.

On the contrary, a negative skew shifts the mode to larger values. Extremely negatively skewed fits suggest that nearly all rates in the field center are equal. In that case resulting spike counts are close to Poisson.

∆ς Σ1D Σ2D α1D α2D

∆ς 1 .42 .35 .27 .34

Σ1D .42 1 .55 .70 .57 Σ2D .35 .55 1 .31 .63 α1D .27 .70 .31 1 .54 α2D .34 .57 .63 .54 1

Table C.1: Correlations between the skewness Σ as a function of the parameters of the BP model fits, the ZIP model parameters α and the difference ∆ς of peak amplitudes in spike-triggered firing maps for short and long delays. The numerical values are the Pearson correlation coefficients (p < .01 for all pairs). The 1D linear track measures shown in the table are the medians over contexts and running directions. Data is shown for N = 99 grid cells for which all the required variables could be measured both in 1D and 2D.

The BP model is preferred over the Poisson model in 83% (likelihood ratio test, p < .001, df = 2) and to the ZINB model in 61% (by direct comparison of the likelihoods) of the cell recordings. For each cell, the BP model fits in 1D and 2D tended to yield similar parameters, which is consistent with firing field drifts occurring in 2D as well as in 1D. To address this question directly, we analyzed the spike-triggered rate maps [5] of grid cells over fixed time windows. In this approach, the animal’s location at each spike is set as the origin of a standard coordinate system. Spikes that occur within a specific time window after the triggering spike are used to build a firing rate map relative to the origin defined by the trigger. These relative firing rate maps are then averaged and collapsed onto a single dimension, which corresponds to a radial distance. If the spatial grid is stable, then the radial spike-triggered firing rate map will have peaks at regularly occurring intervals given by the grid’s period. On the other hand, if a cell’s fields undergo slow field displacement, the peaks in the radial-distance will gradually smear out. So we computed the spike triggered

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Figure C.2: Scatter plot of measures for the quantification of non-stationary tuning curves. A:

Amplitudes ς in the spike-triggered firing rate map at the grid period measured at long delays (up to 15s after trigger) versus short delays (between 60s and 75s after trigger). B: Zero count excess (measured via α from ZIP fits on the linear track) against rate change ∆ς at the grid period in the spike-triggered rate map in the open field. C: Skewness Σ_1D of the mixing distribution computed from BP model fits on the linear track against rate change ∆ς. The correlations in B and C indicate that features of spike count distributions in one context predict drifts of firing fields in a different context.

firing rate maps for successive 15-second intervals after the trigger. If drift occurs, the spatial modulation in the spike-triggered map during later intervals will be more strongly blurred. In particular, we compared the maps for the first 15s after each spike to the maps in the time window from 60s to 75s after each spike.

The amplitude ςearly was quantified as the difference between the first local maximum (grid period) and the average of the minima to the left and to the right of that maximum.

The positions of the extrema were detected for the first 15s window. As the peak did not always persist in the later window, the amplitude ςlate for the 60s to 75s window was measured using the positions of the extrema determined for the early window. If ςlate is close to zero, it means that the peak is smeared out, whereas a ςlate that is larger than ςearly

means the peak has become more prominent. Therefore, large differences ∆ς = ςearly− ς^late indicate a change in the typical relative distances between spike positions on the scale of a minute. While on average ∆ς was close to zero (mean = .014, sem = .028, Wilcoxon-test: U = 2438, p = .9, N = 99) in fact a significant correlation was found between ∆ς

and the skewness of the fitted mixing distributions (see Table C.1). As we pointed out earlier, positive skewness can imply strong jitter. Consequently, jointly drifting firing fields indicate both larger ∆ςas well as positive skewness.

Thus, two very different measures to quantify the non-stationarities of spatial tuning turned out to be correlated. This underscores how grid field displacements yield distinct, but related signatures in the statistics of grid cell spiking and demonstrates the utility of the BP model fits. Such drifts in the spatial representation occur even in cases where trial-averaging yields a clear grid structure to the firing fields. We deduce that such non-stationarities are a major source of trial-to-trial variability in grid-cell spiking in the open field.

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A Spikes and trajectory

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D Linear Track firing rates (Hz)

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E Simulated rates from 2D (Hz)

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A Spikes and trajectory

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D Linear Track firing rates (Hz)

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E Simulated rates from 2D (Hz)

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A Spikes and trajectory

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D Linear Track firing rates (Hz)

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E Simulated rates from 2D (Hz)

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Figure C.3: Each row represents data and model simulation of a different grid cell from the P`erez-Escobar et al. data [150]. A: Spikes (red) and trajectory (grey). B: Collected spike count distribution (n = 5, Fmin = 5Hz) and fitted BP distribution with parameters (p, q, λmax) and the skewness Σ of the mixing distribution. C. Field shape and jitter distribution sketched as suggested by the fitted parameters (copied to each field center). One displacement was drawn per trial to achieve consistent field drift in the case of multiple fields on the track. The field centers were estimated as the positions of local maxima in the tuning curves of the linear track (threshold for local maximum: 3Hz). In the last two rows a bimodal field displacement is suggested from the jitter distribution. D: Firing rates across trials on left runs in the light condition 1. E: Simulated firing rates from the distributions in C.