Statistical specifications

As outlined above (Section 2.3), our recording setup included 12 microelectrodes inserted into both hemispheres of the rat brain, with electrodes located in cortex and striatum of either side. We recorded data from n “ 10 animals that each contributed a highly variable number of trials, yielding 622 recordings (i.e., trials) in total (Table 2.1). One trial of the behavioral task encompassed 1 resting and 4 running levels lasting 1 minute and thus consisting of 12 epochs of 5 seconds length each (Section 2.5).

After preselection of data as described in Section 2.6.1 we were left with a variable number of valid epochs from different animals, electrodes, and recordings from different sessions and thus from partly differing electrode positions. Unfortunately, we were not able to accurately verify the coordinates of the latter. We hence decided to pool the parameter averages (behavioral data, firing rates, spectral measures) obtained for indi-vidual or pairs of electrodes from resting and running levels of each trial first across trials within individual subjects and then across subjects to yield grand average estimates.

Furthermore, we defined threshold criteria with regards to the minimum number of epochs needed to contribute to a valid level average of an individual recording (n“3) and the number of trials needed to contribute to across-trial averages of individual electrodes or electrodes pairs (n “ 3). Recordings and electrodes or electrode pairs not fulfilling these criteria were excluded from further analysis. We did so in order to account for the highly variable number of valid epochs and trials each subject contributed to the data pool and in order to guarantee a minimum level of robustness and comparability between data averages obtained for different trials, electrodes, and electrode pairs.

We computed across-epoch and across-level averages as the mean of the respective sample under the assumption of consistent recording conditions during the course of individual trials. In contrast, averages across trials before the final analysis step were

Subject ID Recordings

a6 115

a7 1

a8 10

a11 100

a12 39

a14 85

a15 37

a16 15

a17 199

a18 21

n“10 n“622

Table 2.1: List of animals implanted and used for recordings under treadmill running, and the number of recordings (i.e., trials) obtained from each individual. The subjects are referred to by means of a unique ID. Note that the mere number of recordings performed may not be representative of the amount of valid data remaining from the respective subject after preselection by means of signal evaluation and data average thresholding.

computed as the median of the respective sample to reduce the possible influence of outliers. Grand averages across all trials from all subjects were again computed as the mean and are shown with one standard error in both positive and negative directions, thereby providing an estimate of the precision of the respective average (Motulsky, 1995).

We expressed the change between rest and running phase-coupling magnitudes (Section 3.6) by taking the mean across median difference z-scores zsccomputed for each trial of each electrode pair:

zsc“ x_run´ xx_resty σrest

?N .

Here, x_run denotes the phase-coupling strength during an average single-trial running level, xx_resty and σ_rest are the mean and standard deviation, respectively, of the phase-coupling strength across all resting levels of an individual electrode pair, andN denotes the number of samples. Z-scores were computed to display both the statistical strength and the sign of the coupling difference between behavioral conditions.

We relied on nonparametric statistical tests (Wilcoxon rank-sum and sign-rank tests) to ensure against possible violations of the assumption of normally distributed data.

Bar plots that depict comparisons of different samples of measured values show the final average as the median with one unit of median absolute deviation.⁹ The latter provides

Chapter 2 Methods 2.7.4

an estimate of the median scatter of data values around the median of the sample. The reason for this is that the ensuing statistical tests are based on ranks of values and hence rely on the median rather than the mean as a measure of the average of the data scatter.

We computed regression coefficients (Spearman’sρ) to test for a significant linear scal-ing of multi-unit firscal-ing rates and spectral peak parameters (peak magnitudes and peak frequencies) with running speed. Throughout the study, we generally assumed a signifi-cance level ofα“0.01for all test statistical procedures. In all computations necessitating multiple tests and not involving permutation statistics (see below), Bonferroni-correction was used to adjust α-levels to the numbernof tests (Motulsky, 1995), according to

α“ 0.01 n .

We tested the statistical significance of spectral estimates by means of comparisons of measured values with those obtained from alternative distributions (Figure 2.17). For raw values of bivariate parameters (coherency, phase, power correlation, phase-amplitude coupling estimates), alternative distributions were obtained by circularly shifting the values of the complex time-frequency transformation matrices (3500time pointsˆ41 fre-quencies; Section 2.7.3.2) of one of the two partners by step size s, where s could be any number randomly drawn from the set of integersI “ t1,2, ...,3500u. Since all of the above parameters in essence quantify the consistency of amplitude or phase relationships between paired signals, destroying the original temporal relationship while preserving their individual temporal structures should yield magnitudes of coupling estimates that are, on average, only as large or even smaller than those obtained from the original data.

To save computation time, we calculated one estimate of the parameter of interest from both real and time-shifted data of each epoch. We then averaged shifted and real values in exactly the same way to obtain level, trial, and electrode pair averages.

For relative estimates (i.e., ratio and difference spectra) of both univariate (i.e., power) and bivariate spectral parameters (see above), we compared the original data with those obtained through averaging of alternative partitions created by randomly permuting the level memberships of data epochs. This approach was based on the assumption that if there was no difference between the distributions of spectral values during both major behavioral states (rest and running), ratio and difference values calculated from rest and running spectra obtained through averaging of correctly labeled epochs should not be significantly different from those obtained through averaging of randomized epochs.

To account for different amounts of valid rest and running epochs present in each trial (n“12andn“48, respectively, at maximum), we stratified their respective numbers to equal half the number of valid epochs of a given level and a given trial. For each level, we

Figure 2.17: Illustration of the principle idea behind time series shift-based statistics.

The upper two left and right plots display different raw striatal LFP time series from a resting rat and the corresponding power spectra, respectively. The lower left plot shows the second time series circularly shifted by a step sizes of 2295points (see main text).

The lower right plot demonstrates the effect of the temporal shift on the phase-coupling strength. While coherence is high between the two LFP signals which exhibit very similar frequency content and comparable temporal structure, the consistent phase relationship is largely destroyed after rotation of the second time series. Note that the power spectrum of the shifted time series is exactly the same as for the measured values, which is why we do not depict it here. We statistically compared the resulting grand average real and shift-based bivariate estimates by means of nonparametric tests in each frequency bin.

calculatedn“10alternative partitions from rest and running epochs and then obtained average alternative distributions by taking their mean. This was done to increase the reliability of estimates based on highly limited numbers of epochs (n“12at maximum for each level). Averaging across levels, trials, and electrodes or electrode pairs was then again performed in exactly the same way for both real and permuted relative spectra.

For modified versions of permutation based statistics in case of some parameters (e.g., cross-hemispheric LFP phase-coupling), see the respective results section (Chapter 3).

We tested the statistical significance of the spectral parameters of interest through non-parametric, paired comparisons (Wilcoxon sign-rank test) of real and shifted or permuted averages obtained for each electrode or electrode pair individually at each frequency, fre-quency pair (in case of power correlation and phase-amplitude coupling estimates), or

Chapter 2 Methods 2.8

frequency-phase bin pair (in case of phase histograms) of interest. These comparisons are tests of the null hypothesis of no difference between the medians of grand average distributions of spectral values obtained from real and time-shifted or permuted data, respectively. We again assumed a general significance level of α “ 0.01 for individual tests taking shifting and permutation procedures to function as implicit controls of the false discovery rate and thus to account for the problem of multiple comparisons to be performed on large numbers of spectral values (n “ 41 for one- and n “ 41ˆ41 or n“61ˆ41for two-dimensional estimates, respectively). For a general introduction into permutation statistics, see Ernst, 2004. For applications of related methods in systems neuroscience research, see Maris and Oostenveld, 2007 and Maris et al., 2007.