LFP methods

For analysis, an interval of interest was identified. The continuous LFP signal was windowed over the interval of interest for each trial. Here, we call xnt the windowed LFP signal for trial n at time t.

LFP power: LFP power spectral density was estimated with a multitaper method. In brief, for each trial, the LFP signal was windowed with each of a number of orthogonal Slepian tapers, and Fourier transforms were estimated. The Fourier transform of the LFP signal xnt with the kth taper, dkt was estimated according to Eq. (¹)

where T is the length of xnt, f is the frequency, and j is the imaginary unit (i.e., −1). The power spectral density for a single trial n, Sxx,nf, was then estimated as a weighted average of auto-spectra across tapers according to Eq. (²)

where f_s is the sampling frequency, K is the number of tapers, and w_k are weights determined by an adaptive algorithm⁸⁷. The power spectral density was then averaged across trials to produce a single estimate of the power spectral density according to Eq. (³)

where N is the number of trials.

We used a time-half-bandwidth product of 2.5, affording us 4 Slepian tapers. We used either 400 ms or 200 ms windows, affording us frequency resolutions of ±6.25 or ±12.5 Hz, respectively. Band-limited power was estimated by summing the power spectral density estimate over the band of interest. Power time signals were estimated by stepping the time window by either 100 ms (400 ms windows) or 50 ms (200 ms windows) and estimating band-limited power centered at each time step. In most cases, we present power time signals and power spectral density as a percentage of baseline power or power spectral density, respectively. Baseline values are estimated as the average value over 500 ms before the target presentation. Power was computed at each LFP recording site individually before averaging across the population.

The bands of interest, 20–30 Hz and 70–120 Hz, were not selected a priori. Instead, these bands were selected empirically early in our study to capture general trends in the power density spectra and then maintained as we collected more data. Note that with a frequency resolution of ±6.25 Hz (400 ms time windows), the band labeled 20–30 Hz actually included information from frequencies from 13.75 to 36.25 Hz. The same is true for measures of synchronization described below. The entire analysis was repeated using the Chronux toolbox (http://chronux.org/⁸⁶). Although estimates were performed at slightly different frequencies, the results were essentially identical.

LFP–LFP synchronization: The degree of synchronization between two LFP signals was quantified with coherence. Like variance, coherence is a measurement across trials. Coherence between two LFP signals, x and y, was estimated according to Eq. (⁴)

where Sxxf and Syyf are the mean power density spectra across trials for LFP signals x and y, respectively, and Sxyf is the mean cross-spectrum across trials for LFP signals x and y. Power spectral densities were estimated with the same multitaper method described above. Cross spectra were computed in a manner similar to power density spectra. The cross-spectrum for a single trial n, Sxy,nf was then estimated as a weighted average of the cross spectra across tapers according to Eq. (⁵)

where Xn,kf and Yn,kf are the Fourier transforms of time series x(t) and y(t), respectively, and Yn,k*f is the complex conjugate of Yn,k*f. The mean cross-spectrum across trials is then estimated according to Eq. (⁶)

where N is the number of trials.

Spike-LFP synchronization: Synchronization between spikes and LFPs was quantified with three different measures: spike-LFP coherence, phase-locking value, and pairwise phase consistency. All three measures were computed over an 800 ms time window just before the go cue. Each measure was computed separately for each movement type. To protect against differences in mean spike rate driving spike-LFP coherence, all analyses were restricted to neurons with at least 500 spikes in each movement condition. Restricting the analysis to exactly 500 (randomly sampled) spikes from each cell for each condition resulted in similar results.

Spike-LFP coherence was estimated using the Chronux toolbox (http://chronux.org/⁸⁶). The methods of the toolbox are briefly summarized here. For each trial, Fourier transforms were computed separately for spike and LFP signals using a multitaper method. For each taper, the LFP and spike signals were windowed by the taper and a Fourier transform was computed using a fast Fourier transform (FFT) algorithm (http://fftw.org⁸⁸). Before computing the FFT of the spikes, the mean spike rate of the trial was subtracted away to remove the DC component. The Fourier transforms were then used to compute coherence values as with LFP–LFP coherence above. We elected to use a time-half-bandwidth product of 12, which afforded the use of 23 Slepian tapers and yielded a frequency resolution of ±15 Hz. Analyses using narrower frequency resolutions revealed the same effects but included additional narrow-band noise. This narrow-band noise likely reflects the fact that we interleaved 10–80 trial types (5 task types and 2 to 8 directions) and therefore had limited numbers of trials per trial type.

Phase-locking values (PLV) were computed for a range of frequencies. For a given frequency, the LFP phase at the time of each spike was estimated with a wavelet transform. Phases were then pooled across trials. PLV was estimated according to Eq. (⁷)

where θ_s is the phase at the time of spike s and S is the number of spikes. Significance was assessed with a Rayleigh test.

Pairwise-phase consistency (PPC) was assessed in a similar manner to PLV. Phases at the time of spikes were obtained using a wavelet transform. However, phases were not pooled across trials. PPC was estimated according to Eq. (⁸)

where N is the total number of trials, and S_n is the number of spikes in trial n, and θ_n,s is the phase at the time of spike s in trial n.

Many neurons have well-defined preferred directions for reach and saccade trials. To test whether LFP also shows preferred directions, one can find the movement direction at each site that produces the strongest modulation, average those strongest responses together, and contrast them with the response obtained for movements in the opposite direction. While such a procedure will capture a tuning if it exists, it is also highly likely to produce a statistically significant differential effect even in the absence of tuning. Appropriate analyses can exclude such artifacts, but we instead ask a simpler question. We restrict the data to sites at which a well-tuned single unit was recorded on the same electrode (113 sites, 43 from MkT and 70 from MkZ) and ask what information is coded by LFP power when each cell’s preferred direction is considered.