Three unbiased estimators of spike-field coherence

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In these notes we show a simplified expression for the Pairwise Phase Consistency (PPC) measure of Vinck et al. (2010). We illustrate it's relation to a bias-corrected spike-field coherence measure from Grasse et al. (2010), and discuss an third notion of spike-field coherence that is intermediate between the two. 

Here, we use the term "event-triggered" rather than "spike-triggered", because we want to apply these measures to neural events other than spikes (e.g. beta-frequency transients in motor cortex).

Pairwise Phase Consistency

Vinck et al. (2010) define Pairwise Phase Consistency (PPC) as the expected dot product between all pairs of (spike-triggered) phase measurements. 

\[\hat\Upsilon = \frac 2 {N(N-1)} \sum_{j=1..N-1} \sum_{k=j+1..N} \cos(\theta_j - \theta_k)\]

There is an alternative way to express PPC that is faster to calculate, and also reveals a relationship between PPC and similar alternatives. 

Motor cortex LFP spatiotemporal dynamics in a cued grasp with instructed delay task

Update: Portions of these notes have now been published in the Journal of Neurophysiology as  "Dissociation between sustained single-neuron spiking and transient β-LFP oscillations in primate motor cortex" and "Phase reorganization leads to transient β-LFP spatial wave patterns in motor cortex during steady-state movement preparation".

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Task-locked modulations in neural activity

The Cued Grasp with Instructed Delay (CGID) task reliably elicits task-locked activity in all three motor areas (M1, PMd, PMv).

• Consistent with prior literature, the movement period of the CGID task is marked by slow motor evoked potentials (Fig. 2), increased single-unit firing rates (Fig. 3), and beta suppression (Fig. 4).
• Beta oscillations are enhanced during the first four seconds of the task, although there are some differences between subjects.
• The average level of beta-LFP synchrony is correlated with beta-LFP power, and varies across phases of the task.
• We find no evidence of task-locked phase resetting of beta LFP oscillations
• The spatiotemporal structure of beta-LFP waves is correlated with amplitude and synchrony, with lower amplitudes reflecting more complex wave structures, and higher amplitudes as more synchronous.

Figure 1: The CGID task reliably elicits evoked potentials, which correlate with beta suppression. In subject S, beta power is strongest in the first second before object presentation. In subject R, beta oscillations are more variable, with somewhat stronger power between the grip and go cues. In both animals, high beta power appears to correspond to periods of higher beta synchrony, and larger phase gradient directionality, a measure of how much LFP activity resembles a plane wave. Conversely, increases in the average magnitude of the Hilbert phase gradient, which summarizes how quickly beta phase changes over the array, and in the number of critical points in the Hilbert phase gradient, which summarizes the complexity of the beta spatiotemporal wave patterns, correspond to periods of beta suppression.