Results

Figure 4.1. Ballistic and sustained brain-computer interface (BCI) cursor movements.

(a) Example single-trial cursor trajectories during BCI center-out task for ballistic and sustained movements.

(b) Distance to the target, as a function of time. Each line corresponds to a trial from session 2. Additional sessions visualized in Supplementary Figure 4.1.

(c) Time to reach target. Each marker corresponds to a trial from session 2. Box plot lines indicate lower quartile, median, and upper quartile, respectively.

4.3.3 Sustained single-neuron and population activity in motor cortex (MC) during sustained BCI movements is shared with ballistic movement encoding

We visualized single-neuron activity across sustained and ballistic trials (Figure 4.2a-c). Because our focus was to better understand the autonomous dynamical systems hypothesis (aDSH), which describes MC, here, we focus on analyzing MC recordings. During ballistic movements, many neurons exhibited phasic firing rate modulation, usually rising with movement onset and falling with target acquisition. During sustained movements, many neurons exhibited the same onset firing rate increase (Figure 4.2a) but instead sustained their firing rate throughout the entire movement (Figure 4.2b-c).

Next, we visualized population activity using principal component analysis (PCA) (Figure 4.2d). Similarly to a previous study (Kaufman et al., 2016), the largest-variance principal component (PC) was largely condition-invariant and reflected movement onset timing. Interestingly, the second- and third-largest-variance PCs corresponded directly to movement direction, regardless of the decoder gain parameter. In other words, population activity during sustained and ballistic movements to the same target took the same initial neural trajectory before diverging. The same projection matrix was used for both conditions, and thus similar responses suggest that ballistic and sustained movements share a common neural substrate.

Figure 4.2. Neural responses in motor cortex (MC) during BCI cursor movements.

(a) Mean firing rate of an example neuron, aligned to Go (left panel) and target acquisition (right panel). Example neuron increased firing rate during movement onset. Most neurons appeared to modulate during movement onset, sustain firing rates during sustained movements, or some combination of onset and sustained preferred direction. 8 targets were used, but only 4 target conditions are shown.

(b) Mean firing rate of an example neuron, which activated during movements to target 5 and sustained firing for sustained movements.

(c) Mean firing rate of an example neuron, which activated during movements to target 3 and sustained firing for sustained movements.

(d) Principal component analysis (PCA) of MC population activity.

4.3.4 Comparing ballistic and sustained BCI movements reveals input-driven dynamics

During movement execution, the initial aDSH study described (quasi-)oscillatory dynamics, akin to a pendulum (Pandarinath et al., 2018a) or spring-mass system (Figure 4.3a). In this hypothetical limit of the system not receiving external inputs, the initial condition and dynamics should fully determine the neural trajectory. They used jPCA, a rotated variant of PCA, to uncover quasi-oscillatory dynamics in the neural activity (Churchland et al., 2012). Briefly, jPCA is a rotated version of PCA first reduces the recording dimensionality to the first 𝑘 = 6 PCs. Within this subspace, a new orthonormal basis of 𝑘 jPCs is calculated, such that the top two jPCs maximize the rotational component. This is mathematically equivalent to fitting the constrained dynamical system:

𝑿̇(𝑡, 𝑐) = 𝑀_{𝑠𝑘𝑒𝑤}𝑿(𝑡, 𝑐)

Equation 4.4 where 𝑿(𝑡, 𝑐) is the 𝑘-dimensional PCA reduction of the population firing rate vectors at time 𝑡 and condition 𝑐. 𝑀_{𝑠𝑘𝑒𝑤} is a skew-symmetric matrix, whose eigenvectors correspond to the jPCs.A corollary to oscillatory dynamics is that ‖𝑿̇(𝑡, 𝑐)‖₂the neural speed should be constant throughout the movement duration (Figure 4.3a). We applied jPCA to MC activity during ballistic BCI movements (Figure 4.3b).

Consistent with ballistic arm reach studies (Churchland et al., 2012), jPCA revealed clear rotational dynamics in the top jPC plane. We compared the jPCA fit to null distribution with the same neural/temporal covariance structure. We generated the covariance-constrained null distribution using tensor maximum entropy (TME) (Elsayed and Cunningham, 2017) and applied to jPCA to the samples. The jPCA fit was significantly better on the true data than the null distribution (p<0.001). Consistent with these results, the cross-validated (Methods) neural population speed

‖𝑿̇(𝑡, 𝑐)‖

2 was large across the duration of the movement (Figure 4.3c).

Next, we applied the neural speed and jPCA analyses to MC activity during sustained BCI movements (Figure 4.3c). Cross-validated neural speed high was during movement onset but returned to baseline levels (close to the ITI baseline) indicating that the neural activity is maintaining a constant stationary representation of intent while sustaining the BCI movement. In the jPCA visualization, this sustained activity appears as a “pause” in the rotational trajectory.

Finally, in Figure 4.3d, we compare directional tuning properties between three windows of time; one from the initial dynamic period (250–450ms after target presentation) and two from the sustained movement period chosen to equate duration and spacing between the three windows. In MC, the within- and across- time window comparisons show different tuning profiles between the early and sustained windows, but consistent coding in the sustained windows. Thus, MC sustained motor intent appears to consist of an initial dynamic phase and a subsequent sustained phase until the target is reached.

Figure 4.3. MC population dynamics during ballistic and sustained BCI movements.

(a) A simple example of an autonomous dynamical system: a spring-mass system.

(b) jPCA shows rotational dynamics begins similarly for ballistic and sustained movements, but freezes in place during sustained movement intent.

(c) Cross-validated neural speed is high at movement initiation and offset. During sustained movement intent, neural speed is slow.

(d) Within- and across-time regression shows changing directional coding dependent on temporal context.