PCA is an unsupervised dimensionality reduction technique, which captures the maximum variance in the data. However, it could be sometimes difficult to observe the different patterns the experiment was aiming for, because the proportion of variance contributed by the experimentally designed dependent variables might not be large enough to be captured by the first few PCs. The neural trajectory in the F5 PCA space is an example of this scenario (Fig. 3.9). The first three PCs explained more than 80% of the total variance, but the trajectories of the different conditions traveled in a same manner within the first-three-PC space. Thus visualization of the inter-conditional differences in area F5 by plotting the first three PCs is suboptimal.
Demixed principal component analysis (dPCA) conquers this problem by decomposing population activity into components based on the designed task parameters (section 2.12). In this study, the two parameters are the condition-dependent and the time-condition-dependent (condition-incondition-dependent) components (Fig.
3.10). The purpose of this technique is to “demixed” the two parameters while maintaining the maximum variance explained in the dimension reduced data. The cumulative variance explained by the demixed principal components (dPCs, red) is similar to the cumulative variance explained by the principal components (PCs, black) in all three areas, confirming the validity of dPCA (Fig. 3.10 A, D and G).
Among the three areas, area AIP had the largest proportion of variance explained by the condition-dependent components (57%, pie chart in Fig. 3.10 B). This can be seen in detail at the first three dPCs (ranked by variance explained, VE), which included two condition-dependent components (#1, 29.5% VE and #3, 11.2% VE) and only one condition-independent component (#2, 29.3%). This is congruent with the neural trajectories obtained from the PCA space, where the neural trajectory pattern in area AIP was the most diverse (Fig. 3.9). When projecting the neural data onto the decoder axes of the condition-dependent dPCs, the five conditions were well separated (Fig. 3.10 C, top row). Similar to the trajectories in PCA space, F1 + F2 (purple) was closer to F2 (red) and F2 + F3 (green) resembled more F3 (yellow).
+ +
Figure 3.10 Demixed principal component analysis of AIP, F5 and M1 population activity (opposite page)
Demixed component analysis of one recording session (Rec9). Area AIP, F5 and M1 were analyzed separately (top, middle and bottom boxes). A) Cumulative variance explained by the first 15 principal components (PCs, black) and demixed principal components (dPCs, red). B) Variance of the individual dPCs. Each bar shows the proportion of total variance, and is composed out of two stacked bars: blue for condition-dependent variance and gray for condition-independent variance. Each bar appears to be single-colored, which signifies nearly perfect demixing. Pie chart shows how the total signal variance is split between dependent and independent. C) First two condition-dependent components (top row) and first two condition-incondition-dependent components (bottom row). In each subplot, the full data are projected onto the respective dPCA decoder axis, so that there are five lines corresponding to five conditions (legend in the bottom left subplot). Trial alignment for dPCA was 250 ms after cue onset and 380 ms before hold onset. Onset of the cue and the hold epoch are marked (fixation and hold epoch in gray). Length of the hold epoch indicates 100 ms. Black lines at the bottom show time intervals during which the conditions can be reliably decoded from single-trial activity (section 2.12). Note that the vertical scale differs across areas. Ordinal number and variance explained in percentage of each component is shown on top of each subplot. D-F) same as A-C for area F5. G-I) same as A-C for area M1.
This was observed in the first two condition-dependent components. The first condition-independent component (Component #2) was modulated by movement and time, while the second one (Component #4) also showed mild visuomotor transition in the early cue epoch (Fig. 3.10 C, bottom row).
The variance explained by the condition-dependent components in area F5 (19%) was similar to two other studies of delayed grasping with precision and power grips (in ‘t Veld, 2016; Michaels and Scherberger, 2018), while this percentage in area AIP (57%) and area M1 (48%) were much higher than from the grasping studies (Fig. 3.10 B, E and H). The first two condition-dependent components in area F5 were Component #3 and Component #5 with 4.8% and 3.7% VE, respectively. However, conditions can be decoded from these two dPCs from the late cue epoch to the end of the trial (Fig. 3.10 F, top row). The first condition-independent component was the largest component (59.1%), modulated strongly by the movement.
Area M1 had similar proportion of variance explained by the condition-dependent and the independent components (Fig. 3.10 H). In the first condition-dependent component (Component #2), the F1 (blue) and F2 (red) lines reached their maximum absolute value (normalized firing rate) shortly before the hold epoch and decreased during the hold epoch, in contrast to area AIP, where the signal was maintained till the end of the trial (Fig. 3.10 C and I). This can also be observed from
the neural trajectories in PCA space where the AIP endpoints were more spread out than the M1 endpoints (Fig. 3.9).
In summary, both PCA and dPCA can be used to visualize the population firing pattern of simultaneously recorded neurons. The differences among area AIP, F5 and M1 can be observed with PCA, and with the first four dPCs in Fig. 3.10, patterns similar to the trajectories in PCA space appeared, since PCA and dPCA are related methods. The advantage of dPCA is to find variance explained by the experimentally designed parameters. This is often important because the population firing rate differences between conditions could be relatively small, compared to common firing rate changes in the trial, for example, during movements. In this study, this is the case for area F5, where the separation in PCA space is suboptimal, while the condition-dependent components obtained from dPCA can clearly show the condition-dependent changes of the population firing rates over time.