The Neural Code for Motor Control in the Cerebellum and Oculomotor Brainstem

Example of an experimental trial and of neural and behavioral variation. A, Superimposed eye and target position for a step-ramp target motion. Dashed and continuous traces show target and eye motion. B, Eye velocity responses during the initiation of pursuit for recordings from a Purkinje cell. Gray and black traces show individual trial responses and the mean response. C, Raster from a typical Purkinje cell during pursuit initiation. Each line shows the response in one behavioral trial, and each tick shows the time of one simple-spike. D, Superimposed eye and target velocity for a step-ramp target motion. Dashed and continuous traces show eye and target motion. E, F, Distributions across the full sample of Purkinje cells of the correlation between actual and predicted magnitudes of the first (E) or second (F) principal component (PC1 and PC2, respectively) of the behavior. G, First two principal components resulting from dimension reduction of the variation in eye velocity at the initiation of pursuit. H, The amount of total behavioral variance explained by leading principal components for the variation in eye speed at the initiation of pursuit. I, J, Each trace shows the weights for an optimized linear estimator ( in Eq. 2) relating the presence of an extra spike at time t on the horizontal axis to the amplitude of the first (I) or second (J) principal component of eye movement variation. Different colors show results for a single Purkinje cell during pursuit at three different target speeds. All times are relative to the onset of target motion.

Prediction of eye speed by a single extra spike in four groups of cerebellar and brainstem neurons. A, B, Superimposed means of eye and target velocity for a typical recording session (A) and average firing rates (B) for the different brainstem neurons. Different colors show the mean firing rate across the different populations, and the gray ribbons show the SEM. C, Distributions across the full samples of different neuron classes of the correlation between actual and predicted magnitudes of the first principal component (PC1) of the pursuit behavior. D, Weight of spikes as a function of time for the best linear estimator. In B, C, and D, black, red, green, and blue show data for floccular Purkinje cells, FTNs, non-FTN vestibular neurons, and abducens neurons. The traces in B were taken from a figure in an earlier paper (Joshua et al., 2013).

Comparison of the predictive value of a single extra spike in real versus simulated spike trains for a floccular Purkinje cell and an FTN. Results for Purkinje cells are in black and FTNs are in red. A, D, Predicted versus actual amplitude of the first principal component (PC) of pursuit behavior based on the optimal linear estimator from the real spike trains. B, E, Predicted versus actual amplitude of the first principal component of pursuit behavior based on the optimal linear estimator from the simulated spike trains. In all four graphs, each symbol shows the results from a single behavioral trial. C, F, Each trace shows the weights for an optimized linear decoder ( in Eq. 2) relating the presence of an extra cell spike at time t on the horizontal axis to the amplitude of the first principal component of eye movement variation. Dark and light curves show results for real data versus simulated spike trains.

Similarity of the predictive value of a single extra spike for real and simulated spike trains across all neurons in our sample. A, B, Each symbol summarizes the results for a single neuron and plots the correlation between the optimal predictor based on simulated spike trains against the correlation based on real spike trains. C, D, Average weights in the optimal linear estimators for the four groups of neurons, plotted as a function of the time of a spike. Continuous and dashed curves show weights based on the real and simulated spike trains, respectively. Throughout the figure, black, red, green, and blue indicate analyses for floccular Purkinje cells, FTNs, non-FTN vestibular neurons, and abducens neurons, respectively.

Effect of discharge variability and overall signal magnitude on the predictive value of a spike train. A, B, Each graph plots the mean and standard deviation of the correlation between the real and predicted magnitude of the first principal component of eye speed based on simulated spike trains. Data are plotted as a function of the coefficient of variation (A) or the gain (B) of the artificial spike train. Red and blue symbols indicate analysis of simulations based on a representative FTN or abducens neuron. C−F, Scatter plots, where each symbol shows the results of analyzing the data of a single neuron, and the different plots show the correlation between the real and predicted magnitude of the first principal component as a function of the coefficient of variation of the spike trains (D), the amplitude of the neural signal (E), and the ratio of the signal amplitude divided by the coefficient of variation (C, F). Different colors show results for different classes of neurons.

Demonstration that computing the best linear estimator with finite datasets causes over-fitting and underestimates the predictive value of a spike train. Each row shows data for a different set of neurons. Left, Each graph contains one symbol for each neuron and shows scatter plots of the correlation of actual and predicted magnitude of the first principal component of the data. Each point plots the correlation obtained using the same number of trials available in the real data on the y-axis and the correlation obtained using 1000 trials of simulated data on the x-axis. Middle and right, Each graph plots the time course of the optimal linear estimator when we used the actual data (middle) or 1000 simulated trials (right). The black curve in each graph shows the average across all neurons and the gray and colored traces show the individual neurons.

Prediction of eye speed by spike counts in windows of different duration. Each graph plots the correlation between the output of different estimators and the amplitude of the first principal component of eye speed. Pink swatches are the lower bounds for the best linear estimator (Eq. 2). Black lines are from the spike count in windows of different duration centered 125 ms after the onset of target motion. The red line labeled “1000 trials” is the estimate of the upper bound of the results from the best linear estimator. Different graphs show data for different neuron types.

Prediction of eye speed by spike counts in floccular Purkinje cells. A, Correlation between actual and predicted amplitude of the first principal component of eye speed as a function of the duration and time of the analysis interval. Different colored traces show results for analysis intervals of different durations. Pink band labeled “Linear” shows the correlation obtained with the best linear estimator. Red line labeled “1000 trials” shows the expected correlation for the best linear estimator if we had acquired 1000 repetitions of the target motion. B, Average firing rate of the population of Purkinje cells as a function of time. Gray bar shows a 20 ms analysis window centered at 140 ms. The gray bar indicates the time of the peak correlation with the first principal component (PC1) in A. C, Number of spike counts in a 5 ms analysis interval as a function of the center of the interval. Different color traces show numbers for different spike counts. D, Averages of eye velocity as a function of the time from onset of target motion for trials in one example Purkinje cell. Different colors show eye velocities for trials with three, four, five, or six spikes in the 41 ms interval centered 140 ms after the onset of target motion. E, Scatter plot showing the z-scored amplitude of the first principle component of eye speed as a function of the number of spikes in the same interval as D. Each symbol shows data from one trial. The regression line defines the sensitivity in units of standard deviations per spike. R = 0.53 for the symbols and the regression equation was y = 0.72x − 3.18. F, Distributions of sensitivity for populations of floccular Purkinje cells, FTN, and abducens neurons.

Comparison of predictions made by spike counts over 40 ms in time versus space. Each panel shows data for a different group of neurons. Within each panel, the traces show the correlation between the actual and predicted amplitude of the first principal component of eye speed as a function of the time at the center of the analysis window. Blue traces show the results for counting spikes in a 40 ms window; the traces end at 180 ms because of the width of the analysis window. Green traces show the result for counting spikes in a 1 ms window across the simulated spike trains of 40 different neurons.