Fig 1.
Simulation of a velocity-tuning based model with variable neuron-kinematic latencies.
(a) Task design of a 13-direction center-out reaching task. The firing of a simulated neuron is plotted around the reach directions. (b) Two example neurons with differing latencies. (c) Principal components (PCs) for a simulated population of 200 neurons (latency SD: 72 ms, movement SD: 56 ms). (d) Exemplar jPCA plane for the first 6 PCs of the simulated population from 0 ms before to 200 ms after neural movement onset (analysis was computed on entire movement). Individual conditions are colored based on their activity at neural movement onset in the first jPC. (e) Proportion of change in neural trajectory explained by rotational dynamics (in all jPCA planes) for various latency offsets and movement durations. A value of 1 indicates that rotational dynamics completely explain the transformation between each time point and its temporal derivative.
Fig 2.
Comparing rotational structure between the representational and the dynamical models.
(a) One of the two oscillatory modes (2.8 Hz) used to generate the simulated muscle activity of all conditions (2.8 Hz and 0.3 Hz). (b) Firing rate of an example neuron of the dynamical model for all 13 conditions. Each neuron is generated from a random combination of the two underlying oscillatory modes and offset for each condition. (c) Rotational dynamics in the first jPCA plane for the observed data. p-value shows results of CMPT for the representational and dynamical models evaluated by the rotational goodness-of-fit ratio (RGR: ). (d) Same as c, but for permuted data without covariance matching. (e) Same as c, but for covariance-matched data. Data is plotted for 200 ms regardless of time period used to generate statistics. Colors are based on the preparatory activity in the first jPC.
Fig 3.
Schematic of recurrent neural network performing center-out reaching.
(a) Schematic of RNN, with input layer, hidden layer, and output layer. The three inputs were a condition-independent hold signal that was released at the go cue and two inputs representing the target angle. The two outputs were a linear combination of the internal neurons and read out velocity in the x and y direction. All weights were modified during training. The network received no feedback from the output layer. (b) Output velocity profiles produced by the RNN compared with target velocity used in training. The normalized error was less than 0.1%. (c) Simulated kinematics produced by integrating the velocity profiles over time, with corresponding targets for illustration.
Fig 4.
Tuning properties of RNN neurons.
(a) Three example units for which the pattern of directional tuning remained highly correlated between the delay period and movement. (b) Same as a, but for example units that have delay tuning that is not correlated with movement activity. (c) Same as a, but for example units that invert their tuning between delay and movement. (d) Preferred reach direction (highest firing) of all 200 units, sorted by preferred direction at go cue. If there was no firing rate difference (< 1e-4) between the preferred direction and non-preferred direction, units were deemed un-tuned and are marked in white. Firing rates are displayed from 0 to the maximum firing rate of each neuron.
Fig 5.
Representational tuning in an RNN for center-out reaching.
(a) Preferred movement direction in Cartesian space of all units, corresponding to the magnitude of bi,2 and bi,3 in Eq 9. (b) Summary of contribution vectors of all individual neurons (one vector each) over the entire movement, with black population vector showing the overall predicted movement direction. (c) Integrating the population vectors in panel b over time traces out a predicted trajectory (solid) that largely matches the actual trajectory (translucent). (d) Mean correlation between condition tuning order at neural movement onset compared to later time points during movement (in steps of 10 ms) for the RNN model and an example PMd/M1 data set presented in Churchland et al. [22]. Insets show full correlation histograms for two time points. (e) Adjusted R-Square obtained by regressing the activity of each neuron (from the go cue to the end of movement 300 ms after go) on a representational cosine model of velocity tuning (Methods). (f) Movement activity of three example neurons and the corresponding velocity based regression fits. The overall fit performance to these units is high (Adjusted R-Square above 0.8), but the regression fails to capture the multiphasic and varied nature of the underlying signal. (g) Time lag between neural activity and velocity, per neuron, obtained from the velocity tuning regression in panel e, showing a large range of values.
Fig 6.
Significant rotational structure in PMd/M1 data and RNN model.
Comparison of rotational dynamics for (a) observed, (b) permuted without covariance matching, and (c) covariance-matched data in the first jPCA plane. p-values in a are from the CMPT for the rotational goodness-of-fit ratio (RGR: ) in all jPCA planes. Conditions and neurons were randomly down-sampled in the PMd/M1 data to match the RNN model. Data is plotted for 200 ms regardless of time period used to generate statistics. Colors are based on the preparatory activity in the first jPC.
Fig 7.
Number of neurons and conditions required for statistically significant rotations.
The CMPT was carried out (500 repetitions) for many subsets of example PMd/M1 data including from 10–120 neurons and 2–20 conditions. (a) Map of p-values for the rotational goodness-of-fit ratio (RGR: ). (b) Map of effect size (difference between observed RGR and mean of permuted distribution, divided by the SD of the permuted distribution). For every permutation, random neurons and conditions were drawn from the example set. Contours show the 0.05 and 0.01 significance levels.