intended velocity ( u k arm movements

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1 Fig. A Complete Brain-Machine Interface B Human Subjects Closed-Loop Simulator ensemble action potentials (n k ) ensemble action potentials (n k ) primary motor cortex simulated primary motor cortex neuroprosthetic device intended velocity (u k ) neuroprosthetic device neural control network device kinematics (x k ) kinect arm movements simulated neural control network user C Synthetic Subjects Closed Loop Simulator ensemble action potentials (n k ) device kinematics (x k ) simulated primary motor cortex neuroprosthetic device linear quadratic controller simulated neural control network device kinematics (x k ) Figure. Closed-loop brain-machine interface (BMI) operation in practice and with two models. (A) Actual BMI system. The subject controls the BMI through an output layer with tens of primary motor cortical neurons, driven by inputs from a larger neural control network, with various recurrent connections. (B) Model system for closed-loop BMI operation based on human subjects. Here, the neural control network is represented by a healthy human subject, observing on-screen cursor kinematics, and adjusting arm movements captured by the Kinect, where arm velocity in the plane orthogonal to the camera represents intended velocity ( u k ). An empirically-derived cosine-tuned point process model of motor cortical neurons converts intended velocity into spiking events from 25 neurons. Actual and decoder-estimated neural parameter values are redrawn at the beginning of every learning period. (C) Model system for closed-loop BMI operation based on a synthetic subject implemented by a linear quadratic controller, modified from the recently described original stochastic optimal control model for closed-loop BMI operation (Lagang & Srinivasan, 23).

2 Fig. 2 Velocity Prediction Density Previous Decoded Velocity Decoded Velocity ReFIT Variants Random Walk Static Estimation Procedure Lockstep Joint Joint Partial. estimate x k+ k+. estimate x k+ k+ Prior on Intended Velocity 2. estimate Θ k+ k+ Feedback to User Figure 2. Naive adaptive control variants with directed and undirected priors. The ReFIT variants,, RW and Static training methods each differ in three elemental ways, as listed in the row labels: joint vs. lockstep estimation, prior on intended velocity (also called the state equation or latent variable model), and the control of visual feedback to the user (cursor movement). The RW uses an undirected prior, where ReFIT-PPF and use different directed priors, as defined in Results. The various training paradigms are explained in detail under Methods.

3 Fig. 3 Velocity Prediction Density Previous Decoded Velocity Decoded Velocity Lockstep RSE/RSE Lockstep RSE/RW Estimation Procedure Joint Lockstep Lockstep. estimate x k+ k+. estimate x k+ k+ Prior on Intended Velocity 2. estimate Θ k+ k+ 2. estimate Θ k+ k+ Feedback to User Figure 3. Naive adaptive control variants to dissect the relative importance of joint estimation versus sensory feedback. To understand the relative contribution of joint estimation and feedback to improved naive adaptive control with, we constructed two control methods. Lockstep RSE/RSE is nearly identical to except that lockstep estimation is used. Lockstep RSE/RW differs from in the use of lockstep estimation, and the determination of cursor movement by a random walk prior (rather than the reach state equation). In its control of feedback (cursor movement), the Lockstep RSE/RW is identical to ReFIT-PPF.

4 Fig. 4 Sample Training Session A Before Training Random Walk Sample Trajectories 2 cm x Velocity (cm/s) y Velocity (cm/s) Sample Trajectory x Velocity Sample Trajectory y Velocity Time (s) B After Training Random Walk Sample Trajectories 2 cm x Velocity (cm/s) y Velocity (cm/s) Sample Trajectory x Velocity Sample Trajectory y Velocity Time (s) C Preferred Direction Estimates ReFIT-PPF Preferred Direction Estimates Random Walk Preferred Direction Estimates Initial Value Final Value True Value Initial Value Final Value True Value Initial Value Final Value True Value Figure 4. Single-learning-session examples of performance under naive adaptive control with directed and undirected priors. (A), (B) Sample trajectories and corresponding velocity profiles (A) early in the training session and (B) late in the training session. (C) Estimates of neuron preferred direction converge to true values with directed priors (ReFIT-PPF, ), but not with undirected priors (RW) on this single learning session. Trajectories result from 25 simulated neurons and 33 ms bin width.

5 Fig. 5A Changes in with Different Types of Naive Adaptive Control Random Walk Figure 5. Success rate and effects of modifications on naïve adaptive control. (A) Changes in success rate with naive adaptive control. Success rates and 95% confidence intervals on success rate were determined for the RW, ReFIT-PPF, and methods using a Bayesian procedure designed for the specific purpose of estimating learning curves (Smith et al., 24). Four subjects participated in 2 learning sessions per method, so each data point is determined by the pooled successes and failures of 48 trials. Black and brown bars drawn near the x -axis represent alternating segments of 4 training trials (black) and test trial (brown). The test trial point is extrapolated from RW performance at test trial.

6 Fig. 5B Effect of Feedback on Naive Adaptive Control Lockstep RSE/RSE Lockstep RSE/RW Figure 5. Success rate and effects of modifications on naïve adaptive control. (B) Effect of feedback on naive adaptive control. Success rates were not significantly different in comparison between Lockstep RSE/RSE and Lockstep RSE/RW methods, which are nearly identical methods, except in the way they apply feedback (cursor control). Lockstep RSE/RSE and feedback methods are identical. Lockstep RSE/RW and ReFIT-PPF feedback methods are identical. Success rates and error bars were determined as in (A). Three new subjects (different from (A)) each participated in 2 learning sessions per method, so each point is determined by the pooled successes and failures of 36 trials. Conventions are unchanged from (A).

7 Fig. 5C Effect of Joint Estimation on Naive Adaptive Control Lockstep RSE/RSE Figure 5. Success rate and effects of modifications on naïve adaptive control. (C) Effect of joint estimation on naive adaptive control. Success rates were significantly different in comparison between and Lockstep RSE/RSE methods, which are nearly identical methods, except in the use of joint versus lockstep estimation, respectively. Success rates and error bars were determined as in (A). Three new subjects (different from (A) or (B)) each participated in 2 learning sessions per method, so each point is determined by the pooled successes and failures of 36 trials. Conventions are unchanged from Figure (A).

8 Fig. 6 A Timescale of Performance Improvements Random Walk B 5 Timescale of Human Adaptation Heading Deviation in Initial Arm Movement (deg) C Deviation from Initial Estimated Preferred Direction Parameter (deg) Timescale of Machine Adaptation Figure 6. Timescales of human learning, machine learning, and BMI performance. (A) Heading deviation as a surrogate for human sensorimotor learning. Heading deviation is the minimum subtended angle between the subject's intended velocity and the straight-line trajectory to target. (B) Changes in estimated preferred direction as a surrogate for machine adaptation. Deviation from initial estimated preferred direction parameter is the minimum subtended angle between the initial estimated preferred direction and the current estimated preferred direction, averaged over all neurons. (C) Success rate, as plotted in Figure 5A, reprinted here for comparison, with timescales of (A) human and (B) machine adaptation. Subjects, trial numbers, and other conventions are unchanged from Figure 5A.

9 Fig. 7 Sample Training Sessions 8/25 Random Neurons A Before Training ReFIT-PPF Static 2 cm 2 cm 2 cm B After Training ReFIT-PPF Static 2 cm 2 cm 2 cm Figure 7. Sample trajectories with the modified human simulator using visuomotor rotation. Position trajectories for, ReFIT-PPF, and Static are plotted (A) before and (B) after training in one new subject. Qualitatively, trajectories appear smoother and more directed than ReFIT-PPF and Static BMI following training. Trials begin at random positions on the outer perimeter with the target at the center. Plotted trajectories have been rotated to start at the top of the perimeter for ease of visual comparison.

10 Fig. 8 A 8/25 Random Neurons Timescale of Performance Improvements Static B 5 Timescale of Human Adaptation Heading Deviation in Initial Arm Movement (deg) C Deviation from Initial Estimated Preferred Direction Parameter (deg) Timescale of Machine Adaptation Figure 8. Timescales of learning for, ReFIT-PPF, and Static under the modified simulator using visuomotor rotation. These curves recapitulate the analysis in Figure 6 under the modified conditions to permit human learning over single learning sessions. Decoder neural parameters were initialized randomly for 8 of 25 neurons, and to pure rotation of preferred direction in the rest. Data is aggregated from 2 new human subjects in the simulator, for a total of 6 learning session per technique.

11 Fig. 9 A 8/25 Random Neurons Timescale of Performance Improvements Lockstep RSE/RSE Lockstep RSE/RW B 5 Timescale of Human Adaptation Heading Deviation in Initial Arm Movement (deg) C Deviation from Initial Estimated Preferred Direction Parameter (deg) Timescale of Machine Adaptation Figure 9. Timescales of learning for Joint versus Lockstep RSE methods, under the modified simulator using visuomotor rotation. These curves recapitulate the analysis in Figure 8 under conditions that permit human learning within a single learning session. As with Figure 8, decoder neural parameters were initialized randomly for 8 of 25 neurons, and to pure rotation of preferred direction in the rest. Data is from the same 2 human subjects used in Figure 8, for a total of 6 learning session per technique.

12 Time To (s) Final Distance to (cm) Final Distance to (cm) Final Distance to (cm) Final Distance to (cm) Final Distance to (cm) Fig. A MID to (cm) MID to (cm) MID to (cm) (i) Mean Integrated Distance to (iii) Mean Integrated Distance to (v) Mean Integrated Distance to (vii) Time to Lockstep RSE/RSE Lockstep RSE/RW (ii) Trajectory Inaccuracy (iv) Trajectory Inaccuracy (vi) Trajectory Inaccuracy Random Walk Lockstep RSE/RSE Static 8/25 Random Neurons B MID to (cm) 3 2 (i) Mean Integrated Distance to (ii) Trajectory Inaccuracy MID to (cm) 3 2 (iii) Mean Integrated Distance to (iv) Trajectory Inaccuracy Lockstep RSE/RSE Lockstep RSE/RW Figure. Other metrics of performance. (A) Mean integrated distance to target (i, iii, v), trajectory inaccuracy (ii, iv, vi), and time to target (vii) as defined in the text, using 4 subjects and conditions from Figure 5. (B) These measures, using subjects and conditions from Figures 8-9.

13 Fig. A Synthetic Subjects Closed-Loop Simulator Sensorimotor Delay = ms Random Walk B Sensorimotor Delay = 267 ms C Sensorimotor Delay = 33 ms Figure. Effect of sensorimotor delay assessed with synthetic subject closed-loop simulator. In contrast to prior analyses, this analysis uses a linear quadratic controller in place of the human subjects, adapted from prior theoretical work (Lagang & Srinivasan, 23). Performance for the, ReFIT-PPF, and RW are compared under (A) zero delay, (B) 267 ms delay, and (C) 33 ms somatosensory delay. Specifically, output neural activity reflects on-screen cursor state from time into the past equal to the specified delay. Sensorimotor delay is the counterpart to delay studied elsewhere (Golub et al., 22) that is introduced by the BMI algorithm itself.

ROSE-HULMAN INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering. Mini-project 3 Tennis ball launcher

ROSE-HULMAN INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering. Mini-project 3 Tennis ball launcher Mini-project 3 Tennis ball launcher Mini-Project 3 requires you to use MATLAB to model the trajectory of a tennis ball being shot from a tennis ball launcher to a player. The tennis ball trajectory model