Motor cortex signals corresponding to the two arms are shared across hemispheres, mixed among neurons, yet partitioned within the population response

Primary motor cortex (M1) has lateralized outputs, yet M1 neurons can be active during movements of either arm. What is the nature and role of activity in the two hemispheres? When one arm moves, are the contralateral and ipsilateral cortices performing similar or different computations? When both hemispheres are active, how does the brain avoid moving the “wrong” arm? We recorded muscle and neural activity bilaterally while two male monkeys (Macaca mulatta) performed a cycling task with one or the other arm. Neurons in both hemispheres were active during movements of either arm. Yet response patterns were arm-dependent, raising two possibilities. First, the nature of neural signals may differ (e.g., be high versus low-level) depending on whether the ipsilateral or contralateral arm is used. Second, the same population-level signals may be present regardless of the arm being used, but be reflected differently at the individual-neuron level. The data supported this second hypothesis. Muscle activity could be predicted by neural activity in either hemisphere. More broadly, we failed to find signals unique to the hemisphere contralateral to the moving arm. Yet if the same signals are shared across hemispheres, how do they avoid impacting the wrong arm? We found that activity related to the two arms occupied distinct, orthogonal subspaces of population activity. As a consequence, a linear decode of contralateral muscle activity naturally ignored signals related to the ipsilateral arm. Thus, information regarding the two arms is shared across hemispheres and neurons, but partitioned at the population level.


Introduction 20
The outputs of motor cortex (M1) are lateralized: most spinal projections influence the 21 contralateral musculature. M1 lesions thus produce contralateral motor deficits (Liu and Rouiller, 22 1999;Murata et al., 2008;Passingham et al., 1983;Vilensky and Gilman, 2002). Similarly, 23 electrical microstimulation activates contralateral musculature (Kwan et al., 1978;Sessle and 24 Wiesendanger, 1982). The degree to which computations within M1 are lateralized versus shared 25 across hemispheres remains less clear. The corpus callosum interconnects M1 across 26 hemispheres, yielding the potential for extensive cooperation (Gould et al., 1986;Jenny, 1979;27 Jones and Wise, 1977). Callosally mediated interactions are readily revealed by paired-pulse 28 TMS protocols and can involve net facilitation or suppression (Ferbert et al., 1992;Hanajima et 29 al., 2001;Meyer et al., 1995). An obvious role for inter-hemispheric cooperation is coordination 30 of bimanual movement (Donchin et al., 1998;Kermadi et al., 1998). Yet there is evidence that 31 unimanual movements also involve sharing of information across hemispheres. 32 Most physiological studies of unimanual movements have focused on activity contralateral to the 33 moving limb, on the grounds that contralateral activity is most functionally relevant and likely to 34 be most prevalent. Yet studies investigating ipsilateral activity have found that it can be robust. 35 Ipsilateral activity is minimal for tasks performed primarily with the digits (Matsunami and 36 Hamada, 1981;Tanji et al., 1988;Aizawa et al., 1990) but prevalent during movements of the 37 upper arm, such as reaching to remove food from a drawer (Kermadi et al., 1998;Kazennikov et 38 al., 1999), or performing center-out reaching movements Steinberg et al., 39 2002;Cisek et al., 2003;Ganguly et al., 2009). 40 While the presence of ipsilateral activity is established, the nature of that activity is less clear. 41 Few studies have directly compared neural response patterns when the same movement is 42 performed by one arm versus the other. In premotor areas, delay-period responses can encode 43 information about an upcoming reach (Cisek et al., 2003) or grasp (Michaels and Scherberger,44 2018) independently of which arm would subsequently move, suggesting that preparatory 45 activity is largely effector independent. However, activity during movement was more effector-46 dependent, for both premotor cortex and M1 (Cisek et al., 2003). Steinberg and colleagues 47 (2002) reported similar single-neuron directional tuning regardless of which arm was moving, 48 yet also found evidence for effector-dependent population-level encoding of direction. 49 If responses are effector-independent (i.e. similar regardless of the moving arm) then the 50 relationship between hemispheres is necessarily simple: both contain the same information, 51 encoded in the same manner. In contrast, if there exist strongly effector-dependent responses, 52 that would raise additional questions. Are 'lower-level' signals (e.g., those describing muscle 53 activity) more prevalent in the contralateral hemisphere? More generally, which signals are 54 shared across hemispheres? How is neural activity structured such that only one arm moves even 55 if both hemispheres are active? 56 We investigated these questions using a novel 'cycling' task, performed with either the left or 57 right arm. We recorded neural activity from both hemispheres simultaneously. In separate 58 sessions we recorded muscle activity bilaterally. Single neurons responded robustly regardless of 59 which arm performed the task. Yet responses were strongly effector-dependent: for a given 60 neuron, response patterns pertaining to the two arms were essentially unrelated. Despite 61 profoundly effector-dependent single-neuron responses, we found no evidence that certain 62 signals were present in one hemisphere but not the other. For example, muscle activity could be 63 decoded equally well from contralateral or ipsilateral neural activity. More broadly, the 64 population response across hemispheres appeared isomorphic; any signal present in one could 65 also be found in the other. Thus, activity in a given hemisphere contains similar information 66 during movement of one arm versus the other, yet that information is distributed very differently 67 across single neurons. This might appear to yield a paradox: how can M1 be robustly active 68 without driving the contralateral arm? A solution emerged when we examined the correlation 69 structure between neurons, which changed dramatically depending on which arm was used. As a 70 result, arm-specific signals were partitioned into orthogonal dimensions, allowing a simple 71 decoder to naturally separate signals related to the two arms. 72 73

Materials and Methods 74
Terminology 75 We adopt the following terminology. For neurons in a given hemisphere, we refer to the 76 contralateral arm as the "driven arm" (reflecting the strong connections to the contralateral spinal 77 cord). We refer to the ipsilateral arm as the "non-driven arm." Thus, for a neuron recorded from 78 the right hemisphere, the left arm is the driven arm and the right arm is the non-driven arm. For 79 the muscles, the driven arm is the arm upon which the muscle acts. Similarly, for a given arm, 80 the contralateral cortex is referred to as the "driving cortex," while the ipsilateral cortex is 81 referred to as the "non-driving cortex." 82 Behavior 83 All animal procedures were approved by the Columbia University Institutional Animal Care and 84 Use Committee. Data were collected from two male monkeys (Macaca mulatta) while they 85 performed a cycling task for juice reward. Experiments were controlled and data collected under 86 computer control (Real-time Target Machine: Speedgoat, Liebfeld, Switzerland). While 87 performing the task, each monkey sat in a custom primate chair with the head restrained via 88 surgical implant. A screen displayed a virtual environment through which the monkey moved. 89 The monkey grasped a custom pedal with each hand, with the hands lightly restrained with tape 90 to keep them in a consistent position on the pedals. The pedal itself was also designed to 91 encourage a consistent hand position, and included a handle and a brace that reduced wrist 92 motion. Pedals turned a crank-shaft attached to a motor (Applied Motion Products, Watsonville, 93 California, USA). A rotary encoder within the motor reported position with 1/8000 cycle 94 precision. The motor used information regarding angular position and its derivatives to provide 95 forces yielding virtual mass and viscosity. 96 Monkeys cycled the pedal to control their position in the virtual environment ( Figure 1A). For a 97 block of twenty consecutive trials, one arm was the "performing arm," and the other was the 98 "non-performing arm." The angular position of the performing arm's pedal was mapped directly 99 to linear position in the virtual world. The non-performing arm's pedal was required to remain 100 within a window (± 0.05 and ± 0.07 cycles for monkey E and F) centered at the bottom of the 101

EMG recording 159
On a separate set of days from the neural recordings, we recorded intramuscular EMG signals 160 from the following muscles: Biceps brachii (long and short head), triceps brachii (medial, long, 161 and lateral heads), deltoid (anterior, lateral, and posterior head), latissimus dorsi, pectoralis, 162 trapezius, and brachioradialis. Pairs of hook-wire electrodes were inserted ~1cm into the belly of 163 the muscle being recorded at the beginning of each session and removed at the end of the 164 session. On each session, 1-3 EMG recordings were made per arm. Electrode voltages were 165 amplified, bandpass filtered (10-500 Hz) and digitized at 1000 Hz (Monkey E) or 30000 Hz 166 (Monkey F). Recordings were not considered further if they contained significant movement 167 artifact or weak signals. Offline, EMG records were high-pass filtered at 40 Hz, rectified, and 168 smoothed with a 25-ms Gaussian. This produced a measure of intensity versus time, which was 169 then averaged across trials. 170

Trial Averaged Firing Rates 171
The spike times of each neuron on each trial were converted to a firing-rate by convolving spikes 172 with a 25-ms Gaussian. To produce trial-averaged firing rates, we first aligned all trials on a 173 common time: the moment when the first half-cycle was completed. This nicely aligned behavior 174 across trials during the first cycle of each trial. However, because trials lasted multiple seconds, 175 small differences in cycling speed could accumulate and cause considerable misalignment of 176 behavior across trials ( Figure 2A). The resulting misalignment of spikes ( Figure 2B) would, if 177 averaged without further alignment, yield an unrepresentative average firing rate (the same 178 problem would impact averages of muscle activity and kinematic variables). Thus, the time-base 179 on each trial was adjusted such that each cycle lasted 500 ms (excluding the first and last half-180 cycles), matching the typical 2 Hz cycling speed ( Figure 2C). This procedure altered the time-181 base of individual trials very modestly, yet maintained appropriate alignment across trials 182 ( Figure 2D), and produced trial-averaged estimates of the firing rate ( Figure 2E) that are 183 representative of what occurred on single trials. 184 Figure 2D shows spike rasters to illustrate improved alignment of neural activity. However, we 185 stress that spikes were always converted to rates before modification of the time-base. Thus, the 186 alignment procedure did not alter the values of the estimate of firing rate; it simply slightly 187 modified when those values occurred. 188 Most analyses of firing rates employed the middle cycles (2-5), excluding the first cycle and the 189 last two cycles. This focused analysis on the steady-state response, rather than on responses 190 associated with starting, stopping, or holding. This aided interpretation in two ways. First,191 muscle activity in the non-performing arm was particularly weak during middle cycles (in 192 contrast, modest activity was occasionally observed when stopping). Focusing on middle cycles 193 largely sidesteps concerns that activity ipsilateral to the performing arm is related to muscle 194 activity in the non-performing arm. Second, we wished to focus key analyses on the rhythmic 195 pattern of firing rate modulation, rather than on overall changes in net firing rate when moving 196 versus not moving. As one example, when predicting muscle activity from neural activity, it is 197 relatively 'easy' to capture the generally elevated activity level during movement, resulting in 198 high " values even if predictions fail to capture cycle-by-cycle activity patterns. We wished to 199 avoid this, and to consider predictions successful only if they accounted for rhythmic response 200 aspects. 201

Single-neuron analyses 202
We wished to compare, for each neuron, the strength of modulation when the driven versus non-203 driven arm performed the task. By modulation, we mean the degree to which a neuron's firing 204 rate varied within cycles, between cycles, and/or between conditions (forward versus backward, 205 and top-start versus bottom-start). We compiled a single firing rate vector, $%&'() , concatenating 206 the firing rate vectors across the four conditions where the driven arm performed the task. 207 $%&'() was thus of size where is the number of conditions and is the number of times 208 during the middle cycles of one condition. We defined $%&'() as the standard 209 deviation of $%&'() , which captures the degree to which the average firing rate varies across 210 time and condition.
)4)5$%&'() was computed analogously. 211 To assess the degree to which a neuron was more strongly modulated when the driven versus 212 non-driven arm performed the task, we computed an arm preference index:

214
This arm preference index is zero if a neuron is equally modulated regardless of the arm used, 215 approaches one if modulation is much larger when using the driven arm, and approaches 216 negative one if modulation is much larger when using the non-driven arm. 217 Firing-rate impact of small movements of the non-performing arm 218 We wished to control for the possibility that neural responses, when the non-driven arm performs 219 the task, might be related to small movements of the non-performing arm. For each trial, we 220 computed the mean (absolute) speed of the non-performing arm. For each condition, we divided 221 trials into those with speeds greater versus slower than the median. We did not apply this 222 analysis if there were fewer than 8 trials for that condition. This could occur if a neuron was well 223 isolated for only part of a recording session. 224 After dividing, we recomputed the mean firing rate for each of the two pools of trials, yielding 225 one firing rate when the non-performing arm moved modestly, and another when it was virtually 226 stationary. For each timepoint, we asked whether these two firing rates were more different than 227 expected given sampling error. This was accomplished via a bootstrap in which trials were 228 divided randomly, rather than based on speed. We performed 1000 such random divisions. 229 Differences were considered significant if they were larger than for 95% of the random divisions. 230

Normalization 231
Because the absolute voltages of EMG traces are largely arbitrary, the scale of muscle activity 232 could be quite different for different muscles. The response of each muscle was therefore 233 normalized by its range. Neural responses were left un-normalized for single-neuron level 234 analyses. However, for population-level analyses, responses were normalized to prevent results 235 from being overly biased toward the properties of a few high-rate neurons. produces 'soft' normalization, and ensures that we don't magnify the activity of very low-rate 242 neurons. We have used this value previously (e.g., Lara et al. 2018 eLife;Russo et al. 2018 243 Neuron) as it strikes a reasonable balance between focusing analysis on all neurons, while still 244 allowing high firing-rate neurons to contribute somewhat more than very low-rate neurons. 245

Population Predictions 246
To predict muscle activity from neural activity, we used Partial Least Squares (PLS) regression 247 (plsregress in MATLAB). For each set of neurons and muscles , PLS regression finds 248 matrices , , that maximize the covariance between and , under the constraint that , 249 are of rank (which must be specified). PLS is similar to Canonical Correlation Analysis, in 250 that it seeks linear transformations of the data that maximize similarity. However, Canonical 251 Correlation Analysis maximizes correlation, and can therefore often be biased toward small 252 dimensions which are coincidentally well-correlated. In contrast, PLS regression maximizes 253 covariance and thus seeks correlated signals that are also high variance. Once is found, is 254 predicted from via standard linear regression. Employing (which has only columns of 255 regressors) rather than (which has hundreds of columns) greatly reduces overfitting. An 256 advantage of PLS regression is that the regularized solution respects not only the correlations in 257 (as for PCA regression) but also the correlations in . 258 All predictions involved the middle cycles (2-5) of movement. We first picked one behavioral 259 condition (e.g. top-start, forward, right hand performing) as our test condition. We then set the 260 training condition to be the behavior with the same cycling direction but the opposite starting 261 pedal position (e.g. bottom-start, forward, right hand performing). We first mean-centered the 262 data, such that average neural and muscle activity was zero for each condition. We then ran PLS 263 regression on the training condition to find a rank-matrix that predicts muscle activity from 264 neural activity. To select the optimal rank, we selected one cycle from our test condition to serve 265 as validation data. We assessed performance on this validation cycle and selected the rank, * , 266 and the corresponding weight matrix, * , that generated the maximal validation " . We assessed 267 prediction performance (generalization) on the remaining cycles of our test data. This procedure 268 was repeated for each behavior and hemisphere. Generalization performance was assessed based 269 on population percent variance explained. We considered N%($ = * , and computed " = 1 − 270 We used a similar procedure to assess how well neural activity could be predicted from neurons 272 in the same or opposite hemisphere. We randomly divided driving-cortex neurons into two Using PLS regression, we calculated generalization performance, $%&'&)A " , when predicting 276 We used the same train/validate/test procedure described above, assessing 278 generalization performance on a held-out condition. We repeated this process with 125 different

Dimensionality Reduction 282
Dimensionality reduction was performed via principle components analysis (PCA). We typically 283 ran PCA on neural data from a sub-set of behavioral conditions. We concatenated neurons' soft-284 normalized FRs from the desired conditions to generate a data matrix A of size ( × , ), where 285 c was the number of conditions, t was the number of timepoints included per condition, and n 286 was the number of neurons. We applied PCA to A to obtain matrices X and V such that = , 287 where X is the projection of the data onto the principal components (PCs) and V contains the 288 weights from neurons to PCs. To project other behavioral conditions into the same space, we 289 could construct a new data matrix ′ using these conditions' FRs. We then multiply by V, such 290 that the new projections ′ are defined by ] = ′ . 291

Trajectory Tangling 292
We assessed trajectory tangling as described in (Russo et al., 2018). To parallel the other analyses 293 of population activity, trajectory tangling was computed for the middle cycles. Neural activity 294 (or muscle activity) was reduced to the top eight dimensions using PCA. We then calculated 295 tangling, ( ), at each time point: 296 where ; is the neural state at time , ̇; is the temporal derivative of the neural state, ‖•‖ is the 298 Euclidean norm, and is a small constant that prevents division by zero (here, set to 10% of the 299 total neural variance across the top eight dimensions). ( ) is large if the neural state at time is 300 close to the neural state at a different time, but the two states have very different derivatives. 301 Trajectory tangling was computed across all times for a given set of conditions -e.g. all 302 conditions where the right arm performed the task. Thus, indexes across all times and 303 conditions for one arm. Trajectory tangling was computed separately for each hemisphere, and 304 for the muscle population in each arm. For a given quantity (e.g., muscle activity) the two 305 distributions (one per arm) for that monkey were combined and the cumulative density was 306

computed. 307
Predicting non-performing arm EMG 308 Above we described our methodology for assessing how well neural activity in a given 309 hemisphere predicts muscle activity in the arm performing the task. We used a similar 310 methodology to address a related but different question: whether a unified linear decoder, based 311 on activity across both hemispheres, can predict muscle activity both when that arm performs the 312 task (and robust EMG needed to be decoded) and when the other arm performs the task (and 313 near-zero EMG should be decoded). As above, we used PLS regression and focused on the 314 middle cycles. For this analysis, we predict EMG activity using all neurons, regardless of 315

hemisphere. 316
We first assessed generalization performance using a 'train-moving' decoder, which was trained 317 using only conditions where the relevant arm performed the task, and was then asked to 318 generalize to conditions where the arm did not move. This is a potentially challenging form of 319 generalization, as the decoder must predict EMG activity in a situation (arm not moving) very 320 different from the situation in which it was trained (arm moving). We also computed 321 generalization performance of a 'train-both' decoder, trained using a set of conditions that 322 included the relevant arm performing and not performing the task. Generalization was to left-out 323 conditions of each type. 324 For the train-moving decoder, we used the following division of training, validation, and testing, 325 conditions: Training: Direction 1, Start Position 1, Arm moving. Validation: Direction 1, Start 326 Position 2, Arm moving (one cycle). Testing: Direction 1, Start Position 2, Other arm moving. 327 For the train-both decoder, we used the following division of training, validation, and testing: 328 Training: Direction 1, Start Position 1, Arm moving and Direction 1, Start Position 1, Other arm 329 Moving. Validation: Direction 1, Start Position 2, Arm moving (one cycle). Testing: Direction 1, 330 Start Position 2, Other arm moving. 331

Results 332
Behavior 333 Two monkeys (E and F) were trained on a cycling task that could be performed with either arm 334 ( Figure 1A). Left and right hands each grasped a pedal. Monkeys performed blocks of left-hand 335 and right-hand trials. Cycling the correct pedal produced motion through the virtual environment. 336 Success required that the non-performing arm be kept still. On each trial, monkeys cycled from 337 one target to another, located seven cycles away ( Figure 1B). Targets were positioned so that 338 cycling started and ended either at the top of the cycle ('top start') or at the bottom of the cycle 339 ('bottom start'). 340 Each combination of starting position and cycling direction was performed for five consecutive 341 trials. The order of the four combinations was consistent within each 20-trial block ( Figure 1C). 342 Monkeys performed an average of 29 and 21 trials per condition per day (monkey E and F, 343 respectively). Monkeys cycled quickly, with a median angular speed of 2.2 Hz and 1.8 Hz 344 (monkey E and F; Figure 1D,E). In contrast, the non-performing hand moved very little (leftmost 345 distributions in Figure 1D,E). Mean angular speed for the non-performing arm was 0.0016 346 cycles/s and 0.024 cycles/s (monkey E and F). 347

Neural and muscle responses 348
We examined the average firing rate of neurons recorded in each hemisphere of M1 (1150 total 349 isolations across two hemispheres and both monkeys). Firing rates were computed after 350 temporally aligning behavior across trials ( Figure 2). Neural responses were typically rhythmic 351 ( Figure 3E-H), and could be nearly sinusoidal ( Figure 3E) or could contain additional higher-352 frequency structure ( Figure 3G). For comparison, we recorded the activity of the major muscles 353 in both arms (48 total recordings). The temporal features of muscle responses ( Figure 3A-D) in 354 many ways resembled those of single-neuron responses. However, muscles and neurons were 355 quite different in the degree to which responses were restricted to movements of a single arm. 356 Muscles exhibited robust activity only when their driven arm performed the task ( Figure 3A-D). 357 E.g., the left anterior deltoid ( Figure 3A) was active when the left arm performed the task (blue) 358 but not when the right arm performed the task (red). While expected, this direct confirmation is 359 important because of the possibility that muscles might have been active in ways that didn't 360 move the pedal (e.g., co-contraction). Such activity could potentially have been substantial, 361 complicating interpretation of neural activity. A few muscles exhibited weak activity when the 362 task was performed by their non-driven arm ( Figure 3A,C). However, this typically occurred 363 only at the end of movement, consistent with tensing to aid stability during stopping. 364 In contrast to the muscles, neurons were typically active throughout the movement, regardless of 365 whether the task was performed with their driven or non-driven arm. A few neurons were active 366 only when cycling with the driven arm ( Figure 3E), and on rare occasions a neuron was active 367 only when cycling with the non-driven arm ( Figure 3H). However, most neurons were active in 368 both situations ( Figure 3F,G). Furthermore, neural responses could be quite different when 369 cycling with the driven versus non-driven arm. Neural response patterns could change in both 370 phase ( Figure 3F) and structure ( Figure 3G) depending on which arm performed the task. 371

Single-neurons are active during movements of either arm 372
To quantify the arm preference of individual neurons, we compared firing-rate modulation when 373 the task was performed with the driven versus non-driven arm. Modulation was assessed as the 374 standard deviation of the firing rate across timepoints, which captures the degree to which 375 activity evolves with time. Average modulation was computed once across all conditions where 376 the driven arm performed the task, and again across all conditions where the non-driven arm 377 performed the task. We analyzed only the middle cycles of movement (excluding the first cycle 378 and the last two cycles). This allowed us to quantify the "steady state" performance of the 379 neurons, without starting and stopping transients, and aided comparison with the muscles. We 380 computed an 'arm preference index': the difference in modulation for the driven versus non-381 driven arm, divided by the sum. This index ranges from -1 to 1, with the extremes indicating 382 complete preference for the non-driven and driven arms respectively. An arm preference index of 383 zero indicates that a neuron was equally responsive regardless of the arm being used. 384 To establish a baseline for comparison, we computed the arm preference index for each muscle. 385 Arm preference indices were typically high for the muscles, confirming that muscles were active 386 primarily when the task was performed with their driven arm. A few muscles showed weak 387 activation regardless of the arm being used, resulting in lower indices. However, most muscles 388 had robust responses, and were much more active when the task was performed with the driven 389 arm. For both monkeys, the median arm preference index was near unity (Figure 4A E: 0.86; F: 390 0.98; blue dots) and the modal response occurred at unity. 391 In contrast, neurons rarely had arm preference indices near unity ( Figure 4B). Instead, the 392 distribution of arm preference indices was centered slightly above zero (median = 0.07 and 0.31 393 for the two monkeys). Thus, neural responses were much more likely than muscle responses to 394 be similar in magnitude regardless of which arm performed the task. Furthermore, many neurons 395 had arm preference indices < 0, indicating stronger modulation when the non-driven arm 396 performed the task (Monkey E: 201/533 neurons; Monkey F: 107/617 neurons). 397 Thus, neurons can be quite active even when the task is performed with their non-driven arm. 398 Might such responses be related to small movements of the driven arm? This explanation is 399 unlikely a priori. As described above, movements of the non-performing arm were small ( Figure  400 1D,E) and corresponding muscle activity was weak ( Figure 4A). In principle, neural responses 401 related to weak muscle activity might be magnified via normalization or some other non-402 linearity. However, such magnification would need to be very strong. To match the median 403 neural arm preference indices, muscle activity in the non-performing arm would need to be 404 magnified by a factor of 12 (monkey E) and 52 (monkey F). Furthermore, magnification cannot 405 account for the finding that neurons commonly had negative arm preference indices, while 406 muscles rarely (monkey E) or never (monkey F) did. 407 We performed an additional control to ask whether neural responses, when the non-driven arm 408 performed the task, were influenced by small movements of the driven arm. Such movements 409 varied across trials ( Figure 5A), allowing us to divide trials into those with movements larger 410 than the median (very modest movement, red) versus lower than the median (nearly stationary, 411 black). Average firing rates were very similar in these two cases, as illustrated for one example 412 neuron in Figure 5B. Differences were significant at only a few moments (black dots, p < 0.05 413 via bootstrap across 1000 resamples). Those differences were small, and occurred at roughly the 414 rate (5%) expected by chance. Across all neurons, 5% of data-points showed significant 415 differences ( Figure 5C,D). This equals the percentage expected by chance, and is thus consistent 416 with no reliable impact of small movements. Applying this same analysis to the (weak) muscle 417 activity in the non-moving arm revealed significant differences at double the chance rate (10% of 418 data-points) and peaking at four times the chance rate near the end of the movement (21% of 419 data-points). 420 In summary, muscles were silent or at most weakly active when the task was performed with 421 their non-driven arm. The weak activity that was present was statistically coupled to small 422 movements of the driven arm. In contrast, neural responses were typically robust during 423 movements of the non-driven arm, were present throughout the movement (not just at the end) 424 and were not statistically linked to small movements of the driven arm. Prior studies have found 425 that neurons can be active when a task is performed with their non-driven arm, although to 426 varying degrees (Cisek et al., 2003;Donchin et al., 1998;Kermadi et al., 1998;Tanji et al., 427 1988). The present findings replicate the finding of weakly lateralized responses in motor cortex, 428 and largely rule out potential explanations based on small residual movements of the driven arm. 429

Neural response patterns are limb-dependent 430
One plausible explanation for weakly lateralized responses is that neural activity encodes higher-431 level, limb-independent features of movement. For example, activity might encode hand 432 velocity, movement goal, or some other quantity, regardless of which limb is moving. 433 Preparatory activity in the more anterior rostral premotor cortex (Cisek et al., 2003) can exhibit 434 largely limb-independent responses. Might this also be true in motor cortex during movement? 435 The two arms performed very similar movements in our task. Thus, limb-independence should 436 be reflected by similar neural responses regardless of the performing arm. Instead, neural 437 responses were strongly limb-dependent. Responses often differed in phase ( Figure 3F) and/or 438 structure ( Figure 3G) depending on which arm performed the task. 439 To provide quantification, for each neuron we computed the correlation between the firing rate 440 patterns when the driven versus non-driven arm performed the task ( Figure 4C). Analysis 441 considered only the middle cycles (2-5). This ensured that high correlations indicate similar 442 response patterns, not simply firing rates that rise non-specifically during movement. On average 443 the correlation was near zero (median correlation: 0.16 and 0.08 for Monkey E and F). Strongly 444 correlated responses were very much the minority: only 18/533 (E) and 9/617 (F) neurons had 445 correlations >= 0.75. Thus, for a given neuron, there was remarkably little relationship between 446 responses when the task was performed with the driven versus non-driven arm. We used a 447 shuffle manipulation to estimate the distribution of correlations if there were in fact no 448 relationship. Each neuron's response was matched with that of another random neuron, yielding 449 a distribution of correlations expected by chance given the range of response patterns present in 450 the data (green). The empirical distribution (black) was only modestly more positive than the 451 chance distribution. 452 Might correlations appear artificially low if responses are weak or noisy? While sampling error 453 will inevitably reduce correlations, this is unlikely to be the source of the low correlations we 454 observed. Cycling evoked particularly strong neural responses with correspondingly small 455 standard errors of the mean firing rate ( Figure 3E-H, envelopes show SEM). We further 456 addressed this concern by computing, for each neuron, the correlation between the firing rate for 457 the top-start versus bottom-start conditions. Behavior was very similar for these two conditions 458 during the middle cycles (after aligning phase), and correlations should thus be high. This was 459 indeed the case ( Figure 4C, orange distributions), confirming that sampling error did not impede 460 the ability to measure high correlations. 461 These results rule out the hypothesis of a representation that is predominantly effector-462 independent. Individual-neurons showed very different responses depending on which arm 463 performed the task -almost as different as if there were no relationship between the activity 464 patterns associated with the two arms. 465

Correlations between neurons are limb-dependent 466
Consider two neurons that have similar response patterns when the task is performed with the 467 driven arm ( Figure 6A-D plots four such example pairs). What occurs when the task is 468 performed with the non-driven arm? From the analysis above, we know that the response pattern 469 of each neuron will change. Does this occur in a coordinated fashion, such that the two neurons 470 remain correlated with one another? This would be consistent with the idea that neurons with 471 related responses 'encode' related features, and continue to do so in new contexts. In fact, this 472 property was rarely observed. We did occasionally observe neurons that were strongly correlated 473 when the driven arm performed the task, and remained strongly correlated when the non-driven 474 arm performed the task ( Figure 6A). Yet it was also common for correlations to invert ( Figure  475 6B), for strong correlations to disappear ( Figure 6C), or for neurons to undergo very different 476 changes in response magnitude ( Figure 6D). 477 We computed correlation matrices to quantify such effects across the population. To aid 478 visualization, we ordered neurons to group responses that were similar when the task was 479 performed with the driven arm, resulting in a block structure ( Figure 6E-F, left). We asked 480 whether this correlation structure remained similar when the task was performed with the non-481 driven arm. Note that it is possible for the correlation matrix to remain identical, even if every 482 neuron changes its response, so long as correlated neurons remain correlated. Instead, the 483 correlation structure was dramatically altered. As a result, the original ordering no longer groups 484 neurons with similar response properties (right column of Figure 6E-F). 485 This change in correlation structure was not due to correlations being largely spurious, as could 486 occur if estimated firing rates were noisy. To investigate this possibility, we asked whether the 487 correlation structure differed between conditions where cycling started at the top of the cycle 488 rather than at the bottom. The correlation structure was very similar in these two cases (compare 489 middle and left columns of Figure 6E-F). This finding rules out the possibility that correlations 490 are unstable simply because they are spurious, and demonstrates that not just any change in the 491 task results in a change in the correlation structure. Changing the starting position had relatively 492 little impact, while changing the performing arm had a dramatic impact. 493 Each matrix in Figure 6E-F corresponds to a given condition (a starting position and cycling 494 direction). We wished to summarize, across all such conditions, the degree to which correlations 495 are or aren't preserved when the task is performed with one arm versus the other. To do so, for 496 each condition and each pair of neurons we plotted their firing-rate correlation when the non-497 driven arm performed the task versus their correlation when the driven arm performed the task 498 ( Figure 6G). This is equivalent to plotting the values of the non-driven-arm correlation matrix 499 (right column of Figure 6E,F) versus the corresponding values of the driven-arm correlation 500 matrix (left column). Preserved correlations would yield diagonal structure. In fact, there was 501 little tendency for correlated neurons to remain correlated, or for anti-correlated neurons to 502 remain anti-correlated. The 'meta-correlation' was 0.1 and 0.05 (monkey E and F). Thus, if two 503 neurons responded similarly when the driven arm performed the task, this said little regarding 504 whether those neurons would respond similarly when the non-driven arm performed the task. 505

Population activity is isomorphic across hemispheres 506
The above results demonstrate that both individual-neuron responses and their correlation 507 structure are very different depending on which arm is employed to perform the task. One 508 potential explanation is that very different signals are present: perhaps muscle-like signals when 509 employing the driven arm versus more abstract signals when employing the non-driven arm. An 510 alternative explanation is that many of the same signals are present, yet are reflected differently 511 at the level of individual neurons. We have argued that motor cortex carries both muscle-like 512 signals and non-muscle-like signals fundamental to the underlying computations (Churchland et 513 al., 2012;Russo et al., 2018). However, those experiments examined only the driving cortex; it 514 remains unknown which signals are shared with the non-driving motor cortex. 515 We first asked whether muscle-like signals are present in both hemispheres. We trained a 516 regularized linear decoder to predict performing-arm muscle activity based on neural activity. 517 We assessed generalization to a held-out condition, repeating this procedure for each condition. 518 Both the driving and non-driving cortex accurately predicted muscle activity ( Figure 7A). Across 519 all conditions, generalization " was high for both the driving and non-cortex ( Figure 7B, 520 generalization performance computed across all muscles). Generalization performance was lower 521 for the non-driving cortex, but this was a small effect and was significant for only one monkey (p 522 = 0.15 and p = 0.016 for monkey E and F, two-sided Wilcoxon signed rank test across 8 523 conditions). We thus saw no evidence that signals related to muscle activity were absent in the 524 non-driving cortex. 525 Might other signals be restricted to the driving cortex? Rather than attempting to infer specific 526 candidate signals, we developed a method to address this question generically ( Figure 7C, top). 527 We randomly divided driving-cortex neurons into two groups. Because the division is random 528 and the number of neurons large, each group should reflect approximately the same set of 529 signals. Thus, one group should be able to accurately predict activity in the other group. This was 530 indeed the case ( Figure 7C, 'Driving Predicts'). We next generated a size-matched group of 531 neurons from the non-driving cortex, and asked whether their activity could be used to predict 532 activity in the driving cortex. If all signals are present in both hemispheres, it should be possible 533 to predict driving-cortex activity based on non-driving cortex activity. Conversely, if a major 534 signal is missing in the non-driving hemisphere, prediction would be compromised. 535 Driving cortex activity was predicted almost as well, based on activity in the opposite 536 hemisphere, as it had been based on activity within the same hemisphere ( Figure 7C, thin lines 537 show results for 1000 random divisions). For Monkey E, the difference was non-significant (p = 538 0.16, bootstrap test across 1000 resamples). For Monkey F, the difference was significant but 539 small: a loss of 5% of the variance explained (p = 0.001, bootstrap across 1000 resamples). Thus, 540 we saw no evidence for large signals that are present in the driving cortex but absent in the non-541 driving cortex. 542

Neural trajectory tangling is low for both hemispheres 543
We recently described a major difference between M1 population activity and downstream 544 muscle activity (Russo et al., 2018). Only M1 avoids 'trajectory tangling,' defined as the 545 occurrence of similar population states with very different derivatives. Trajectory tangling 546 becomes high if the population trajectory crosses itself, or if the trajectory for one condition 547 traverses near that for another condition but travels in a different direction. Pattern-generating 548 recurrent networks are noise-robust only if trajectory tangling is low, suggesting an explanation 549 for why low trajectory tangling was observed in M1. It is unknown whether the non-driving 550 cortex participates (via callosal connections) in pattern generation. We therefore wondered 551 whether the non-driving cortex would show similarly low trajectory tangling. Notably, it is 552 possible for a cortical area to be active during cycling yet have high trajectory tangling; this was 553 true of proprioceptive primary somatosensory cortex (Russo et al., 2018). 554 We computed the tangling index (as used in Russo et al. 2018) for every time during the middle 555 cycles, across all conditions. We did so for population activity in the driving and non-driving 556 cortex, and also for the muscle populations, and compared the resulting distributions. The 557 muscles often showed high trajectory tangling, revealed by a long right tail in the cumulative 558 distributions ( Figure 8A,B, black lines). The driving cortex displayed consistently low trajectory 559 tangling: cumulative distributions (blue) plateaued early. This replicates prior results: trajectory 560 tangling is much lower for the driving cortex than for the downstream muscle population. 561 Notably, this is true even though single-neuron and single-muscle responses are superficially 562

similar. 563
Trajectory tangling was also low for the non-driving cortex (red). For monkey E, tangling was 564 slightly higher in the non-driving versus driving cortex (468 ± 201 versus 420 ± 153; mean ± 565 S.D.) while the opposite was true for monkey F (374 ± 105 versus 430 ± 146). Thus, trajectory 566 tangling is similarly low for both cortices, with only small and inconsistent differences. 567 Critically, for both the driving and non-driving cortex, neural trajectory tangling was much lower 568 than muscle trajectory tangling. The latter averaged 2296 ± 1766 for Monkey E and 4392 ± 2950 569 for Monkey F. Taken together with the results above, we found little hemispheric difference 570 regarding either the major signals or the organization of population trajectories. 571

Neural activity occupies different dimensions for movements of different arms 572
If similar signals are present regardless of which arm moves, how does the brain avoid moving 573 the wrong arm? In confronting this question, we took inspiration from recent work suggesting 574 that only some neural dimensions in motor cortex are 'muscle potent'; activity in those 575 dimensions produces output that will influence the muscles. Other dimensions are 'muscle null'; 576 activity in those dimensions has no direct outgoing impact on muscle activity (Druckmann and 577 Chklovskii, 2012;Kaufman et al., 2014). The presence of output-null dimensions is natural (and 578 typically inevitable) when patterns are generated by a recurrent network with more internally-579 connected neurons than output neurons. We wondered whether this principle might apply to the 580 present case. We considered all recorded neurons, across both hemispheres, as a unified 581 population. We asked whether signals related to the movement of each arm are partitioned in a 582 manner that could allow signals related to one arm to naturally avoid impacting the other arm. 583 We used principal component analysis (PCA) to find neural dimensions that best explain 584 activity. We applied PCA once for conditions where the right arm performed the task ('right-585 performing' conditions) and again for conditions where the left arm performed the task ('left-586 performing' conditions). A 'right-arm' space was defined by the PCs found for the right-587 performing conditions. A 'left-arm' space was defined analogously. We were interested in what 588 occurred in the right-arm space when the left arm performed the task, and vice versa. 589 Importantly, in both cases PCA considered the responses of the same unified population of 590 neurons (all recorded neurons across both hemispheres). 591 The right-arm space captured (by construction) population activity when the right arm performed 592 the task. This can be appreciated in Figure 9A,C by the large near-circular trajectories for the 593 right-performing conditions (red). The rapid rise in cumulative variance accounted for ( Figure  594 9B,D, red) reveals that a small number of right-arm PCs successfully captured most of the 595 variance for the right-performing conditions. In contrast, the right-arm space did not effectively 596 capture variance for the left-performing conditions. Left-performing neural trajectories are small 597 when projected onto the right-arm PCs, with little clear structure ( Figure 9A,C, blue). The 598 cumulative percent variance accounted for (left-performing conditions projected onto the right-599 arm PCs) rose slowly ( Figure 9B,D, blue). Analogous results were found when analyzing the 600 left-arm space ( Figure 9E-H). 601 Thus, relatively little variance was captured when activity for left-performing conditions was 602 projected onto the right-arm space, and vice versa. Averaged across all such conditions, the top 5 603 PCs explained 7 ± 1% (monkey E) and 2 ± 0.5% (monkey F) of the variance (mean ± std. 604 computed across conditions). This was in contrast to the large amount of variance captured when 605 conditions were projected onto the top 5 PCs of their 'own' space: 80 ± 2% (monkey E) and 80 ± 606 2% (monkey F). This asymmetry was not simply due to projecting onto a space built from the 607 same data versus other data. For example, the top 5 PCs based on top-start conditions captured 608 almost as much variance during bottom-start conditions (63 ± 4% for monkey E and 66 ± 2% for 609 monkey F) as during top-start conditions (79 ± 2% and 79 +/-2%). 610 Thus, neural responses related to the two arms occupy nearly orthogonal subspaces. Dimensions 611 that robustly capture activity when one arm performs the task do not continue to do so when the 612 other arm performs the task. This could occur trivially if neurons fall into two groups: neurons 613 that respond only when the right arm performs the task, and neurons that perform only when the 614 left arm performs the task. However, as documented above, no such separation was present. The 615 distribution of arm preference indices was unimodal with a median near zero ( Figure 4B) 616 indicating that most neurons responded regardless of which arm performed the task. The 617 orthogonality of dimensions is instead related to the finding that the correlation structure depends 618 strongly on which arm performs the task ( Figure 6). As such, this is intrinsically a population-619 level finding that could not have been inferred from analyses focused on individual neurons. 620

Linear decoders naturally separate signals related to the two arms 621
The separation of activity into orthogonal subspaces may allow descending control of one arm to 622 naturally ignore signals related to the other arm. To test the plausibility of this hypothesis, we 623 trained linear decoders to predict, based on the activity of the entire neural population, muscle 624 activity for a given arm. The decoder was trained only using conditions where that arm 625 performed the task. For example, the decoder was trained to predict muscle activity in the right 626 arm while the right arm performed the task. Restricted to this situation, decoders performed well, 627 predicting a median of 91% (Monkey E) and 93% (Monkey F) of the variance on held-out 628 conditions. Examples of predicted muscle activity ( Figure 10A,D, orange traces at top) are 629 shown for one muscle for each monkey. We then assessed generalization to conditions where the 630 other arm performed the task. For example, a decoder that fit right-arm muscle activity (trained 631 only on right-performing conditions) was asked to generalize and predict right-arm muscle 632 activity during left-performing conditions. Does the prediction stay relatively flat, accurately 633 capturing the absence of muscle activity? Or does the decoder become contaminated by signals 634 related to the performing arm? 635 Decoders accurately generalized, and predicted little modulation of muscle activity in the non-636 performing arm. For the two example muscles shown, the predicted muscle activity was 637 relatively flat, in agreement with the lack of modulation of the empirical muscle activity ( Figure  638 10A,D, bottom, compare orange and black traces). This was true across muscles and conditions: 639 decoded muscle activity was only weakly modulated for conditions when the task was performed 640 with the other arm (Figure 10 B,E, orange), even though the decoder was not trained on such 641 conditions and even though the neural activity upon which the decoder was based was similarly 642 modulated regardless of the performing arm. The ability of the decoder to ignore such activity 643 was inherited from the orthogonality of subspaces described above. When trained using right-644 arm conditions, decoders naturally employ right-arm dimensions. Because those dimensions are 645 largely unoccupied when the left arm performs the task, the decode shows minimal modulation. 646 Due to these properties, decoders naturally produce predicted muscle activity with positive arm-647 preference indices ( Figure 10C, orange histograms). These distributions are right-shifted relative 648 to those for the neural activity upon which decoding was based (red histogram). Thus, the 649 structure of population activity ensures that a decoder, trained to extract activity related to one 650 arm, will naturally tend to ignore activity related to the other arm. 651 Yet while decoders tended to naturally ignore activity related to the 'wrong' arm, this feature 652 was imperfect: small amounts of residual modulation were still present ( Figure 10A,D, orange 653 trace at bottom) leading to arm-preference indices smaller than those of the muscles. Of course, 654 one would expect improved ability to segregate activity if a decoder is trained to do so: i.e., to 655 predict muscle activity both when the muscle is strongly modulated (when the relevant arm 656 performs the task) and also when it is not (when the other arm performs the task). This was 657 indeed the case (Figure 10, green). Note that these decoders still had to generalize to left-out 658 conditions; they simply had the benefit of training data that included both left-and right-659 performing conditions. 660 There is thus no paradox in the absence of muscle activity in the non-performing arm, despite 661 robust neural activity across both hemispheres. Signals related to the two arms are separated into 662 different neural subspaces. As a result, even simple linear decoders naturally separate signals 663 related to one arm versus the other. 664

Discussion 665
We found that neural signals related to movements of the right and left arms are anatomically 666 intertwined. Signals were mixed across hemispheres; when one arm moved, neurons in both 667 hemispheres were modulated. Signals were also mixed across neurons; most neurons responded 668 when both their driven and their non-driven arm performed the task. Individual neurons 669 responded very differently depending on which arm was moving. Yet at the level of the 670 population, both hemispheres contained similar information. Surprisingly, we did not find signals 671 that were strongly present in the driving cortex but absent in the non-driving cortex. This was 672 true even for muscle-like signals, which could be decoded similarly well from either hemisphere. 673 Despite this intermixing, signals corresponding to the two arms were highly separable at the 674 level of neural dimensions. Activity related to the left arm occupied a set of dimensions nearly 675 orthogonal to the dimensions occupied by activity related to the right arm. As a result, even 676 simple linear decoders could read out commands for one arm while ignoring commands for the 677 other arm. 678

Separation of information across dimensions is a common feature of cortical activity 679
Our results contribute to an increasingly broad set of studies reporting that neural activity related 680 to different computations or task parameters is often separated across neural dimensions, instead 681 of at the level of brain areas or individual neurons. During reaching, dimensions carrying 682 preparatory activity are orthogonal to dimensions carrying muscle-related signals (Kaufman et 683 al., 2014), and more generally to all dimensions occupied by movement-related activity (Elsayed 684 et al., 2016). Activity related to the timing of movement initiation and activity related to which 685 movement is being generated also occupy orthogonal dimensions . In 686 sensory decision making, different aspects of the task (e.g., stimulus versus task time, color 687 versus direction of the stimulus, or auditory versus visual cues) are integrated in different neural 688 dimensions in prefrontal cortex (Machens et al., 2010;Mante et al., 2013) and parietal cortex 689 (Raposo et al., 2014). Separating neural activity into separate dimensions for separate 690 computations may thus be a general strategy used by many brain regions. A potential advantage 691 of this strategy is that signals are able to interact (e.g., in the case where movements of two limbs 692 are coordinated) yet can still be read out separately (Druckmann and Chklovskii, 2012;Kaufman 693 et al., 2014) 694

Comparison with prior studies of lateralization in M1 695
Our finding that individual neurons often respond during movements of either arm is in broad 696 agreement with prior primate recording studies. A majority of these studies describe intermixing 697 of right-and left-arm responses in the activity of individual M1 neurons (Cisek et al., 2003;698 Donchin et al., 2002698 Donchin et al., , 1998Kermadi et al., 1998;Steinberg et al., 2002). However, a handful of 699 these studies report a much smaller percentage of ipsilateral (non-driving) neural responses 700 (Aizawa et al., 1990;Tanji et al., 1988). These studies examined neural responses in the hand 701 area of M1 during small finger movements. The hand area of M1 has fewer callosal connections 702 and fewer ipsilateral projections than the arm area of M1 has (Jenny, 1979;Jones and Wise, 703 1977;Rouiller et al., 1994). Thus, a likely explanation for varied prior results is that hand-related 704 neural computations are more divided by hemisphere, while arm-related computations are largely 705 intermixed. An alternative explanation is that it is simply easier to move one hand while keeping 706 the other still, resulting in greater neural segregation due to better experimental control over 707 lateralization of muscle activity. Our results support the first explanation; muscle activity was 708 weak in the non-driven arm, and the very small movements of that arm had no statistical impact 709 on neural activity. 710 Of prior studies, two explicitly compared neural response properties (e.g., preferred directions) 711 during unimanual movements of each arm. Cisek et al. (2003) found that neurons in M1 had 712 limb-dependent preferred directions, yet Steinberg et al. (2002)  Responses during performance with one versus the other arm were weakly correlated at the 717 single-neuron level, and occupied nearly orthogonal subspaces at the population level. 718

Possible reasons for ipsilateral motor cortical activity 719
There exist multiple reasons why motor cortex might be active when the non-driving arm 720 performs the task. A straightforward possibility is that motor cortex employs an abstract limb-721 independent representation of movement. However, this hypothesis is unlikely given the strongly 722 limb-dependent nature of responses. Alternatively, the two cortices may process different but 723 complementary information. This hypothesis is also unlikely; we found no large signals that 724 were present in the driving cortex but absent in the non-driving cortex. 725 It is also possible that activity ipsilateral to the moving arm may relate to uncrossed descending 726 connections (Kuypers, 1981;Rosenzweig et al., 2009). For example, activity in the right motor 727 cortex could exist to drive, via uncrossed connections, muscle activity in the right arm when that 728 arm performs the task. Our results are in principle consistent with this hypothesis. However, 729 prior studies have found little evidence for a robust relationship between M1 and ipsilateral 730 muscle activations. Intracortical microstimulation readily produces contralateral muscle 731 responses (Kwan et al., 1978;Sessle and Wiesendanger, 1982), yet very rarely generates 732 ipsilateral muscle responses (Aizawa et al., 1990). Intracellular recordings of motoneurons reveal 733 no monosynaptic evoked potentials from ipsilateral corticospinal tract stimulation and spike-734 triggered EMG effects are present only for contralateral muscles (Soteropoulos et al., 2011). For 735 these reasons, we suspect that uncrossed projections are unlikely to be the primary reason that 736 the non-driving motor cortex is active. 737 Another possibility is that activity in the non-driving cortex is produced by an efference copy of 738 signals generated and employed by the driving cortex, which are conveyed to allow coordination 739 between the limbs. Many -perhaps most -movements require coordination across the midline. 740 Given the near-ubiquitous need for coordination, it may be that efference copy signals are simply 741 conveyed by default, and ignored if they are not needed. Our results are consistent with this 742 possibility, and argue that if it is correct, then the relevant efference copy must be quite 743 complete. That is, the driving cortex must convey the majority of the signals it generates, rather 744 than (for example) just the output signals. 745 A final, and intriguing possibility, is that motor cortical computations are largely distributed 746 across both hemispheres (Li et al., 2016). In the extreme, neurons in the non-driving cortex 747 might simply be viewed the way we view most neurons in the driving cortex; they can contribute 748 to the computation even if they are one or more synapses from the cortico-spinal neurons that 749 will convey the output. This hypothesis is appealing because it could explain the finding that all 750 major signals appear to be shared between hemispheres. More generally, if a randomly chosen 751 neuron from the non-driving cortex has responses that are nearly indistinguishable from a neuron 752 chosen from the driving cortex, perhaps our default assumption should be that they are 753 participating in the same computation. While appealing, it is unclear if this hypothesis can be 754 reconciled with the finding that motor cortex inactivation principally affects the contralateral 755 limbs (Glees and Cole, 1950;Liu and Rouiller, 1999;Passingham et al., 1983). If both 756 hemispheres participate in controlling both arms, one would expect a more bilateral deficit. A 757 possible, but highly speculative, resolution is that the network is sufficiently robust that it can 758 still function when many neurons are inactivated, so long as the output neurons can still convey 759 the necessary commands. 760

Ipsilateral arm signals in other brain regions 761
Cortical areas, subcortical areas, and the spinal cord all contribute to the control of dexterous 762 movements. Indeed, other studies comparing contralateral and ipsilateral movements have found 763 that not only M1, but also the dorsal premotor cortex (Cisek et al., 2003;Kermadi et al., 2000;764 Tanji et al., 1988), ventral premotor cortex (Michaels and Scherberger, 2018), Supplemental 765 Motor Area Gribova et al., 2002;Kazennikov et al., 1999;Kermadi et al., 766 2000Kermadi et al., 766 , 1998Tanji et al., 1988), Cingulate Motor Area (Kermadi et al., 2000), and the Posterior 767 Parietal Cortex (Kermadi et al., 2000) all contain neurons which respond to movements of the 768 ipsilateral arm. Furthermore, there are circuits in the brainstem and spinal cord which 769 specifically support the generation of coordinated, rhythmic movements like locomotion 770 (Duysens and de Crommert, 1998). Other brain regions, such as the Anterior Intraparietal Area, 771 encode movement parameters in a largely effector-independent manner (Michaels and 772 Scherberger, 2018), suggesting that the relationship of motor and visuo-motor areas to ipsilateral 773 movements may vary depending on their role in motor computation. In general, movement 774 generation is the result of the action of a broad, interconnected network of brain and spinal 775 regions. We focused on M1 because it is the cortical region that, based on anatomy and 776 microstimulation results, seemed most likely to have a lateralized representation of movement. 777 Yet even in M1 we failed to find evidence of strongly lateralized activity. 778

Summary 779
Neural signals related to movements of either arm were mixed, both within hemispheres and 780 within single neurons. However, signals related to the two arms were naturally partitioned into 781 different neural dimensions. This underscores the computational usefulness of leveraging 782 different dimensions for different computations; signals can be shared across a wide population 783 yet still be readily separated by downstream regions. Our results argue that motor cortex shares 784 highly detailed information cross-cortically, suggesting that control may span both hemispheres 785 even if output commands originate primarily from the contralateral hemisphere. 786 Average firing rate calculated after the second alignment step. Black: Mean firing rate. Gray 812 shading: standard error across trials. 813 the mean angular speed of the non-performing arm. This distribution is shown for one condition, 832 recorded on one day. Trials were divided into those with mean speed less than (gray) or greater 833 than (red) the median (vertical dashed line). (B) Firing rate of one example neuron for these two 834 groups: trials with speeds less than (black) and greater than (red) the median. Envelopes show 835 standard errors of the mean. Black dots at top indicate times when the two rates were 836 significantly different (p<0.05). Plotting conventions as in Fig. 3. (C) Percentage of neurons 837 (black trace) showing a statistically significant difference (p<0.05) in firing rate for trials with 838 speeds less than versus greater than the median. Differences occurred roughly as often as 839 expected by chance (red line at 5%). Gray box denotes the time of movement. Each tick mark 840 delineates a cycle. Data are for monkey E. Analysis is based on 426 neurons. (D) As in C but for 841 Monkey F. Analysis is based on 479 neurons. 842 (blue) conditions, with PCs found using only right-arm-performing data. The neural population 932 includes all recorded neurons across both hemispheres. Analysis considers data from the middle 933 cycles when cycling forward, for both top-start and bottom-start conditions. Data are for monkey 934 E. (B) Cumulative variance explained for right-arm-performing (red) and left-arm-performing 935 (blue) conditions, using PCs found from right-arm-performing data only. (C,D) As above but for 936 Monkey F. (E-H) As in A-D, but with PCs found using only left-arm-performing data. All data 937 shown are for analyses performed on forward cycling conditions. 938