Rotational dynamics in motor cortex are consistent with a feedback controller

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1. 

   Version published to 10.7554/elife.67256 on eLife

   Nov 3, 2021
2. 

   eLife

   Aug 6, 2021

   Author Response:

   Reviewer #1 (Public Review):

   This paper aims to address the question of whether the rotational dynamics in motor cortex may be due to sensory feedback signals rather than to recurrent connections and autonomous dynamics as is typically assumed. This is indeed a question of importance in neural control of movement.

   The authors employ both analyses of motor cortical data and simulation analyses where a neural network is trained to perform a motor task. For the simulations, the authors use a neural network model of a brain performing arm control tasks. Importantly, in addition to the task goals, the brain also receives delayed sensory feedback from the muscle activity and kinematics of the simulated arm. The brain is modeled either using a stack of two recurrent neural networks (RNN) or using two non-recurrent neural …

   Author Response:

   Reviewer #1 (Public Review):

   This paper aims to address the question of whether the rotational dynamics in motor cortex may be due to sensory feedback signals rather than to recurrent connections and autonomous dynamics as is typically assumed. This is indeed a question of importance in neural control of movement.

   The authors employ both analyses of motor cortical data and simulation analyses where a neural network is trained to perform a motor task. For the simulations, the authors use a neural network model of a brain performing arm control tasks. Importantly, in addition to the task goals, the brain also receives delayed sensory feedback from the muscle activity and kinematics of the simulated arm. The brain is modeled either using a stack of two recurrent neural networks (RNN) or using two non-recurrent neural network layers to investigate the importance of autonomous recurrent dynamics. The authors use this framework to simulate the brain performing two tasks: 1) posture perturbation task, where the arm is perturbed by external loads and has to return to original posture, and 2) delayed center-out reach task. In both tasks, the authors apply jPCA to units of the trained network, simulated muscle activity, and simulated kinematics and investigate their rotational dynamics. They find that when using an RNN in the brain model, both the RNN layers and kinematics show rotational dynamics but the muscle activity does not. Interestingly, these conclusions for both tasks also hold when networks without recurrent connections are used instead of the RNNs. Also importantly, the rotational dynamics also exist in the sensory feedback signals about the limb state (e.g. joint position, velocity). These results suggest that recurrent dynamics are not necessary for the emergence of rotational dynamics in population activity, rather sensory feedback can also achieve the same.

   The authors perform similar jPCA analyses on monkey motor cortical (MC) or somatosensory cortical activity during the same two tasks and find largely consistent results. As with simulations, neural population activity and kinematics show rotational dynamics but muscle activity, which is explored only in the posture task, does not. Importantly, population activity in both motor and somatosensory cortices shows rotational dynamics. This observation is more consistent with the view that rotational dynamics emerge due to inter-region communications and processing of sensory feedback and planning, rather than autonomous dynamics within the motor cortex.

   The approach of the paper is interesting and valuable and the questions being addressed are very important to the field. To further improve the paper and the analyses, there are several major comments that should be addressed to fully support the conclusions and clarify the results:

   Major:

   1. In the Methods, the authors explain how they model a non-recurrent network as follows: "We also examined networks where we removed the recurrent connections from each layer by effectively setting Whh, Woo to zero for the entire simulation and optimization (NO-REC networks)". However, if this is the only modification, it still leaves recurrent elements in the network. For example, if we set W_{hh} to zero, equation 2 will be:

   h_{t+1} = (1-a) * h_t + a * tanh(W_{sh} * s_t + b_h)

   where a is a constant scalar (seems to be equal to 0.5). This is indeed still a recurrent neural network since h_{t+1} depends on h_t. If their explanation in the Methods is accurate, then the current approach restricts the recurrent dynamics to be a specific linear dynamic (i.e. "h_{t+1} = (1-a) * h_t + …") but does not fully remove them. The second layer is also similar (equation 3) and will still have recurrent linear dynamics even if W_{oo} is set to 0. To be able to describe networks as non-recurrent, the first terms in equations 2 and 3 (that is (1-a)h_t and (1-a)o_t) should also be set to 0. This is critical as an important argument in the paper is that non-recurrent networks can also produce rotational dynamics, so the networks supporting that argument must be fully non-recurrent. Perhaps the authors have already done this but just didn't explain it in the Methods, in which case they should clarify the Methods. However, if the current Method description is accurate, they should rerun their NO-REC simulations by also setting the fixed linear recurrent components (that is (1-a)h_t and (1-a)o_t) to zero as explained above to have a truly non-recurrent model.

   We thank the reviewer for raising this important concern. We have re-simulated the NO-REC network while removing the dynamics related to the leaky-integration component. This removal did not impact the network’s ability to perform the tasks and yielded virtually identical neural dynamics (see Figure 8). Throughout the Results we have updated the figures for the NO-REC network to the network without the leak-integration component.

   1. Assuming my comment in 1 is addressed and the results stay similar, the authors show in simulations that even without recurrent dynamics (referred to as the NO-REC case), rotational dynamics are observed in the simulated brain during both tasks (Figure 8). This result is used to suggest that the sensory feedback is what causes the rotational dynamics in the brain model in this case. However, I think to fully demonstrate the role of feedback, additional simulations are also needed where the sensory feedback is removed from the brain model. In other words, what would happen if recurrent and non-recurrent brain models are trained to perform the tasks but are not provided with the sensory feedback (only receive task goals)? One would expect the recurrent model to still be able to perform the task and autonomously produce similar rotational dynamics (as has been shown in prior work), but the non-recurrent model to fail in doing the task well and in showing rotational dynamics. I think adding such simulations without the feedback signals would really strengthen the paper and help its message.

   We apologize if the network architecture was not clear. In the case of the NO-REC network the only way they can generate the time-varying signals needed for the tasks is through sensory feedback. The network simply will not work without recurrent AND sensory feedback. For the posture task there are no additional inputs since it only receives sensory feedback. For the reaching task the task-goal input is static and the GO cue turns off on a timescale considerably shorter (~20ms) than the reach duration. Thus, the REC network would always perform better than the NO-REC network when sensory feedback was removed as the NO-REC network cannot generate any dynamics. We have now included in the Results the following statement. "Note, by removing the recurrent connections these networks can only generate time-varying outputs by exploiting the time-varying sensory inputs from the limb." (line 345-347).

   We have also now included simulations to highlight how REC networks that receive sensory feedback are able to generalize better to scenarios with increased motor noise than REC networks where sensory feedback is either completely removed (reaching task) or only provided at the beginning of the trial (posture task) (Figure S8). Thus, sensory feedback makes REC networks more robust in less predictable scenarios.

   We agree that this could be an interesting manipulation and have now included manipulations of the sensory feedback delays. We considered three separate delays, 0ms, 50ms and 100ms and found that there was a dependence on the rotational frequency of the top jPC plane with greater delays resulting in a general reduction in frequency (see now Supplementary Figure 10). There was less effect of delay on fit qualities to the constrained and unconstrained dynamical system. This has been added to the Results section (line 423-446).

   We simulated this scenario and found the answer to be rather complex and we have added these results to the supplementary material. The network's behavioural performance in the perturbation posture task is similar to the previous networks with joint-based feedback. However, the dynamics in the output layer are not the same with a clear reduction in how well the dynamics are described as rotational (Figure S11A-B).

   Oddly, rotational dynamics could still be observed in the input layer dynamics (data now shown) and the kinematic signals when they were converted to a cartesian reference frame (Figure S11D-E). Furthermore, rotational dynamics could emerge in the output layer if we used a different initialization method for the network weights. We initialized weights from a uniform distribution bound from ±1/√N, where N is the number of units. In contrast, previous studies have initialized network weights using a Gaussian distribution with standard deviation equal to g/√N where g is constant larger than 1. This alternative initialization scheme encourages strong intrinsic dynamics often needed for autonomous RNN models (Sussillo et al., 2015). We found networks initialized with this method and trained on the perturbation posture task exhibited stronger rotational dynamics with fits to the constrained and unconstrained dynamical systems of 0.5 and 0.88, respectively (Figure S11C-D). When examining the reaching task, we found similar results (Figure S11F-K). When initialized with a uniform distribution, fit quality for the constrained and unconstrained dynamical systems were 0.4 and 0.77, respectively (Figure S11F-G), which were smaller than for the joint-based feedback (Figure 7B, constrained R2=0.7, unconstrained R2=0.83). Qualitatively, the dynamics were different when the network was initialized with a Gaussian distribution (Figure S11H), however fit qualities were comparable between the two initialization methods (Figure S11 I). There was also a noticeable reduction in the fit quality for the kinematic signals particularly for the constrained dynamical system (Figure S11K, constrained R2=0.36, unconstrained R2=0.77). These findings have been added to the Results

   1. A measure of how well each trained network is able to perform the task should be provided. For example, is the non-recurrent network able to perform the tasks as accurately as the recurrent models? The authors could use an appropriate measure, for example average displacement in the posture task and time-to-target in the center-out task, to objectively quantify task performance for each network. Another performance measure could be the first term of the loss in equation 5. Also, plots of example trials that show the task performance should be provided for the non-recurrent networks (for example by adding to Figure 8), similar to how they are shown for the recurrent models in Figures 2 and 6.

   We have now presented and quantified the NO-REC network behavioural performance. Kinematics for the NO-REC network are shown in Figure S7A-C and E-G which are comparable to the REC network. Furthermore, quantifying the maximum displacement during the posture task yielded no obvious differences between the NO-REC and REC networks (Figure S7D). For the reaching task, the time-to-target was noticeably more variable and tended to be slower for the NO-REC network (Figure S7H). These observations have been added to the Results.

   1. An important observation is that rotational dynamics also exist in the sensory signals about the limb state. This may imply that the task structure that dictates the limb state and thus the associated sensory feedback may play an important role in the rotations without the recurrent connections. While the present study will be a valuable addition regardless of what the answer is, this is an important point to address: What is the role of the task structure in producing rotational dynamics? In both the posture task and the center-out task, the task instruction instructs subjects to return to the initial movement 'state' by the end of the trial: in the posture task the simulated arm needs to return to the original posture upon disturbance, and in the center out task the arm needs to start from zero velocity and settle at the target with zero velocity. Is this structure what's causing the rotational dynamics? This is an important question both for this paper and for the field and the authors have a great simulation setup to explore it. For example, what happens if the task instructions u* instruct the arm to follow a random trajectory continuously, instead of stopping at some targets? With a simulated tracking task like this, one could eliminate obvious cases of return-to-original-state from the task. Would the network still produce rotational dynamics? Of course, I don't expect the authors to collect experimental monkey data for such new tasks, rather to just change the task instructions in their numerical simulations to explore the dependence of observed rotational dynamics on the task structure. I think this will help the message of the paper and can be very useful for the field.

   We agree that a tracking task would be an interesting manipulation and have simulated this with the REC and NO-REC networks (Figure 9). Here, we trained up the network to reach from the starting position and track a target moving radially at a constant velocity for the rest of the trial (1.2seconds). Thus, the network has to move the limb at a constant velocity. We found there was a consistent reduction in how well the network’s dynamics (constrained R2=0.13, unconstrained R2=0.3) were described as rotational when compared to the previous reaching task (Figure 7, constrained R2=0.7, unconstrained R2=0.83). Also, note that this reduction in rotational dynamics remained even when we initialized the network weights using a Gaussian distribution (see Essential revision 2.3). These simulations have been added to the Results section.

   1. It would be beneficial if the authors could elaborate in the discussion on intuitive explanations of why sensory feedback can produce rotational dynamics even with no internal recurrent dynamics in the brain model. To me, it seems like sensory feedback is providing a path for recurrence to exist in the overall brain-arm system, so the non-recurrent neural networks can learn to exploit that path to effectively implement some recurrent dynamics. Some intuitive explanations like this will be helpful for readers.

   The main reason why rotational dynamics emerge in sensory feedback is due to the phase offset between the joint position and velocity as changes first occur in the velocity followed by position (see pendulum example Pandarinath et al., 2018a also DeWolf et al., 2016; Susilaradeya et al., 2019). This phase offset is maintained across reach directions and gives rise to the orderly rotational dynamics observed in the kinematic signals (DeWolf et al., 2016; Pandarinath et al., 2018a; Susilaradeya et al., 2019; Vyas et al., 2020). Furthermore, the tracking task disrupted this phase relationship and thus the rotational dynamics were substantively reduced in the network models. This text has been added to the Discussion (lines 519-526).

   1. One main result in data from non-human primates is that there exist rotations also in the somatosensory cortex not just in motor cortex. A more thorough discussion of prior work on rotational dynamics or lack thereof across brain regions and behavioral tasks is important to add here. For example, besides the works cited by the authors, there are other works such as (Kao et al., 2015; Gao et al., 2016; Remington et al., 2018; Stavisky et al., 2019; Aoi et al., 2020; Sani et al., 2021) that discuss or show rotational dynamics in various brain regions and behavioral tasks and should be cited and discussed.

   We have cited the above papers and included in the Discussion the following paragraph (lines 537-549) “Importantly, findings of rotational dynamics in cortical circuits are not trivial. Activity in the supplementary motor area does not exhibit rotational dynamics during reaching (Lara et al., 2018). The hand area of MC also does not exhibit rotational dynamics during grasping-only behaviour (Suresh et al., 2020), though it does exhibit rotational dynamics during reach-to-grasp (Abbaspourazad et al., 2021; Rouse and Schieber, 2018) which may reflect the reaching component of the behaviour. More broadly there is a growing body of work characterizing cortical neural dynamics across different behavioural tasks which have revealed rotational (Abbaspourazad et al., 2021; Aoi et al., 2020; Libby and Buschman, 2021; Remington et al., 2018; Sohn et al., 2019; Stavisky et al., 2019), helical (Russo et al., 2020), stationary (Machens et al., 2010), and ramping dynamics (Finkelstein et al., 2021; Kaufman et al., 2016; Machens et al., 2010) and these dynamics appear to support various classes of computations. Thus, finding rotational dynamics across the fronto-parietal circuit in our study is not trivial."

   1. The authors state that "In contrast, rotational dynamics appear to be absent in… MC activity during grasping driven by sensory inputs (Suresh et al., 2020)." There are other papers that study dynamics during reach-grasps and still finds rotational dynamics and modes (Abbaspourazad et al., 2021; Vaidya et al., 2015) and should be cited and discussed. The recent paper on naturalistic reach-grasps (Abbaspourazad et al., 2021) also argues for the involvement of a large-scale network in these movements, which further supports the authors' interpretation that "This interpretation of motor control emphasizes that the objective of the motor system is to attain the behavioural goal and this requires feedback processed by a distributed network." A discussion of this point made in this recent paper in the intro/discussion is important. Finally, there is a recent paper that argues for the input-driven nature of motor cortex (Sauerbrei et al., 2020) and is cited/discussed by the authors but briefly and mainly in the discussion. I think given the relevance of this recent paper to the core message here, it should also be briefly discussed in the introduction to better set up the work.

   We agree with the reviewer that there are discrepancies between the motor cortical dynamics reported by Suresh et al. 2020 and Abbaspourazad et al., 2021 during grasping tasks. This difference may reflect differences in task as in Suresh et al. 2020 the monkeys grasped objects whereas in Abbaspourazad et al., 2021 monkeys had to reach and grasp objects. Thus, rotations may reflect the reaching component of the behaviour. This has been elaborated on in the Discussion which now reads (lines 539-542) “The hand area of MC also does not exhibit rotational dynamics during grasping-only behaviour (Suresh et al., 2020), though it does exhibit rotational dynamics during reach-to-grasp (Abbaspourazad et al., 2021; Rouse and Schieber, 2018; Vaidya et al., 2015) which may reflect the reaching component of the behaviour.”.

   We have also briefly mentioned the findings by Sauerbrei et al. 2020 in the Introduction which now reads (line 79-81) “Lastly a recent study demonstrates that motor cortical dynamics are driven by inputs coming from motor thalamus (Sauerbrei et al., 2020)."

   Minor:

   1. The Methods are clear and comprehensive, but just to make understanding of the simulation setup easier, it would help to have a diagram of the computation graph for the recurrent and non-recurrent networks that shows their number of units, activations/nonlinearities, RNN cell type, etc., added as supplementary figure.

   We agree that this is useful and have added it to Figure 1

   1. Again, to help more clearly convey the simulations, it would help to show the task goals (x*) that are inputs to the simulated brain for example trials in each task (for example added to Figures 2 and 6).

   We agree that this is useful and have added it to Figure 1

   1. Similar to how VAF is shown on top of all plots of jPC planes, it would be helpful to have the rotation frequency for each jPC plane noted next to it. Currently it is difficult to find the jPC frequency associated with each plot from the text.

   We agree and have added it to the appropriate figures

   1. I am a bit surprised by how different the null distributions are for modeling muscle activity (Figure 3F) and kinematics (Figure 3H). The null distribution is simply the R2 for a constrained or unconstrained dynamic model fit to a subsampled version of the neural activity. The only difference between the null distributions in Figure 3F and Figure 3H seems to be the downsampled dimension, which for muscle activity is 6 and for kinematics is 4 (per equation 1). Any insight will be welcome as to why down sampling the population activity to 4 (Figure 3H) results in so much worse R2 compared with down sampling it to 6 (Figure 3F)?

   We thank the reviewer for raising this concern. Originally, we had applied PCA to reduce the dimensionality of the kinematic signals from 4 dimensions to 2, and the muscle signals from 6 to 4. We realize now that to be more conservative in our significance testing, we should use the full dimensionality of the kinematic and muscle signals. As such, we have changed the figures throughout to reflect this.

   References:

   Abbaspourazad, H., Choudhury, M., Wong, Y.T., Pesaran, B., Shanechi, M.M., 2021. Multiscale low-dimensional motor cortical state dynamics predict naturalistic reach-and-grasp behavior. Nature Communications 12, 607. https://doi.org/10.1038/s41467-020-20197-x

   Aoi, M.C., Mante, V., Pillow, J.W., 2020. Prefrontal cortex exhibits multidimensional dynamic encoding during decision-making. Nature Neuroscience 1-11. https://doi.org/10.1038/s41593-020-0696-5

   Gao, Y., Archer, E.W., Paninski, L., Cunningham, J.P., 2016. Linear dynamical neural population models through nonlinear embeddings, in: Lee, D.D., Sugiyama, M., Luxburg, U.V., Guyon, I., Garnett, R. (Eds.), Advances in Neural Information Processing Systems 29. Curran Associates, Inc., pp. 163-171.

   Kao, J.C., Nuyujukian, P., Ryu, S.I., Churchland, M.M., Cunningham, J.P., Shenoy, K.V., 2015. Single-trial dynamics of motor cortex and their applications to brain-machine interfaces. Nature Communications 6, 7759. https://doi.org/10.1038/ncomms8759

   Remington, E.D., Narain, D., Hosseini, E.A., Jazayeri, M., 2018. Flexible Sensorimotor Computations through Rapid Reconfiguration of Cortical Dynamics. Neuron 98, 1005-1019.e5. https://doi.org/10.1016/j.neuron.2018.05.020

   Sani, O.G., Abbaspourazad, H., Wong, Y.T., Pesaran, B., Shanechi, M.M., 2021. Modeling behaviorally relevant neural dynamics enabled by preferential subspace identification. Nature Neuroscience 24, 140-149. https://doi.org/10.1038/s41593-020-00733-0

   Stavisky, S.D., Willett, F.R., Wilson, G.H., Murphy, B.A., Rezaii, P., Avansino, D.T., Memberg, W.D., Miller, J.P., Kirsch, R.F., Hochberg, L.R., Ajiboye, A.B., Druckmann, S., Shenoy, K.V., Henderson, J.M., 2019. Neural ensemble dynamics in dorsal motor cortex during speech in people with paralysis. eLife 8, e46015. https://doi.org/10.7554/eLife.46015

   Vaidya, M., Kording, K., Saleh, M., Takahashi, K., Hatsopoulos, N.G., 2015. Neural coordination during reach-to-grasp. Journal of Neurophysiology 114, 1827-1836. https://doi.org/10.1152/jn.00349.2015

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3. 

   eLife

   Aug 6, 2021

   Evaluation Summary:

   The authors use numerical simulations and analyses of neural data from non-human primates to investigate whether rotational dynamics in motor cortical population activity which are typically attributed to recurrent connections can alternatively be explained by sensory feedback alone. They find that neural networks performing the same tasks will produce rotational dynamics even without any internal recurrent units. Overall, this paper examines an important question in the motor control field. The authors should clarify in more detail how the case with no recurrent dynamics has been simulated and address/discuss the role of task structure in their conclusions. Once the authors address issues associated with precisely how they eliminated recurrence from their simulations, the results that rotational dynamics are not …

   Evaluation Summary:

   The authors use numerical simulations and analyses of neural data from non-human primates to investigate whether rotational dynamics in motor cortical population activity which are typically attributed to recurrent connections can alternatively be explained by sensory feedback alone. They find that neural networks performing the same tasks will produce rotational dynamics even without any internal recurrent units. Overall, this paper examines an important question in the motor control field. The authors should clarify in more detail how the case with no recurrent dynamics has been simulated and address/discuss the role of task structure in their conclusions. Once the authors address issues associated with precisely how they eliminated recurrence from their simulations, the results that rotational dynamics are not necessarily generated autonomously due to recurrent connections will be a valuable and important addition to the ongoing debate about the nature of these rotational dynamics.

   (This preprint has been reviewed by eLife. We include the public reviews from the reviewers here; the authors also receive private feedback with suggested changes to the manuscript. The reviewers remained anonymous to the authors.)

   Read the original source
4. 

   eLife

   Aug 6, 2021

   Reviewer #1 (Public Review):

   This paper aims to address the question of whether the rotational dynamics in motor cortex may be due to sensory feedback signals rather than to recurrent connections and autonomous dynamics as is typically assumed. This is indeed a question of importance in neural control of movement.

   The authors employ both analyses of motor cortical data and simulation analyses where a neural network is trained to perform a motor task. For the simulations, the authors use a neural network model of a brain performing arm control tasks. Importantly, in addition to the task goals, the brain also receives delayed sensory feedback from the muscle activity and kinematics of the simulated arm. The brain is modeled either using a stack of two recurrent neural networks (RNN) or using two non-recurrent neural network layers to …

   Reviewer #1 (Public Review):

   This paper aims to address the question of whether the rotational dynamics in motor cortex may be due to sensory feedback signals rather than to recurrent connections and autonomous dynamics as is typically assumed. This is indeed a question of importance in neural control of movement.

   The authors employ both analyses of motor cortical data and simulation analyses where a neural network is trained to perform a motor task. For the simulations, the authors use a neural network model of a brain performing arm control tasks. Importantly, in addition to the task goals, the brain also receives delayed sensory feedback from the muscle activity and kinematics of the simulated arm. The brain is modeled either using a stack of two recurrent neural networks (RNN) or using two non-recurrent neural network layers to investigate the importance of autonomous recurrent dynamics. The authors use this framework to simulate the brain performing two tasks: 1) posture perturbation task, where the arm is perturbed by external loads and has to return to original posture, and 2) delayed center-out reach task. In both tasks, the authors apply jPCA to units of the trained network, simulated muscle activity, and simulated kinematics and investigate their rotational dynamics. They find that when using an RNN in the brain model, both the RNN layers and kinematics show rotational dynamics but the muscle activity does not. Interestingly, these conclusions for both tasks also hold when networks without recurrent connections are used instead of the RNNs. Also importantly, the rotational dynamics also exist in the sensory feedback signals about the limb state (e.g. joint position, velocity). These results suggest that recurrent dynamics are not necessary for the emergence of rotational dynamics in population activity, rather sensory feedback can also achieve the same.

   The authors perform similar jPCA analyses on monkey motor cortical (MC) or somatosensory cortical activity during the same two tasks and find largely consistent results. As with simulations, neural population activity and kinematics show rotational dynamics but muscle activity, which is explored only in the posture task, does not. Importantly, population activity in both motor and somatosensory cortices shows rotational dynamics. This observation is more consistent with the view that rotational dynamics emerge due to inter-region communications and processing of sensory feedback and planning, rather than autonomous dynamics within the motor cortex.

   The approach of the paper is interesting and valuable and the questions being addressed are very important to the field. To further improve the paper and the analyses, there are several major comments that should be addressed to fully support the conclusions and clarify the results:

   Major:

   1. In the Methods, the authors explain how they model a non-recurrent network as follows: "We also examined networks where we removed the recurrent connections from each layer by effectively setting Whh, Woo to zero for the entire simulation and optimization (NO-REC networks)". However, if this is the only modification, it still leaves recurrent elements in the network. For example, if we set W_{hh} to zero, equation 2 will be:

   h_{t+1} = (1-a) * h_t + a * tanh(W_{sh} * s_t + b_h)

   where a is a constant scalar (seems to be equal to 0.5). This is indeed still a recurrent neural network since h_{t+1} depends on h_t. If their explanation in the Methods is accurate, then the current approach restricts the recurrent dynamics to be a specific linear dynamic (i.e. "h_{t+1} = (1-a) * h_t + ...") but does not fully remove them. The second layer is also similar (equation 3) and will still have recurrent linear dynamics even if W_{oo} is set to 0. To be able to describe networks as non-recurrent, the first terms in equations 2 and 3 (that is (1-a)*h_t and (1-a)*o_t) should also be set to 0. This is critical as an important argument in the paper is that non-recurrent networks can also produce rotational dynamics, so the networks supporting that argument must be fully non-recurrent. Perhaps the authors have already done this but just didn't explain it in the Methods, in which case they should clarify the Methods. However, if the current Method description is accurate, they should rerun their NO-REC simulations by also setting the fixed linear recurrent components (that is (1-a)*h_t and (1-a)*o_t) to zero as explained above to have a truly non-recurrent model.

   1. Assuming my comment in 1 is addressed and the results stay similar, the authors show in simulations that even without recurrent dynamics (referred to as the NO-REC case), rotational dynamics are observed in the simulated brain during both tasks (Figure 8). This result is used to suggest that the sensory feedback is what causes the rotational dynamics in the brain model in this case. However, I think to fully demonstrate the role of feedback, additional simulations are also needed where the sensory feedback is removed from the brain model. In other words, what would happen if recurrent and non-recurrent brain models are trained to perform the tasks but are not provided with the sensory feedback (only receive task goals)? One would expect the recurrent model to still be able to perform the task and autonomously produce similar rotational dynamics (as has been shown in prior work), but the non-recurrent model to fail in doing the task well and in showing rotational dynamics. I think adding such simulations without the feedback signals would really strengthen the paper and help its message.

   2. A measure of how well each trained network is able to perform the task should be provided. For example, is the non-recurrent network able to perform the tasks as accurately as the recurrent models? The authors could use an appropriate measure, for example average displacement in the posture task and time-to-target in the center-out task, to objectively quantify task performance for each network. Another performance measure could be the first term of the loss in equation 5. Also, plots of example trials that show the task performance should be provided for the non-recurrent networks (for example by adding to Figure 8), similar to how they are shown for the recurrent models in Figures 2 and 6.

   3. An important observation is that rotational dynamics also exist in the sensory signals about the limb state. This may imply that the task structure that dictates the limb state and thus the associated sensory feedback may play an important role in the rotations without the recurrent connections. While the present study will be a valuable addition regardless of what the answer is, this is an important point to address: What is the role of the task structure in producing rotational dynamics? In both the posture task and the center-out task, the task instruction instructs subjects to return to the initial movement 'state' by the end of the trial: in the posture task the simulated arm needs to return to the original posture upon disturbance, and in the center out task the arm needs to start from zero velocity and settle at the target with zero velocity. Is this structure what's causing the rotational dynamics? This is an important question both for this paper and for the field and the authors have a great simulation setup to explore it. For example, what happens if the task instructions u* instruct the arm to follow a random trajectory continuously, instead of stopping at some targets? With a simulated tracking task like this, one could eliminate obvious cases of return-to-original-state from the task. Would the network still produce rotational dynamics? Of course, I don't expect the authors to collect experimental monkey data for such new tasks, rather to just change the task instructions in their numerical simulations to explore the dependence of observed rotational dynamics on the task structure. I think this will help the message of the paper and can be very useful for the field.

   4. It would be beneficial if the authors could elaborate in the discussion on intuitive explanations of why sensory feedback can produce rotational dynamics even with no internal recurrent dynamics in the brain model. To me, it seems like sensory feedback is providing a path for recurrence to exist in the overall brain-arm system, so the non-recurrent neural networks can learn to exploit that path to effectively implement some recurrent dynamics. Some intuitive explanations like this will be helpful for readers.

   5. One main result in data from non-human primates is that there exist rotations also in the somatosensory cortex not just in motor cortex. A more thorough discussion of prior work on rotational dynamics or lack thereof across brain regions and behavioral tasks is important to add here. For example, besides the works cited by the authors, there are other works such as (Kao et al., 2015; Gao et al., 2016; Remington et al., 2018; Stavisky et al., 2019; Aoi et al., 2020; Sani et al., 2021) that discuss or show rotational dynamics in various brain regions and behavioral tasks and should be cited and discussed.

   6. The authors state that "In contrast, rotational dynamics appear to be absent in... MC activity during grasping driven by sensory inputs (Suresh et al., 2020)." There are other papers that study dynamics during reach-grasps and still finds rotational dynamics and modes (Abbaspourazad et al., 2021; Vaidya et al., 2015) and should be cited and discussed. The recent paper on naturalistic reach-grasps (Abbaspourazad et al., 2021) also argues for the involvement of a large-scale network in these movements, which further supports the authors' interpretation that "This interpretation of motor control emphasizes that the objective of the motor system is to attain the behavioural goal and this requires feedback processed by a distributed network." A discussion of this point made in this recent paper in the intro/discussion is important. Finally, there is a recent paper that argues for the input-driven nature of motor cortex (Sauerbrei et al., 2020) and is cited/discussed by the authors but briefly and mainly in the discussion. I think given the relevance of this recent paper to the core message here, it should also be briefly discussed in the introduction to better set up the work.

   Minor:

   1. The Methods are clear and comprehensive, but just to make understanding of the simulation setup easier, it would help to have a diagram of the computation graph for the recurrent and non-recurrent networks that shows their number of units, activations/nonlinearities, RNN cell type, etc., added as supplementary figure.

   2. Again, to help more clearly convey the simulations, it would help to show the task goals (x*) that are inputs to the simulated brain for example trials in each task (for example added to Figures 2 and 6).

   3. Similar to how VAF is shown on top of all plots of jPC planes, it would be helpful to have the rotation frequency for each jPC plane noted next to it. Currently it is difficult to find the jPC frequency associated with each plot from the text.

   4. I am a bit surprised by how different the null distributions are for modeling muscle activity (Figure 3F) and kinematics (Figure 3H). The null distribution is simply the R2 for a constrained or unconstrained dynamic model fit to a subsampled version of the neural activity. The only difference between the null distributions in Figure 3F and Figure 3H seems to be the downsampled dimension, which for muscle activity is 6 and for kinematics is 4 (per equation 1). Any insight will be welcome as to why down sampling the population activity to 4 (Figure 3H) results in so much worse R2 compared with down sampling it to 6 (Figure 3F)?

   References:

   Abbaspourazad, H., Choudhury, M., Wong, Y.T., Pesaran, B., Shanechi, M.M., 2021. Multiscale low-dimensional motor cortical state dynamics predict naturalistic reach-and-grasp behavior. Nature Communications 12, 607. https://doi.org/10.1038/s41467-020-20197-x

   Aoi, M.C., Mante, V., Pillow, J.W., 2020. Prefrontal cortex exhibits multidimensional dynamic encoding during decision-making. Nature Neuroscience 1-11. https://doi.org/10.1038/s41593-020-0696-5

   Gao, Y., Archer, E.W., Paninski, L., Cunningham, J.P., 2016. Linear dynamical neural population models through nonlinear embeddings, in: Lee, D.D., Sugiyama, M., Luxburg, U.V., Guyon, I., Garnett, R. (Eds.), Advances in Neural Information Processing Systems 29. Curran Associates, Inc., pp. 163-171.

   Kao, J.C., Nuyujukian, P., Ryu, S.I., Churchland, M.M., Cunningham, J.P., Shenoy, K.V., 2015. Single-trial dynamics of motor cortex and their applications to brain-machine interfaces. Nature Communications 6, 7759. https://doi.org/10.1038/ncomms8759

   Remington, E.D., Narain, D., Hosseini, E.A., Jazayeri, M., 2018. Flexible Sensorimotor Computations through Rapid Reconfiguration of Cortical Dynamics. Neuron 98, 1005-1019.e5. https://doi.org/10.1016/j.neuron.2018.05.020

   Sani, O.G., Abbaspourazad, H., Wong, Y.T., Pesaran, B., Shanechi, M.M., 2021. Modeling behaviorally relevant neural dynamics enabled by preferential subspace identification. Nature Neuroscience 24, 140-149. https://doi.org/10.1038/s41593-020-00733-0

   Stavisky, S.D., Willett, F.R., Wilson, G.H., Murphy, B.A., Rezaii, P., Avansino, D.T., Memberg, W.D., Miller, J.P., Kirsch, R.F., Hochberg, L.R., Ajiboye, A.B., Druckmann, S., Shenoy, K.V., Henderson, J.M., 2019. Neural ensemble dynamics in dorsal motor cortex during speech in people with paralysis. eLife 8, e46015. https://doi.org/10.7554/eLife.46015

   Vaidya, M., Kording, K., Saleh, M., Takahashi, K., Hatsopoulos, N.G., 2015. Neural coordination during reach-to-grasp. Journal of Neurophysiology 114, 1827-1836. https://doi.org/10.1152/jn.00349.2015

   Read the original source
5. 

   eLife

   Aug 6, 2021

   Reviewer #2 (Public Review):

   In this interesting study, Kalidindi et al. investigate the dynamics of responses in cortical areas during motor control.

   In a landmark paper, Churchland, Cunningham ... Shenoy suggested surprisingly that the dynamics of peri-movement related responses in neurons could be well described by a dynamical system with imaginary components to their eigen values and resulting in rotational structure. One possibility is that MC is an autonomous and rotational dynamical system with such imaginary eigen values. Since Churchland et al. 2012, and a companion paper Sussillo et al. 2015 originally appeared, there have been many discussions on how to interpret this result. For instances sequences of neural activity could lead to rotations, or smoothness in time, or nonnormal dynamical systems etc.

   Here, in this new study …

   Reviewer #2 (Public Review):

   In this interesting study, Kalidindi et al. investigate the dynamics of responses in cortical areas during motor control.

   In a landmark paper, Churchland, Cunningham ... Shenoy suggested surprisingly that the dynamics of peri-movement related responses in neurons could be well described by a dynamical system with imaginary components to their eigen values and resulting in rotational structure. One possibility is that MC is an autonomous and rotational dynamical system with such imaginary eigen values. Since Churchland et al. 2012, and a companion paper Sussillo et al. 2015 originally appeared, there have been many discussions on how to interpret this result. For instances sequences of neural activity could lead to rotations, or smoothness in time, or nonnormal dynamical systems etc.

   Here, in this new study the authors argue through a convincing combination of modeling and analysis that even without local recurrent connections in motor cortex, one could generate rotational structure in the dynamics. This paper is another attempt to understand these rotations and tries to argue that perhaps it is misleading to view the MC as a purely autonomical dynamical system and that feedback is an integral part of the process of generating these dynamics. I particularly liked the use of data from 5 monkeys in the paper and also the various perturbations to the model and the fact that the model has realistic feedback.

   These arguments are based on training a 2-stage RNN with feedback and use it to solve a task where they perturb the arm. They then analyze these networks and neural data and show 2 key results: Rotations can occur in non-motor cortices and recurrence is not necessary for rotational structure. Both of these results are important and deepen our understanding of rotational structure in areas involved in motor control.

   Weakness emerges from the fact that feedback in itself is a form of recurrence and not something that is mystically leading to rotations. This was something that Sussillo et al. 2015 made plain: "it should be stressed that, although our model reveals a robust dynamical solution to the problem of producing multiphasic EMG, the scope of the recurrent circuitry, cortical, central and/or feedback, supporting those dynamics remains an open question". In my opinion, the lack of a common framework in the paper for understanding local recurrence and longer time scale feedback reduces the impact of the study, because without such a framework, it feels like a slew of observations.

   These results build on Churchland et al. 2012, Sussillo et al. 2015, Sauerbrei et al. 2020, Suresh et al. 2020 and provide insight into how and when does rotational structure emerge in motor cortex. In my opinion, the study is interesting and provides food for thought and robust discussion for the many researchers working on motor control.

   Read the original source
6. 

   eLife

   Aug 6, 2021

   Reviewer #3 (Public Review):

   The authors trained monkeys to perform a posture perturbation task and showed that monkey motor cortex (MC) activity exhibits significant rotational dynamics during movement. These findings suggests that near-autonomous dynamics may not be the only explanation for rotational dynamics in the MC, as sensory inputs to the MC are likely important for solving this task. To validate this idea, the authors trained recurrent neural networks (RNNs) to actuate a two-link arm model and perform a similar posture perturbation task. Importantly, these RNNs receive delayed sensory feedback about about the state of the arm and the muscle activity produced. They found that the trained RNNs and the sensory inputs provided to these RNNs both exhibited rotational dynamics, while the corresponding muscle activity did not. This …

   Reviewer #3 (Public Review):

   The authors trained monkeys to perform a posture perturbation task and showed that monkey motor cortex (MC) activity exhibits significant rotational dynamics during movement. These findings suggests that near-autonomous dynamics may not be the only explanation for rotational dynamics in the MC, as sensory inputs to the MC are likely important for solving this task. To validate this idea, the authors trained recurrent neural networks (RNNs) to actuate a two-link arm model and perform a similar posture perturbation task. Importantly, these RNNs receive delayed sensory feedback about about the state of the arm and the muscle activity produced. They found that the trained RNNs and the sensory inputs provided to these RNNs both exhibited rotational dynamics, while the corresponding muscle activity did not. This suggests that sensory inputs could be the source of rotational dynamics in the RNNs. Indeed, the authors found that monkey somatosensory cortex also exhibited rotational dynamics during the posture perturbation task. Moreover, feedforward networks trained to perform the task also exhibited rotational dynamics, suggesting that recurrent connections are necessary for the emergence of rotational dynamics.

   This work provides an alternative explanation for the emergence of rotational dynamics in the motor cortex (MC). In doing so, the authors (1) dispel the idea that the only explanation for rotational dynamics is that the MC is a near-autonomous dynamical system and (2) show that rotational dynamics in the MC is consistent with the well-supported view that the MC uses sensory feedback for online motor control. The central thesis of this study is well-supported by both analyses of neural recordings and simulation experiments. In particular, the authors showed that RNNs driven by sensory inputs could also exhibit rotational dynamics. While this fact alone does not prove that sensory inputs induce rotational dynamics in the MC, it certainly shows that sensory feedback could be a potential mechanism. Overall, this is an important contribution that connects recent electrophysiological studies of population dynamics in the MC with the long-standing view of the MC as a feedback controller.

   Read the original source
7. 

   Version published to 10.1101/2020.11.17.387043v1 on bioRxiv

   Nov 20, 2020