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Movement is governed by rotational neural dynamics in spinal motor networks

Abstract

Although the generation of movements is a fundamental function of the nervous system, the underlying neural principles remain unclear. As flexor and extensor muscle activities alternate during rhythmic movements such as walking, it is often assumed that the responsible neural circuitry is similarly exhibiting alternating activity1. Here we present ensemble recordings of neurons in the lumbar spinal cord that indicate that, rather than alternating, the population is performing a low-dimensional ‘rotation’ in neural space, in which the neural activity is cycling through all phases continuously during the rhythmic behaviour. The radius of rotation correlates with the intended muscle force, and a perturbation of the low-dimensional trajectory can modify the motor behaviour. As existing models of spinal motor control do not offer an adequate explanation of rotation1,2, we propose a theory of neural generation of movements from which this and other unresolved issues, such as speed regulation, force control and multifunctionalism, are readily explained.

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Fig. 1: Neuronal population activity in the lumbar spinal cord has rotational dynamics.
Fig. 2: Rotational dynamics emerges in the BSG model.
Fig. 3: Network control of amplitude in the BSG model.
Fig. 4: Modulation of period in the BSG model.
Fig. 5: Two motor programs in model and experiment.

Data availability

Data are available through a link to an online repository, which can be found on the laboratory web page (https://berg-lab.net/) or on reasonable request from the corresponding authors.

Code availability

The Python code that was used for simulating the BSG network is available in an online repository (https://github.com/BergLab/BSG). The Matlab code for analysing the experimental data is available on the laboratory web page (https://berg-lab.net/) or on reasonable request from the corresponding authors.

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Acknowledgements

This work was supported by The Independent Research Fund Denmark and Mobilex (number 8020-00436B, DFF–1333-00226A), the Carlsberg Foundation (number CF18-0845) and the Lundbeck Foundation (number R366-2021-233).

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Authors and Affiliations

Authors

Contributions

R.W.B. conceived the original experiments. P.C.P. set up and performed the experiments, collected the data and analysed some of the data. M.V., H.L. and R.W.B. conceived the original theory. H.L. performed the model simulations. H.L. and R.W.B. designed and developed the theory, analysed the experimental data and wrote the manuscript.

Corresponding authors

Correspondence to Henrik Lindén or Rune W. Berg.

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The authors declare no competing interests.

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Extended data figures and tables

Extended Data Fig. 1 Firing rate is rhythmic with rotational population dynamics across trials in the lumbar spinal motor network during rhythmic movement.

a, The firing rates of 3 sample units (black) with their spike times indicated as blue dots. A sinus function was fitted to the firing rate (red) and the mean square error is indicated (bottom right). b, More sample units, with the mean square error indicated to the left. c, Distribution of mean square errors for the population (n = 214). The mean square error is unitless, since the firing rates were high pass filtered and normalized (maximal firing is 1). d, The firing rates (normalized, color coded) of 214 spinal neurons in laminae VII-VIII as a function of time and sorted according to phase with respect to the nerve activity (hip flexor). Eight consecutive trials from same experiment with a 5 min pause in between each. e, The phase distribution across the neuronal population. f, The population activity has rotational dynamics, as demonstrated by the circular motion of the first two PCs. The PCs were calculated by the data of one trial (trial 3, "*") and the applied to the rest of the trials. The sorting of neurons was according to their phase relation with representative nerve for one trial (also trial 3, "*") and this order was maintained for the rest of the trials. Bottom scale bars represent 1000.

Extended Data Fig. 2 Rotational population dynamics in a spinal motor network across animals.

a, The rhythmic firing rates in populations of spinal neurons in laminae VII-VIII shown in colors as a function of time and sorted according to phase with respect to a nerve (hip flexor). A representative trial from 5 experiments of approximately 10 s demonstrate similar sequential/rotational population activity. Animal used in Extended Data Fig. 1 is marked "*". b, The corresponding distribution of neurons having preferred phases among the population of rhythmic neurons. c, Population activity represented by first two PCs exhibit rotational dynamics. Scale bars: 250. d, Cumulative explained variance by principal components, indicating the population dynamics is low-dimensional, i.e. most of the variance is captured by few components.

Extended Data Fig. 3 Neuronal population trajectories in PC-space have lower tangling than the corresponding motor nerve trajectories.

a, Illustration that during rotational dynamics the points in the trajectory that move in opposite direction are also far apart, i.e. they have low tangling (left), whereas during alternation the points of the trajectory that move in opposite direction are also close, i.e. have high tangling (right). b, The ratio of tangling metric of the PC trajectory of the nerves (Qnerve) to that of the network (Qnetwork). This ratio is close to 100%, which indicates most trials and animals had a larger tangling of the motor nerves than the network. (N = 11, data set 1; N = 10, data set 2; N = 10, data set 3; N = 4, data set 4; N = 3, data set 5). c–g, Sample trials from 5 different data sets. Left is shown the phase sorted firing rate activity (top) and the associated nerves (bottom). The nerves were rectified and low-pass filtered (red) on temporal scale matching the firing rates. The PCs of network (middle left) and nerves (middle right, green). Scales of PCs are variance normalized. The tangling metric (Q) for the nerve PCs (in 3 dimensions) is calculated as a function of time (t) through the trial and plotted versus that for the network. The ratio of points below the x = y–line (pale blue) is indicated in percent and form one point in panel (b). Note that the nerve trajectories more resemble "alternation" whereas the network more resembles "rotation"-scheme of (a).

Extended Data Fig. 4 Rotational ensemble activity within the excitatory and inhibitory sub-populations in the BSG-model.

a, Activation of the motor circuit by descending drive. b, The firing rates of 10 sample excitatory neurons oscillate because of the descending input. (c) Sorting the excitatory neurons according to phase of firing rates reveals a sequential activity like the previously observed for all neurons. d-e, Activity and similar sorting of the inhibitory sub-populations reveals similar sequential and rotational dynamics within that sub-population. f, The network eigenmode for the whole network: Each dot represent both the phase (the polar angle) and the peak firing rate (the radius) for a given neuron (n = 200). g-h, Similar plot for the excitatory and inhibitory populations. i-k, the distribution of phases in linear histograms for all neurons (i), excitatory (j) and inhibitory neurons (k). To be compared with experimental distributions (Extended Data Figs. 1 and 2).

Extended Data Fig. 5 BSG-model: Correlation between descending drive and radius of rotation as well as amplitude of nerve output without affecting the period.

a, For low neuronal gain (top), the eigenvalue spectrum does not have any eigenvalues that cross the stability line (broken vertical line). As the gain increases (downward direction) the spectrum expands, and eigenvalues cross the stability line. For larger gain the eigenvalues cross the stability line farther. b, The associated population dynamics (sorted firing rates) exhibit oscillation of increasing magnitude as the drive increase. c, The rotational dynamics also has a radius that increases with increasing drive. d, The resulting motor nerve output is also increasing in amplitude. e, Descending drive (gain) versus the population firing rate (RMS), radius of rotation in PC space, f, and amplitude of nerve output (flexor RMS), g. h, the radius of rotation (PC1 RMS) vs. the nerve amplitude (flexor RMS).

Extended Data Fig. 6 Radius of rotation correlates with nerve output in experiment.

a, Sample trial where the population activity was divided up in pieces with the corresponding nerve output b. c, The PC manifolds had rotation with varying radius. d-f, other pieces with same organization. g, The RMS of the nerve activity versus the RMS of the first two PCs for various pieces of activity had a significant correlation (F-statistic of rejection of no trend at p«0.01). h, The R2 values for all animal tested (n=5). *: F-statistic of rejection of null hypothesis of zero correlation at p«0.01. f, Scale bar: 1000.

Extended Data Fig. 7 Representation of one behavior in the subspace of another behavior.

a, The variance captured by the projection of the network dynamics onto the first three PCs of another trial (green) normalized by the variance captured by the PCs of its own dynamics. Orange: the subspace-overlap of a different behavior. Independent samples: N= Same behavior/ different behavior, N = 5/6, data set 1; N = 3/3, data set 2; N=8/10, data set 3; N = 3/4, data set 4; N = 2/3, data set 5). Whisker plots represent min and max values. Box plots represent median −25% and +75% quartiles. b, the subspace representation of the nerve activity of same behavior (green) and a different behavior (orange). N=Same behavior / different behavior: N = 5/6, data set 1; N = 3/3, data set 2; N = 8/10, data set 3; N = 3/4, data set 4; N = 2/3, data set 5). Whisker plots represent min and max values. Box plots represent median −25% and +75% quartiles. c, Nerve overlap plotted against the network overlap. A large overlap in nerve output is associated with a large overlap in network overlap. Gray line represents a linear fit, red region represents 95% confidence. d-aa, The flexor/extensor nerve output from the BSG-network. e, The sorted neuronal population firing rate (n = 400 neurons) with rotational dynamics. (cc) Color map of the population firing rate. dd, Mean (red) and variance of the population activity. e, same organization as in (d), but for experimental data. Animal no. 3 trial 8.

Extended Data Fig. 8 Multifunctionalism in the BSG-model.

a, Five examples of specific activation/modulation of selected neurons in the network ("activation profiles"). The top profile has an even distribution, whereas all the below profiles have selective modulation of specific neurons. b, The ensemble activity as a result of the activation profile show a sequential activity, with similar but not identical sequence of activity. c, The first two PCs, based on the top activation profile, all exhibit rotational dynamics, albeit with different radius and trajectories. (d) the output motor patterns associated with the different activation profiles and ensemble activities.

Extended Data Fig. 9 "Deletions" both in the experimental data and in the BSG model.

a, Six trials (1–6) shown with the neuronal firing rates sorted according to phase (color map, top) and the 6 nerves (bottom) during a motor behavior (pocket scratching). The absence of a burst, i.e. a deletion, was observed in the hip extensor nerve recording (red dots) whereas the hip flexor (bottom trace) seems to continue and combine two cycles although with a small decrease. Regular bursts are also indicated (blue dots). The corresponding trajectories in PC-space are shown with the corresponding dots matching the time in the nerve activity. A selected period around the occurrence of one delete is indicated in the nerve traces (red vertical lines). The corresponding time in the trajectory is also indicated in red. Note that deletions tend to occur at smaller radius of the rotation and the population firing rates (color map) are dimmer at those instances. b, A proper motor behavior devoid of "deletions" can be produced by the balanced sequence generator despite receiving a varying input (top). The firing rates for the sorted neuronal population (middle), and the resulting motor nerve output pattern (bottom). c, The appropriate motor program shown in (b) is achieved by a selective gain-modulation, i.e. gain-profile (y-axis), across the neuronal population (x-axis). d, Population activity from (b) represented by in PC-space by the first two components. e, When the varying input transiently becomes too low at a certain phase the nerve cycle is absent, i.e. a "deletion" has occurred (red dots). The firing rates of the neuronal population will be lower at these instances and hence appear dimmer in the color map (middle). A consequent absence of a burst in the nerve is seen (nerve 3, compare red and blue dots). f, the PC-trajectories corresponding to (e) indicated as 1,2 and 3. The temporary distortion of the trajectory at a particular phase is associated with a deletion (red dots). Red parts of the trajectory represent the period between vertical red bars indicated in the nerve activities (d–f). Compare with (a).

Extended Data Fig. 10 Reconstruction of nerve output based on linear decoding of neuronal population activity.

a, A linear decoder function was estimated using a training set consisting of 9 trials of same behavior. Top: color coded firing rates for the neuronal population (sorted according to phase) with 9 trials concatenated. Bottom: the rectified and low-pass filtered motor nerve output of 6 nerves. b, A trial, that was not included in the training set, is used for validation of the linear decoder. Top: the firing rates of the population, like (a). Bottom: The nerve output of 3 selected nerves (rectified and LP-filtered in green, nerves 3, 4 and 6). The reconstructed standard deviations of the nerves (orange) are multiplied by white Gaussian noise to imitate nerve output (gray). c, the correlation between predicted and actual nerve output for the six nerves (individual dots) are shown for two different motor behaviors (right and left pocket scratching) across the 5 experiments. The median value across all nerves and experiments is R = 0.6. All 5 data sets had correlations, which were found significantly different than zero using a t-test of Pearson linear correlation, p << 0.01, N = 6. d, Training set consisting of 9 bouts, ie. trials of different motor behaviors, which is used to train a linear decoder function. Top: color coded firing rates for the neuronal population (sorted according to phase) with 9 concatenated bouts. Bottom: the rectified and low-pass filtered motor nerve output of 6 nerves. e, two trials, that was not included in the training set, contained instances of "deletions". Top: the firing rates of the population, like (a). Bottom: The nerve output of 3 selected nerves (rectified and LP-filtered in green, nerves 3, 4 and 6). The reconstructed standard deviations of the nerves (orange) are multiplied by white Gaussian noise to imitate nerve output (gray). f, the correlation between predicted and actual nerve output for the six nerves (individual dots) are shown for two different motor behaviors (right and left pocket scratching) across the 5 experiments. The median value across all nerves and experiments is R = 0.6. All 6 correlations were found significantly different than zero using a t-test of Pearson linear correlation, p << 0.01, N = 1300 temporal-measurements.

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Lindén, H., Petersen, P.C., Vestergaard, M. et al. Movement is governed by rotational neural dynamics in spinal motor networks. Nature 610, 526–531 (2022). https://doi.org/10.1038/s41586-022-05293-w

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