When you swing a tennis racket, muscles in your arm contract in a specific sequence. For this to happen, millions of neurons in your brain and spinal cord must fire to make those muscles contract. If you swing the racket a second time, the same muscles in your arm will contract again. But the firing pattern of the underlying neurons will probably be different. This phenomenon, in which different patterns of neural activity generate the same outcome, is called neural redundancy.

Neural redundancy allows a set of neurons to perform multiple tasks at once. For example, the same neurons may drive an arm movement while simultaneously planning the next activity. But does performing a given task constrain how often different patterns of neural activity can be produced? If so, this would limit whether other tasks could be carried out at the same time. To address this, Hennig et al. trained macaque monkeys to use a brain-computer interface (BCI). This is a device that reads out electrical brain activity and converts it into signals that can be used to control another device. The key advantage of a BCI is that the redundant activity patterns are precisely known. The monkeys learned to use their brain activity, via the BCI, to move a cursor on a computer screen in different directions.

The results revealed that monkeys could only produce a limited number of different patterns of brain activity for a given BCI cursor movement. This suggests that the ability of a group of neurons to multitask is restricted. For example, if the same set of neurons is involved in both planning and performing movements, then an animal’s ability to plan a future movement will depend on the one it is currently performing.

BCIs can help patients who have suffered stroke or paralysis. They enable patients to use their brain activity to control a computer or even robotic limbs. Understanding how the brain controls BCIs will help us improve their performance and deepen our knowledge of how the brain plans and performs movements. This might include designing BCIs that allow users to multitask more effectively.

Neural circuits relay information from one population of neurons to another. This relay involves successive stages of downstream neurons reading out the activity of upstream neurons. In many cases, the same activity in the downstream population can be produced by different population activity patterns in the upstream population, a phenomenon termed neural redundancy. Redundancy is ubiquitous in neural computation, from sensory input to motor output. For example, during a task where subjects need to discriminate the color of a stimulus while ignoring its orientation (Mante et al., 2013), population activity patterns corresponding to the same color but different orientations are read out equivalently, and are therefore redundant. There is mounting evidence that redundancy in readouts may provide various computational benefits. For example, neural redundancy may allow us to prepare movements without executing them (Kaufman et al., 2014; Elsayed et al., 2016), enable stable computation despite unstable neural dynamics (Driscoll et al., 2017; Druckmann and Chklovskii, 2012; Murray et al., 2017) and allow the central nervous system to filter out unwanted noise (Moreno-Bote et al., 2014).

To fully utilize the proposed benefits of neural redundancy, the population activity should be allowed to freely vary, as long as the readout of this activity remains consistent with task demands. This would allow the population activity to perform computations that are not reflected in the readout. However, a commonly held assumption is that neural activity might also be constrained by energetics: All things being equal, if two population activity patterns are read out equivalently, the brain should prefer the pattern that requires less energy to produce (Laughlin et al., 1998; Barlow, 1969; Levy and Baxter, 1996). These two lines of reasoning raise the following questions: What principles guide the production of redundant neural activity patterns? Are there constraints on which redundant activity patterns can be produced? If so, this may limit the extent to which neural circuits can exploit the proposed computational benefits of redundancy.

Redundancy has been studied extensively in motor control (Lashley, 1933; Bernstein, 1967), albeit in terms of muscular redundancy rather than neural redundancy. During arm movements, different combinations of muscle activity can lead to the same arm kinematics, meaning these different muscle activity patterns are redundant. Previous work on this muscle redundancy problem has identified two principles guiding the selection of redundant muscle activity. First, because muscle contraction requires energy in the form of ATP, the selected muscle activity should require minimum energy relative to the other redundant options (Thoroughman and Shadmehr, 1999; Huang et al., 2012; Fagg et al., 2002). Second, a minimal intervention strategy has been proposed in which subjects control only the aspects of muscle activity that influence the task outcome, and allow for variability in the aspects of muscle activity that do not influence the task outcome (Scholz and Schöner, 1999; Todorov and Jordan, 2002; Valero-Cuevas et al., 2009). To generate movements, the brain not only needs to deal with muscle redundancy, but also neural redundancy, which has been less studied.

One way in which neural redundancy can arise is when there are more elements (neurons or muscles) upstream than downstream. During arm movements, the activity of around thirty muscles in the arm and hand is controlled by tens of thousands of neurons in the spinal cord (Gray, 1918; Feinstein et al., 1955). Those neurons are in turn influenced by millions of neurons in the primary motor cortex and other motor areas (Ettema et al., 1998; Lemon, 2008). Thus, the neural control of arm movement is redundant (Figure 1A), in that different population activity patterns can generate the same movement (Rokni et al., 2007; Ajemian et al., 2013). Can the principles of muscular redundancy inform our understanding of neural redundancy?

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A common challenge in studying neural redundancy is that it is typically not known which neural activity patterns are redundant, because we do not know how downstream neurons or muscles read out information. In this study we overcome this problem by leveraging a brain-computer interface (BCI), in which the activity of dozens of neurons is read out as movements of a cursor on a computer screen (Figure 1B) (Taylor et al., 2002; Carmena et al., 2003; Hochberg et al., 2006; Ganguly and Carmena, 2009; Gilja et al., 2012; Hauschild et al., 2012; Sadtler et al., 2014). A key advantage of a BCI is that the readout of the population activity (termed the BCI mapping) is fully known and defined by the experimenter (Golub et al., 2016). This allows us to determine precisely the redundant population activity patterns, which are those that move the cursor in exactly the same way. To illustrate this, consider a simplified example where the activity of two neurons controls a 1D cursor velocity (Figure 1C). The two dark green activity patterns produce the same cursor movement (${\displaystyle {\mathbf{v}}_1}$), and the two light green patterns produce a different movement ($v_2$). We can decompose any population activity pattern into two orthogonal components: output-potent activity and output-null activity (Figure 1C, black axes) (Kaufman et al., 2014; Law et al., 2014). The output-potent component determines the cursor’s movement, whereas the output-null component has no effect on the cursor. Two population activity patterns are redundant if they have the same output-potent activity, but different output-null activity (e.g. the dark green square and circle on the '$v_1$' dotted line in Figure 1C). The question we address here is, which redundant population activity patterns are preferred by the nervous system? To answer this, we assessed the distribution of output-null activity produced during each cursor movement (Figure 1D), and compared it to what we would expect to observe under each of several candidate hypotheses for explaining neural redundancy.

We trained three Rhesus macaques to perform a brain-computer interface task in which they controlled the velocity of a cursor on a computer screen by volitionally modulating neural activity in primary motor cortex. To understand the principles guiding the selection of redundant neural activity, we compared the observed distributions of output-null activity to those predicted by three different hypotheses. The first two hypotheses we considered were inspired by studies of muscle redundancy. First, by analogy to minimum energy principles (Thoroughman and Shadmehr, 1999; Huang et al., 2012; Fagg et al., 2002), neural activity may minimize unnecessary spiking (Barlow, 1969; Levy and Baxter, 1996). Second, by analogy to the minimal intervention strategy (Scholz and Schöner, 1999; Todorov and Jordan, 2002; Valero-Cuevas et al., 2009), output-null activity might be uncontrolled (i.e. output-potent activity is modified independently of output-null activity) because neural variability in this space has no effect on cursor movement. Third, we considered the possibility that the distribution of redundant activity may be coupled with the task-relevant activity, so that producing particular activity patterns in output-potent dimensions requires changing the distribution of activity in output-null dimensions.

We tested all hypotheses in terms of their ability to predict the distribution of output-null activity, given the output-potent activity. Hypotheses were tested within the space in which the population activity naturally resides, termed the intrinsic manifold (Sadtler et al., 2014). The results of Sadtler et al. (2014) indicate that neural activity cannot readily leave this manifold, and more recent results demonstrate that neural activity is further constrained by a neural repertoire within the intrinsic manifold (Golub et al., 2018). However, a repertoire defines only a set of population activity patterns, and not how often different activity patterns within the repertoire are produced. Therefore, to understand the principles governing the selection among redundant population activity patterns, we focused on predicting the distribution of redundant population activity within the intrinsic manifold and neural repertoire.

We found strong evidence for the third hypothesis, that redundant activity is coupled with task-relevant activity. This indicates that neural redundancy is resolved differently than muscular redundancy. Furthermore, the output-null space should not be thought of as a space in which neural activity can freely vary to carry out computations without regard to the output-potent activity. Instead, the distribution of output-null activity is constrained by the corresponding output-potent activity. If the required output-potent activity is defined by the task demands, this can constrain how the output-null activity can vary, and correspondingly the computations that can be carried out in the output-null space.

To study the selection of redundant neural activity, we used a BCI based on 85–94 neural units recorded using a Utah array in the primary motor cortex in each of three Rhesus macaques. Animals modulated their neural activity to move a computer cursor in a 2D center-out task (see Materials and methods; Figure 1—figure supplement 1). At the beginning of each experiment, we identified the 10 dimensions of the population activity that described the largest activity modulations shared among the neural units, termed the intrinsic manifold (Sadtler et al., 2014). A two-dimensional subspace of the 10-dimensional intrinsic manifold was mapped to horizontal and vertical cursor velocity and was therefore output-potent, while the eight orthogonal dimensions were output-null. Our goal was to predict the joint distribution of the observed neural activity in this eight-dimensional output-null space.

We tested several hypotheses for the selection of redundant neural activity using the following logic. First, we predicted the distributions of output-null activity expected under each hypothesis. All hypotheses’ predictions were consistent with the observed behavior (i.e. the output-potent activity), and we ensured that none of these predictions required unrealistic firing rates when combined with the output-potent activity. Next, we compared the predicted distributions to the observed distributions of output-null activity to determine which hypothesis provided the best match to the observed distributions. We built the observed distributions of output-null activity as follows: At each time step during the BCI task, we assigned the recorded population activity pattern to one of eight bins corresponding to the direction of cursor movement (${\displaystyle 0^{\circ },{45}^{\circ },{90}^{\circ }}$0°, 45°, 90°, etc.) produced by that neural activity. We binned by the cursor movement because we are studying the population activity that is redundant for a given cursor movement direction. For each bin, we projected the corresponding population activity patterns onto the eight output-null dimensions of the intrinsic manifold. The black histograms in Figure 2, Figure 3, and Figure 4 show the marginal distributions in the first three output-null dimensions (ordered by variance accounted for). The colored histograms in Figure 2, Figure 3, and Figure 4 are the predicted output-null distributions built under each hypothesis, which we compared to the observed distributions. The ensuing three subsections describe each hypothesis, and compare how well the corresponding predicted distributions matched the observed distributions.

Figure 2

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Histograms of predictions and data, as depicted in Figure 2B–C and Figure 2E–F.

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Histograms of predictions and data, as depicted in Figure 3B–C and Figure 3E–F.

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Histograms of predictions and data, as depicted in Figure 4B–C and Figure 4E–F.

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During each experiment, animals controlled two different BCI mappings (i.e. the two mappings had different output-potent subspaces). The first mapping was an ‘intuitive’ one that required no learning for proficient control. The second mapping was a within-manifold perturbation (see Materials and methods). For the second mapping, we analyzed the trials after the behavioral performance reached asymptote. Each hypothesis predicted the distribution of output-null activity that the animal would produce under the second mapping. To form its prediction, a hypothesis could utilize the output-potent activity observed during the second mapping, as well as all neural activity recorded under control of the first mapping. This technique allowed us to avoid circularity in our results because we built the hypothesized distributions using the first behavioral context and evaluated those predictions in the second. Additionally, because animals learned to use the BCI mappings through trial and error, it is possible that the animals’ assumptions about the output-null dimensions do not align perfectly with the actual output-null dimensions of the BCI mapping. To control for this, we estimated the animal’s internal model of the BCI mapping (Golub et al., 2015). The results in the main text are based on this internal model, and we show in supplemental figures that all results still hold when using the actual BCI mapping.

Task-transfer hypotheses accurately predict output-null activity

Thus far, the hypothesis that best predicts the observed output-null activity is the one that uses previously observed activity to generate its predictions (Uncontrolled-empirical). This motivated us to consider more refined hypotheses that make use of this previously observed activity to generate predictions.

We first considered the hypothesis that in order to produce a desired movement, the subject selects neural activity as if he were still using the previous mapping, and corrects this activity only to ensure task success (Figure 4A, Persistent Strategy). Conceptually, when the subject wants to move the cursor in a particular direction using the current BCI mapping, he starts with the population activity patterns that he used to move the cursor in that direction under an earlier mapping (Figure 4A, light blue shading). Because this activity will not move the cursor in the same way that it did under the previous mapping, this activity is modified along the output-potent dimensions of the current mapping (Figure 4A, red arrows), reflecting the minimal intervention principle (Todorov and Jordan, 2002; Valero-Cuevas et al., 2009; Diedrichsen et al., 2010). This is similar to the Uncontrolled-empirical hypothesis in that we assume activity in output-null dimensions can be corrected independently of the activity in output-potent dimensions. However, instead of sampling from the entire distribution of previously observed output-null activity at each time step, here we only sample from the subset of this activity observed when subjects needed to move the cursor in the same direction as the current time step. The predictions of this hypothesis (Figure 4B–C) differed from the observed output-null activity by 17.4% ±0.7% (mean ± SE) across sessions.

The principle of minimal intervention posits that output-null activity can change independently from output-potent activity. Here we examine this assumption in detail. Previous work has found that the characteristic ways in which neurons covary (i.e. the dimensions of the intrinsic manifold) persist even under different BCI mappings, perhaps owing to underlying network constraints (Sadtler et al., 2014). All hypotheses we consider here are evaluated within the intrinsic manifold, and thus respect these constraints on population variability. Because the dimensions of the intrinsic manifold capture the variability among the neurons, it is plausible that the activity along different dimensions of the intrinsic manifold can vary independently, consistent with the minimal intervention principle. By contrast, in the next hypothesis we consider the possibility that activity along different dimensions exhibit dependencies.

We considered the hypothesis that the distribution of activity in output-null dimensions would be predictably coupled with the activity in output-potent dimensions, even under a different BCI mapping when those dimensions were not necessarily potent and null. Under this hypothesis (Figure 4D, Fixed Distribution), given the output-potent activity, the distribution of the corresponding output-null activity remains the same as it was under a different BCI mapping (Figure 4D, blue frequency distribution), even if this activity was not output-null under the other mapping. This hypothesis predicts that neural activity patterns are ‘yoked’ across dimensions, such that producing particular activity in output-potent dimensions requires changing the distribution of activity in output-null dimensions. The histograms of output-null activity predicted by the Fixed Distribution hypothesis were a striking visual match to the recorded activity, and accurately predicted the dependence of these distributions on the cursor direction (Figure 4E–F). Overall, these predictions differed from the observed output-null activity by only 13.4% ±0.5% (mean ± SE) across sessions.

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Histogram, mean, and covariance errors of all hypotheses for all sessions, as depicted in Figure 5 and Figure 5—figure supplement 1.

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The Fixed Distribution hypothesis yielded a lower histogram error than all other hypotheses across sessions from three different animals (Figure 5A). In total, the Fixed Distribution hypothesis had the lowest histogram error in 41 of 42 sessions. The histogram error metric does not explicitly capture the degree to which hypotheses predicted the mean output-null activity, or any correlations that exist across output-null dimensions. We therefore assessed how well the predictions captured the mean and covariance of observed data in all output-null dimensions jointly (see Materials and methods). In agreement with our findings for histogram error, the mean (Figure 5B) and covariance (Figure 5C) of output-null activity was best predicted by the Fixed Distribution hypothesis, with an average mean error of 23.5 ± 1.4 spikes/s (mean ± SE) and an average covariance error of 1.4 ± 0.1 (mean ± SE in arbitrary units; see Materials and methods). These error metrics offer further evidence that the Fixed Distribution hypothesis provides a good match to the output-null distribution, as measured by the agreement between the first and second moments of the two distributions. Because these error metrics rely on a limited number of trials, they should not be compared relative to zero error. We estimated the smallest histogram, mean, and covariance errors achievable by any hypothesis, given the limited number of samples available to estimate the true output-null distributions (see Materials and methods, and gray regions in Figure 5). The errors of Fixed Distribution were exceedingly close to the lowest achievable error given the number of samples available (see Materials and methods). Next, we found that the Fixed Distribution hypothesis achieved the lowest prediction errors among all hypotheses when data for each monkey was considered individually (Figure 5—figure supplement 1). We repeated our analyses to predict output-null activity produced during the first mapping using activity observed during the second mapping (Figure 5—figure supplement 2). We also predicted output-null activity using the actual BCI mapping rather than the animal’s internal model to define the output-null dimensions (Figure 5—figure supplement 3). Both analyses yielded results similar to those in Figure 5.

Predicting changes in neural variability when activity becomes output-null

So far we have shown that the Fixed Distribution hypothesis provides a better explanation for the structure of output-null activity than hypotheses incorporating constraints on firing rates or the minimal intervention principle. We next sought stronger evidence for the Fixed Distribution hypothesis by assessing our predictions in the particular dimensions of population activity where it is least likely to hold. Because cursor velocity is a two-dimensional quantity, all but two dimensions of population activity for each BCI mapping are output-null. Thus, given two different BCI mappings, most dimensions will be output-null under both mappings, and so most components of the population activity have no reason to change from one mapping to the other. Therefore, we assessed whether our results held in dimensions of population activity that were output-potent during the first mapping, but output-null during the second mapping (see Materials and methods). These are the dimensions in which one would expect to see the most changes in the population activity between the first and second mappings.

Our hypotheses make distinct predictions about how the variance of activity should change if a dimension is output-potent under the first mapping and becomes output-null under the second mapping. For example, according to the Minimal Firing and Minimal Deviation hypotheses, the variance of activity will collapse in dimensions that are output-null because unnecessary spiking is undesirable. Thus, if a dimension becomes output-null, variance in this space should exhibit a marked decrease. On the other hand, the Uncontrolled hypotheses predict that, when conditioned on the cursor movement, variance will expand when the activity is output-null. This occurs because variability in this dimension will no longer affect cursor movement, and would therefore no longer need to be suppressed. Finally, the Fixed Distribution hypothesis posits that the same distributions of output-null activity will be observed regardless of whether a dimension was previously output-potent or output-null, and so this hypothesis predicts that there will be little to no change in the variance of activity in a particular dimension under the two mappings.

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Observed and predicted change in covariance across sessions, as depicted in Figure 6C.

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We asked whether the variance of population activity decreased, increased, or remained the same in dimensions that changed from being output-potent to output-null (Figure 6A). Critically, we computed the variance of activity after first binning by the corresponding angle in the output-potent dimensions of the second mapping. This was done so that the neural activity in each bin would all result in similar cursor movements under the second mapping, and is identical to the procedure used previously to assess the errors of the hypotheses’ predictions. Notably, binning in this way means that each bin may contain activity corresponding to different cursor movements under the first mapping, and so one might expect that in each bin the activity recorded under the first mapping would be more heterogeneous than the activity recorded under the second mapping.

We observed that the variance of population activity recorded under the first and second mappings was remarkably similar in the dimensions that changed from output-potent to output-null, even though these activity patterns usually corresponded to different cursor movements under the two mappings (Figure 6B). Thus, the variance of activity did not change much when an output-potent dimension became output-null, in agreement with the predictions of the Fixed Distribution hypothesis. To quantify these observations, we computed the average change in variance in each session (see Materials and methods). Across sessions, we found that the variance of observed activity showed a small but significant decrease when it became output-null (Figure 6C, ‘Data’) (t-test, $p<0.001$). This is in contrast to the predictions of the Minimal Firing and Minimal Deviation hypotheses, which predicted much larger decreases.

The observed change in variance lies closest to the predictions of the Fixed Distribution hypothesis. In fact, we observed that the Fixed Distribution hypothesis also predicted a slight decrease in variance in dimensions that became output-null (Figure 6C, ‘Fixed Distribution’) (t-test, $p<0.001$). This slight predicted change in variance occurs because the distributions of activity in the output-potent dimensions of the second mapping are different under the first and second mappings. Because the Fixed Distribution hypothesis predicts a fixed conditional distribution of output-null activity given the output-potent activity, slightly different sets of output-potent activity will result in a slightly different distribution of the corresponding output-null activity.

These analyses show that, contrary to the predictions of the minimal firing and uncontrolled hypotheses, the variance of population activity did not change dramatically in dimensions that were output-potent under the first mapping and output-null under the second mapping. We also assessed whether the reverse was true—if the variance of activity changed in dimensions that began as output-null and became output-potent. To measure this, we repeated the above analyses after predicting output-null activity produced during the first mapping using the activity observed under the second mapping (as in Figure 5—figure supplement 2). We found that the activity showed little to no change in variance in these dimensions (t-test, $p>0.5$), in agreement with the predictions of Fixed Distribution (Figure 6—figure supplement 1).

Importantly, the agreement between the observed output-null activity and the predictions of the Fixed Distribution hypothesis in these analyses indicates that our ability to accurately predict the distribution of output-null activity is not merely a result of most activity being output-null under both mappings. Instead, the distribution of output-null activity remains consistent with the Fixed Distribution hypothesis even in the output-null dimensions that were previously output-potent.

In Figure 6C, the observed output-null activity showed a larger decrease in variance than the predictions of the Fixed Distribution hypothesis, at least in the 2D subspace of output-null activity that was output-potent during the first mapping. This slight decrease in variance is in the direction of the predictions of Minimal Firing and Minimal Deviation. If this decrease in variance is to be explained by Minimal Firing or Minimal Deviation principles, we would expect that the observed mean output-null activity would also move in the direction of the predictions of Minimal Firing and Minimal Deviation, relative to what is predicted by Fixed Distribution. To see if this was the case, we first computed the distance of the observed mean output-null activity from the mean predicted by Minimal Deviation for each movement direction, and compared this to the distance of the mean output-null activity predicted by Fixed Distribution from the mean predictions of Minimal Deviation (Figure 6—figure supplement 2A). We did not find evidence that the observed mean output-null activity was closer to the mean predicted by Minimal Deviation than was the mean predicted by Fixed Distribution (one-sided Wilcoxon signed rank test, $p>0.5$; see Figure 6—figure supplement 2B and Materials and methods). Repeating the analysis with Minimal Firing instead of Minimal Deviation yielded similar results (one-sided Wilcoxon signed rank test, $p>0.5$). Thus, while we observed a slight decrease in the variance of output-null activity in dimensions that changed from output-potent to output-null, we did not find any evidence that the mean output-null activity moved in the direction of the predictions of Minimal Firing or Minimal Deviation.

Recent work has suggested that neural redundancy may be exploited for various computations (Druckmann and Chklovskii, 2012; Kaufman et al., 2014; Moreno-Bote et al., 2014; Elsayed et al., 2016; Driscoll et al., 2017; Murray et al., 2017). However, if the activity in output-null dimensions is constrained by the output-potent activity, then this may limit the ability of output-null activity to perform computations without affecting the readout. Here, we studied neural redundancy in the primary motor cortex using a BCI, where it is known exactly which population activity patterns are redundant, meaning they produce an identical cursor movement. We generated predictions of the distributions of output-null neural activity for subjects performing a BCI cursor control task, and compared them to the distributions observed in our experiments. We found that hypotheses inspired by minimal firing and minimal intervention principles, drawn from theories of muscle coordination, did not accurately predict the observed output-null activity. Instead, we found that the distribution of output-null activity was well predicted by the activity in the two output-potent dimensions. This coupling between the output-potent and output-null activity implies that, when output-potent activity is used to satisfy task demands, there are constraints on the extent to which neural circuits can use redundant activity to perform additional computations.

Our results indicate that the way in which neural redundancy is resolved is different from how muscle redundancy is resolved. There have been several prevalent proposals for how muscle redundancy is resolved, including minimal energy, optimal feedback control (OFC), and habitual control. Models incorporating minimal energy principles have helped to explain observed gait (McNeill Alexander and McNeill, 2002) and arm reaches (Thoroughman and Shadmehr, 1999; Huang et al., 2012; Fagg et al., 2002; Farshchiansadegh et al., 2016). By analogy, it has been proposed that the brain may prefer an ‘economy of impulses’ (Barlow, 1969; Softky and Kammen, 1991; Levy and Baxter, 1996), resolving neural redundancy by minimizing the production of action potentials. However, we found that minimal energy principles in terms of firing rates do not play a dominant role in the selection of output-null neural activity. Given that metabolic activity can decrease without corresponding changes in firing rates (Picard et al., 2013), the brain may implement minimal energy principles without influencing the way neural redundancy is resolved.

OFC posits that motor control signals are selected to minimize a cost function that depends on task requirements and other factors, such as effort or delayed reward. OFC models have been widely used to explain muscle activity during motor tasks (Todorov, 2004; Scott, 2004; Diedrichsen et al., 2010). Our results for neural activity differ in two important respects from OFC predictions with standard cost functions involving task requirements and effort. First, those implementations of OFC predict that variability in task-irrelevant dimensions should be higher than variability in task-relevant dimensions, a concept often referred to as the ‘uncontrolled manifold’ (Scholz and Schöner, 1999). We found that the variability of neural activity did not increase in dimensions that went from being task-relevant to task-irrelevant (Figure 6C). Second, those implementations of OFC predict a ‘minimal intervention’ strategy, whereby activity in task-relevant dimensions is corrected independently of activity in task-irrelevant dimensions (Todorov and Jordan, 2002; Valero-Cuevas et al., 2009; Diedrichsen et al., 2010). Three of the hypotheses we tested incorporate this minimal intervention principle: Uncontrolled-uniform, Uncontrolled-empirical, and Persistent Strategy. None of these hypotheses predicted neural activity in task-irrelevant dimensions as accurately as did the Fixed Distribution hypothesis, which predicts that the distributions of task-relevant and task-irrelevant activity are yoked. Overall, our work does not rule out the possibility that OFC is appropriate for predicting neural activity. First, it may be possible to design a cost function such that OFC predictions are consistent with the findings presented here. Second, one could consider applying OFC with the control signal being the input to M1 (e.g. PMd activity), rather than the control signal being M1 activity (as we have done here) or muscle activity (where OFC has been traditionally applied). This could induce coupling between the output-potent and output-null dimensions of the M1 activity, and thereby yield predictions that are consistent with the findings presented here.

It has also been proposed that muscle recruitment is habitual rather than optimal, such that muscle recruitment under altered dynamics is a rescaled version of that under normal control (de Rugy et al., 2012). The results for habitual control are similar to what we found for neural activity, in that (1) we could predict activity from previously observed activity, and (2) we observed a tight coupling of the distributions of task-relevant and task-irrelevant activity (in contrast to minimal intervention). However, the results for habitual control are different from our findings in that we found that subjects appear to use the same distribution of activity in each of two different BCI mappings, whereas different (overlapping) subsets of muscle activation patterns were used under different conditions in de Rugy et al. (2012).

Given how many dimensions of population activity there are (in this case, 10), it is somewhat surprising that conditioning on only the two output-potent dimensions could provide so much explanatory power for predicting the distribution in the remaining neural dimensions. This suggests that many of the dimensions of population activity are coupled, that is, changing the activity along some dimensions may also lead to changes along other dimensions, even though those dimensions are mutually orthogonal. During arm movement control, output dimensionality and presumably the neural dimensionality are larger than in our BCI setup. We speculate that during arm movements, many of the null dimensions will remain coupled with the potent dimensions, thereby yielding results similar to what we found here. Future work could examine whether animals can be trained to uncouple dimensions, as well as the effects of larger output-potent dimensionality on redundancy, by repeating our analyses with a higher-dimensional effector, such as a multiple degree-of-freedom robotic limb (e.g. Wodlinger et al., 2015).

The results presented here are related to, and go beyond, those in Golub et al. (2018). Although the two studies analyzed data from the same experiments, they ask distinct questions. Golub et al. (2018) focused on explaining the changes in population activity underlying behavioral learning. By contrast, in the present work we seek to determine the constraints on activity in the task-irrelevant (i.e. output-null) dimensions. In other words, while Golub et al. (2018) focused on explaining the changes leading to behavioral learning, we focus here on the principles other than behavior that constrain population activity. As a result, all hypotheses we consider in the present work make predictions consistent with the observed behavior in the output-potent dimensions.

Golub et al. (2018) found that the amount of learning animals showed was consistent with a fixed neural repertoire of population activity patterns being reassociated to control the second BCI mapping. The repertoire of population activity refers to the set of population activity patterns that were observed, whereas here we focused on the distribution, which describes how often the animals produced different activity patterns. In other words, the finding of a fixed repertoire is a statement about the support of the distribution of population activity, whereas here we found that the distribution of population activity can be predicted in output-null dimensions, given the output-potent activity. Because many different distributions of neural activity can be constructed from a fixed repertoire, the present results represent a stronger constraint on population activity than that shown in Golub et al. (2018). Indeed, the majority of the hypotheses we tested were consistent with a fixed neural repertoire, and thus cannot be disambiguated based on our prior work. This is evidenced by the predicted distributions largely overlapping with the support of the actual data distributions (Figures 2–4). The two hypotheses that were not fully consistent with a fixed repertoire are the Minimal Firing and Uncontrolled-uniform hypotheses. However, in the context of predicting the distribution of activity in redundant dimensions, these hypotheses represent interesting cases worth considering (i.e. where population activity either obeys minimal firing constraints, or that the output-null activity is fully unstructured, respectively), and so we included these hypotheses to cover these possibilities.

It is interesting to consider the relationship between arm movements and BCI cursor movements (Orsborn et al., 2014; Vyas et al., 2018). If the dimensions responsible for moving the arm overlap with both the output-potent and output-null dimensions of the BCI, this might explain the coupling we observe between the output-potent and output-null dimensions. However, in these experiments, the animal’s arm was not moving during BCI control (see Extended Data Figure 5 in Sadtler et al., 2014). Thus, the activity we study here resides within the arm’s output-null dimensions. This implies that in our recordings the arm’s output-potent dimensions do not overlap with either the output-potent or the output-null dimensions of the BCI, and so arm movements (or the lack thereof) are unlikely to explain the coupling we observed between the output-potent and output-null dimensions of the BCI. Overall, being unaware of extra output-potent dimensions would likely make the predictions of the Fixed Distribution hypothesis worse, not better. The reason for this is as follows. The Fixed Distribution hypothesis predicts that the distribution of activity in output-null dimensions depends upon the corresponding output-potent activity. Under this hypothesis, the more we know of the output-potent activity, the better we can predict the output-null distribution. If there is an output-potent dimension that we have not accounted for in our analyses, accounting for this dimension would likely improve our predictions. The fact that we were able to accurately predict the output-null distributions (13% histogram error on average, with the lowest possible error being 7%) without knowing all the potent dimensions is then evidence that these extra potent dimensions, if they exist, would not provide substantial additional predictive power.

In this work, we define a set of population activity patterns as redundant if they all result in the same readout in downstream areas. This definition of redundancy comes from early work on motor control (Bernstein, 1967; Sporns and Edelman, 1993), where it was noted that different motor signals can result in the same movement kinematics. This is related to but distinct from the information-theoretic definition of redundancy (Schneidman et al., 2003; Latham et al., 2005; Averbeck et al., 2006). In the information-theoretic case, redundancy describes the extent to which correlations among neurons limit decoding accuracy for different stimuli. This is distinct from the type of redundancy studied here, defined as the existence of multiple population activity patterns corresponding to the same readout. For example, by the information-theoretic definition, a system may have no redundancy (e.g. the population activity allows one to perfectly decode the encoded variable), but there may still be multiple population activity patterns that refer to this same encoded variable.

We found that the distribution of output-null activity could be well predicted using activity recorded under a different BCI mapping. Two factors of our experimental design are particularly relevant when interpreting this result. First, we used a balanced center-out task design in which subjects made roughly equal numbers of movements in each direction. If we had, for example, required far more leftward than rightward movements, this would have altered the distribution of joint activity and skewed the estimates of output-null activity during the second mapping. Second, this study focused on short timescales, where we predicted output-null activity within one to two hours of subjects learning a new BCI mapping. On this timescale, the motor system must be able to rapidly learn a variety of different mappings between neural activity and behavior, and thus, a variety of different sets of redundant activity. An interesting avenue for further research would be to determine if the constraints we observe on neural redundancy remain over longer timescales. Given repeated practice with the same BCI mapping across days and weeks (Ganguly and Carmena, 2009), it is possible that there are different and perhaps fewer constraints on neural redundancy than what we found here.

We have tested six specific hypotheses about how neural redundancy is resolved. These hypotheses cover a spectrum of how strongly the activity in output-null dimensions is constrained, with the minimal firing hypotheses being the most constrained, the minimal intervention hypotheses being the least constrained, and the Fixed Distribution hypothesis lying in between. Although the hypotheses we tested are not exhaustive, the best hypothesis (Fixed Distribution) yielded predictions of the distributions of output-null activity whose marginal histograms differed from the data by only 13% on average (Figure 4F), where we estimated the lowest error possible to be 7% on average. Further improvements to the prediction accuracy may be possible by incorporating additional constraints, such as dynamics (Shenoy et al., 2013). It should be stressed that our focus here was on predicting the distribution of output-null activity. Future work can assess whether output-null activity can be predicted on a time-step-by-time-step basis.

The central premise of the null space concept is that some aspects of neural activity are read out by downstream areas (output-potent) while other aspects are not (output-null) (Kaufman et al., 2014). This idea is related to the study of noise correlations, where it was recognized that activity fluctuations that lie outside of a stimulus encoding space (i.e. ‘stimulus-null’) are not detrimental to the stimulus information encoded by the neurons (Averbeck et al., 2006; Moreno-Bote et al., 2014). Studies have also shown that structuring neural activity in an appropriate null space can allow for multiplexing of different types of information (Mante et al., 2013; Raposo et al., 2014), as well as stable behavior (Leonardo, 2005; Rokni et al., 2007; Ajemian et al., 2013) and stable working memory (Druckmann and Chklovskii, 2012; Murray et al., 2017) in the presence of time-varying neural activity. Additionally, the existence of output-null dimensions in the motor system may facilitate motor learning (Moorman et al., 2017; Ranganathan et al., 2013; Singh et al., 2016) or allow for motor preparation (Kaufman et al., 2014; Elsayed et al., 2016) or novel feedback processing (Stavisky et al., 2017) without causing overt movement. Our work suggests that there may be limits on the extent to which output-null activity might be leveraged for neural computation. The coupling we observe between the distributions of output-null and output-potent activity suggests that output-null activity is not modified independently of output-potent activity. This coupling may cause activity fluctuations in a stimulus-null space to influence the downstream readout, or limit one’s ability to plan the next movement without influencing the current movement. Moving forward, an important direction for understanding the computations performed by different brain areas is to find out which aspects of the neural activity are read out (Pagan et al., 2013; Kaufman et al., 2014) and to understand how the dependencies like those identified in this study impact the computations being performed.

 To understand how our results might change if we recorded from more neural units, we assessed the dimensionality and shared variance of population activity with a varying number of units (Williamson et al., 2016) (Appendix 1—figure 1). For each session, we fit factor analysis (FA) models (as defined in Equation 1) to subsets of varying numbers of units and identified the number of factors needed to maximize the cross-validated data likelihood. This resulted in estimates of the model parameters $L$ and $\Psi$. As in Williamson et al. (2016), dimensionality was defined as the number of eigenvectors of $L{L}^{\top }$ needed to explain 95% of the shared variance. Concretely, if the eigenvalues of $L{L}^{\top }$ are ${\lambda }_1,{\lambda }_2,\mathrm{...},{\lambda }_D$, then $d_{shared}$ is the smallest $J$ such that $\left({\displaystyle {\sum }_{i=1}^J}{\lambda }_i\right)/\left({\displaystyle {\sum }_{i=1}^D}{\lambda }_i\right)\ge 0.95$. Note that the absolute dimensionality depends on the method (% shared variance threshold) and criterion (threshold = 95%) used for assessing dimensionality. This is the same method used in Williamson et al. (2016), but differs slightly from the method used in Sadtler et al. (2014). We found that the dimensionality of the population activity increased with the number of units (Appendix 1—figure 1A). 

 As in Williamson et al. (2016), we computed the percentage of each neural unit’s activity variance that was shared with other recorded units (% shared variance). We calculated the average percent shared variance across neurons as follows: 

where $L_k$ is the row of $L$ corresponding to unit $k$. We found that the % shared variance initially increased with the number of units, then reached an asymptote, such that the % shared variance was similar with 30 and 85 units (Appendix 1—figure 1B). The results in Appendix 1—figure 1A–B imply that the top $\sim$10 dimensions explain nearly all of the shared variance, and that additional dimensions identified by recording from more units explain only a small amount of additional shared variance. Thus, recording from more units beyond the $\sim$85 units that we recorded in these experiments is not likely to reveal additional dimensions with substantial shared variance. We next measured the principal angles between modes identified using 30 units with those identified using 85 units (Appendix 1—figure 1C) (Bjorck and Golub, 1973). Modes were defined as the eigenvectors of the shared covariance matrices corresponding to units from the 30-unit set (i.e. the eigenvectors of $L{L}^{\top }$ where $L$ includes only the rows corresponding to the same 30 units). To restrict the analysis to the number of modes used to estimate the intrinsic manifold, only the ten modes explaining the most shared variance were included in the principal angle calculations. The small principal angles between modes identified using 30 and 85 units indicate that the dominant modes remained largely unchanged when using more units, in agreement with Williamson et al. (2016). These modes define the intrinsic manifold, the space within which we perform all of our analyses in the current work. Thus, recording from more units beyond the $\sim$85 units that we recorded in these experiments is not likely to substantially change the results reported in this work.

Appendix 1—figure 1

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Source data files have been provided for Figures 2-6. Code for analysis has been made available at https://github.com/mobeets/neural-redundancy-elife2018, with an MIT open source license (copy archived at https://github.com/elifesciences-publications/neural-redundancy-elife2018).

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Author details

1. Jay A Hennig

   1. Program in Neural Computation, Carnegie Mellon University, Pittsburgh, United States
   2. Center for the Neural Basis of Cognition, Carnegie Mellon University, Pittsburgh, United States
   3. Machine Learning Department, Carnegie Mellon University, Pittsburgh, United States

   Contribution

   Software, Formal analysis, Writing—original draft, Writing—review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0001-7982-8553

2. Matthew D Golub

   1. Center for the Neural Basis of Cognition, Carnegie Mellon University, Pittsburgh, United States
   2. Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, United States

   Contribution

   Conceptualization, Writing—review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-4508-0537

3. Peter J Lund

   1. Center for the Neural Basis of Cognition, Carnegie Mellon University, Pittsburgh, United States
   2. Machine Learning Department, Carnegie Mellon University, Pittsburgh, United States
   3. Department of Biomedical Engineering, Carnegie Mellon University, Pittsburgh, United States

   Contribution

   Conceptualization, Writing—review and editing

   Competing interests

   No competing interests declared

4. Patrick T Sadtler

   1. Center for the Neural Basis of Cognition, Carnegie Mellon University, Pittsburgh, United States
   2. Department of Bioengineering, University of Pittsburgh, Pittsburgh, United States

   Contribution

   Conceptualization, Writing—review and editing, Performed the animal experiments

   Competing interests

   No competing interests declared

5. Emily R Oby

   1. Center for the Neural Basis of Cognition, Carnegie Mellon University, Pittsburgh, United States
   2. Department of Bioengineering, University of Pittsburgh, Pittsburgh, United States

   Contribution

   Writing—review and editing, Performed the animal surgeries, Performed the animal experiments

   Competing interests

   No competing interests declared

6. Kristin M Quick

   1. Center for the Neural Basis of Cognition, Carnegie Mellon University, Pittsburgh, United States
   2. Department of Bioengineering, University of Pittsburgh, Pittsburgh, United States

   Contribution

   Conceptualization, Writing—review and editing

   Competing interests

   No competing interests declared

7. Stephen I Ryu

   1. Department of Neurosurgery, Palo Alto Medical Foundation, California, United States
   2. Department of Electrical Engineering, Stanford University, California, United States

   Contribution

   Writing—review and editing, Performed the animal surgeries

   Competing interests

   No competing interests declared

8. Elizabeth C Tyler-Kabara

   1. Center for the Neural Basis of Cognition, Carnegie Mellon University, Pittsburgh, United States
   2. Department of Physical Medicine and Rehabilitation, University of Pittsburgh, Pittsburgh, United States
   3. Department of Neurological Surgery, University of Pittsburgh, Pittsburgh, United States

   Contribution

   Writing—review and editing, Performed the animal surgeries

   Competing interests

   No competing interests declared

9. Aaron P Batista

   1. Center for the Neural Basis of Cognition, Carnegie Mellon University, Pittsburgh, United States
   2. Department of Bioengineering, University of Pittsburgh, Pittsburgh, United States

   Contribution

   Conceptualization, Supervision, Writing—original draft, Writing—review and editing

   Contributed equally with

   Byron M Yu and Steven M Chase

   Competing interests

   No competing interests declared

10. Byron M Yu

    1. Center for the Neural Basis of Cognition, Carnegie Mellon University, Pittsburgh, United States
    2. Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, United States
    3. Department of Biomedical Engineering, Carnegie Mellon University, Pittsburgh, United States

    Contribution

    Conceptualization, Supervision, Writing—original draft, Writing—review and editing

    Contributed equally with

    Aaron P Batista and Steven M Chase

    Competing interests

    No competing interests declared

    "This ORCID iD identifies the author of this article:" 0000-0003-2252-6938

11. Steven M Chase

    1. Center for the Neural Basis of Cognition, Carnegie Mellon University, Pittsburgh, United States
    2. Department of Biomedical Engineering, Carnegie Mellon University, Pittsburgh, United States

    Contribution

    Conceptualization, Supervision, Writing—original draft, Writing—review and editing

    Contributed equally with

    Aaron P Batista and Byron M Yu

    For correspondence

    schase@cmu.edu

    Competing interests

    No competing interests declared

    "This ORCID iD identifies the author of this article:" 0000-0003-4450-6313

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