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Conceived and designed the experiments: KT. Performed the experiments: KT. Analyzed the data: KT. Contributed reagents/materials/analysis tools: KT MO. Wrote the paper: KT MO.

The authors have declared that no competing interests exist.

Many redundancies play functional roles in motor control and motor learning. For example, kinematic and muscle redundancies contribute to stabilizing posture and impedance control, respectively. Another redundancy is the number of neurons themselves; there are overwhelmingly more neurons than muscles, and many combinations of neural activation can generate identical muscle activity. The functional roles of this neuronal redundancy remains unknown. Analysis of a redundant neural network model makes it possible to investigate these functional roles while varying the number of model neurons and holding constant the number of output units. Our analysis reveals that learning speed reaches its maximum value if and only if the model includes sufficient neuronal redundancy. This analytical result does not depend on whether the distribution of the preferred direction is uniform or a skewed bimodal, both of which have been reported in neurophysiological studies. Neuronal redundancy maximizes learning speed, even if the neural network model includes recurrent connections, a nonlinear activation function, or nonlinear muscle units. Furthermore, our results do not rely on the shape of the generalization function. The results of this study suggest that one of the functional roles of neuronal redundancy is to maximize learning speed.

There are overwhelmingly more neurons than muscles in the motor system. The functional roles of this neuronal redundancy remains unknown. Our analysis, which uses a redundant neural network model, reveals that learning speed reaches its maximum value if and only if the model includes sufficient neuronal redundancy. This result does not depend on whether the distribution of the preferred direction is uniform or a skewed bimodal, both of which have been reported in neurophysiological studies. We have confirmed that our results are consistent, regardless of whether the model includes recurrent connections, a nonlinear activation function, or nonlinear muscle units. Additionally, our results are the same when using either a broad or a narrow generalization function. These results suggest that one of the functional roles of neuronal redundancy is to maximize learning speed.

In the human brain, numerous neurons encode information about external stimuli, e.g., visual or auditory stimuli, and internal stimuli, e.g., attention or motor planning. Each neuron exhibits different responses to stimuli, but neural encoding, especially in the visual and auditory cortices, can be explained by the maximization of stimulus information

A critical problem exists in the relationship between motor cortex neurons and output units: the neuronal redundancy problem, or overcompleteness, which refers to the fact that the number of motor cortex neurons far exceeds the number of output units. Many different combinations of neural activities can therefore generate identical outputs. Neurophysiological and computational studies have revealed that the motor cortex exhibits neuronal redundancy

One of these types of redundancy is muscle redundancy: many combinations of muscle activities can generate identical movements. The functional roles of this muscle redundancy include impedance control to achieve accurate movements

Similar to the muscle and kinematic redundancies, neuronal redundancy likely has functional roles in motor control and learning. However, the functional roles of this redundancy are unclear. Here, using a redundant neural network, we investigate these functional roles by varying the number of model neurons while holding the number of output units constant. This manipulation allows us to control the degree of neuronal redundancy because, if a neural network includes a large number of neurons and a small number of output units, many different combinations of neural activities can generate identical outputs. It should be noted that we used a redundant neural network model that can explain neurophysiological motor cortex data

Initially, a linear model with a fixed decoder was used. Analytical calculations revealed that neuronal redundancy is a necessary and sufficient condition to maximize learning speed. This maximization is invariant whether the distribution of PDs is unimodal

Neuronal redundancy is defined as the dimensional gap between the number of neurons

In this study, we discuss the relationship between neuronal redundancy and learning speed by assuming adaptation to either a visuomotor rotation or a force field. These tasks are simulated by using a rotational perturbation

In the case of a fixed decoder,

In this case, the squared error can be calculated recursively as

Analytical calculations can yield necessary and sufficient conditions to maximize learning speed (see the

We numerically confirmed the above analytical results.

(A): Learning speed when

The question remains whether it is necessary for FD and FA to be distributed uniformly, so we assume that the values

(A): Scatter plot of

Some neurophysiological studies have suggested that the distribution of PD is a skewed bimodal

We have analytically elucidated the relevance of neuronal redundancy to learning speed only when

(A): Bar graphs and error bars depict sample means and standard deviations both of which are calculated using the results from 1000 sets of

Although we have revealed that neuronal redundancy maximizes learning speed when

(A): Learning speed when

In addition, we investigated whether neuronal redundancy or neuron number is important when

The generality of our results should be investigated because we analyzed only linear and feed-forward networks, but neurophysiological experiments have suggested the existence of recurrent connections

In addition, we used only deterministic gradient descent, so the generality regarding the learning rule needs to be investigated. In fact, previous studies have suggested that stochastic gradient methods are more biologically relevant than deterministic ones

Although our results have strong generality, there is still an open question regarding the robustness of noise: does neuronal redundancy maximize learning speed even in the presence of neural noise? Actually, neural activities show trial-to-trial variation

(A): Variance of the learning curve when

In many situations, learning in one context is generalized to different contexts, such as different postures

(A): Learning speed when

We have quantitatively demonstrated that neuronal redundancy maximizes learning speed. The larger the dimensional gap grows between the number of neurons and the number of constrained tasks, the faster learning speed becomes. This maximization does not depend on whether the PD distribution is unimodal or bimodal, the decoder is fixed or adaptable, the network is linear or nonlinear, the task is linear or nonlinear, or the learning rule is stochastic or non-stochastic. Additionally, we have shown that neuronal redundancy has another important functional role: it provides robustness in response to neural noise. Furthermore, neuronal redundancy maximizes learning speed in a manner independent of the shape of the generalization function. These results strongly support the generality of our results.

Neuronal redundancy maximizes learning speed because only

At first glance, our results may seem inconsistent with the results of Werfel et al.

Neuronal redundancy plays another important role: generating robustness in response to neural noise (

Our study assumed the following task: participants move their arms towards one of

The neural population generates a force of

If the error occurs between

Equation (13) yields the following update rule of squared error:

Equation (16) requires that the larger the eigenvalues become, the faster the learning speed becomes and the smaller the residual error becomes (

What kind of conditions can simultaneously satisfy equations (19) and (20)? The only answer is sufficient neuronal redundancy, i.e.,

The above analytical calculations hold even when

When

When

In the Importance of Neuronal Redundancy section, the neural network generates the output

Equation (13) yields the following update rule for motor commands:

Because the shape of the generalization function depends on the task, we need to confirm the generality of our results with regard to the shape of the generalization function. To simulate various shapes of generalization functions, we used the von-Mises function

We conducted 100 baseline trials with

For all of the statistical tests, we used the Wilcoxon sign rank test. It should be noted that the

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We thank D. Nozaki, Y. Sakai, Y. Naruse, K. Katahira, T. Toyoizumi, and T. Omori for their helpful discussions.