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# Chance, long tails, and inference in a non-Gaussian, Bayesian theory of vocal learning in songbirds

Edited by Terrence J. Sejnowski, Salk Institute for Biological Studies, La Jolla, CA, and approved July 23, 2018 (received for review July 21, 2017)

## Significance

Skilled behaviors are learned through a series of trial and error. The ubiquity of such processes notwithstanding, current theories of learning fail to explain how the speed and the magnitude of learning depend on the pattern of experienced sensory errors. Here, we introduce a theory, formulated and tested in the context of a specific behavior—vocal learning in songbirds. The theory explains the observed dependence of learning on the dynamics of sensory errors. Furthermore, it makes additional strong predictions about the dynamics of learning that we verify experimentally.

## Abstract

Traditional theories of sensorimotor learning posit that animals use sensory error signals to find the optimal motor command in the face of Gaussian sensory and motor noise. However, most such theories cannot explain common behavioral observations, for example, that smaller sensory errors are more readily corrected than larger errors and large abrupt (but not gradually introduced) errors lead to weak learning. Here, we propose a theory of sensorimotor learning that explains these observations. The theory posits that the animal controls an entire probability distribution of motor commands rather than trying to produce a single optimal command and that learning arises via Bayesian inference when new sensory information becomes available. We test this theory using data from a songbird, the Bengalese finch, that is adapting the pitch (fundamental frequency) of its song following perturbations of auditory feedback using miniature headphones. We observe the distribution of the sung pitches to have long, non-Gaussian tails, which, within our theory, explains the observed dynamics of learning. Further, the theory makes surprising predictions about the dynamics of the shape of the pitch distribution, which we confirm experimentally.

Learned behaviors—reaching for an object, talking, and hundreds of others—allow the organism to interact with the ever-changing surrounding world. To learn and execute skilled behaviors, it is vital for such behaviors to fluctuate from iteration to iteration. Such variability is not limited to inevitable biological noise (1, 2), but rather a significant part of it is controlled by animals themselves and is used for exploration during learning (3, 4). Furthermore, learned behaviors rely heavily on sensory feedback. The feedback is needed, first, to guide the initial acquisition of the behaviors and then to maintain the needed motor output in the face of changes in the motor periphery and fluctuations in the environment. Within such sensorimotor feedback loops, the brain computes how to use the inherently noisy sensory signals to change patterns of activation of inherently noisy muscles to produce the desired behavior. This transformation from sensory feedback to motor output is both robust and flexible, as demonstrated in many species in which systematic perturbations of the feedback dramatically reshape behaviors (1, 5⇓⇓–8).

Since many complex behaviors are characterized by both tightly controlled motor variability and robust sensorimotor learning, we propose that, during learning, the brain controls the distribution of behaviors. In contrast, most prior theories of animal learning have assumed that there is a single optimal motor command that the animal tries to produce and that, after learning, deviations from the optimal behavior result from the unavoidable (Gaussian) downstream motor noise. Such prior models include the classic Rescorla–Wagner (RW) model (9), as well as more modern approaches belonging to the family of reinforcement learning (10⇓–12), Kalman filters (13, 14), or dynamical Bayesian filter models (15, 16). In many of these theories, the variability of the behavior is intrinsic to motor exploration and deliberately controlled, but the distribution of this variability is not itself shaped by the animal’s experience (17⇓–19). Such theories have addressed many important experimental questions, such as evaluating the optimality of the learning process (13, 20⇓⇓–23), accounting for multiple temporal scales in learning (7, 13, 24, 25), identifying the complexity of behaviors that can be learned (26), and pointing out how the necessary computations could be performed using networks of spiking neurons (12, 27⇓⇓⇓–31).

However, despite these successes, most prior models that assume that the brain aims to achieve a single optimal output have been unable to explain some commonly observed experimental results. For example, since such theories assume that errors between the target and the realized behavior drive changes in future motor commands, they typically predict large behavioral changes in response to large errors. In contrast, experiments in multiple species report a decrease in both the speed and the magnitude of learning with an increase in the experienced sensory error (6, 22, 32, 33). One can rescue traditional theories by allowing the animal to reject large errors as “irrelevant”—unlikely to have come from its own actions (22, 34). However, such rejection models have not yet explained why the same animals that cannot compensate for large errors can correct for even larger ones, as long as their magnitude grows gradually with time (6, 33).

Here we present a theory (Fig. 1) of a classic model system for sensorimotor learning—vocal adaptation in a songbird—in which the brain controls a probability distribution of motor commands and updates this distribution by a recursive Bayesian inference procedure. The distribution of the song pitch is empirically heavy tailed (Fig. 2*C*), and the pitch variability is much smaller in a song directed at a female vs. the undirected song, suggesting that the variability and hence its non-Gaussian tails are deliberately controlled. Thus, our model does not make the customary Gaussian assumptions. The focus on learning and controlling (non-Gaussian) distributions of behavior allows us to capture successfully all of the above-described nonlinearities in learning dynamics and, furthermore, to account for previously unnoticed learning-dependent changes in the shape of the distribution of the behavior.

## Results

### Biological Model System.

Vocal control in songbirds is a powerful model system for examining sensorimotor learning of complex tasks (36). The phenomenology we are trying to explain arises from experimental approaches to inducing song plasticity (33). Songbirds sing spontaneously and prolifically and use auditory feedback to shape their songs toward a “template” learned from an adult bird tutor during development. When sensory feedback is perturbed (see below) using headphones to shift the pitch (fundamental frequency) of auditory feedback (33), birds compensate by changing the pitch of their songs so that the pitch they hear is closer to the unperturbed one. As shown in Fig. 2*A*, the speed of the compensation and its maximum value, which is measured as a fraction of the pitch shift and referred to as the magnitude of learning hereafter, decrease with the increasing shift magnitude, so that a shift of three semitones results in near-zero fractional compensation. Crucially, the small compensation for large perturbation does not reflect the limited plasticity of the adult brain since imposing the perturbation gradually, rather than instantaneously, results in a large compensation (Fig. 2*B*).

### Data.

We use experimental data collected in our previous work (8, 33) to develop our mathematical model of learning. As detailed in ref. 39, we used a virtual auditory feedback system (8, 40) to evoke sensorimotor learning in adult songbirds. For this, miniature headphones were custom fitted to each bird and used to provide online auditory feedback in which the pitch (fundamental frequency) of the bird’s vocalizations could be manipulated in real time, with a loop delay of roughly 10 ms. In addition to providing pitch-shifted feedback, the headphones largely blocked the airborne transmission of the bird’s song from reaching the ear canals, thereby effectively replacing the bird’s natural airborne auditory feedback with the manipulated version. Pitch shifts were introduced after a baseline period of at least 3 d in which birds sang while wearing headphones but without pitch shifts. All pitch shifts were implemented relative to the bird’s current vocal pitch and were therefore “correctable” in the sense that if the bird changed its vocal pitch to fully compensate for the imposed pitch shift, the pitch of auditory feedback heard through the headphones would be equal to its baseline value. All data were collected during undirected singing (i.e., no female bird was present).

### Mathematical Model.

To describe the data, we introduce a dynamical Bayesian filter model (Fig. 1*A*). We focus on just one variable learned by the animal during repeated singing—the pitch of the song syllables. Even though the animal learns the motor command and not the pitch directly, we do not distinguish between the produced pitch ϕ and the motor command leading to it because the latter is not known in behavioral experiments. We set the mean “baseline” pitch sung by the animal as

However, while an instantaneous output of the motor circuit in our model is a scalar value of the pitch, the state of the motor learning system at each time step is a probability distribution over motor commands that the animal expects can lead to the target motor behavior. This is in contrast to the more common assumption that the state of the learning system is a scalar, usually the mean behavior, which is then corrupted by the downstream noise (34). Thus, at time t, the animal has access to the prior distribution over plausible motor commands,

After producing a vocalization, the bird then senses the pitch of the produced song syllable through various sensory pathways. Besides the normal airborne auditory feedback reaching the ears, which we can pitch shift, information about the sung pitch may be available through other, unmanipulated pathways. For example, efference copy may form an internal short-term memory of the produced specific motor command (42). Additionally, proprioceptive sensing presumably also provides unshifted information (43). Finally, unshifted acoustic vibrations might be transmitted through body tissue in addition to the air, as is thought to be the case in studies that use pitch shifts to perturb human vocal production (44, 45).

We denote all feedback signals as

Finally, the animal expects that the motor command needed to produce the target pitch with no error may change with time because of slow random changes in the motor plant. In other words, in the absence of new sensory information, the animal must increase its uncertainty about which command to produce with time (this is a direct analogue of increase in uncertainty of the Kalman filter without new measurements). Such increase in uncertainty is given by

We choose *A* and *B*). Since the bird sings hundreds of song bouts daily, we now use the law of large numbers and replace the unknown sensory feedback for individual vocalizations by its expectation value **1** as

As illustrated in Fig. 1*B*, such Bayesian filtering behaves differently for Gaussian and heavy-tailed likelihoods and propagators. Indeed, if the two likelihoods are Gaussians, their product is also a Gaussian centered between them. In this case, the learning speed of an animal is linear in the error Δ, no matter how large this error is, which conflicts with the experimental results in songbirds and other species (5, 8, 22, 36). Similarly, if the two likelihoods have long tails, then when the error is small, their product is also a single-peaked distribution as in the Gaussian case. However, when the error size Δ is large, the product of such long-tailed likelihoods is bimodal, with evidence peaks at the shifted and the unshifted values, with a valley in the middle. Since the prior expectations of the animal are developed before the sensory perturbation is turned on, they peak near the unshifted value. Multiplying the prior by the likelihood then leads to suppression of the shifted peak and hence of large error signals in animal learning.

In Eq. **2**, there are three distributions to be defined: **2** with no perturbation, *Materials and Methods*). To constrain the selection of specific functional forms of the distributions, we point out that the error in sensory feedback obtained by the animal is a combination of many noisy processes, including both sensing itself and the neural computation that extracts the pitch from the auditory input and then compares it to the target pitch. By the well-known generalized central limit theorem, the sum of these processes is expected to converge to what are known as Lévy alpha-stable distributions, often simply called stable distributions (47) (*Materials and Methods*). If the individual noise sources have finite variances, the stable distribution will be a Gaussian. However, if the individual sources have heavy tails and infinite variances, then their stable distribution will be heavy tailed as well (Cauchy distribution is one example). Most stable distributions cannot be expressed in a closed form, but they can be evaluated numerically (*Materials and Methods*). Here we assume symmetric stable distributions, truncated at *Materials and Methods*). Such distributions are characterized by three parameters: the stability parameter α (measuring the proportion of the probability in the tails), the scale or width parameter γ, and the location or the center parameter μ (the latter can be predetermined to be 0, Δ, or the previous time-step value in our case). For three distributions

### Fits to Data.

We fitted the set of six parameters of our model simultaneously to all of the data shown in Fig. 2. Our dataset consists of 23 individual experiments across five experimental conditions: four constant pitch-shift learning curves and one gradual, staircase-shift learning curve (see *Materials and Methods* for details). As mentioned previously, birds learn the best (larger and faster compensation) for smaller perturbations, here 0.5 semitone (Fig. 2*A*). In contrast, for a large 3-semitone perturbation, the birds do not compensate at all within the 14 d of the experiment. However, the birds are able to learn and compensate large perturbations when the perturbation increases gradually, as in the staircase experiment in Fig. 2*B*. Importantly, the baseline distribution (Fig. 2*C*) has a robust non-Gaussian tail, supporting our model. We note that our six-parameter model fits are able to simultaneously describe all of these data with a surprising precision, including their most salient features: dependence of the speed and the magnitude of the compensation on the perturbation size for the constant and the staircase experiments, as well as the heavy tails in the baseline distribution.

### Predictions.

Mathematical models are useful to the extent that they can predict experimental results not used to fit them. Quantitative predictions of qualitatively new results are particularly important for arguing that the model captures the system’s behavior. To test the predictive power of our model, we used it to predict the dynamics of higher-order statistics of pitches during learning, rather than using it to simply predict the mean behavior. We first use the model to predict time-dependent measures of the variability (SD in this case) of the pitch. As shown in Fig. 3 *A*–*E*, our model correctly predicted time-dependent behaviors in the SD in both single-shift (Fig. 3 *A*–*D*) and staircase-shift experiments (Fig. 3*E*) with surprising accuracy. We stress again that no new parameter fits were done for these curves. Potentially even more interesting, Fig. 3*F* shows that our model is capable of predicting unexpected features of the probability distribution of pitches, such as the asymmetric and bimodal structure of the pitch distribution at the end of the staircase-shift experiment. This bimodal structure is predicted by our theory, since the theory posits that the (bimodal) likelihood distribution (Fig. 1*B*, *Bottom*) will iteratively propagate into the observable pitch distribution (the prior). The existence of the bimodal pitch distribution in the data therefore provides strong evidence in support of our theory. Importantly, this phenomenon can never be reproduced by models based on animals learning a single motor command with Gaussian noise around it, rather than a heavy-tailed distribution of motor commands.

## Discussion

We introduced a mathematical framework within the class of observation–evolution models (35) for understanding sensorimotor learning: a dynamical Bayesian filter with non-Gaussian (heavy-tailed) distributions. Our model describes the dynamics of the whole probability distribution of the motor commands, rather than just its mean value. We posit that this distribution controls the animal’s deliberate exploration of plausible motor commands. The model reproduces the learning curves observed in a range of songbird vocal adaptation experiments, which classical behavioral theories have not been able to do to date. Further, also unlike the previous models, our approach predicts learning-dependent changes in the width and shape of the distribution of the produced behaviors.

To further increase the confidence in our model, we show analytically (*Materials and Methods*) that traditional linear models with Gaussian statistics (13) cannot explain the different levels of compensation for different perturbation sizes. While we cannot exclude that birds would continue adapting if exposed to perturbations for longer time periods and would ultimately saturate at the same level of adaptation magnitude, the Gaussian models are also argued against by the shape of the pitch distribution, which shows heavy tails (Figs. 2*C* and 3*F*), and by our ability to predict not just the mean pitch, but the whole pitch distribution dynamics during learning.

An important aspect of our dynamical model is its ability to reproduce multiple different timescales of adaptation (Fig. 2 *A* and *B*) using a nonlinear dynamical equation with just a single timescale of the update (1 d). As with other key aspects of the model, this phenomenon results from the non-Gaussianity of the distributions used and specifically the many timescales built into a single long-tailed kernel of uncertainty increase over time. This is in contrast to other multiscale models that require explicit incorporation of many timescales (7, 13). While multiple timescales could be needed to account for other features of the adaptation, our model clearly avoids this for the present data. We expect that an extension of our model to include multiple explicit timescales will account for individual differences across animals, for the dynamics of acquisition of the song during development, and for the slight shift of the peak of the empirical distribution in Fig. 3*F* from

Previous analyses of the speed and magnitude of learning in the Bengalese finch have noted that both depend on the overlap of the distribution of the natural variability at the baseline and at the shifted means (25, 39): Small overlaps result in slower and smaller learning, so that different overlaps lead to different timescales. However, these prior studies have not provided a computational mechanism or a learning–theoretic explanation of why or how such overlap might determine the dynamics of learning. Our dynamical inference model provides such a computational mechanism.

We have chosen the family of so-called Lévy alpha-stable distributions to provide the central ingredient of our model: the heavy tails of the involved probability distributions. In general, a symmetric alpha-stable distribution has a relatively narrow peak in the center and two long fat tails, and this might provide some valuable qualitative insights into how the nervous system processes sensory inputs. For example, a narrow peak in the middle of the likelihood function suggests that the brain puts a high belief in the sensory feedback. However, the heavy tails say that it also puts certain weight (nearly constant) on the probability of very large errors outside of the narrow central region. We have verified that the actual choice of the stable distributions is not crucial for our modeling. For example, one could instead take each likelihood as a power-law distribution or as a sum of two Gaussians with equal means, but different variances. The latter might correspond to a mixture of high (narrow Gaussian) and low (wide Gaussian) levels of certainty about sensory feedback, potentially arising from variations in environmental or sensory noise or from variations in attention. As shown in *Materials and Methods*, different choices of the underlying distributions result in essentially the same fits and predictions. This suggests that the heavy tails themselves, rather than their detailed shape, are crucial for the model.

Another extension of this work would be to use this framework to account for interindividual differences in behavior and neural activity (48, 49), which are not easily addressable given current experimental limitations. Like in many other behavioral modeling studies (13, 34), the present version of our model can fit only an average animal (because of our need to aggregate large datasets to accurately estimate behavioral distributions), making our results semiquantitative with respect to the statistics of any particular individual. Nevertheless, our framework represents a class of model that can explain a broader range of qualitative results than previous efforts, including, in particular, the shape of the distribution of exploratory behaviors.

Finally, while we used Bengalese finches as the subject of this study, nothing in the model relies on the specifics of the songbird system. Our approach might, therefore, also be applied to studies of sensorimotor learning in other model systems, and we predict that any animal with a heavy-tailed distribution of motor outputs should exhibit similar phenomenology in its sensorimotor learning. Exploring whether the model allows for such cross-species generalizations is an important topic for future research, as are questions of how networks of neurons might implement such computations (50⇓⇓–53).

## Materials and Methods

### Experiments.

The data used are taken from the experiments in ref. 33 and are described in detail there. Briefly, subjects were nine male adult Bengalese finches (females do not produce song) aged over 190 d. Lightweight headphones and microphones were used to shift the perceived pitches of birds’ own songs by different amounts, and the pitch of the produced song was recorded. For each day, only data from 10 AM to 12 PM are used. The same birds were used in multiple (but not all) pitch-shift experiments separated by at least 32 d. Changes in vocal pitch were measured in semitones, which are a relative unit of the fundamental frequency (pitch) of each song syllable:

### Stable Distributions.

A probability distribution is said to be stable if a linear combination of two variables distributed according to the distribution has the same distribution up to location and scale (47). By the generalized central limit theorem, the probability distributions of sums of a large number of i.i.d. random variables with infinite variances tend to be stable distributions (47). A general stable distribution does not have a closed-form expression, except for three special cases: Lévy, Cauchy, and Gaussian. A symmetric stable variable x can be written in the form

Note that even though we take the propagator and the likelihood distributions as stable distributions in our model, their iterative application (effectively, a product of many likelihood distributions iterated with a convolution with the kernel), as well as truncation, results in finite variance predictions, allowing us to compare predicted variances of the behavior with experimentally measured ones.

### Fitting.

Our model consists of three truncated stable distributions, one for each of the two likelihood functions resembling the feedback modalities and a third for the propagation kernel. We use truncation to ensure biological plausibility: Neither extremely large errors nor extremely large pitch changes are physiologically possible. We truncate the distributions to the range **1**). The prior (and accordingly the posterior) is a discrete distribution with a resolution of 1,600 bins covering equidistantly the entire support of the distribution (**2**. As described earlier in Mathematical Model, the initial prior is given as the steady-state distribution after repeatedly applying Eq. **2** to a uniform distribution. In other words, the prior is not an independent variable in the model, but it is determined self-consistently by the likelihoods and the kernel. The resulting prior distribution is compared with the empirical distribution to compute the quality of fit according to the objective functions described below. Furthermore we point out that it is essential to have a non-Gaussian kernel: With a Gaussian kernel the baseline (*D*. Similarly, the likelihood functions must be long tailed; otherwise the combined likelihood would not be bimodal as found empirically in Fig. 3*F*, and large perturbations would not be rejected—even if the prior was long tailed (e.g., due to a long-tailed kernel).

We construct an objective function that is a sum of terms representing the quality of fit for each of the three datasets to our disposition: the *A*), the *B*), and the log-likelihood of the observed baseline pitch probability distribution (Fig. 2*C*). Using Eq. **3** to define the error of the mean, we calculate

The objective function landscape is not trivial in this case, and there is not a single best set of parameters. Fig. 4 illustrates this by showing the quality of fit as a function of each pair of

#### Choice of the Shape of Distributions.

For Figs. 2 and 3 in the main text, we have chosen stable distributions for

#### Linear Dependence on Pitch Shift in a Kalman Filter with Multiple Timescales.

We emphasized that traditional learning models cannot account for the nonlinear dependence of the speed and the magnitude of learning on the error signal. Here we show this for one such common model, originally proposed by Körding et al. (13). This Kalman filter model belongs to the family of Bayes filters, which are dynamical models describing the temporal evolution of the probability distribution of a hidden state variable (can be a vector or a scalar) and its update using the Bayes formula for integrating information provided by observations, which are conditionally dependent on the current state of the hidden variable. The specific attributes of a Kalman filter within the general class of Bayes filters (35) are the linearity of the temporal evolution of the hidden state (the pitch ϕ for the birds, but referred to as disturbances d in ref. 13 and hereon), the linear relation between the measurements (observations) and the hidden variable, and the Gaussian form of the measurement noise and the distribution of disturbances.

One can argue that Kalman filter models with multiple timescales may be able to account for the diversity of learning speeds in our pitch-shift experiments. We explore this in the context of an experimentally induced constant shift Δ to one disturbance d in the Kalman filter model with multiple timescales from ref. 13. If there is a constant shift Δ, equation 3 in ref. 13 takes the form**9**, and Q is the covariance matrix of the intrinsic (temporal evolution) noise.

From Eq. **11** we see that **9** in Eq. **10** we get

Generalizing the result to n disturbances with different timescales, we get the following equations at steady state:

## Acknowledgments

We are grateful to the NVIDIA Corporation for supporting our research with donated Tesla K40 graphics processing units. This work was partially supported by NIH Brain Research through Advancing Innovative Neurotechnologies (BRAIN) Initiative Theory Grant 1R01-EB022872, James S. McDonnell Foundation Grant 220020321, NIH Grant NS084844, and NSF Grant 1456912.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: ilya.nemenman{at}emory.edu.

Author contributions: B.Z., S.J.S., and I.N. designed research; B.Z., D.H., and I.N. developed the theory; S.J.S. and I.N. managed research; B.Z., D.H., I.P., S.J.S., and I.N. performed research; B.Z. analyzed data; and B.Z., D.H., I.P., S.J.S., and I.N. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Data deposition: The code and data reported in this paper have been deposited on GitHub and are available at https://github.com/EmoryUniversityTheoreticalBiophysics/NonGaussianLearning.

Published under the PNAS license.

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