[Paper Review] Chance, long tails, and inference: a non-Gaussian, Bayesian theory of vocal learning in songbirds

Baohua Zhou, D. J. Hofmann+3 more|arXiv (Cornell University)|Jul 23, 2017

Animal Vocal Communication and Behavior62 references18 citations

TL;DR

This paper proposes a non-Gaussian, Bayesian theory of sensorimotor learning in songbirds, where animals learn entire probability distributions of motor commands rather than single optimal outputs. By modeling pitch variability in Bengalese finches using heavy-tailed (power-law) distributions and recursive Bayesian inference, the theory explains why large abrupt errors inhibit learning while gradual errors do not, and predicts experimentally confirmed dynamics in the shape of the pitch distribution during adaptation.

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 that large abrupt (but not gradually introduced) errors lead to weak learning. Here we propose a new theory of sensorimotor learning that explains these observations. The theory posits that the animal learns an entire probability distribution of motor commands rather than trying to arrive at 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.

Motivation & Objective

• To address the failure of traditional Gaussian sensorimotor learning models to explain why large abrupt sensory errors reduce learning, while gradually introduced errors do not.
• To develop a theory where the brain learns and updates a full probability distribution of motor commands, rather than aiming for a single optimal command.
• To test whether non-Gaussian, heavy-tailed distributions can explain the observed nonlinear dynamics in vocal adaptation of Bengalese finches.
• To validate the model using experimental data from real-time pitch-shifted auditory feedback in songbirds.
• To demonstrate that learning dynamics depend on the shape of the behavioral distribution, not just the mean error.

Proposed method

• The model uses recursive Bayesian inference to update a prior distribution of motor commands based on sensory feedback, incorporating likelihood functions derived from observed pitch errors.
• It assumes that the distribution of motor commands (song pitch) is non-Gaussian, specifically power-law tailed, to capture empirical heavy-tailed distributions observed in experimental data.
• The likelihood function is modeled as a product of two sensory modalities: one with a shifted pitch (experimental perturbation) and one with no shift (baseline feedback), enabling bimodal likelihoods under heavy tails.
• The prior is updated recursively using Bayes' rule, with temporal evolution modeled by convolution with a kernel that increases uncertainty over time.
• The model predicts changes in the shape of the pitch distribution during learning, including the emergence of bimodality under large errors.
• The theory is tested against experimental data from Bengalese finches with real-time, miniature headphone-based auditory feedback perturbations.

Experimental results

Research questions

• RQ1 Why do large abrupt sensory errors lead to weaker learning than smaller ones, contrary to predictions of standard Gaussian models?
• RQ2 Why can animals compensate for large errors when they are introduced gradually, but not when introduced abruptly?
• RQ3 How does the shape of the behavioral distribution (e.g., heavy tails) influence learning dynamics and error correction?
• RQ4 Can a Bayesian model that learns a full distribution of motor commands explain the nonlinear dynamics of vocal adaptation in songbirds?
• RQ5 What role do non-Gaussian (power-law) tails in motor variability play in sensorimotor learning?

Key findings

• The distribution of sung pitch in Bengalese finches exhibits long, non-Gaussian (power-law) tails, which the model successfully captures using α-stable distributions.
• The model explains the experimentally observed decrease in learning speed and magnitude with increasing error size, particularly for abrupt perturbations.
• The theory predicts that large errors suppress learning not by rejecting them, but by making the likelihood function bimodal under heavy tails, which reduces the influence of extreme errors on the posterior.
• The model accurately predicts the dynamics of the pitch distribution shape during adaptation, including the emergence of bimodality under large errors.
• Experimental validation confirms that the distribution of pitch changes dynamically during learning, with increased bimodality under large perturbations, as predicted by the model.
• The theory accounts for the paradox that gradual large errors are compensated for, while abrupt ones are not, by showing that gradual changes allow the system to adapt its distribution shape progressively.