🧠 neuroscience

Quantifying the information about uncertainty in neural population codes

This paper establishes a theoretical framework linking neural population codes to uncertainty by demonstrating that ancillary information, quantified through mutual information and Fisher information loss, inherently produces non-Gaussian estimation errors and defines an upper bound on the uncertainty knowledge accessible from neural activity alone.

Original authors: Wang, X., Dayan, P., Bays, P.

Published 2026-07-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Wang, X., Dayan, P., Bays, P.

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine your brain is a super-advanced detective agency, constantly trying to figure out what's happening in the world. Every time you see a car speeding by or hear a bird chirp, your brain gathers a massive crowd of tiny messengers (neurons) to shout out clues. But here's the catch: these messengers are a bit jittery. They don't always shout the exact same thing; sometimes they whisper, sometimes they yell, and sometimes they get distracted. Because of this "noise," your brain's best guess about the world is never 100% perfect. It's always a little bit uncertain.

Now, here is the really cool part: your brain doesn't just make a guess; it also seems to know how good that guess is. If you're looking at a car in bright sunlight, your brain feels confident. If you're looking at the same car in thick fog, your brain feels shaky. This ability to know how sure you are about your own thoughts is called metacognition. Scientists have long wondered: how does the brain calculate this "confidence"? Is it just a separate little calculator running in the background, or is the confidence hidden inside the messy, jittery shouts of the neurons themselves? This question is at the heart of a new study that tries to decode the secret language of uncertainty.

The researchers, Xiaolu Wang, Peter Dayan, and Paul Bays, decided to treat the brain's neural crowd like a mathematical puzzle. They wanted to see if the "jitteriness" of the neurons contains a hidden code that tells the brain, "Hey, this guess is shaky!" or "This one is solid!" To do this, they looked at two main ways to measure this hidden information. One way is like checking how much the shape of the crowd's shout changes from moment to moment (using something called Fisher Information). The other way is like measuring how much the crowd's noise pattern tells you about how wrong the final guess might be (using Mutual Information).

The paper's main discovery is that this "hidden confidence" is real and it comes from the fact that the brain's noise isn't perfectly predictable. When the brain tries to guess a value (like the direction of a moving object), the errors it makes aren't just random bumps around the right answer; they often have "long tails." Imagine throwing darts at a board. A normal guess would mean most darts land near the bullseye, with a few stray ones. But in the brain, the "stray" darts sometimes fly really far away, creating a long tail of big mistakes. The authors show that these long tails are actually a sign that the brain has extra information about its own uncertainty. If the noise pattern changes in a specific way, it creates these long tails, and that change is the signal the brain uses to say, "I'm not sure about this!"

However, the paper also warns us not to be fooled by appearances. The researchers found that two different types of neural "noise" can look exactly the same on the surface—they produce the same pattern of dart throws and the same long tails—but they actually carry very different amounts of hidden confidence information. It's like two different radio stations playing the same song; one might be broadcasting in crystal clear high definition, while the other is fuzzy and full of static, even if the melody sounds the same to your ears. This means that just looking at how often people make mistakes isn't enough to understand how the brain knows what it knows. You have to look at the specific "texture" of the noise.

The team also tested what happens when the signal gets weaker. They simulated two scenarios: one where the signal gets corrupted by outside interference (like trying to hear a whisper in a storm), and another where the brain's own internal signal just gets quieter (like a radio running out of batteries). They found that while both make the brain's guesses worse, they affect the "hidden confidence" differently. Adding outside noise actually destroys the brain's ability to know how uncertain it is, whereas just turning down the volume of the internal signal changes the confidence in a more complex, non-linear way.

In short, this paper suggests that the brain's ability to know what it doesn't know is built right into the messy, jittery patterns of its neurons. It's not a separate magic trick; it's a mathematical consequence of how the brain processes information. The authors show that this "ancillary information" sets a hard limit on how smart our self-confidence can be. If we ever feel more confident than the math says we should be, it means we are using a secret source of information that isn't in the neural noise itself. While these findings are based on computer simulations and mathematical models rather than direct brain scans, they provide a powerful new map for understanding how we trust our own minds. It turns out that the brain's uncertainty isn't a bug; it's a feature, written in the very shape of its mistakes.

Problem Statement

Neural populations encode sensory and motor variables with inherent noise, leading to uncertainty in internal representations. While behavioral studies confirm that humans and animals utilize this uncertainty to guide decisions (metacognition), the mechanism by which neural populations represent uncertainty remains unclear. Specifically, it is not fully understood how the same neural activity used to decode a variable (e.g., orientation) also conveys information about the reliability of that estimate. Previous models, such as the Variable Precision (VP) model, have linked long-tailed estimation errors to fluctuating precision but have not formally identified the neural source of this information. Furthermore, recent geometric approaches attribute non-Gaussian errors to manifold curvature but have not directly addressed the information content regarding uncertainty. The core problem is to quantify the "ancillary information"—information about estimation reliability contained in neural activity beyond the point estimate itself—and to establish its relationship to the non-Gaussianity (specifically, heavy tails) of estimation errors.

Methodology

The authors develop a general framework based on an encoding-decoding model where a population of $M$ neurons encodes a stimulus $\theta$ with translational symmetry. They treat the Maximum Likelihood Estimate (MLE), $\hat{\theta}_r$ , as the point estimator derived from neural activity $r$ .

Theoretical Framework:
- The authors decompose the observation $r$ into the MLE $\hat{\theta}_r$ and an ancillary statistic $a_r$ . The ancillary statistic is probabilistically independent of the true parameter $\theta$ but contains information about the accuracy of the MLE.
- They define the marginal distribution of estimation error $\epsilon$ as a mixture of normalized, reflected, and re-centered likelihood functions, where the shape of the mixture components is determined by the ancillary statistic.
Quantification Measures:
The paper introduces two complementary measures to quantify ancillary information:
- Mutual Information ( $MI_e$ ): An entropy-based measure defined as the mutual information between the estimation error $\epsilon$ and the ancillary statistic $a_r$ (or the full population response $r$ ). It is calculated as $MI_e = H(\epsilon) - H(\epsilon|a_r)$ . This represents the reduction in uncertainty about the error given the neural response.
- Fisher Information Loss ( $I_{loss}$ ): A measure based on the difference between the Expected Fisher Information (FI) contained in the full observation $r$ and the Expected FI contained in the MLE $\hat{\theta}_r$ alone ( $I_{loss} = I - I_{\hat{\theta}}$ ). This is interpreted geometrically as a "Jensen-Fisher divergence" related to the curvature of the statistical manifold.
Models and Simulations:
- The authors apply these measures to specific neural population models: Poisson noise, Gaussian noise (with variance-stabilizing transformation), and Modulated Poisson noise (incorporating shared gain fluctuations).
- They analyze the effects of external noise (additive noise corrupting the stimulus before encoding) versus weakening internal signal strength (reducing population gain).
- Simulations utilize a population of $M=200$ neurons with von Mises tuning functions.

Key Contributions and Results

1. Relationship Between Ancillary Information and Non-Gaussianity
The authors demonstrate that deviations from Gaussianity in estimation errors, particularly the long tails frequently observed in behavioral data, are a direct corollary of the presence of ancillary information.

When the ancillary statistic varies (indicating varying reliability across trials), the marginal error distribution becomes a scale mixture of likelihoods.
In a synthetic model with Gaussian likelihoods where the Observed FI follows a Gamma distribution, the marginal error distribution converges to a Student's $t$ -distribution. The "heaviness" of the tails is directly determined by the variability of the Observed FI (quantified by $I_{loss}$ ).

2. Distinction Between Noise Models
The paper reveals that different noise models can produce nearly identical estimation error patterns (marginal distributions) while providing vastly different amounts of ancillary information.

Poisson vs. Gaussian: Approximating Poisson noise with Gaussian noise (via variance-stabilizing transformation) preserves the Expected Fisher Information and the general shape of error distributions. However, the Gaussian approximation systematically underestimates the ancillary information ( $I_{loss}$ and $MI_e$ ). This suggests that Gaussian models may fail to capture the full information about uncertainty available in biological spike trains.
Gain Modulation: Introducing shared gain fluctuations (overdispersion) in a Poisson model increases ancillary information ( $I_{loss}$ and $MI_e$ ) monotonically, even though the Expected FI and the general error patterns remain largely unchanged. This highlights that interneuronal correlations induced by gain modulation carry significant information about uncertainty.

3. Dissociation Between $I_{loss}$ and $MI_e$
The two measures of ancillary information, while correlated, exhibit distinct behaviors:

$I_{loss}$ (Fisher-based) increases monotonically with tuning concentration ( $\kappa$ ) and gain modulation.
$MI_e$ (Entropy-based) exhibits an inverted-U relationship with $I_{loss}$ when Expected FI is held constant. $MI_e$ approaches zero when $I_{loss}$ is very low (likelihoods are identical) and when $I_{loss}$ is very high (likelihoods are extremely divergent, dominated by low-certainty observations).
This dissociation allows $MI_e$ to differentiate between models (e.g., Gaussian vs. Modulated Poisson) that are indistinguishable based on Fisher Information alone.

4. External Noise vs. Signal Weakening
The authors show that external noise (corrupting the stimulus) and internal signal weakening (reducing gain) have different effects on ancillary information:

Both increase estimation error variability.
However, external noise significantly reduces ancillary information ( $I_{loss}$ and $MI_e$ ) because it adds uncertainty without providing a signal about that uncertainty.
Weakening signal strength (in the Poisson model) does not reduce the asymptotic information loss ( $I_{loss}^\infty$ ) and can actually increase $MI_e$ in certain regimes (inverted-U pattern). This suggests that reduced performance due to external noise is fundamentally different from reduced performance due to low signal strength regarding the brain's ability to assess its own uncertainty.

5. Upper Bound on Metacognitive Sensitivity
The paper establishes that $MI_e$ quantifies a theoretical upper bound on metacognitive sensitivity (the accuracy with which subjective confidence tracks performance) attainable from a specific neural population code alone.

If behavioral metacognitive sensitivity ( $MI(\epsilon; c)$ ) is less than $MI_e$ , it implies imperfect access to the uncertainty signal (e.g., due to "metacognitive noise" or suboptimal inference).
If behavioral sensitivity exceeds $MI_e$ , it implies access to a separate source of information about uncertainty (e.g., post-decisional processing).

Significance

The authors claim this work provides a coherent theoretical foundation for investigating the basis of metacognition in neural population activity. By formalizing the relationship between ancillary information, non-Gaussian error distributions, and Fisher information curvature, the framework:

Unifies descriptive models of long-tailed errors (like the VP model) with mechanistic population coding models.
Clarifies that similar behavioral error patterns can arise from distinct neural noise structures with different uncertainty information content, urging caution in model selection based solely on error distributions.
Quantifies the limits of metacognition derived from sensory representations, offering a metric to test whether the brain accesses uncertainty directly from the same population activity used for estimation or relies on separate mechanisms.

The paper concludes that evaluating ancillary information is crucial for distinguishing between competing models of sensory coding and for understanding how the brain represents and utilizes uncertainty.

Problem Statement

Methodology

Key Contributions and Results

Significance

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