The geometry of uncertainty in population codes

@misc{geometry-of-uncertainty,
  author = {Mara Ellison Voss},
  title = {The geometry of uncertainty in population codes},
  howpublished = {Mara Ellison Voss},
  year = {2025},
  url = {https://mouwumou.github.io/Offprint/blog/geometry-of-uncertainty/}
}

Voss, M. E. (2025). The geometry of uncertainty in population codes. In Mara Ellison Voss.

Voss, Mara Ellison. “The Geometry of Uncertainty in Population Codes.” Mara Ellison Voss, 2025.

Voss, Mara Ellison. 2025. “The Geometry of Uncertainty in Population Codes.” In Mara Ellison Voss. Preprint.

There is a comfortable story we tell about neural codes: a stimulus arrives, a population of neurons fires, and somewhere in that firing is a point — the brain's estimate of what is out there. It is a good story. It is also, I think, incomplete, because it leaves no room for the thing the brain spends most of its energy on: not knowing.

This post is about where uncertainty might actually live in a population code, and why I have come to believe it lives on a curved surface.

A point is not enough#

Suppose a population of $N$ neurons responds to a scalar stimulus $s$ with mean rates $\mathbf{f}(s) \in \mathbb{R}^N$. As $s$ varies, $\mathbf{f}(s)$ traces out a one-dimensional curve — the tuning manifold. Decoding, in the point story, is just finding the closest point on that curve.

But a Bayesian observer does not report a point; it reports a posterior. And a posterior has width. So the real question is: where does width go?

The claim I want to defend is that the manifold is not a curve but a ribbon, and that the local width of the ribbon encodes uncertainty:

$$\operatorname{Var}[s \mid \mathbf{r}] \;\approx\; \frac{1}{\;\mathbf{f}'(s)^{\top}\,\Sigma^{-1}(s)\,\mathbf{f}'(s)\;}$$

The denominator is the linear Fisher information. When the tuning curve is steep and the noise is small, information is high and the ribbon is thin. Where the curve flattens, the ribbon widens — and curvature is what couples these two regimes together.

An afternoon experiment#

You do not need a two-photon microscope to get intuition for this. Here is a synthetic population you can simulate:

import numpy as np

def population(s, N=200, gain=8.0, noise=0.4):
    """Bell-shaped tuning curves tiled across the stimulus range."""
    centers = np.linspace(-np.pi, np.pi, N)
    rates = gain * np.exp(np.cos(s - centers) - 1.0)   # von Mises tuning
    return rates + noise * np.sqrt(rates) * np.random.randn(N)

# sweep the stimulus and collect responses
S = np.linspace(-np.pi, np.pi, 400)
R = np.stack([population(s) for s in S])

Project R onto its top three principal components and you will see the ribbon: a closed loop that is thin where tuning curves overlap densely and bulges where they thin out. Uncertainty is not stored in a variable. It is stored in the shape.

The representation and its uncertainty are not two things. They are one curved thing, read two ways.

Why curvature, specifically#

Flat manifolds are boring in exactly the way that matters here: on a flat sheet, information is constant everywhere, and there is no room for the code to say "I am less sure here than there." Curvature is the degree of freedom that lets a fixed number of neurons allocate certainty non-uniformly across the stimulus range.

This framing is developed formally in (Voss et al., 2025), with the decoding stress-test methodology of (Voss et al., 2024) as the diagnostic.

This reframes a lot of old results. Adaptation, attention, and expectation all reshape tuning curves — and if the ribbon picture is right, reshaping tuning curves is the same operation as reallocating uncertainty.

What I am unsure about#

Plenty. The linear-Fisher approximation breaks down for wide posteriors, the noise covariance $\Sigma(s)$ is doing a lot of quiet work in that formula, and I have said nothing about time. But the shape-first framing has changed how I read data, and that is usually the sign of a hypothesis worth chasing.

If you run the experiment above and see something different, I would genuinely like to know.

References

1. Voss, M. E., Nakamura, R., & Feldmann, D. (2025). The geometry of uncertainty: curved manifolds in cortical population codes. Nature Neuroscience. https://doi.org/10.1038/s41593-025-01234-5
2. Voss, M. E., Feldmann, D., & Ortega, S. (2024). When do linear readouts fail? A stress test for neural decoding. In arXiv.