I was following your team’s recent brainstorming on Object Deformations (specifically the “Bent Cup” and “Egg vs. Circle” problem). I was reviewing 2025/06 - Brainstorming Around Behavior and Deformations . There was mention of feeling like there was a “simpler conceptual idea” missing to explain how reference frames handle continuous distortion.
The missing concept might be Topology (specifically Homeomorphism).
If Cortical Grid Cells are mapping a Hyperbolic Manifold (like Escher’s Circle Limit IIICircle Limit III - Wikipedia) rather than a rigid Euclidean grid, then “Deformation” is mathematically identical to “Movement.”
In a Hyperbolic metric, a shape naturally distorts (shrinks/bends) as it moves relative to the observer.
The brain doesn’t need a separate “Distortion Map.” The Grid Cells themselves handle the stretch because the coordinate system itself is flexible (curved).
Is it possible the “Thousand Brains” are using Conformal Mapping to normalize these inputs? This would solve the “Balloon Logo” problem instantly.
This is a really interesting idea! I haven’t thought about topology in a while. I think the last time I thought about it was in the context of “morphological classes”, i.e. how can we group different kinds of mugs (e.g. mug with slightly different shapes, or plain mug and mug with logo) into a single “class”. I didn’t go very far though, just that we can’t use homeomorphism since that would group mugs and donuts together.
So based on the Continuous Attractor Network (CAN) model of the grid cells (specifically this paper titled “Toroidal topology of population activity in grid cells”), I took this to mean that grid cells represent a space that is: (1) locally Euclidean, (2) globally periodic, and (3) flat toroidal manifold. However, this is just a model, and I’d be curious to see if there are any experimental evidence that grid cells may be representing a hyperbolic space with negative curvature.
I think conformal mapping is an interesting idea… I think there is conformal mapping going on from retinal coordinates to V1 cortical coordinates. Could you explain more what you mean by:
Is it possible the “Thousand Brains” are using Conformal Mapping to normalize these inputs?
If a balloon inflates non-uniformly, the deformation can be no longer represented as a conformal map, right? (I think I’m still thinking about morphological classes than object deformations, so my question may not matter).
Thanks @hlee! Non-uniform inflation is technically quasi-conformal bounding angle distortion rather than preserving it perfectly. To restate my hypothesis in this language:
Might the 1000 brains be voting on the Beltrami Coefficient (the “stress” of the map)? If the geometric cost to normalize the deformation is low enough (Quasi-Conformal), the columns vote “Yes, it’s the same object.” The brain is measuring the stretch, not just the shape.
This sounds interesting. It’s not an area I’m familiar with, so I’d like to understand it better. How universal are these properties across a given map (e.g. the Beltrami Coefficient you refer to)? One of the things about distortions is that they can vary in different parts of an object, and in different ways. This means that when testing hypotheses, including ensuring we perform appropriate path integration through our learned model, we need to account for distortions in different ways at different points in the map. While a metric like the “stress” between a learned map and incoming observations can be a useful measure, what we really want is some way to correctly make reasonable predictions as we move. Do you see a connection there?
I should note that typically when this is the case (i.e., that changes can be different at different locations in a map), then we think about resolving it through compositionality (e.g. a logo with multiple bends in it can be learned by multiple associations, where each one is associated with a different rotation). This is more difficult though with continuous deformations, like a wobbly logo.
The core philosophical idea here is Topology over Parameterization. So not teaching the ANN how to fix distortion, instead create Geometry where distortion is mathematically impossible. enforce metric tensor of the sphere.
physics engine to correct the distortion at every single time step:
def project_to_tangent(z, v):
"""
Riemannian Geometry: Projects a raw velocity vector 'v' onto the
tangent space of the sphere at point 'z'.
Math: v_tan = v - <v, z> * z
"""
# Calculate the inner product (scalar projection)
dot = (v * z).sum(dim=-1, keepdim=True)
# Subtract the radial component to ensure orthogonality
return v - dot * z
the mechanism for real-time metric tensor modulation could be Cholinergic Control of Theta Frequency. Under this model, the brain would not have to ‘calculate’ a new map. It merely modulates the Gain (ACh) and Frequency (Theta), and the interference pattern naturally stretches to fit the new geometry. That would be the biological implementation of a dynamic Riemannian metric.
Literature support for this view:
Burgess, Barry, & O’Keefe (2007) - “An oscillatory interference model of grid cell firing.” (Hippocampus). modifying the “clock speed” (theta frequency) of the network automatically stretches or shrinks the “ruler” (grid scale) without requiring synaptic rewiring
Barry et al. (2012) - “Grid cell firing patterns signal environmental novelty by expansion.” (PNAS). when animals enter a novel environment, grid scales expand by ~40% while theta frequency drops. biological implementation of a dynamic metric tensor.
Keinath, Epstein, & Balasubramanian (2018) - “Environmental deformations dynamically shift the grid cell spatial metric.” (eLife). as an animal moves through a deformed space, the phase of the grid shifts dynamically relative to the nearest wall, creating a shearing effect (continuous deformation)
This is a really interesting idea! I haven’t thought about topology in a while. I think the last time I thought about it was in the context of “morphological classes”, i.e. how can we group different kinds of mugs (e.g. mug with slightly different shapes, or plain mug and mug with logo) into a single “class”. I didn’t go very far though, just that we can’t use homeomorphism since that would group mugs and donuts together. 