September topics:

Evidence for something like “brain algebra”.

How does the human brain combine visual concepts? Suppose you have records of your brain activity when (A) seeing a cow, and when (B) seeing a cornfield. Is there some sense in which the brain activity generated by seeing a cow in a cornfield is A + B? Does brain activity manifest compositionality?

An August 22 article in Communications Biology explores this question, using functional magnetic resonance imaging (fMRI) records of brain activity. An fMRI treats the brain as dissected into tiny cubes (side length around .75mm) and records the instantaneous blood flow through each cube. The resulting image is a 3-dimensional array of numbers; mathematically, this can be thought of as a single high-dimensional vector. Of course, this is only a shadow of the information processing going on. Nevertheless, the authors of the study successfully trained computers to (roughly) link fMRI records to visual input, using fMRI data for four subjects in the Natural Sciences Dataset. (Each subject’s data consisted of 24,980 readings corresponding to 8,859 images.)

For each subject, the authors generated pseudo-fMRI data by summing two vectors, A and B. A was one of the fMRI images from the dataset, while B was a synthetic perturbation based on one of twelve “semantic concepts”: winter, summer, man, woman, night, day, empty, crowded, indoor, outdoor, happy and sad.

Creating the pseudo-fMRI vector was a multi-step undertaking. First, the authors created an image vector matching each semantic concept. To do this, they created text descriptions of the images in their dataset using the software CLIP (Contrastive Language–Image Pre-training). To extract an image representing “winter”, say, they identified the 100 images with descriptions best matching “winter”, and then averaged and scaled down the fMRI records corresponding to these images. Finally, they added the resulting fMRI vector (B) to the fMRI record corresponding to the original image (A) to create the pseudodata A + B.

Finally, they selected the image best matching A + B using an AI algorithm trained on that subject’s image-fMRI pairs.

Top: Scene of two people shaking hands. Middle: fMRI data corresponding to the initial scene, and pseudo-fMRI data perturbed by semantic concepts "winter" and "summer". Bottom: Image of people walking through a snowy scene, and image of people in a sunny park.
Reconstitution of images from perturbed brain patterns. The fMRI recorded from a subject exposed to a scene is perturbed by superposition with (left) the edited average of a collection of fMRIs of winter scenes recorded from the same subject and (right) the same, but with summer scenes. The two new pseudo-fMRIs are decoded by an AI algorithm trained on the entire corpus of that subject’s recorded fMRIs. Image: Modified version of Fig. 2 of Commun Biol 8, 1263 (2025), used under a CC BY-NC-ND license. The article presents eight examples of one scene with all twelve possible perturbations.

To gauge the significance of this process the authors generated a text description of the final image. The final text was statistically compatible with compositionality—the text for A + B was correlated both with the text of A, and with the semantic concept behind B.

Stochastic geometry of the extracellular matrix.

A living tissue is made up of cells, but there is more. Molecules secreted by the cells self-assemble into a kind of scaffolding that holds the tissue together. This is the “extracellular matrix” (ECM). It is specific to each tissue and determines characteristics like stiffness and dimensionality. A PNAS paper from August develops geometric tools for investigating ECM self-assembly, about which little is known.

All figures below are from PNAS 122 (33) e2425759122, used under a CC by 4.0 Deed license.

An adult female specimen of the alga Volvox carteri has $\sim2000$ somatic (non-reproductive) cells evenly distributed on its boundary. Below them lie 16 gonidia, where the next generation of algae is incubated. Image (lightly edited) from Fig. 1A.

The researchers genetically modified the alga Volvox carteri to make its ECM fluorescent, allowing them to photograph the ECM throughout the alga’s development. As the alga grows, the geometry of the ECM changes significantly.

The analysis focuses on the central zone (CZ) of the ECML, the structure just below the surface of the alga.

A schematic showing a series of compartments. The surface the cell inside each compartment is labeled CZ1, the space between the cell and compartment wall is CZ2, and the outer surface of each compartment is labeled CZ3.
Schematic section normal to the boundary of the outer layers of V. carteri in the first stage of its growth with details of the ECM, including the components CZ1, CZ2, and CZ3 of its central zone. Fig. 1B, lightly edited.

The ECM consists of compartments, each containing one cell. In the first stage of the alga’s growth, the CZ3 compartments form a polygonal tessellation of the surface. By one of the last stages, the compartments look like a family of tangent bubbles.

A1: A near-hexagonal lattice in green, with magenta cells inside each hexagonal compartment. B1: The compartments now appear more circular, and the cells are smaller.
Closeups of CZ3 during Stage I (A1) and Stage IV (B1) of the growth of V. carteri. Superimposed on the (green) fluorescent record of the ECM are images taken in a wavelength that makes the chlorophyll in the somatic cells flouresce magenta. Note the change in scale: the ECM has grown with respect to the cells. Figs. 5A1 and 5B1.

To give a quantitative record of this evolution, the authors track the shape and the geometric moments of each compartment throughout the growth cycle of V. carteri. These include quantities like the area, center of mass, covariance matrix, as well as measurements derived from these quantities.

Figure 7 in the PNAS article gives graphic/schematic representations of quantitative factors. Parts of the figure are reproduced here. Another part represents the deviation of a CZ3 compartment with center of mass ${\bf x}$ from the corresponding cell ${\bf C}$ of the Voronoi tessellation: ${\bf C}$ consists of all the points which are closer to ${\bf x}$ than to any other CZ3 center of mass.

A. Superimposed on an image of Stage II is part of the segmentation of the surface into CZ3 compartments; they are colored dark to light by size (area). The black square singles out one typical compartment to be used in the rest of this analysis.
B. Schematic for the areas of the cell and of the compartment, showing the center of mass of the cell. C. The aspect ratio symbolized by the major and minor axes of the approximating ellipse, i.e. the eigen-directions of the covariance matrix.

The authors use these measures to distinguish the maturation of the Volvox ECM from other similar natural processes, for example the hydration of foams. They suggest that Volvox could be used as a model multicellular organism for studying natural structures that are not under direct control by cells. Their detailed analysis of the geometry of the ECM evolution could point to specific points in time and space where self-assembly is taking place.

—Tony Phillips, Stony Brook University

Read other recent Tony’s Take columns.