Neural population geometry
Finding the low-dimensional structure in the activity of many neurons at once.
Recordings from many neurons are high-dimensional, but the activity they carry often lies on a much lower-dimensional manifold. Population geometry is the study of that structure: the reduced set of coordinates along which a population's activity actually varies, and how those coordinates relate to behaviour and stimulus.
The methods range from linear projection to latent-variable models. Principal component analysis finds the directions of greatest variance exactly, through the eigenvectors of the covariance. Gaussian-Process Factor Analysis goes further, inferring smooth low-dimensional trajectories from single trials by placing a Gaussian-process prior on the latent time courses — a model that couples dimensionality reduction with temporal smoothing.
Both are implemented here as exact, deterministic kernels rather than approximations. The dimensionality lesson computes PCA through a Jacobi eigen-decomposition; the Gaussian-process lesson solves the GP through a Cholesky factorisation. Each ends at the studio's own sealed benchmark measurement for the corresponding kernel, so the reader moves from the method to its bit-exact evidence.
The vocabulary of the topic.
- Low-dimensional manifold
- The reduced surface on which high-dimensional population activity actually lies.
- PCA
- Principal component analysis — variance-maximising linear projection via the covariance eigenvectors.
- GPFA
- Gaussian-Process Factor Analysis — smooth single-trial latent trajectories from a GP prior.
- Latent trajectory
- The inferred low-dimensional time course underlying the observed spikes.
Studios working on this.
Each runs standalone in its own repository and federates its evidence through the platform.
Run it, don't take it on faith.
The runnable lessons and sealed measurements this site ships for the topic.
Read it at source, in order.
Canonical references for the topic, ordered from the foundations to current work. Every one was verified at source — a DOI resolves through doi.org, a standard through its issuer — so each link goes to the real record.
Start with the foundations
- 1
The standard GP reference, freely available; the RBF kernel and Cholesky solve used in the GP lesson.
How it works
- 2
GPFA — the smoothing latent-variable model this site implements exactly in a lesson.
Put to work
- 3
The autoencoder view of dimensionality reduction, for contrast with linear PCA.
Current work
- 4
The review of population-geometry methods, PCA through GPFA.
Related topics.
Where this topic connects to the rest of the federation's work.