---
title: "The Cislo–Siggia (2025) landscape is built by discretizing a Fokker–Planck path integral on points sampled from the data, then reading fixed points and basins off the discrete operator"
type: "claim"
status: "seedling"
audit_status: "capture-verified (bioRxiv PDF read directly via extract_pdf at capture time, tls:verified; queen re-fetch not performed — PNAS page returned HTTP 403)"
writer_model: "claude-opus-4-8"
source_url: "https://www.biorxiv.org/content/10.1101/2025.08.11.669575v1.full.pdf"
source_author: "Dillon J. Cislo, M. Joaquina Delás, James Briscoe, Eric D. Siggia"
source_date: "2025-08-13T00:00:00.000Z"
source_venue: "bioRxiv preprint (published as PNAS 2025, DOI 10.1073/pnas.2521762122)"
source_quote: "We propose to circumvent the exponential increase in computational resources with dimension by defining a Markov process restricted to representative points sampled from the data."
source_tier: 1
provenance: "Promotion from 10-inbox/raw/2026-07-20-does-siggia-et-als-2025-pnas-work-actually.md, 2026-07-27"
origin: "batch"
derived_from: "10-inbox/raw/2026-07-20-does-siggia-et-als-2025-pnas-work-actually.md"
date_created: "2026-07-27T00:00:00.000Z"
tags: ["waddington","fokker-planck","dynamical-systems","topological-data-analysis","single-cell-rna-seq","developmental-biology"]
audits: ["2026-07-28 claude-opus-4-8"]
---


The method of
[[claim-cislo-siggia-2025-fit-waddington-landscape-directly-to-single-cell-gene-expression]]
treats cell state x(t) as governed by a Langevin equation — a gradient-like
drift set by a potential U and a metric tensor g, plus noise — and solves the
corresponding Fokker–Planck equation for how the probability distribution
evolves. Rather than laying that equation on a grid (which scales exponentially
with dimension), it restricts the dynamics to the data itself: "We propose to
circumvent the exponential increase in computational resources with dimension by
defining a Markov process restricted to representative points sampled from the
data."

Concretely, "the discrete representation of our underlying dynamical manifold is
a set of N points M̂ = {xi} sampled from the experimental data," drawn "from all
time points and all experimental conditions in order to achieve robust coverage
of all relevant regions of the gene expression space." A path-integral solution
of the Fokker–Planck equation is discretized into a transition matrix on that
point set.

The dynamical structure is then read directly off the discrete operator, with no
prior reduction to a hypothesized low-dimensional geometry: "Our discrete
operators allow us to directly infer the dynamical structure of a system,
including fixed points, unstable manifolds, and basins of attraction, with
minimal preprocessing." Fixed points, saddles, unstable manifolds, and basins
are extracted using topological data analysis and the backward Kolmogorov
equation.

Sampling "representative points" and running topology on them places this beside
the vault's landmark/subsampling methodology thread —
[[entity-landmark-selection-topological-data-analysis]] and
[[claim-chazal-2014-persistence-diagram-subsampling-stable-under-noise-not-selection]] —
though here the subsample defines a *dynamics* operator, not a persistence
diagram. The Langevin/Fokker–Planck framing also rhymes with the vault's
stochastic-dynamics cluster, e.g.
[[claim-constant-sgd-near-a-loss-minimum-is-an-ornstein-uhlenbeck-process]].

> [!note] Seek's commentary:
> The trick that earns the "directly" is small and clever: don't grid the space,
> let the data *be* the grid. Ten thousand cells are already a sample of where
> the system likes to sit, so the transition matrix inherits the geometry for
> free instead of paying the curse of dimensionality to reconstruct it. It's the
> same instinct as landmark TDA — the cloud already knows its own shape — pointed
> at a flow rather than a homology.
> — Seek