Unmatched spatial transcriptomics snapshots get a multiscale drift model: stMIND splits local and nonlocal cues to pin an axolotl regeneration window at 10–15 days
Synopsis
Spatial transcriptomics sequencing is destructive, so each time point yields unaligned cell snapshots and the intermediate trajectories and cross-scale intercellular interactions stay unobserved. stMIND decomposes each cell's expression drift into a cell-intrinsic program, a local graph-difference operator over relative expression differences and spatial geometry within a radius, and a mass-weighted nonlocal mean-field attention operator outside it, with neighborhoods co-evolving as cells migrate. On axolotl telencephalon regeneration and zebrafish embryogenesis benchmarks it reaches the highest gene-expression accuracy, anatomical consistency and spatial coherence; ablations show both operators are needed, and the learned fields locate a regenerative transition at 10–15 days post-injury.
Interpretation
Time-series spatial transcriptomics can now reconstruct continuous gene-expression and spatial-migration trajectories from unaligned section snapshots, without requiring the same cells to be tracked longitudinally. Earlier dynamic optimal-transport and flow-based generative models inferred continuous trajectories from time-series single-cell data, but their drift parameterizations did not explicitly separate local from nonlocal intercellular interactions; the spatial methods STORIES and stVCR added spatial geometry yet still did not split the two scales. On two time-series spatial transcriptomics atlases, axolotl telencephalon regeneration and zebrafish embryogenesis, stMIND achieved the lowest mean gene-expression errors among all five methods on the short-horizon extrapolation tasks, at both the axolotl 30 DPI and zebrafish 18 hpf endpoints.
Expression drift is written as a sum of three terms: a network term depending only on the focal cell's own expression state, a local term inside a radius, and a nonlocal term outside it, with the two interaction terms acting on disjoint context sets updated as cells migrate. The nonlocal term extends the interacting mean-field neural stochastic differential equation attention, aggregating distant cells through mass-weighted keys and values, which acts as a learned integral operator over the exterior empirical measure; the local term aggregates expression differences between neighbors and the focal cell, with attention modulated by a learned encoding of relative displacement, so its response vanishes when a neighborhood is transcriptionally homogeneous. Training couples trajectory kinetic energy, optimal transport in joint expression–position space and spatial occupancy matching, and adds an endpoint orthogonality penalty that discourages collinearity between the intrinsic drift and each interaction drift, making the three-term decomposition identifiable.
Separating local from nonlocal operators directly changes how well cellular microenvironments are reconstructed, and lets the learned drift fields be read by scale as regenerative dynamics reorganize. In a six-way ablation on the axolotl forecasting benchmark, the nonlocal-only and global-only variants improved on the intrinsic-only variant but neither matched the full model's microenvironment fidelity. The full model had the lowest microenvironment metric at both 30 and 60 DPI among the five spatially informed variants, and its within-radius Moran's I also exceeded the global-only variant's, while the local-only variant likewise retained strong spatial autocorrelation.
Perspective
The demonstrated scope is held-out forecasting on two time-series spatial transcriptomics atlases: axolotl telencephalon regeneration fitted at 2, 5, 10, 15 and 20 DPI and evaluated at 30 and 60 DPI, and zebrafish embryogenesis fitted at 3.3, 5.25, 10 and 12 hpf and evaluated at 18 and 24 hpf, with metrics reported as mean and standard deviation over five random seeds.
On long-horizon extrapolation stMIND ranks second in marginal expression accuracy, trailing the non-spatial scIMF at both axolotl 60 DPI and zebrafish 24 hpf, which the source attributes to the trade-off of simultaneously enforcing tissue transport and spatial organization. Nonlocal attention subsamples context cells in mini-batches to cut cost, and that subsampling may omit informative long-range context; stVCR had higher Moran's I at zebrafish 18 hpf, while at 24 hpf stVCR did not finish within the computational time limit. The regenerative-transition reading is explicitly stated as a testable hypothesis about how cellular context at different spatial scales contributes to cell-state reorganization.