Identifying Neural Dynamics Using Nonlinear Interventional State Space Models
Authors: Hanna M. Tolle, Carles Balsells-Rodas, Yixin Wang, Pedro A. M. Mediano, Amin Nejatbakhsh
Organizations: Department of Computing, Imperial College London, London, UK · Department of Statistics, University of Michigan, Ann Arbor, USA · Center for Neural Science, New York University, New York, USA
State-space models (SSMs) recover low-dimensional latent dynamics from high-dimensional observations, but latents are generally not identifiable from observational data alone. Interventional state-space models (iSSMs) use perturbations to break observational symmetries and identify parameters up to permutation, scaling, and shift. However, prior work restricts identifiability to linear latent dynamics and piecewise-linear emissions, excluding many behaviors characteristic of neural circuits. We extend iSSMs to a substantially broader model class: we prove that (i) affine identifiability holds under general injective emission functions and Gaussian latent transitions, and (ii) with interventions, identifiability up to permutation, scaling, and shift is preserved under general nonlinear latent dynamics. We validate our theory on simulated FitzHugh-Nagumo dynamics, and then demonstrate two applications: adjudicating between competing models of decision-making; and predicting responses to unseen micro-stimulation in macaque and human neural recordings, with proof-of-concept generalization to a held-out stimulation site in the latter. Finally, simulations yield practical guidance for designing stimulation protocols that improve identifiability.
Figures & tables
Figure 1: Overview: ingredients for identifying latent dynamics in state-space models. (a) iSSM leverages do -interventions to identify latent dynamics up to permutation, coordinate-wise scaling, and shift. (b) Two independent failure modes prevent SSMs from identifying the true dynamics g∗ : identification error ϵ1 , resolved by interventions, and hypothesis class mismatch ϵ2 , resolved by a nonlinear model class. (c) Latent trajectory recovery across model class (rows) and data regime (columns): only a nonlinear model trained on interventional data recovers the true trajectory (gray).
Figure 2: Nonlinear iSSM identifies FitzHugh-Nagumo latent dynamics. (a,b) Open-loop rollout and aligned latent trajectories on a test trial: true (black), linear iSSM (blue), nonlinear iSSM (magenta); mean ± s.d. across 10 seeds. Gray lines mark interventions. (c) Latent flow fields from the seed with highest Jacobian similarity to ground truth. Gray/black: true flow (arrows) and trajectory (line); blue/magenta: inferred linear/nonlinear flow and trajectories. (d) Median inferred input map B after permutation-scale alignment. (e) Latent-dimension sweep over D∈{1,2,3,4} shows that model selection picks out the true dimension ( D=2 ).
Figure 3: Nonlinear iSSM identifies a broader class of decision-making dynamics. Latent flow fields and trajectories for linear (left, blue) and nonlinear (right, magenta) iSSMs trained on interventional data from point-, line-, and bistable-well systems. Ground truth: gray arrows and black lines. Panels show best seed by Jacobian similarity; check marks indicate correct qualitative recovery.
Figure 4: Nonlinear iSSM outperforms linear iSSM on macaque prefrontal cortex recordings. (a,b) Example reconstructions and rollouts: observations (black), linear iSSM (blue), nonlinear iSSM (magenta); mean ± s.d. across seeds. Gray lines mark interventions. (c,d) Reconstruction and rollout accuracy on held-out control and stimulation trials. Asterisks: p<0.05 .
Figure 5: Nonlinear iSSM predicts human iEEG stimulation responses more accurately, including at a held-out site. (a) Axial view: recording contacts, stimulation sites, annotated epileptogenic contacts, and the held-out site. (b) Rollouts for the held-out site trial: observations (black), linear iSSM (blue), nonlinear iSSM (magenta); mean ± s.d. across seeds. Grey line: stimulation pulse. (c,d) Reconstruction and rollout accuracy on held-in and held-out site trials. Asterisks: p<0.05 .
Figure 6: Numerous and diverse interventions improve latent recovery. (a) Latent accuracy improves with more training interventions. (b) Diverse amplitudes and irregular timing yield higher latent accuracy; boxes: 10 seeds, *** p<0.001 main effect. (c) Latent accuracy versus rollout (black) and reconstruction (gray) accuracy; gray dashed line: identity.
Appendix figures & tables6 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 7: Nonlinear iSSM identifies Van Der Pol oscillator latent dynamics. (a,b) Open-loop rollout and aligned latent trajectories on a test trial: true (black), linear iSSM (blue), nonlinear iSSM (magenta); mean ± s.d. across 10 seeds. Gray lines mark interventions. (c) Latent flow fields from the seed with highest Jacobian similarity to ground truth. Gray/black: true flow (arrows) and trajectory (line); blue/magenta: inferred linear/nonlinear flow and trajectories. (d) Median inferred input map B after permutation-scale alignment. (e) Latent-dimension sweep over D∈{1,2,3,4} shows that model selection picks out the true dimension ( D=2 ).
Figure 8: Accuracy versus latent dimension D . (a) Reconstruction and (b) rollout accuracy for linear (blue) and nonlinear (magenta) iSSMs; mean ± s.e.m. across 10 seeds. Asterisks mark significant differences ( p<0.05 , FDR-corrected).
Figure 9: Accuracy versus latent dimension D . (a) Reconstruction and (b) rollout accuracy for linear (blue) and nonlinear (magenta) iSSMs; mean ± s.e.m. across 10 seeds. Asterisks mark significant differences ( p<0.05 , FDR-corrected).
Figure 10: Virtual stimulation of clinically annotated sites. (a) Peak and (b) sustained network-wide response ∥Δy∥ to virtual stimulation of healthy ( n=51 ) and epileptogenic ( n=26 ) bipolar pairs, under the linear and the nonlinear iSSM. Mann-Whitney U tests, FDR-corrected over the two readouts and two models; ∗∗∗ : p<0.001 .
Figure 11: Intervention-design trends replicate on point and line attractors. Top row (a-c): point attractor; bottom row (d-f): line attractor. Panel layout matches Fig. 6 : (a,d) latent accuracy versus intervention count; (b,e) amplitude-timing factorial (10 seeds); (c,f) latent accuracy versus rollout (black) and reconstruction (gray); dashed line: identity.
Figure 12: Latent accuracy under softened interventions and model comparison. Sweep over the intervention hardness parameter α of the data-generating process, from a purely additive drive ( α=0 ) to a hard clamp ( α=1 ), comparing nonlinear iSSM to two baselines. (a) Bistable well (App. J.1 ). (b) FitzHugh-Nagumo replication (App. J.2 ). Asterisks indicate α -regimes where iSSM significantly outperforms both baselines ( p<0.05 ).
Systems and Control Laboratory, HUN-REN Institute for Computer Science and Control, 1111 Budapest, Hungary · Control Systems Group, Eindhoven University of Technology, 5612 Eindhoven, The Netherlands
Center for Neurocomputation and Machine Intelligence, Wu Tsai Institute, Department of Biomedical Engineering, Yale University, New Haven, CT 06510, USA