The identifiability problem
Here is the discomfort at the heart of the whole enterprise. Fit a latent dynamical model and you recover a state and a dynamics matrix — but apply an invertible change of coordinates to the latent space and you get a different-looking model that predicts the observed spikes exactly as well. The likelihood cannot tell them apart. This is the identifiability problem: the data constrain the model only up to a large equivalence class of transformations.
The similarity-transform ambiguity: any invertible T relabels the latent coordinates, changing A, C, and the state while leaving the likelihood of the observed spikes identical. Nonlinear models admit even larger symmetry groups.
The consequence is sobering: a recovered latent axis, on its own, has no privileged meaning; a rotation rate or attractor location may be an artifact of the parameterization, not a fact about cortex. Progress here comes from adding structure the data alone cannot supply — sparsity, known inputs, behavioral supervision, or auxiliary variables that break the symmetry, as in some contrastive formulations shown to be identifiable under stated assumptions.
What is behaviorally relevant?
Even a perfectly identifiable model faces a second question: most neural variance has nothing to do with the task. A generative model trained to reconstruct spikes will happily spend its capacity on the dominant-but-irrelevant signals. Preferential subspace identification (PSID) confronts this by explicitly decomposing the dynamics into a subspace that predicts behavior and a residual that does not — dissociating behaviorally-relevant dynamics from the rest.
The PSID split: neural activity yₜ is generated by a behaviorally-relevant latent x⁽ᵇ⁾ and a residual x⁽ʳ⁾, but behavior zₜ depends only on x⁽ᵇ⁾. Prioritizing that subspace yields cleaner, lower-dimensional, behavior-linked dynamics.
Comparing dynamics across brains and models
If individual latent axes are not comparable, how do you ask whether two animals — or a brain and a trained network — use the same dynamics? Dynamical similarity analysis (DSA) answers by comparing the vector fields themselves in a way invariant to the nuisance transformations above, going beyond geometry-only tools like representational similarity analysis. It is the beginning of a principled comparative science of dynamics, and a partial answer to the identifiability worry: some properties are invariant even when the coordinates are not.
Causality: the hardest frontier
Every method so far is observational — it fits activity you passively recorded. But a flow field inferred from observation is a correlational summary; it cannot, by itself, tell you that a state causes the next, only that they co-occur. Counterfactual and causal neural modeling aims higher: to predict what activity would have been under an intervention that never happened. Testing such claims ultimately requires perturbation — the write-side tools of optogenetics and two-photon holographic stimulation that let you nudge the state and check whether the model predicted the consequence.
Honest limits and where it is going
Two further limits deserve naming. First, dynamics drift: the latent dynamics are non-stationary across days, so a model fit today may be stale tomorrow — the problem the drift-robust track exists to solve, often by stabilizing the latent geometry rather than the neurons. Second, there is an irreducible variability floor: not all neural variance is signal, and no model can explain trial-to-trial noise that is genuinely stochastic. Knowing where that floor sits keeps expectations honest.
Where is it going? Toward models that are simultaneously expressive, identifiable under stated assumptions, causally testable, and stable over time — today no single method is all four. The most promising direction unifies the threads of this track: large pretrained dynamical backbones (foundation models) that transfer across subjects, interrogated with interpretable structure and validated by closed-loop perturbation, edging toward a genuine neural digital twin. That is aspiration, not achievement — but it is a disciplined one, and it is what makes this the most intellectually alive corner of the field.