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Deep Dynamics: RNNs, Neural ODEs, and Structured State Spaces

The generation that made latent dynamics famous: expressive nonlinear models that infer single-trial trajectories at spike-time resolution — and the new interpretability debt they run up.

LFADS: dynamics via RNN generators

LFADS (Latent Factor Analysis via Dynamical Systems) instantiates the seq-VAE from Guide 2 with a recurrent network as the generator. From a whole trial of spikes it infers a single initial condition plus a stream of inferred inputs, then a trained RNN rolls those forward into a smooth firing-rate estimate. Its headline result was striking: recovering clean, single-trial neural trajectories where trial-averaging had previously been mandatory — a genuine leap in what could be seen in one trial.

A deep sequential-VAE reads a trial's spikes into an inferred initial state, then a learned nonlinear generator produces the latent trajectory and firing rates — the LFADS template.

Conceptually, LFADS reifies the dynamical-systems hypothesis into a fittable model: the trajectory it recovers is the population state, and the RNN's learned update is the flow F. The flow field it implies can even be probed to ask what attractors and computations the network settled on.

Neural ODEs: continuous-time flows

A discrete RNN takes fixed steps; but the brain runs in continuous time, and spikes arrive irregularly. The neural ordinary differential equation answers this by letting a neural network parameterize the derivative of the latent state, and integrating it with a numerical solver. The flow becomes explicit and continuous, and the model handles irregular sampling natively — a clean match to the \dot{x}=F(x) picture from Guide 1.

\frac{dz(t)}{dt} = f_{\theta}\!\bigl(z(t)\bigr), \qquad z(t_1) = z(t_0) + \int_{t_0}^{t_1} f_{\theta}\!\bigl(z(t)\bigr)\,dt

A neural ODE: a network f_θ defines the flow, and the latent state is obtained by integrating it. Learning f_θ is learning the continuous-time dynamical law directly, rather than a discrete transition matrix.

Structured state spaces and long context

RNNs struggle to carry information across long windows. A recent line — the deep / structured state-space model (S4) — revives the classical continuous linear state-space equation but parameterizes it so a very long sequence can be processed as either a fast convolution or a linear recurrence, with special initializations that let the state remember far into the past. For neural data with slow context and long dependencies, these models are an increasingly attractive, efficient backbone.

x'(t) = \mathbf{A}\,x(t) + \mathbf{B}\,u(t), \qquad y(t) = \mathbf{C}\,x(t) + \mathbf{D}\,u(t)

The linear state-space equation at the heart of S4 — the same object as the classical LDS. The advance is in how A is structured and initialized so the recurrence stays stable and remembers over very long horizons.

Transformers and self-supervised alternatives

Not everyone models an explicit flow. The Neural Data Transformer drops recurrence entirely and treats a population time series as a sequence to be modelled by attention, learning representations by masking and reconstructing held-out neurons and time bins. Contrastive methods like CEBRA take yet another route, shaping an embedding so that behaviorally or temporally similar states sit close together — no generative dynamics required.