Decomposed Linear Dynamical Systems (dLDS) for learning the latent components of neural dynamics

TL;DR

Decomposed Linear Dynamical Systems (dLDS) introduce a manifold-aware, sparse-dictionary approach to neural population dynamics, enabling non-stationary and nonlinear patterns to be represented as combinations of simple, interpretable linear regimes. By replacing rigid switches with continuous, time-varying coefficients over a dictionary of dynamic operators, the method achieves smooth transitions, demixing of overlapping subnetworks, and scalable modeling of multiple subsystems. The authors develop a dictionary-learning framework with variational EM and demonstrate strong performance across synthetic benchmarks (speed, rotations, stability shifts, Lorenz, FHN) and real data (C. elegans neural activity), often outperforming or matching switched models with far fewer parameters. The work provides a versatile, interpretable tool for uncovering latent neural dynamics, with potential for broader applications in high-dimensional time series where multiple overlapping dynamical modes operate concurrently. Overall, dLDS offers a principled trade-off between interpretability and expressivity, enabling demixed, smoothly evolving dynamics on a neural manifold and practical analysis of complex brain data.

Abstract

Learning interpretable representations of neural dynamics at a population level is a crucial first step to understanding how observed neural activity relates to perception and behavior. Models of neural dynamics often focus on either low-dimensional projections of neural activity, or on learning dynamical systems that explicitly relate to the neural state over time. We discuss how these two approaches are interrelated by considering dynamical systems as representative of flows on a low-dimensional manifold. Building on this concept, we propose a new decomposed dynamical system model that represents complex non-stationary and nonlinear dynamics of time series data as a sparse combination of simpler, more interpretable components. Our model is trained through a dictionary learning procedure, where we leverage recent results in tracking sparse vectors over time. The decomposed nature of the dynamics is more expressive than previous switched approaches for a given number of parameters and enables modeling of overlapping and non-stationary dynamics. In both continuous-time and discrete-time instructional examples we demonstrate that our model can well approximate the original system, learn efficient representations, and capture smooth transitions between dynamical modes, focusing on intuitive low-dimensional non-stationary linear and nonlinear systems. Furthermore, we highlight our model's ability to efficiently capture and demix population dynamics generated from multiple independent subnetworks, a task that is computationally impractical for switched models. Finally, we apply our model to neural "full brain" recordings of C. elegans data, illustrating a diversity of dynamics that is obscured when classified into discrete states.

Figures (12)

• Figure 1: Decomposed dynamical system model.A: Trajectories along the manifold are guided by local DOs. In neuroscience, we indirectly observe the latent manifold state $\bm{x}_t$ through the observation model $\bm{D}$. The space of transports $\{\bm{g}_l\}_{l=1:L}$ can be learned directly or through a discretized approximation $\{\bm{f}_m\}_{m=1:M}$. B: The decomposed linear dynamical systems includes an observation model, a dynamics model, and hierarchical variables $\bm{c}_t$ that control the non-stationarity in the dynamics. These dynamics coefficients can be structured (top), e.g., one fixed active coefficient at a time results in switching between discrete states, whereas enabling flexibility in the coefficient's value can enable scaled dynamics, sparsely structured dynamics, or even more arbitrarily distributed dynamics.
• Figure 2: Synthetic linear systems, examples of efficient representation.A: The generated path from ground truth 2D spiral system, colored by speed in phase space (top) and unrolled over time (bottom). B: The inferred rSLDS discrete states (top) and the inferred dLDS coefficients (bottom). rSLDS models speed changes with three discrete states and incorrectly groups the two fastest speeds together while dLDS changes coefficients on a single dictionary element. C: dLDS learns a single dictionary element (left column) that can be reused while rSLDS learns redundant systems (right column). Smooth transitions between DOs represent different paths on a spherical manifold:D: The generated path on the ground truth 3D sphere colored to visualize progression through time. E: Possible ground truth rotations. Multiple traces show dynamics and a single trace is highlighted (red) for better visibility. F: Convex combinations of learned DOs $g_1$ and $g_3$ allow for smooth transitions along continuum of rotated systems. G: dLDS learns two DOs that can be combined to represent all paths on the sphere while rSLDS must learn each angle of rotation separately.
• Figure 3: A,B: dLDS captures changes in system stability.A: Comparison of the generated coefficients (blue) versus the dLDS recovered coefficients (red), and the rSLDS coefficients (green). B: The ground truth dynamics over time (blue, top), versus the recovered dynamics by dLDS (middle, red) and rSLDS (bottom, green). C,D,E,F: dLDS can model smooth transitions in discrete-time dynamics.C: Schematic behavior of dLDS and switching dynamics in modeling transitions between linear dynamics. Switching systems jump between dynamic states producing sharp trajectories while dLDS can smoothly change the DO coefficients to capture gradual transitions in the system. D: The ground truth dynamics (top, blue), compared to the recovered dynamics by dLDS (middle, red) and rSLDS (right, green). Each column corresponds to a different axis. E: Comparison of the generated coefficients (blue), the dLDS recovered coefficients (red), and the rSLDS coefficients (green). F: The ground-truth DOs (top, blue) versus the DOs recovered by dLDS (middle, red) and rSLDS (bottom, green).
• Figure 4: dLDS identifies independently evolving groups from combined time series.A: The ground truth DOs displaying the block structure of the data (top), DOs recovered by dLDS (middle), DOs recovered by rSLDS (bottom). Framed dLDS DOs have a perfect correlation to the "true" DOs for both populations (cyan, maroon). B: The ground truth coefficients (top), dLDS recovered coefficients (middle) and rSLDS recovered coefficients (bottom). dLDS can accurately recover the structure of the ground truth and nullifies redundant coefficients, while rSLDS combines dynamics across the blocks. C: True generated dynamics (left), dLDS reconstruction (middle), and rSLDS reconstruction (right). D: The data reconstruction correlations for dLDS (red) and rSLDS (green) with the ground truth. dLDS achieves perfect reconstruction. E: The correlations of each true DO (rows) with each recovered DO (columns). dLDS (left) recovers all true DOs (each row presents exactly one black cell), while rSLDS tends to combine DOs across the two independent groups.
• Figure 5: Comparison between the unregularized dLDS (top row) and the sparse dLDS model (bottom row) for the FitzHugh-Nagumo oscillator.A & F: The temporal evolution of the membrane voltage (blue-green) and of the dynamics coefficients. Three points of interest were highlighted to showcase our model's capability to capture various behaviors with only 2 basis components. These points of interest include 1) the action potential (AP) repolarization ($t=100$), 2) the AP peak ($t=332$), and 3) hyper-polarization ($t=733$). B & G: Comparison between the coefficients' space of the unregularized and the regularized case. For the unregularized dLDS model, the model coefficients ($\bm{c}_t$) can occupy any location in space, and need not be on the axes. By contrast, when adding regularization to the model, sparse coefficients lie near the axes. C & H: The learned reconstructed dynamics $\bm{F}_t$ at the time points of interest, highlighting dLDS's ability to infer more distinct phases than the number of sub-dynamics ($\bm{f_i}$)— a capability that is not available to linear or switching models. D & I: Stream-plots of the basis operators learned by dLDS ($\bm{f}_1$ and $\bm{f}_2$). E: The phase-space plot of the FHN model (v-w space), with time points of interest highlighted.