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Probabilistic Decomposed Linear Dynamical Systems for Robust Discovery of Latent Neural Dynamics

Yenho Chen, Noga Mudrik, Kyle A. Johnsen, Sankaraleengam Alagapan, Adam S. Charles, Christopher J. Rozell

TL;DR

This work proposes a probabilistic approach to latent variable estimation in decomposed models that improves robustness against dynamical noise and introduces an extended latent dynamics model to improve robustness against system nonlinearities.

Abstract

Time-varying linear state-space models are powerful tools for obtaining mathematically interpretable representations of neural signals. For example, switching and decomposed models describe complex systems using latent variables that evolve according to simple locally linear dynamics. However, existing methods for latent variable estimation are not robust to dynamical noise and system nonlinearity due to noise-sensitive inference procedures and limited model formulations. This can lead to inconsistent results on signals with similar dynamics, limiting the model's ability to provide scientific insight. In this work, we address these limitations and propose a probabilistic approach to latent variable estimation in decomposed models that improves robustness against dynamical noise. Additionally, we introduce an extended latent dynamics model to improve robustness against system nonlinearities. We evaluate our approach on several synthetic dynamical systems, including an empirically-derived brain-computer interface experiment, and demonstrate more accurate latent variable inference in nonlinear systems with diverse noise conditions. Furthermore, we apply our method to a real-world clinical neurophysiology dataset, illustrating the ability to identify interpretable and coherent structure where previous models cannot.

Probabilistic Decomposed Linear Dynamical Systems for Robust Discovery of Latent Neural Dynamics

TL;DR

This work proposes a probabilistic approach to latent variable estimation in decomposed models that improves robustness against dynamical noise and introduces an extended latent dynamics model to improve robustness against system nonlinearities.

Abstract

Time-varying linear state-space models are powerful tools for obtaining mathematically interpretable representations of neural signals. For example, switching and decomposed models describe complex systems using latent variables that evolve according to simple locally linear dynamics. However, existing methods for latent variable estimation are not robust to dynamical noise and system nonlinearity due to noise-sensitive inference procedures and limited model formulations. This can lead to inconsistent results on signals with similar dynamics, limiting the model's ability to provide scientific insight. In this work, we address these limitations and propose a probabilistic approach to latent variable estimation in decomposed models that improves robustness against dynamical noise. Additionally, we introduce an extended latent dynamics model to improve robustness against system nonlinearities. We evaluate our approach on several synthetic dynamical systems, including an empirically-derived brain-computer interface experiment, and demonstrate more accurate latent variable inference in nonlinear systems with diverse noise conditions. Furthermore, we apply our method to a real-world clinical neurophysiology dataset, illustrating the ability to identify interpretable and coherent structure where previous models cannot.
Paper Structure (35 sections, 4 theorems, 31 equations, 7 figures, 5 tables, 1 algorithm)

This paper contains 35 sections, 4 theorems, 31 equations, 7 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Probabilistic Decomposed Linear Dynamical Systems
  4. Time-varying offset term
  5. Probabilistic Time-Informed Sparsity
  6. Inference and Learning
  7. Results
  8. Synthetic Dynamical Systems
  9. Simulated Motor Cortex Data in a Reaching Task
  10. Clinical Neurophysiology Data
  11. Conclusion
  12. Appendix / supplemental material
  13. p-dLDS Algorithm
  14. Latent Variable Inference
  15. Lemma \ref{['prop:offset_solution']} Derivation
  16. ...and 20 more sections

Key Result

Lemma 1

Let the transition between any two state vectors $\bm{x}_t, \bm{x}_{t+1} \in \mathbb{R}^N$ be defined by the linear dynamics matrix $\bm{F}_t \in \mathbb{R}^{N\times N}$ and the dynamics offset $\bm{b}_t \in \mathbb{R}^N$. For any $\lambda > 0$, the objective, is minimized when $\bm{F}_t = \bm{0}$ and $\bm{b}_t = \bm{x}_{t+1} - \bm{x}_{t}$.

Figures (7)

  • Figure 1: (A) Graphical model of dLDS. (B) p-dLDS includes hierarchical variables for probabilistic sparse inference and reparameterizes the latent space to include a time-varying offset term.
  • Figure 2: Probabilistic model and offset term reduce estimation errors.(A) Example trial from the NASCAR experiment colored by the true switching labels (not provided during training). Each track segment has a random speed $\tau$. (B) Inferred state space, colored by discrete state or dominant coefficients. p-dLDS identifies correct track segments. (C) Example trial from the Lorenz experiment. The speed ramps according to the time intervals $\Delta \tau$ in an ODE solver. (D) Inferred state space, colored by the dominant coefficients. The time-varying offset term allows p-dLDS coefficients to switch according to the true speed and accurately model the two fixed points in the opposing lobes.
  • Figure 3: p-dLDS efficiently captures changes in dynamics.(A) Latent factors are computed from empirical EMG of the reaching experiment in depasquale2023centrality. Dynamics are characterized by a preparatory and movement phase. (B) Synthetic spikes and LFPs are generated using the wslfp package johnsen2023cleojohnsen24wslfp(C) The trial-averaged coefficients for p-dLDS smoothly vary with reaching angle. DO 1 captures preparatory dynamics while DO 3 captures movement dynamics. (D) Confusion matrix for linear classification of reach directions. p-dLDS predictions closely align to true diagonal.
  • Figure 4: Learned system discovers coherent structure in clinical data.(A, D) LFP data was collected on patients watching videos with different emotional content. (B, E) LFP spectrograms are 40-dimensional signals where each channel represents a particular frequency. (C, F) Inferred states and coefficients shows that rSLDS and dLDS exhibit unpredictable switching behavior. In contrast, p-dLDS captures smooth coefficients and identifies DOs that align with the trial's emotional content. The learned patterns broadly generalize to the held-out data.
  • Figure 5: Empirically-Derived Reach Experiment.(A) 1,200 neurons are randomly placed into a 5 mm by 10 mm by 1 mm region. Electrodes are placed in a grid centered in this region (B) Spiking activity for a subset of neurons in an example trial produced from a factor-based spiking network. (C) First 15 channels in a simulated multi-channel LFP recording. Preparatory and Movement phases are marked by the dotted lines.
  • ...and 2 more figures

Theorems & Definitions (6)

  • Lemma 1
  • Lemma 2
  • Lemma 2
  • proof
  • Lemma 2
  • proof