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Modeling Latent Neural Dynamics with Gaussian Process Switching Linear Dynamical Systems
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Proposes gpSLDS, a GP-SDE latent dynamics model with a smoothly switching linear kernel and a partially optimized ELBO for hyperparameter learning, as an interpretable, uncertainty-aware alternative to rSLDS.
Problem
Latent neural dynamics models must be both expressive for nonlinear dynamics and interpretable; rSLDS produces artifactual oscillations near regime boundaries, lacks posterior uncertainty over dynamics, and does not enforce smoothness.
Approach
A novel Smoothly Switching Linear (SSL) GP kernel, a sum over J linear kernels multiplied by softmax partition kernels from logistic-regression decision boundaries, placed as a prior on dynamics in a GP-SDE, yielding smooth locally linear flows with posterior uncertainty; plus a modified vEM objective that maximizes the ELBO over inducing-variable posteriors analytically before updating hyperparam
Key results
- On synthetic data, gpSLDS and GP-SDE with RBF kernel consistently outperform rSLDS in MSE of latents and dynamics across varying numbers of trials.
- On hypothalamic calcium imaging, gpSLDS finds high-probability slow points consistent with the line attractor reported for rSLDS, and outperforms rSLDS in forward simulation R^2 though slightly below the RBF kernel after 500 steps.
- The modified vEM objective recovers the true decision boundary x1=0 where standard vEM fails (shown in appendix experiments).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use gpSLDS with the SSL kernel instead of rSLDS | Synthetic neural dynamics · Synthetic data | latent trajectory recovery | Helped |
| Use gpSLDS with the SSL kernel instead of rSLDS | Synthetic neural dynamics · Synthetic data | dynamics recovery | Helped |
| Use gpSLDS with the SSL kernel instead of GP-SDE with an RBF kernel | Synthetic neural dynamics · Synthetic data | dynamics recovery | Helped |
| Use a continuous-time model instead of rSLDS | Synthetic neural dynamics · Synthetic data | latent and dynamics MSE | Helped |
| Use gpSLDS with the SSL kernel instead of rSLDS | Hypothalamic neural recordings · Hypothalamic aggression data | in-sample forward simulation R² | Helped |
| Use gpSLDS with the SSL kernel instead of GP-SDE with an RBF kernel | Hypothalamic neural recordings · Hypothalamic aggression data | in-sample forward simulation R² | Mixed |
| Optimize kernel hyperparameters using the partially optimized ELBO rather than standard coordinate-ascent vEM (ablation) | gpSLDS · Synthetic data | decision-boundary estimation accuracy | Helped |
Surprising findings
- The RBF-kernel GP-SDE slightly outperforms gpSLDS in forward simulation on the hypothalamic data after about 500 steps, though gpSLDS still beats rSLDS.
Limitations
- Memory cost grows exponentially with latent dimension because Gauss-Hermite quadrature approximates SSL kernel expectations, limiting use to about 3 latent dimensions.
- Requires smaller discretization steps than rSLDS for stable inference.
- Variational inference may underestimate posterior variance.
Experimental setup
Datasets: synthetic Poisson LDS data hypothalamic calcium imaging (aggression) LIP spiking data (decision making)
Benchmarks: synthetic latent and dynamics recovery hypothalamic forward simulation R^2
Metrics: MSE between true and inferred latents and dynamics forward simulation R^2
Baselines: rSLDS GP-SDE with RBF kernel standard vEM for GP-SDE
Builds on: GP-SDE recurrent switching linear dynamical system variational EM for GP-SDEs
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