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Variational Latent Gaussian Process for Recovering Single-Trial Dynamics from Population Spike Trains

Yuan Zhao, Il Memming Park

arXiv:1604.03053v5stat.MLq-bio.NC

TL;DR

Single-trial neural dynamics are difficult to recover because spike trains are noisy and activity need not repeat in a time-locked way. vLGP combines a history-dependent point-process model with smooth Gaussian-process latent trajectories. In V1 recordings, it outperforms GPFA and PLDS in capturing spatiotemporal structure at 1 ms bins and recovers stimulus-related structure and noise fluctuations.

  • Problem

    Neural activity can vary across trials rather than follow a stereotyped, time-locked path, making recovery of hidden dynamics from individual trials important.

    Source: §1 · p. 2

  • Method

    vLGP combines a history-dependent point-process observation model with smooth Gaussian-process latent processes and uses variational inference for the posterior.

    Source: §1 · pp. 2–3, §2 · p. 3, §3 · p. 5

  • Results

    At 1 ms bins, vLGP captures spatiotemporal neural structure better than GPFA and PLDS, while V1 latents encode stimulus orientation and temporal modulation on a continuous manifold and capture noise fluctuations.

    Source: §4.4 · p. 19, §5 · p. 21

  • Takeaways & Limitations

    Single-trial dimensionality reduction separates neural signals into a small number of smooth temporal signals and their contributions to the population signal.

    Source: §5 · p. 21

    Recovering very fast-changing latent trajectories may be difficult, especially when overall firing rates are very low.

    Source: §5 · p. 21

Abstract

from arXiv · show

When governed by underlying low-dimensional dynamics, the interdependence of simultaneously recorded population of neurons can be explained by a small number of shared factors, or a low-dimensional trajectory. Recovering these latent trajectories, particularly from single-trial population recordings, may help us understand the dynamics that drive neural computation. However, due to the biophysical constraints and noise in the spike trains, inferring trajectories from data is a challenging statistical problem in general. Here, we propose a practical and efficient inference method, called the variational latent Gaussian process (vLGP). The vLGP combines a generative model with a history-dependent point process observation together with a smoothness prior on the latent trajectories. The vLGP improves upon earlier methods for recovering latent trajectories, which assume either observation models inappropriate for point processes or linear dynamics. We compare and validate vLGP on both simulated datasets and population recordings from the primary visual cortex. In the V1 dataset, we find that vLGP achieves substantially higher performance than previous methods for predicting omitted spike trains, as well as capturing both the toroidal topology of visual stimuli space, and the noise-correlation. These results show that vLGP is a robust method with a potential to reveal hidden neural dynamics from large-scale neural recordings.

1 Introduction

Single-trial latent trajectories are important because neural activity can vary across trials rather than follow a stereotyped, time-locked path. The paper introduces vLGP to infer these trajectories while accommodating nonlinear dynamics and spike-train observations.

1 Introduction

  • Recovering dynamics from individual trials matters because trial-to-trial variation can appear in behavior and distinguish neural implementations that averages cannot disambiguate.Such variation may include error trials, broad reaction-time distributions, and changes of mind.Source: §1 · p. 2
  • vLGP uses a general Gaussian process prior for latent dynamics and a history-dependent point-process observation model, rather than linear dynamics or an instantaneous Gaussian observation.The non-conjugate model requires approximate posterior inference.Source: §1 · pp. 2–3
  • The method uses a Gaussian process posterior approximation and maximizes a lower bound on the marginal likelihood to infer latent trajectories.Source: §1 · p. 3
  • The paper evaluates vLGP on simulated systems and applies it to V1 recordings to recover noise correlations and the topology of population responses to drifting orientations.Source: §1 · p. 3

2 Generative model

The generative model explains population spike trains through shared latent processes and neuron-specific spike histories. Smoothness is imposed on the latent processes through a Gaussian process prior.

2 Generative model

  • Each neuron's spike train is modeled as a point process with conditional intensity, using a parametric form for the intensity.Source: §2 · p. 3
  • Neurons share latent processes but also depend directly on their own observed spike histories through neuron-specific weights.The model attributes remaining trial-to-trial variability to shared latent processes or individual point-process noise.Source: §2 · p. 3
  • The model assumes a small number of latent dimensions relative to neurons and limits self-history influence to the specified past bins.The history vector includes a bias term and p time-step history values.Source: §2 · p. 3
  • The model allows external covariates to be added to the point-process regression, similar to the self-history terms.Source: §2 · p. 3
  • A Gaussian process prior encodes the assumed temporal smoothness of each latent dimension.The manuscript uses a squared-exponential covariance function for general smoothness over time.Source: §2 · p. 4

3 Variational inference

Variational inference approximates the intractable posterior and fits the model by maximizing a lower bound on the marginal likelihood. The method updates latent trajectories, neuron weights, and GP hyperparameters, using low-rank covariance approximations and temporal subsampling for efficiency.

3 Variational inference

  • The variational method approximates the intractable posterior and maximizes the ELBO, a lower bound on marginal likelihood, through coordinate-wise parameter updates.The posterior approximation factorizes into Gaussian distributions across latent dimensions.Source: §3 · pp. 5–6

3.1 Posterior over the latent process

  • Latent posterior means are updated with Newton-Raphson; the Hessian is negative definite, making this update a convex optimization given the other arguments and firing rates.Source: §3.1 · p. 6
  • A truncated incomplete Cholesky factor approximates the large prior covariance matrix, providing a compact representation and numerical stability.The stated per-iteration complexity is O(Tr^2L) in time and O(Ts) in space, with s = max(rL, pN).Source: §3 · p. 6, §3.1 · p. 8

3.2 Weights

  • Neuron weights α and β are updated with Newton-Raphson, and both Hessians are negative definite.Source: §3.2 · p. 9

3.3 Hyperparameters

  • Because the marginal likelihood is intractable, GP hyperparameters are optimized by maximizing the variational lower bound with parameters and posterior fixed.For long time series, the method randomly samples shorter temporal subsamples to reduce the cost of hyperparameter updates.Source: §3.3 · pp. 9–10

4 Results

The results validate vLGP under its assumed model, compare it with GPFA and PLDS on simulations, and apply it to V1 recordings.

4.1 Convergence

  • As neuron count grows, posterior latent means converge to the true latents; as time bins grow, parameters converge to the true weights.Learned timescales also scatter around ground truth across a wide range of true timescales.Source: §4.1 · p. 11

4.2 Evaluation

  • Evaluation uses leave-one-neuron-out spike prediction and Spearman rank correlation to assess prediction and latent recovery, respectively.For prediction, the held-out neuron is inferred from the remaining neurons in a test trial using training-trial weights.Source: §4.2 · p. 11

4.3 Simulation

  • vLGP achieves the largest latent rank correlation and runs an order of magnitude faster than PLDS, while GPFA converges fastest.The comparison tracks rank correlation against running time.Source: Figure 4 · p. 14
  • vLGP infers smooth latent processes close to the true trajectories in both Lorenz and linear-dynamics simulations.PLDS lacks smoothness, while GPFA’s smooth inference deviates from the true LDS latent.Source: §4.3 · pp. 13–14

4.4 V1 population recording

  • vLGP is consistently best in both predictive log-likelihood and predictive R2 on the V1 comparisons.GPFA’s predictive log-likelihood is not a fair comparison because it can predict negative means.Source: Figure 5 · p. 16
  • In V1, uncertainty in inferred latents increases during the off-stimulus half, when firing rates are lower.Single-trial latent variation reflects visible trial-to-trial spike-train variability.Source: §4.4 · p. 18
  • The V1 population encoding of orientation and temporal phase has a toroidal topology in the inferred trajectories.The projected orientation trajectories preserve the stimulus-space topology.Source: §4.4 · pp. 18–19
  • vLGP explains more noise correlation than GPFA and PLDS during both stimulus and off-stimulus periods.The results support its capture of both time-locked signal and trial-to-trial population fluctuations.Source: §4.4 · p. 19

5 Discussion

vLGP recovers low-dimensional latent dynamics from single-trial recordings and captures V1 stimulus structure on a continuous manifold. The discussion highlights its comparative performance, possible extensions, and potential limitations.

5 Discussion

  • Single-trial dimensionality reduction separates neural activity into a small number of smooth temporal signals and their relative contributions to the population signal.The authors describe latent-trajectory analysis as well suited to decision-making, attention, and memory recall, which they characterize as intrinsically low-dimensional computations.Source: §5 · pp. 19, 21
  • vLGP uses a generalized linear model observation and outperforms GPFA and PLDS in capturing neural spatiotemporal structure at 1-ms bins, while running faster than PLDS.It recovers low-dimensional latent dynamics from high-dimensional time series.Source: §5 · pp. 19, 21
  • In V1, inferred latents encode orientation and temporal modulation on a continuous manifold, with mean trajectories forming an approximate torus.The plotted trajectories use the first three singular vectors; orientation is color-coded and cycle averages indicate oscillation phase.Source: Figure 8 · p. 20, §5 · p. 21
  • The current method uses a smoothness prior, but other GP kernels and covariates such as stimuli or local field potential could be incorporated.The authors suggest spectral kernels for periodic latent processes and internal neural oscillations.Source: §5 · p. 21
  • Potential weaknesses include the variational approximation, latent-process assumptions, numerical shortcuts, and difficulty recovering fast dynamics at low firing rates.Hyperparameter subsampling may miss rare but important spikes when spike trains are sparse.Source: §5 · p. 21
  • The authors see potential for studying how neural computations are implemented as dynamics and are applying the method to sensorimotor decision-making.They note that how the neural system implements the normative computation remains unclear.Source: §5 · p. 21
  • An open-source Python implementation of vLGP is available under the MIT license.Source: §5 · p. 21
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