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Linear dynamical neural population models through nonlinear embeddings
Yuanjun Gao, Evan Archer, Liam Paninski, John P. Cunningham
TL;DR
Existing neural population models often use linear generative observations, limiting their ability to capture nonlinear structure in large-scale neural activity. This paper introduces fLDS, combining linear latent dynamics with flexible neuron-specific nonlinear rates and variational inference. In simulations and two neural datasets, fLDS achieves better predictive performance with more compact and informative latent representations than competing models.
Problem
Existing linear observation models can fail to recover low-dimensional neural structure when the data-generating process is nonlinear.
Method
fLDS combines a latent linear dynamical system with an arbitrary continuous per-neuron rate function, fitted using variational inference with a temporally correlated approximate posterior.
Results
fLDS yields higher prediction performance and more compact, informative latent representations than state-of-the-art neural population models across simulations and two neural datasets.
Takeaways & Limitations
The model retains linear latent dynamics while allowing nonlinear observation functions, preserving a low-dimensional latent representation for neural population data.
Takeaways & Limitations
The current framework assumes latent linear dynamics; future work proposes more flexible latent dynamics such as Gaussian-process or nonlinear phase-space priors.
Abstract
from arXiv · showhide
A body of recent work in modeling neural activity focuses on recovering low-dimensional latent features that capture the statistical structure of large-scale neural populations. Most such approaches have focused on linear generative models, where inference is computationally tractable. Here, we propose fLDS, a general class of nonlinear generative models that permits the firing rate of each neuron to vary as an arbitrary smooth function of a latent, linear dynamical state. This extra flexibility allows the model to capture a richer set of neural variability than a purely linear model, but retains an easily visualizable low-dimensional latent space. To fit this class of non-conjugate models we propose a variational inference scheme, along with a novel approximate posterior capable of capturing rich temporal correlations across time. We show that our techniques permit inference in a wide class of generative models.We also show in application to two neural datasets that, compared to state-of-the-art neural population models, fLDS captures a much larger proportion of neural variability with a small number of latent dimensions, providing superior predictive performance and interpretability.
1 Introduction
Modern experiments record increasingly large neural populations, motivating low-dimensional dynamical models for tractable analysis. fLDS extends latent linear dynamics with neuron-specific nonlinear observation functions and variational inference, yielding stronger predictive and representational performance than existing population models.
- Motivation: Recordings now span hundreds to tens of thousands of neurons, motivating statistical methods for high-dimensional population activity.Low-dimensional dynamical trajectories offer tractable analysis and visualization of these data.
- Limitations of prior models: Prior latent-trajectory models use linear dynamics with linear or restricted nonlinear observation models, which can miss nonlinear low-dimensional structure.Such model mismatch can hamper recovery when neurons respond nonlinearly to a shared latent stimulus.
- fLDS: fLDS lets each neuron’s firing rate be an arbitrary smooth nonlinear function of a shared latent state with linear dynamics.This creates a nonlinear embedding of neural time series into a linear-dynamical state space.
- Inference: A variational inference scheme with a temporally correlated approximate posterior enables inference for this nonlinear model class.The approach is designed for a large class of latent-LDS generative models.
- Results: Across simulation and two neural datasets, fLDS provides higher prediction performance with a more compact and informative latent representation than state-of-the-art population models.The reported benefits include improved predictive performance and interpretability.
2 Notation and overview of neural data
Neural population recordings are represented as spike-count vectors collected over time and repeated trials. Each observation records the activity of n neurons in a time bin, with trials indexing repeated experimental conditions.
- Population observations: A population observation x_t is a length-n vector whose ith entry counts spikes from neuron i in a time bin.Time is discretized into bins of duration Δt.
- Repeated trials: Repeated trials r index multiple recordings of the same experiment, reflecting variability in spike responses under identical conditions.The notation distinguishes time, neuron, and trial indices.
- Data organization: The full dataset is organized as time-by-neuron matrices for each trial and a collection of those matrices across trials.For trial r, x^r contains T time points and n neurons.
3 Review of latent LDS neural population models
Latent LDS population models represent neural activity with low-dimensional linear dynamics coupled to an observation model. Different noise distributions and rate functions produce distinct models, while non-Gaussian or nonlinear observations generally require approximate inference.
- Latent factors: Latent factor models infer low-dimensional, time-evolving trajectories that capture population variability.The latent dimension m is much smaller than the neuron count n.
- Latent dynamics: LDS-based models describe an unobserved m-dimensional state evolving through linear dynamics that modulate neural population activity.The dynamics use a linear matrix A and state-noise covariances.
- Observation model: The observation model maps the latent state through a deterministic rate function f and a noise model P_λ.Choosing different rate functions and noise distributions yields different latent LDS models.
- Inference: Gaussian noise with a linear rate function recovers the classical Kalman filter and permits closed-form conjugate inference.Non-Gaussian noise or nonlinear rates break conjugacy and require approximate inference.
- Example models: The PLDS combines Poisson observations with an element-wise exponential of a linear latent transformation.The GCLDS instead uses a generalized count distribution with a linear rate function.
- Example models: The generalized count model can flexibly represent both under-dispersed and over-dispersed count distributions.Its normalizing constant appears in the generalized count probability specification.
4 Nonlinear latent variable models for neural populations
fLDS retains latent linear dynamics while allowing each neuron’s firing rate to depend on the latent state through a separate nonlinear function. Because nonlinear observations make exact inference intractable, the paper uses variational inference with a temporally correlated approximate posterior.
- Generative model: fLDS retains the latent LDS but gives each neuron a separate nonlinear dependence on the latent state through a continuous rate function.The rate function is represented with a neural network, and the model parameters include its weights and biases.
- Inference: The model is non-conjugate because both the nonlinear observation function and the noise model can make posterior and marginal-likelihood computation intractable.The paper therefore uses stochastic variational inference to learn generative parameters and infer latent variables.
- Inference: Variational inference approximates the intractable posterior with a tractable distribution and optimizes the evidence lower bound.The approximation is conditioned on observations and is learned jointly with the generative model.
- Optimization: AEVB reparameterizes latent samples as transformations of simple random variables, enabling unbiased stochastic gradient estimation for the variational objective.The method applies the estimator to minibatches consisting of randomly selected trials.
- Approximate posterior: The approximate posterior uses a block-tridiagonal precision matrix, allowing rich temporal correlations while keeping computational complexity linear in trial length T.This structure is intended to capture the smooth, correlated trajectories typical of LDS posteriors.
5 Experiments
Experiments compare fLDS variants with linear and alternative inference models in simulations and macaque neural datasets. Nonlinear fLDS models recover nonlinear responses, achieve strong held-out prediction with fewer latent dimensions, and reveal structured neural manifolds.
- Simulation experiments: AEVB inference gives PLDS and GCLDS predictive results comparable to specialized EM-based methods, while PfLDS and GCfLDS retain comparable performance despite more complex generative models.The comparison reports predictive log likelihood and computation time across Bernoulli, Poisson, and negative-binomial simulations.
- Simulation experiments: PfLDS recovers the nonlinear grid-cell latent trajectory and firing rates, whereas PLDS fails to recover both; held-out PLL is -0.581 for PfLDS versus -0.622 for PLDS.The paired PLL difference is statistically significant (p < 10^-6).
- Applications to experimentally-recorded neural data: In macaque V1, PfLDS captures nonlinear firing responses and produces a cleaner 2D latent representation than PLDS.Both models recover rotational structure, but PfLDS places nonlinear features in the observation model.
- Applications to experimentally-recorded neural data: PfLDS matches PLDS predictive performance with 3 latent dimensions where PLDS requires approximately 10, yielding a more compact representation.The comparison uses one-step-ahead held-out prediction evaluated by MSE and NLL reduction.
- Applications to experimentally-recorded neural data: Across the full V1 dataset, PfLDS with 4 latent dimensions uncovers a toroidal trajectory structure across grating orientations.The trajectory is obtained by projecting the latent representation onto its first three principal components.
6 Discussion and Conclusion
fLDS extends latent linear dynamical models with a flexible nonlinear observation model for high-dimensional neural populations. The framework recovers low-dimensional structure more effectively than linear-rate models and supports prediction and interpretation.
- fLDS extends latent linear dynamical system models with a flexible nonlinear observation model for high-dimensional neural population data.
- Linear rate functions can fail to capture true low-dimensional structure even when population activity is governed by low-dimensional linear dynamics.
- fLDS recovers low-dimensional structure while providing better predictive performance and more interpretable latent-variable representations.
- Unlike many manifold-learning methods, the approach uses temporal structure, an explicit generative model, and count-valued observations.
- Future work includes replacing the latent linear dynamical assumption with more flexible latent dynamics.
Supplementary material: Linear dynamical neural population
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A Temporally correlated approximate posterior
The recognition model uses observation-dependent nonlinear functions within a temporally correlated approximate posterior. Its Gaussian-factor representation yields a block-tridiagonal inverse covariance structure.
- The recognition model qφ(z|x) is used in auto-encoder variational inference for fitting fLDS.
- Observation-dependent nonlinear functions parameterize the posterior mean and precision-related terms.
- The posterior factors form a Gaussian functional representation with block-tridiagonal inverse covariance.
B Neural network structure for generative model and approximate posterior
The generative model and approximate posterior are parameterized with feed-forward neural networks using two tanh hidden layers.
- The generative function fψ is parameterized by a feed-forward neural network with two 60-node tanh hidden layers.
- For PfLDS, an exponential output transformation ensures positive firing rates.
- The approximate posterior networks use shared two-hidden-layer tanh representations for its mean and precision-related functions.
C Description of the video
The video visualizes fLDS latent trajectories for macaque V1 responses across grating orientations, revealing circular orientation-specific paths and a toroidal projection of the full dataset.
- A 4-dimensional PfLDS models macaque V1 activity from 300ms to 1200ms after stimulus onset.The video plots the first three principal components of inferred latent trajectories, using 10 training trials per grating orientation.
- Trajectories for individual directions are shown alongside spike rasters from associated trials.Examples include grating directions of 0°, 60°, and 120°.
- Neural activity for each grating orientation forms a circle in latent space, matching the periodicity of the sinusoidal stimulus and neural activity.The stimulus temporal frequency is 6.25Hz.
- The full dataset forms a torus in latent space, consistent with the stimulus structure.