Decoding neural population activity involves transforming high-dimensional, noisy spike trains into meaningful, low-dimensional representations. While linear methods provide a baseline, they often fail to capture the intricate, non-linear dynamics inherent in neural circuits. The goal is to uncover the underlying latent manifold—a smooth, low-dimensional surface on which the neural state evolves over time. This requires models that can infer these continuous trajectories while accounting for both neural variability and temporal dependencies.

Gaussian Process Factor Analysis (GPFA) extends factor analysis by imposing a temporal smoothness prior on the latent variables. Instead of treating each time point independently, GPFA models each latent dimension as a sample from a Gaussian Process (GP). This is a powerful way to enforce that neural trajectories should be smooth, reflecting the biophysical constraints of neural systems.

The choice of kernel, $K$, is critical as it defines the temporal correlation structure of the latent trajectories. A common choice is the squared exponential kernel, which assumes that points close in time are highly correlated, with the correlation decaying as a function of temporal distance. The model parameters, including the kernel hyperparameters, factor loadings $C$, and noise covariance $R$, are typically fit using the Expectation-Maximization algorithm. GPFA's Bayesian nature provides a full posterior distribution over the latent trajectories, offering a principled way to handle uncertainty.

While GPFA excels at finding smooth trajectories, its assumption of linear dynamics is a significant limitation. Neural systems are fundamentally non-linear. Latent Factor Analysis via Dynamical Systems (LFADS) addresses this by using a deep learning framework to model neural activity as arising from a non-linear dynamical system. It leverages a sequential variational autoencoder (VAE) architecture to infer these dynamics directly from single-trial spike train data.

The LFADS model consists of several key components:

• Encoder: A bidirectional RNN that processes the entire spike train sequence for a single trial to infer the initial state ($g_0$) of the generator.
• Generator: An autonomous RNN that evolves from the initial state $g_0$ to produce a sequence of hidden states, representing the latent dynamics.
• Factors: A linear layer maps the generator's state at each time step to the low-dimensional latent factors ($z_t$).
• Rates: Another linear layer maps the generator's state to the inferred log firing rates of each neuron.
• Spike Generation: The observed spikes are modeled as a Poisson process conditioned on the inferred firing rates.

Training is performed by optimizing the Evidence Lower Bound (ELBO), a standard objective for VAEs. This process simultaneously learns the parameters of the encoder, the generator, and the linear mappings. A key innovation in LFADS is the use of a learned prior distribution for the generator's initial state, which improves regularization and allows the model to better generalize across trials. The result is a powerful tool for de-noising neural data and inferring single-trial dynamics that would be obscured by trial-averaging.

By moving from linear-Gaussian assumptions to non-linear dynamics learned by recurrent neural networks, these methods provide a far more flexible and powerful framework for understanding how neural populations compute.