State Space Analysis

How can we characterise the statistical structure of a neural population beyond the firing rate of individual neurons?

In this hands-on session, we will use the pairwise maximum-entropy (Ising) model to describe the joint activity of small neural populations represented as binary spike patterns. We will work with simulated data where the ground-truth parameters are known. We will begin by exploring the model’s natural parameters: neuron biases and pairwise couplings. We learn how they shape the distribution over spike patterns. We will then fit the model using an EM algorithm and assess how well pairwise interactions account for population variability using thermodynamic quantities such as the entropy ratio and KL divergence.

In the second part, we extend the model to non-stationary data. A state-space formulation treats the parameters as a hidden process evolving over time and estimates them with a Kalman filter and backward smoother. We will track dynamic correlations, quantify posterior uncertainty, and examine how thermodynamic signatures change when neural interactions fluctuate.

The aim is to build intuition for what the model parameters represent, when pairwise interactions matter, and what the algorithm can and cannot resolve given limited data.

Tools

• SSLL: state-space log-linear model for estimating time-varying spike-train correlations