[Paper Review] Successes and failures of simple statistical physics models for a network of real neurons

Leenoy Meshulam, Jeffrey L. Gauthier+3 more|arXiv (Cornell University)|Dec 29, 2021

Gene Regulatory Network Analysis4 citations

TL;DR

This study tests maximum entropy models based on pairwise correlations to predict collective neural activity in a mouse hippocampus network of over 1,000 neurons. It finds that these simple statistical physics models achieve near-perfect quantitative agreement with experimental data for spatially contiguous local subgroups of 100 neurons, but fail for randomly sampled distant subgroups, demonstrating that model success depends on spatial contiguity and local network structure.

ABSTRACT

Biological networks exhibit complex, coordinated patterns of activity. Can these patterns be captured precisely in simple models? Here we use measurements of simultaneous activity in 1000+ neurons in the mouse brain to test the validity of models grounded in statistical physics. When cells are dense samples from a small region, we find extremely detailed quantitative agreement between theory and experiment; sparse samples from larger regions lead to model failures. These results show we can aspire to more than qualitative agreement between simplifying theoretical ideas and the detailed behavior of a complex biological system.

Motivation & Objective

• To evaluate whether simple maximum entropy models based on pairwise correlations can quantitatively predict collective neural activity in real biological networks.
• To investigate whether the success of such models depends on spatial organization of neurons in the network.
• To determine whether quantitative agreement between theory and experiment is achievable in complex biological systems when using minimal assumptions.
• To test whether model failure in certain subgroups reflects limitations of the model or properties of the underlying neural network.
• To assess whether effective fields derived from the model accurately predict single-neuron activity states across different network subgroups.

Proposed method

• Use of maximum entropy models constrained only by pairwise correlations $ C_{ij} = ig angle $ to predict higher-order statistical structure in neural population activity.
• Application of the Ising model framework with parameters $ h_i $ and $ J_{ij} $ fitted to match empirical mean firing rates and pairwise correlations.
• Computation of effective fields $ h_i^{ ext{eff}} = h_i + igsum_{j eq i} J_{ij} ho_j $ to predict single-neuron activity probabilities.
• Binarization of calcium fluorescence traces into $ ho_i \in \{0,1\} $ to represent neural spiking activity at 1/30 s resolution.
• Systematic selection of neuron subgroups: local (dense, small radius) vs. distant (sparse, large radius), matched in mean activity and correlation distributions.
• Use of Kullback–Leibler divergence to ensure statistical equivalence between subgroups during progressive expansion of sampling regions.

Experimental results

Research questions

• RQ1 Can maximum entropy models based solely on pairwise correlations quantitatively predict higher-order statistical features of neural population activity in real biological networks?
• RQ2 Does the success of such models depend on the spatial contiguity of the sampled neurons in the network?
• RQ3 Is the failure of the model in certain subgroups due to model oversimplification or to intrinsic network properties?
• RQ4 To what extent do effective fields derived from the model accurately predict the probability of individual neuron activity?
• RQ5 Can the model distinguish network states that drive activity or silence of individual neurons based on their effective fields?

Key findings

• For spatially contiguous local subgroups of 100 neurons, the maximum entropy model predicted all tested higher-order statistics (including $ C_{ijk} $) within experimental measurement error.
• Prediction errors for local subgroups were comparable to measurement errors, indicating near-perfect quantitative agreement with data.
• For randomly sampled distant subgroups from larger regions, prediction errors were significantly larger, especially for low-frequency and small-magnitude higher-order correlations.
• Effective field predictions matched observed activity probabilities extremely well in local subgroups, but failed to distinguish active from silent states in distant subgroups.
• The distribution of effective fields showed clear separation between active and silent neurons in local subgroups, but not in distant subgroups, indicating a breakdown in predictive power.
• These results were consistently reproduced across 30 different local/distant subgroup pairs in three different mice, confirming that model performance depends on spatial sampling rather than experimental variability.