Shot noise generated by subpopulations of neural networks

TL;DR

This paper derives an analytical expression for the spectral density of shot noise generated by subpopulations in neural networks, accounting for realistic parameter distributions and finite-size effects.

Contribution

It introduces a generalized nesting method to accurately model subpopulation shot noise spectra, improving upon previous Lorentzian-based approaches.

Findings

Derived an analytical formula for subpopulation shot noise spectral density.

Validated the formula with numerical simulations showing excellent agreement.

Demonstrated the importance of non-trivial spectral mixtures in hierarchical neural networks.

Abstract

While recent advances in next-generation neural mass models provide exact descriptions of densely coupled neural populations in the thermodynamic limit, populations in vivo remain strictly finite in size. Finite-size effects introduce stochastic fluctuations whose impact on network dynamics depends on their spectral content. Furthermore, coupling between different populations is typically sparse, meaning that only a small, random subset of neurons from one population projects connections to another. This subset (a subpopulation) produces an output signal that is inherently noisy. Given that the subpopulation constitutes only a fraction of the full population, its shot noise differs from that of the whole population in both intensity and spectral shape. In the present work, we analyze these differences and demonstrate that they depend non-trivially on subpopulation size. Using a generalization of our nesting method, we derive an analytical expression for the power spectral density of subpopulation shot noise, which shows excellent agreement with direct numerical simulations. Unlike many previous studies that rely on mathematically convenient but unrealistic Lorentzian distributions (with diverging moments), our approach accounts for more realistic, non-Lorentzian distributions of local neuron parameters using a previously developed reduction technique. These results provide a foundation for a new class of stochastic mean-field models for hierarchical neural networks. Such models can now incorporate the correct, size-dependent frequency spectrum of subpopulation shot noise. Crucially, this spectrum is not a simple scaled version of the full population's noise. Instead, it arises from a non-trivial mixture of two distinct spectral components. This is essential for networks with dense local connectivity and sparse inter-population connectivity.