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Transport-based mean field theory for spiking neural network population dynamics. Derives approximate macroscopic firing rate evolution from Fokker-Planck transport solutions rather than steady-state assumptions. Use when: studying SNN population
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| name | transport-mean-field-snn-dynamics |
| description | Transport-based mean field theory for spiking neural network population dynamics. Derives approximate macroscopic firing rate evolution from Fokker-Planck transport solutions rather than steady-state assumptions. Use when: studying SNN population |
Title: Approximate Macroscopic Dynamics of Spiking Neural Networks Based on Solutions to the Transport Equation
Authors: Wilten Nicola, Sue Ann Campbell (University of Calgary, University of Waterloo)
arXiv: 2605.14319 (May 14, 2026)
Categories: q-bio.NC, math.DS
Derives a transport-based mean field theory for firing rate fluctuations in populations of coupled integrate-and-fire neurons. Unlike prior approaches that assume asynchronous steady-state solutions to the Fokker-Planck equation, this work uses transport solutions to the advection equation, linking initial voltage distributions to time-varying population firing rates through a closed-form operator.
Traditional mean field approaches assume asynchronous or constant-flux steady states. This work derives firing rate dynamics from the transport (advection) solution of the Fokker-Planck system, assuming:
This captures how firing rate fluctuations emerge from dynamic interaction between time-varying inputs, initial voltage distributions, and network coupling.
The population density rho(v,t) evolves as:
d rho/dt = -d/dv[mu(t) * rho] + (sigma^2 / 2) * d^2 rho/dv^2 + boundary conditions
For slow inputs and excitation-driven regime:
d rho/dt + v * grad(rho) = 0 (transport equation)
J(t) = integral of mu(t)*rho(v_th,t) dv (instantaneous flux)
See scripts/transport_meanfield.py for full implementation.
| Approach | Assumption | Captures |
|---|---|---|
| Async steady-state | Constant flux | Equilibrium only |
| Linear response | Small perturbations | Near-equilibrium |
| Transport mean field | Slow inputs, excitation-driven | Full transient dynamics |
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