| 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
|
name: transport-mean-field-snn-dynamics
description: "Transport-based mean field theory for spiking neural network population dynamics. Derives firing rate fluctuations from initial voltage distributions using transport solutions to the Fokker-Planck/advection equation, applicable to any 1D integrate-and-fire neuron model. ArXiv 2605.14319 (Nicola & Campbell, 2026)."
tags: [snn, mean-field, neural-dynamics, transport-equation, fokker-planck, firing-rate]
arxiv_id: "2605.14319"
date: "2026-05-14"
Transport Mean Field Theory for SNN Population Dynamics
Paper Reference
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
Core Contribution
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.
Core Insight
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:
- Time-varying inputs are slow relative to neuronal dynamics
- Neurons operate in the excitation-driven regime
This captures how firing rate fluctuations emerge from dynamic interaction between time-varying inputs, initial voltage distributions, and network coupling.
Mathematical Framework
Fokker-Planck System for IF Neurons
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
Transport Approximation
For slow inputs and excitation-driven regime:
- Solve the advection equation: d rho/dt = -d/dv[mu(t) * rho]
- The instantaneous firing rate (flux) at threshold v_th: J(t) = mu(t) * rho(v_th, t)
- Initial voltage distribution determines transient dynamics
Key Equations
d rho/dt + v * grad(rho) = 0 (transport equation)
J(t) = integral of mu(t)*rho(v_th,t) dv (instantaneous flux)
Implementation Steps
- Define initial voltage distribution rho(v, 0) - can be Gaussian, uniform, or empirical
- Compute transport solution via method of characteristics
- Derive flux at threshold boundary
- Iterate with coupling: J(t) feeds back into mu(t) for recurrent networks
Python Implementation Sketch
See scripts/transport_meanfield.py for full implementation.
Applications
- Population coding: Understanding how neural populations encode time-varying stimuli
- Working memory: How initial state distributions affect sustained activity
- Oscillatory dynamics: Emergence of collective rhythms from coupled populations
- Network stability: When does coupling amplify vs. suppress fluctuations?
Key Differences from Existing Approaches
| 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 |
Activation Keywords
- transport mean field
- SNN population dynamics
- Fokker-Planck neural
- firing rate fluctuations
- integrate-and-fire mean field
- neural transport equation
- macroscopic neural dynamics
- arXiv 2605.14319
References
- Nicola, W. & Campbell, S.A. (2026). Approximate Macroscopic Dynamics of Spiking Neural Networks Based on Solutions to the Transport Equation. arXiv:2605.14319 [q-bio.NC, math.DS].
- Related: Brunel (2000) async state theory; Fourcaud & Brunel (2002) linear response
