transport-mean-field-snn-dynamics

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:

1. Time-varying inputs are slow relative to neuronal dynamics
2. 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:

1. Solve the advection equation: d rho/dt = -d/dv[mu(t) * rho]
2. The instantaneous firing rate (flux) at threshold v_th: J(t) = mu(t) * rho(v_th, t)
3. 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

1. Define initial voltage distribution rho(v, 0) - can be Gaussian, uniform, or empirical
2. Compute transport solution via method of characteristics
3. Derive flux at threshold boundary
4. 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