transport-mean-field-snn-dynamics

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