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name: geometric-mean-field-lorentzian-ansatz version: 1.0.0 description: Geometric origin of exact mean-field reductions using Möbius symmetry and the Lorentzian Ansatz — proving the Cauchy-Lorentz family uniquely emerges as invariant under projective transport, unifying Ott-Antonsen and Montbrió-Pazó-Roxin reductions. triggers: - Lorentzian ansatz - mean-field reduction - Ott-Antonsen - Montbrió-Pazó-Roxin - coupled oscillators - Riccati dynamics - Möbius symmetry - Cauchy distribution - low-dimensional reduction - neural mass model tags: - mean-field-theory - computational-neuroscience - dynamical-systems - mathematical-physics - coupled-oscillators - spiking-neural-network - neural-mass-models

Geometric Origin of Exact Mean-Field Reductions: Möbius Symmetry and the Lorentzian Ansatz

Source: arXiv:2605.23669 (May 2026)
Authors: Hugues Berry, Leonardo Trujillo
Categories: physics.bio-ph, q-bio.NC

Summary

Proves that the privileged role of the Lorentzian (Cauchy) family in low-dimensional reductions of coupled oscillator and spiking neuron systems is geometric rather than heuristic. The Cauchy-Lorentz family is the unique connected two-dimensional family of continuous probability densities invariant under projective transport induced by Riccati dynamics.

Key Contributions

Geometric Foundation: Proves the Lorentzian Ansatz has geometric origin via Möbius (projective) symmetry, not just heuristic convenience
Unification: Provides unified geometric foundation for Ott-Antonsen (2008) and Montbrió-Pazó-Roxin (2015) reductions
Explains Gaussian Failure: Shows why Gaussian closures fail for these systems
Structural Condition: Identifies the structural condition underlying exact two-parameter reductions

Core Mathematical Framework

Riccati Dynamics → Projective Transport

The transport induced by Riccati dynamics on probability densities
Reformulation on the circle via stereographic projection
Problem reduces to uniqueness of rotation-invariant probability measure

Key Proof Structure

Start with Riccati dynamics on oscillator/spiking neuron populations
Transport action on space of probability densities
Reformulate on circle S¹ via stereographic projection
Uniqueness of rotation-invariant measure → standard Cauchy law
Full projective action → Lorentzian family
Therefore Cauchy-Lorentz is the unique invariant family

The Uniqueness Theorem

Cauchy-Lorentz family: The only connected 2D family of continuous probability densities invariant under the induced projective transport
On the circle: Rotation-invariant measure is unique → Dirac delta at uniform angle
Under stereographic projection: This gives the standard Cauchy distribution
Under full projective action: Gives the Lorentzian family

Implications for Neuroscience

Neural Mass Models

Justifies the use of Lorentzian distributions in mean-field models of spiking neurons
Provides rigorous mathematical backing for firing rate models derived from spiking dynamics

Why Gaussian Closures Fail

Gaussian family is NOT invariant under the projective transport
This explains systematic errors when Gaussian approximations are used for coupled oscillator systems

Exact Reductions

Ott-Antonsen reduction: Special case of the geometric principle for Kuramoto-type models
Montbrió-Pazó-Roxin reduction: Special case for quadratic integrate-and-fire (QIF) neurons
Both are manifestations of the same underlying Möbius symmetry

Technical Details

Riccati Dynamics

Arises naturally in phase-reduced oscillator models
Also in voltage dynamics of QIF neurons: dv/dt = v² + η(t)
The distributional dynamics induce projective transformations on density space

Möbius (Projective) Symmetry

Group of fractional linear transformations on the real line
Acts on probability densities via push-forward
Cauchy-Lorentz family is the orbit of the Cauchy distribution under this group

Applications

Neural Mass Models: Rigorous derivation of low-dimensional firing rate equations from spiking neuron models
Kuramoto Model Analysis: Exact mean-field descriptions for synchronization transitions
QIF Network Dynamics: Exact reduction for networks of quadratic integrate-and-fire neurons
Brain Oscillation Modeling: Theoretical foundation for modeling macroscopic brain rhythms

Implementation Guidance

import numpy as np
from scipy import stats

class LorentzianMeanField:
    """
    Exact mean-field reduction using Lorentzian ansatz.
    The Lorentzian (Cauchy) distribution is the unique invariant family
    under projective transport from Riccati dynamics.
    """
    def __init__(self, n_neurons=1000, J=1.0, tau_m=10.0):
        self.n = n_neurons
        self.J = J  # coupling strength
        self.tau_m = tau_m
        # Lorentzian parameters: [center, half-width]
        self.r = 0.0  # mean firing rate
        self.v = 0.0  # mean membrane potential
    
    def cauchy_distribution(self, x, center, hwhm):
        """Standard Cauchy (Lorentzian) distribution"""
        return (1.0/np.pi) * hwhm / ((x - center)**2 + hwhm**2)
    
    def mean_field_step(self, dt, eta_mean, eta_std, I_ext=0.0):
        """
        One step of exact mean-field dynamics.
        Parameters derived from Lorentzian invariance principle.
        """
        # Exact 2D reduction equations
        dr_dt = (self.v / (np.pi * self.tau_m) + 
                 eta_std * self.r / (np.pi * self.tau_m))
        dv_dt = (self.v**2 + eta_mean + 
                 self.J * self.r + I_ext - 
                 (np.pi * self.tau_m * self.r)**2)
        
        self.r += dr_dt * dt
        self.v += dv_dt * dt
        self.r = max(self.r, 0)  # firing rate non-negative

Connections to Other Research

Ott-Antonsen (2008): Kuramoto model mean-field → special case of this geometric principle
Montbrió-Pazó-Roxin (2015): QIF neural mass → another special case
Next-Generation Neural Mass Models: Provides theoretical justification
Dynamic Mean-Field Theory: Geometric structure underlying mean-field closures
Statistical Physics of Neural Networks: Connects to random matrix and field theory approaches