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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

1. Geometric Foundation: Proves the Lorentzian Ansatz has geometric origin via Möbius (projective) symmetry, not just heuristic convenience
2. Unification: Provides unified geometric foundation for Ott-Antonsen (2008) and Montbrió-Pazó-Roxin (2015) reductions
3. Explains Gaussian Failure: Shows why Gaussian closures fail for these systems
4. 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

1. Start with Riccati dynamics on oscillator/spiking neuron populations
2. Transport action on space of probability densities
3. Reformulate on circle S¹ via stereographic projection
4. Uniqueness of rotation-invariant measure → standard Cauchy law
5. Full projective action → Lorentzian family
6. 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

1. Neural Mass Models: Rigorous derivation of low-dimensional firing rate equations from spiking neuron models
2. Kuramoto Model Analysis: Exact mean-field descriptions for synchronization transitions
3. QIF Network Dynamics: Exact reduction for networks of quadratic integrate-and-fire neurons
4. 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