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Population Level Activity in Large Random Neural Networks

James MacLaurin, Moshe Silverstein, Pedro Vilanova

TL;DR

It is proved that repeated applications of this operator must converge to the limiting dynamics (thus yielding a relatively efficient means of numerically simulating the limiting equations), and the convergence of the double empirical measure is determined.

Abstract

We determine limiting equations for large asymmetric `spin glass' networks. The initial conditions are not assumed to be independent of the disordered connectivity: one of the main motivations for this is that allows one to understand how the structure of the limiting equations depends on the energy landscape of the random connectivity. The method is to determine the convergence of the double empirical measure (this yields population density equations for the joint distribution of the spins and fields). The limiting dynamics is expressed in terms of a fixed point operator. It is proved that repeated applications of this operator must converge to the limiting dynamics (thus yielding a relatively efficient means of numerically simulating the limiting equations,

Population Level Activity in Large Random Neural Networks

TL;DR

It is proved that repeated applications of this operator must converge to the limiting dynamics (thus yielding a relatively efficient means of numerically simulating the limiting equations), and the convergence of the double empirical measure is determined.

Abstract

We determine limiting equations for large asymmetric `spin glass' networks. The initial conditions are not assumed to be independent of the disordered connectivity: one of the main motivations for this is that allows one to understand how the structure of the limiting equations depends on the energy landscape of the random connectivity. The method is to determine the convergence of the double empirical measure (this yields population density equations for the joint distribution of the spins and fields). The limiting dynamics is expressed in terms of a fixed point operator. It is proved that repeated applications of this operator must converge to the limiting dynamics (thus yielding a relatively efficient means of numerically simulating the limiting equations,
Paper Structure (18 sections, 25 theorems, 214 equations)

This paper contains 18 sections, 25 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. Outline of Model and Main Results
  3. Assumptions on the Initial Conditions
  4. Case 1: Connectivity-Dependent Initial Conditions
  5. Case 2: Connectivity-Independent Initial Conditions
  6. Main Result
  7. An Example System that Satisfies the Conditions of Section \ref{['subsection connectivity dependent initial conditions']}
  8. Proof Outline
  9. Large Deviations of the Uncoupled System
  10. Exponential Tightness
  11. The Coupled System (with connectivity-dependent initial conditions)
  12. Connectivity-Independent Initial Conditions
  13. Proofs
  14. Regularity Estimates and Compactness
  15. Large Deviations of the Uncoupled System
  16. ...and 3 more sections

Key Result

Theorem 1

The mapping $\Phi$ is well-defined for all $\mu \in \mathcal{U}$. Furthermore there exists a unique probability measure $\xi \in \mathcal{P}( \mathcal{C}([0,T],\mathbb{R}^M)^2 )$ such that with unit probability, $\xi$ is the unique measure such that $\Phi(\xi) = \xi$. Furthermore, where $\xi^{(n+1)} = \Phi( \xi^{(n)})$ and $\xi^{(1)}$ is any measure in $\mathcal{U}$.

Theorems & Definitions (46)

  • Theorem 1
  • remark 1
  • Lemma 2
  • proof
  • Theorem 3
  • Lemma 4
  • Lemma 5
  • Corollary 6
  • Lemma 7
  • Lemma 8
  • ...and 36 more