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Large Deviations of a Network of Neurons with Dynamic Sparse Random Connections

James MacLaurin

Abstract

In this work we determine a process-level Large Deviation Principle (LDP) for a model of interacting particles indexed by a lattice $\mathbb{Z}^d$. The connections are random, sparse and unscaled, so that the system converges in the large size limit due to the probability of a connection between any two particles decreasing as the system size increases. The particles are also subject to noise (such as independent Brownian Motions). The method of proof is to assume a process-level (or Level 3) LDP for the double-layer empirical measure for the noise and connections, and then apply a series of transformations to this to obtain an LDP for the process-level empirical measure of our system. Although it is not explicitly necessary, we expect that most applications of this work should involve an assumption of stationarity of the probability law for the noise and connections under translations of the lattice, so that the system converged to an ergodic probability law in the large size limit. This work synthesizes the theory of large-size limits of interacting particles with that of random graphs and matrices. It should therefore be relevant to neuroscience and social networks theory in particular.

Large Deviations of a Network of Neurons with Dynamic Sparse Random Connections

Abstract

In this work we determine a process-level Large Deviation Principle (LDP) for a model of interacting particles indexed by a lattice . The connections are random, sparse and unscaled, so that the system converges in the large size limit due to the probability of a connection between any two particles decreasing as the system size increases. The particles are also subject to noise (such as independent Brownian Motions). The method of proof is to assume a process-level (or Level 3) LDP for the double-layer empirical measure for the noise and connections, and then apply a series of transformations to this to obtain an LDP for the process-level empirical measure of our system. Although it is not explicitly necessary, we expect that most applications of this work should involve an assumption of stationarity of the probability law for the noise and connections under translations of the lattice, so that the system converged to an ergodic probability law in the large size limit. This work synthesizes the theory of large-size limits of interacting particles with that of random graphs and matrices. It should therefore be relevant to neuroscience and social networks theory in particular.

Paper Structure

This paper contains 22 sections, 13 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Outline of Model and Preliminary Definitions
  3. Outline of Model and Main Result
  4. Assumptions
  5. An Example Application: A Fitzhugh-Nagumo Neural Network with Gibbsian Random Connectivity and Learning
  6. Model of Interactions
  7. Proof of Theorem \ref{['Theorem Main LDP']}
  8. Proof:
  9. Proof:
  10. Proof:
  11. Existence and continuity of $\tilde{\Psi}^m$
  12. Proof:
  13. Proof:
  14. Proof:
  15. Proof:
  16. ...and 7 more sections

Key Result

Theorem 1

Let the law of $\hat{\mu}^n(U)$ be $\Pi^n \in \mathcal{P}(\mathcal{P}_S(\mathcal{T}^{\mathbbm{Z}^d}))$. Under the assumptions outlined in Section Section Assumptions, $(\Pi^n)_{n\in\mathbbm{Z}^+}$ satisfy a Large Deviation Principle with good rate function $I$ (i.e. $I$ has compact level sets). That For all open subsets $O$ of $\mathcal{P}_S(\mathcal{T}^{\mathbbm{Z}^d})$, The rate function is wh

Theorems & Definitions (13)

  • Theorem 1
  • Corollary 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • ...and 3 more