## Abstract

We study the asymptotic law of a network of interacting neurons when the number of neurons becomes infinite. Given a completely connected network of neurons in which the synaptic weights are Gaussian correlated random variables, we describe the asymptotic law of the network when the number of neurons goes to infinity. We introduce the process-level empirical measure of the trajectories of the solutions to the equations of the finite network of neurons and the averaged law (with respect to the synaptic weights) of the trajectories of the solutions to the equations of the network of neurons. The main result of this article is that the image law through the empirical measure satisfies a large deviation principle with a good rate function which is shown to have a unique global minimum. Our analysis of the rate function allows us also to characterize the limit measure as the image of a stationary Gaussian measure defined on a transformed set of trajectories.

Original language | English (US) |
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Pages (from-to) | 4701-4743 |

Number of pages | 43 |

Journal | Entropy |

Volume | 17 |

Issue number | 7 |

DOIs | |

State | Published - 2015 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- General Physics and Astronomy

## Keywords

- Correlated synaptic weights
- Firing rate neurons
- Good rate function
- Large deviations
- Neural networks
- Spectral representations
- Stationary gaussian processes
- Stationary measures