The mean-field limit of a network of Hopfield neurons with correlated synaptic weights

Olivier Faugeras 1, 2 James Maclaurin 3 Etienne Tanré 1
1 TOSCA - TO Simulate and CAlibrate stochastic models
CRISAM - Inria Sophia Antipolis - Méditerranée , IECL - Institut Élie Cartan de Lorraine : UMR7502
2 MATHNEURO - Mathématiques pour les Neurosciences
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : We study the asymptotic behaviour for asymmetric neuronal dynamics in a network of Hopfield neurons. The randomness in the network is modelled by random couplings which are centered Gaussian correlated random variables. We prove that the annealed law of the empirical measure satisfies a large deviation principle without any condition on time. We prove that the good rate function of this large deviation principle achieves its minimum value at a unique Gaussian measure which is not Markovian. This implies almost sure convergence of the empirical measure under the quenched law. We prove that the limit equations are expressed as an infinite countable set of linear non Markovian SDEs.
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Submitted on : Monday, May 13, 2019 - 3:12:11 PM
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  • HAL Id : hal-02000172, version 1
  • ARXIV : 1901.10248

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Olivier Faugeras, James Maclaurin, Etienne Tanré. The mean-field limit of a network of Hopfield neurons with correlated synaptic weights. 2019. ⟨hal-02000172⟩

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