  The present work aims at proving mathematically that a neural network
inspired by biology can learn a classification task thanks to local
transformations only. In this purpose, we propose a spiking neural network
named CHANI (Correlation-based Hawkes Aggregation of Neurons with
bio-Inspiration), whose neurons activity is modeled by Hawkes processes.
Synaptic weights are updated thanks to an expert aggregation algorithm,
providing a local and simple learning rule. We were able to prove that our
network can learn on average and asymptotically. Moreover, we demonstrated that
it automatically produces neuronal assemblies in the sense that the network can
encode several classes and that a same neuron in the intermediate layers might
be activated by more than one class, and we provided numerical simulations on
synthetic dataset. This theoretical approach contrasts with the traditional
empirical validation of biologically inspired networks and paves the way for
understanding how local learning rules enable neurons to form assemblies able
to represent complex concepts.
