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.

翻译：本研究旨在从数学上证明，一种受生物学启发的神经网络仅通过局部变换即可学习分类任务。为此，我们提出了一种名为CHANI（基于相关性的生物启发霍克斯神经元聚合）的脉冲神经网络，其神经元活动由霍克斯过程建模。突触权重通过专家聚合算法进行更新，提供了一种局部且简单的学习规则。我们证明了该网络在平均意义下能够渐进地学习。此外，我们证明了网络能自动形成神经元集群，即网络能够编码多个类别，且中间层的同一神经元可能被多个类别激活，并在合成数据集上进行了数值模拟。这一理论方法有别于传统生物启发网络的经验验证，为理解局部学习规则如何使神经元形成能够表示复杂概念的集群开辟了道路。