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Cortical plasticity is one of the main features that enable our ability to learn and adapt in our environment. Indeed, the cerebral cortex self-organizes itself through structural and synaptic plasticity mechanisms that are very likely at the basis of an extremely interesting characteristic of the human brain development: the multimodal association. In spite of the diversity of the sensory modalities, like sight, sound and touch, the brain arrives at the same concepts (convergence). Moreover, biological observations show that one modality can activate the internal representation of another modality when both are correlated (divergence). In this work, we propose the Reentrant Self-Organizing Map (ReSOM), a brain-inspired neural system based on the reentry theory using Self-Organizing Maps and Hebbian-like learning. We propose and compare different computational methods for unsupervised learning and inference, then quantify the gain of the ReSOM in a multimodal classification task. The divergence mechanism is used to label one modality based on the other, while the convergence mechanism is used to improve the overall accuracy of the system. We perform our experiments on a constructed written/spoken digits database and a DVS/EMG hand gestures database. The proposed model is implemented on a cellular neuromorphic architecture that enables distributed computing with local connectivity. We show the gain of the so-called hardware plasticity induced by the ReSOM, where the system's topology is not fixed by the user but learned along the system's experience through self-organization.