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Throughout evolution the brain has mastered the art of processing real-world inputs through networks of interlinked spiking neurons. Synapses have emerged as key elements that, owing to their plasticity, are merging neuron-to-neuron signalling with memory storage and computation. Electronics has made important steps in emulating neurons through neuromorphic circuits and synapses with nanoscale memristors, yet novel applications that interlink them in heterogeneous bio-inspired and bio-hybrid architectures are just beginning to materialise. The use of memristive technologies in brain-inspired architectures for computing or for sensing spiking activity of biological neurons8 are only recent examples, however interlinking brain and electronic neurons through plasticity-driven synaptic elements has remained so far in the realm of the imagination. Here, we demonstrate a bio-hybrid neural network (bNN) where memristors work as "synaptors" between rat neural circuits and VLSI neurons. The two fundamental synaptors, from artificial-to-biological (ABsyn) and from biological-to- artificial (BAsyn), are interconnected over the Internet. The bNN extends across Europe, collapsing spatial boundaries existing in natural brain networks and laying the foundations of a new geographically distributed and evolving architecture: the Internet of Neuro-electronics (IoN).