Formal Neuron Models: Delays Offer a Simplified Dendritic Integration for Free

We firstly define an improved version of the spiking neuron model with dendrites introduced in [8] and we focus here on the fundamental mathematical properties of the framework. Our main result is that, under few simplifications with respect to biology, dendrites can be simply abstracted by delays. Technically, we define a method allowing to reduce neuron shapes and we prove that reduced forms define equivalence classes of dendritic structures sharing the same input/output spiking behaviour. Finally, delays by themselves appear to be a simple and efficient way to perform an abstract dendritic integration into spiking neurons without explicit dendritic trees. This overcomes an explicit morphology representation and allows exploring many equivalent configurations via a single simplified model structure.