A difficult classification for neurons without dendrites

Neurons are capable, when the summation of excitatory inputs in dendrites is locally non-linear, to perform linearly non-separable classifications. What is the minimum set of vectors which can be linearly non-separable? How well neurons without dendrite approximate these classifications? The present paper tackles these problems which could strengthen or weaken the impact of dendrites on computation. To address this question we use exhaustive parameter searches and receiver-operator characteristics (ROC). This well-known tool in engineering measures how well a classification can be approximated, we use exhaustive search to list all possible classifications a neuron, without dendrites, can do and we quantify how close they approximate a linearly non-separable classification. We demonstrate that a neuron model without dendrite poorly approximate this classification because the linear neuron model without dendrites cannot maximizes simultaneously both hits and false positive. This result demonstrates a sharp contrast between artificial neuron models taking into account or not dendrite, as these difficult classifications are easy to implement in a neuron model with a passive dendrite. This work suggests that even the most simple artificial neuron model should take into account non-linear summation of excitatory inputs.