Synaptic and somatic operators for fuzzy neurons: which t-norms to choose?

Fuzzy (logic) neurons have triangular norms as synaptic operators and triangular s-norms as somatic operators. MIN and MAX are often the chosen t-norm/s-norm pair. This choice is mainly due to the simplicity of the calculations and to the fact that a layer of MAX-MIN neurons implements the widely-used MAX-MIN composition. From a neural networks perspective (i.e. having in mind the ability to use the learning and adaptation mechanisms used with classic neuron models), other t-norms may be more suitable for defining fuzzy neurons. A set of three conditions was chosen to reflect the suitability of t-norms for implementing synaptic and somatic operators. The conditions reflect the need to allow gradient-descent learning, parametric adaptation and the ability to reduce to the MAX-MIN model as a particular case. A set of 12 t-norm pairs was analyzed to asses how well they match the chosen criteria. Only two t-norm pairs satisfy all the criteria, one of them (the fundamental t-norm pair, also known as Frank´s t-norms) having the advantage of covering the product-probabilistic sum case. The fundamental t-norms are chosen to define the fundamental fuzzy neuron