Spiking and saturating dendrites differentially expand single neuron computation capacity

The integration of excitatory inputs in dendrites is non-linear: multiple excitatory inputs can produce a local depolarization departing from the arithmetic sum of each input's response taken separately. If this depolarization is bigger than the arithmetic sum, the dendrite is spiking; if the depolarization is smaller, the dendrite is saturating. Decomposing a dendritic tree into independent dendritic spiking units greatly extends its computational capacity, as the neuron then maps onto a two layer neural network, enabling it to compute linearly non-separable Boolean functions (lnBFs). How can these lnBFs be implemented by dendritic architectures in practise? And can saturating dendrites equally expand computational capacity? To adress these questions we use a binary neuron model and Boolean algebra. First, we confirm that spiking dendrites enable a neuron to compute lnBFs using an architecture based on the disjunctive normal form (DNF). Second, we prove that saturating dendrites as well as spiking dendrites also enable a neuron to compute lnBFs using an architecture based on the conjunctive normal form (CNF). Contrary to the DNF-based architecture, a CNF-based architecture leads to a dendritic unit tuning that does not imply the neuron tuning, as has been observed experimentally. Third, we show that one cannot use a DNF-based architecture with saturating dendrites. Consequently, we show that an important family of lnBFs implemented with a CNF-architecture can require an exponential number of saturating dendritic units, whereas the same family implemented with either a DNF-architecture or a CNF-architecture always require a linear number of spiking dendritic unit. This minimization could explain why a neuron spends energetic resources to make its dendrites spike.