On the Complexity of Learning for Spiking Neurons with Temporal Coding

Spiking neurons are models for the computational units in biological neural systems where information is considered to be encoded mainly in the temporal patterns of their activity. In a network of spiking neurons a new set of parameters becomes relevant which has no counterpart in traditional neural network models: the time that a pulse needs to travel through a connection between two neurons (also known as delay of a connection). It is known that these delays are tuned in biological neural systems through a variety of mechanisms. In this article we consider the arguably most simple model for a spiking neuron, which can also easily be implemented in pulsed VLSI. We investigate the VC dimension of networks of spiking neurons where the delays are viewed as programmable parameters and we prove tight bounds for this VC dimension. Thus we get quantitative estimates for the diversity of functions that a network with fixed architecture can compute with different settings of its delays. In particular, it turns out that a network of spiking neurons with $k$ adjustable delays is able to compute a much richer class of functions than a threshold circuit with $k$ adjustable weights. The results also yield bounds for the number of training examples that an algorithm needs for tuning the delays of a network of spiking neurons. Results about the computational complexity of such algorithms are also given.