### On the complexity of learning for spiking neurons with temporal coding

**Authors: Wolfgang Maass, and Michael Schmitt**

**Source: ***Information & Computation* Vol. **153**, No. 1,
1999, 26 - 46.

**Abstract.**
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
Vapnik-Chervonenkis (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.

©Copyright 1999, Academic Press