The computational power of neural networks depends on properties of the real numbers used as weights. We focus on networks restricted to compute in polynomial time, operating on boolean inputs. Previous work has demonstrated that their computational power happens to coincide with the complexity classes P and P/poly, respectively, for networks with rational and arbitrary real weights. Here we prove that the crucial concept that characterizes this computational power is the Kolmogorov complexity of the weights, in the sense that, for each bound on this complexity, the networks can solve exactly the problems in a related nonuniform complexity class located between P and P/poly. By proving that the family of such nonuniform classes is infinite, we show that neural networks can be classified into an infinite hierarchy of different computing capabilities.