Critical dynamics of randomly assembled and diluted threshold networks.

The dynamical behavior of a class of randomly assembled networks of binary threshold units subject to random deletion of connections is studied based on the annealed approximation suitable in the thermodynamic limit. The dynamical phase diagram is constructed for several forms of the probability density distribution of nonvanishing connection strengths. The family of power-law distribution functions rho0(x)=(1-alpha)/(2|x|alpha) is found to play a special role in expanding the domain of stable, ordered dynamics at the expense of the disordered, "chaotic" phase.

Higher-order neural networks, Polyà polynomials, and Fermi cluster diagrams.

The problem of controlling higher-order interactions in neural networks is addressed with techniques commonly applied in the cluster analysis of quantum many-particle systems. For multineuron synaptic weights chosen according to a straightforward extension of the standard Hebbian learning rule, we show that higher-order contributions to the stimulus felt by a given neuron can be readily evaluated via Polyà's combinatoric group-theoretical approach or equivalently by exploiting a precise formal analogy with fermion diagrammatics.