Universal computation using localized limit-cycle attractors in neural networks

Neural networks are dynamical systems that compute with their dynamics. One example is the Hopfield model, forming an associative memory which stores patterns as global attractors of the network dynamics. From studies of dynamical networks it is well known that localized attractors also exist. Yet, they have not been used in computing paradigms. Here we show that interacting localized attractors in threshold networks can result in universal computation. We develop a rewiring algorithm that builds universal Boolean gates in a biologically inspired two-dimensional threshold network with randomly placed and connected nodes using collision-based computing. We aim at demonstrating the computational capabilities and the ability to control local limit cycle attractors in such networks by creating simple Boolean gates by means of these local activations. The gates use glider guns, i.e., localized activity that periodically generates "gliders" of activity that propagate through space. Several such gliders are made to collide, and the result of their interaction is used as the output of a Boolean gate. We show that these gates can be used to build a universal computer.

Self-organized criticality and adaptation in discrete dynamical networks

It has been proposed that adaptation in complex systems is optimized at the critical boundary between ordered and disordered dynamical regimes. Here, we review models of evolving dynamical networks that lead to self-organization of network topology based on a local coupling between a dynamical order parameter and rewiring of network connectivity, with convergence towards criticality in the limit of large network size $N$. In particular, two adaptive schemes are discussed and compared in the context of Boolean Networks and Threshold Networks: 1) Active nodes loose links, frozen nodes aquire new links, 2) Nodes with correlated activity connect, de-correlated nodes disconnect. These simple local adaptive rules lead to co-evolution of network topology and -dynamics. Adaptive networks are strikingly different from random networks: They evolve inhomogeneous topologies and broad plateaus of homeostatic regulation, dynamical activity exhibits $1/f$ noise and attractor periods obey a scale-free distribution. The proposed co-evolutionary mechanism of topological self-organization is robust against noise and does not depend on the details of dynamical transition rules. Using finite-size scaling, it is shown that networks converge to a self-organized critical state in the thermodynamic limit. Finally, we discuss open questions and directions for future research, and outline possible applications of these models to adaptive systems in diverse areas.