Recent advances in the neuromorphic operation of atomic switches as individual synapse-like devices demonstrate the ability to process information with both short-term and long-term memorization in a single two terminal junction. Here it is shown that atomic switches can be self-assembled within a highly interconnected network of silver nanowires similar in structure to Turing's B-Type unorganized machine, originally proposed as a randomly connected network of NAND logic gates. In these experimental embodiments, complex networks of coupled atomic switches exhibit emergent criticality similar in nature to previously reported electrical activity of biological brains and neuron assemblies. Rapid fluctuations in electrical conductance display metastability and power law scaling of temporal correlation lengths that are attributed to dynamic reorganization of the interconnected electro-ionic network resulting from induced non-equilibrium thermodynamic instabilities. These collective properties indicate a potential utility for real-time, multi-input processing of distributed sensory data through reservoir computation. We propose these highly coupled, nonlinear electronic networks as an implementable hardware-based platform toward the creation of physically intelligent machines.