Author Summary Biological neuronal networks exhibit well-defined properties such as hierarchical structures and scale-free topologies, as well as a high degree of local clustering and short path lengths between nodes. These structural properties are intimately connected to the observed long-range temporal correlations in the network dynamics. Fabrication of artificial networks with similar structural properties would facilitate brain-like (“neuromorphic”) computing. Here we show experimentally that percolating networks of nanoparticles exhibit similar long-range temporal correlations to those of biological neuronal networks and use simulations to demonstrate that the dynamics arise from an underlying scale-free network architecture. We discuss similarities between the biological and percolating systems and highlight the potential for the percolating networks to be used in neuromorphic computing applications. Abstract Biological neuronal networks are the computing engines of the mammalian brain. These networks exhibit structural characteristics such as hierarchical architectures, small-world attributes, and scale-free topologies, providing the basis for the emergence of rich temporal characteristics such as scale-free dynamics and long-range temporal correlations. Devices that have both the topological and the temporal features of a neuronal network would be a significant step toward constructing a neuromorphic system that can emulate the computational ability and energy efficiency of the human brain. Here we use numerical simulations to show that percolating networks of nanoparticles exhibit structural properties that are reminiscent of biological neuronal networks, and then show experimentally that stimulation of percolating networks by an external voltage stimulus produces temporal dynamics that are self-similar, follow power-law scaling, and exhibit long-range temporal correlations. These results are expected to have important implications for the development of neuromorphic devices, especially for those based on the concept of reservoir computing.