The plasmodium of slime mold Physarum polycephalum behaves as an amorphous reaction-diffusion computing substrate and is capable of apparently 'intelligent' behavior. But how does intelligence emerge in an acellular organism? Through a range of laboratory experiments, we visualize the plasmodial cytoskeleton-a ubiquitous cellular protein scaffold whose functions are manifold and essential to life-and discuss its putative role as a network for transducing, transmitting and structuring data streams within the plasmodium. Through a range of computer modeling techniques, we demonstrate how emergent behavior, and hence computational intelligence, may occur in cytoskeletal communications networks. Specifically, we model the topology of both the actin and tubulin cytoskeletal networks and discuss how computation may occur therein. Furthermore, we present bespoke cellular automata and particle swarm models for the computational process within the cytoskeleton and observe the incidence of emergent patterns in both. Our work grants unique insight into the origins of natural intelligence; the results presented here are therefore readily transferable to the fields of natural computation, cell biology and biomedical science. We conclude by discussing how our results may alter our biological, computational and philosophical understanding of intelligence and consciousness.