In parallel to the size increase of classic Artificial Neuronal Network in the industry, there has been another academic race to the biggest network to represent a (human) brain. For instance, the European Human Brain Project generated simulations of at least entire parts of the brain like the visual cortex thanks to its own supercalculators. The current state of the art of such representations and simulations relies on Spiking Neural Networks (Integrate-and-fire neurons) with learning rules that are interpreted as neuronal plasticity in the models.
We have shown with Alexandre Muzy, Cyrille Mascart and Gilles Scarella, that injecting more randomness in the neuronal models can make them much more easy to compute. In particular, one can use only one single processor of a laptop to reach the size of small monkey brains.
Another and more exploratory approach consists in injecting even more randomness to simulate part of the brain as an open physical system, where we do not need to simulate the whole to have the state of a smaller part.