The intrinsic dynamics of Josephson Junctions naturally form a good candidate for modelling biological neurons. Following basic neuroscientific principles, we demonstrate that a network of Niobium-Neurons with a truncated synapse model can behave like a neural network of excitatory and inhibitory neurons in the human brain. We developed an extended Niobium Neuron Soma model that includes multiple outward connections using single-flux-quantum building blocks and scaling as well as sign switching of the created current spikes using a modified Josephson comparator. This neuron exhibits features typical for leaky integrate-and-fire neurons with equal-strength outward connections necessary to build all-to-all or sparsely connected networks, two crucial connectivity types to represent neural circuits in the human brain. The resulting network shows input rate dependent state switching between asynchronous and synchronous firing and is a first demonstration towards the study of large-scale brain dynamics with Josephson Junction based technology. This bears the potential of brain network simulations with computation speeds orders of magnitude faster than both semiconductor realizations and biological neural networks.