The "edge of chaos" phase transition in artificial neural networks is of renewed interest in light of recent evidence for criticality in brain dynamics. Statistical mechanics traditionally studied this transition with connectivity $k$ as the control parameter and an exactly balanced excitation-inhibition ratio. While critical connectivity has been found to be low in these model systems, typically around $k=2$, which is unrealistic for natural systems, a recent study utilizing the excitation-inhibition ratio as the control parameter found a new, nearly degree independent, critical point when connectivity is large. However, the new phase transition is accompanied by an unnaturally high level of activity in the network. Here we study random neural networks with the additional properties of (i) a high clustering coefficient and (ii) neurons that are solely either excitatory or inhibitory, a prominent property of natural neurons. As a result, we observe an additional critical point for networks with large connectivity, regardless of degree distribution, which exhibits low activity levels that compare well with neuronal brain networks.

10 pages 8 figures, revised version