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We study how the dynamics of a class of discrete dynamical system models for neuronal networks depends on the connectivity of the network. Specifically, we assume that the network is a directed Erdos--R\'enyi random graph and analytically derive scaling laws for the average lengths of the attractors and transients. In contrast to earlier results that were reported in [D. Terman, S. Ahn, X. Wang, and W. Just, Phys. D, 237 (2008), pp. 324--338], here we focus on the connection probabilities near the phase transition where the most complex dynamics is expected to occur.

Lengths of Attractors and Transients in Neuronal Networks with Random Connectivities