droste2013analytical
Analytical investigation of self-organized criticality in neural networks
Felix Droste, Anne-Ly Do and Thilo Gross
J. R. Soc. Interface. 10, 20120558, 2013
Dynamical criticality has been shown to enhance information processing in dynamical systems, and there is evidence for self-organized criticality in neural networks. A plausible mechanism for such self-organization is activity dependent synaptic plasticity. Here, we model neurons as discrete-state nodes on an adaptive network following stochastic dynamics. At a threshold connectivity, this system undergoes a dynamical phase transition at which persistent activity sets in. In a low dimensional representation of the macroscopic dynamics, this corresponds to a transcritical bifurcation. We show analytically that adding activity dependent rewiring rules, inspired by homeostatic plasticity, leads to the emergence of an attractive steady state at criticality and present numerical evidence for the system's evolution to such a state.
Figure 1: Bifurcation diagram for the static network. Plotted is the steady state density of firing neurons $$[F]$$ over the network’s mean degree $$k$$. The solid line marks stable steady states of the static system, the dashed line unstable ones. At $$k = k_{\rm c} \approx 5.6$$, the inactive steady state loses stability in a transcritical bifurcation. The respective transition from an inactive to an active phase is already observed in individual-based simulations with $$N = 10^6$$ neurons (circles). Note that the critical k is nicely predicted by the link-level approximation Eqn. (1), but underestimated by a MCA at mean-field level (dotted lines).