Self-organized criticality in neural networks from activity-based rewiring

Neural systems process information in a dynamical regime between silence and chaotic dynamics. This has lead to the criticality hypothesis, which suggests that neural systems reach such a state by self-organizing toward the critical point of a dynamical phase transition. Here, we study a minimal neural network model that exhibits self-organized criticality in the presence of stochastic noise using a rewiring rule which only utilizes local information. For network evolution, incoming links are added to a node or deleted, depending on the node's average activity. Based on this rewiring-rule only, the network evolves toward a critical state, showing typical power-law-distributed avalanche statistics. The observed exponents are in accord with criticality as predicted by dynamical scaling theory, as well as with the observed exponents of neural avalanches. The critical state of the model is reached autonomously without the need for parameter tuning, is independent of initial conditions, is robust under stochastic noise, and independent of details of the implementation as different variants of the model indicate. We argue that this supports the hypothesis that real neural systems may utilize such a mechanism to self-organize toward criticality, especially during early developmental stages.

Figure 1

(a) Time series of the average in-connectivities and branching parameter for $N=1000,\beta =10,W=1000$, starting from a completely unconnected network. After a transient period, the average connectivities and the branching parameter become stationary. The branching parameter fluctuates around $〈\lambda 〉=1.10\pm 0.11$, indicating possible criticality. (b) Evolution starts with an average connectivity of $〈k〉=2$ for activating and inhibiting links. Even though having a very different initial configuration, the system evolves toward a similar steady state as found in panel (a).

Figure 2

(a) The average connectivity of activating incoming links $〈k_{+}〉$ for different values of the averaging length $W$ and the inverse temperature $\beta$. The white dotted line is the upper bound of $W$ given by Eq. (4). (b) The average branching parameter $〈\lambda 〉$ for different values of the averaging window length $W$ and the inverse temperature $\beta$. The average branching parameter is close to a typical value near one over a broad range of $(\beta ,W)$. The data was obtained by averaging over 30 000 evolution steps, system size is $N=1000$.

Figure 3

Avalanche statistics and collapse of avalanche profiles. (a) Avalanche duration distribution. (b) Avalanche size distribution. (c) Average avalanche size over avalanche duration. (d) Collapse of avalanche shapes. The curves show the Hamming distance during avalanches of lengths between 5 and 30. $N=2000,W=1000,\beta =10$, data from $6\times {10}^6$ avalanches.

Figure 4

Scaling of the avalanche size distribution with increasing system size $N$. Each distribution is obtained from ${10}^6$ avalanches. During one avalanche each node can only contribute once to the avalanche size. Parameters: $\beta =10$, $W=1000$.