Scaling behavior in probabilistic neuronal cellular automata

We study a neural network model of interacting stochastic discrete two-state cellular automata on a regular lattice. The system is externally tuned to a critical point which varies with the degree of stochasticity (or the effective temperature). There are avalanches of neuronal activity, namely, spatially and temporally contiguous sites of activity; a detailed numerical study of these activity avalanches is presented, and single, joint, and marginal probability distributions are computed. At the critical point, we find that the scaling exponents for the variables are in good agreement with a mean-field theory.

Figure 1

(a) Boolean Lyapunov exponent $\Lambda$ for ${\mathcal{T}}_{le}={10}^4$ and $n={10}^3$, (b) spontaneous activity $F_0$ averaged for ${10}^3$ time steps, and (c) susceptibility $\chi$ with control parameter $\lambda$ for different values of $\mu$ for ${10}^6$ nodes, $\eta =0$, and Moore neighborhood.

Figure 2

Plot of avalanche size distribution $P\left(S\right)$ for different $\lambda$ for ${10}^7$ avalanche events.

Figure 3

Joint size-duration probability distribution $P(T,S)$ for total nodes, $N={1024}^2$, using log binned data.

Figure 4

The data collapse of $P_T\left(S\right)$ curves for the joint size-duration distribution. Each $P_{T_0}\left(S\right)$ curve is the intersection of the joint size-duration distribution $P(T,S)$ with the $T=T_0$ plane. The collapse width is $w=4.81\times {10}^{-2}$.