Self-Tuned Critical Anti-Hebbian Networks

It is widely recognized that balancing excitation and inhibition is important in the nervous system. When such a balance is sought by global strategies, few modes remain poised close to instability, and all other modes are strongly stable. Here we present a simple abstract model in which this balance is sought locally by units following “anti-Hebbian” evolution: all degrees of freedom achieve a close balance of excitation and inhibition and become “critical” in the dynamical sense. At long time scales, a complex “breakout” dynamics ensues in which different modes of the system oscillate between prominence and extinction; the model develops various long-tailed statistical behaviors and may become self-organized critical.

Figure 1

Relaxation of the real parts of the eigenvalues of $W$. For clarity of illustration, $N=20$. At short times ($\approx 1$) all eigenvalues with positive real parts relax to having negative real parts; they typically overshoot and flip sign in doing so. On a scale given by $\alpha ={10}^{-3}$, all real parts relax to the vicinity of the real axis. Beyond this scale, all eigenvalues fluctuate around the real axis.

Figure 2

Zooming in the rightmost portion of Fig. 1. Starting from an antisymmetric matrix, the eigenvalues fluctuate around the instability line with a time scale $\approx \sqrt{\alpha }$.

Figure 3

Statistical criticality in our model. Top row: Globally coupled (unrestricted $W$). Bottom row: Nodes arranged in one dimension with periodic boundary conditions; only entries of $W$ up to third nearest neighbor are allowed to be nonzero. First column: A display of the spatiotemporal dynamics. Second column: The distribution of the number of simultaneously active units in the dynamics (blue) and in surrogate data (red); compare to 10, where the argument was made this distribution indicates that even though the two-point correlations between individual neurons are small, the system has globally correlated states. Third column: Sizes of avalanches (blue) versus surrogate data (red); note in the 1D case the power-law distribution of avalanche sizes, while the globally coupled ($\infty \mathrm{D}$) case shows a piece of a power law followed by a large lump of rather large avalanches (as clearly visible in the spatiotemporal plot). Fourth column: Marginal distribution of the values of $x$ (invariant under surrogation). Surrogation is carried out by scrambling the pixels of the spatiotemporal plot.