Criticality in the one-dimensional Kohonen neural map
István Csabai, Tamás Geszti, Gábor Vattay · Physical Review A · 1992
In the marginally stable ordered state of Kohonen's feature-map neural-network model the distribution of fluctuating distances between neighboring cells is found to be a self-similar Weierstrass-Mandelbrot-type function with a nontrivial scaling exponent. The relationship among the quantities describing the distribution is discussed in terms of a balance between a deterministic multiplicative and a stochastic additive process, described by a Perron-Frobenius operator. Two regions of the Kohonen learning parameter \ensuremath{\alpha}, separated by ${\mathrm{\ensuremath{\alpha}}}_{\mathit{c}}$\ensuremath{\approxeq}0.63, differ in the character of both fluctuation and ordering, the latter being fastest close to ${\mathrm{\ensuremath{\alpha}}}_{\mathit{c}}$.