  During the recent few years, in response to empirical findings suggesting
scale-free self-organisation phenomena emerging in complex nervous systems at a
mesoscale level, there has been significant search for suitable models and
theoretical explanations in neuroscientific literature, see the recent survey
by Bullmore and Sporns (2009). In Piekniewski and Schreiber (2008) we have
developed a simple and tractable mathematical model shedding some light on a
particular class of the afore-mentioned phenomena, namely on mesoscopic level
self-organisation of functional brain networks under fMRI imaging, where we
have achieved a high degree of agreement with existing empirical reports. Being
addressed to the neuroscientific community, our work Piekniewski and Schreiber
(2008) relied on semi-rigorous study of information flow structure in a class
of recurrent neural networks exhibiting asymptotic scale-free behaviour and
admitting a description in terms of the so-called winner-take-all dynamics. The
purpose of the present paper is to define and study these winner-take-all
networks with full mathematical rigour in context of their asymptotic spectral
properties, well known to be of interest for neuroscientific community. Our
main result is a limit theorem for spectra of the spike-flow graphs induced by
the winner-take-all dynamics. We provide an explicit characterisation of the
limit spectral measure expressed in terms of zeros of Bessel's J-function.
