During the recent few years, in response to empirical findings suggesting scalefree 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 & Sporns (2009). In Piekniewski & 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 & Schreiber (2008) relied on semirigorous 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.