In this paper, we provide theoretical and numerical analysis of a geometric activity flow network model which is aimed at explaining mathematically the scale-free functional graph self-organization phenomena emerging in complex nervous systems at a mesoscale level. In our model, each unit corresponds to a large number of neurons and may be roughly seen as abstracting the functional behavior exhibited by a single voxel under functional magnetic resonance imaging (fMRI). In the course of the dynamics, the units exchange portions of formal charge, which correspond to waves of activity in the underlying microscale neuronal circuit. The geometric model abstracts away the neuronal complexity and is mathematically tractable, which allows us to establish explicit results on its ground states and the resulting charge transfer graph modeling functional graph of the network. We show that, for a wide choice of parameters and geometrical setups, our model yields a scale-free functional connectivity with the exponent approaching 2, which is in agreement with previous empirical studies based on fMRI. The level of universality of the presented theory allows us to claim that the model does shed light on mesoscale functional self-organization phenomena of the nervous system, even without resorting to closer details of brain connectivity geometry which often remain unknown. The material presented here significantly extends our previous work where a simplified mean-field model in a similar spirit was constructed, ignoring the underlying network geometry.
Abstract-Stochastic recurrent neural networks have been successful in many applications [8], [16], [3]and their position in the literature is now quite strong, in particular in context of networks aimed at mimicking the thermodynamic behavior of complex physical systems, where a number of theoretical tools and motivations are available, originating from the area of statistical mechanics. As prominent examples of such networks one can invoke Hopfield-type content-addressed associative memories and Boltzmann machines, which are often interpreted within the framework of the spin-glass theory. Yet there are many unsolved questions in theories of such systems. Whereas a lot is known about the characteristics of Hopfield networks learning random independent patterns, less is known about their behavior in the presence of strong correlations, especially in non-zero temperature systems. In this paper we consider non-zero low temperature large-size locally Hopfield neural networks in finite loading regime, learning patterns admitting strong and regular correlation structure (chosen to be periodic for simplicity 1 ). Our aim being to find conditions for pattern stability and successful retrieval as depending on the imposed correlation structure, we propose to apply the so-called Pirogov-Sinai theory from statistical mechanics to describe the geometry of pattern stability regions. We check that this theory does match with the experimental data and we investigate some further interesting phenomena that occur in such systems.
D r a f t D r a f t D r a f t D r a f t D r a f t D r a f t D r a f t D r a f t D r a f t D r a f t D r a f t D r a f t [7], [8] we believe that the edge of these two dynamically growing disciplines might be an interesting field for research. In this paper we introduce a simplified model of spike flow network which in some details resembles a recurrent neural network with stochastic dynamics. We argue that within this setup a scale-free network structure emerges as a natural consequence of model structuring principles and we provide numerical evidence supporting this claim. Further in the paper we investigate a number of interesting properties of this network and discuss some consequences of this result.
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