Abstract
Self-organization and pattern formation in network-organized systems emerges from the collective activation and interaction of many interconnected units. A striking feature of these non-equilibrium structures is that they are often localized and robust: only a small subset of the nodes, or cell assembly, is activated. Understanding the role of cell assemblies as basic functional units in neural networks and socio-technical systems emerges as a fundamental challenge in network theory. A key open question is how these elementary building blocks emerge, and how they operate, linking structure and function in complex networks. Here we show that a network analogue of the Swift-Hohenberg continuum model—a minimal-ingredients model of nodal activation and interaction within a complex network—is able to produce a complex suite of localized patterns. Hence, the spontaneous formation of robust operational cell assemblies in complex networks can be explained as the result of self-organization, even in the absence of synaptic reinforcements.
Highlights
Pattern formation in reaction-diffusion systems[1,2] has emerged as a mathematical paradigm to understand the connection between pattern and process in natural and sociotechnical systems[3]
The simplicity and robustness of the proposed single-species pattern-forming mechanisms suggest that analogous dynamics may explain localized patterns of activity emerging in many network-organized natural and socio-technical systems
Our model can be understood as a network analogue of the Swift-Hohenberg continuum model[49,50,51], and is able to produce a complex suite of localized patterns
Summary
We restrict our analysis to the simplified case of symmetric networks, but our main results can be generalized to other network topologies, including directed[21] and multiplex[19,20] networks. The currents, Ii, represent the excitatory/inhibitory interactions among nodes in the network The structure of these nodal interactions is one of the key pattern forming mechanisms in the present model. Stationary solutions u of Eq (4) satisfy f (ui, μ) = 0, where the nodal state of activation is equal for all nodes in the network, ui = u, ∀ i = 1, ..., N. For positive values of μ the trivial stationary solution is stable with respect to uniform small random perturbations (solid line) while for negative values of μ this state becomes unstable (dotted line). In the stable regime, input stimuli may trigger localized patterns of activation
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