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Theoretical neuroscience of self‐organized criticality: from formal approaches to realistic models

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Levina, A., Herrmann, J., & Geisel, T. (2014). Theoretical neuroscience of self‐organized criticality: from formal approaches to realistic models. In D. Plenz, E. Niebur, & H. Schuster (Eds.), Criticality in Neural Systems (pp. 417-436). Weinheim, Germany: Wiley-VCH. doi:10.1002/9783527651009.ch20.

Cite as: https://hdl.handle.net/21.11116/0000-0003-9690-0
The search for models of self‐organized criticality in neural networks started before the first experiments demonstrated examples of the critical brain. One of the early models, the Eurich model, predicted critical exponents and various dynamical regimes that were experimentally observed, but failed to describe a mechanism for the self‐organization of criticality. In contrast to simultaneous attempts, the LHG model did not only describe a route to criticality in neural systems but also turned out to be simple enough for analytical treatment. Interestingly, it showed also greater biological plausibility than the Eurich model. When the synapses in the network obey a realistic dynamics, the critical region in the parameter space increases and becomes infinite in the large system limit. This effect of depressive synapses can be interpreted as a feedback control mechanism that drives the system toward the critical state. If also facilitation is included in the synaptic dynamics the critical region extends even more. In the latter case an analytical treatment is still possible and reveals an interesting type of stationary state consisting of self‐organized critical phase and a subcritical phase that has not been described earlier. The phases are connected by first‐ and second‐order phase transitions which form a cusp bifurcation. Switching between phases can be induced by synchronized activity or by activity deprivation. Having the model established, we ask how the network topology, synaptic homeostasis, neural leakage, and long‐term learning affect the critical behavior of the network. We demonstrate that all main topology types (random, small‐world, scale‐free) permit critical avalanches. We conclude with a discussion of astonishing fact that various types of adaptivity in neural systems appear to cooperate in order to enable robust critical behavior.