Neural networks display characteristics of critical dynamics in the neural activities [Eurich et al 2002]. This theoretical prediction was confirmed by the power-law statistics for the size of avalanches of neural activity in real neurons [1], where the critical behavior is re-approached even after a substantial perturbation of the parameters of the system. These findings provide evidence for the presence of self-organized criticality (SOC) [2], a key concept to describe the emergence of complexity in nature. It is present in a large number of seemingly unrelated phenomena such as piling of granular media, plate tectonics and stick–slip motion. Particularly in neural systems, however, it raises questions concerning the underlying mechanisms of the regulation towards the critical state and the functional role of the phenomenon.

For a globally coupled network of rather simple model neurons parameter special values have been identified that give rise to a power-law distribution of the responses to weak external stimulation [Eurich et al 2002]. More recently, we have provided numerical evidence [Levina et al 2006] that a realistic neurotransmitter dynamics at the synapses of the network leads to a phase transition towards a critical regime such that a nearly
critical behavior is present for a broad range of the maximal synaptic efficiency.

Probability distributions of the avalanche size L in the neural activity of a globally coupled network with dynamical synapses. In a finite system (N=300) we observe different dynamical regimes for an increasing maximal synaptic efficiency α: sub-critical (green), critical (red), and super-critical (blue). With increasing system size the critical regime extends more and more and covers eventually the full range above a certain value of α.

The quality of fit of the power law (see Fig. above) is given as a function of the synaptic strength α0 for static synapses (red) and as a function of the maximal synaptic efficiency α for dynamic synapses (blue). The inset shows the width of the interval where the fitting error is below 0.005. For the dynamic synapses this interval increases for increasing system size and has been shown analytically to diverge.

A breakthrough was achieved by the analytical treatment of the interaction between the neural activity and the synaptic transmitter dynamics [Levina et al 2007]. In this study we showed that the self-organized critical behavior is typical in the sense of self-tuning of parameters in systems towards critical effective values. It is particularly interesting that a biologically realistic synaptic dynamics causes the neuronal avalanches to turn from an exceptional phenomenon into a typical and robust self-organized critical behavior, with the only condition that the total resources of neurotransmitters exceed a certain minimal value.

A suggestive explanation of the functional role of self-organized criticality results from an effectiveness constraint. If the system adjusts its parameters such that each spike causes on average one of the target neurons to become subsequently active and, on the other hand, the total activity is to remain bounded and stationary, then it can be proved within the framework of branching theory that the resulting activity distribution is indeed critical. Moreover, this consideration allowed us to derive a learning rule from first principles that guides the system towards criticality and resembles biological learning rules [Levina et al 2007"].

[1] J. Beggs and D. Plenz, J Neurosci. 23, 11167 (2003).
[2] P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987).

Selected Publications: