Self-organization without conservation: Are neuronal avalanches generically critical?

Recent experiments on cortical neural networks have revealed the existence of well-defined avalanches of electrical activity. Such avalanches have been claimed to be generically scale invariant - i.e.power law distributed - with many exciting implications in neuroscience. Recently, a self-organized model has been proposed by Levina, Herrmann and Geisel to explain this empirical finding. Given that (i) neural dynamics is dissipative and (ii) there is a loading mechanism progressively 'charging' the background synaptic strength, this model/dynamics is very similar in spirit to forest-fire and earthquake models, archetypical examples of non-conserving self-organization, which have recently been shown to lack true criticality. Here we show that cortical neural networks obeying (i) and (ii) are not generically critical; unless parameters are fine-tuned, their dynamics is either subcritical or supercritical, even if the pseudo-critical region is relatively broad. This conclusion seems to be in agreement with the most recent experimental observations. The main implication of our work is that, if future experimental research on cortical networks were to support the observation that truly critical avalanches are the norm and not the exception, then one should look for more elaborate (adaptive/evolutionary) explanations, beyond simple self-organization, to account for this.