  The critical brain hypothesis receives increasing support from recent
experimental results. It postulates that the brain is at a critical point
between an ordered and a chaotic regime, sometimes referred to as the "edge of
chaos." Another central observation of neuroscience is the principle of
excitation-inhibition balance: Certain brain networks exhibit a remarkably
constant ratio between excitation and inhibition. When this balance is
perturbed, the network shifts away from the critical point, as may for example
happen during epileptic seizures. However, it is as of yet unclear what
mechanisms balance the neural dynamics towards this excitation-inhibition ratio
that ensures critical brain dynamics. Here we introduce a simple yet
biologically plausible toy model of a self-organized critical neural network
with a self-organizing excitation to inhibition ratio. The model only requires
a neuron to have local information of its own recent activity and changes
connections between neurons accordingly. We find that the network evolves to a
state characterized by avalanche distributions following universal scaling laws
typical of criticality, and to a specific excitation to inhibition ratio. The
model connects the two questions of brain criticality and of a specific
excitation/inhibition balance observed in the brain to a common origin or
mechanism. From the perspective of the statistical mechanics of such networks,
the model uses the excitation/inhibition ratio as control parameter of a phase
transition, which enables criticality at arbitrary high connectivities. We find
that network clustering plays a crucial role for this phase transition to
occur.
