  Phase transitions and critical behavior are crucial issues both in
theoretical and experimental neuroscience. We report analytic and computational
results about phase transitions and self-organized criticality (SOC) in
networks with general stochastic neurons. The stochastic neuron has a firing
probability given by a smooth monotonic function $\Phi(V)$ of the membrane
potential $V$, rather than a sharp firing threshold. We find that such networks
can operate in several dynamic regimes (phases) depending on the average
synaptic weight and the shape of the firing function $\Phi$. In particular, we
encounter both continuous and discontinuous phase transitions to absorbing
states. At the continuous transition critical boundary, neuronal avalanches
occur whose distributions of size and duration are given by power laws, as
observed in biological neural networks. We also propose and test a new
mechanism to produce SOC: the use of dynamic neuronal gains -- a form of
short-term plasticity probably in the axon initial segment (AIS) -- instead of
depressing synapses at the dendrites (as previously studied in the literature).
The new self-organization mechanism produces a slightly supercritical state,
that we called SOSC, in accord to some intuitions of Alan Turing.

Adaptation and Self-Organizing Systems;Neurons and Cognition