  We present a theoretical framework using quorum-percolation for describing
the initiation of activity in a neural culture. The cultures are modeled as
random graphs, whose nodes are excitatory neurons with kin inputs and kout
outputs, and whose input degrees kin = k obey given distribution functions pk.
We examine the firing activity of the population of neurons according to their
input degree (k) classes and calculate for each class its firing probability
\Phi_k(t) as a function of t. The probability of a node to fire is found to be
determined by its in-degree k, and the first-to-fire neurons are those that
have a high k. A small minority of high-k classes may be called "Leaders", as
they form an inter-connected subnetwork that consistently fires much before the
rest of the culture. Once initiated, the activity spreads from the Leaders to
the less connected majority of the culture. We then use the distribution of
in-degree of the Leaders to study the growth rate of the number of neurons
active in a burst, which was experimentally measured to be initially
exponential. We find that this kind of growth rate is best described by a
population that has an in-degree distribution that is a Gaussian centered
around k = 75 with width {\sigma} = 31 for the majority of the neurons, but
also has a power law tail with exponent -2 for ten percent of the population.
Neurons in the tail may have as many as k = 4, 700 inputs. We explore and
discuss the correspondence between the degree distribution and a dynamic
neuronal threshold, showing that from the functional point of view, structure
and elementary dynamics are interchangeable. We discuss possible geometric
origins of this distribution, and comment on the importance of size, or of
having a large number of neurons, in the culture.
