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 neurons with kin inputs and kout outputs, and whose input degrees kin (noted 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 sub-network 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 75 with width 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 4700 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.