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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., Comment: Keywords: Neuronal cultures, Graph theory, Activation dynamics, Percolation, Statistical mechanics of networks, Leaders of activity, Quorum. http://www.weizmann.ac.il/complex/tlusty/papers/FrontCompNeuro2010.pdf

Several experimental studies show the existence of leader neurons in population bursts of 2D living neural networks. A leader neuron is, basically, a neuron which fires at the beginning of a burst (respectively network spike) more often that we expect by looking at its whole mean neural activity. This means that leader neurons have some burst triggering power beyond a simple statistical effect. In this study, we characterize these leader neuron properties. This naturally leads us to simulate neural 2D networks. To build our simulations, we choose the leaky integrate and fire (lIF) neuron model. Our lIF model has got stable leader neurons in the burst population that we simulate. These leader neurons are excitatory neurons and have a low membrane potential firing threshold. Except for these two first properties, the conditions required for a neuron to be a leader neuron are difficult to identify and seem to depend on several parameters involved in the simulations themself. However, a detailed linear analysis shows a trend of the properties required for a neuron to be a leader neuron. Our main finding is: A leader neuron sends a signal to many excitatory neurons as well as to a few inhibitory neurons and a leader neuron receives only a few signals from other excitatory neurons. Our linear analysis exhibits five essential properties for leader neurons with relative importance. This means that considering a given neural network with a fixed mean number of connections per neuron, our analysis gives us a way of predicting which neuron can be a good leader neuron and which cannot. Our prediction formula gives us a good statistical prediction even if, considering a single given neuron, the success rate does not reach hundred percent., Comment: 25 pages, 13 figures, 2 tables

Eytan and Marom recently showed that the spontaneous burst activity of rat neuron cultures includes `first to fire' cells that consistently fire earlier than others. Here we analyze the behavior of these neurons in long term recordings of spontaneous activity of rat hippocampal and rat cortical neuron cultures from three different laboratories. We identify precursor events that may either subside (`small events') or can lead to a full-blown burst (`pre-bursts'). We find that the activation in the pre-burst typically has a first neuron (`leader'), followed by a localized response in its neighborhood. Locality is diminished in the bursts themselves. The long term dynamics of the leaders is relatively robust, evolving with a half-life of 23-34 hours. Stimulation of the culture can temporarily alter the leader distribution, but it returns to the previous distribution within about 1 hour. We show that the leaders carry information about the identity of the burst, as measured by the signature of the number of spikes per neuron in a burst. The number of spikes from leaders in the first few spikes of a precursor event is furthermore shown to be predictive with regard to the transition into a burst (pre-burst versus small event). We conclude that the leaders play a r\^ole in the development of the bursts, and conjecture that they are part of an underlying sub-network that is excited first and then act as nucleation centers for the burst.