Abstract
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 Φk(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 interconnected 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 k = 75 with width σ = 31 for the majority of the neurons, but also has a power law tail with exponent −2 for 10% 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.
Highlights
Development of connectivity in a neuronal network is strongly dependent upon the environment in which the network grows: cultures grown in a dish will develop very differently from networks formed in the brain
Of Results In summary, we have shown here that the experimental situation of an exponential transient followed by super-exponential growth can be well described in our model of an in-degree distribution that is k−2 at high k but is Gaussian for the majority of the neurons
The basic argument we presented for the number of input connections relates it to the multiplication of the area of the dendritic tree by the density of the axons that cross through this area
Summary
Development of connectivity in a neuronal network is strongly dependent upon the environment in which the network grows: cultures grown in a dish will develop very differently from networks formed in the brain. We have previously shown (Cohen et al, 2010) that graph theory and statistical mechanics are useful in unraveling properties of the network in a rat hippocampal culture, mostly because of the statistical nature of the connectivity. With these tools we have been able to understand such phenomena as the degree distribution of input connections, the ratio of inhibitory to excitatory neurons and the input cluster size distribution (Breskin et al, 2006; Soriano et al., 2008). We observed that the appearance of a fully connected network coincides precisely with the time of birth (Soriano et al, 2008)
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