The branching parameter will serve to scale all connection strengths (Fig. 6C). Second, the distribution of transmission probabilities from each unit may be controlled from sharp (where only one connection has a transmission probability of 1 and all the other connections are 0) to homogeneous (where all connection strengths have equal transmission probabilities equal to 1/N)(Fig. 6D). There are many different types of distributions that could be used here, but for simplicity we will only consider distributions that are defined by an exponential function:
pij = Ae-B*j, where A is a scaling constant that keeps the sum of the transmission probabilities equal to o, and B is the exponent that determines how sharp (B large) or flat (B small) the distribution will be.
How well can a simple model like this capture features from actual data? In experiments with organotypic cortical cultures, Beggs and Plenz (Beggs JM and D Plenz, 2003) found that suprathreshold local field potential activity (Fig. 2A) at one electrode was, on average, followed by activity in one other electrode in the next time step. When the model is tuned to have a branching parameter o = 1.0, it reproduces this result faithfully. This should not be too surprising, though, since it is well known that for a branching process, o gives the expected number of descendants from a single ancestor (Harris TE, 1989). What is somewhat less expected is that the distribution of "avalanche" sizes produced by the model also closely matches the distribution from the data. Here, the avalanche size is just the total number of electrodes activated in one spatio-temporal pattern of activity. Representative patterns of activity are shown in Fig. 2D and 7B, and consist of consecutively active frames that are bracketed in time by inactive frames. When the probability of an avalanche is plotted against its size, the result is a power law, as shown in Fig. 7A. Power law distributions are often found in complex systems and can be used to describe domain sizes in magnets during a phase transition (Stanley HE,
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