Fig. 6. Network model with tunable connectivity. A, Network initially has all-to-all connectivity, but selected connection strengths can be set to zero. A network with N = 8 units is shown. B, Each unit i has a set of transmission probabilities: {pij ,Pik,---PiN} that determines connection strengths. C, The sum of the transmission probabilities emanating from a given unit i will determine the branching parameter o for that unit. D, The distribution of transmission probabilities can be made sharp or flat by adjusting the exponent B. The normalization constant, A, makes the probabilities sum to o. As discussed in the text, tuning the branching parameter o or the distribution exponent B can influence network dynamics

Fig. 6. Network model with tunable connectivity. A, Network initially has all-to-all connectivity, but selected connection strengths can be set to zero. A network with N = 8 units is shown. B, Each unit i has a set of transmission probabilities: {pij ,Pik,---PiN} that determines connection strengths. C, The sum of the transmission probabilities emanating from a given unit i will determine the branching parameter o for that unit. D, The distribution of transmission probabilities can be made sharp or flat by adjusting the exponent B. The normalization constant, A, makes the probabilities sum to o. As discussed in the text, tuning the branching parameter o or the distribution exponent B can influence network dynamics

Activity can propagate from unit i to unit j through a connection that has a transmission probability pij that is constrained to be between zero and one (Fig. 6B). Transmission is simple and works like this: If a unit i is active, then unit j will become active in the next time step if a randomly drawn number is less than the transmission probability pij. In other words, unit i will transmit to unit j with probability pij. Unlike traditional integrate-and-fire neuron models, these units do not sum all of the incoming activity and then fire if this sum is over a threshold. They simply fire if one of the units connected to them is active and if transmission between them is successful. Given this arrangement, activity in the model typically originates spontaneously at one or a few units and then propagates through connections to other units in the network. While this model may seem too simplistic, it actually does a good job of reproducing phenomena observed in the data, as will be explained more below. If a parsimonious model can successfully capture the main features of the data, then this suggests that network dynamics may be governed by a few simple principles (Haldeman C and JM Beggs, 2005).

The connectivity may be tuned in one of two ways. First, the sum of the transmission probabilities emanating from each unit may be scaled from 0 (where each pij = 0) to N (where each pij = 1). Let us define the branching parameter, o, as this sum:

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