Criticality in probabilistic models of spreading dynamics in brain networks: Epileptic seizures
Fig 7
A-1 to H-1: Behavior of the order parameter (normalized spread size) around the point of maximum fluctuations for the mean-field approximation of a system with N = 215. Each plot was obtained by keeping E fixed and varying w around the point of maximum fluctuations wm which was obtained by numerical evaluation of the standard deviation of spread sizes. Each point was obtained from the number of nodes recruited to seizure (spread size) in one realization of the mean-field dynamics. For each w we plot 50 realizations. Two distinct behaviors are observed: (1) a continuous crossover (without a singularity in the derivatives of the order parameter) for values of E close to zero, and (2) a clear discontinuous transition with a jump for E < −1.5 × 10−6. The shift from continuous to discontinuous behavior is expected to pass through a critical point of a (critical) phase transition. Panels A-2 to H-2: Transition between unimodal and bimodal probability distributions of seizure spread size. Probability distributions of normalized spread size, obtained from 300 to 500 stochastic realizations, are shown at the point of maximum fluctuations wm and different values of the excitability E. A clear transition from unimodal to bimodal distributions is observed. In all simulations N = 215. The locations of the above continuous and discontinuous, as well as unimodal and bimodal, regimes in the control parameter space (w, E) are shown in Fig 8I and 8J with more detail.