Criticality in probabilistic models of spreading dynamics in brain networks: Epileptic seizures
Fig 8
A-D): Marginal Probability density functions of normalized spread sizes s/N. Values are shown for different control parameters (E, w) near the critical point. Presented data were obtained from mean-field simulations of the probabilistic model for different system sizes of N = 215, 216, 217, 218. A Unimodal distribution in agreement with a Gaussian, observed for a point at the upper part (with respect to the critical point location) of the boundary between no-spread and spread phases (E = 0, w = 5.5 × 10−5). B,C Closer to the critical point (E = −10−6, w = 7.67 × 10−5 in (B) and E = −1.1 × 10−6, w = 7.67 × 10−5 in (C)), the distributions become skewed with large variance. E The distributions become bimodal near the lower part of the phase boundary (E = −2.00 × 10−6, w = 7.88 × 10−5). (E-H): Joint Probability density functions of normalized spread sizes s/N and duration of seizures D. Values are plotted as heat maps for control parameters (E, w) near the critical point. The parameters in panels (E,F,G,H) are respectively the same as in panels (A,B,C,D). Data were obtained from mean-field simulations of the model with system size N = 218. E Unimodal distribution, which is roughly in agreement with a Gaussian probability density function (but slightly skewed in the duration coordinate), is found on the upper part (with respect to the critical point location) of boundary between no-spread and spread phases. Duration and size of seizures appear to be uncorrelated. F,G Near the critical point stronger correlation between spread size and duration of seizures is observed and the distribution exhibits a wider peak and stronger correlation in the two dimensional space of (D, s/N) in G. H Moving near the boundary lower to the critical point, the joint distribution becomes bimodal with two distinct modes. The locations of the above unimodal and bimodal regimes in the control parameter space (w, E) are shown in the next panel with more detail. (I-J): Details of the phase diagram near the critical point. I Parameters are shown in the (w, E) space. Red dots denote the points at which the variability of spread size across realizations is maximized (wm, E) in Fig 7. Black dots denote the points for which we plotted the probability density functions of normalized spread size (s/N) and the joint probability density of duration (D) and spread sizes in Fig.8A-H. J The black line indicates the boundary between the no-spread and spread phases of the order parameter. The blue line indicates the location of points of maximum fluctuations in the order parameter. Between the red dashed lines we observe bimodality in the probability distribution of the order parameter. The arrow above the critical point indicates a continuous crossover from small spread to large spread sizes. The arrow below the critical point indicates a transition with a discontinuity in the order parameter, i.e. it is not differentiable at that point. Passing through the critical point results in a continuous transition that is expected to exhibit a singularity in the derivative of the order parameter in the thermodynamic limit. We investigated this expected property via finite-size scaling analysis in Figs 9 and 10. Despite the apparent very small region where the above transition from discontinuous to continuous behavior happens, we emphasize that different choices of parameters and their scaling can constrain the seizure spread activity to this small region. For example, based on Eq 21, we note that a choice of smaller EZ excitability (Eez) level can constrain the spread phase to a very small region around the critical point.