Fig 1.
Dynamics of the probabilistic model.
An example of seizure spreading dynamics in a patient-specific Epileptor network (subject P1) based on an 84-node parcellation of brain areas (Desikan-Killiany Atlas; see also Materials and methods) and the corresponding dynamics in the proposed probabilistic model. A Seizure spread observed in a simulation of a patient-specific Epileptor network model with global coupling strength set to w = 0.45, and excitability levels set to x0 = −2.173 and x0,ez = −1.8 for the surrounding and epileptogenic nodes, respectively (Materials and methods). The seizure starts at the EZ node (red; node 61; onset time at 0), and then spreads to all other nodes. A postical refractory period follows after seizure termination in each node. (The “spikes” preceding the seizure onset correspond to interictal spikes). Pink dots show rescaled seizure onset times obtained by simulating the proposed probabilistic model shown in panel B. B Seizure spread in a simulation of the proposed model. Parameters of the model are a = 0.46, b = 0.0021, c = 1.3, d = 0.05, w = 0.45, E = −0.112, Eez = 0.0026. Seizure onset and offset times in each node are shown by a circle and a diamond, respectively. Panels C-F show the evolution of different dynamical variables and transition rates of node 73 (shown in blue in panel B). C Evolution of the state variable xi for i = 73. Before seizure onset xi = −1 (susceptible state); during the seizure xi = 1 (seizure state) and after seizure termination xi = 0 (refractory state). D Evolution of variables ui and vi defined in Eqs 8 and 9. E Evolution of the fraction of active (seizing) nodes in the system and zi(t) as defined in Eq 7. F Evolution of transition rates: f(zi + Ei) for transitioning from the susceptible to the seizure state, and for transitioning from the seizure to the refractory state. Blue dashed lines indicate seizure onset and offset times. For examples of a small spread size see Figs B and C in S1 Text.
Fig 2.
A comparison of the seizure-spread phase diagrams from patient-specific Epileptor networks and proposed probabilistic model.
Each panel shows a phase diagram in the space of excitability and global interaction weight. Phase diagrams of the Epileptor network models are shown on the top row for 5 different patient-specific networks. The corresponding diagrams for the probabilistic model are shown on the bottom row. The regions colored as green, yellow, and blue correspond to the phases no-spread (i.e. the seizure remains localized to the epileptogenic zone), spread, and no-seizure (i.e. even the epileptogenic zone does not go into seizure, indicating a strong restrain effect of the surrounding nodes). The colorbar indicates the average spread size across stochastic realizations. The yellow region consists mostly of full spread cases, as illustrated in in Fig 1A and 1B by the time series from both the Epileptor network and probabilistic model network models. Cases of partial small spread are observed closer to the phase boundaries. Note that, while the horizontal axes of top and bottom panels are the same, we set the vertical axes of top panels to be centered at x0,b for the phase diagrams of Epileptor model in the top row to be comparable with those of the proposed probabilistic model in the bottom row.
Fig 3.
The proposed probabilistic model captures the spread timing in patient-specific Epileptor network models.
The plots show the results for patient-specific network P1, EZ-61. A A stochastic realization of an Epileptor network simulation in this patient-specific network. The vertical axis specifies the node index in the patient-specific network while the horizontal axis is time centered at the seizure onset time in the EZ node (node 61). Red diamonds specify the expected seizure onset time predicted by the model. B Linear relation between mean seizure onset times in Epileptor networks versus the mean onset time in the model for all points in phase space in which we observe full spread (yellow area in Fig 2). We note that, while a linear relation between the seizure onset times is observed for all points in phase space, the slope of the line varies depending on w and E. We rescaled all the lines to align with the diagonal. C Seizure-onset ordering in Epileptor networks versus the proposed model for all points in phase space in which we observe full spread. Red dots specify the 95 percentile of the data.
Fig 4.
Temporal evolution of average spread size and fraction of active (seizing) nodes: Example for patient-specific network P1, EZ = 61.
Each curve contains data from 20 different stochastic realizations. The right vertical axis (red) shows the mean normalized spread size ψ = 〈s/N〉, where s is the number of nodes in the surround to which a seizure has spread to. The left panel corresponds to simulations of the patient-specific epileptor network model. The middle panel relates to the simulations of the proposed probabilistic model with the temporal Gillespie algorithm. The right panel shows simulations based on a derived mean-field dynamics (see below and Materials and methods, Mean-field dynamics). In this mean-field approximation, derived under the assumption of Erdős-Rényi random network connectivity, we used an average connectivity weight and an average interaction delay. For this figure, both were computed from the corresponding patient-specific network.
Fig 5.
The proposed probabilistic model captures well the size of the fluctuations (standard deviation) in spread size across different stochastic realizations observed at the transition boundary between no-spread and spread phases.
In each panel the gray scale colorbar shows the standard deviation of seizure spread across 30 different realizations. The axis of each panel follows the same convention as for the axis in Fig 2. The top row panels are for patient-specific Epileptor network simulations and the bottom row for the probabilistic model.
Fig 6.
Phase diagrams derived from mean-field approximations.
Top row: phase diagrams were obtained from exact continuous time simulations (temporal Gillespie algorithm) of the proposed probabilistic model on ER networks of different sizes N = 1024, 4096, 8192. Red curves indicate the boundary between no-spread and spread phases estimated via mean-field finite-size corrected approximations. Black curves denote the boundary separating the no-seizure phase from the other two phases derived from the mean-field approximations. Bottom row: phase diagrams obtained by the simulation of the mean-field dynamics approximation in discrete-time (Eqs 26, 27 and 28; see also Materials and methods). Red and black curves are the transition boundaries derived from the mean-field approximation without the finite-size correction. As the network size grows, the agreement between the two (top and bottom) phase diagrams improves as expected.
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.
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.
Fig 9.
Power-law divergence of stochastic fluctuations in spread size near the critical point.
We used finite-size scaling analysis over four different network sizes of 213, 214, 215, 216. A The standard-deviation σ of the fluctuations as a function of w (fixed E = Ec) near the critical point (wc ≈ 6.7610−5, Ec ≈ 1.0010−6). The inset shows the power-law divergence of σ at its maximum and the corresponding scaling σm ∼ N0.66(1). B,C Power-law behavior of σ shown on log-scale for w approaching the critical point from below with corresponding scaling and exponent estimated as
, and from above with corresponding scaling σ+ ∼ (w − wc)−γ and exponent estimated as
, respectively. D The standard-deviation σ of the fluctuations as a function of E (fixed w = wc) near the critical point. The inset shows the power-law divergence of σ at its maximum and the corresponding scaling σm ∼ N0.68(1). G,H Power-law behavior of σ(E) shown on log-scale for E approaching the critical point from below with corresponding scaling
and exponent estimated as
, and from above with corresponding scaling σ+ ∼ (E − Ec)−α and exponent estimated as
.
Fig 10.
Power-law divergence of response functions and
near the critical point.
We used finite-size scaling analysis over four different network sizes of 213, 214, 215, 216. A The expected value of the normalized spread size, ψ = 〈s/N〉, as a function of w (fixed E) near the critical point (wc ≈ 6.76 10−5, Ec ≈ 1.00 10−6). The inset zooms the view around the critical point. B The response χw plotted as a function of w (for fixed E = Ec). The inset shows the divergence of the maximum response χw,m as a function of χw,m ∼ N0.16(1). C,D The log-scale plots show the power-law behavior of the response function χw as w approaches the critical point from below with corresponding scaling χw− ∼ (wc − w)β′−1 and exponent estimated as , and from above with corresponding scaling χw+ ∼ (w − wc)β−1 and exponent estimated as
. E The expected value of the normalized spread size, as a function of E near the critical point (for fixed w = wc). F The response χE plotted as a function of E (for fixed w = wc). The inset shows the divergence of the maximum response χw,m as a function of χw,m ∼ N0.18(1). G,H The log-scale plots show the power-law behavior of the response χE as E approaches the critical point from below with corresponding scaling χE− ∼ (Ec − E)1/δ′−1 and exponent estimated as
, and from above with corresponding scaling χE+ ∼ (E − Ec)1/δ−1 and exponent estimated as
.
Table 1.
The terms including σ denote the standard deviation (size of fluctuations) of spread size and their dependence on excitability (E) and connectivity strength (w) in the probabilistic model. The terms including ψ relate to the order parameter (normalized spread size) and their dependence on excitability and connectivity strength. The numbers in parentheses after each value represent the error in the last digit, e.g 1.63(3) = 1.63 ± 0.03.