Table 1.
Optimization problems for various configurations of the VEP model.
The seizure propagation depends on the interplay between node dynamics (excitability) and network coupling (structure). The signals generated by VEP models on the brain region level are called source signals (see Eq 1). The measured signals from the electrodes are called sensor signals. To map the simulated sources from the brain regions to the sensors, the electromagnetic forward problem needs to be solved (see Eqs 2 and 3). Due to the sparse placement of electrodes, the gain (lead-field matrix) used for mapping the source to the sensors is not of full rank.
Fig 1.
This method employs a parallel cooperative scheme based on a master-worker strategy. An example of solution propagation flow is illustrated: when a worker obtains a good solution, it is shared with the rest of the workers through the master. Additionally, the master implements an adaptation mechanism to replace the settings of those workers with poor performance (as registered in a scoreboard), helping to improve their performance.
Fig 2.
Best solutions obtained by SaCeSS using 12 processors for the deterministic VEP model with weak coupling at source and sensor levels (problems 1 and 2).
(A) The simulated fast activities across brain regions at source level used for optimization. No propagation is observed due to weak coupling. (B) The true and estimated trajectories at phase-plane for three node types as Epileptogenic Zone (EZ, in red), Propagation Zone (PZ, in yellow), and Healthy Zone (HZ, in green). (C) The envelope of the simulated SEEG signals at the sensor level. (D) The confusion matrix indicates 100% accuracy in the estimation of three node types (EZ/PZ/HZ) based on SEEG signals.
Fig 3.
Best solutions obtained by SaCeSS using 12 processors for deterministic VEP model with strong coupling at source and sensor levels (problems 3 and 4).
A) The simulated fast activities across brain regions at source level used for optimization. Due to strong coupling, the seizure propagates from epileptogenic zone (in red) to other brain regions (in yellow). (B) The phase-plane displayed both the actual and predicted trajectories for three node categories as EZ, PZ, and HZ. (C) The simulated SEEG signals at the sensor level. (D) The confusion matrix that illustrates 100% accuracy for the estimation based on SEEG signals.
Fig 4.
Best solutions obtained by SaCeSS using 12 processors for stochastic VEP model with stiff equations at source and sensor levels (problems 5 and 6).
A) The simulated fast activities across brain regions with large sizure lenght (due to large time scale separation τ) at source level used for optimization. Due to strong coupling, the seizure propagates from EZ (in red) to PZ (in yellow). (B) The true and estimated trajectories in phase-plane for EZ, PZ, and HZ. (C) The simulated SEEG signals at the sensor level. (D) The confusion matrix that illustrates that one of the PZ is mis-classified as HZ.
Table 2.
Results for different parallel GO methods considering the set of VEP problems.
Each solver was executed 10 times (different parallel jobs), with different stopping times depending on the problem. Resulted obtained using 12 parallel processors in the FT3 supercomputer. Here, fx stands for the cost function defined as the root mean square error between observed and generated data.
Fig 5.
Scalability analysis of SaCeSS on the Finisterrae III supercomputer.
Each convergence curve represents the best run for the sequential method (eSS) and for SaCeSS using 6, 12, and 24 processors. The boxplots illustrate the spread in the solutions obtained in repeated runs (using different number of parallel workers for the case of SaCeSS).
Fig 6.
Scalability analysis of SaCeSS with different numbers of brain regions.
The four groups of convergence curves (differentiated by color) represent runs for SaCeSS using 12 processors solving problem 1, VEP with weak coupling at the source level, having different number of brain regions (NN): 42, 84, 162, and 400. The boxplots illustrate the spread in the solutions obtained in repeated runs.
Fig 7.
Comparison of SaCeSS with other parallel optimizers.
Each convergence curve represents the best run for SaCeSS, asynPDE, and PS-CMA-ES, all using 12 processors. The boxplots illustrate the dispersion in the solutions obtained with the aforementioned solvers. Each colored spot within the boxplots represents the solution cost obtained by a single run for each solver.
Fig 8.
Comparison between the original and the fine-tuned SaCeSS.
Convergence curves for original (orange) and fine-tuned SaCeSS (blue), plus boxplots illustrating the solutions spread.
Fig 9.
Uncertainty quantification of the parameters estimated using SaCeSS: Dispersion as parallel coordinates plots.
On the y-axis, the time-varying distribution percentiles are manifested as shaded red bands encircling a central black line, which signifies the median. The x-axis represents the parameters in the VEP models: the level of epileptogenicity ηi at each brain region and the global coupling K.
Table 3.
Scalability of SaCeSS in an supercomputing infrastructure (FT3) and in a desktop workstation (DELL Precision 5820).
SaCeSS was executed 10 times using different number of processors, varying the stopping time for each problem. In the case of one processor, the sequential enhanced scatter search (eSS) solver was used.