Macroscopic models of epilepsy can deliver surprisingly realistic EEG simulations. In the present study, a prolific series of models is evaluated with regard to theoretical and computational concerns, and enhancements are developed. Specifically, we analyze three aspects of the models: (1) Using dynamical systems analysis, we demonstrate and explain the presence of direct current potentials in the simulated EEG that were previously undocumented. (2) We explain how the system was not ideally formulated for numerical integration of stochastic differential equations. A reformulated system is developed to support proper methodology. (3) We explain an unreported contradiction in the published model specification regarding the use of a mathematical reduction method. We then use the method to reduce the number of equations and further improve the computational efficiency. The intent of our critique is to enhance the evolution of macroscopic modeling of epilepsy and assist others who wish to explore this exciting class of models further.
Significant attention has been given to computational models of epilepsy that simulate the electroencephalogram (EEG) at the level of a neuronal population [
Most macroscopic models used in computational neuroscience today are derived, to some extent, from one of three seminal formulations: Freeman [
Wendling et al. have been the most prolific in using the basic approach of Lopes da Silva, with at least 17 different studies during the years 2000–2013. A key feature of their approach is the incorporation of synaptic interactions between specific groups of neurons. This permits the study of a broad class of mechanisms for epileptogenesis that depend on the levels of network excitation and inhibition. Most of their models are direct extensions of the previous work of Jansen et al. [
The earliest model of Wendling et al. used the same structure as Jansen et al. and many of the same parameter values [
In the present work, the modeling approach of Wendling et al. is critiqued with regard to theoretical and computational concerns, and enhancements are developed. Specifically, we analyze three aspects of the models: (1) Using dynamical systems analysis, we demonstrate and explain the presence of direct current potentials in the simulated EEG that were previously undocumented. (2) We explain how the system was not ideally formulated for numerical integration of stochastic differential equations. A reformulated system is developed to support proper methodology. (3) We explain an unreported contradiction in the published model specification regarding the use of a mathematical reduction method. We then use the method to reduce the number of equations and further improve the computational efficiency.
A basic diagram of the earliest model [
Core models. (a) Initial model [
Figure
Subgroups are connected using multiplier constants, labeled
The full equations are shown below, using the original variable indexing [
Model parameters from Wendling et al. [
Param. | Interpretation | Value |
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Average excitatory synaptic gain | 5 mV |
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Average slow inhibitory synaptic gain | See Figure |
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Average fast inhibitory synaptic gain | See Figure |
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Dendritic average time constant in the feedback excitatory loop | 100 s−1 |
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Dendritic average time constant in the slow feedback inhibitory loop | 50 s−1 |
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Somatic average time constant in the fast feedback inhibitory loop | 350 s−1 |
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Mean number of synaptic contacts in the excitatory feedback loop | 135, |
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Mean number of synaptic contacts in the slow feedback inhibitory loop |
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Mean number of synaptic contacts in the fast feedback inhibitory loop |
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Mean number of synaptic contacts between slow and fast inhibitory interneurons |
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Dynamical systems analysis was performed, based on the above equations, using an approach described previously [
All simulations were performed using MATLAB (The MathWorks, Natick, MA). Numerical integration was done using a fixed-step forward Euler method. Source code for simulations will be made available publicly on ModelDB (
As noted in Results, certain simulations used a stochastic forward Euler numerical integration method [
We analyze three aspects of the models: (1) Using dynamical systems analysis, we demonstrate and explain the presence of direct current potentials in the simulated EEG that were previously undocumented. (2) We explain how the system was not ideally formulated for numerical integration of stochastic differential equations. A reformulated system is developed to support proper methodology. (3) We explain an unreported contradiction in the published model specification regarding the use of a mathematical reduction method. We then use the method to reduce the number of equations and further improve the computational efficiency.
Models are not expected to be perfect replications, but knowledge of the underlying inaccuracies is critical to proper usage [
Example of DC offset in the model. (Left) Recreation of simulation of five different dynamical behaviors shown in Wendling et al. [
The DC offset was present in a predecessor model [
We used dynamical systems analysis to study the observed DC offset values. To the best of our knowledge, the present study is the first to perform such analysis on the model. For an accessible treatment of dynamical systems theory, see Strogatz [
Using the approach of Grimbert and Faugeras [
Figure
Table
Equilibrium points and their stabilities for specific phases.
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1 | 45 | Stable | −0.124 | 0.008 | 6.097 | 5.882 | 0.339 | 0.174 |
Unstable | 2.526 | 0.031 | 11.777 | 8.962 | 0.290 | 0.266 | ||
Unstable | 5.087 | 0.094 | 30.864 | 25.749 | 0.028 | 0.763 | ||
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2 | 38 | Stable | 1.018 | 0.014 | 7.037 | 5.600 | 0.419 | 0.166 |
Unstable | 1.781 | 0.022 | 8.553 | 6.358 | 0.415 | 0.188 | ||
Unstable | 5.416 | 0.105 | 31.220 | 25.768 | 0.036 | 0.764 | ||
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3 | 37 | Unstable | 5.466 | 0.106 | 31.254 | 25.750 | 0.037 | 0.763 |
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4 | 8 | Stable | 10.004 | 0.226 | 31.500 | 19.258 | 2.238 | 0.571 |
Eigenvalues for Phase 1,
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−0.124 | 2.53 | 5.09 |
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Stability | Stable | Unstable | Unstable |
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Eigenvalues for Phase 2,
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1.02 | 1.78 | 5.42 |
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Stability | Stable | Unstable | Unstable |
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Eigenvalues for Phase 3 (
Phase 3 |
Phase 4 |
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5.47 | 10.00 |
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Stability | Stable | Unstable |
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In Figure
Of particular interest in the above analysis is the DC offset that is different for each phase. High-pass filtering is commonly used in EEG acquisition to eliminate DC and improve dynamic range. DC offset in the physiological EEG is still poorly understood, but studies have shown that it coincides with ictal activity in some circumstances. Contrary to the simulation in Figure
Clearly, the model was not designed to accurately simulate the DC shift because electrochemical effects are not directly accounted for. Electrochemical changes are the most likely mechanism responsible for the DC offset that is seen in the EEG. To the best of our knowledge, only one epilepsy model has specifically addressed EEG offset [
Note that Labyt et al. [
The literature indicates that the model has historically been simulated as a set of ordinary differential equations. This was verified by source code that is publicly available on ModelDB (
An undesirable consequence is that, as the integration step size becomes smaller (e.g., 1 ms, 0.1 ms, and 0.01 ms), the accuracy of the simulated output actually decreases. Figure
Examples of using the 4th-order Runge-Kutta method, as in Wendling et al. [
We addressed the issue by reformulating (
Examples of using forward Euler for SDEs. (a) For integration step size of 1 ms, the variance
Lastly, we present a discrepancy in the published models that, to the best of our knowledge, has never been addressed. In Wendling et al. [
Mathematically equivalent models. (a) Jansen et al. [
The reduction is reasonable considering that both subgroups share the same input and that the PSP block is modeled as a linear transfer function. The end result is a reduction in the required number of equations. For unknown reasons, a similar reduction was not applied to the dual output paths from inhibitory interneurons whose output was defined as
Revised model with additional mathematical reduction. The former
A reduced set of equations is provided below that contains three major revisions. First, the input to the pyramidal cells now uses the multiplier
We critiqued a prolific computational modeling approach that has been used for the study of epilepsy. We evaluated three aspects of the models with regard to theoretical and computational concerns, and we developed enhancements to the model formulation. None of the issues that were raised invalidate the published results. However, we feel they are important considerations for other researchers to utilize the models effectively.
First, we demonstrated that a previously unreported DC offset is present in the model and that the offset varies for different parameter configurations. As explained previously, the presence of a DC offset is a well-known characteristic of the physiological EEG that is typically ignored. However, the model was not designed to incorporate any supposed mechanism for this phenomenon. Though the model produces an output that is interpreted as a voltage, the reactive-diffusive process of extracellular ion flow is in no way described by the system. We used dynamical systems analysis to show how the DC offset in the model can be predicted from the equations. Though another model has specifically addressed DC offset [
Second, we described how numerical integration methods may significantly affect the results. Using the published method, the voltage amplitude of the simulated EEG was greatly affected by the integration step size. Methods appropriate for SDEs require a separation of stochastic and deterministic terms. From a practical perspective, this affects whether results are reproducible by other researchers. We provided a reformulation of the equations in order to separate the stochastic and deterministic terms, and we described how this formulation would be implemented using a forward Euler integration method.
Note that there are additional numerical methods available for SDEs. For example, a stochastic Runge-Kutta method exists [
Third, we discussed a mathematical reduction that led to a contradiction between system diagrams and the equations in the literature. We further proposed a modification to improve the efficiency of the equations by applying the same mathematical reduction to a different part of the model. Though the reduction is mathematically equivalent to the longer form, it is an important conceptual modification because it contradicts actual physiological structure.
The intent of our critique is to enhance the evolution of macroscopic modeling of epilepsy and assist others who wish to explore this exciting class of models further. Just as modeling is only one research tool among many, macroscopic modeling is merely one of many levels of modeling that are needed to study a system like the brain that exhibits complexities at many temporal and spatial scales. Microscopic models of large networks may require significant computing power, but macroscopic models can usually be simulated using common office computing equipment. As we have demonstrated here, low-dimensional models also allow for rigorous mathematical analysis in order to better understand the mechanisms behind dynamical behavior. These advantages can benefit epilepsy research as well as neuroscience in general.
Source code for simulations will be made available publicly on ModelDB (
The authors declare that there are no competing interests regarding the publication of this paper.
The authors thank the reviewers for helpful suggestions.