Modeling the effects of anesthesia on the electroencephalogram

Changes to the electroencephalogram (EEG) observed during general anesthesia are modeled with a physiological mean field theory of electrocortical activity. To this end a parametrization of the postsynaptic impulse response is introduced which takes into account pharmacological effects of anesthetic agents on neuronal ligand-gated ionic channels. Parameter sets for this improved theory are then identified which respect known anatomical constraints and predict mean firing rates and power spectra typically encountered in human subjects. Through parallelized simulations of the eight nonlinear, two-dimensional partial differential equations on a grid representing an entire human cortex, it is demonstrated that linear approximations are sufficient for the prediction of a range of quantitative EEG variables. More than 70 000 plausible parameter sets are finally selected and subjected to a simulated induction with the stereotypical inhaled general anesthetic isoflurane. Thereby 86 parameter sets are identified that exhibit a strong “biphasic” rise in total power, a feature often observed in experiments. A sensitivity study suggests that this “biphasic” behavior is distinguishable even at low agent concentrations. Finally, our results are briefly compared with previous work by other groups and an outlook on future fits to experimental data is provided.

Figure 1

A schematic illustration of the Liley et al. [8, 9] model. Two distinct macrocolumns with their excitatory $\left(E\right)$ and inhibitory $\left(I\right)$ neuron populations are shown, as well as their short- and long-range cortical and extracortical connections. The labels correspond to Eqs. (1, 2, 3, 4, 5, 6).

Figure 2

(a) Response $R$ of the inhibitory subpopulation to an inhibitory impulse depending on time and concentration of isoflurane according to Eqs. (9, 16, 17) and Tables . (b) The grid lines now represent $R_0$ of Eq. (7) with ${\phantom{\mid }{\gamma }_{lk}\mid }_{R_0}={\phantom{\mid }3.1462∕{\zeta }_{lk}\left(c\right)\mid }_R$. $R$ of (a) is displayed for comparison as a solid surface. (c) The response $R$ of the excitatory subpopulation after an excitatory impulse. (d) The responses integrated over time, approximately proportional to the total charge transferred.

Figure 3

Dependence of the largest Lyapunov exponent of Eqs. (1, 2, 3, 4, 5, 6) in the homogeneous case [36]. The units of the axes are $1{\mathrm{ms}}^{-1}$. This illustrates the “fat fractal” dependence of model dynamics on parameters $p_{ee}$ and $p_{ei}$.

Figure 4

(a) One second of $h_e$ data from a full grid simulation using the parameters of Table . The base shows the entire grid ($32\times 32$ values, interpolated) at one point in time; the vertical slices show the development with time (500 samples). (b) Radial spectrum of these data ($24\times 21$ values, interpolated) in decibels. (c) and (d) show the same as (a) and (b), respectively, but for a fully anesthetized state ($c=0.81\mathrm{m}M$ isoflurane).

Figure 5

Excitatory $S_e\left(h_e\right)$ and inhibitory $S_i\left(h_i\right)$ firing rates during a full grid simulation with the parameters of Table and changing isoflurane concentration, with $c\left(1\mathrm{MAC}\right)=0.243\mathrm{m}M$. The bands indicate the mean and standard deviation over all grid points.

Figure 6

Normalized power spectra obtained through full grid simulations with 27 PSO parameter sets.

Figure 7

Normalized eigenspectra using Eq. (38) as thick and Eq. (32) with $k=1.24{\mathrm{cm}}^{-1}$ as thin lines, respectively, in comparison with the average power spectra ($3.072\mathrm{s}$, detrended, Hanning window) from the grid with Table parameters.

Figure 8

(a) Eigenspectra of 73 454 “linear” parameter sets. The spectra are displayed along the vertical axis at $0.25\mathrm{Hz}$ frequency resolution with gray scale indicating their normalized values ($0\leftrightarrow \text{black}$ to $1\leftrightarrow \text{white}$). The spectra are sorted along the horizontal axis according to their $\alpha$-peak frequency. (b) Corresponding eigenspectra at 2 MAC isoflurane; the order along the horizontal axis from (a) has been maintained, but now set numbers are shown. Example: there are 2853 eigenspectra with $10\mathrm{Hz}$ $\alpha$ peak frequency [“9.75” to “10” in (a), “11 689” to “14 542” in (b)].

Figure 9

The bands display the 16% (lower edge), 50% median (middle line), and 84% (upper edge) quantile of all 73 454 (light) and 86 selected (dark) parameter sets. The selection criterion is marked in (a). Results for the set of Table , one of the 86, are shown by a solid black line (eigenspectrum) and stars (grid simulation). The plots show the following variable’s dependence on isoflurane concentration $c$: (a) total $(0–60\mathrm{Hz})$ power normalized to that at $c=0\mathrm{m}M$; (b) $\delta$ $(0–4\mathrm{Hz})$, (c) $\theta$ $(4–8\mathrm{Hz})$, (d) $\alpha$ $(8–13\mathrm{Hz})$, (e) $\beta$ $(13–30\mathrm{Hz})$, (f) $\gamma$ $(30–60\mathrm{Hz})$ fraction of power; (g) 50% median, (h) 90%, (i) 95% spectral edge frequency.

Figure 10

Bands (16–84 % quantile, middle line is median) showing the change with isoflurane concentration for the two least damped eigenvalues. Frequency $\mid \mathrm{Im}\left({\lambda }_2\right)\mid ∕\left(2\pi \right)=0$ for all 86 “biphasic” sets used.

Figure 11

Mean and standard deviation of the $\alpha$ pole frequency $(\mathrm{Re}\partial {\omega }^{*}∕\partial {\widehat{q}}_j)$ and damping $(-\mathrm{Im}\partial {\omega }^{*}∕\partial {\widehat{q}}_j)$ sensitivities of 86 parameter sets which showed a strong “biphasic” response, in comparison to 86 others which did not (see Table ). Differences in means were uniformly highly significant with $p⪡0.001$, except for ${\gamma }_{ee}$ damping, as determined by a large sample distribution-free method [44].

Figure 12

Comparison of the means and standard deviations of (a) ${\mathrm{SEF}}_{90}$, (b) total power normalized to that at $c=0\mathrm{m}M$, (c) $\delta$, (d) $\theta$, (e) $\alpha$, and (f) $\beta$ band fraction of power of data from 12 patients in Schwender et al. [54] (error bars) with eigenspectra predictions from 12 parameter sets each in Table (dark bands) and Table (light bands), respectively. The parameter sets have been chosen to match the data better than a random selection would, but do not represent a proper fit.