^{1–3}Traditionally, the explanation for this phenomenon has been sought at a molecular or synaptic level of description, by finding differences between general anesthetic drugs that commonly precipitate seizure activity and those drugs that do not cause seizures. It has been suggested that proconvulsant drugs (such as enflurane) may (1) act to decrease the amplitude of miniature inhibitory postsynaptic currents

^{4}or (2) elicit greater calcium-induced presynaptic mobilization of excitatory neurotransmitters

^{5}than anticonvulsant drugs (such as isoflurane or thiopentone). These descriptions are qualitative. Are these observations of true causative mechanisms of seizure genesis, or are they simply observations of epiphenomena? It is not clear exactly how these observations at synaptic and molecular scales would quantitatively result in repetitive synchronous widespread burst firing of cortical neurons—the signature of seizure activity.

^{6}In this article, we describe a mathematical model of cerebral cortical function (the so-called mean-field model). In our model, we are able to input the known molecular-scale effects of anesthetic drugs and see how they alter the output (a “pseudoencephalogram”) of the computer-simulated “pseudocortex.” We incorporate the values of inhibitory postsynaptic potential (IPSP) amplitude and duration that have been previously published and studied in detail for isoflurane and enflurane, and we compare the output from simulations run on our theoretical mathematical model with various experimental and clinical observations that have been previously reported in the scientific literature. We find that subtle changes in the shape of the IPSP—induced by enflurane—are sufficient to cause the model of the cerebral cortex to undergo a sudden change in behavior from a general anesthetic state (in which neuronal firing is suppressed) into a seizure-like state—manifest as oscillation between neuronal silence and supramaximal neuronal firing. We use the example of enflurane-induced seizures as a dramatic demonstration of the application of mean-field models of cortical dynamics to link molecular and macroscopic descriptions of nervous system phenomena.

#### Modeling Cortical Activity

^{5}neurons. Its advantages over the neuron-by-neuron methods are as follows:

*via*a plethora of mechanisms (

*e.g.*, intracellular, ligand-gated, metabotropic receptors, or even

*via*the glia). In most cases, the end effect on the activity of the neuron is similar for all of these mechanisms and can be included in the mean-field model simply as a change in the sigmoid curve that describes the soma potential–to–firing rate relation—the average irritability of the neurons.

#### Description of the Mean-field Model

*et al.*

^{7}A synopsis is presented in the appendix to this article. It builds on a previous body of work by Liley, Wright, Robinson, Steyn-Ross, and others.

^{8–13}It has been used recently by Kramer

*et al.*

^{14}to study generalized seizure states in relation to epilepsy. We were at some pains to use realistic, physiologically derived values for the parameters of the model, as described in our previous articles. However, the question of accurate parametrization is still an open one because the measurements are typically derived from reported studies of single neurons, which are then assumed to be representative of average of the neural population. In brief, the model is formulated as a set of coupled differential equations in time and space that describe the time evolution of the mean soma potential of cortical “macrocolumns” of approximately 10

^{5}inhibitory and excitatory cortical neurons. The scalp electroencephalogram is an image of the time evolution of the summed mean soma potential of a large number of macrocolumns. In this way, the output from the theoretical model is a “pseudoelectroencephalogram” that can be directly compared with experimentally derived data that have been previously reported in the scientific literature. The differential equations also explicitly model action potential rates from cortical neuronal input. The distant input is modeled with a wave equation, and local input is modeled as an instantaneous feedback (no reverberating cortex–thalamus loops are included). The shapes and time durations of the excitatory postsynaptic potentials (EPSPs), and IPSPs, are modeled with a second-order differential equation in time. Reversal potentials are used to modulate the resulting input to the cell membrane, thus constraining the mean membrane potentials to lie within certain bounds. A sigmoidal function is used to relate mean action potential rate to mean membrane potential in both excitatory and inhibitory neuronal populations. Subcortical neuronal input is included through a random noise term. Specifically, we used these equations to model the cortex in two spatial dimensions as a 50 × 50-cm two-dimensional slab using periodic boundary conditions (approximately the area of unspecialized association cortex in humans). The quantitative effects of isoflurane and enflurane on IPSPs were incorporated using the data derived from Banks and Pearce

^{4}in a manner similar to that of Bojak and Liley.

^{15}

#### Model Predictions

##### Enflurane *versus* Isoflurane

Fig. 1 Image Tools |
Fig. 2 Image Tools |

*versus*the time-integrated IPSP magnitude (x-axis). In essence, both enflurane and isoflurane cause an increase in total synaptic charge transfer (the area under each IPSP) of between 100 and 200%. This results in the cortex making a transition from an active state (fig. 2, area A)—which is identified as being the awake state—to a quiescent (anesthetized) state (fig. 2, area B).

^{10}Figure 2b shows this transition more clearly by explicitly plotting the third dimension (the excitatory mean neuronal soma potential) along the z-axis. The trajectories followed by the model cortex in response to increasing concentrations of each anesthetic agent (enflurane = dashed line, isoflurane = solid line) are shown in figure 2. The start of each trajectory (bottom left) is zero anesthetic concentration, and the end of each trajectory (top right) is approximately 2 minimum alveolar concentration (MAC)

_{50}. The point at which the anesthetic drug concentration equals 1 MAC

_{50}is shown by

*X*s near the apex of each trajectory. It can be seen that both isoflurane and enflurane cause the cortex to move from the awake state (area A) to the anesthetized state (area B). However, there is a region (area C) where the cortex undergoes a sudden change in behavior to the unstable seizure state. Mathematically, the technical term for this is a subcritical Hopf bifurcation, to enter a limit-cycle oscillation. The figure shows that the enflurane trajectory has a greater probability of crossing, or entering, this unstable region, because enflurane reduces the IPSP magnitude more than isoflurane for a given IPSP time constant (

*i.e.*, for an equal IPSP area, the shape of the enflurane-affected IPSP is “flatter” than that for isoflurane). When in this state (area C), the cortex alternates (approximately 5/s) abruptly between zero firing and supramaximal firing states, similar to that observed during seizures.

^{16,17}The changes in stability of the cortical model were initially derived from analysis of the eigenvalues of the equations and then checked by the computer simulations. In addition to states similar to spike-wave seizures, spiral waves were also sometimes observed in the model when the trajectory crossed an unstable region. The shape of the IPSP is critical. Therefore, for a relative IPSP charge-transfer of two, the cortex could be either in a state of coma, if the IPSP was short and of high amplitude (fig. 2, position shown by diamond and subgraph ii), or in a seizure state, if the IPSP was long and of low amplitude (fig. 2, position shown by square and subgraph i). The model also predicts the propensity of enflurane to induce seizures in almost all cases at higher concentrations of above approximately 2%—where the trajectory turns back into the middle of the unstable region—as the total IPSP charge decreases in the presence of a slow IPSP decay time. This is because the effect of enflurane to decrease the peak amplitude of the IPSP is greater than its effect in prolonging the decay time. These results are in agreement with experimental observation

^{18}and the theoretical work of Liley and Bojak.

^{19}

##### Interactions with Thiopentone

^{20}Thiopentone prolongs the IPSP decay time but has no effect on the peak amplitude of the IPSP

^{21}and therefore would generate a trajectory that travels up and rightward. From inspection of figure 2, it can be seen that the effect of adding thiopentone to a low concentration of enflurane could shift the enflurane trajectory up and to the right, thus causing it to be more likely to enter the oscillatory seizure state (the lower part of area C). Conversely, at higher concentrations of enflurane (> 2%), thiopentone has an anticonvulsant effect because it would shift the trajectory to the right, away from reentering area C.

##### Other Proconvulsant and Anticonvulsant Drug Interactions

*N*-methyl-d-aspartate antagonist drugs effectively prevent or abort enflurane-induced convulsions.

^{22}Simulations of

*N*-methyl-d-aspartate blockade with an EPSP reduced in both area and time by 25% give just 19–21% of 1,000 simulated enflurane paths as crossing the unstable region, compared with 57–60% of enflurane paths without the

*N*-methyl-d-aspartate blockade.

*N*-methyl-d-aspartate blocker) and with benzodiazepines (potent antiepileptic drugs that are known to increase tonic chloride currents).

^{23}

^{24}(modeled as a 25% increase in the area of the EPSP) cause a great increase in the area of the unstable region (fig. 3b, area C), such that it dominates the phase plane. Likewise, decreased tonic inhibition—such as may be induced by physostigmine or 4-amino-pyridine

^{25}—may be modeled as depolarization of the resting membrane potential of the cortical neurons. This effect will greatly increase the propensity for the development of seizures (note large area C in fig. 3d).

#### Discussion

^{26,27}It may be surmised that other drugs and pathologic processes that slow or broaden the dendritic summation will have similar proseizure effects.

*e.g.*, the various voltages; the size, number, and time courses of the synaptic events; see appendix) are derived as far as possible from data that have been experimentally verified in the peer-reviewed literature. However, there exists some uncertainty as to the exact values. We justify our choice of parameter values from three arguments: (1) The experimental measurements of neuron-by-neuron parameters that are reported in the literature seem to vary widely with time and context. Therefore, narrow limits in the variability of individual parameters are probably less physiologically critical than relations between groups of parameters. (2) The fact that the cortex can continue to function, to some degree, in the face of significant neurobiologic disturbances (

*e.g.*, histamine or muscarinic blockade by drugs) argues that multiple parallel redundant pathways exist. (3) The cortex exhibits some degree of adaptive self-organization, which would hugely constrain the effective size of the parameter space.

*via*the indirect effect of changes in neuromodulatory systems on the parameter values of the cortical neurons. There is strong experimental evidence that thalamocortical circuits are not essential for the development of seizures.

^{17}We also assume that the only effect of the anesthetic drugs is on the IPSP, and we model the cortex as a simple two-dimensional sheet, without taking into consideration three-dimensional layering or variations between different regions within the cortex. These and other issues related to seizure onset and termination (such as second-order changes in chloride and potassium balance across the neuronal cellular membrane) may be investigated in more detail in future models. For example, the exaggerated frontal electroencephalographic effects of common γ-aminobutyric acid–mediated anesthetic drugs would clearly indicate heterogeneity in parameters between frontal and occipital lobes. Similarly, most of the corticocortical axons interact with inhibitory interneurons and apical pyramidal cell dendrites in the superficial layers of the cortex (cortical layers 1 and 2), whereas axonal input from the thalamus occurs predominantly near the pyramidal cell bodies (cortical layers 3–5). However, even our simple model seems to have adequate explanatory power to describe most of the observed phenomena, because these phenomena (anesthesia and seizures) are global features of cortical function.

#### Appendix: Formulation of the Mean-field Model of Cortical Interactions

*e*) and inhibitory interneurons (subscript

*i*). We have used the convention of a→b, as meaning the direction of transmission in the synaptic connections is from the presynaptic nerve a to postsynaptic nerve b. The subscript

*sc*indicates random subcortical input that is independent of the cortical membrane potential. The variables

*Va*and Φ

*ab*vary in time and space. The time evolutions of the mean neuronal soma membrane potential (

*Va*) in each population (excitatory [

*e*] and inhibitory [

*i*]) of neurons in response to synaptic input (ρ

*a*ψ

*ab*Φ

_{ab}) are given by the following set of equations:

Equation (Uncited) Image Tools |
Equation (Uncited) Image Tools |

*a*are the time constants of the neurons, ρ

*a*are the strength of the postsynaptic potentials (proportional to the total charge transferred), ψ

*ab*are the weighting functions that allow for the effects of reversal potentials and are described by the equation

_{ab}are the synaptic input spike-rate densities (flux) that are described by the following equations:

Equation (Uncited) Image Tools |
Equation (Uncited) Image Tools |
Equation (Uncited) Image Tools |

Equation (Uncited) Image Tools |

*ab*are the synaptic rate constants,

*N*

^{α}is the number of long-range connections, and

*N*

^{β}is the number of local, intramacrocolumn connections. The mean axonal velocity is given by ν, and the characteristic length (the length at which the connectivity decays to 1/

*e*) is 1/Λ ea . The spatial interactions among and within the macrocolumns are described by the two equations

Equation (Uncited) Image Tools |
Equation (Uncited) Image Tools |

Equation (Uncited) Image Tools |
Equation (Uncited) Image Tools |
Table 1 Image Tools |

*a*describes the inflection point voltage, and ς

*a*describes the SD of the threshold potential. The parameters and ranges used in our simulations are shown in table 1 (

*e*and

*i*refer to values assigned to excitatory and inhibitory cell populations).