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Perioperative Medicine  |   May 2009
Volatile Anesthetic Action in a Computational Model of the Thalamic Reticular Nucleus
Author Affiliations & Notes
  • Allan Gottschalk, M.D., Ph.D.
    *
  • Sam A. Miotke, M.S.
  • * Associate Professor, † Research Associate.
Article Information
Perioperative Medicine / Central and Peripheral Nervous Systems / Pharmacology
Perioperative Medicine   |   May 2009
Volatile Anesthetic Action in a Computational Model of the Thalamic Reticular Nucleus
Anesthesiology 5 2009, Vol.110, 996-1010. doi:10.1097/ALN.0b013e31819db923
Anesthesiology 5 2009, Vol.110, 996-1010. doi:10.1097/ALN.0b013e31819db923
ALTHOUGH it is known that the volatile anesthetics (VAs) modulate activity at a large number of molecular sites,1,2 it has yet to be determined which of these sites are responsible for general anesthesia (GA) and how action at one or more of these sites leads to the general anesthetic state. Making this determination is further complicated because GA is not a single well-defined phenomenon, but an amalgam of clinically useful effects. These include, in the order they are generally observed as VA concentration is increased, amnesia, unconsciousness, immobility, and blunting of autonomic reflexes. Therefore, the relevancy of a particular molecular site or regions of the central nervous system may depend on which property of GA is being considered.
The property of GA that is most frequently considered is immobility, and this property is generally used to rank VA potency using the minimum alveolar concentration (MAC) of VAs at which 50% of subjects remain immobile in response to a surgical stimulus.3 As a means of identifying the molecular site of action of the VAs, it was postulated that the concentration–effect curves of the molecular site(s) responsible for GA should match the concentration–effect curve at the systems level.4 Because the VA concentrations that produce immobility exert relatively modest effects on voltage-gated ion channels,5–7 this argument seemed to eliminate any of the voltage-gated ion channels as contributors to GA and helped to focus attention on ligand-gated channels such as γ-aminobutyric acid type A (GABAA) receptors, which are modulated by both VAs8–10 and intravenous anesthetics (IAs).11–14 Such an argument would seem to have even greater validity for unconsciousness that occurs at VA concentrations (MAC-awake) well below MAC (e.g.  , 0.25 MAC for isoflurane15 and 0.59 MAC for halothane,15 though there is some variability in these determinations16–20), where VA effects on ligand-gated channels would also be considerably less. This raises the issue of how small VA effects on ion channels at clinically relevant VA concentrations below MAC can produce meaningful systems level effects. Although computational modeling of neural activity has previously demonstrated how relatively small VA effects can profoundly alter model behavior,21 this observation has yet to be examined in models relevant to specific features of GA, or those that simultaneously consider VA action at multiple sites.
It is not fully established which networks and which types of network behavior are relevant to the mechanism(s) of GA. However, electroencephalograhic,22 functional imaging,23 and pharmacologic studies24,25 during GA make it clear that the brain is far from quiescent during GA and that GA may be the consequence of anesthetic action at specific anatomic sites within the central nervous system. Within the brain, thalamic activity is altered by the IAs26,27 and VAs,23,28 which is consistent with the hypothesized role of the thalamus in sleep and consciousness.29,30 Overall, these studies demonstrate that during administration of clinically relevant concentrations of IAs and VAs, brain electrical activity continues but often becomes more synchronous, and this can manifest itself in slowing of electroencephalographic activity.
If, in fact, the thalamus does play a role in VA induction of unconsciousness, it is reasonable to hypothesize that this is due to known VA actions at T-type Ca2+channels6 and/or GABAAreceptors,8–10 which are anesthetic-sensitive voltage- and ligand-gated ion channels that dominate, respectively, many intrinsic and network properties of thalamic neurons.29,31–33 However, as already emphasized, at MAC-awake concentrations of the VAs, these effects are limited. This raises the possibility of interaction between multiple VA sites of action if anesthetic action at these channels in the thalamus contributes to loss of consciousness. VA action at both voltage- and ligand-gated thalamic channels is consistent with the putative contribution of the thalamus to consciousness in that isoflurane is more potent than halothane at both T-type Ca2+channels and GABAAreceptors,6,8,34,35 and isoflurane, though less potent than halothane with respect to immobility, is, in fact, more potent than halothane with respect to MAC-awake.15–20 
To better assess how established VA effects on ion channels can impact thalamic function at VA concentrations known to affect consciousness, a computational model of the isolated thalamic reticular nucleus (RTN) was examined. Because the characteristic behavior of the RTN plays an important role in regulating thalamic oscillations36 and is dominated by the behavior of T-type Ca2+channels and GABAAreceptors,29,32,33,37 the impact of VA action on network behavior of the RTN was examined for these sites individually and in concert.
Materials and Methods
Neural Model
A model of the RTN (fig. 1) was generated from a single compartment model of individual neurons based on previous models.32,33,37 In addition to a leak current, membrane currents included the slow T-type Ca2+channel present in neurons of the RTN, a fast Na+channel, and a delayed rectifier K+channel. Networks of individual neurons were connected with GABAAsynapses using a random pattern of connectivity.
Fig. 1. Schematic of computational model (  A  ) and representative examples of membrane potentials from individuals network neurons (  B  ). Single compartment models of neurons of the reticular nucleus of the thalamus are coupled by inhibitory γ-aminobutyric acid type A (GABAA) synapses. Each neuron contains a T-type Ca2+channel, a fast Na+channel, a delayed rectifier K+channel, a leak current, and a membrane capacitance. The Ca2+current modulates internal Ca2+concentration, which, in turn, modulates Ca2+reversal potential. Each of the 100 neurons in the model is randomly coupled, on average, to 85% of the other neurons in the network (excluding self) with a fixed value of GABAAconductance. Neural activity generally consists of bursts of action potentials riding on the depolarization produced by the T-type Ca2+channel. Bursting occurs in response to postinhibitory rebound of the membrane potential.  32,33,42 Each action potential in a burst generates a synaptic pulse of GABA. Note how some neurons are in phase whereas others are not. Also note the presence of subthreshold oscillations in membrane potential due to inhibitory input from other neurons in the network. 
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Fig. 1. Schematic of computational model (  A  ) and representative examples of membrane potentials from individuals network neurons (  B  ). Single compartment models of neurons of the reticular nucleus of the thalamus are coupled by inhibitory γ-aminobutyric acid type A (GABAA) synapses. Each neuron contains a T-type Ca2+channel, a fast Na+channel, a delayed rectifier K+channel, a leak current, and a membrane capacitance. The Ca2+current modulates internal Ca2+concentration, which, in turn, modulates Ca2+reversal potential. Each of the 100 neurons in the model is randomly coupled, on average, to 85% of the other neurons in the network (excluding self) with a fixed value of GABAAconductance. Neural activity generally consists of bursts of action potentials riding on the depolarization produced by the T-type Ca2+channel. Bursting occurs in response to postinhibitory rebound of the membrane potential.  32,33,42 Each action potential in a burst generates a synaptic pulse of GABA. Note how some neurons are in phase whereas others are not. Also note the presence of subthreshold oscillations in membrane potential due to inhibitory input from other neurons in the network. 
×
The voltage in each neuron was modeled using the relation
where Ciis the membrane capacitance of the i-th neuron, Viis the membrane potential of the i-th neuron, Iinject-iis the injection current to the i-th neuron, Iion-iis the sum of ionic currents due to the leak and voltage-gated channels in the i-th neuron, and the IGABAA−ikare the GABAAcurrents from any other neurons that make synaptic contacts with the i-th neuron. The ionic currents are modeled using the general formula
where Iion-ijis the j-th ionic current in the i-th neuron, gijis the maximum conductance of that current, mijis the corresponding activation variable, hijis corresponding inactivation variable, pjis the power associated with activa-tion of the j-th ionic current, qjis the power associated with inactivation of the j-th ionic current, and Vrev-ijis the corresponding reversal potential. The general form for both the activation and inactivation variables is
where α(Vi) is the opening rate of the corresponding variable and β(Vi) is the corresponding closing rate, which are generally voltage dependent. In general, the reversal potential is fixed for each of the ions except for Ca2+, whose reversal potential is a function of the internal Ca2+concentration. Here, the reversal potential for Ca2+is given by the Nernst equation,
where zCa= 2 is the charge on the Ca2+ion, F = 9.648 × 104C mol−1is the Faraday constant, R = 8.315 J K−1mol−1is the gas constant, T is the temperature in K, [Ca2+]in-iis the internal Ca2+concentration in the i-th neuron in millimoles, [Ca2+]outis the external Ca2+concentration in millimoles, and ln[·] is the natural logarithm. Modulation of the internal Ca2+concentration by the Ca2+current is given by
where h is the thickness in μm of the internal shell whose concentration is of interest, c is the equilibrium value of [Ca2+]in-i, k is a constant to permit the units to be correct, and τCais the time constant governing the achievement of the equilibrium concentration.
For networks coupled by GABAAsynaptic interactions, each spike generated by a presynaptic neuron produced a fixed pulse of GABA of 0.5 mm and 0.3 ms duration at the postsynaptic neuron, increasing the concentration of GABA in the synapse.33 The kinetics governing activation of the GABAAreceptor becomes
where [GABA] is the concentration of GABA in the synapse, and α and β are the rate constants governing opening and closing of the GABAAreceptor. Typically, in the absence of anesthetic effects, α= 20 ms−1mm−1and β= 0.16 ms−1.33 
The specific form of the rate parameters and exponents of the activation variables of equation 2for the slow T-type Ca2+, Na+, and K+ion channels have been specified previously along with the model parameters described subsequently.33 The membrane capacitance was 1 μF/cm2. The baseline Ca2+conductance used was 3 mS/cm2, the Na+conductance used was 200 mS/cm2, the K+conductance used was 20 mS/cm2, and the leak conductance used was 0.05 mS/cm2. The reversal potentials were VNa= 50 mV, VK=−100 mV, VL=−90 mV, and VCl=−80 mV (reversal potential of GABAAcurrent). The external Ca2+concentration was assumed to be 2 mm. The resting internal Ca2+concentration was assumed to be 2.4 × 10−4mm at equilibrium, and τCawas 5 ms. The surface area of the neuron was 1.41887 × 10−4cm2, the thickness of the internal shell h was 0.1 μm, and the constant k was 0.1. As described in the Results, the remaining parameters were tuned to provide model behavior suitable for examining alterations in network synchrony by the VAs (e.g.  , initial state neither perfectly synchronous nor asynchronous). The synaptic conductance of the inhibitory GABAAsynapses was set to 0.2/NConnμS, where NConnis the expected number of connections in the randomly connected network of N neurons. The number of neurons was 100 with 85% connectivity so that NConnwas 85. A bias current of 0.12 nA was applied to all neurons to simulate exogenous excitatory input. All simulations were performed at 36°C. Examples of the behavior of representative neurons from this model of the RTN before introducing VAs are given in figure 1along with the model schematic.
Anesthetic Effects
Multiple VA effects on both the voltage- and ligand-gated ion channels are known.1,2 The nature and magnitude of the interaction can be highly specific to the channel and can vary with the anesthetic. The Ca2+channel seems to be the most sensitive of the voltage-gated ion channels considered in the current model.4,5 Although the VAs seem to affect the dynamics of activation and inactivation of the L- and N-type Ca2+channels,5,38 these features are minimally affected for the T-type channels.6 Consequently, VA action at these channels is modeled by modulating the conductance as a function of VA concentration using the Hill equation to fit existing experimental data. Therefore, the current in the channel is modified by multiplication with the function
which is one when the anesthetic concentration [A] is zero, and goes to zero as [A] continues to increase. Here, n is the Hill exponent and [A]1/2is the semisaturation constant, the value of [A] where φ([A]) = 1/2. Examples of φ([A]) for halothane and isoflurane are shown in figure 2A. VA decrements in the currents produced by the fast Na+and the delayed rectifier K+are known,5 but they are small and have had little impact when incorporated into a simplified computational model of the CA3 hippocampal neuron.21 In addition to its direct effect on the ion channel, VA modulation of Ca2+channel activity affects internal Ca2+concentration, which modulates the Ca2+reversal potential.
Fig. 2. Volatile anesthetic effects on T-type Ca2+channels and γ-aminobutyric acid type A (GABAA) receptors for both isoflurane (  solid line  ) and halothane (  dashed line  ). The normalized decrease in T-type Ca2+channel conductance is shown as a function of volatile anesthetic concentration in  A  . The values of the Hill exponent and semisaturation constant used in  equation 7are, respectively, n = 2.3 (±0.15) and [A]1/2= 0.30 (±0.01) for isoflurane, and n = 2.0 (±0.10) and [A]1/2= 0.66 (±0.03) for halothane.  6 The relative increase in the time constant governing deactivation of the GABAAreceptor as anesthetic concentration is increased is shown in  B  . Here, the maximum prolongation of the time constant, the Hill exponent, and the semisaturation constant used in  equation 8are, respectively, fmax= 4.7 (±0.50), n = 2.7 (±1.2), and [A]1/2= 0.32 (±0.05) for isoflurane,  8 and fmax= 2.1 (±0.51), n = 1.5 (±1.0), and [A]1/2= 0.36 (±0.26) for halothane.  8,34,35 The decrease in the amplitude of the inhibitory postsynaptic potential (IPSP) at the GABAAreceptor as anesthetic concentration is increased is shown in  C  . The values of the Hill exponent, semisaturation constant, and maximum IPSP attenuation used in  equation 9are, respectively, n = 2.6 (±0.80), [A]1/2= 0.79 (±0.24), and ymax= 0.56 (±0.13) for isoflurane,  8 and n = 1.9 (±1.0), [A]1/2= 1.0 (±0.36), and ymax= 0.72 for halothane, where the value for ymaxfor halothane was obtained from a single point.  8,35 
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Fig. 2. Volatile anesthetic effects on T-type Ca2+channels and γ-aminobutyric acid type A (GABAA) receptors for both isoflurane (  solid line  ) and halothane (  dashed line  ). The normalized decrease in T-type Ca2+channel conductance is shown as a function of volatile anesthetic concentration in  A  . The values of the Hill exponent and semisaturation constant used in  equation 7are, respectively, n = 2.3 (±0.15) and [A]1/2= 0.30 (±0.01) for isoflurane, and n = 2.0 (±0.10) and [A]1/2= 0.66 (±0.03) for halothane.  6 The relative increase in the time constant governing deactivation of the GABAAreceptor as anesthetic concentration is increased is shown in  B  . Here, the maximum prolongation of the time constant, the Hill exponent, and the semisaturation constant used in  equation 8are, respectively, fmax= 4.7 (±0.50), n = 2.7 (±1.2), and [A]1/2= 0.32 (±0.05) for isoflurane,  8 and fmax= 2.1 (±0.51), n = 1.5 (±1.0), and [A]1/2= 0.36 (±0.26) for halothane.  8,34,35 The decrease in the amplitude of the inhibitory postsynaptic potential (IPSP) at the GABAAreceptor as anesthetic concentration is increased is shown in  C  . The values of the Hill exponent, semisaturation constant, and maximum IPSP attenuation used in  equation 9are, respectively, n = 2.6 (±0.80), [A]1/2= 0.79 (±0.24), and ymax= 0.56 (±0.13) for isoflurane,  8 and n = 1.9 (±1.0), [A]1/2= 1.0 (±0.36), and ymax= 0.72 for halothane, where the value for ymaxfor halothane was obtained from a single point.  8,35 
×
Like many of the IAs, some of the VAs can also prolong the time constant (τGABAA= 1/β for β in equation 6) governing deactivation of the GABAAreceptor.8,34,35 The Hill equation has been used to characterize these effects, so that β in equation 6is adjusted by dividing it by the expression
where fmaxis the maximum possible increase in the time constant governing deactivation of the GABAAreceptor. Examples of the increase τGABAAfor halothane and isoflurane are shown in figure 2B.
The VAs can also decrease amplitude of the inhibitory postsynaptic potential (IPSP) generated by GABA at the GABAAreceptor. This can be addressed in much the same way that reductions in other ionic currents were addressed, but by modifying equation 7to reflect that the amplitude of the IPSP does not go to zero as VA concentration increases,17,23 using the following equation:
where ymaxis the relative IPSP amplitude at maximum attenuation. Examples of the reductions in the IPSPs generated at the GABAAreceptor due to VAs are given in figure 2C. From this, it can be seen that at lower VA concentrations VA reductions in IPSP amplitude are minimal. Therefore, at the GABAAreceptor, the VA effect of decreasing the rate of receptor deactivation is dominant.
Computer Simulations and Analysis
Simulations of single neurons and small networks were performed using Mathematica (Wolfram Research, Champaign, IL)39 on a personal computer based on an Intel Pentium IV processor (Dell, Inc., Round Rock, TX) running Windows XP (Microsoft Corporation, Redmond, WA), and larger network simulations were performed using the neural simulation language GENESIS1using a computer based on a Xenon Quad-Core processor (Western Scientific, San Diego, CA) with the Linux operating system (Red Hat, Raleigh, NC).
Network activity was initiated by stimulating each neuron with a single randomly generated inhibitory current pulse with an occurrence time uniformly distributed over 1 s and with an amplitude uniformly distributed between 0 and 0.2 nA. After this initial second, network behavior was permitted to evolve for an additional 5 s before generating data for analysis. When estimating properties of network behavior from multiple simulations with the same parameters, simulations were performed using a different random seed to generate both network connectivity and the random pulse sequence used to initiate network activity.
Overall network behavior was quantified in a number of ways. Raster plots were generated, which graphically indicate the membrane potential of all neurons in the network over time by converting the membrane potential of each to a grayscale variable and simultaneously plotting all of these. Network synchrony can be further visualized by examining the relative phase of each neuron in the network. This was accomplished using Fourier analysis.41 The Fourier transform of the membrane potential from each neuron was computed from 3 s of data, and the dominant frequency of each was determined from the location of the maximum peak in the power spectrum. Then, for this frequency, the phase was computed from the Fourier transform. The results for the entire network were summarized in a histogram. The average membrane potential was computed to simulate the field potential of an appropriately located depth electrode by simply averaging the individual potentials of all neurons within the network. The autocorrelation function of the averaged membrane potential was also determined. A coherence parameter (χ2) computed from the same 3 s of data was used to quantify network synchrony with a single parameter.42 This is done using the following coherence function:
where
where Vi(t) is the membrane potential of the i-th neuron, and T is the duration of the interval under consideration. χ2Can range between zero (no synchrony) and one (perfect synchrony), and is the ratio of variation of the overall average membrane potential around the overall mean to the average of the individual membrane potentials around their individual means.
Results
Joint Sensitivity to the Volatile Anesthetics
Using the data from figure 2, as the concentrations of the VAs isoflurane and halothane are increased, a locus for each is described in the plane defined by relative decrements in conductance of T-type Ca2+conductance and increases in the time constant (τGABAA) governing the rate of GABAAreceptor deactivation (fig. 3). Note the proximity of the clinically determined values for MAC-awake (the minimum alveolar concentration at which 50% of subjects will recover consciousness) for these two VAs. This is consistent with the hypothesis that action at these sites may be associated with VA effects on consciousness in brain regions, such as the thalamus, whose behavior is dominated by T-type Ca2+channels and inhibitory GABAAinteractions.
Fig. 3. Loci of volatile anesthetic effects for decrements in conductance of T-type Ca2+channels and increase in the time constant (τGABAA) governing γ-aminobutyric acid type A (GABAA) receptor deactivation for both isoflurane and halothane as volatile anesthetic concentration is increased.  Symbols  (•= isoflurane, ○= halothane) correspond to increments of 0.02 mm volatile anesthetic concentration.  Arrows  denote locations of MAC-awake (0.075 mm = 0.25 MAC for isoflurane and 0.12 mm = 0.59 MAC for halothane),  15 where MAC is the minimum alveolar concentration at which 50% of subjects remain immobile to incision (0.3 mm for isoflurane and 0.2 mm for halothane). The fractional decrease in T-type Ca2+conductance is shown on the  horizontal axis  , and the fractional increase of the time constant (τGABAA) governing GABAAreceptor deactivation is shown on the  vertical axis  . The  origin  corresponds to an anesthetic concentration of zero. 
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Fig. 3. Loci of volatile anesthetic effects for decrements in conductance of T-type Ca2+channels and increase in the time constant (τGABAA) governing γ-aminobutyric acid type A (GABAA) receptor deactivation for both isoflurane and halothane as volatile anesthetic concentration is increased.  Symbols  (•= isoflurane, ○= halothane) correspond to increments of 0.02 mm volatile anesthetic concentration.  Arrows  denote locations of MAC-awake (0.075 mm = 0.25 MAC for isoflurane and 0.12 mm = 0.59 MAC for halothane),  15 where MAC is the minimum alveolar concentration at which 50% of subjects remain immobile to incision (0.3 mm for isoflurane and 0.2 mm for halothane). The fractional decrease in T-type Ca2+conductance is shown on the  horizontal axis  , and the fractional increase of the time constant (τGABAA) governing GABAAreceptor deactivation is shown on the  vertical axis  . The  origin  corresponds to an anesthetic concentration of zero. 
×
Volatile Anesthetic Action on Biophysical Model of the RTN
Volatile anesthetic action at both T-type Ca2+channels and GABAAreceptors as quantified in figure 2can increase the level of synchrony in the biophysical model of the RTN at clinically relevant VA concentrations. This is shown in figure 4for both isoflurane and halothane at VA concentrations of twice MAC-awake. In figure 4, the raster plots depict the firing pattern for all 100 neurons in the network over time under control conditions and in the presence of anesthetic effects attributable to 0.15 mm isoflurane and 0.24 mm halothane. This demonstrates how relatively modest VA effects can noticeably alter the behavior of this model but does not indicate whether the alterations in model behavior are due to VA effects at the T-type Ca2+channel, the GABAAreceptor, or both.
Fig. 4. Synchronization of model network activity by the volatile anesthetic actions of  figure 2. Raster plots depicting the activity of all 100 neurons in the network are shown for control conditions (  A  ) and at twice minimum alveolar concentration (MAC)-awake for two different volatile anesthetics (0.15 mm isoflurane [  B  ] and 0.24 mm halothane [  C  ]). Volatile anesthetic effects included decreasing T-type Ca2+channel conductance and an increase in the time constant (τGABAA) governing γ-aminobutyric acid type A (GABAA) receptor deactivation. The activity of each of the 100 neurons in the network is depicted over time by continuously mapping membrane potential to grayscale intensity, where darker intensity denotes larger values of membrane potential. Note the ability of clinically relevant volatile anesthetic concentrations to synchronize network activity (  e.g.  , more membrane activity at a given phase) and that pattern of activity seems more organized for isoflurane (  B  ) than for halothane (  C  ). 
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Fig. 4. Synchronization of model network activity by the volatile anesthetic actions of  figure 2. Raster plots depicting the activity of all 100 neurons in the network are shown for control conditions (  A  ) and at twice minimum alveolar concentration (MAC)-awake for two different volatile anesthetics (0.15 mm isoflurane [  B  ] and 0.24 mm halothane [  C  ]). Volatile anesthetic effects included decreasing T-type Ca2+channel conductance and an increase in the time constant (τGABAA) governing γ-aminobutyric acid type A (GABAA) receptor deactivation. The activity of each of the 100 neurons in the network is depicted over time by continuously mapping membrane potential to grayscale intensity, where darker intensity denotes larger values of membrane potential. Note the ability of clinically relevant volatile anesthetic concentrations to synchronize network activity (  e.g.  , more membrane activity at a given phase) and that pattern of activity seems more organized for isoflurane (  B  ) than for halothane (  C  ). 
×
The contribution of the individual VA effects can be evaluated by considering their impact on model behavior individually and collectively. This is done and further quantified in figure 5for the same VA concentrations as in figure 4. Here, network behavior for each condition is depicted by the average membrane potential, which is the model equivalent of the local field potential or electroencephalogram. Average membrane potential was generated by averaging the individual membrane potentials from all neurons in the network. Despite the obvious alterations in synchrony of the model neurons in figure 4, the alterations in the average membrane potential are subtle and are not further revealed by autocorrelation (not shown). However, other measures shown in figure 5demonstrate the extent to which neural activity becomes more organized. The histogram shown in each panel indicates the number of neurons that are active for a given relative phase of the network's fundamental frequency of activity. Increased synchrony would be indicated by a greater number of neurons firing at the same or similar phases. It is convenient to quantify the degree of network synchrony with a single parameter using the coherence parameter described in the Materials and Methods.42 An increase in network synchrony is indicated in figure 5by slowing and increased amplitude of the oscillations in average membrane potential, a greater number of neurons at a similar phase, and an increase in the coherence (χ2). An increase in the synchrony of model activity was observed for VA effects on T-type Ca2+channels alone, GABAAreceptors alone, and both T-type Ca2+channels and GABAAreceptors. However, at least for the VA concentrations depicted in figure 5, synchronous activity is greatest when VA effects are applied to both sites. Note also in figure 5that the coherence values for both isoflurane and halothane are greater than for control conditions and that for isoflurane is greater than that for halothane, all of which is consistent with the appearance of the raster plots of figure 4.
Fig. 5. Characteristics of the network synchronization by volatile anesthetics depicted in  figure 4for both 0.15 mm isoflurane (  A  ) and 0.24 mm halothane (  B  ). Model behavior is shown for control (no volatile anesthetic,  upper left panel  ), VA-induced decrease in T-type Ca2+channel conductance only (  lower left panel  ), volatile anesthetic–induced increase in the time constant (τGABAA) governing γ-aminobutyric acid type A (GABAA) receptor deactivation only (  upper right panel  ), and actions at both of these sites (  lower right panel  ). For each condition, the field potential of the network, a numeric measure of network synchrony (coherence, χ2), and a histogram of relative phase are shown. Note how volatile anesthetic actions at either the voltage-gated site or ligand-gated site leads to increasingly organized network behavior with each of these measures, and that organized behavior seems to be the greatest for action at both sites. Note also that although the histogram for halothane when both channels are considered demonstrates greater similarity of phase in the histogram than isoflurane, the χ2value is lower. Here, the level of synchrony as reflected by χ2is more consistent with what is seen in  figure 4and emphasizes the importance of using a synchrony measure such as χ2that considers both phase and amplitude at all frequencies, as highlighted in the Discussion in the section Characterizing Network Behavior. Scales for both voltage and time are given in the  upper left panel  of each. See text for details. 
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Fig. 5. Characteristics of the network synchronization by volatile anesthetics depicted in  figure 4for both 0.15 mm isoflurane (  A  ) and 0.24 mm halothane (  B  ). Model behavior is shown for control (no volatile anesthetic,  upper left panel  ), VA-induced decrease in T-type Ca2+channel conductance only (  lower left panel  ), volatile anesthetic–induced increase in the time constant (τGABAA) governing γ-aminobutyric acid type A (GABAA) receptor deactivation only (  upper right panel  ), and actions at both of these sites (  lower right panel  ). For each condition, the field potential of the network, a numeric measure of network synchrony (coherence, χ2), and a histogram of relative phase are shown. Note how volatile anesthetic actions at either the voltage-gated site or ligand-gated site leads to increasingly organized network behavior with each of these measures, and that organized behavior seems to be the greatest for action at both sites. Note also that although the histogram for halothane when both channels are considered demonstrates greater similarity of phase in the histogram than isoflurane, the χ2value is lower. Here, the level of synchrony as reflected by χ2is more consistent with what is seen in  figure 4and emphasizes the importance of using a synchrony measure such as χ2that considers both phase and amplitude at all frequencies, as highlighted in the Discussion in the section Characterizing Network Behavior. Scales for both voltage and time are given in the  upper left panel  of each. See text for details. 
×
The results of figure 5are expanded in figure 6, where coherence is computed as a function of the relative conductance of the T-type Ca2+channel with respect to its baseline, relative increases in τGABAA, and as a function of anesthetic concentration for isoflurane and halothane. The results from figures 6C and Dare rescaled in figures 6E and Fto depict network synchrony as a function of MAC fraction and to demonstrate the similarity of the coherence parameter for isoflurane and halothane for VA concentrations corresponding to MAC-awake.
Fig. 6. Network coherence (χ2) as channels in the model are modulated one type at a time, and when these effects are considered together for the volatile anesthetics isoflurane and halothane. Network coherence is shown for decrements in the conductance of the T-type Ca2+channel (  A  ), and as the time constant (τGABAA) governing the rate of γ-aminobutyric acid type A (GABAA) receptor deactivation is increased (  B  ). Network coherence was also determined as a function of isoflurane concentration (  C  ) and halothane concentration (  D  ) using the combined effects of each volatile anesthetic on the T-type Ca2+conductance and the time constant (τGABAA) governing the rate GABAAreceptor deactivation that is depicted in  figures 2 and 3. Each  point  is the mean (±SEM) value of χ2obtained from either 6 (  B  ) or 12 (  A, C,  and  D  ) simulations differing only in their randomly assigned pattern of connectivity and timing of initial current pulses to the network (see Materials and Methods). Each curve was obtained from a least-squares fit of the simulation data to the equation f(x) = V/(1 + e−g(x)) + k, where g(x) = a0+ a1x + a2x2+… adxd, where the degree d was either 5 (  A–C  ) or 4 (  D  ). The data from  C  and  D  are plotted jointly in  E  but expressed in multiples of minimum alveolar concentration (MAC) for both isoflurane (MAC = 0.3 mm) and halothane (MAC = 0.2 mm), which is rescaled (  F  ) to demonstrate the similarity of the coherence for each volatile anesthetic's MAC-awake concentration (0.25 MAC for isoflurane and 0.59 MAC for halothane). Here, χ2was 0.325 for isoflurane and 0.335 for halothane at the MAC-awake concentrations of these volatile anesthetics. Note that the control value of χ2is 0.295. For MAC-awake concentrations of the volatile anesthetics acting on one channel at a time, isoflurane increases χ2to 0.304 when affecting the GABAAreceptor alone and 0.320 when affecting the T-type Ca2+channel alone. Similarly, halothane increases χ2to 0.316 when acting only at either the GABAAreceptor or T-type Ca2+channel. 
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Fig. 6. Network coherence (χ2) as channels in the model are modulated one type at a time, and when these effects are considered together for the volatile anesthetics isoflurane and halothane. Network coherence is shown for decrements in the conductance of the T-type Ca2+channel (  A  ), and as the time constant (τGABAA) governing the rate of γ-aminobutyric acid type A (GABAA) receptor deactivation is increased (  B  ). Network coherence was also determined as a function of isoflurane concentration (  C  ) and halothane concentration (  D  ) using the combined effects of each volatile anesthetic on the T-type Ca2+conductance and the time constant (τGABAA) governing the rate GABAAreceptor deactivation that is depicted in  figures 2 and 3. Each  point  is the mean (±SEM) value of χ2obtained from either 6 (  B  ) or 12 (  A, C,  and  D  ) simulations differing only in their randomly assigned pattern of connectivity and timing of initial current pulses to the network (see Materials and Methods). Each curve was obtained from a least-squares fit of the simulation data to the equation f(x) = V/(1 + e−g(x)) + k, where g(x) = a0+ a1x + a2x2+… adxd, where the degree d was either 5 (  A–C  ) or 4 (  D  ). The data from  C  and  D  are plotted jointly in  E  but expressed in multiples of minimum alveolar concentration (MAC) for both isoflurane (MAC = 0.3 mm) and halothane (MAC = 0.2 mm), which is rescaled (  F  ) to demonstrate the similarity of the coherence for each volatile anesthetic's MAC-awake concentration (0.25 MAC for isoflurane and 0.59 MAC for halothane). Here, χ2was 0.325 for isoflurane and 0.335 for halothane at the MAC-awake concentrations of these volatile anesthetics. Note that the control value of χ2is 0.295. For MAC-awake concentrations of the volatile anesthetics acting on one channel at a time, isoflurane increases χ2to 0.304 when affecting the GABAAreceptor alone and 0.320 when affecting the T-type Ca2+channel alone. Similarly, halothane increases χ2to 0.316 when acting only at either the GABAAreceptor or T-type Ca2+channel. 
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Synergy of VA Effects at Multiple Targets
To characterize the nature and extent of VA effects at T-type Ca2+channels and GABAAreceptors across a range of VA concentrations, an isobologram for each VA can be generated for the model43,44 using the coherence measure in equation 10as the quantifiable drug effect. In this particular instance, the process of generating an isobologram is somewhat restricted because both drug effects are not manipulated separately by varying independently the concentration of two different drugs. Instead, both effects vary jointly as a function of anesthetic concentration as shown in figure 3, and joint effects are restricted to the loci shown there. However, the computational approach illustrated in figures 5 and 6demonstrates how VA effects on the model can be examined individually and then jointly for the combinations permitted by each VA as its concentration is increased.
The data in figures 3 and 6were used to generate the isoboles which collectively form the isobolograms shown in figure 7. For each anesthetic concentration along the loci of figure 3, the coherence of the network was determined from the data given in figures 6C and D. Then, the relative attenuation of the T-type Ca2+conductance and relative prolongation in the time constant for GABAAreceptor deactivation that would produce the same level of network coherence were determined using the equations fit in figure 6A and B. The line connecting these three points is an isobole, and generating these for multiple anesthetic concentrations produced the isobolograms of figure 7.
Fig. 7. Isobolograms for isoflurane (  A  ) and halothane (  B  ) using network synchronization as the drug effect for combinations of decreases in T-type Ca2+channel conductance and increases in the time constant (τGABAA) governing γ-aminobutyric acid type A (GABAA) receptor deactivation that correspond to the actions of the volatile anesthetics. The isobolograms were generated using the data of  figure 6and the loci of  figure 3to define pharmacologically relevant trajectories. The line emerging from the origin is redrawn from  figure 3, and is the locus of the combined effects of the volatile anesthetic on both the T-type Ca2+channel and GABAAreceptor as volatile anesthetic concentration is increased. For each given volatile anesthetic concentration, the coherence was determined. Isoboles were then generated by connecting this point with points on each axis where modulation of that channel, by itself, would lead to the same network behavior as measured by the coherence. For isoflurane (  A  ), isoboles were generated for increments of 0.025 mm on the left and 0.0125 mm on the right. For halothane (  B  ), isoboles were generated for increments of 0.04 mm on the left and 0.02 mm on the right. In all panels,  arrows  depict the volatile anesthetic concentrations corresponding to minimum alveolar concentration (MAC)-awake. Note the regions of synergy (concave with respect to origin), additivity (linear with respect to origin), and antagonism (convex with respect to origin). Isoboles corresponding to MAC-awake concentrations of isoflurane (  solid lines  ) and halothane (  dashed lines  ) are depicted in  C  . Here, χ2was 0.325 for isoflurane and 0.335 for halothane at MAC-awake. See text for details. 
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Fig. 7. Isobolograms for isoflurane (  A  ) and halothane (  B  ) using network synchronization as the drug effect for combinations of decreases in T-type Ca2+channel conductance and increases in the time constant (τGABAA) governing γ-aminobutyric acid type A (GABAA) receptor deactivation that correspond to the actions of the volatile anesthetics. The isobolograms were generated using the data of  figure 6and the loci of  figure 3to define pharmacologically relevant trajectories. The line emerging from the origin is redrawn from  figure 3, and is the locus of the combined effects of the volatile anesthetic on both the T-type Ca2+channel and GABAAreceptor as volatile anesthetic concentration is increased. For each given volatile anesthetic concentration, the coherence was determined. Isoboles were then generated by connecting this point with points on each axis where modulation of that channel, by itself, would lead to the same network behavior as measured by the coherence. For isoflurane (  A  ), isoboles were generated for increments of 0.025 mm on the left and 0.0125 mm on the right. For halothane (  B  ), isoboles were generated for increments of 0.04 mm on the left and 0.02 mm on the right. In all panels,  arrows  depict the volatile anesthetic concentrations corresponding to minimum alveolar concentration (MAC)-awake. Note the regions of synergy (concave with respect to origin), additivity (linear with respect to origin), and antagonism (convex with respect to origin). Isoboles corresponding to MAC-awake concentrations of isoflurane (  solid lines  ) and halothane (  dashed lines  ) are depicted in  C  . Here, χ2was 0.325 for isoflurane and 0.335 for halothane at MAC-awake. See text for details. 
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The isobolograms for isoflurane and halothane, separately and together, are shown in figure 7. An isobole forming a straight line indicates additivity, whereas isoboles concave with respect to the origin indicate some degree of synergism and those convex with respect to the origin indicate some degree of antagonism.43 For the VAs, there is no single pattern of interaction individually or collectively. What is most apparent is that there is meaningful interaction over the range of simulations studied that is largely additive.
Mechanism of Interaction
The T-type Ca2+channel conductance contributes to the intrinsic properties of individual neurons, and the GABAAreceptor is responsible for network interactions. However, figure 6Aand the isobolograms of figure 7demonstrate that VA modulation of Ca2+channel conductance can also affect synchronization of the model network. Reductions in T-type Ca2+conductance will reduce membrane depolarization by the Ca2+channel and permit the inhibition produced by a given level of GABAAactivation to produce a relatively greater reduction in membrane potential than when it is applied while the Ca2+current is larger. The significance of this with respect to network synchronization is shown in figure 8, where a reduction in the T-type Ca2+conductance allows a GABA pulse to increase the resultant delay in the firing of a single neuron firing without network influence. The conductance from a single synapse from a presynaptic neuron in the network corresponds to approximately 0.002 μS. As shown in figure 8, as the GABAAreceptor conductance increases, so does the delay due to a single GABA pulse. By enhancing the effectiveness of an inhibitory pulse in preventing the firing of a postsynaptic neuron, the window in time where neural activity is more probable becomes smaller, making it more likely that activity will occur at more similar phases and synchronize. This effect exerts a greater impact on network synchrony once several neurons start to fire in concert (simulated in fig. 8by increases in conductance of the GABAAreceptor). When these results are considered within a larger network, it can be seen that reductions in T-type Ca2+conductance make the inhibitory interactions in the network effectively stronger, leading to greater network synchrony. However, the effect on synchronization of decreasing the T-type Ca2+conductance is constrained. As shown in figure 9, the average number of spikes per burst decreases as the T-type Ca2+conductance is decreased. This decrease in the burst length decreases the inhibition generated in the postsynaptic neuron because each action potential in the burst generates a synaptic pulse of GABA. However, some of the decrease in burst length as the T-type Ca2+conductance decreases is overcome by simultaneous arrival of multiple inhibitory pulses as network synchrony increases. Network coherence as the T-type Ca2+conductance is decreased in figure 6Acould be interpreted as a balance of the effects shown in figures 8 and 9.
Fig. 8. Decreasing T-type Ca2+conductance potentiates delay until next burst produced by a pulse of γ-aminobutyric acid (GABA) in individual model neurons isolated from the network. In the example in  A  , a burst consisting of a single spike is shown at the top for a single unperturbed neuron excited by steady current injection. Onset of the burst is delayed by a pulse of GABA (0.5 mm GABA, 0.3 ms duration, with GABAAconductance of 0.01 μS), applied at the  arrow  under control conditions (  left  ) and further delayed when the T-type Ca2+conductance is reduced to 0.8 of its control value. In  A  , the time axis was adjusted so that each neuron has a cycle length of 1.0 and the pulse was applied at 0.9, where 0 corresponds to the peak of the first spike. The delay as the T-type Ca2+conductance is decreased is systematically explored in  B  and  C  for increasing levels of the GABAAconductance. Here, 0.002 μS corresponds to approximately the conductance of a single model synapse. Absolute delays in burst onset with a pulse of GABAAare shown in  B  along with inset depicting how cycle length increases with decrements in T-type Ca2+conductance. Relative delays in cycle length are shown in  C  . Note how decrements in T-type Ca2+conductance increase delays, as does an increase in GABAAconductance, as would occur from multiple presynaptic pulses occurring in unison. 
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Fig. 8. Decreasing T-type Ca2+conductance potentiates delay until next burst produced by a pulse of γ-aminobutyric acid (GABA) in individual model neurons isolated from the network. In the example in  A  , a burst consisting of a single spike is shown at the top for a single unperturbed neuron excited by steady current injection. Onset of the burst is delayed by a pulse of GABA (0.5 mm GABA, 0.3 ms duration, with GABAAconductance of 0.01 μS), applied at the  arrow  under control conditions (  left  ) and further delayed when the T-type Ca2+conductance is reduced to 0.8 of its control value. In  A  , the time axis was adjusted so that each neuron has a cycle length of 1.0 and the pulse was applied at 0.9, where 0 corresponds to the peak of the first spike. The delay as the T-type Ca2+conductance is decreased is systematically explored in  B  and  C  for increasing levels of the GABAAconductance. Here, 0.002 μS corresponds to approximately the conductance of a single model synapse. Absolute delays in burst onset with a pulse of GABAAare shown in  B  along with inset depicting how cycle length increases with decrements in T-type Ca2+conductance. Relative delays in cycle length are shown in  C  . Note how decrements in T-type Ca2+conductance increase delays, as does an increase in GABAAconductance, as would occur from multiple presynaptic pulses occurring in unison. 
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Fig. 9. Role of T-type Ca2+conductance on burst length. As conductance decreases, the average number of spikes in a burst also decreases from those shown in the examples of  figure 1Band less directly in  figure 4A, effectively decreasing the strength of inhibitory network interactions because each spike produces a synaptic pulse of γ-aminobutyric acid (GABA) in the model. Note that as the number of spikes per burst plateaus, but decreases in Ca2+conductance continue to influence entrainment properties of the network  (fig. 8), there is a steep increase in the influence of Ca2+conductance on network coherence (  fig. 6A), and the isobolograms (  fig. 7) depict some synergistic behavior. The SE of the means from three separate simulations is 0.03 spikes or less for each of the points shown. 
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Fig. 9. Role of T-type Ca2+conductance on burst length. As conductance decreases, the average number of spikes in a burst also decreases from those shown in the examples of  figure 1Band less directly in  figure 4A, effectively decreasing the strength of inhibitory network interactions because each spike produces a synaptic pulse of γ-aminobutyric acid (GABA) in the model. Note that as the number of spikes per burst plateaus, but decreases in Ca2+conductance continue to influence entrainment properties of the network  (fig. 8), there is a steep increase in the influence of Ca2+conductance on network coherence (  fig. 6A), and the isobolograms (  fig. 7) depict some synergistic behavior. The SE of the means from three separate simulations is 0.03 spikes or less for each of the points shown. 
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Sensitivity of Model Assumptions
Because the results already presented could depend heavily on the assumptions and parameters that underlie the model, a number of these were examined. These included the level of bias current exciting each neuron, use of a randomly distributed bias current, the average fraction of neurons to which each neuron is connected, the conductance of the GABAAreceptor, the presence of the small conductance Ca2+-dependent K+channel (SK), and assumptions about VA effects on the ion channels. The bias current was adjusted so that the network would oscillate at approximately 10 Hz with the neurons displaying appropriate individual bursting patterns.33 For the network parameters used in the current study, network synchrony as determined by the coherence parameter introduced earlier varied biphasically with the bias current. As the amplitude of the bias current increased around the value of 0.12 nA, network synchrony increased with the magnitude of the bias current. When the bias current reached approximately 0.15 nA, network synchrony began to decline. Use of a random distribution of bias current amplitude altered network synchrony in no substantial way. As connectivity increased, so did network synchrony, and the value of 0.85 used for the fraction of total neurons to which each was connected represented a balance between starting with a poorly synchronized network and one that was so organized that VA-induced changes would be difficult to observe. Similarly, an increase in GABAAreceptor conductance effectively enhanced connectivity and, therefore, network synchrony. The value used in the current simulations represents a balance between asynchronous network behavior and one whose activity in the control state is already highly organized. Inclusion of SK did not appreciatively alter network behavior when VA effects were introduced. Therefore, as in a number of other models of the RTN, it was not incorporated.32,33 Furthermore, inclusion of the VA modulation of IPSP amplitude as depicted in figure 2Cdid not appreciatively alter network behavior over the range of VA concentrations studied and, therefore, was not further considered in the detailed studies.
Discussion
The simulations presented here demonstrate how relatively modest VA effects at T-type Ca2+channels and GABAAreceptors, individually and jointly, can alter the behavior of a computational model of the RTN. That effects at multiple sites can individually lead to similar alterations in network behavior may help to explain why point mutations of the GABAAreceptor, while largely abolishing some animal behaviors in the presence of IAs, minimally affected those observed in the presence of VAs.45 These VA effects enhanced the synchrony of the neurons in the network and, when considered jointly, the combination was largely additive with some deviations toward mild synergism or antagonism depending on VA concentration and the choice of VA. As demonstrated in figure 8, VA-induced decrements in the T-type Ca2+current potentiated the effects of GABA. This interaction was limited as VA concentration increased because this same VA effect on the T-type Ca2+current led to reductions in burst length, as shown in figure 9, and reduced the amount of inhibition generated by that burst. Overall, the VAs isoflurane and halothane, despite substantially different potencies with respect to immobility and consciousness, produced similar levels of network synchrony at VA concentrations corresponding to clinical measures of MAC-awake in humans.
These computational studies were motivated by the role of the thalamus in consciousness and evidence that thalamic behavior is altered at VA concentrations capable of producing unconsciousness.23,46–48 Implicit in the hypothesis that VAs produce unconsciousness by modulation of thalamic activity is the assumption that thalamic neurons are particularly vulnerable to the modulation of ion channel behavior produced by VAs. Thalamic neurons are somewhat unique in that their intrinsic behavior is dominated by low-voltage–activated Ca2+channels.32 GABAAinhibitory interactions contribute to network behavior, particularly in the RTN.29 Therefore, it is consistent with the hypothesis of thalamic involvement in VA-induced unconsciousness that VAs modulate the activity of both T-type Ca2+channels and GABAAreceptors (fig. 2). This hypothesis is further supported by the observation, also illustrated in figure 2, that VA sensitivity of the key ion channels of the thalamus is, overall, greater for isoflurane than for halothane. Although halothane is more potent with respect to immobility (MAC), isoflurane is more potent with respect to consciousness (MAC-awake), particularly when MAC-awake is expressed in terms of MAC (0.25 for isoflurane and 0.59 for halothane).15 Therefore, as shown in figure 3, MAC-awake concentrations of the two VAs lie close to one another when considering the joint VA effect on T-type Ca2+channels and GABAAreceptors. Nonetheless, as also seen in figure 3, the relative modulation of the thalamic ion channels at MAC-awake concentrations differs somewhat between the two VAs, suggesting that their joint impact on thalamic behavior should be considered. When this was done by quantifying the synchrony of network activity within the model RTN, the impact on network behavior was similar for the two VAs. Collectively, these results are consistent with clinical and laboratory data that identify the thalamus as a site of VA-induced unconsciousness, and that VA modulation of both voltage- and ligand-gated ion channels contributes.
The isobolographic analysis of figure 7demonstrated that the joint effect of the VAs on network synergy was largely additive, with some small deviations toward synergism or antagonism. As shown in figure 8, the ability of VA-induced decrements in the T-type Ca2+current to enhance network synchrony seems to lie in their ability to augment the impact of inhibitory pulses mediated by GABAAreceptors at times when the Ca2+current was active. Qualitative differences in the isobolograms likely reflects the different balance in effects on T-type Ca2+and GABAAcurrents illustrated for the two VAs in figures 2 and 3.
Values for MAC-awake differ somewhat as a function of the study and target population in those studies. For example, just as with MAC, MAC-awake decreases with age, though maintaining a similar proportion to MAC.49 The values used in the current study for MAC-awake (0.25 MAC for isoflurane and 0.59 MAC for halothane)15 have the merit of being determined in a single study. These values are similar to those from other studies (e.g.  , 0.55 MAC for halothane16 and 0.317–0.364 MAC for isoflurane17–20). The slightly smaller value for halothane and slightly larger ones for isoflurane than those used in the current study would lead to even greater similarity in the values of network coherence obtained for MAC-awake, as can be seen in figure 6F, where network coherence varies monotonically with VA concentration.
Suitability of Network Model
When compared with the entire brain or even the most detailed thalamocortical models,50 the model used in the current study is relatively simple. Enhancements might include interactions with other brain regions, multicompartment models of individual neurons that replicate the detailed dendritic structure of the neurons and the spatial distribution of ion channels over those dendrites, and a full complement of known ion channels. Although all models represent a compromise between absolute reality, availability of data to configure the model, solubility of the model, and the outstanding question the model is intended to answer, each of these candidate modifications are of potential significance.
The RTN modulates and, in turn, is modulated by a number of other brain regions,29 and is an important regulator of thalamic oscillations.36 The RTN inhibits thalamic relay neurons and receives excitatory input from them. Thalamic relay neurons project the majority of the input from sensory organs to the cerebral cortex, leading to excitation of the cerebral cortex, which then provides excitatory feedback to both the RTN and thalamic relay neurons. Apart from its impact on network interactions, excitation of thalamic neurons due to feedback from the cortex could substantially alter the intrinsic behavior of individual neurons by increasing membrane potential so that the T-type Ca2+channels can no longer de-inactivate. Therefore, it should not be surprising that models including all of these structures have demonstrated important interactions,51 that functional imaging demonstrates that sedative levels of VAs alter interactions between these structures,23,47 and that recordings from these structures during anesthesia demonstrate the importance of corticothalamic feedback loops.52 
Even if feedback from other brain structures did not substantially alter thalamic behavior, a more comprehensive model would still be important. First, thalamic electrical activity, as opposed to cortical electroencephalographic activity, is not generally accessible during clinical practice. Therefore, clinical correlates of model predications could require cortical electroencephalograms generated by a model that incorporated corticothalamic interactions. Second, even though the current model demonstrated an enhancement of network synchrony with increasing anesthetic concentrations and identified the level of model synchrony associated with MAC-awake concentrations of the VAs, it was not immediately clear why this level of synchrony might be associated with unconsciousness. A fundamental qualitative change in model behavior as VA concentration is increased, the bifurcation seen in more abstract models of anesthetic action,53 might only be observable in a model with a cortical component driven by an increasingly synchronized thalamus.
Although represented in the current model as a single compartment, neurons are complex structures with elaborate dendritic arbors, often requiring models with hundreds of compartments to capture this level of detail. Thalamic neurons are no exception, with models containing up to several hundred compartments.31,32 Such models have demonstrated the importance of dendritic anatomy in replicating the details of bursting in thalamic neurons. However, much of this detail can be captured in models with as few as three compartments using algorithms designed to generate reduced models from the anatomically more complex ones.32 Some of the fidelity of the three-compartment models to physiologically determined burst trajectories is lost when such models are reduced to a single compartment.32 Nonetheless, such models do replicate a considerable portion of the known physiology, particularly what is necessary to assess their contribution to network behavior. One of the major differences, though of limited impact with respect to the current study, which is driven by relative VA-induced decrements in conductances, is that larger values of the T-type Ca2+conductance are necessary in the reduced models. One potential limitation of using anatomically detailed models of individual neurons, apart from the computational burden this imposes, is that this requires additional assumptions regarding connectivity of proximal and distal compartments of their respective dendritic trees. Such detailed models would also require assumptions regarding the distribution of conductances along the dendritic tree.
Apart from the anatomic detail of individual neurons and their connections, the role of ion channels not included in the simulations needs to be considered. With respect to voltage-gated channels, the current model is somewhat parsimonious, using a T-type Ca2+channel, with fast Na+and K+channels to generate spikes. This combination of channels has been shown to be sufficient to generate realistic bursting behavior in models of individual neurons, particularly in multicompartment models.32,33 The small Ca2+-activated K+channel (SK) is known to contribute to the behavior of reticular neurons by producing an afterhyperpolarization that facilitates de-inactivation of the T-type Ca2+channels.29 It is conceivable that VA modulation of Ca2+entry via  the T-type Ca2+channels could decrease internal Ca2+concentration, leading to a decrease in the current produced by SK. However, inclusion of SK did not substantially alter network behavior for the VA concentrations studied here and, therefore, it was not included. Another conductance that is present in reticular neurons54 and depends on intracellular Ca2+concentration, but has slower kinetics than SK, is the calcium-activated nonspecific cation current.55 Through this channel, an increase in intracellular Ca2+concentration leads to depolarization and a shift from bursting to tonic spiking as the T-type Ca2+channel becomes inactivated. Again, this channel was not incorporated because large network simulations have not found it to be necessary,33 and because SK with an opposite but faster effect was also not incorporated. Importantly, neither of these conductances are known to be directly modulated by VAs.
Another class of ion channel not incorporated is the hyperpolarization-activated cyclic nucleotide-gated (HCN) channels, which have recently been shown to be present in reticular neurons.56 Because these channels seem to play a role in integrating signals from spatially disparate locations along the dendritic tree,57 their inclusion would likely be necessary when considering multicompartment models with realistic dendritic anatomy. Halothane is known to affect HCN channels,58 including HCN2, an isoform known to be present on reticular neurons.56,59 However, for the concentrations used in the current study, the effects of halothane on HCN2 are small,58 but of undetermined impact on model behavior. Members of the two-pore domain K+channel family can also be localized to the reticular nucleus60 and, of those, several are modulated by VAs.61,62 In contrast to VA effects on the T-type Ca2+channel, VAs increase K+conductance of the two-pore domain K+channels. However, the impact on network activity of VAs increasing the hyperpolarizing current produced by these channels may be complex, as indicated by recent experiments demonstrating how these channels can actually increase neural firing.63 
Although GABAAreceptors were incorporated in the model, extrasynaptic GABAAreceptors are also known to reside in the thalamus.64,65 In contrast to GABAAreceptors at the synapse, extrasynaptic GABAAreceptors generate tonic inhibitory currents.64–66 These inhibitory currents are enhanced by the IAs67,68 and the VA isoflurane.69 Therefore, inclusion of extrasynaptic GABAAreceptors in the model would introduce an inhibitory current that would increase with introduction of many types of IAs and VAs. This inhibitory current would effectively reduce the level of excitatory bias current in the model or, at higher levels, even produce a net inhibitory effect. As indicated in Results, this would actually reduce network synchrony because it would tend to reduce spiking during a burst, leading to less inhibition of other members of the network. However, because the response to the excitatory bias current is biphasic, for higher levels of excitatory input network the addition of an inhibitory current will cause network synchrony to increase. At higher levels of excitatory input, inhibition reduces burst frequency, making it easier to synchronize the network,70 particularly when the anesthetics are simultaneously increasing the deactivation time of the synaptic GABAAreceptors. These issues may also be important when considering the excitatory feedback to the RTN from thalamocortical relay neurons and the cortex.
VA Modulation of Ion Channel Behavior
The current studies are heavily dependent on data regarding VA modulation of multiple ion channels by multiple VAs. Of the data used in the current study, the greatest degree of uncertainty lay in the effect of halothane on GABAAreceptor activity because, as described in the figure and accompanying text, the function in figure 2was obtained from limited data pooled from multiple sources.8,34,35 It is conceivable that better data could lead to even greater similarity in behavior of isoflurane and halothane at MAC-awake concentrations. One aspect of VA effects on GABAAreceptors that was not incorporated into the principal simulations was the decrease in the amplitude of the IPSP produced by GABA and depicted in figure 2C. This effect differed little from unity over the range of anesthetics studied in the simulations presented here. When both VA-induced increases in the time constant governing deactivation of the GABAAchannel and the decrease in IPSP amplitude were simultaneously incorporated, for the range of VA concentrations under consideration, the impact on model coherence was not discernable from results obtained when only the effect related to the rate of GABAAdeactivation was included. Some of this relates to the fact that although equivalent from the standpoint of net inhibitory charge transfer, the product of the decrement in IPSP amplitude and increases in the time constant governing channel deactivation, these two effects are not equivalent in their impact on network behavior. Increases in the time constant governing GABAAreceptor deactivation is more effective at increasing network synchrony than an increase in IPSP amplitude that is equivalent with respect to inhibitory charge transfer (A. Gottschalk, M.D., Ph.D., and S. A. Miotke, M.S., Baltimore, Maryland, unpublished computer simulations, 2007), which is consistent with more general studies of how synaptic decay time affects network synchronization.70 However, net charge transfer may be more relevant for tonic, as opposed to phasic, inhibitory input as might occur from extrasynaptic GABAAreceptors in the thalamus.69 
Whether the results governing network behavior, like those in the upper right panels of figure 5and that of figure 6B, are sufficient to account for the behavior of the IAs that act principally at the GABAAreceptor remains to be established. Computational approaches have already demonstrated the importance of inhibitory interactions in network synchronization71–73 and established that increases in the time constant for GABAAreceptor deactivation can alter network behavior in computational models of the hippocampus,74 sensory cortex,72 and RTN.42 However, prolongation in the time constant for GABAAreceptor deactivation can also reduce synchronization,21,73 particularly in networks with higher intrinsic frequency.70 The data of figure 6Bindicate that a prolongation in the time constant for GABAAreceptor deactivation of approximately 50% achieves the same level of network synchrony as seen for MAC-awake concentrations of the VAs considered in this study. Although the available direct experimental data on GABAAkinetics at IA concentrations associated with loss of consciousness are sparse, this observation from the model is consistent with the relative changes in GABAAkinetics imputed from recordings in hippocampal slice preparations.74 In these studies, ED50concentrations of propofol (2.8 μg/ml)75 and sodium thiopental (15.6 μg/ml)76 for loss of consciousness are associated with an approximately 50% increase in the time constant for GABAAreceptor deactivation. However, the IAs do more than prolong GABAAdeactivation. Like the VAs, the IAs can decrease the amplitude of GABAAreceptor IPSPs.77,78 The quantitative extent that decreases in IPSP amplitude associated with the IAs affect network behavior was not determined, but these effects would be expected to decrease network synchrony to some extent. Importantly, other IA effects could conceivably play a critical role, as suggested by animal studies where point mutations of the GABAAreceptor abolish the response to noxious stimuli and reduce but do not abolish the loss of righting reflex.45 For example, propofol may inhibit SK activity in neurons of the RTN,79 leading to enhanced excitability of these neurons. Propofol can also block HCN channels in neurons of the RTN, enhancing the impact of postsynaptic potentials at the soma, but also interfering with the integrative properties of the dendritic network.59 Therefore, to more fully account for IA effects, model enhancements may require not only the inclusion of additional channels such as SK and one or more HCN subtypes, but also a more realistic dendritic arbor to address the impact of HCN modulation on the integrative properties of the dendrites.
Characterizing Network Behavior
To construct the isobolograms and to evaluate the effect of parameter variations in general, it was necessary to be able to characterize network behavior with a single continuous parameter with some degree of physiologic relevance. The choice of network synchrony and the specific synchrony measure require explanation. It is synchronization of the electrical activity of multiple neurons that leads to the organized pattern of electrical activity seen with electroencephalography and with depth electrodes. Such organization contributes to the slowing and increased amplitude of the waveform that characterizes the electroencephalogram as anesthetic depth is increased.22 However, loss of synchronized activity among neurons of higher-frequency ranges (e.g.  , gamma) could also contribute to slowing of the electroencephalogram and is known to occur in hippocampal slice with IAs.80 The process whereby anesthetic-induced alterations in the organization of thalamic activity could alter the cortical electroencephalogram is complex and seems to depend on corticothalamic feedback loops.48,51 Therefore, an appreciation for this process will minimally require more elaborate models that incorporate these elements. Nonetheless, although many variations are known, synchronization of neural activity with administration of general anesthetics is an empiric observation.
There are many theoretical and practical issues involved in the choice of a synchrony measure which preclude the existence of an ideal measure.81 One, used in many theoretical studies of coupled sinusoidal oscillators, examines the phase of each oscillator by representing it as a point on the unit circle in the complex plane.82 Network synchrony is determined by essentially taking the average distance from the center of the unit circle. This straightforward approach underlies the histograms of relative phase given in figure 5. However, there are several potential pitfalls. The first is that highly organized subgroups that are 180° out of phase will seem no more organized than a system with random phase. Such issues can be readily addressed, but examination of the histograms of relative phase in the current study indicated that this was not an overwhelming issue. The second is that by considering phase only, a large number of low-amplitude signals could dominate the synchrony measure. The synchrony measure used in the current study42 effectively considers both phase and amplitude information about the individual neural trajectories. Moreover, it does so over the entire frequency range of the neuron's trajectory, rather than just at a dominant or otherwise chosen frequency. Eventually, rather than using such abstract measures, models should be directly comparable with experimentally determined electroencephalograms.
Conclusions
Despite limitations on the data characterizing the interaction of multiple VAs at multiple receptors, the imprecision of clinical endpoints such as MAC-awake, the use of a somewhat parsimonious model of a single thalamic nucleus, and the use of an abstract measure to characterize model behavior, a somewhat coherent picture emerges from the foregoing studies. These studies demonstrated how relatively small, but established, VA effects at multiple sites in a neurobiologically relevant location of the central nervous system could combine in a predominately additive fashion to alter model behavior at VA concentrations corresponding to MAC-awake. The appropriateness of the choice of model, the adequacy of the data used to drive the model, and the utility of the measure of model behavior are supported by the ability to obtain a similar effect with the model when using MAC-awake concentrations of VAs with markedly different clinical potencies. It is expected that the picture painted by modeling studies will become clearer and more relevant as additional laboratory data about VA interaction with ion channels become available, models are expanded to consider additional anesthetic effects due to both VAs and IAs, and such models generate outputs that can readily be contrasted with data obtained in vivo  .
The authors thank Peter A. Goldstein, M.D. (Associate Professor, Department of Anesthesiology, Weill Cornell Medical College, New York, New York), and Marek A. Mirski, M.D., Ph.D. (Associate Professor, Department of Anesthesiology and Critical Care Medicine, Johns Hopkins Hospital, Baltimore, Maryland), for their many valuable suggestions.
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Fig. 1. Schematic of computational model (  A  ) and representative examples of membrane potentials from individuals network neurons (  B  ). Single compartment models of neurons of the reticular nucleus of the thalamus are coupled by inhibitory γ-aminobutyric acid type A (GABAA) synapses. Each neuron contains a T-type Ca2+channel, a fast Na+channel, a delayed rectifier K+channel, a leak current, and a membrane capacitance. The Ca2+current modulates internal Ca2+concentration, which, in turn, modulates Ca2+reversal potential. Each of the 100 neurons in the model is randomly coupled, on average, to 85% of the other neurons in the network (excluding self) with a fixed value of GABAAconductance. Neural activity generally consists of bursts of action potentials riding on the depolarization produced by the T-type Ca2+channel. Bursting occurs in response to postinhibitory rebound of the membrane potential.  32,33,42 Each action potential in a burst generates a synaptic pulse of GABA. Note how some neurons are in phase whereas others are not. Also note the presence of subthreshold oscillations in membrane potential due to inhibitory input from other neurons in the network. 
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Fig. 1. Schematic of computational model (  A  ) and representative examples of membrane potentials from individuals network neurons (  B  ). Single compartment models of neurons of the reticular nucleus of the thalamus are coupled by inhibitory γ-aminobutyric acid type A (GABAA) synapses. Each neuron contains a T-type Ca2+channel, a fast Na+channel, a delayed rectifier K+channel, a leak current, and a membrane capacitance. The Ca2+current modulates internal Ca2+concentration, which, in turn, modulates Ca2+reversal potential. Each of the 100 neurons in the model is randomly coupled, on average, to 85% of the other neurons in the network (excluding self) with a fixed value of GABAAconductance. Neural activity generally consists of bursts of action potentials riding on the depolarization produced by the T-type Ca2+channel. Bursting occurs in response to postinhibitory rebound of the membrane potential.  32,33,42 Each action potential in a burst generates a synaptic pulse of GABA. Note how some neurons are in phase whereas others are not. Also note the presence of subthreshold oscillations in membrane potential due to inhibitory input from other neurons in the network. 
×
Fig. 2. Volatile anesthetic effects on T-type Ca2+channels and γ-aminobutyric acid type A (GABAA) receptors for both isoflurane (  solid line  ) and halothane (  dashed line  ). The normalized decrease in T-type Ca2+channel conductance is shown as a function of volatile anesthetic concentration in  A  . The values of the Hill exponent and semisaturation constant used in  equation 7are, respectively, n = 2.3 (±0.15) and [A]1/2= 0.30 (±0.01) for isoflurane, and n = 2.0 (±0.10) and [A]1/2= 0.66 (±0.03) for halothane.  6 The relative increase in the time constant governing deactivation of the GABAAreceptor as anesthetic concentration is increased is shown in  B  . Here, the maximum prolongation of the time constant, the Hill exponent, and the semisaturation constant used in  equation 8are, respectively, fmax= 4.7 (±0.50), n = 2.7 (±1.2), and [A]1/2= 0.32 (±0.05) for isoflurane,  8 and fmax= 2.1 (±0.51), n = 1.5 (±1.0), and [A]1/2= 0.36 (±0.26) for halothane.  8,34,35 The decrease in the amplitude of the inhibitory postsynaptic potential (IPSP) at the GABAAreceptor as anesthetic concentration is increased is shown in  C  . The values of the Hill exponent, semisaturation constant, and maximum IPSP attenuation used in  equation 9are, respectively, n = 2.6 (±0.80), [A]1/2= 0.79 (±0.24), and ymax= 0.56 (±0.13) for isoflurane,  8 and n = 1.9 (±1.0), [A]1/2= 1.0 (±0.36), and ymax= 0.72 for halothane, where the value for ymaxfor halothane was obtained from a single point.  8,35 
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Fig. 2. Volatile anesthetic effects on T-type Ca2+channels and γ-aminobutyric acid type A (GABAA) receptors for both isoflurane (  solid line  ) and halothane (  dashed line  ). The normalized decrease in T-type Ca2+channel conductance is shown as a function of volatile anesthetic concentration in  A  . The values of the Hill exponent and semisaturation constant used in  equation 7are, respectively, n = 2.3 (±0.15) and [A]1/2= 0.30 (±0.01) for isoflurane, and n = 2.0 (±0.10) and [A]1/2= 0.66 (±0.03) for halothane.  6 The relative increase in the time constant governing deactivation of the GABAAreceptor as anesthetic concentration is increased is shown in  B  . Here, the maximum prolongation of the time constant, the Hill exponent, and the semisaturation constant used in  equation 8are, respectively, fmax= 4.7 (±0.50), n = 2.7 (±1.2), and [A]1/2= 0.32 (±0.05) for isoflurane,  8 and fmax= 2.1 (±0.51), n = 1.5 (±1.0), and [A]1/2= 0.36 (±0.26) for halothane.  8,34,35 The decrease in the amplitude of the inhibitory postsynaptic potential (IPSP) at the GABAAreceptor as anesthetic concentration is increased is shown in  C  . The values of the Hill exponent, semisaturation constant, and maximum IPSP attenuation used in  equation 9are, respectively, n = 2.6 (±0.80), [A]1/2= 0.79 (±0.24), and ymax= 0.56 (±0.13) for isoflurane,  8 and n = 1.9 (±1.0), [A]1/2= 1.0 (±0.36), and ymax= 0.72 for halothane, where the value for ymaxfor halothane was obtained from a single point.  8,35 
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Fig. 3. Loci of volatile anesthetic effects for decrements in conductance of T-type Ca2+channels and increase in the time constant (τGABAA) governing γ-aminobutyric acid type A (GABAA) receptor deactivation for both isoflurane and halothane as volatile anesthetic concentration is increased.  Symbols  (•= isoflurane, ○= halothane) correspond to increments of 0.02 mm volatile anesthetic concentration.  Arrows  denote locations of MAC-awake (0.075 mm = 0.25 MAC for isoflurane and 0.12 mm = 0.59 MAC for halothane),  15 where MAC is the minimum alveolar concentration at which 50% of subjects remain immobile to incision (0.3 mm for isoflurane and 0.2 mm for halothane). The fractional decrease in T-type Ca2+conductance is shown on the  horizontal axis  , and the fractional increase of the time constant (τGABAA) governing GABAAreceptor deactivation is shown on the  vertical axis  . The  origin  corresponds to an anesthetic concentration of zero. 
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Fig. 3. Loci of volatile anesthetic effects for decrements in conductance of T-type Ca2+channels and increase in the time constant (τGABAA) governing γ-aminobutyric acid type A (GABAA) receptor deactivation for both isoflurane and halothane as volatile anesthetic concentration is increased.  Symbols  (•= isoflurane, ○= halothane) correspond to increments of 0.02 mm volatile anesthetic concentration.  Arrows  denote locations of MAC-awake (0.075 mm = 0.25 MAC for isoflurane and 0.12 mm = 0.59 MAC for halothane),  15 where MAC is the minimum alveolar concentration at which 50% of subjects remain immobile to incision (0.3 mm for isoflurane and 0.2 mm for halothane). The fractional decrease in T-type Ca2+conductance is shown on the  horizontal axis  , and the fractional increase of the time constant (τGABAA) governing GABAAreceptor deactivation is shown on the  vertical axis  . The  origin  corresponds to an anesthetic concentration of zero. 
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Fig. 4. Synchronization of model network activity by the volatile anesthetic actions of  figure 2. Raster plots depicting the activity of all 100 neurons in the network are shown for control conditions (  A  ) and at twice minimum alveolar concentration (MAC)-awake for two different volatile anesthetics (0.15 mm isoflurane [  B  ] and 0.24 mm halothane [  C  ]). Volatile anesthetic effects included decreasing T-type Ca2+channel conductance and an increase in the time constant (τGABAA) governing γ-aminobutyric acid type A (GABAA) receptor deactivation. The activity of each of the 100 neurons in the network is depicted over time by continuously mapping membrane potential to grayscale intensity, where darker intensity denotes larger values of membrane potential. Note the ability of clinically relevant volatile anesthetic concentrations to synchronize network activity (  e.g.  , more membrane activity at a given phase) and that pattern of activity seems more organized for isoflurane (  B  ) than for halothane (  C  ). 
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Fig. 4. Synchronization of model network activity by the volatile anesthetic actions of  figure 2. Raster plots depicting the activity of all 100 neurons in the network are shown for control conditions (  A  ) and at twice minimum alveolar concentration (MAC)-awake for two different volatile anesthetics (0.15 mm isoflurane [  B  ] and 0.24 mm halothane [  C  ]). Volatile anesthetic effects included decreasing T-type Ca2+channel conductance and an increase in the time constant (τGABAA) governing γ-aminobutyric acid type A (GABAA) receptor deactivation. The activity of each of the 100 neurons in the network is depicted over time by continuously mapping membrane potential to grayscale intensity, where darker intensity denotes larger values of membrane potential. Note the ability of clinically relevant volatile anesthetic concentrations to synchronize network activity (  e.g.  , more membrane activity at a given phase) and that pattern of activity seems more organized for isoflurane (  B  ) than for halothane (  C  ). 
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Fig. 5. Characteristics of the network synchronization by volatile anesthetics depicted in  figure 4for both 0.15 mm isoflurane (  A  ) and 0.24 mm halothane (  B  ). Model behavior is shown for control (no volatile anesthetic,  upper left panel  ), VA-induced decrease in T-type Ca2+channel conductance only (  lower left panel  ), volatile anesthetic–induced increase in the time constant (τGABAA) governing γ-aminobutyric acid type A (GABAA) receptor deactivation only (  upper right panel  ), and actions at both of these sites (  lower right panel  ). For each condition, the field potential of the network, a numeric measure of network synchrony (coherence, χ2), and a histogram of relative phase are shown. Note how volatile anesthetic actions at either the voltage-gated site or ligand-gated site leads to increasingly organized network behavior with each of these measures, and that organized behavior seems to be the greatest for action at both sites. Note also that although the histogram for halothane when both channels are considered demonstrates greater similarity of phase in the histogram than isoflurane, the χ2value is lower. Here, the level of synchrony as reflected by χ2is more consistent with what is seen in  figure 4and emphasizes the importance of using a synchrony measure such as χ2that considers both phase and amplitude at all frequencies, as highlighted in the Discussion in the section Characterizing Network Behavior. Scales for both voltage and time are given in the  upper left panel  of each. See text for details. 
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Fig. 5. Characteristics of the network synchronization by volatile anesthetics depicted in  figure 4for both 0.15 mm isoflurane (  A  ) and 0.24 mm halothane (  B  ). Model behavior is shown for control (no volatile anesthetic,  upper left panel  ), VA-induced decrease in T-type Ca2+channel conductance only (  lower left panel  ), volatile anesthetic–induced increase in the time constant (τGABAA) governing γ-aminobutyric acid type A (GABAA) receptor deactivation only (  upper right panel  ), and actions at both of these sites (  lower right panel  ). For each condition, the field potential of the network, a numeric measure of network synchrony (coherence, χ2), and a histogram of relative phase are shown. Note how volatile anesthetic actions at either the voltage-gated site or ligand-gated site leads to increasingly organized network behavior with each of these measures, and that organized behavior seems to be the greatest for action at both sites. Note also that although the histogram for halothane when both channels are considered demonstrates greater similarity of phase in the histogram than isoflurane, the χ2value is lower. Here, the level of synchrony as reflected by χ2is more consistent with what is seen in  figure 4and emphasizes the importance of using a synchrony measure such as χ2that considers both phase and amplitude at all frequencies, as highlighted in the Discussion in the section Characterizing Network Behavior. Scales for both voltage and time are given in the  upper left panel  of each. See text for details. 
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Fig. 6. Network coherence (χ2) as channels in the model are modulated one type at a time, and when these effects are considered together for the volatile anesthetics isoflurane and halothane. Network coherence is shown for decrements in the conductance of the T-type Ca2+channel (  A  ), and as the time constant (τGABAA) governing the rate of γ-aminobutyric acid type A (GABAA) receptor deactivation is increased (  B  ). Network coherence was also determined as a function of isoflurane concentration (  C  ) and halothane concentration (  D  ) using the combined effects of each volatile anesthetic on the T-type Ca2+conductance and the time constant (τGABAA) governing the rate GABAAreceptor deactivation that is depicted in  figures 2 and 3. Each  point  is the mean (±SEM) value of χ2obtained from either 6 (  B  ) or 12 (  A, C,  and  D  ) simulations differing only in their randomly assigned pattern of connectivity and timing of initial current pulses to the network (see Materials and Methods). Each curve was obtained from a least-squares fit of the simulation data to the equation f(x) = V/(1 + e−g(x)) + k, where g(x) = a0+ a1x + a2x2+… adxd, where the degree d was either 5 (  A–C  ) or 4 (  D  ). The data from  C  and  D  are plotted jointly in  E  but expressed in multiples of minimum alveolar concentration (MAC) for both isoflurane (MAC = 0.3 mm) and halothane (MAC = 0.2 mm), which is rescaled (  F  ) to demonstrate the similarity of the coherence for each volatile anesthetic's MAC-awake concentration (0.25 MAC for isoflurane and 0.59 MAC for halothane). Here, χ2was 0.325 for isoflurane and 0.335 for halothane at the MAC-awake concentrations of these volatile anesthetics. Note that the control value of χ2is 0.295. For MAC-awake concentrations of the volatile anesthetics acting on one channel at a time, isoflurane increases χ2to 0.304 when affecting the GABAAreceptor alone and 0.320 when affecting the T-type Ca2+channel alone. Similarly, halothane increases χ2to 0.316 when acting only at either the GABAAreceptor or T-type Ca2+channel. 
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Fig. 6. Network coherence (χ2) as channels in the model are modulated one type at a time, and when these effects are considered together for the volatile anesthetics isoflurane and halothane. Network coherence is shown for decrements in the conductance of the T-type Ca2+channel (  A  ), and as the time constant (τGABAA) governing the rate of γ-aminobutyric acid type A (GABAA) receptor deactivation is increased (  B  ). Network coherence was also determined as a function of isoflurane concentration (  C  ) and halothane concentration (  D  ) using the combined effects of each volatile anesthetic on the T-type Ca2+conductance and the time constant (τGABAA) governing the rate GABAAreceptor deactivation that is depicted in  figures 2 and 3. Each  point  is the mean (±SEM) value of χ2obtained from either 6 (  B  ) or 12 (  A, C,  and  D  ) simulations differing only in their randomly assigned pattern of connectivity and timing of initial current pulses to the network (see Materials and Methods). Each curve was obtained from a least-squares fit of the simulation data to the equation f(x) = V/(1 + e−g(x)) + k, where g(x) = a0+ a1x + a2x2+… adxd, where the degree d was either 5 (  A–C  ) or 4 (  D  ). The data from  C  and  D  are plotted jointly in  E  but expressed in multiples of minimum alveolar concentration (MAC) for both isoflurane (MAC = 0.3 mm) and halothane (MAC = 0.2 mm), which is rescaled (  F  ) to demonstrate the similarity of the coherence for each volatile anesthetic's MAC-awake concentration (0.25 MAC for isoflurane and 0.59 MAC for halothane). Here, χ2was 0.325 for isoflurane and 0.335 for halothane at the MAC-awake concentrations of these volatile anesthetics. Note that the control value of χ2is 0.295. For MAC-awake concentrations of the volatile anesthetics acting on one channel at a time, isoflurane increases χ2to 0.304 when affecting the GABAAreceptor alone and 0.320 when affecting the T-type Ca2+channel alone. Similarly, halothane increases χ2to 0.316 when acting only at either the GABAAreceptor or T-type Ca2+channel. 
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Fig. 7. Isobolograms for isoflurane (  A  ) and halothane (  B  ) using network synchronization as the drug effect for combinations of decreases in T-type Ca2+channel conductance and increases in the time constant (τGABAA) governing γ-aminobutyric acid type A (GABAA) receptor deactivation that correspond to the actions of the volatile anesthetics. The isobolograms were generated using the data of  figure 6and the loci of  figure 3to define pharmacologically relevant trajectories. The line emerging from the origin is redrawn from  figure 3, and is the locus of the combined effects of the volatile anesthetic on both the T-type Ca2+channel and GABAAreceptor as volatile anesthetic concentration is increased. For each given volatile anesthetic concentration, the coherence was determined. Isoboles were then generated by connecting this point with points on each axis where modulation of that channel, by itself, would lead to the same network behavior as measured by the coherence. For isoflurane (  A  ), isoboles were generated for increments of 0.025 mm on the left and 0.0125 mm on the right. For halothane (  B  ), isoboles were generated for increments of 0.04 mm on the left and 0.02 mm on the right. In all panels,  arrows  depict the volatile anesthetic concentrations corresponding to minimum alveolar concentration (MAC)-awake. Note the regions of synergy (concave with respect to origin), additivity (linear with respect to origin), and antagonism (convex with respect to origin). Isoboles corresponding to MAC-awake concentrations of isoflurane (  solid lines  ) and halothane (  dashed lines  ) are depicted in  C  . Here, χ2was 0.325 for isoflurane and 0.335 for halothane at MAC-awake. See text for details. 
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Fig. 7. Isobolograms for isoflurane (  A  ) and halothane (  B  ) using network synchronization as the drug effect for combinations of decreases in T-type Ca2+channel conductance and increases in the time constant (τGABAA) governing γ-aminobutyric acid type A (GABAA) receptor deactivation that correspond to the actions of the volatile anesthetics. The isobolograms were generated using the data of  figure 6and the loci of  figure 3to define pharmacologically relevant trajectories. The line emerging from the origin is redrawn from  figure 3, and is the locus of the combined effects of the volatile anesthetic on both the T-type Ca2+channel and GABAAreceptor as volatile anesthetic concentration is increased. For each given volatile anesthetic concentration, the coherence was determined. Isoboles were then generated by connecting this point with points on each axis where modulation of that channel, by itself, would lead to the same network behavior as measured by the coherence. For isoflurane (  A  ), isoboles were generated for increments of 0.025 mm on the left and 0.0125 mm on the right. For halothane (  B  ), isoboles were generated for increments of 0.04 mm on the left and 0.02 mm on the right. In all panels,  arrows  depict the volatile anesthetic concentrations corresponding to minimum alveolar concentration (MAC)-awake. Note the regions of synergy (concave with respect to origin), additivity (linear with respect to origin), and antagonism (convex with respect to origin). Isoboles corresponding to MAC-awake concentrations of isoflurane (  solid lines  ) and halothane (  dashed lines  ) are depicted in  C  . Here, χ2was 0.325 for isoflurane and 0.335 for halothane at MAC-awake. See text for details. 
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Fig. 8. Decreasing T-type Ca2+conductance potentiates delay until next burst produced by a pulse of γ-aminobutyric acid (GABA) in individual model neurons isolated from the network. In the example in  A  , a burst consisting of a single spike is shown at the top for a single unperturbed neuron excited by steady current injection. Onset of the burst is delayed by a pulse of GABA (0.5 mm GABA, 0.3 ms duration, with GABAAconductance of 0.01 μS), applied at the  arrow  under control conditions (  left  ) and further delayed when the T-type Ca2+conductance is reduced to 0.8 of its control value. In  A  , the time axis was adjusted so that each neuron has a cycle length of 1.0 and the pulse was applied at 0.9, where 0 corresponds to the peak of the first spike. The delay as the T-type Ca2+conductance is decreased is systematically explored in  B  and  C  for increasing levels of the GABAAconductance. Here, 0.002 μS corresponds to approximately the conductance of a single model synapse. Absolute delays in burst onset with a pulse of GABAAare shown in  B  along with inset depicting how cycle length increases with decrements in T-type Ca2+conductance. Relative delays in cycle length are shown in  C  . Note how decrements in T-type Ca2+conductance increase delays, as does an increase in GABAAconductance, as would occur from multiple presynaptic pulses occurring in unison. 
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Fig. 8. Decreasing T-type Ca2+conductance potentiates delay until next burst produced by a pulse of γ-aminobutyric acid (GABA) in individual model neurons isolated from the network. In the example in  A  , a burst consisting of a single spike is shown at the top for a single unperturbed neuron excited by steady current injection. Onset of the burst is delayed by a pulse of GABA (0.5 mm GABA, 0.3 ms duration, with GABAAconductance of 0.01 μS), applied at the  arrow  under control conditions (  left  ) and further delayed when the T-type Ca2+conductance is reduced to 0.8 of its control value. In  A  , the time axis was adjusted so that each neuron has a cycle length of 1.0 and the pulse was applied at 0.9, where 0 corresponds to the peak of the first spike. The delay as the T-type Ca2+conductance is decreased is systematically explored in  B  and  C  for increasing levels of the GABAAconductance. Here, 0.002 μS corresponds to approximately the conductance of a single model synapse. Absolute delays in burst onset with a pulse of GABAAare shown in  B  along with inset depicting how cycle length increases with decrements in T-type Ca2+conductance. Relative delays in cycle length are shown in  C  . Note how decrements in T-type Ca2+conductance increase delays, as does an increase in GABAAconductance, as would occur from multiple presynaptic pulses occurring in unison. 
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Fig. 9. Role of T-type Ca2+conductance on burst length. As conductance decreases, the average number of spikes in a burst also decreases from those shown in the examples of  figure 1Band less directly in  figure 4A, effectively decreasing the strength of inhibitory network interactions because each spike produces a synaptic pulse of γ-aminobutyric acid (GABA) in the model. Note that as the number of spikes per burst plateaus, but decreases in Ca2+conductance continue to influence entrainment properties of the network  (fig. 8), there is a steep increase in the influence of Ca2+conductance on network coherence (  fig. 6A), and the isobolograms (  fig. 7) depict some synergistic behavior. The SE of the means from three separate simulations is 0.03 spikes or less for each of the points shown. 
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Fig. 9. Role of T-type Ca2+conductance on burst length. As conductance decreases, the average number of spikes in a burst also decreases from those shown in the examples of  figure 1Band less directly in  figure 4A, effectively decreasing the strength of inhibitory network interactions because each spike produces a synaptic pulse of γ-aminobutyric acid (GABA) in the model. Note that as the number of spikes per burst plateaus, but decreases in Ca2+conductance continue to influence entrainment properties of the network  (fig. 8), there is a steep increase in the influence of Ca2+conductance on network coherence (  fig. 6A), and the isobolograms (  fig. 7) depict some synergistic behavior. The SE of the means from three separate simulations is 0.03 spikes or less for each of the points shown. 
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