RESEARCH ARTICLE Effects of the anesthetic agent propofol on neural populations Axel Hutt �� Andre Longtin Received: 4 February 2009 / Revised: 29 August 2009 / Accepted: 31 August 2009 �� Springer Science+Business Media B.V. 2009 Abstract The neuronal mechanisms of general anesthesia are still poorly understood. Besides several characteristic features of anesthesia observed in experiments, a promi- nent effect is the bi-phasic change of power in the observed electroencephalogram (EEG), i.e. the initial increase and subsequent decrease of the EEG-power in several fre- quency bands while increasing the concentration of the anaesthetic agent. The present work aims to derive ana- lytical conditions for this bi-phasic spectral behavior by the study of a neural population model. This model describes mathematically the effective membrane potential and involves excitatory and inhibitory synapses, excitatory and inhibitory cells, nonlocal spatial interactions and a finite axonal conduction speed. The work derives conditions for synaptic time constants based on experimental results and gives conditions on the resting state stability. Further the power spectrum of Local Field Potentials and EEG gen- erated by the neural activity is derived analytically and allow for the detailed study of bi-spectral power changes. We find bi-phasic power changes both in monostable and bistable system regime, affirming the omnipresence of bi-spectral power changes in anesthesia. Further the work gives conditions for the strong increase of power in the d-frequency band for large propofol concentrations as observed in experiments. Keywords General anesthesia Neural fields EEG Power spectrum Introduction General anesthesia (GA) is an indispensible tool in today���s medical surgery. In the optimal case, it leads to the patients immobility, amnesia and unconsciousness, i.e. lack of awareness towards external stimuli (Orser 2007 John and Prichep 2005). Although GA is omnipresent in recent medicine, its underlying mechanisms and the molecular action of anesthetic agents (AA) have been a long-standing mystery. One of the major obstacles towards its under- standing is the occurrence of different effects. For instance, immobility is assumed to be generated in the spinal cord (Rampil and King 1996), and the dorsolateral prefrontal cortex and the thalamus are affected during amnesia (Veselis et al. 1997). Similarly the underlying mechanism of the loss of consciousness and its spatial location is unknown though some studies point out the importance of the thalamus (Carstens and Antognini 2005 Alkire et al. 2008 Stienen et al. 2008). The present work focusses on the loss of consciousness (LOC) and aims to model cor- responding experimental findings. To learn more about the effects of AAs, the pharmaco- kinetics of AA have attracted some attention in the last decades (Forrest et al. 1994 Dutta et al. 1997 Franks 2008), i.e. the binding of the agent molecule to the blood and the effective concentration at the neural site. It has been shown that the speed of the AAs experimental administra- tion strongly affects the blood concentration and the effect- site concentration of the AAs. In other words the blood concentration of AA and its concentration at the effect site in the neural tissue may be different and may obey different A. Hutt (&) INRIA CR Nancy - Grand Est, CS20101, 54603 Villers-ls-Nancy Cedex, France e-mail: axel.hutt@loria.fr A. Longtin Department of Physics, University of Ottawa, 150 Louis Pasteur, Ottawa, ON K1N-6N5, Canada 123 Cogn Neurodyn DOI 10.1007/s11571-009-9092-2

temporal dynamics. These differences may yield hysteresis effects in the anesthetic action (Dutta et al. 1997). More recent studies examined the direct action of AA on single neurons (Antkowiak 1999 Franks and Lieb 1994) and synaptic and extrasynaptic receptors (Franks 2008 Hem- mings Jr. et al. 2005 Orser 2007 Bai et al. 1999 Alkire et al. 2008). In this context one of the most important findings is the AAs weakening action on excitatory synaptic receptors and the enhancement of inhibitory synaptic activ- ity. For instance, the AA ketamine inhibits synaptic NMDA- receptors, while the AA propofol enhances the action of inhibitory GABAA synapses (Franks 2008). In addition to these studies of microscopic actions, much research has been devoted to macroscopic effects of AA, such as the cardiovascular response of subjects to AAs (Mustola et al. 2003 Musizza et al. 2007) and the power spectrum of the subjects��� resting electroencephalogram (EEG) as a function of the blood concentration of AAs (Forrest et al. 1994 Dutta et al. 1997 Kuizenga et al. 2001 Fell et al. 2005 Han et al. 2005). The resting EEG power spectrum especially reflects the anesthetic action in a characteristic way and permits the classification of the depth of anesthesia by so-called monitors, see e.g. the review of Antkowiak (2002). These monitors are also used to pinpoint the LOC. Considering the action of the AA propofol, increasing its blood concentration first increases and then decreases the spectral power in most frequencies up to the gamma-range (0-40 Hz). This bi-phasic behavior is characteristic for GA and has been observed both in rats (Dutta et al. 1997) and in human subjects (Kuizenga et al. 1998 Yang et al. 1995). Interestingly some studies reported LOC during the power increase in the EEG (Kuizenga et al. 1998, 2001), while most monitors use the decay-phase of the bi-phasic power changes as indicator for LOC. The present work aims to describe mathematically this bi-phasic behavior by a neuronal population model. Our work focusses on the action of propofol, which is a widely-applied anesthetic agent (Marik 2004). It affects the cognitive abilities of subjects, such as the response to auditory stimuli (Kuizenga et al. 2001) or pain (Andrews et al. 1997). It acts mainly on GABAA receptors and hence changes the response of inhibitory synapses, while NMDA- and non-NMDA excitatory receptors are insignificantly affected. Increasing the blood concentration of propofol increases the charge transfer in synaptic GABAA-receptors and increases the decay time constant of their synaptic response function (Kitamura et al. 2002). We point out that the present work is not limited to the action of propofol and may be applied to the action of other anesthetic agents. The question arises whether the resulting anesthetic effect originates from the action of a population of neurons in a single brain area or whether GA is a network effect, i.e. results from the interaction of several brain areas. In the following we discuss briefly this question. On one hand it is well-known that single brain areas play an important role, such as the thalamus (Carstens and Antognini 2005 Alkire et al. 2008) which generates spindle waves close to the point of LOC. Since the thalamus is the gateway for sensory information in the brain, GA appears as a network effect mainly triggered by the thalamic action. On the other hand GABAA-receptors play an important role in the anesthetic action and are present in most cortical areas and some subcortical areas. Hence there is no unique action site of propofol this may relate to the fact that the spatial location of the anesthetic action is still unknown, see e.g. studies on cortical neurons (McKernan et al. 1997) and thalamic relay neurons (Ying and Goldstein 2005). Consequently GA may represent an unspecific action on neural populations. This view is fostered by invitro experiments on cortical slices while applying anesthetic agents. Such experiments showed that the firing rates of neurons decreased during the administration of an increased concentration of the AA (Antkowiak 1999, 2002) similar to neural effects observed in invivo experiments. These findings indicate that anes- thetic effects may occur in a single brain area and network interactions might not be necessary for their occurrence. Moreover the presence of a global heterogeneous network involving brain areas with specific actions may result in an EEG with spatially localized activity regions. However John and Prichep (2005) measured the EEG during the administration of propofol and found no spatial structure. Consequently these findings indicate that the anesthetic action is rather unspecific to brain areas and it is reasonable to treat a single brain area as a first approximation. Besides the experimental studies, previous theoretical studies on GA assumed single neuron populations, i.e. single brain areas, and have reproduced successfully the characteristic EEG-power spectrum changes observed in experiments. These studies have explained the bi-phasic behavior in the EEG power spectrum by different mecha- nisms. Steyn-Ross et al. support the idea that the bi-phasic spectrum and the LOC result from a first-order phase tran- sition in the population (Steyn-Ross and Steyn-Ross 1999 Steyn-Ross et al. 2001b, 2004). In this context the phase transition of first order reflects a sudden disappearance of the system���s resting state accompanied by a jump to another resting state. The associated pre-jump increase in state activity has been interpreted as the sudden loss of con- sciousness as observed in experiments. In contrast, Liley et al. (Bojak and Liley 2005 Liley and Bojak 2005) showed in an extensive numerical study of a slightly different model that such a phase transition is not necessary to reproduce bi-phasic power changes, but did not suggest a mechanism for the occurrence of LOC. Moreover Molaee-Ardekani et al. introduced the idea of slow adaptive firing rates which Cogn Neurodyn 123

explains the bi-phasic spectrum and LOC without a phase transition (Molaee-Ardekani et al. 2007). The present work studies a neural population not embedded in a larger net- work and which is subjected to uncorrelated fluctuations. Consequently we aim to answer the question whether an isolated neural population is sufficient to model the biphasic behavior in the EEG-power spectrum. In contrast to the previous studies, we introduce a less complex neural pop- ulation model which allows for a thorough analytical study. The latter theoretical studies (Steyn-Ross and Steyn- Ross 1999 Bojak and Liley 2005 Molaee-Ardekani et al. 2007) are based on the model of Liley et al. (1999), whose basic elements we discuss briefly in the following. The model considers a continuous spatial mean-field of neurons in one or two spatial dimensions, synapses and axonal connections the synapses and neurons (Bojak and Liley 2005) may be excitatory and inhibitory. This mean-field represents the spatial mean in a neural population descrip- tion and thus averages the spiking activity of single neurons using a sigmoidal population firing rate. The firing activity is assumed to spread diffusively via a damped activity wave along the axonal trees and terminates at pre-synaptic ter- minals. The wave speed of this axonal wave is set to the mean axonal conduction speed. At the synaptic terminals the incoming pre-synaptic activity evokes the temporal synaptic response on the dendritic trees according to the dynamics of a single synapse, i.e. treating the membrane as an RC-circuit with a time-dependent conductance, see e.g. (Koch 1999). This model neglects the spatial extension of dendritic trees and assumes a volume conduction mecha- nisms for the spread along axonal fibers. The model considered in the present work is similar to previous models (Foster et al. 2008) in several aspects such as the model of Liley et al. (1999) but differs in some important aspects. In contrast to the Liley-model our model considers a one-dimensional spatial domain and the popu- lation of synapses on dendritic trees [on average *7,800 synapses on each dendritic tree in rat cortex (Koch 1999)] and the passive activity spread on dendrites (Agmon-Sir and Segev 1993). Considering the propagation delay of evoked synaptic activity along dendritic branches, previous studies showed that the temporal synaptic response on the dendritic trees smears out temporally (Koch 1999 Smetters 1995 Agmon-Sir and Segev 1993). Consequently the synaptic response arriving at the soma differs from that at a single synapse. To cope with the various delay distributions caused by the spatial distribution of synapses on the dendritic branches, the present model considers an average synaptic population response which obeys an average synaptic response function, see (Freeman 1992 Gerstner and Kistler 2002) and section ������Methods������ in the present work. This model assumption contrasts to the Liley-model, that con- siders the dynamics of a single synapse to describe the population dynamics. In addition the present work models the activity transmission along axonal trees by taking into account the spatial probability density of axonal connec- tions. This contrasts to the Liley-model, that considers a volume conduction mechanism for the activity spread along the axonal branch. It has been shown in previous theoretical studies that the choice of the axonal connection probability functions can significantly alter spatio-temporal dynamics of the neural population (Hutt 2008 Hutt and Atay 2005 Laing and Troy 2003 Bressloff 2001 Bressloff et al. 2002 Coombes 2005). This model of axonal activity spread has been shown to extend the damped activity wave considered in the model of Liley et al. (Coombes et al. 2007 Hutt 2007) to nonlocal interactions. Moreover, the model pre- sented here is mathematically less complex than the Liley- model since it has less parameters. This aspect allows for an analytical treatment of the model and, consequently, the analytical derivation of conditions for physiological parameters. To obtain dynamical criteria for the occurrence of anesthetic effects, and hence learn more about their importance and the underlying dynamics, the present work aims to extract some analytical relations between physio- logical parameters. To achieve this goal, the subsequent section introduces the model and discusses the chosen physiological parameters. Section ������Results������ extracts a condition on synaptic time scales from experimental data, and gives conditions on the number of resting states and their linear stability. In addition, that section derives the power spectrum of Local Field Potentials and EEG ana- lytically and investigates the conditions for bi-phasic behavior in EEG. Finally the discussion section ������Discus- sion������ summarizes the results obtained and gives an outlook onto future work. Methods The model considers an ensemble of neurons on a meso- scopic spatial scale in the range of cortical hypercolumns, i.e. on a spatial scale of some millimeters. It considers two types of neurons, namely pyramidal cells and interneurons. The former cell type typically excites other cells by excit- atory synapses, and thus the pyramidal cell is called an excitatory cell. In contrast, interneurons are known to inhibit other cells by inhibitory synapses and are called inhibitory cells. Consequently, taking into account excit- atory and inhibitory cells involves the treatment of excit- atory and inhibitory synapses. Moreover, both types of synapses may occur on dendritic branches of both cell types. In the following, we consider excitatory synapses (abbreviated by e) at excitatory (E) and inhibitory cells (I) in addition to inhibitory synapses (i) at both cell types. Cogn Neurodyn 123