Analysis of stochastic fluctuations in responsiveness is a critical step toward personalized anesthesia
Abstract
Traditionally, drug dosing is based on a concentrationresponse relationship estimated in a population. Yet, in specific individuals, decisions based on the populationlevel effects frequently result in over or underdosing. Here, we interrogate the relationship between populationbased and individualbased responses to anesthetics in mice and zebrafish. The anesthetic state was assessed by quantifying responses to simple stimuli. Individual responses dynamically fluctuated at a fixed drug concentration. These fluctuations exhibited resistance to state transitions. Drug sensitivity varied dramatically across individuals in both species. The amount of noise driving transitions between states, in contrast, was highly conserved in vertebrates separated by 400 million years of evolution. Individual differences in anesthetic sensitivity and stochastic fluctuations in responsiveness complicate the ability to appropriately dose anesthetics to each individual. Identifying the biological substrate of noise, however, may spur novel therapies, assure consistent drug responses, and encourage the shift from populationbased to personalized medicine.
eLife digest
Every year, millions of patients undergo general anesthesia for complex or lifesaving surgeries. In the vast majority of cases, the drugs work as intended. But a minority of patients take longer than expected to regain consciousness after anesthetic, and a few wake up during the surgery itself. It is unclear what causes these unintended events.
When choosing an anesthetic dose for each patient, physicians rely on data from large clinical studies. These studies expose many patients to different doses of an anesthetic drug. At higher doses, fewer and fewer patients remain conscious. This enables physicians to identify the dose at which an average person will lose consciousness. But this approach ignores the difference between the response of an individual and that of the population as a whole. At the population level, the likelihood of a patient being awake decreases smoothly as the concentration of anesthetic increases. But within that population, each individual patient can only ever show a binary response: awake or not awake.
To compare anesthetic effects on individuals versus populations, McKinstryWu, Wasilczuk et al. exposed mice to a commonly used anesthetic called isoflurane. During prolonged exposure to a constant dose of the drug, each mouse was sometimes unconscious and sometimes awake. These fluctuations in responsiveness seemed to occur at random. Exposing zebrafish to propofol, an anesthetic that works via a different mechanism, had a similar effect.
Notably, the responses of both species to anesthesia showed a phenomenon known as inertia. If an individual was unresponsive at one point in time, they were likely to still be unresponsive when assessed again after three minutes. The amount of inertia was similar in mice and zebrafish. This suggests that the mechanism responsible for inertia has remained unchanged over more than 400 million years of evolution.
The results reveal similarities between how individuals respond to anesthetics and how individual anesthetic molecules act on cells. When a molecule binds to its receptor protein on a cell, the receptor fluctuates spontaneously between active and inactive states. Studying how individuals respond to drugs could thus provide clues to how the drugs themselves work. Future studies should explore the biological basis of fluctuations in anesthetic responses. Understanding how these arise will help us tailor anesthetics to individual patients.
Introduction
One of the great promises of personalized medicine is the delivery of a maximally efficacious and minimally harmful dose of appropriate medication to every patient (Fitzgerald et al., 2006). Traditionally, dosing decisions are based on the relationship between drug concentration and the magnitude of effect observed in a population expressed as the sigmoidal doseresponse curve (Goodman, 1996). An implicit assumption of this approach is that population averages adequately reflect the processes operating within each individual patient. This is not always true. For many drug classes such as antiepileptics, anesthetics, and antiarrhythmics, the response is binary at the level of an individual—the desired effect is either present or not (Löscher, 2011; Sunderam et al., 2001; Jürgens et al., 2003; Sonner, 2002). The populationbased doseresponse curve, in contrast, is a smooth graded function of drug concentration. Therefore, in order to deliver on the promise of optimal drug dosing at the individual level, the relationship between individual binary responses and the populationbased graded estimates of drug potency have to be more rigorously defined (KochWeser, 1975).
Many examples of graded populationlevel responses coexisting with discrete, binary individual responses are seen in biophysics. A priori, one might have expected that the current flow through a single ion channel molecule would simply be a scaled version of the current recorded in a cell containing many such ion channels. Yet, this is not the case (Sakmann and Neher, 1984). While at the whole cell level, current varies smoothly as a function of voltage and time (Hodgkin and Huxley, 1952), each ion channel molecule stochastically switches between conductive and nonconductive states (Fukushima, 1982; Hoshi et al., 1990; Aldrich et al., 1983). Ligandgated ion channels also stochastically fluctuate between discrete conductive and nonconductive states at a constant concentration of a ligand (Brickley et al., 1999; Swanson et al., 1996; Papke et al., 1989). Stochastic transitions between discrete states extend to higher levels of organization. For example, synaptic transmission is mediated by stochastic vesicular fusion events (del Castillo and Katz, 1954; Kuffler and Yoshikami, 1975; Rizzoli and Betz, 2005). At the cellular level, stochastic fluctuations in gene expression and protein abundance are observed even in a clonal population of cells (Elowitz et al., 2002; McAdams and Arkin, 1997; Newman et al., 2006; Raser and O'Shea, 2004). For instance, stochastic forces influence differentiation patterns across identical cells and give rise to drastically different proteomic responses of clonal cancer cells to drugs (Cohen et al., 2008; Losick and Desplan, 2008). Thus, stochastic fluctuations among discrete states even in a constant environment are the rule at the microscopic level in many, if not all, biological processes.
Detailed quantitative models of stochastic switching between different states of an individual molecule (Hille, 2001; Colquhoun and Hawkes, 1995) are required to forge the relationship between discrete microscopic events and graded responses in the population. Macroscopic constructs such as pharmacologic efficacy depend upon druginduced changes in discrete states of single molecules as well as stochastic switches among them. Yet, drug efficacy does not uniquely specify responses of individual molecules (Colquhoun, 1998). Thus, while discrete responses at the individual level can be linked to the smooth populationlevel responses using techniques from statistical mechanics, individual responses cannot be readily inferred from populationlevel analyses.
Here, we asked whether stochastic fluctuations among discrete states, similar to those postulated to act at the level of individual molecules, organelles, and cells influence the responses of intact multicellular organisms exposed to a constant drug concentration. For this purpose, we chose to focus on anesthetics. Anesthesia is thought to consist of four basic components: amnesia, immobility, analgesia, and unconsciousness. Responses to stimuli are typically used to quantify the various components of anesthesia. At lower anesthetic doses that produce light sedation, a patient can respond to salient verbal commands and less noticeable auditory commands (Wong et al., 2014), albeit with longer latency. When the state of general anesthesia is attained, the patient is unresponsive even to a painful surgical stimulus (Miller, 2014). Clinical assessment of anesthetic depth ultimately collapses to binary outcomes: amnestic or not, immobile or not, and conscious or not. We consequently studied binary responses to a simple stimulus at a fixed anesthetic concentration across individuals to determine whether they are anesthetized or not. These allornone responses can be assessed by determining whether a human patient reacts to a simple verbal command (Sanders et al., 2017; Russell, 2013; Flaishon et al., 1997). The concentration of anesthetic at which 50% of patient lose their ability to respond to verbal commands is known at MACawake (Franks, 2008). In rodent literature, the righting reflex (RR) is typically used to establish whether a mouse is awake or not (Figure 1A) (Wasilczuk et al., 2018; Franks, 2006). MACawake and the EC_{50} for loss of RR are closely correlated across mechanistically distinct anesthetics (Franks, 2008). Discrete responses to simple stimuli can be used in other organisms such as larval zebrafish to assess anesthetic potency (Yang et al., 2018). In contrast to discrete measures used in individual subjects, anesthetic potency at a population level is expressed as a smooth sigmoid function (Miller, 2014). Traditionally, the dichotomy between the graded populationlevel response on the one hand, and the binary responses of individuals on the other, has been interpreted as intersubject variability (Sonner et al., 2000). For instance, at the population level half maximal effective concentration (EC_{50}), it is assumed that half of all the subjects will be able to consistently respond to the stimulus while the other half will consistently fail to do so (Figure 1B). Yet, this is not the only possibility. It is possible that at EC_{50}, each individual subject will respond to 50% of stimuli (Figure 1C). Finally, it is possible that the probability of response is influenced by the state of the subject at the moment when the stimulus is applied (Figure 1D).
To distinguish among these possibilities, we exposed both mice and zebrafish to fixed concentrations of two mechanistically distinct anesthetics and repeatedly tested their responsiveness. The results unequivocally demonstrate that at a fixed drug concentration, each individual mouse or zebrafish stochastically switches between being responsive and unresponsive. The probability of response depends upon the subject’s state. Therefore, stochastic state switching is apparently present at all organizational levels, from receptor to cell to whole animal. Unlike identical ion channel molecules, however, individual animals both in highly genetically inbred and across outbred populations exhibited dramatic variability in parameters that describe the stochastic switching between the responsive and unresponsive states. The interindividual variability in parameters of the stochastic model fit to each individual was highly structured in both mice and in zebrafish. One manifestation of this structure is that while the overall sensitivity to anesthetics varied among individuals, the amount of noise that drives state switches was the same across individuals exposed to the same anesthetic concentration.
As a result of the intersubject variability and stochastic intrasubject fluctuations in responsiveness, the populationlevel concentration response cannot be used to reliably determine the probability that any given individual will (or will not) be anesthetized. Detailed quantification of stochastic forces that shape the withinsubject fluctuations in responsiveness and betweensubject variability lays the foundation for the construction of more informative stochastic models that can reconcile binary responses of individual organisms with populationbased measures of drug potency. These models can in turn be used to deliver on the promise of personalized medicine to deliver the appropriate dose of medication tailored to each individual patient.
Results
Individual responses to anesthetics fluctuate in a statedependent fashion at a fixed drug concentration
The presence of the righting reflex (RR) during continuous isoflurane administration was used as the measure of responsiveness in mice. After 2 hr of equilibration at 0.6% isoflurane, the probability of an intact RR was 44 ± 6% (mean ± SD across trials). This did not correspond to 44% of mice being consistently responsive and the rest being consistently unresponsive, as is commonly assumed. Instead, in every mouse, the outcome of the RR test fluctuated over time at a fixed anesthetic concentration (Figure 2A) while the population response probability at 0.6% isoflurane remained stable over time (Figure 2B). Individual fluctuations in RR at concentrations deviating from EC_{50} were less frequent, but nevertheless were reliably observed at concentrations below 0.9% (Figure 2—figure supplement 1).
In an analogous experiment using a mechanistically distinct anesthetic, propofol, we determined the responsiveness in larval zebrafish using the startle reflex (SR) triggered by mechanical stimulation. Individual larval zebrafish demonstrated fluctuating responses to identical tap stimuli (Figure 2C). Zebrafish exposed to no propofol had a significantly higher response probability to the tap stimulus (Figure 2—figure supplement 2A, U = 55, n_{E3} = n_{3µM}=360 trials, p<0.0001). Drug concentration remained constant in the propofol exposures (Figure 2—figure supplement 2B). The response probability across the population remained constant (45 ± 5% mean ± SD across trials) for three hours after an initial hour of equilibration in 3 μM propofol (Figure 2D).
Effective concentration, defined as the concentration of the drug required to produce an effect of a given intensity, is a universally used populationbased measure of drug potency (Goodman, 1996). We sought to apply this measure to each individual. At the individual level, effective concentration is equivalent to the average acrosstrial probability of observing a response to a stimulus at a given drug concentration. Therefore, we compared the experimental results in Figure 2A and C to simulations of a Bernoulli process (Materials and methods) constructed such that the probability of positive RR (or SR) was identical on each trial and the same as that observed experimentally for a population (Figure 3A). In mice for instance, this simulation was constructed such that the probability of responding to a stimulus is 44% on every trial. As expected, both simulations (Figure 3A) and the experimental results (Figure 3B) have a similar overall response probability (45 ± 3% and 44 ± 6% mean ± SD for simulations and mouse experiments, respectively). The median response probability in simulations was not statistically different from experimental observations in both mice (Figure 3C, U = 197.5, n_{sim} = n_{exp} = 20, p=0.95) and in zebrafish (Figure 3D, U = 2276, n_{sim} = n_{exp} = 72, p=0.34). Note, however, that the experimentally observed switches between positive and negative RR occur less frequently than in the simulation (Figure 3E, U = 48.5, n_{sim} = n_{exp} = 20, p<0.0001). This was also true of zebrafish (Figure 3F, U = 300.5, n_{sim} = n_{exp} = 72 p<0.0001).
To further quantify this resistance to state transitions, we compared the probability of becoming unresponsive after responding to a stimulus on a previous trial, P(UR) (Figure 3B, red arrow), to the probability of failing to respond on two consecutive trials, P(UU) (Figure 3B, purple arrow). In both mice (Figure 3G, U = 103, n_{P(UR)} = n_{P(UU)}=20, p<0.001) and zebrafish (Figure 3H, U = 501, n_{P(UR)} = n_{P(UU)}=72, p<0.0001), the probability of being unresponsive on the next trial was significantly higher if the animal was found to be unresponsive on the preceding trial. Hence, fluctuations in responsiveness under constant anesthetic concentration are inconsistent with a Bernoulli process. Therefore, while effective concentration is a useful measure of populationbased drug potency, it cannot be adequately applied to an individual. Specifically, effective concentration cannot account for the apparent resistance to state transitions.
Individual differences are highly structured
We now turn to the relationship between trialtotrial fluctuations and population level variability. One common assumption is that observing a single mouse over long period of time is equivalent to observing a snapshot of the population of mice. In other words, an experiment on each individual mouse can be thought of as a different realization of the same process. In an apparent departure from this assumption, we observed high interindividual variability in responsiveness in mice and in zebrafish. For instance, at 0.6% isoflurane some mice were able to right themselves on fewer than 20% of trials, while other mice within the same highly inbred population, exposed to the same anesthetic concentration, during the same experiment, were able to right themselves on ~70% of trials (Figure 3C). Experimentally observed interindividual variability was significantly higher than in simulations constrained to have the same number of trials (Figure 3C, F(1, 38)=27.8, p<0.0001 for mice, Figure 3D, F(1,142)=52.5, p<0.0001 for zebrafish, BrownForsythe test). This implies that the observed intersubject variability in responsiveness is unlikely to be solely due to finite sample size. Rather, anesthetic sensitivity can differ significantly between individuals.
We then sought to determine how interindividual variability is reflected in the parameters of a model of trialtotrial fluctuations in responsiveness fit to each animal individually (Materials and methods). The dwell times in responsive and unresponsive states for both mice (Figure 4—figure supplement 1A) and zebrafish (Figure 4—figure supplement 1B) were approximately exponential. No significant autocorrelations in fluctuations were observed (Figure 4—figure supplement 1C,D). Thus, switching between states of responsiveness and unresponsiveness in each individual can be well approximated by a twostate transition probability matrix (Materials and methods). Because the sum of transition probabilities in each row of this matrix is exactly one, the twostate transition probability matrix is completely specified by knowing the two transition probabilities along the diagonal, P(UU) and P(RR). The plane spanned by these two diagonal transition probabilities, therefore, is the parameter space for models of stochastic fluctuations in responsiveness in mice and in zebrafish (Figure 4A). Movement along the xaxis to the right, results in decrease in the overall probability of response (e.g. from I to II and from III to IV, Figure 4A). Conversely, movement up along the yaxis results in the increase in the overall probability of response (e.g. from III to I and from IV to II, Figure 4A). Moving along the dotted lines within the parameter space does not affect the overall probability of being responsive – that is systems II and III have the same overall probability of response. The key difference between II and III is the number of transitions between the responsive and the unresponsive state. In contrast, movement from I to IV does not affect the sum of P(UU) and P(RR). As a result, while the overall probability of responsiveness in I and IV is different, the amount of noise that drives state transitions between the responsive and the unresponsive states is the same. To realize why this is the case, we can visualize the twobytwo transition probability matrix as an energy landscape with two wells which correspond to the responsive and the unresponsive states (Proekt and Hudson, 2018). Conservation of the sum of P(UU) and P(RR) reflects the fact that the sum of the depths of the wells is conserved.
To characterize interindividual variability, we estimated transition probability matrices for each animal individually. This yields a single estimate of P(UU) and P(RR) for each individual. The simplest model of interindividual variability is that the transition probability matrix, M, for each individual is a random sample taken independently from the distribution of P(UU) and P(RR). This null hypothesis corresponds to a cloud of points in Figure 4B. Yet, in stark departure from this prediction, the observed joint distribution of transition probabilities in individual mice lies on a diagonal (Figure 4C). Therefore, P(UU) and P(RR) are strongly negatively correlated among individuals (R^{2} = −0.85, p<0.0001, Pearson’s R). The same relationship was observed in larval zebrafish (Figure 4E, R^{2} = −0.77, p<0.0001, Pearson’s R). A similar strong negative correlation is observed when comparing decay rates of the responsive and unresponsive states estimated from individual dwell time distributions fit to single exponential decay functions (dwell times from Figure 4—figure supplement 1A,B, R^{2}_{mouse} = −0.64, p=0.002, Pearson’s R, R^{2}_{zebrafish} = −0.71, p<0.0001, Pearson’s R).
Note that P(UU) and P(RR) quantify the propensity of the system to stay in its previously observed state. The fact that their sum is constant across individuals implies that the amount of noise that drives state transitions between responsive and unresponsive states was consistent in all individuals exposed to a fixed anesthetic concentration. Yet, sensitivity to anesthetics measured as the overall probability of responding to a stimulus varied broadly in the same population of individuals. Altogether these results indicate that, while transitions between states of responsiveness are noisedriven and therefore unpredictable, the amount of noise is tightly controlled in all individuals. Too little noise would result in individuals being trapped in a single state, whereas too much noise would overpower the intrinsic dynamics of the brain, leading to a noise dominated process characterized by rapid state switching.
Individual differences in drug sensitivity complicate decoding of drug concentration from drug responses
Sigmoid doseresponse curves are onetoone functions—knowing drug concentration is sufficient to estimate probability of a response in a population. Critically, the converse is also true; knowing the probability of a response in a population is sufficient to determine the concentration of the drug to which this population is exposed. While this onetoone relationship may hold for a population, we sought to determine whether the large interindividual variability (Figures 4–5) complicates this onetoone relationship at the level of an individual. We compared individual transition probability matrices estimated at 0.6% and 0.3% isoflurane. Note that here, mice exposed to 0.6% isoflurane were distinct from mice exposed to 0.3% isoflurane. Individual matrices estimated for 0.3% isoflurane, much like those estimated at 0.6% isoflurane, exhibited strong negative correlations (R^{2} = −0.86, p<0.0001, Pearson’s R) between the diagonal elements of the transition probability matrices (Figure 6A). At the level of the population, there were statistically significant differences between righting probability observed at the two anesthetic concentrations (Figure 6B, U = 54, n_{0.3% iso} = 18, n_{0.6% iso} = 20, p<0.001). Note, however, that there is significant overlap (57%) between the two distributions. As a consequence of this overlap, it is not possible to reliably determine whether an individual was exposed to 0.6% or 0.3% isoflurane over a large range of individual anesthetic sensitivities (Figure 6C). The same phenomenon was observed for individual zebrafish exposed to medium alone or 3 μM propofol (Figure 6—figure supplement 1). This observation is in stark contrast to the populationbased measures of anesthetic potency. Hill slopes of the populationbased doseresponse curves for anesthetics are some of the steepest among clinically useful drugs (Friedman et al., 2010; Joiner et al., 2013). Thus, small increases in drug concentration are expected to result in a dramatic change in the probability that an individual will be anesthetized. Yet, as demonstrated here, there is considerable overlap in sensitivity to anesthetics. This illustrates the fundamental inconsistency between populationbased and individualbased measures of drug potency.
Because results in Figure 6 were obtained in two separate cohorts of mice each exposed to a single anesthetic concentration, we sought to determine whether it is possible to reliably infer drug concentration from behavioral responses for the same individual exposed to two drug concentrations. To address this, we exposed a separate cohort of 20 mice on eight occasions, four times each to 0.4% and 0.7% isoflurane. Strong negative correlations between P(UU) and P(RR) existed at both 0.4% isoflurane (R^{2} = −0.69, p<0.0001, Pearson’s R) and 0.7% isoflurane (R^{2} = −0.79, p<0.0001, Pearson’s R). Populationlevel response probability at 0.4% and 0.7% isoflurane differed significantly (Figure 7A, U = 7, n_{0.4% iso} = n_{0.7% iso} = 20, p<0.0001). Overlap of response probabilities at the population level between different isoflurane concentrations did exist (18%), but was approximately three times smaller than that observed across separate mouse cohorts exposed to 0.3% and 0.6% isoflurane (Figure 6A). Overlap at the individual level varied widely, from close to zero to nearly complete (Figure 7B,C, Figure 7—figure supplement 1). Thus, observing the same individual exposed to different drug concentrations, improves the reliability of distinguishing between drug concentrations on the basis of righting probability. Comparing different, albeit highly genetically similar individuals, exposed to different isoflurane concentrations increases the response variability and therefore decreases the reliability of classification.
Neuronal network modeling of stochastic state switching confirms that noise driving state transitions is conserved among individuals
In Figures 4 and 6, we observe a strong negative correlation between the two conditional probabilities that express the tendency of staying in the previously observed state (P(UU) and P(RR)). To investigate the origins of this observation, we constructed a simple mathematical model that can explain this striking correlation. The Markov process defined by a twostate transition probability matrix can be thought of as a discrete approximation of a continuous system that fluctuates stochastically between two stable states. To illustrate how such a multistable system can be embodied in the brain, we consider a simple network consisting of two neuronal populations (α and β Figure 8A). The two neuronal populations inhibit each other and excite themselves. MorenoBote et al. (2007) demonstrate that activity of such networks can be parameterized as a function of difference in activity of the two neuronal populations (Figure 8B). Because of selfexcitation and mutual inhibition, the network exhibits two stable activity patterns: one where activity of α dominates, and the other, where activity of β dominates. We operationally define these stable network patterns as awake and anesthetized respectively. The likelihood of every activity pattern of the network is expressed by an ‘energy function’ (Figure 8B). The more likely activity patterns are associated with lower energy and the less likely activity patterns are associated with higher energy. In the absence of noise such networks stay in one of the two stable states indefinitely. When noise is added to the system, it switches between the two stable network configurations stochastically. While the amount of noise changes the frequency of switching between the two states (Figure 8C), it does not have a dramatic effect on the shape of the overall distribution of states. In both cases the distribution of states of the system is bimodal akin to what we observe in the startle reflex data in the zebrafish (Figure 2—figure supplement 2C).
To model the effects of anesthetics on different individuals, we assume that anesthetics activate sleepactive β neurons and inhibit wakeactive α neurons. To account for the strong negative correlation between P(UU) and P(RR), we assume that the degree of anestheticinduced excitation and inhibition is correlated. Thus, individual differences in anesthetic sensitivity can be modeled by modulating the anesthetic effect on the network. In an attempt to model the results in Figures 4 and 6, we simulate the behavior of such networks with various amounts of noise. We then approximate the data of the kind seen in Figure 8C obtained for different anesthetic sensitivities and noise levels using a Markov model akin to that deployed for analysis of righting and startle reflex throughout this work. To accomplish this, the continuous fluctuations in the state of the network are binarized using a threshold (Figure 8C) into the ‘responsive’ and the ‘unresponsive’ states. The transition probability matrix is then estimated from these binary timeseries. The results of this analysis are shown in Figure 8D. When the noise amounts are small, the system tends to stay in its previously observed state. Thus, for lownoise simulations, the points are found around the periphery of the plane spanned by P(UU) and P(RR). When the noise is exceedingly high, the dynamics of the system become independent of the shape of the energy landscape and tends toward the point (0.5, 0.5) in the plot in Figure 8C. This would be expected if the system behaved like a Bernoulli process at EC_{50}. In order to reproduce the results in Figures 4 and 6, the amount of noise needs to be tuned and maintained at an approximately the same level across individuals. In the simulations, this corresponds to Gaussian noise with mean 0 and standard deviation (σ)=0.4. These simulations support our conjecture that while individuals are different in terms of anesthetic sensitivity, the amount of noise that drives fluctuations between the responsive and the unresponsive states is conserved. Note, that in the units of trial number, the dwell time distributions were similar in fish and in mice. However, in mice the trials were spaced 3 minutes apart whereas in fish the intertrial interval was 30 seconds. Thus, in units of seconds, mice dwelled signficantly longer in the awake and the unresponsive state than zebrafish. In order to account for this difference in the characteristic dwell times, the time scale of fluctuations shown in Figure 8C would have to be scaled by a speciespecific diffusion constant (Methods).
Discussion
Here, we demonstrate that even when an animal is exposed to a fixed drug concentration, its response fluctuates stochastically. The probability of response depends on the state of the animal at the moment of stimulation. Specifically, the fluctuations exhibit inertia – the animal is more likely to be stuck in its current state than to transition between states of responsiveness. In contrast to most drugs that are administered either intravenously or orally, the route of administration of many anesthetics is inhalational. This allows us to assure that the drug concentration in the animal is at equilibrium with the ambient concentration in a closed chamber (Friedman et al., 2010). We utilize this significant experimental advantage to focus on the dynamics of responses within each individual at a constant drug concentration. However, stochastic fluctuations are not unique to a volatile anesthetic. An intravenous anesthetic, propofol, administered to zebrafish equilibrated with a fixed concentration bath, was also associated with dynamic fluctuations in responsiveness. The stochastic processes that govern the fluctuations observed under two mechanistically distinct anesthetics were remarkably similar between mice and zebrafish. It is curious to note that while the dwell times in the responsive and the unresponsive state in mice and in zebrafish were similarly correlated, the absolute duration of these states varied signficantly between mice and zebrafish. The duration of responsive and unresponsive states was approximately similar to the duration of sleep episodes in these two species (Zhang et al., 2007; Yokogawa et al., 2007). It is possible that this relationship between duration of sleep episodes and the duration of the episodes of unresponsiveness arises because similar neurobiological mechanisms are responsible for both transitions between sleep and wakefulness and fluctuations between the responsive and the unresponsive states observed under anesthesia.
Our observations of stochastic switching between responsive and unresponsive states are consistent with previous findings showing that at a fixed anesthetic concentration, spectral characteristics of electrical activity within thalamocortical networks switch stochastically between several discrete activity patterns (Hudson et al., 2014; Clement et al., 2008). Similar observations have been made using electroencephalography (EEG) of patients under anesthesia (Chander et al., 2014; Li et al., 2019; Vlisides et al., 2019). Our findings in this study suggest that such stochastic fluctuations have a behavioral counterpart expressed as fluctuating ability to respond to a stimulus. It has been suggested previously that anesthetics stabilize neuronal dynamics in humans (Alonso et al., 2014; Tagliazucchi et al., 2016) and in nonhuman primates (Solovey et al., 2015; Alonso et al., 2019). Stabilization of neuronal dynamics may contribute to the behavioral inertia observed in fluctuations in responsiveness of both mice and zebrafish. This resistance to state transitions may contribute (Proekt and Hudson, 2018) to anesthetic hysteresis (Friedman et al., 2010; Joiner et al., 2013; Kuizenga et al., 2018; Warnaby et al., 2017) – a leftshift in a doseresponse curve for emergence relative to induction of anesthesia that has been observed across taxa from Drosophila to humans.
Stochastic responses at a fixed drug concentration help explain several otherwise puzzling phenomena in clinical anesthesiology. On the one hand, Hill slopes of doseresponse curves for anesthetics are reported between 10 and 40 (Miller, 2014); some of the steepest of the clinically used drugs. This is traditionally interpreted as a sign of low intersubject variability in responsiveness. Nonetheless, approximately ten percent of patients transiently wake up during surgery as measured by their ability to respond to a verbal command (Sanders et al., 2017; Russell, 2013; Gaskell et al., 2017). Luckily, incidence of awareness accompanied by postoperative recall is much less frequent (Avidan et al., 2008). However, even episodes of awareness associated with recall and high risk of developing posttraumatic stress disorder (Osterman et al., 2001) are not reliably detected by the existing EEGbased monitors of anesthetic state (Avidan et al., 2011). While anesthetics are thought to impart dosedependent effects on neuronal activity (Stanski et al., 1984; Katoh et al., 1998), only weak correlations are observed between EEG characteristics and anesthetic concentration (Whitlock et al., 2011). Indeed, both interindividual differences (Whitlock et al., 2011) and fluctuations of EEG measures of depth of anesthesia at a constant drug concentration (Bloom et al., 2005) obscure the relationship between drug concentration and the observed state of the EEG. It is likely that the traditional populationbased approach of estimating the relationship between the concentration of the drug and the response dramatically underrepresents the within and intersubject variability. Indeed, under most circumstances, each animal is tested only once at each drug concentration. The repeated testing of each animal under a fixed drug concentration is more akin to the clinical scenario where the patient is exposed to many surgical stimuli that can elicit stochastic fluctuations in the level of consciousness.
The Hill slope value observed in our study was approximately an order of magnitude lower than the previously published data (Friedman et al., 2010; Joiner et al., 2013). This relative decrease in Hill slope reflects high interindividual variability and stochastic fluctuations in each individual. There are several significant differences between our experimental paradigm and those explored in the previous work. Previous work, guided by purely pharmacokinetic considerations, used shorter isoflurane exposures. In contrast, here we measured anesthetic responsiveness after a twohour equilibration time. Under most circumstances, Markov processes lead to a single equilibrium distribution of states (Roman, 1989). The time it takes to reach this equilibrium, however, is dictated by, among other things, the degree of resistance to state transitions. Thus, one reason for the discrepancy between our Hill slope estimate and that published previously is that our analysis assures that the population of animals is at a behavioral steady state. In order to reliably anesthetize an individual after a short exposure, higher concentrations of anesthetics are necessary. Indeed, our EC_{50} estimate (0.54–0.55% isoflurane) is significantly lower than that published in the previous work (0.9% for induction and 0.83% for emergence) (Friedman et al., 2010). Another fundamental difference between our approach and that used in the previous work is that we tested mice repeatedly. In contrast, in the previous work, mice are typically tested only once at each concentration. Repeated testing revealed trialtotrial fluctuations. These fluctuations increase the apparent variability in responsiveness, thereby decreasing the apparent Hill slope of the population.
The detailed mechanism of these stochastic fluctuations is not known as anesthetics exert effects on many molecular targets distributed broadly throughout the brain and spinal cord. A number of lines of evidence converge on the fact that anesthetics at least in part hijack the sleepwake circuitry by exciting sleeppromoting and inhibiting wakepromoting neurons (Zhang et al., 2015; Moore et al., 2012; Vazey and AstonJones, 2014; Nelson et al., 2002). Switching between sleep and wake is thought to be mediated by the reciprocal inhibition between these neuronal populations. Networks consisting of reciprocally inhibitory neuronal populations tend to exhibit selfreinforcing behavior. Once sleepactive neurons activate beyond a certain threshold, they shut down the wakeactive neurons thereby decreasing their inhibitory effects and further strengthen mutual excitation amongst the sleepactive neurons (Saper et al., 2005). The converse happens during wakefulness. In the absence of perturbations, the network consisting of such mutually inhibitory selfreinforcing neuronal populations will remain in the same state indefinitely. Once sufficient noise is added, however, the system will stochastically switch between consolidated states of sleep and wakefulness. Theoretical investigations of bistable neuronal networks suggest that activity within such mutually inhibitory neuronal populations can be well approximated by a diffusion on an energy landscape with two potential wells; one for each stable activity pattern (MorenoBote et al., 2007). Anesthetics can therefore be thought of as stabilizing (deepening) the wells associated with unresponsiveness and destabilizing the wells associated with wakefulness. The degree to which these states are stabilized by clinically relevant doses of anesthetics is apparently insufficient to consistently keep all animals in the anesthetized state. Hence, stochastic fluctuations may be responsible for episodes of awareness that occur during surgeries.
The importance of noise in systems near a bifurcation between two stable behaviors is not limited to circuits that control sleep and wakefulness (Chialvo, 2010; Destexhe and Contreras, 2006). Neuronal architectures similar to those involved in sleep and wakefulness are thought to play a role in diverse processes such as sensory perception (MorenoBote et al., 2007), decision making (Wong and Wang, 2006), seizure generation (Suffczynski et al., 2004; Fröhlich et al., 2010), and working memory (Camperi and Wang, 1998) to name a few. Mathematically similar phenomena govern transitions between normal sinus rhythm and arrhythmias (Kim et al., 2009). Thus, it is likely that stochastic fluctuations between distinct responses observed under constant drug concentration are not unique to anesthetics.
To determine whether a bistable system with noise can explain the striking correlations between P(UU) and P(RR) observed herein, we simulated such a neuralnetworkinspired system with parametrically varied amount of noise. The results of this simulation show that in order to obtain such correlations, the amount of noise must be conserved among individuals. Another issue addressed by the modeling approach is the binary nature of behavioral responses in each individual. In zebrafish, this binarization was motivated by the observation of a bimodal distribution of distances travelled by each zebrafish after the tap stimulus. This bimodality naturally suggests that the responses fall into two distinct classes. Yet, similar measurements are much more difficult to perform for the righting reflex. Thus, it is not clear from behavioral observations of RR alone, whether the underlying processes that give rise to a response on a RR trial are continuous or discrete. Furthermore, even if the underlying processes are discrete it is not a priori clear that only two states are present. We observed that dwell times in both the responsive and the unresponsive state are wellapproximated by a single exponential distribution. Thus, we only find empirical evidence for two states. Longer recordings may be required in future work to determine whether other states are needed to completely describe the data. The two state Markov model used to analyze behavioral responses, can be seen as a natural discretization of the continuous bistable system exemplified by our modeling approach. For most reasonable choices of threshold, the continuous system and its discrete approximation will yield similar results.
In some ways, our observations of stochastic switching between different states at a fixed drug concentration are similar to those well known for single receptor molecules (Hoshi et al., 1990; Papke et al., 1989; Hille, 2001; Colquhoun and Hawkes, 1995; Tank et al., 1982). There is an important distinction, however, between molecular scale state transitions and those observed at the behavioral level in animals. At the molecular scale, differences between receptors are largely immaterial—the same stochastic model can be used to describe state transitions in all receptors of a particular kind. Thus, observing a single receptor molecule over time is sufficient to determine the response of a population of such receptors. Within intact multicellular organisms, in contrast, we show that the transition probabilities were significantly different amongst highly genetically similar (Uchimura et al., 2015) and genetically outbred individuals exposed to the same anesthetic concentration. Observation of one individual on many trials does not equate to observing a population of individuals. Indeed, at population level EC_{50}, some animals were three times as likely to be responsive as other individuals from the same population. Furthermore, these individual differences were consistent across time in each individual.
The interindividual variability in the transition probabilities that govern switching between responsive and unresponsive states was constrained such that the sum of the diagonal elements was a constant. This constraint is evolutionarily conserved across vertebrates from zebrafish to mice. Note that the sum of the diagonal elements in a square matrix is known as the trace and is equal to the sum of its eigenvalues. The largest eigenvalue of a transition probability matrix for a reversible Markov process is 1 (Levin and Peres, 2017). The fact that the trace is approximately the same in all individuals under similar experimental conditions implies that the spectral gap of the 2 × 2 transition probability matrix defined as, λ_{1}−λ_{2}, where λ_{n} is the n^{th} eigenvalue is also a constant conserved among individuals. The spectral gap of the matrix sets its mixing time, or the time it takes for a system starting out in a random distribution of states to approach its equilibrium distribution. In this particular case, the equilibrium distribution is the overall probability that a given animal will be able to respond to a stimulus. Boltzmann equation asserts that diffusive systems of the kind used in our model come to a single equilibrium distribution that depends just on the energy function. The time it takes to reach this equilibrium distribution of states, however, depends on the amount of noise. The more noise is added to the system, the quicker it reaches the equilibrium distribution. Thus, there is a fundamental relationship between the conservation of the spectral gap across individuals and the tightly controlled noise in a continuous system characterized by a two well potential. In a clear departure from the predictions made by the populationlevel doseresponse curve, our findings indicate that the equilibrium probability of responding to a stimulus at a given drug concentration is different for distinct individuals. The time it takes to reach this behavioral equilibrium, however, is conserved amongst animals. This strongly implies that the noise which drives state transitions between responsive and unresponsive states is tightly biologically controlled. A similar observation has been made for stochastic influences on gene expression (Elowitz et al., 2002).
On the one hand, our results may be seen as disappointing for the clinical practice of anesthesiology. Because of the strong influence of stochastic forces, it does not appear possible to keep a subject reliably in an anesthetized state without exposing them to the potentially dangerous high concentrations of anesthetics associated with subsequently impaired cognition (Whitlock et al., 2014; Chan et al., 2013; Fritz et al., 2016). Yet, the fact that variability in transition probabilities is constrained offers a possible novel avenue for improvement by developing therapies specifically aimed at state stabilization. Selective stabilization of the unconscious state could provide a solution to minimize the risk of spontaneously shifting into a brain state where awareness is possible, without requiring drug concentrations prone to adverse effects. How might this be achieved? One possibility is to target the mechanisms that are known to affect the stability of the sleepwake circuitry. For instance, interference with orexinergic signaling destabilizes both sleep and wake states thereby increasing the frequency of spontaneous transitions between sleep and wakefulness (Mochizuki et al., 2004). Interestingly, interference with orexin signaling also affects responses to anesthetics (Kelz et al., 2008; Shirasaka et al., 2011; Tose et al., 2009; Dong et al., 2009). Detailed investigation of the synergy between anesthetics and modification of noise inherent in neuronal networks that control sleep and wakefulness may help develop novel therapies that will allow clinicians better control over the state of each individual patient.
Materials and methods
Animals
Studies were approved by the Institutional Animal Care and Use Committee at the University of Pennsylvania and were conducted in accordance with National Institutes of Health guidelines. For larval zebrafish (Danio rerio) experiments, Tübingen long fin wildtype zebrafish were mated (Zebrafish International Resource Center, OR), and the embryos raised for 5 days in constant darkness. At 5 days postfertilization (dpf), the larvae were used for the experiments (n = 120). For mouse experiments, inbred male wildtype C57Bl/6 mice (Jackson Laboratories, ME) aged 16–24 weeks (n = 60) were used in righting reflex behavioral assays. One group of 20 mice was exposed to 0.6% and 0.9% isoflurane. A second set of 20 mice were exposed to 0.3% isoflurane. A third set of 20 mice were exposed to 0.4% and 0.7% isoflurane.
Isoflurane exposure with Righting Reflex evaluation
Request a detailed protocolAll mice were acclimatized to sealed, temperaturecontrolled, 200 mL cylindrical chambers with 100% oxygen flowing at 200 mL/minute, as previously described (Sun et al., 2006). This flow rate ensures isoflurane chamber equilibration within 5 min (Figure 2—figure supplement 3). Mice were exposed to 0.90%, 0.7%, 0.60%, 0.4% or 0.30% isoflurane in 100% oxygen for 4 hr, beginning at ZT12ZT14, corresponding to the period of maximal activity and wakefulness. Chamber isoflurane concentrations in all assays were confirmed using a Riken FI21 refractometer (AM Bickford, NY.). As tolerance to repeated isoflurane exposures does not occur (Smith et al., 1979), mice were exposed to each isoflurane concentration a total of four times over the course of 3 weeks. The presence or absence of the righting reflex (RR) was checked every 3 min, starting after 2 hr of isoflurane exposure to assure pharmacokinetic equilibration. RR was assessed as described previously (Sun et al., 2006). A mouse was considered to have an intact righting reflex if it was able to restore its upright posture twice in a row after being turned on its back by rotating the anesthetic chamber without interrupting anesthetic delivery. Otherwise, the righting reflex was considered to be absent. Forty RR assessments were performed per animal per exposure. In total, 160 RR assessments were performed on each mouse at each anesthetic concentration.
Propofol exposure with Startle Reflex evaluation in zebrafish
Request a detailed protocolAt 5 days post fertilization (dpf), individual larvae were placed into a 96well glass plate (JG Finneran, 500 μL volume wells). Seventytwo zebrafish were exposed to 3 μM propofol in E3 medium, while 48 zebrafish were exposed to E3 alone (Kaufman et al., 2009). Larvae were then placed into the DanioVision (Noldus, Leesburg VA) behavioral system. Startle reflex experiments were performed at a maximal intensity tap every 30 s for 4 hr. This choice of interstimulus interval was based upon the lack of habituation to acoustic and vibrational stimuli in larval zebrafish when these stimuli are delivered every 15 s (Burgess and Granato, 2007; Woods et al., 2014). The histogram of the total distance travelled in the first second after each stimulus was bimodal suggesting an allornone response (Figure 2—figure supplement 2C). A threshold of 0.4 mm (approximately 1/10 of a 5 dpf zebrafish’s body length), was chosen on the basis of this histogram to distinguish between responsive and unresponsive startle reflex assessments (Kimmel et al., 1995).
Empirical estimation of transition probability matrices
Request a detailed protocolThe outcomes of the behavioral response assays, righting reflex (RR) and startle reflex (SR), are binary. The output is a series of zeros and ones (one for intact RR or SR and zero otherwise). The simplest model that describes this time series is a stochastic Markov process. Markov models relate the state of the system at time t to the state of the system observed at the previous time step t − 1, as X_{t} = MX _{t−1}, where the 2x2 transition probability matrix, M, constructed as shown below is the fundamental quantity of interest.
In the above matrix, a denotes the probability that the mouse determined to be responsive on trial one will stay responsive on the next trial. b denotes the probability that an animal found to be unresponsive on trial one will stay unresponsive on the next behavioral assessment. Note that because the sum of probabilities in a row of M must be 1, probabilities of transitions between the awake and the anesthetized state are completely determined by finding a and b in a system with just two states. The simplest possibility is that the probability of being in a particular state at time t is the same for all trials (i.e. independent of the state of the system) (Roman, 1989). This corresponds to a scenario where, b = 1 − a. This particular type of process is called a Bernoulli process. Conversely, the future outcome of a behavioral trial may depend on the outcome of the previous trial. One manifestation of this historydependence is inertia; the system tends to persist in a particular state. Inertia increases as the diagonal elements (a and b) of the transition probability matrix approach one. a and b were empirically estimated from the binary sequence of behavioral outcomes. For instance, to calculate a, we computed the fraction of trials where the animal responded to the stimulus on two consecutive trials. Formally, this corresponds to P(R_{t}R_{t−1}) or P(RR).
Comparison between Bernoulli process and behavioral response
Request a detailed protocolBy definition, a Bernoulli process is a stochastic process in which the probability of a given outcome (positive RR or SR) is the same for every trial. This is equivalent to adapting the concept of effective concentration to an individual. For a given effective concentration, ec, computed as the overall probability of positive RR or SR, the Bernoulli process can be expressed as the following transition probability matrix.
M_{b} is completely specified by a single experimentally determined parameter, ec. For the purposes of simulations, we experimentally determined drug potency as the average response probability across all RR or SR trials and simulated the Bernoulli process given by M_{b}. M_{b}was used to simulate 80 experiments consisting of 40 trials each to mimic experimental conditions in mice (72 experiments each consisting of 360 trials were simulated for the zebrafish experiments). Initial states were randomized in each simulated experiment. To verify that the simulated Bernoulli process and experimental observations give rise to similar drug potency, we compared the simulated and observed ec. To test the hypothesis that behavioral observations exhibit inertia or resistance to state transitions, we compared the observed probability of state transitions, $\frac{P(UR)+P(RU)}{P(RR)+P(UU)+P(UR)+P(RU)}$, in the observed and the simulated time series. As an additional test of adequacy for describing experimentally observed time series by a Bernoulli process, we compared the tendency of the system to stay in the unresponsive state, P(UU), to probability of becoming unresponsive after being responsive on the previous trial, P(UR).
Interindividual variability
Request a detailed protocolThe transition probability matrix for a twostate Markov process has two free parameters. A transition probability matrix estimated for each specific individual can therefore be represented by a point on a plane spanned by the diagonal elements of the matrix. Principal component analysis (PCA) was used to capture the maximal interindividual variance between transition probability matrices. To determine whether the transition probability matrix is a consistent trait of each individual animal, we determined the correlation in position of each individual along the first principal component (PC) across time. Statistical significance of this correlation was assessed using a permutation test (10,000 permutations).
Decoding drug concentration from individual responsiveness
Request a detailed protocolTo quantify how reliably the drug concentration can be inferred from the observed behavioral responses of each individual, we applied Bayes’ theorem:
This analysis was performed to determine how reliably one can distinguish between exposures to 0.6% and 0.3% isoflurane in mice or 0 vs. 3 μM propofol in zebrafish.
Since mice exposed to 0.6% and 0.3% isoflurane were from two separate populations, we wanted to further investigate whether it is possible to infer drug concentration from behavioral responses within an individual exposed to two drug concentrations. For this purpose, we exposed a separate cohort of twenty mice on eight occasions, four times each to 0.4% and 0.7% isoflurane. Here, the collective trials at each concentration were pooled together for each individual (160 trials), and a bootstrap resampling technique with replacement was used to generate a response probability distribution for each concentration for each individual mouse. 1000 bootstraps consisting of 20 randomly chosen trials were taken, where the average response probability of each bootstrap was recorded. Probability distribution functions were then fit to each mouse’s bootstrapped response probabilities for both isoflurane concentrations. Overlap between the two concentrations was computed for each mouse by finding the union of the two distributions. At the population level, response probability distributions were computed based on average response probabilities for each animal, and overlap was computed in a similar fashion.
Hill slope estimation from steady state population response probabilities
Request a detailed protocolThe population response probabilities observed in Figure 2B and Figure 2—figure supplement 1 suggest a shallower doseresponse as compared to previously published populationbased doseresponse curves (Friedman et al., 2010; Joiner et al., 2013; Sun et al., 2006). In order to compute the aggregate doseresponse, we fit sigmoid curves to jackknifed subsamples across all tested isoflurane concentrations. Data from one animal at each concentration was removed for each subsample, and the remaining subset of data was fitted to a sigmoidal curve:
where R is the righting probability, Iso is the isoflurane concentration and the two paramters EC_{50} and H are the halfmaximal effect concentration and the Hill slope respectively. Fits were calculated such that the sum of the squared error was minimized between the subsample values and sigmoid. We constrained the fit such that the sigmoid value was 1 at 0.0% isoflurane, and 0 at 0.9% isoflurane. Mean and 95% confidence intervals were calculated for the EC_{50} and the Hill slope.
Modeling of a bistable neuronal network
Request a detailed protocolThe modeling approach here is essentially identical to that in Proekt and Hudson (2018) and MorenoBote et al. (2007). Activity patterns of the bistable neuronal network consisting of two neurons can be parametrized as a function of the difference between activity of α neurons and β neurons x = Activity_{α}Acivity_{β}. To express the fact that mutual inhibition and selfexcitation give rise to the winnertakeall behavior, MorenoBote et al. (2007) used the following energy function that we adapt here:
This function has two energy minima x=(−1, 1) corresponding to βdominated (‘anesthetized’) and αdominated (‘awake’) activity states, respectively. To add the effects of anesthetics we modified this function as follows (Proekt and Hudson, 2018)
where i is the anesthetic concentration i ∈ [0, 1] in arbitrary units. Note that in this equation, the effect of anesthetic on stabilizing the anesthetized state is proportional to that destabilizing the awake state. This was done to reflect the results shown in Figures 4 and 6 which show strong negative correlation between P(UU) and P(RR) in both mice and zebrafish. Individual differences in righting probability were simulated by changing i. This results in deepening the energy well in the vicinity of x=1 and making the well in the vicinity of x = 1 proportionally shallower. Time evolution of the state of the network, x, was simulated as diffusion on an energy function using the standard approach:
where D is the diffusion constant (assumed to be one in all simulations for simplicity). The first term in the equation reflects the energy gradient. This assures that the system tends towards energy minima. The second term, ε, is zero mean Gaussian noise. To modulate the amount of noise in the system, the standard deviation, σ, of noise was altered parametrically. Simulations of this diffusion equation were performed for 1,000,000 time steps. While x varies continuously, the distribution of x is bimodal with peaks near the energy troughs x=(−1, 1). To compare simulations to our experimental results on RR and SR modeled using a twostate Markov process, we binarized x such that x > 0 was classified as ‘anesthetized’ and x ≤ 0 was classified as ‘awake’. To further mimic the fact that RR and SR are performed intermittently, the binarized time series from the simulation was decimated 100 fold. This binary time series was modeled by a twostate transition probability matrix using the same methods as for the analysis of the RR and SR. The results of these simulations for different anesthetic concentrations and σ’s is shown in Figure 8D.
Statistical analyses
Request a detailed protocolAnalyses were performed using custom code written in Matlab using the Statistics and Machine Learning toolboxes (Mathworks, MA). Steady state population response probabilities were confirmed through Pearson correlation coefficient analysis between the trial index and the response probability averaged across animals. R^{2} values approaching zero indicate that the population is near a steady state. Statistical comparisons of medians were performed using the nonparametric MannWhitney U test. The BrownForsythe test for equal variances was used to compare variability in simulated and experimental response probabilities. p<0.05 were considered statistically significant for all comparisons.
Data availability
All data generated or analysed during this study are included in the manuscript and supporting files. Raw data and source data files have been provided as Supplementary files.
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Article and author information
Author details
Funding
National Institutes of Health (K08 GM123317)
 Andrew R McKinstryWu
National Institutes of Health (R01 GM124023)
 Alexander Proekt
National Institutes of Health (R01 GM088156)
 Max B Kelz
National Institutes of Health (R01 GM107117)
 Max B Kelz
National Institutes of Health (T32 GM112596)
 Max B Kelz
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Acknowledgements
We are grateful to Andrew Hudson, Connor Brennan, Sarah Reitz, Adeeti Aggarwal, and Brenna Shortal for their valuable discussion and suggestions about the data.
This work was supported by the National Institute of Health grants K08 GM123317, R01 GM124023, R01 GM088156, R01 GM107117, T32 GM112596, and by the Department of Anesthesiology and Critical Care at the University of Pennsylvania.
Ethics
Animal experimentation: Studies were approved by the Institutional Animal Care and Use Committee protocol (803844) at the University of Pennsylvania and were conducted in accordance with National Institutes of Health guidelines.
Copyright
© 2019, McKinstryWu et al.
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
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