All experimental biologists face a daily conundrum: on the

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1 Variability, compensation, and modulation in neurons and circuits Eve Marder 1 Department of Biology and Volen Center, Brandeis University, Waltham, MA Edited by Donald W. Pfaff, The Rockefeller University, New York, NY, and approved December 14, 2010 (received for review September 22, 2010) I summarize recent computational and experimental work that addresses the inherent variability in the synaptic and intrinsic conductances in normal healthy brains and shows that multiple solutions (sets of parameters) can produce similar circuit performance. I then discuss a number of issues raised by this observation, such as which parameter variations arise from compensatory mechanisms and which reflect insensitivity to those particular parameters. I ask whether networks with different sets of underlying parameters can nonetheless respond reliably to neuromodulation and other global perturbations. At the computational level, I describe a paradigm shift in which it is becoming increasingly common to develop families of models that reflect the variance in the biological data that the models are intended to illuminate rather than single, highly tuned models. On the experimental side, I discuss the inherent limitations of overreliance on mean data and suggest that it is important to look for compensations and correlations among as many system parameters as possible, and between each system parameter and circuit performance. This second paradigm shift will require moving away from measurements of each system component in isolation but should reveal important previously undescribed principles in the organization of complex systems such as brains. circuit dynamics neuronal variability neuronal homeostasis dynamic clamp All experimental biologists face a daily conundrum: on the one hand, we know that all individual biological organisms, be they lobsters, cats, or humans, are distinct individuals. On the other hand, we must do experiments on multiple individuals to ensure the reliability of our results. As biologists wishing to understand how the function of a cell, a circuit, or a brain depends on the properties of its constituent processes, we confront two issues: (i) all our data come with associated measurement error (often difficult to assess), and (ii) there is considerable natural variability in the populations we study. Consequently, we conventionally rely on statistics calculated from populations to assure ourselves that our measurements are reliable. Most commonly, we report mean data, with the underlying assumption that these means capture something akin to a platonic ideal of the individual neurons or animals whose properties were measured. Although this strategy has been enormously useful over the years, it has many limitations, some of which I discuss below. Multiple Solutions and Failure of Averaging Despite the widespread use of means, computational work has shown unambiguously some of the dangers and confounds that can come from exclusive reliance on mean data (1 3). One example of this comes from a study in which several thousand model neurons were generated by randomly picking the maximal conductances of the five different ionic conductances in the model (2). Fig. 1 shows three examples of single-spike bursters chosen from a population of 164 single-spike bursters generated in this study. Although each of the three depicted neurons shows very similar electrical behavior, their underlying conductances are quite different. Specifically, neuron 1 has a high Na + conductance and a low delayed rectifier K + conductance, neuron 3 has a low Na + conductance and a high delayed rectifier K + conductance, and neuron 2 has low values of both (Fig. 1A). This illustrates an important general principle, namely, that similar electrophysiological behavior can arise from widely disparate sets of underlying conductances (4 15). Fig. 1B shows a model neuron constructed from the mean values of the Na + and K + conductances of all the single-spike bursters. Unlike all the individual neurons whose values were averaged to calculate the mean conductances, the resulting model was a three-spike burster. Thus, the model developed using the mean data shows behavior different from all the individuals from which the mean conductances were calculated (2). This occurs because the single-spike bursters are found in a concave region in parameter space (2) (blue dots plotted in Fig. 1C). Models constructed from mean data need not necessarily fail; indeed, they may do so relatively infrequently. Nonetheless, it is important to be cognizant that when trying to understand how a biological system s behavior arises from the values of its underlying processes, mean data may be insufficient. Instead, it would be vastly preferable, to the extent to which it is technically feasible, to measure, in the same preparation, as many parameters as possible. Moreover, if there are nonlinear relationships among system components, these will not necessarily be obvious if the components are measured singly (2, 4, 9, 11). Multiple Solutions to Producing Similar Circuit Performance: Models of the Pyloric Rhythm Fig. 1 illustrates that similar activity patterns can be produced by different sets of underlying conductances at the level of a single neuron. The same principle holds at the level of circuit performance (8), as is shown in Fig. 2. The pyloric rhythm of the crustacean stomatogastric ganglion (STG) is a triphasic motor pattern in which three neuron types, the pyloric dilator (PD), lateral pyloric (LP), and pyloric (PY) neurons, fire in a stereotyped and repeating sequence (16, 17). Although the frequencies of the pyloric rhythms measured in vitro under the same conditions can vary, the phase relationships, or the timing of the activities of the three neurons, are quite constant in different preparations (11, 16). Prinz et al. (8) generated more than 20 million model three-cell networks, by varying the synaptic strengths and intrinsic properties within the network. Of these, about 400,000 produced patterns of activity similar to those of the biological pyloric rhythm networks (8). Fig. 2 shows two of these model pyloric rhythm networks that are producing almost identical motor patterns (Upper) but with disparate sets of synaptic and intrinsic conductances (Lower). Note that many of the underlying conductances are substantially different in these two This paper results from the Arthur M. Sackler Colloquium of the National Academy of Sciences, Quantification of Behavior held June 11 13, 2010, at the AAAS Building in Washington, DC. The complete program and audio filesofmostpresentationsare available on the NAS Web site at Author contributions: E.M. wrote the paper. The author declares no conflict of interest. This article is a PNAS Direct Submission. 1 marder@brandeis.edu. PNAS Early Edition 1of7

2 Fig. 2. Disparate circuit parameters can produce similar network activity. (Upper) Traces from two distinct model networks with similar network activity. (Lower) Synaptic and membrane conductance values for those networks. These two networks have markedly different parameter values despite having very similar activity. AB (Anterior Burster) (Reprinted from ref. 8.) In the remaining sections of this paper, I discuss recent experimental work that addresses each of these issues. Fig. 1. Averaging fails to capture the relationships among conductances. (A) (Left) Voltage traces of three one-spike burster model neurons. (Right) Histograms of the Na and K d conductances in those model neurons. (B) Voltage trace and conductance values of a model neuron with the mean Na and K d conductances. This model produces three spikes per burst. (C) Onespike bursters (blue) lie close to the axes of parameter space, whereas the average of the one-spike bursters lies outside the space of one-spike bursters. The point labeled 4 is a two-spike burster, and the point labeled 5 is a four-spike burster. The points labeled 1, 2, and 3 are the models shown in A. The oval shows the one SD covariance ellipse (2).g max (maximal conductance) (Used with permission from ref. 2.) cases. This raises a number of issues that need to be answered in direct biological experiments: i) How variable are conductance densities, intrinsic properties, and synaptic strengths in biological neurons and networks? In other words, how tightly tuned are neurons of the same cell type within an animal and across animals? How tightly regulated are the strengths of the same synapse in different animals? ii) Which parameter variations reflect the fact that a given property of the system s performance is not sensitive to that parameter, so that the parameter need not be tightly controlled, and which parameter differences reflect a series of compensatory mechanisms associated with patterns of correlations in parameters? iii) Can networks with different sets of underlying parameters nonetheless respond reliably to neuromodulation and other global perturbations? Variation in Biological Neurons and Networks There is an ever-increasing body of work showing that the values of synaptic and intrinsic conductances measured in neurons of the same cell type or among identified neurons vary as much as two- to sixfold across cells and across animals (2, 4, 9 11, 18 22). These data come from voltage-clamp measurements of isolated currents and synapses. Likewise, the expression levels seen for mrnas for various ion channels vary in the same range (20, 23 26). Additionally, a recent study finds that variation in neuronal excitability across neurons in the same population may serve to increase the information transfer by a circuit (27). Synaptic Strength Does Not Always Matter It is frequently assumed that changes in synaptic strength will alter the performance of the networks in which those connections are found. In many studies of synaptic plasticity, much is made of changes in synaptic strength of 30 50%. Indeed, changes of this amount may often result in functional changes in network performance. Nonetheless, there are circumstances in which changes in synaptic strength may have little or no effect on network performance (28, 29). This can be seen dramatically if one looks at the effects of inputs to a neuronal oscillator (28, 30). It is often useful to determine the influence of an input to an oscillator by measuring its phase-response curve (PRC), which captures the response of the oscillator to perturbations at different times in the oscillator s cycle (31 34). Fig. 3 shows families of PRCs made using the dynamic clamp to inject artificial synaptic conductances of varying strengths into the PD neuron of the lobster STG (28). The raw data traces show the effects of varying the strength of the synaptic inhibition both early (Fig. 3A) and late (Fig. 3B) in the PD neuron s cycle. Fig. 3C shows the full PRCs for synaptic conductances from 20 1,000 ns. Note that synaptic inhibition early in the cycle advances the time at which the following burst occurs, synaptic inhibition late in the cycle delays the time at which the following burst occurs, and synaptic inhibition midcycle has little functional effect. This demonstrates a well-known feature of many PRCs, that the efficacy of an input to an oscillator depends on its phase. 2of7 Marder

3 Parameter Compensation and Correlations When a neuron fires in response to a synaptic input depends on the strength and time course of its synaptic input and on its intrinsic membrane conductances. As a consequence of this, similar changes in network performance can result from changes in different network parameters (22, 35). Therefore, in principle, in all circuits, there will be sets of underlying parameters that are consistent with the production of similar circuit performance. One of the most straightforward indications of parameter compensations is genetic knockout of channel genes that have little phenotype (36, 37). Fig. 4 shows data from a study in which mouse cortical neurons were studied biophysically after the potassium channel gene Kv4.2 was deleted. Little or no change in electrophysiological phenotype was seen because of compensatory up-regulation in other K + channel subunits. Genetic overexpression experiments also reveal the existence of compensations. For example, injection of the mrna encoding the transient outward current (I A ) into lobster PD neurons resulted in large transient outward currents but no apparent change in the burst properties of the PD neurons (38). This was a surprising result at first, because much previous pharmacological work would have predicted that significant changes would result from an enhanced I A (39, 40). This puzzle was resolved by the finding that the overexpression of I A was accompanied by a compensating up-regulation of the hyperpolarization-activated inward current (I H ) (38, 41). Interestingly, strong correlations in the expression of I A and I H mrna are also seen in crab STG neurons (11, 20, 25), and it has been suggested that the specific patterns of correlations seen in identified neurons may be a signature of their identity (25, 26, 42, 43). Compensation at the circuit level can be seen in a recent study (44) in which identified neurons from the crab STG were coupled to a model neuron using the dynamic clamp (45) to form hybrid circuits (Fig. 5). Fig. 5A shows a schematic of the experimental design, in which artificial inhibitory synapses were made Fig. 3. Saturation of the effects of increasing synaptic strength. The dynamic clamp was used to create an artificial inhibitory synaptic conductance in a PD neuron. (A) Inhibitory conductances of various amplitudes injected into the PD cell early in its phase produced a phase advance. (B) Inhibitory conductances injected into the PD cell late in its phase produced a phase delay. (C) Full PRCs plotted for the injections of pulses of 20 ns (blue), 50 ns (green), 200 ns (red), and 1, 000 ns (black) into the PD neuron. Between phases of 0.4 and 0.6, an inhibitory conductance causes no phase shift. At earlier and later phases, increasing the strength of the input from small (blue) to moderate (red) increases the phase shift, but no additional increase is seen when the strength is further increased (black). P, period; L/P, latency/ period. From Thirumalai et al. (28). (Used with permission from ref. 28.) Fig. 3C illustrates another, equally important, feature of these PRCs: the effect of increasing the inhibitory synaptic conductance on the oscillator saturates (28, 30), so that once the synaptic input has reached a certain level, further increases in its amplitude produce no additional change in the PRC (seen as the overlap in the red and black dots in Fig. 3). In the case shown in this figure, once the conductance reaches a certain strength, additional increases in the strength of the synaptic input will have no further effect on the postsynaptic neuron, even at the phases at which the presynaptic neuron s actions are effective. Because the form of the PRC displayed by a given neuron depends on its intrinsic membrane currents, neurons with different underlying conductances may have different PRCs, and therefore different sensitivities to presynaptic inputs. Fig. 4. Some channel knockouts reveal compensation and demonstrate that there are multiple sets of conductances consistent with the production of the same output. (A) Action potential profiles of cortical pyramidal neurons from WT and mice with a knockout of Kv4.2 (Kv4.2 / ) are almost identical. (B) Accompanying phase diagrams are indistinguishable. dv/dt. (C) Although the total peak outward current densities were the same for WT and knockout mice (Left), I A was eliminated in knockout mice (Center). However, I K and I SS were up-regulated in the knockout mice to compensate (Right). I K, slowly decaying Kv currents; I SS, noninactivating Kv currents. [Reprinted with permission from ref. 37 (Copyright 2008, John Wiley and Sons).] Marder PNAS Early Edition 3of7

4 Fig. 5. Intrinsic properties determine network output. (A) Schematic of hybrid network containing a Morris Lecar (ML) model cell (64) and a biological LP cell. (B) Output of uncoupled cells. Different values for synaptic strength and h-conductance can produce an LP-dominated network (C, green), a modeldominated network (D, blue), or a half-center network (E, red) in which the model and biological cells burst in alternation. (F) Target half-center behavior produced by coupling three different biological neurons to the same model neuron. (G) Parameter combinations that produced target half-center activity in 12 preparations. (Reprinted from ref. 44.) between the model neuron and a biological neuron. Additionally, the dynamic clamp was used to add an artificial I H to the biological neuron. In isolation, 12 LP neurons showed significant variability in six different measures of intrinsic excitability (e.g., threshold, input resistance, spike frequency in response to injected current) (44). In these experiments, because the strengths of the inhibitory synapses and I H were varied, the resulting two-cell networks produced a variety of behaviors. The firing properties of the uncoupled cells are shown in Fig. 5B. Fig. 5C shows the case in which the biological neuron completely suppressed the model s activity. Fig. 5D shows a case in which the model completely suppressed the biological neuron s activity, and Fig. 5E shows a case in which the model and biological neuron fired in alternating bursts of activity. Despite the large variability in the excitability of the isolated LP neurons, a target network performance was found in each experiment with a combination of synaptic and I H parameters (Fig. 5F). The parameters that gave the target performance for each experiment are plotted in Fig. 5G. These data show that a target network performance can be achieved despite variability in the properties of a single neuron or a single synapse, by compensations of other circuit parameters. It is interesting to note that there is a two- to threefold range in the synaptic and I H conductances needed to compensate for the variability in the initial biological neurons. It is sometimes intuitively obvious that there are sets of currents that can compensate for each other, such as with the K + currents illustrated in Fig. 4. Slightly less intuitive are the compensations illustrated in Fig. 5, where the strength of an inhibitory synaptic input is balanced by the amount of I H. Other compensations that involve many voltage-dependent currents may not be intuitively obvious or simple. 4of7 Marder

5 Fig. 6. Serotonin has a variable effect on reciprocal inhibitory networks formed using the dynamic clamp from two GM cells. (Upper) Serotonin increased the oscillator frequency. (Lower) Serotonin decreased the oscillator frequency. Horizontal line on traces indicates 50 mv. (Scale bars: vertical, 20 mv; horizontal, 10 s.)(reprinted from ref. 57.) Can Circuits with Different Underlying Structures Respond Reliably to Perturbation? There must be perturbations that will differentiate among circuits with different sets of underlying parameters even if they produce similar behaviors (46). Therefore, it is interesting to ask how reliably different individual animals respond to the kinds of perturbations that they routinely see in their lifetimes. All neuronal circuits are modulated by many amines and peptides (47 49). Responses of individual neurons, synapses, and circuits to neuromodulation are often quite variable (50, 51). In some instances, it is clear that the effect of the modulator depends on some obvious feature of the state of the preparation (52 56). In some instances, this variability arises from unknown sources and is likely a consequence of differing underlying cell or circuit parameters. This is illustrated in the example shown in Fig. 6, in which two gastric mill (GM) neurons from the crab STG were connected using the dynamic clamp. The top two traces in Fig. 6 show the two GM neurons in one experiment, and the third and fourth traces show the two GM neurons in a different experiment, whose initial frequency was slower than that of the first. In the first experiment, serotonin increased the frequency of the alternating bursts of activity; in the second experiment, serotonin decreased the frequency of the alternating bursts of activity (57). Among many networks, although both responses to serotonin were seen, the mean response to serotonin was a highly significant increase in frequency. This illustrates that even highly reliable neuromodulatory responses of a population may hide individuals with anomalous responses (57), presumably because those individuals have a set of underlying parameters that, when modulated, produce a different response (4). Neuromodulatory substances affect specific sets of individual synaptic and intrinsic conductances at specific sites in neuronal circuits. In contrast, in cold-blooded animals, all the biological processes governing neuronal excitability are affected by the temperature of their environment. Thus, temperature is a global perturbation that influences all circuit parameters. Because the temperature dependencies of all biological processes are different, maintaining robust behavior in response to temperature perturbations is not obvious. Fig. 7 shows the effects of changing temperature on the crab pyloric rhythm (58). Fig. 7 A and B shows extracellular recordings from the same preparation at 7 C and 19 C. Note that although the frequency of the pyloric rhythm substantially increased, the normal triphasic rhythm was preserved. This is shown in pooled data for frequency in Fig. 7C and for pyloric rhythm phase relationships in Fig. 7D. The preservation of phase across a large temperature range requires mechanisms that maintain firing phase, although the frequency changes dramatically. The problem of maintaining phase as frequency changes has been studied for a long time (59 61) and depends on the interaction of several membrane currents. Therefore, it is remarkable that despite the fact that neurons and synapses in the pyloric rhythm are variable (11, 20, 25), the effects of temperature on both the frequency and phase of the component pyloric network neurons were extremely reliable. This demonstrates that the parameter variability existing across individuals is nonetheless consistent with reliable responses to common environmental perturbations. It also illustrates that although Fig. 7. Pyloric rhythm frequency changes as a function of temperature but remains phase-invariant. (A) Pyloric rhythm at 7 C. (B) Same preparation as in A but at 19 C. (C) Semilog plot of the network frequency as a function of temperature shows a Q 10 of Q 10 (the change in output resulting from a 10 change in temperature). (D) Phase relationships of firing of the pyloric network neurons remain constant over a large temperature range remain constant. (Reprinted from ref. 58.) Marder PNAS Early Edition 5of7

6 motor pattern frequency commonly fluctuates, the phase relationships of the constituent neurons must be maintained to produce a functional behavior. Therefore, it is possible that there is strong environmental pressure for animals to have found mechanisms that support temperature compensation of phase (58). Conclusions The performance of brain circuits depends on the complex interaction between the intrinsic properties of brain neurons and their synaptic connections. Much work in cellular electrophysiology and biophysics has gone into describing in detail how each synaptic and intrinsic conductance behaves as a function of voltage, ligand binding, time, and history of activity. This has led to an ever-increasing use of computational models to allow us to understand the interactions between system components and circuit function. Two paradigm shifts are a consequence of this work: (i) It is now feasible, and useful in many cases, to construct families of models rather than single highly tuned models to capture the behavior of individual neurons and circuits (4 8, 12, 13, 22, 62, 63), and (ii) the importance of trying to measure as many system components as possible within an individual is now evident. To the extent to which it is experimentally feasible, it is important to measure both the system s behavior and the properties of several of its component properties to look for correlations between them and the performance of the circuit and/or animal (2, 9, 11, 42). This realization should alter the design of much experimental work designed to shape our understanding of how the properties of circuits depend on their underlying structure. Eventually, understanding the rules that govern correlations and compensations in the nervous system will allow us to understand the diversity of healthy individuals and why specific brain disorders occur. ACKNOWLEDGMENTS. I thank Gabrielle Gutierrez for help with figure and manuscript preparation.this work was supported by National Institutes of Health Grant MH46742 and the James D. McDonnell Foundation. 1. Beer RD, Chiel HJ, Gallagher JC (1999) Evolution and analysis of model CPGs for walking: II. General principles and individual variability. J Comput Neurosci 7: Golowasch J, Goldman MS, Abbott LF, Marder E (2002) Failure of averaging in the construction of a conductance-based neuron model. J Neurophysiol 87: Foster WR, Ungar LH, Schwaber JS (1993) Significance of conductances in Hodgkin- Huxley models. J Neurophysiol 70: Goldman MS, Golowasch J, Marder E, Abbott LF (2001) Global structure, robustness, and modulation of neuronal models. 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