Variability, compensation, and modulation inneurons and circuitsEve Marder1

Department of Biology and Volen Center, Brandeis University, Waltham, MA 02454

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 thataddresses the inherent variability in the synaptic and intrinsicconductances in normal healthy brains and shows that multiplesolutions (sets of parameters) can produce similar circuit perfor-mance. I then discuss a number of issues raised by this observa-tion, such as which parameter variations arise from compensatorymechanisms and which reflect insensitivity to those particularparameters. I ask whether networks with different sets of un-derlying parameters can nonetheless respond reliably to neuro-modulation and other global perturbations. At the computationallevel, I describe a paradigm shift in which it is becoming in-creasingly common to develop families of models that reflect thevariance in the biological data that the models are intended toilluminate rather than single, highly tuned models. On theexperimental side, I discuss the inherent limitations of overrelianceon mean data and suggest that it is important to look forcompensations and correlations among as many system parame-ters as possible, and between each system parameter and circuitperformance. This second paradigm shift will require moving awayfrom measurements of each system component in isolation butshould reveal important previously undescribed principles in theorganization of complex systems such as brains.

circuit dynamics | neuronal variability | neuronal homeostasis | dynamicclamp

All experimental biologists face a daily conundrum: on theone hand, we know that all individual biological organisms,

be they lobsters, cats, or humans, are distinct individuals. On theother hand, we must do experiments on multiple individuals toensure the reliability of our results. As biologists wishing to un-derstand how the function of a cell, a circuit, or a brain dependson the properties of its constituent processes, we confront twoissues: (i) all our data come with associated measurement error(often difficult to assess), and (ii) there is considerable naturalvariability in the populations we study. Consequently, we con-ventionally rely on statistics calculated from populations to assureourselves that our measurements are reliable. Most commonly,we report mean data, with the underlying assumption that thesemeans capture something akin to a “platonic ideal” of the in-dividual 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 AveragingDespite the widespread use of means, computational work hasshown unambiguously some of the dangers and confounds thatcan come from exclusive reliance on mean data (1–3). One ex-ample of this comes from a study in which several thousand modelneurons were generated by randomly picking the maximal con-ductances of the five different ionic conductances in the model(2). Fig. 1 shows three examples of single-spike bursters chosenfrom a population of 164 single-spike bursters generated in thisstudy. Although each of the three depicted neurons shows verysimilar electrical behavior, their underlying conductances arequite different. Specifically, neuron 1 has a high Na+ conductanceand 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 animportant general principle, namely, that similar electrophysio-logical behavior can arise from widely disparate sets of underlyingconductances (4–15).Fig. 1B shows a model neuron constructed from the mean

values of the Na+ and K+ conductances of all the single-spikebursters. Unlike all the individual neurons whose values wereaveraged to calculate the mean conductances, the resulting modelwas a three-spike burster. Thus, the model developed usingthe mean data shows behavior different from all the individualsfrom which the mean conductances were calculated (2). Thisoccurs because the single-spike bursters are found in a concaveregion 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 isimportant to be cognizant that when trying to understand howa biological system’s behavior arises from the values of its un-derlying processes, mean data may be insufficient. Instead, itwould be vastly preferable, to the extent to which it is technicallyfeasible, to measure, in the same preparation, as many parame-ters as possible. Moreover, if there are nonlinear relationshipsamong system components, these will not necessarily be obviousif the components are measured singly (2, 4, 9, 11).

Multiple Solutions to Producing Similar Circuit Performance:Models of the Pyloric RhythmFig. 1 illustrates that similar activity patterns can be produced bydifferent sets of underlying conductances at the level of a singleneuron. The same principle holds at the level of circuit perfor-mance (8), as is shown in Fig. 2. The pyloric rhythm of thecrustacean stomatogastric ganglion (STG) is a triphasic motorpattern in which three neuron types, the pyloric dilator (PD),lateral pyloric (LP), and pyloric (PY) neurons, fire in a stereo-typed and repeating sequence (16, 17). Although the frequenciesof the pyloric rhythms measured in vitro under the same con-ditions can vary, the phase relationships, or the timing of theactivities of the three neurons, are quite constant in differentpreparations (11, 16). Prinz et al. (8) generated more than 20million model three-cell networks, by varying the synapticstrengths and intrinsic properties within the network. Of these,about 400,000 produced patterns of activity similar to those ofthe biological pyloric rhythm networks (8). Fig. 2 shows two ofthese model pyloric rhythm networks that are producing almostidentical motor patterns (Upper) but with disparate sets of syn-aptic and intrinsic conductances (Lower). Note that many of theunderlying conductances are substantially different in these two

This paper results from the Arthur M. Sackler Colloquium of the National Academy ofSciences, “Quantification of Behavior” held June 11–13, 2010, at the AAAS Building inWashington, DC. The complete program and audio files of most presentations areavailable on the NAS Web site at www.nasonline.org/quantification.

Author contributions: E.M. wrote the paper.

The author declares no conflict of interest.

This article is a PNAS Direct Submission.1E-mail: marder@brandeis.edu.

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cases. This raises a number of issues that need to be answered indirect biological experiments:

i) How variable are conductance densities, intrinsic proper-ties, and synaptic strengths in biological neurons and net-works? In other words, how tightly tuned are neurons ofthe same cell type within an animal and across animals?How tightly regulated are the strengths of the same syn-apse in different animals?

ii) Which parameter variations reflect the fact that a givenproperty of the system’s performance is not sensitive tothat parameter, so that the parameter need not be tightlycontrolled, and which parameter differences reflect a seriesof compensatory mechanisms associated with patterns ofcorrelations in parameters?

iii) Can networks with different sets of underlying parametersnonetheless respond reliably to neuromodulation andother global perturbations?

In the remaining sections of this paper, I discuss recent ex-perimental work that addresses each of these issues.

Variation in Biological Neurons and NetworksThere is an ever-increasing body of work showing that the valuesof synaptic and intrinsic conductances measured in neurons ofthe same cell type or among identified neurons vary as much astwo- to sixfold across cells and across animals (2, 4, 9–11, 18–22).These data come from voltage-clamp measurements of isolatedcurrents and synapses. Likewise, the expression levels seen formRNAs for various ion channels vary in the same range (20, 23–26). Additionally, a recent study finds that variation in neuronalexcitability across neurons in the same population may serve toincrease the information transfer by a circuit (27).

Synaptic Strength Does Not Always MatterIt is frequently assumed that changes in synaptic strength willalter the performance of the networks in which those con-nections are found. In many studies of synaptic plasticity, much ismade of changes in synaptic strength of 30–50%. Indeed,changes of this amount may often result in functional changes innetwork performance. Nonetheless, there are circumstances inwhich changes in synaptic strength may have little or no effect onnetwork performance (28, 29). This can be seen dramatically ifone looks at the effects of inputs to a neuronal oscillator (28, 30).It is often useful to determine the influence of an input to anoscillator by measuring its phase-response curve (PRC), whichcaptures the response of the oscillator to perturbations at dif-ferent times in the oscillator’s cycle (31–34). Fig. 3 shows familiesof PRCs made using the dynamic clamp to inject artificial syn-aptic conductances of varying strengths into the PD neuron ofthe lobster STG (28). The raw data traces show the effects ofvarying the strength of the synaptic inhibition both early (Fig.3A) and late (Fig. 3B) in the PD neuron’s cycle. Fig. 3C showsthe full PRCs for synaptic conductances from 20–1,000 nS. Notethat synaptic inhibition early in the cycle advances the time atwhich the following burst occurs, synaptic inhibition late in thecycle delays the time at which the following burst occurs, andsynaptic inhibition midcycle has little functional effect. Thisdemonstrates a well-known feature of many PRCs, that the ef-ficacy of an input to an oscillator depends on its phase.

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 Kd conductances in those model neurons. (B)Voltage trace and conductance values of a model neuron with the mean Naand Kd conductances. This model produces three spikes per burst. (C) One-spike bursters (blue) lie close to the axes of parameter space, whereas theaverage of the one-spike bursters lies outside the space of one-spike bur-sters. The point labeled 4 is a two-spike burster, and the point labeled 5 isa four-spike burster. The points labeled 1, 2, and 3 are the models shown inA. The oval shows the one SD covariance ellipse (2).gmax (maximal conduc-tance) (Used with permission from ref. 2.)

Fig. 2. Disparate circuit parameters can produce similar network activity.(Upper) Traces from two distinct model networks with similar network ac-tivity. (Lower) Synaptic and membrane conductance values for those net-works. These two networks have markedly different parameter values despitehaving very similar activity. AB (Anterior Burster) (Reprinted from ref. 8.)

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Fig. 3C illustrates another, equally important, feature of thesePRCs: the effect of increasing the inhibitory synaptic conduc-tance on the oscillator saturates (28, 30), so that once the syn-aptic input has reached a certain level, further increases in itsamplitude produce no additional change in the PRC (seen as theoverlap in the red and black dots in Fig. 3). In the case shown inthis figure, once the conductance reaches a certain strength,additional increases in the strength of the synaptic input willhave no further effect on the postsynaptic neuron, even at thephases at which the presynaptic neuron’s actions are effective.Because the form of the PRC displayed by a given neurondepends on its intrinsic membrane currents, neurons with dif-ferent underlying conductances may have different PRCs, andtherefore different sensitivities to presynaptic inputs.

Parameter Compensation and CorrelationsWhen a neuron fires in response to a synaptic input depends onthe strength and time course of its synaptic input and on its in-

trinsic membrane conductances. As a consequence of this, sim-ilar changes in network performance can result from changes indifferent network parameters (22, 35). Therefore, in principle, inall circuits, there will be sets of underlying parameters that areconsistent with the production of similar circuit performance.One of the most straightforward indications of parameter

compensations is genetic knockout of channel genes that havelittle phenotype (36, 37). Fig. 4 shows data from a study in whichmouse cortical neurons were studied biophysically after the po-tassium channel gene Kv4.2 was deleted. Little or no change inelectrophysiological phenotype was seen because of compensa-tory up-regulation in other K+ channel subunits.Genetic overexpression experiments also reveal the existence

of compensations. For example, injection of the mRNA encod-ing the transient outward current (IA) into lobster PD neuronsresulted in large transient outward currents but no apparentchange in the burst properties of the PD neurons (38). This wasa surprising result at first, because much previous pharmaco-logical work would have predicted that significant changes wouldresult from an enhanced IA (39, 40). This puzzle was resolved bythe finding that the overexpression of IA was accompanied bya compensating up-regulation of the hyperpolarization-activatedinward current (IH) (38, 41). Interestingly, strong correlations inthe expression of IA and IH mRNA are also seen in crab STGneurons (11, 20, 25), and it has been suggested that the specificpatterns of correlations seen in identified neurons may be a sig-nature 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 cou-pled to a model neuron using the dynamic clamp (45) to formhybrid circuits (Fig. 5). Fig. 5A shows a schematic of the exper-imental design, in which artificial inhibitory synapses were made

Fig. 3. Saturation of the effects of increasing synaptic strength. The dy-namic clamp was used to create an artificial inhibitory synaptic conductancein a PD neuron. (A) Inhibitory conductances of various amplitudes injectedinto the PD cell early in its phase produced a phase advance. (B) Inhibitoryconductances injected into the PD cell late in its phase produced a phasedelay. (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. Betweenphases of 0.4 and 0.6, an inhibitory conductance causes no phase shift. Atearlier and later phases, increasing the strength of the input from small(blue) to moderate (red) increases the phase shift, but no additional increaseis 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. 4. Some channel knockouts reveal compensation and demonstrate thatthere are multiple sets of conductances consistent with the production ofthe same output. (A) Action potential profiles of cortical pyramidal neuronsfrom WT and mice with a knockout of Kv4.2 (Kv4.2−/−) are almost identical.(B) Accompanying phase diagrams are indistinguishable. dV/dt. (C) Althoughthe total peak outward current densities were the same for WT andknockout mice (Left), IA was eliminated in knockout mice (Center). However,IK and ISS were up-regulated in the knockout mice to compensate (Right). IK,slowly decaying Kv currents; ISS, noninactivating Kv currents. [Reprinted withpermission from ref. 37 (Copyright 2008, John Wiley and Sons).]

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between the model neuron and a biological neuron. Addition-ally, the dynamic clamp was used to add an artificial IH to thebiological neuron. In isolation, 12 LP neurons showed significantvariability in six different measures of intrinsic excitability (e.g.,threshold, input resistance, spike frequency in response toinjected current) (44). In these experiments, because thestrengths of the inhibitory synapses and IH were varied, theresulting two-cell networks produced a variety of behaviors. Thefiring properties of the uncoupled cells are shown in Fig. 5B. Fig.5C shows the case in which the biological neuron completelysuppressed the model’s activity. Fig. 5D shows a case in which themodel completely suppressed the biological neuron’s activity,and Fig. 5E shows a case in which the model and biologicalneuron fired in alternating bursts of activity. Despite the largevariability in the excitability of the isolated LP neurons, a targetnetwork performance was found in each experiment with

a combination of synaptic and IH parameters (Fig. 5F). Theparameters that gave the target performance for each experi-ment are plotted in Fig. 5G. These data show that a target net-work performance can be achieved despite variability in theproperties of a single neuron or a single synapse, by compensa-tions of other circuit parameters. It is interesting to note thatthere is a two- to threefold range in the synaptic and IH con-ductances needed to compensate for the variability in the initialbiological neurons.It is sometimes intuitively obvious that there are sets of cur-

rents that can compensate for each other, such as with the K+

currents illustrated in Fig. 4. Slightly less intuitive are the com-pensations illustrated in Fig. 5, where the strength of an in-hibitory synaptic input is balanced by the amount of IH. Othercompensations that involve many voltage-dependent currentsmay not be intuitively obvious or simple.

Fig. 5. Intrinsic properties determine network output. (A) Schematic of hybrid network containing a Morris–Lecar (ML) model cell (64) and a biological LPcell. (B) Output of uncoupled cells. Different values for synaptic strength and h-conductance can produce an LP-dominated network (C, green), a model-dominated network (D, blue), or a half-center network (E, red) in which the model and biological cells burst in alternation. (F) Target half-center behaviorproduced by coupling three different biological neurons to the same model neuron. (G) Parameter combinations that produced target half-center activity in12 preparations. (Reprinted from ref. 44.)

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Can Circuits with Different Underlying Structures RespondReliably to Perturbation?There must be perturbations that will differentiate among cir-cuits with different sets of underlying parameters even if theyproduce similar behaviors (46). Therefore, it is interesting to askhow reliably different individual animals respond to the kinds ofperturbations that they routinely see in their lifetimes.All neuronal circuits are modulated by many amines and

peptides (47–49). Responses of individual neurons, synapses, andcircuits to neuromodulation are often quite variable (50, 51). Insome instances, it is clear that the effect of the modulatordepends on some obvious feature of the state of the preparation(52–56). In some instances, this variability arises from unknownsources and is likely a consequence of differing underlying cellor circuit parameters. This is illustrated in the example shown inFig. 6, in which two gastric mill (GM) neurons from the crabSTG were connected using the dynamic clamp. The top twotraces in Fig. 6 show the two GM neurons in one experiment, and

the third and fourth traces show the two GM neurons in a dif-ferent experiment, whose initial frequency was slower than that ofthe first. In the first experiment, serotonin increased the fre-quency of the alternating bursts of activity; in the second exper-iment, serotonin decreased the frequency of the alternatingbursts of activity (57). Among many networks, although both re-sponses to serotonin were seen, the mean response to serotoninwas a highly significant increase in frequency. This illustrates thateven highly reliable neuromodulatory responses of a populationmay hide individuals with anomalous responses (57), presumablybecause those individuals have a set of underlying parametersthat, when modulated, produce a different response (4).Neuromodulatory substances affect specific sets of individual

synaptic and intrinsic conductances at specific sites in neuronalcircuits. In contrast, in cold-blooded animals, all the biologicalprocesses governing neuronal excitability are affected by thetemperature of their environment. Thus, temperature is a globalperturbation that influences all circuit parameters. Because thetemperature dependencies of all biological processes are differ-ent, maintaining robust behavior in response to temperatureperturbations is not obvious. Fig. 7 shows the effects of changingtemperature on the crab pyloric rhythm (58). Fig. 7 A and Bshows extracellular recordings from the same preparation at 7 °Cand 19 °C. Note that although the frequency of the pyloricrhythm substantially increased, the normal triphasic rhythm waspreserved. This is shown in pooled data for frequency in Fig. 7Cand for pyloric rhythm phase relationships in Fig. 7D. Thepreservation of phase across a large temperature range requiresmechanisms that maintain firing phase, although the frequencychanges dramatically. The problem of maintaining phase as fre-quency changes has been studied for a long time (59–61) anddepends on the interaction of several membrane currents.Therefore, it is remarkable that despite the fact that neurons andsynapses in the pyloric rhythm are variable (11, 20, 25), the effectsof temperature on both the frequency and phase of the compo-nent pyloric network neurons were extremely reliable. Thisdemonstrates that the parameter variability existing across indi-viduals is nonetheless consistent with reliable responses to com-mon 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 Abut at 19 °C. (C) Semilog plot of the network frequency as a function of temperature shows a Q10 of 2.32. Q10 (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.)

Fig. 6. Serotonin has a variable effect on reciprocal inhibitory networksformed using the dynamic clamp from two GM cells. (Upper) Serotonin in-creased the oscillator frequency. (Lower) Serotonin decreased the oscillatorfrequency. Horizontal line on traces indicates −50 mV. (Scale bars: vertical,20 mV; horizontal, 10 s.)(Reprinted from ref. 57.)

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motor pattern frequency commonly fluctuates, the phase rela-tionships of the constituent neurons must be maintained to pro-duce a functional behavior. Therefore, it is possible that there isstrong environmental pressure for animals to have found mech-anisms that support temperature compensation of phase (58).

ConclusionsThe performance of brain circuits depends on the complex in-teraction between the intrinsic properties of brain neurons andtheir synaptic connections. Much work in cellular electrophysi-ology and biophysics has gone into describing in detail how eachsynaptic and intrinsic conductance behaves as a function ofvoltage, ligand binding, time, and history of activity. This has ledto an ever-increasing use of computational models to allow us tounderstand the interactions between system components andcircuit function. Two paradigm shifts are a consequence of thiswork: (i) It is now feasible, and useful in many cases, to constructfamilies 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 asmany system components as possible within an individual is nowevident. To the extent to which it is experimentally feasible, it isimportant to measure both the system’s behavior and the prop-erties of several of its component properties to look for corre-lations between them and the performance of the circuit and/oranimal (2, 9, 11, 42). This realization should alter the design ofmuch experimental work designed to shape our understanding ofhow the properties of circuits depend on their underlyingstructure. Eventually, understanding the rules that govern cor-relations and compensations in the nervous system will allow usto understand the diversity of healthy individuals and why spe-cific brain disorders occur.

ACKNOWLEDGMENTS. I thank Gabrielle Gutierrez for help with figure andmanuscript preparation.This work was supported by National Institutes ofHealth Grant MH46742 and the James D. McDonnell Foundation.

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