# What the fly’s nose tells the fly’s brain

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Contributed by Charles F. Stevens, June 2, 2015 (sent for review March 15, 2015; reviewed by Larry F. Abbott)

## Significance

This work is significant for three reasons. First, a new way the brain represents information is identified, one in which the information present in a large population of neurons can be gotten from a small subset of any neurons rather that from a subset of certain specific neurons. Second, the nature of the combinatorial code used to specify odor identity is identified, and this same type of code may be used in other contexts. Third, the way fly olfaction works can potentially provide a model for understanding the function of important vertebrate brain regions that appear to share the same neural circuit architecture.

## Abstract

The fly olfactory system has a three-layer architecture: The fly’s olfactory receptor neurons send odor information to the first layer (the encoder) where this information is formatted as combinatorial odor code, one which is maximally informative, with the most informative neurons firing fastest. This first layer then sends the encoded odor information to the second layer (decoder), which consists of about 2,000 neurons that receive the odor information and “break” the code. For each odor, the amplitude of the synaptic odor input to the 2,000 second-layer neurons is approximately normally distributed across the population, which means that only a very small fraction of neurons receive a large input. Each odor, however, activates its own population of large-input neurons and so a small subset of the 2,000 neurons serves as a unique tag for the odor. Strong inhibition prevents most of the second-stage neurons from firing spikes, and therefore spikes from only the small population of large-input neurons is relayed to the third stage. This selected population provides the third stage (the user) with an odor label that can be used to direct behavior based on what odor is present.

A hallmark of at least three major brain structures found in essentially all vertebrates—cerebellum, hippocampus, and olfactory system—is an architecture with three stages of information processing (Fig. 1). In the first stage (the encoder), information arriving from other brain areas is assembled into a combinatorial code and relayed, with a massive expansion of neuron number, to the second stage. This code is “broken” by the second stage (the decoder), and passed to the third stage, where the desired pieces of decoded information are selected for use in other brain regions.

The first proposal for the operation of this three-stage processing architecture was made by Marr (1), over four decades ago, to explain the function of the cerebellum, and I shall refer to the first two (encoder/decoder) stages of the architecture as the Marr motif. According to Marr, the encoder stage provides the pattern (neuronal activity compiled in precerebellar nuclei) that is relayed to cerebellar granule cells (the decoder stage). In granule cells, the pattern provided by the precerebellar neurons is separated by spreading the information over many more neurons (there are many more granule cells than precerebellar neurons), and by quieting most of the granule cells with strong inhibition from Golgi inhibitory neurons. These inhibitory neurons collect the output of many granule cells and feed it back to them. Finally, in the third stage (Purkinje cells), some parts of the separated pattern relayed by the granule cells are selected for labeling by concurrent climbing fiber activity that adjusts the strength of synapses conveying the chosen signals to Purkinje cells. Whenever labeled signals happened to recur, the modified synapses cause selected Purkinje cells change their firing rates and provide an output based on the tagged signals.

Although Marr’s proposal has been enormously influential, we still do not understand the combinatorial code (Marr’s pattern) generated by the precerebellar neurons, nor do we know how it is decoded by the granule cells (Marr’s pattern separation). What appears to be this same three-stage architecture is used by *Drosophila* for the first three levels of its olfactory system (2), but the fly system is much smaller, simpler, and more completely understood than any vertebrate version. Here I exploit the simplicity and extensive knowledge about the fly olfactory system to learn the properties of the combinatorial odor code, and how it is decoded. My hope is that what is learned from the fly will help us understand the similar three-level architecture in vertebrates.

## Results

I will consider (see Fig. 1) only the first two stages (the Marr motif) of the fly’s three-stage architecture. The first stage is the antennal lobe (corresponding to the encoder stage precerebellar nuclei) and the second stage consists of Kenyon cells of the mushroom body (corresponding to the decoder stage cerebellar granule cells). The basic outline of the process is quite simple: the antennal lobe collects information from olfactory receptor neurons and reformats this information into a combinatorial code, which is sent to the mushroom body. The mushroom body then converts the combinatorially coded odor into a unique output that can be used by the fly to identify the odor.

My discussion is divided into five sections. In the first section, I present information about the fly’s olfactory system that is needed in the later sections, and in the second section I describe how olfactory information is represented in the main cell type of the mushroom body, the Kenyon cells. The third section identifies properties of the combinatorial code constructed in the antennal lobe, and these properties are used in the fourth section to explain how a unique Kenyon cell output is generated for each odor. In the last section I identify some idealizations made in the earlier sections and explain the consequences of departures from these idealizations.

### Structure and Function of the Fly Olfactory System.

Information about the fly’s odor environment is delivered to the first stage of the three-stage olfactory system architecture, the antennal lobe, by axons of olfactory receptor neurons (ORNs) (3). Each of these ORNs expresses just one of the adult fly’s ∼50 odorant receptors (the actual best current estimate is 54; ref. 4), and all ORNs expressing that receptor converge on a structure in the antennal lobe called a glomerulus. Each of the about 50 glomeruli is a spherical neuropil structure with three main components: (*i*) ORN axon terminals, (*ii*) dendrites of several projection neurons that send modified odor information from ORN terminals to Kenyon cells in the mushroom body, and (*iii*) circuitry for modifying the ORN input to give the odor code the right properties. These modifications include lateral inhibition from other glomeruli onto ORN axon terminals to adjust the firing rate of projection neurons so they are mostly independent of odor concentration (a gain control mechanism; refs. 5⇓–7, 9, 10), and other mechanisms to spread information about the odor environment more evenly over all of the glomeruli (3, 6, 7, 9, 10). Projection neurons have a background firing rate of about 5 Hz. The projection neurons from a glomerulus do not increase their rate much for most odors, but a lot (>300 Hz) for some odors (8). Generally, different odors produce the highest projection neuron firing rates for different glomeruli but, of course, the same odor always produces the same average response in the same glomeruli. Projection neurons in the antennal lobe send their output to two major brain regions, the lateral horn (11, 12) (where evolution has designed circuits to deal with specific odors) and the mushroom body (where the fly can learn to recognize arbitrary odors). In the following, I will not consider the lateral horn, but will focus on the mushroom body, which contains the second stage of the Marr motif.

The mushroom body (13, 14) is a large (for the fly brain) structure with a neuropil calyx that contains the dendrites of many intrinsic neurons, the Kenyon cells (the Kenyon cell bodies are grouped just next to the calyx), and with two elongated lobes, one vertical and one horizontal, to which Kenyon cell axons project. The Kenyon cells and their associated circuitry constitute the second stage of the Marr motif (corresponding to the cerebellar granule cells), and the lobes contain the third stage (corresponding to the cerebellar Purkinje cells) of the three-stage architecture. This third stage will not be considered in most of the following discussion.

All axon terminals of antennal lobe projection neurons are found in special structures, called microglomeruli (15), in the mushroom body calyx. Each microglomerulus contains three main components: (*i*) a single large axon terminal from an antennal lobe projection neuron, (*ii*) dendritic structures, called claws, from about 10 Kenyon cells, each of which is postsynaptic to the single projection neuron terminal, and (*iii*) GABAergic terminals from a single anterior paired lateral (APL) neuron that also form synapses on Kenyon cell claws. Each Kenyon cell has about six claws (16), so one Kenyon cell collects information from only about 6 of the 50 glomeruli. The single APL neuron (only one on each side of the brain) sends its dendrites into the mushroom body lobes to collect the output from all of the Kenyon cell axons, and this APL neuron inhibits all Kenyon cells by sending its axons to the calycal microglomeruli to terminate on Kenyon cell dendrites (18⇓–20).

Three experimental observations will be important in the following discussion. First, the 50 glomeruli send odor information to about 2,000 Kenyon cells, so there is a 40-fold expansion from antennal lobe to mushroom body. Second, each Kenyon cell collects odor information from only about 10% of the glomeruli, and it takes this sample randomly (16, 17). Third, most odors do not produce synaptic currents in most Kenyon cells, but even when Kenyon cells do receive synaptic input on presentation of an odor, only about 5% of those Kenyon cells fire spikes reliably (21); this sparsity of firing is the result of the winner-take-all circuit formed by the inhibitory APL neuron (18).

### A Distributed Representation of Olfactory Information in the Mushroom Body.

In this section I will argue that the mushroom body uses a distributed representation of odor information rather than the more familiar localized representation. I make the distinction between these two ways information is represented by comparing the mammalian visual cortex (V1) with the mushroom body. V1 has a retinotopic map of the visual world but, on a finer scale, this visual area has a pinwheel organization where neurons with the same orientation preference are found along the spokes of the pinwheel, and neurons with the same receptive field size (spatial frequency) are arranged in concentric rings around the pinwheel center (note that V1 neurons are jointly selective for orientation and spatial frequency) (22). This means that all of the information about one tiny patch of the visual world can be found in the neurons making up one pinwheel. I have just described a localized representation: a specific group of cells can give you a specific part of all information available.

In contrast, I will argue that, for the mushroom body, one can get all of the information available about the fly’s odor environment from a specific number of any Kenyon cells, no matter which ones are chosen. In this sense, the information is distributed because the number of neurons, not their identity, is what matters.

I pointed out above that each Kenyon cell receives input from six glomeruli selected at random (16). To see why this gives a distributed representation of the odor information provided by the 50 glomeruli, I need to formalize this statement to make a connection with a new field of mathematics and computer science called compressed sensing (23).

We start with an empty list **s** with places for 50 numbers. Call this list **s** the signal, and fill it with the 50 projection neuron firing rates generated, at some particular time, by all of the glomeruli. Now create an empty table

Now multiply the signal **s** by the sensing matrix **r**. This **r** is a 2,000-long list of olfactory inputs (measured as spikes per second) received by each Kenyon cell:**s** (first by first, second by second, etc.) and then adding up all of the products and putting the sum in the first entry of **r** will contain the total number of spikes per second that each Kenyon cells is receiving from the antennal lobe glomeruli.

What I just described is called a random projection, and if one looks at any 50 of the 2,000 entries in the Kenyon cell synaptic input list **r**, one can always get back the original signal **s**. That is, all of the information the fly has about its odor environment is available to any 50/2,000 Kenyon cells.

To actually get back olfactory information in **s** (the projection neuron firing rate produced in each glomerulus) from **r** (the total olfactory input into each Kenyon cell), pick any 50 Kenyon cells (50 rows of **R**) and extract their inputs from the list **r** and their corresponding 50 rows from the large matrix **R**. This procedure gives a new shorter 50-long list

Actually, things are somewhat better than I just described. The main result from the new field of compressed sensing is that, depending on how much redundancy there is in the signal **s**, all of the information about the odor represented by **r** is, in fact, present in fewer, generally many fewer, than 50 entries in **r** (24). Based on experience from many types of signals, one would expect that any one or two dozen entries of the 2,000-long list **r** of Kenyon cell inputs would show everything the fly can know about the odor present.

The reason for using the random projection to give a 40-fold expansion (from 50 glomeruli to 2,000 Kenyon cells) is to provide many different representations of the odor for the Kenyon cells to pick between to find an output that best characterizes the odor and is very likely to be different from the Kenyon cell output for almost any other odor. If only 20 of the 2,000 numbers in **r** are needed to define the odor, the expansion is 100-fold rather than 40-fold, and the Kenyon cells are given more representations, and thus more chances to find the representation that best characterizes the odor. The collection of Kenyon cells that reliably fire in response to a particular odor constitutes the neural ‘tag’ associated with that odor. How the Kenyon cells choose an odor’s tag is the subject of the following two sections.

We know that the odor information in the antennal lobe is indeed mapped to the Kenyon cells by a random projection (16), but is it true, as just predicted, that the antennal lobe response to an odor can be recovered from any small sample of Kenyon cell activity? Campbell et al. (25) showed that behavioral performance to odor presentations could be predicted from 25 randomly selected Kenyon cells out of a larger population of neurons whose activity in response to odors had been measured using calcium imaging. Thus, as predicted, the odor information is distributed over a large number of neurons, and this information can be recovered from a randomly selected small subset of the entire population.

### Properties of the Combinatorial Odor Code.

We know that the Kenyon cells decode the combinatorial odor code—that is, they provide the tag associated with any odor—because flies have been shown to use the output of Kenyon cells to learn odors that have been paired with reward or punishment (26). The previous section demonstrated that odors are represented in a distributed way over the population of Kenyon cells, but to see how these cells provide a (close to) unique output for each odor, we need to know properties of the combinatorial code produced by the antennal lobe. That is the task undertaken in this section.

The purpose of the odor code generated by the antennal lobe is to let the fly identify as many odors as possible. The combinatorial odor code, then, that best accomplishes this should provide the greatest survival value for the fly. In other words, the code should be the most informative one. According to information theory (27), the best combinatorial odor code should be the one with maximum entropy. Entropy is precisely defined and roughly has to do with how many distinct odors can be coded for by the antennal lobe. Maximum entropy codes are known to have two properties used in the following paragraphs.

A single glomerulus and its projection neurons constitute an information channel, and I now focus on such a channel. We know that, if an odor is selected at random, our glomerulus usually will not respond much, but for some odors it will respond well, and sometimes, for odors that bind best to its odorant receptor protein, the odor response for that glomerulus will be maximal. For the *k*th glomerulus, then, the probability density *r* should be in a class of well-known functions that are maximum entropy (first property of a maximum entropy code; ref. 27). We know that the gain control mechanism in the antennal lobe sets the average firing rate across all glomeruli (6, 7, 9). A likely probability density function, then, would be an exponential

The total entropy of the combinatorial code is the sum of the contributions from each of the 50 glomeruli (entropy is additive) and, according to information theory, this total entropy is maximized if all of the channels have the same probability density function (second property of a maximum entropy code; ref. 27). I therefore drop the subscript *k* that identifies the glomerulus, and call the common maximum entropy density function

Notice that the fact that all glomeruli have odor responses that obey the same probability density function does not mean that all glomeruli have the same rate *r* for a randomly selected odor. On the contrary, each glomerulus usually responds differently for every odor and always the same way for the same odor. The function

The density function

### Decoding the Maximum Entropy Code by Kenyon Cells.

With the properties of the combinatorial odor code just described, I can calculate what fraction of the Kenyon cell population is depolarized by how much in response to an odor chosen at random. My description is, however, only a probabilistic one, so I have no idea which glomeruli are responding to the odor and which Kenyon cells are depolarized by what amount. This calculation will be the first step in understanding how a (close to) unique response to an odor is generated.

My final result of this calculation is summarized here: The probability that a Kenyon cell has a particular depolarization *v* is approximately Gaussian, and this means that Kenyon cells with the largest depolarizations are rare. The second step in understanding the “trick” Kenyon cells use to tag odors is to recognize that an inhibitory feedback from the Kenyon cells onto themselves permits only the rare, most depolarized Kenyon cells to fire spikes in response to an odor. Two different, randomly selected odors, then, are unlikely to cause the same Kenyon cells to fire spikes, so the tags for two different odors rarely overlap.

Experiments have shown that the average Kenyon cell depolarization, produced by activation at a single claw, is *v* is the Kenyon cell depolarization (mV), *r* is the projection neuron firing rate (hertz) and *v* and *r* can be used to eliminate *r* in favor of *v* in the probability density *v* with a probability^{−1} (remember that the mean firing rate of olfactory projection neurons is *u*, and the second claw gives a depolarization *v* is *u*. The function *v* mV with a probability density *j*, the mean of *v* is *j* increases, the γ density approaches a Gaussian, and

The expected fraction of the Kenyon cells with each possible depolarization *v* for a random odor is described by

Often, one will be able to find a pair of odors that bind to odorant receptor proteins in very similar ways. This particular odor pair would activate very similar patterns of Kenyon cell firing and it would be difficult, or perhaps impossible, for the fly to discriminate between the two odors. If that particular odor pair needed to be discriminated, presumably evolution would have selected odorant receptors that would cause activation of different Kenyon cell pairs. Because the theory presented here is a statistical one, this sort of question cannot be addressed.

### Consequences of Variability in Kenyon Cell Properties.

In the earlier sections, I assumed three things: (*i*) all Kenyon cells have six claws, (*ii*) all Kenyon cells are interchangeable and sample all glomeruli with the same probability, and (*iii*) all synapses made by projection neurons onto Kenyon cells have the same strength. In fact, the number of Kenyon cell claws ranges from around 1 to 11 (16), three classes of Kenyon cells with different properties and different targets for their axons are known (14, 30), and the synaptic strength varies considerably from one synapse to the next (28), whereas I used the mean value. Relaxing these simplifying assumptions changes my results quantitatively, but does not alter the main conclusions. Some consequences of the first two of these simplifications are described here and, in the *Supporting Information*, I discuss nonuniform sampling of glomeruli by different Kenyon cells classes, and address the assumption that all projection neurons make synapses with the same strength on Kenyon cells.

I noted earlier that streams of parallel Kenyon cell axons run vertically and horizontally to define the two lobes of the mushroom body. The vertical lobe is divided into two sublobes called α and

Caron et al. (16) published counts of the number of claws for all three Kenyon cell types. The average number of claws, 5.75 (range = 2–8), was not different between the *A* where they are fitted to binomial distributions. Both distributions have a probability of 0.715 for adding a claw, and the non-γ Kenyon cells have at most 8 claws, whereas the γ Kenyon cells have a maximum of 11.

In the preceding section, I used the observed exponential distribution of antennal lobe projection neuron firing rates to calculate the probability density function for the depolarization of Kenyon cells. In this earlier calculation, however, I assumed that all Kenyon cells have six claws, a value close to the one observed for the non-γ class. In that section, I found that, for a Kenyon cell with *c* claws, the density function is*v* of a Kenyon cell with *c* claws, and λ = 0.87 mV^{−1}. When the number of claws *c* follows a binomial distribution *N* = 8 for non-γ Kenyon cells and *v*. The corresponding cumulative probability distribution is

Of the ∼2,000 Kenyon cells, 1/3 are of the γ type and 2/3 are of the non-γ *v* caused by any odor is *B* (solid line), where it can be seen to be close to a normal distribution (the light dotted curve superimposed) for depolarizations greater than about 5 mV. This distribution describes the depolarization of all Kenyon cells for any odor, and the winner-take-all circuit restricts the actual firing to those Kenyon cells that happen to have depolarizations greater than about 10 mV. These particular Kenyon cells would then provide the neural tag for the odor.

Because the non-γ and γ types of Kenyon cells have different numbers of claws, I have also included the probability distributions for the depolarization for both types separately in Fig. 3*B*. The dark dotted curve on the left of Fig. 3*B* is the distribution for the non-γ class

## Discussion

The final picture of the fly olfactory system is quite simple. The antennal lobe formulates the most informative odor code possible (technically a maximum entropy code), based on information from ORNs and from constraints (like gain control) provided by the antennal lobe circuitry. Because of the structure of this code, most projection neurons fire slowly, but a very few projection neurons respond with high firing rates in a small fraction of the glomeruli. Each Kenyon cell in the mushroom body combines information from about six glomeruli, selected at random, and those cells that best (in a statistical sense) characterize the odor are ones with largest depolarizations. The output of the mushroom body, then, is based on a winner-take-all circuit that permits firing only of the most depolarized Kenyon cells. Thus, the Kenyon cells that are permitted to fire spikes provide tags for a pair of randomly selected odors that are unlikely to overlap.

My motivation for this work was to learn how one example of the Marr motif works and, to identify the Marr motif in the fly, I relied on what appeared to be analogous architectures across vertebrate and insect brain regions. I am not, however, the first to notice these same parallels between insect and vertebrate brains; indeed, similarities between the mushroom bodies and vertebrate olfactory system, hippocampus, and cerebellum have long been discussed (31⇓–33). To be explicit about the parallels I identify, I include Table 1.

To apply the conclusions I have reached to other systems (cerebellum, vertebrate olfaction, and the hippocampus), two steps are required. First, it must be verified that the combinatorial code generated by the various brain regions is maximum entropy, and that the decoding step uses distributed information and a winner-take-all circuit to establish statistically nonoverlapping tags for distinct inputs. This task will likely be difficult, and will presumably require many small steps. Second, even if it turned out that the identified Marr motif structures generally work the same way, the actual brain structures that appear to use this motif all have quite different functions, and we must discover what specializations and embellishments are used on top of a generic architecture to account for what makes, for example, the cerebellum different from the hippocampus, or the olfactory bulb different from the antennal lobe. Whatever is finally established about structures that seem to use the Marr motif, I hope that the framework I presented here will be useful for studying them.

In the main text, I initially treated all Kenyon cells (KCs) as if they were interchangeable, but then relaxed this requirement to take account of the fact that the mushroom body has three classes of KCs defined according to which lobe they send their axons. The vertical lobe is divided into two sublobes called α and

## Nonuniform Sampling of Glomeruli by KCs

I initially assumed that all KCs sampled six glomeruli randomly with equal probabilities so that every row of

The relative frequency for sampling a glomerulus is replotted [from Carron et al. figure S2 (16)] in my Fig. S1 for the non-γ and γ KC types. My Fig. S1 is based on histograms in the Carron et al. figure S2 with data from their figure reordered to run from largest to smallest relative frequencies. Data in my Fig. S1*A* is based on 278 PN synapses onto 84 *A* on 354 PN synapses onto 102 γ KCs. Note that the numbers *n* assigned to glomeruli are different between my Fig. S1 *A* and *B* (each is arranged from largest to smallest relative frequency) and both are different from the standard order in Carron et al. figure S2.

The descriptive theory (smooth lines in Fig. S1) is a geometric distribution normalized to range from glomerulus number *n* will be sampled by a KC, is*n* is the glomerulus index (the glomeruli are numbered from 1 to 54 in order of decreasing probabilities) and *a* is a parameter that determines the probability *a* for the KC type. Note that rarely the same glomerulus may be sampled twice (or, very rarely, more than twice) by the same KC.

## Strength of PN Synapses onto Kenyon Cells

In the main text, I assumed that the sensing matrix (also called connection matrix)

The data described here appeared in figure 6S of Gruntman and Turner (28). They optically stimulated a small set of antennal lobe projection neurons and then recorded the membrane potential from KCs that received synaptic input from only a single claw. For 16 such neurons, they measured the amplitude of the first excitatory synaptic potential (EPSP) elicited by the stimulus. Gruntman and Turner provided me with their original list of EPSP amplitudes for each of the 16 KCs that is used in the following analysis.

In Fig. S2*A* I have plotted a cumulative histogram of the mean EPSP amplitudes, and fitted it with a straight line with an ordinate value that ranges from 0 to 1 and EPSP amplitudes that range from 0.588 to 1.76 mV. Thus, the synaptic strength of these synapses is uniformly distributed over a threefold range [0.59,1.8] with a probability density of 0.85. The coefficient of variation for these 16 claw synapses is constant, so a possible source for the variability is the number of acetylcholine receptors per synapse.

Because one would expect the EPSP amplitude distribution for all claw synapses to be described by the same function, I have divided the observed EPSP amplitudes by the mean for that claw, plotted the cumulative histogram for all 253 EPSPs from the 16 KCs in Fig. S2*B*, and I also carried out a sanity check by showing that the observed mean of 253 scaled EPSP amplitudes = 1. The claw synapses are constructed with separated fingers of the claw having a single active zone (15) and the average number of such separated active zones is 4. The simplest consistent basis for the observed variability would be for each active zone to produce an exponentially distributed contribution to the total EPSP amplitude. Because there are four active zones per KC synapse on a single claw, the EPSP amplitude should have a *B* is this function.

Thus, the sensing matrix should not have zeros and six randomly placed (according to the distributions in Fig. S1) ones per row, but rather zeros and nonzero entries at locations chosen at random from a binomial distribution with a probability of adding a claw of 0.715 and the appropriate mean number of claws for the KC type for that row. The nonzero entries in each row are drawn from a uniform probability density over the range [0.69,2.1] with a probability density of 0.71 that replaces the ones of the earlier sensing matrix. Note that the range of uniformly distributed EPSP amplitudes is a different from the observed range because the grand mean of all EPSP amplitudes was used when converting the total input spike rate into depolarization with the linear relation between total rate and mean depolarization.

As a biophysical theory for the distribution of synaptic strengths and the corresponding EPSP amplitudes, what I have just laid out is unsatisfactory because identification for sources in the variation of EPSP amplitudes from claw to claw is too unconstrained. As a description of the data, however, it is very accurate as far as it goes. The problem is that the types of KCs that provided the sample upon which Fig. S2 is based are unknown, too few KCs with a single claw activation are available, and those that are available represent activation by only a few types of glomeruli. The value of the description is that it shows what the actual sensing matrix should look like, and indicates what types of experiments would ideally be needed to find the actual sensing matrix. It also identifies one source in the variability of olfactory system functioning that is the basis for the observed variability in fly behavior in response to odors.

## Acknowledgments

I am indebted to Drs. Sophie Aimon, Kenta Asahina, Chih-Ying Su, and John Thomas for many helpful comments on an early draft, to Dr. Vikas Bhandawat who sent me original data for his figure S2 (ref. 8) upon which my Fig. 2 is based, and to Drs. Eyal Gruntman and Glenn Turner who sent me their original data for figure 6S (ref. 28) upon which much of the *Supporting Information* is based. This work was supported by National Science Foundation under Grants PHY-1444273 and PHY-1066393 and by the hospitality of the Aspen Center for Physics, where much of the work was done.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: stevens{at}salk.edu.

Author contributions: C.F.S. designed research, performed research, contributed new reagents/analytic tools, analyzed data, and wrote the paper.

Reviewers included: L.F.A., Columbia University.

The author declares no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1510103112/-/DCSupplemental.

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