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The author has declared that no competing interests exist.

The olfactory system faces the difficult task of identifying an enormous variety of odors independent of their intensity. Primacy coding, where the odor identity is encoded by the receptor types that respond earliest, might provide a compact and informative representation that can be interpreted efficiently by the brain. In this paper, we analyze the information transmitted by a simple model of primacy coding using numerical simulations and statistical descriptions. We show that the encoded information depends strongly on the number of receptor types included in the primacy representation, but only weakly on the size of the receptor repertoire. The representation is independent of the odor intensity and the transmitted information is useful to perform typical olfactory tasks with close to experimentally measured performance. Interestingly, we find situations in which a smaller receptor repertoire is advantageous for discriminating odors. The model also suggests that overly sensitive receptor types could dominate the entire response and make the whole array useless, which allows us to predict how receptor arrays need to adapt to stay useful during environmental changes. Taken together, we show that primacy coding is more useful than simple binary and normalized coding, essentially because the sparsity of the odor representations is independent of the odor statistics, in contrast to the alternatives. Primacy coding thus provides an efficient odor representation that is independent of the odor intensity and might thus help to identify odors in the olfactory cortex.

Humans can identify odors independent of their intensity. Experimental data suggest that this is accomplished by representing the odor identity by the earliest responding receptor types. Using theoretical modeling, we here show that such a primacy code outperforms alternative encodings and allows discriminating odors with close to experimentally measured performance. This performance depends strongly on the number of receptors considered in the primacy code, but the receptor repertoire size is less important. The model also suggests a strong evolutionary pressure on the receptor sensitivities, which could explain observed receptor copy number adaptations. By predicting psycho-physical experiments, the model will thus contribute to our understanding of the olfactory system.

The olfactory system identifies and discriminates odors for solving vital tasks like navigating the environment, identifying food, and engaging in social interactions. These tasks are complicated by the enormous variety of odors, which vary in composition and in the concentrations of their individual molecules. In particular, the olfactory system needs to separately recognize the odor identity (what is there?) and the odor intensity (how much is there?). For instance, the identity is required to decide whether to approach or avoid an odor source, whereas the intensity information is important for localizing it. It is not understood how these two odor properties are separated and how odors are discriminated reliably.

Odors are comprised of chemicals that bind to and excite olfactory receptors in the nose in mammals and on antenna in insects. Each receptor responds to a wide range of odors and each odor activates many receptor types. The resulting combinatorial code allows to distinguish odor identities [

The olfactory bulb contains neural clusters called glomeruli, which each receive input from a specific receptor type [

In this paper, we consider a simple model of primacy coding and investigate how well it represents complex odors. In particular, we identify how much information is transmitted to the cortex and how well this information can be used to perform typical olfactory tasks, like identifying the addition of a target odor to a background or discriminating odor mixtures. Our statistical approach allows linking parameters of the primacy code to results from typical psychophysical experiments. We show that primacy coding provides a robust and compact representation of the odor identity over a wide range of odors, independent of the odor intensity, and that it outperforms other simple coding schemes. However, this good performance of the olfactory system hinges on tuned receptor sensitivities, which suggests that there is a strong selective pressure to adjust the sensitivities on evolutionary and shorter timescales.

We describe odors by concentration vectors _{i} ≥ 0 of all ligands that can be detected by the olfactory receptors. The number _{L} of possible ligands is at least _{L} = 2300 [_{i} are zero.

In experiments, the olfactory system is typically characterized by presenting odors with particular statistics, e.g., by choosing mixtures from a given ligand library. Although such experiments allow to characterize the olfactory system in a part of odor space, we ultimately want to understand how the system performs in its natural environment. Unfortunately, the statistics of natural odors are difficult to measure [_{i} to appear in an odor. For simplicity, we neglect correlations in their appearance, so the mean number _{i} _{i}. To model the broad distribution of ligand concentrations, we choose the concentration _{i} of ligand _{i} and standard deviation _{i} if the ligand is present. Consequently, the mean concentration of a ligand in any odor reads 〈_{i}〉 = _{i} _{i} and the associated variance is _{env}(_{i} = _{i} = _{i} =

Odors are detected by an array of receptors in the nasal cavity in mammals and on the antenna in insects. The receptor array consists of _{R} different receptor types, which each are expressed many times. Typical numbers are _{R} ≈ 50 in flies [_{R} ≈ 300 in humans [_{R} ≈ 1000 in mice [_{n} of glomerulus _{ni} denotes the effective sensitivity of glomerulus _{ni} is proportional to the copy number of receptor type

(A) Schematic picture of the signal processing in the olfactory bulb: An odor comprised of many ligands excites the olfactory receptors and the signals from all receptors of the same type are accumulated in respective glomeruli. Under primacy coding, the glomeruli with the strongest (earliest) excitations encode the odor composition, whereas the odor intensity could be encoded separately. (B) Excitations of _{R} = 16 glomeruli for an arbitrary odor. The _{C} = 4 glomeruli with the highest excitations, above the threshold

The sensitivity matrix _{ni} could in principle be determined by measuring the response of each glomerulus to each possible ligand. However, because the numbers of receptor and ligand types are large, this is challenging and only parts of the sensitivity matrix have been measured, e.g., in humans [_{ni} is chosen independently from the same log-normal distribution, which is parameterized by its mean

The odor representation on the level of glomeruli excitations _{n} depends strongly on the odor intensity, which is quantified by the total concentration _{tot} = ∑_{i} _{i}_{tot} and thus provides a concentration-invariant representation.

In our simple model of primacy coding, odors are encoded by the identity of the _{C} glomeruli that respond first. For simplicity, we here neglect the order in which they respond, in contrast to rank coding [_{C} glomeruli with the largest excitation, which is known as the primacy set [

The primacy set can be represented by a binary activity vector _{n} = 1 implies that glomerulus _{n} = 0 denotes an inactive glomerulus not belonging to the primacy set. Since the active glomeruli have the highest excitation, they can be identified using an excitation threshold

Physiologically, the activities _{n} could be encoded by projection neurons in insects and mitral and tufted cells in mammals. These neurons receive excitatory input from one glomerulus [_{C}_{C}/_{R} of all glomeruli is activated. Moreover, _{tot}, implying concentration-invariance. This is because multiplying the concentration vector _{n} and the threshold

In the binary representation given by

To see whether primacy coding encodes odor information efficiently [_{C}/_{R}. Since our model is deterministic, _{env}(_{C}, _{R}, and λ. Since further processing in the downstream regions of the brain can only reduce the amount of information,

In an optimal receptor array, each output _{env}(_{C} active receptor types are permissible. The resulting representations would be optimal if each receptor type was activated with a probability 〈_{n}_{C}/_{R} and all types were uncorrelated, cov(_{n}, _{m}) = 0 for _{R} ≫ _{C}).

We start by analyzing the information _{max} given by _{max} implies that the information grows linearly with the primacy dimension _{C}, but only logarithmically with the number _{R} of receptor types. Consequently, the number of distinguishable signals, _{S} = 2^{I}_{C}, but the dependence on the repertoire size is weaker; see _{C}, our model thus predicts that the transmitted information in mice is only twice that of flies, although mice possess about 20 times as many receptor types. However, the number of discriminable signals changes by many orders of magnitudes, since it scales exponentially with

(A) The maximally transmitted information _{max} (solid lines) given by ^{7}, error smaller than symbol size) obtained from ensemble averages of _{L} = 512, _{max} as a function of _{C} for several _{R}. The right axis indicates the maximal number of distinguishable signals, _{S} = 2^{I}_{max} when half the receptor types are removed as a function of _{C} for various _{R}.

The logarithmic scaling of the transmitted information _{R} could explain why the ability of rats to discriminate odors is not significantly affected when half the olfactory bulb is removed in lesion experiments [_{C} bits; see _{R} ≈ 1000; see _{R} ≈ 50. Our model thus predicts that lesion experiments have a much more severe effect on the performance of animals with smaller receptor repertoires.

Taken together, this first analysis already suggests that the primacy code provides a robust odor representation, which is sparse, concentration-invariant, and depends only weakly on the details of the receptor array. However, for this representation to be useful to the animal, it needs to allow solving typical olfactory tasks.

Typical olfactory tasks include detecting a ligand in a distracting background, detecting the addition of a ligand to a mixture, as well as discriminating similar mixtures. All these tasks involve discriminating odors with common ligands, implying that the associated primacy sets are correlated. This correlation can be quantified by the expected Hamming distance

To build an intuition for this analysis, we start by considering two uncorrelated odors. In this case, each receptor type has an expected activity of 〈_{n} 〉 = _{C}/_{R}, implying the distance _{*} = 2_{C}(1 − _{C} _{R}^{−1}) and _{C} = 4 and _{R} = 50). The discriminability increases strongly with _{C}, while the receptor repertoire size _{R} has a much weaker effect in the typical case _{C} ⪡ _{R}, similar to the scaling of the information _{*} marks the upper bound for the discriminability

One simple task where odors are correlated is the detection of a target odor in a distracting background. To understand when a target can be detected, we analyze how the primacy set _{t} is added to a background ligand at concentration _{b}. Because of concentration-invariance, the result only depends on the relative target concentration _{t}/_{b}. _{t}/_{b}) and when more receptor types participate in the primacy code (larger _{C}). Conversely, the repertoire size _{R} has only a weak influence, similar to the cases discussed above; see _{t}/_{b}) are more difficult to discriminate with larger receptor repertoires.

(A-B) Probability _{t} to a background ligand at concentration _{b} can be detected using primacy coding as a function of _{t}/_{b} for (A) various _{C} at _{R} = 300 and (B) various _{R} at _{C} = 8. Numerical simulations (dots; sample size: 10^{5}) are compared to the theoretical prediction (lines) obtained using the statistical model; see _{R} = 32 (green line) and _{R} = 128 (orange line); see _{R} = 32 (upper panel) and _{R} = 128 (lower panel) excitation realizations (vertical bars) drawn from the same excitation distribution (black lines). The orange bars indicate the primacy set consisting of the _{C} = 4 largest excitations. (A–C) Remaining parameters are given in

The fact that increasing the receptor repertoire size _{R} can impede the detection of the target odor can be understood in a simplified statistical model, where we consider ensemble averages over sensitivity matrices; see _{C} receptor types with the largest excitations, _{R}, essentially because the distribution of the glomeruli excitation _{n} has a heavy tail, so that sampling more excitations leads to larger gaps between the largest excitations. These larger gaps in the excitations reduce the likelihood that adding the target changes the order of the excitations and thus the primacy set. Consequently, it is more difficult to detect the target using larger repertoires. Taken together, these arguments suggest that increasing the receptor repertoire is only beneficial if the primacy dimension

So far, we considered simple odors consisting of single ligands. However, realistic odors are comprised of many different ligands and target odors thus also need to be detected in backgrounds of many distracting ligands. Not surprisingly, experiments in humans [_{correct} of giving the correct answer is related to the probability _{correct} to the ones predicted by our model to restrict its parameters.

For simplicity, we first ask whether the primacy set changes when a ligand is added to a background, which is necessary for discriminating the background with the target from the background without it. This analysis will provide a theoretical upper bound for the performance, allowing us to restrict model parameters. In particular, we use ensemble averages over sensitivity matrices to compute _{correct} for the addition of a single ligand to a background consisting of _{C}. Conversely, whether the receptor repertoire size of humans (_{R} = 300; _{R} = 1000, _{C}, although we cannot exclude the possibility that the decoding in higher regions of the brain is much more efficient in mice than in humans, e.g., because they were trained better.

(A,B) Probability _{C}. The right axes display the expected fraction _{correct} of correct responses in the respective go/no-go experiment. Theoretical predictions (colored symbols and lines) are compared to experimental data (black symbols and lines) for (A) humans (_{R} = 300, data from [_{R} = 1000, data from [_{B} ligands of a mixture of _{B}/_{C} at _{R} = 300, ^{2}/^{2} = 0. The inset shows _{B}/_{C} = 8, _{R} = 300, and ^{2}/^{2} = 10. (A–C) Remaining parameters are given in

A surprising finding of this analysis is that target odors can be detected more reliably when the background at a given total concentration _{b} consists of many ligands. This can be seen by comparing single-ligand backgrounds (_{t}/_{b} = 1/_{C} = 8, we find _{t}/_{b} ≈ 0.2 in the single-ligand case, while the ratio can be much smaller (1/

To consider the discrimination of similar odors that have common ligands, we next consider odors that each contain _{B} of them. Such odors are uncorrelated (_{*}) when they do not share any ligands (_{B} = 0) and they are identical (_{B} =

So far, we only discussed how well odors can be discriminated, but in reality it is often necessary to identify individual odors in mixtures. To identify ligands, a decoder must compare odor representations to stored patterns. For simplicity, we here only consider a perfect decoder, which associates each representation

We start by considering mixtures of _{L} ligands are distinguishable, which is the case when _{L} = 1000 ligands, they could do this for mixtures of at most _{C} = 8. Note that this is merely an upper bound for the actual performance, since the calculation assumes that the olfactory system is optimized to identify ligands at one particular concentration, whereas natural odors contain ligands at various relative concentrations.

(A) Maximal number of ligands, _{R} for _{C} = 8. (B) Number _{S} of ligands that could be distinguished when a ligand of concentration _{t} is added to a background of _{b} determined from numerical (dots) and analytical (lines) ensemble averages for various _{R} and _{C} = 8. (A,B) Remaining parameters are given in

To see how concentration variations affect the odor identification, we next use the previously calculated mean distances _{t} to a mixture of _{b} can be estimated as _{R} − _{C} receptor types that were inactive for the background mixture and became active when the target was added.

The discussion of the identification of odors is limited by the simple description of the decoder in our model. We thus derive upper bounds for the performance, assuming that a mapping of all possible odor combinations to different representations _{ni} and the actual performance might thus lie well below the bounds derived here. However, in realistic olfactory system, the time-course of receptor activation might provide additional information about which ligands are present. For instance, odors from multiple sources might not fully mix and thus arrive in distinguishable whiffs [

We showed that primacy coding contains sufficient information to perform typical olfactory tasks with experimentally measured accuracy. Although this provides some support for primacy coding, alternative encoding schemes might also be consistent with experimental data. To elucidate this, we next compare primacy coding to two alternatives, which are also based on the simple model described by Eqs (

To see how binary and normalized coding compare to primacy coding, we calculate the probability

Comparison of the primacy code (blue; _{C} = 8) to a normalized code (black) and a binary code (gray). In the normalized code, glomeruli are active when their excitation exceeds _{R}〈_{n}

The example of the normalized code shows that it is not sufficient to study how well different coding schemes can solve olfactory tasks, but one also needs to consider how useful this code is to the downstream decoder. Without modeling the decoder in detail, we here just propose that sparser codes are preferable since they imply fewer firing neurons, which saves energy and simplifies the downstream processing. In fact, sparse coding is typical for sensory information [_{n}〉 of activated glomeruli and _{C}. Taken together, primacy coding outperforms both binary coding and normalized coding essentially because the sparsity of the representation is independent of the presented odors and can thus be adjusted to be useful and efficient over the whole range of possible odors.

The three models discussed here differ in how the statistics of the odor _{tot} affects the mean excitation 〈_{n}〉 and therefore the sparsity 〈_{n}〉. This clearly prevents the response from being useful over a wide concentration range. This dependence on the odor intensity is removed in normalized coding, but the variance of the excitations _{n} still depends on the odor statistics, e.g. larger mixtures imply smaller variations in _{n}. This is problematic since it implies the excitations of fewer glomeruli exceed the fixed threshold in normalized coding, so the sparsity 〈_{n}〉 and the usefulness decrease [

So far, we calculated the transmitted information and tested the discrimination performance of primacy coding under the assumption that all receptor types behave similarly. In fact, we established that the maximal information is achieved when all receptor types are activated with equal probability _{C}/_{R}. However, neither the receptor sensitivities nor the odors themselves are distributed equally in realistic situations. Variations in these quantities affect the transmitted information and thus the usefulness of the primacy code. For instance, the transmitted information decreases if a single receptor is activated less often than all the others; see

(A) Information _{max} given by _{1}〉 of the first receptor type while all others are unchanged for _{C} = 8. Dotted lines indicate _{max}(_{C}, _{R} − 1). Inset: Same data for the activity rescaled by _{C}/_{R}_{1} of the first receptor type for _{R} = 16, _{C} = 4, and _{n} = 1 for _{n} as a function of var(^{2} for various _{C} at _{R} = 20. The inset shows that the scaled information _{max} collapses as a function of _{R} = 10, 20 and _{C} = 2, 4, 6 for different widths ^{2} for various _{R} and _{C}. (B–D) Shown are numerical simulations with _{L} = 512,

The effect of varying receptor sensitivities can be studied in our model of primacy coding by discussing more general sensitivities matrices. We consider _{n}, which modulates the uniform sensitivity matrix _{n} = 1 for _{R}.

To investigate the effect of heterogeneous sensitivities, we start by varying the sensitivity factor of one receptor type while keeping all others untouched, i.e., we change _{1} while keeping _{n} = 1 for _{1} = 0, the first receptor type will never become active, the array behaves as if this type was not present, and the transmitted information is approximately _{max}(_{C}, _{R} − 1). This value is lower than the maximally transmitted information _{max}(_{C}, _{R}) reached for the symmetric case _{1} = 1. However, the associated information loss Δ_{max}(_{C}, _{R}) − _{max}(_{C}, _{R} − 1) ≈ _{C}/(_{R} ln 2) is relatively small in large receptor arrays (_{R} ≫ _{C}); see _{1} = 1 and the receptors will thus be active more often than the others. In the extreme case of _{1} → ∞, the first receptor type will always be active and thus not contribute any information. Since this receptor type would always be part of the primacy set, the information transmitted by the remaining receptor types is approximately _{max}(_{C} − 1, _{R} − 1), which is smaller than _{max}(_{C}, _{R} − 1) in the typical case _{R} ≫ _{C}. Consequently, an overly active receptor type can be worse than not having this type at all under primacy coding.

The fact that overly sensitive receptors are detrimental to the transmitted information is also visible in numerical simulations. _{1}. As qualitatively argued above, _{1} = 1 and it is slightly lower for smaller _{1} since the receptor type is active less often. In contrast, for _{1} > 1, _{1} = 0 for _{1} ≳ 1.5. These data suggest that it would be better to remove receptor types that exhibit a 50% higher sensitivity than the other types.

To see whether overly sensitive receptor types are also detrimental when all types have varying sensitivities, we next considering sensitivity factors _{n} distributed according to a lognormal distribution. Numerical results shown in _{n}) of the sensitivity factors. In fact, a variation of var(_{n})/〈_{n}〉^{2} = 0.5 already implies a reduction of the transmitted information by almost 50% for small concentration variations _{max} given in _{C} and _{R}, suggesting that this analysis also holds for realistic receptor repertoire sizes. Note that the reduced transmitted information also implies poorer odor discrimination performance; see

We analyzed a simple model of neural representations of olfactory stimuli, where odors are identified by the _{C} strongest responding receptor types. This version of primacy coding provides a sparse representation of the odor identity, which is independent of the odor intensity. We showed using numerical simulations and a statistical model that the primacy dimension _{C} strongly affects the transmitted information and the discriminability of odors. Interestingly, already for small values of _{C} ≲ 10, the typical olfactory discrimination tasks can be carried out with performances close to experimentally measured ones. Conversely, the number _{R} of receptor types does not strongly affect the coding capacity and the discriminability of similar odors, in accordance with lesion experiments. Our model even indicates that lowering _{R} can improve the identification of a target ligand in a background.

The advantage of our simple model is that we can analyze its behavior in depth and explicitly link the statistical properties of the olfactory system to data from psycho-physical experiments. In particular, we predict how likely two different odors drawn from a particular statistics can be distinguished. For instance, our model implies that target odors are easier to detect if disturbing backgrounds consist of many ligands. We generally find that representations are sensitive to the relative concentration of ligands in mixtures and that dilute components are basically completely shadowed. Conversely, for fixed ligand concentrations mixtures can typically be discriminated very well. However, identifying the individual ligands in mixture is only possible for mixtures with few components. In any case, our results suggest that the primacy code formed in the olfactory bulb is more useful to identify odors in the subsequent olfactory cortex than simple alternatives, essentially because the statistics of the representations are independent of the odor statistics.

Our model predicts that receptors are only useful if their likelihood to respond to incoming odors is similar. This is because receptor types that are overly sensitive and respond strongly to many odors could dominate other types and thus degrade the total information. In fact, having a receptor type that is 50% more sensitive than others, and thus responds about three times as often, can lead to less transmitted information than when this type is absent. This observation is related to the primacy hull discussed in [

Our results raise the question why mice have 20 times as many receptor types than flies although the transmitted information under primacy coding is only increased by a factor of 2 (see

We discussed the simplest version of primacy coding with a minimal receptor model and a constant primacy dimension _{C} implemented by a hard threshold. This model neglects the complex interactions of ligands at the olfactory receptors, which can affect perception [

All numerical simulations are based on ensemble averages over sensitivity matrices _{ni}. The elements of _{ni} are drawn independently from a log-normal distribution with _{env}(_{L} ligands are present using a Bernoulli distribution with probability _{L} and then independently drawing their concentration from a log-normal distribution with mean _{C} receptors with the highest excitation calculated from ^{5} and 10^{7} times, respectively.

In order to obtain deeper insights into the numerical results, we also develop analytical approximations using a statistical description of all involved quantities, which is based on accounting for the means and variances of the respective distributions. For instance, the statistics of the output _{n} are also well approximated by a log-normal distribution with mean _{n} exceeds the threshold _{env}(

The constraint _{n}_{C}/_{R}, so that the mean threshold reads
^{−1} is the inverse function of _{n}, since 〈_{n}〉. This situation is comparable to simple normalized representations resulting from the threshold _{n}〉, where

The expected difference between excitations corresponding to a given odor _{n} are distributed identically when considering all odors according to _{env}(_{(n)} at rank

The joint distribution of _{(n)} and _{(m)}, 1 ≤ _{R}, reads [

Consequently, the distribution of the difference Δ_{(n)} − _{(n−1)} of consecutive excitations is

Hence, the expected difference 〈Δ_{ΔE}(_{R} − _{C} − 1) d

The expected number ^{t} is added to some background ^{b} reads
_{on} is the probability that a receptor type that was inactive for ^{b} is turned on by the perturbation ^{t} and _{off} is the probability that a receptor type that was active is turned off. Both probabilities depend on the excitation thresholds ^{(1)} and ^{(2)} associated with the odors ^{b} and ^{b} + ^{t}, respectively, which can be estimated from _{on} follows from the probability that the excitation was at the value ^{(1)} and the additional excitation by the target brings the total excitation above ^{(2)},
^{j} describe the excitation statistics of the target (^{(1)} and ^{(2)} depend on _{R}, so the distance _{R}, in contrast to the case of normalized representations [

We use Eqs (_{t} is added to a background ligand at concentration _{b}. The associated statistics of the excitations obey

The third case of correlated odors that we discuss in the main text concerns two odor mixtures of equal size _{B} of the ligands. In this case, the excitation threshold _{xor} that a receptor type is excited by one mixture but not the other as
^{j} need to be evaluated for the excitations associated with the _{B} ligands that are the same (_{B} ligands that are different (_{R}_{xor} and we recover _{*} for unrelated mixtures (_{B} = 0) and _{B} =

The calculated distances _{C}} with a mean equal to ^{−d/2} in the limit _{C} ≫ 1.

In the case where the primacy sets _{M} groups with all elements within a group appearing with the same probability, we can write the information _{m} is the number of elements within group _{m} is the probability that group _{m} _{m} = 1. In the simple case of one receptor type with deviating statistics, we have _{M} = 2 with
_{n}_{C} − 〈_{1}〉)/(_{R} − 1) for _{1} = 0, _{max}(_{C}, _{R} − 1), whereas the maximum _{max}(_{C}, _{R}) is reached for _{1} = _{C}/_{R}. The information decreases for larger _{1} and eventually reaches values lower than _{max}(_{C}, _{R} − 1) when _{1} = _{R}/(_{R} − 1), we find
_{R} ≫ _{C} of large repertoires, so

Probability _{C}. (A) ^{2}/^{2} of the ligand concentration distribution at _{R} = 300 and _{C} = 8. (B,C) _{C} at ^{2}/^{2} = 10. Ensemble averages of the model (colored symbols) are compared to experimental data (black symbols and lines) for (B) humans (_{R} = 300, data from [_{R} = 1000, data from [

(EPS)

(A–B) Probability _{t} to a background ligand at concentration _{b} changes the primacy set _{t}/_{b} for (A) various _{C} at _{R} = 300 and (B) various _{R} at _{C} = 8. (C) Probability _{C} and _{R} = 300. (A–C) Shown are numerical simulations (dots; sample size: 10^{5}) for _{L} = 512, _{ni} behave similarly.

(EPS)

I thank Michael P. Brenner, Michael Lässig, Venkatesh N. Murthy, and Mikhail Tikhonov for helpful discussions.