Adaptation of olfactory receptor abundances for efficient coding
Abstract
Olfactory receptor usage is highly heterogeneous, with some receptor types being orders of magnitude more abundant than others. We propose an explanation for this striking fact: the receptor distribution is tuned to maximally represent information about the olfactory environment in a regime of efficient coding that is sensitive to the global context of correlated sensor responses. This model predicts that in mammals, where olfactory sensory neurons are replaced regularly, receptor abundances should continuously adapt to odor statistics. Experimentally, increased exposure to odorants leads variously, but reproducibly, to increased, decreased, or unchanged abundances of different activated receptors. We demonstrate that this diversity of effects is required for efficient coding when sensors are broadly correlated, and provide an algorithm for predicting which olfactory receptors should increase or decrease in abundance following specific environmental changes. Finally, we give simple dynamical rules for neural birth and death processes that might underlie this adaptation.
https://doi.org/10.7554/eLife.39279.001eLife digest
A mouse’s nose contains over 10 million receptor neurons divided into about 1,000 different types, which detect airborne chemicals – called odorants – that make up smells. Each odorant activates many different receptor types. And each receptor type responds to many different odorants. To identify a smell, the brain must therefore consider the overall pattern of activation across all receptor types. Individual receptor neurons in the mammalian nose live for about 30 days, before new cells replace them. The entire population of odorant receptor neurons turns over every few weeks, even in adults.
Studies have shown that some types of these receptor neurons are used more often than others, depending on the species, and are therefore much more abundant. Moreover, the usage patterns of different receptor types can also change when individual animals are exposed to different smells. Teşileanu et al. set out to develop a computer model that can explain these observations.
The results revealed that the nose adjusts its odorant receptor neurons to provide the brain with as much information as possible about typical smells in the environment. Because each smell consists of multiple odorants, each odorant is more likely to occur alongside certain others. For example, the odorants that make up the scent of a flower are more likely to occur together than alongside the odorants in diesel. The nose takes advantage of these relationships by adjusting the abundance of the receptor types in line with them. Teşileanu et al. show that exposure to odorants leads to reproducible increases or decreases in different receptor types, depending on what would provide the brain with most information.
The number of odorant receptor neurons in the human nose decreases with time. The current findings could help scientists understand how these changes affect our sense of smell as we age. This will require collaboration between experimental and theoretical scientists to measure the odors typical of our environments, and work out how our odorant receptor neurons detect them.
https://doi.org/10.7554/eLife.39279.002Introduction
The sensory periphery acts as a gateway between the outside world and the brain, shaping what an organism can learn about its environment. This gateway has a limited capacity (Barlow, 1961), restricting the amount of information that can be extracted to support behavior. On the other hand, signals in the natural world typically contain many correlations that limit the unique information that is actually present in different signals. The efficientcoding hypothesis, a key normative theory of neural circuit organization, puts these two facts together, suggesting that the brain mitigates the issue of limited sensory capacity by eliminating redundancies implicit in the correlated structure of natural stimuli (Barlow, 1961; van Hateren, 1992a). This idea has led to elegant explanations of functional and circuit structure in the early visual and auditory systems (see, e.g. Laughlin, 1981; Atick and Redlich, 1990; Van Hateren, 1993; Olshausen and Field, 1996; Simoncelli and Olshausen, 2001; Fairhall et al., 2001; Lewicki, 2002; Ratliff et al., 2010; Garrigan et al., 2010; Tkacik et al., 2010; Hermundstad et al., 2014; Palmer et al., 2015; Salisbury and Palmer, 2016). These classic studies lacked a way to test causality by predicting how changes in the environment lead to adaptive changes in circuit composition or architecture. We propose that the olfactory system provides an avenue for such a causal test because receptor neuron populations in the mammalian nasal epithelium are regularly replaced, leading to the possibility that their abundances might adapt efficiently to the statistics of the environment.
The olfactory epithelium in mammals and the antennae in insects are populated by large numbers of olfactory sensory neurons (OSNs), each of which expresses a single kind of olfactory receptor. Each type of receptor binds to many different odorants, and each odorant activates many different receptors, leading to a complex encoding of olfactory scenes (Malnic et al., 1999). Olfactory receptors form the largest known gene family in mammalian genomes, with hundreds to thousands of members, owing perhaps to the importance that olfaction has for an animal’s fitness (Buck and Axel, 1991; Tan et al., 2015; Chess et al., 1994). Independently evolved large olfactory receptor families can also be found in insects (Missbach et al., 2014). Surprisingly, although animals possess diverse repertoires of olfactory receptors, their expression is actually highly nonuniform, with some receptors occurring much more commonly than others (Rospars and Chambille, 1989; IbarraSoria et al., 2017). In addition, in mammals, the olfactory epithelium experiences neural degeneration and neurogenesis, resulting in replacement of the OSNs every few weeks (Graziadei and Graziadei, 1979). The distribution of receptors resulting from this replacement has been found to have a mysterious dependence on olfactory experience (Schwob et al., 1992; Santoro and Dulac, 2012; Zhao et al., 2013; Dias and Ressler, 2014; Cadiou et al., 2014; IbarraSoria et al., 2017): increased exposure to specific ligands leads reproducibly to more receptors of some types, and no change or fewer receptors of other types.
Here, we show that these puzzling observations are predicted if the receptor distribution in the olfactory epithelium is organized to present a maximally informative picture of the odor environment. Specifically, we propose a model for the quantitative distribution of olfactory sensory neurons by receptor type. The model predicts that in a noisy odor environment: (a) the distribution of receptor types will be highly nonuniform, but reproducible given fixed receptor affinities and odor statistics; and (b) an adapting receptor neuron repertoire should reproducibly reflect changes in the olfactory environment; in a sense it should become what it smells. Precisely such findings are reported in experiments (Schwob et al., 1992; Santoro and Dulac, 2012; Zhao et al., 2013; Dias and Ressler, 2014; Cadiou et al., 2014; IbarraSoria et al., 2017).
In contrast to previous work applying efficientcoding ideas to the olfactory system (Keller and Vosshall, 2007; McBride et al., 2014; Zwicker et al., 2016; Krishnamurthy et al., 2017), here we take the receptor–odorant affinities to be fixed quantities and do not attempt to explain their distribution or their evolution and diversity across species. Instead, we focus on the complementary question of the optimal way in which the olfactory system can use the available receptor genes. This allows us to focus on phenomena that occur on faster timescales, such as the reorganization of the receptor repertoire as a result of neurogenesis in the mammalian epithelium.
Because of the combinatorial nature of the olfactory code (Malnic et al., 1999; Stopfer et al., 2003; Stevens, 2015; Zhang and Sharpee, 2016; Zwicker et al., 2016; Krishnamurthy et al., 2017) receptor neuron responses are highly correlated. In the absence of such correlations, efficient coding predicts that output power will be equalized across all channels if transmission limitations dominate (Srinivasan et al., 1982; Olshausen and Field, 1996; Hermundstad et al., 2014), or that most resources will be devoted to receptors whose responses are most variable if input noise dominates (van Hateren, 1992a; Hermundstad et al., 2014). Here, we show that the optimal solution is very different when the system of sensors is highly correlated: the adaptive change in the abundance of a particular receptor type depends critically on the global context of the correlated responses of all the receptor types in the population—we refer to this as contextdependent adaptation.
Correlations between the responses of olfactory receptor neurons are inevitable not only because the same odorant binds to many different receptors, but also because odors in the environment are typically composed of many different molecules, leading to correlations between the concentrations with which these odorants are encountered. Furthermore, there is no way for neural circuitry to remove these correlations in the sensory epithelium because the candidate lateral inhibition occurs downstream, in the olfactory bulb. As a result of these constraints, for an adapting receptor neuron population, our model predicts that increased activation of a given receptor type may lead to more, fewer or unchanged numbers of the receptor, but that this apparently sporadic effect will actually be reproducible between replicates. This counterintuitive prediction matches experimental observations (Santoro and Dulac, 2012; Zhao et al., 2013; Cadiou et al., 2014; IbarraSoria et al., 2017).
Olfactory response model
In vertebrates, axons from olfactory neurons converge in the olfactory bulb on compact structures called glomeruli, where they form synapses with dendrites of downstream neurons (Hildebrand and Shepherd, 1997); see Figure 1a. To good approximation, each glomerulus receives axons from only one type of OSN, and all OSNs expressing the same receptor type converge onto a small number of glomeruli, on average about two in mice to about 16 in humans (Maresh et al., 2008). Similar architectures can be found in insects (Vosshall et al., 2000).
The anatomy shows that in insects and vertebrates, olfactory information passed to the brain can be summarized by activity in the glomeruli. We treat this activity in a firingrate approximation, which allows us to use available receptor affinity data (Hallem and Carlson, 2006; Saito et al., 2009). This approximation neglects individual spike times, which can contain important information for odor discrimination in mammals and insects (Resulaj and Rinberg, 2015; DasGupta and Waddell, 2008; Wehr and Laurent, 1996; Huston et al., 2015). Given data relating spike timing and odor exposure for different odorants and receptors, we could use the time from respiratory onset to the first elicited spike in each receptor as an indicator of activity in our model. Alternatively, we could use both the timing and the firing rate information together. Such data is not yet available for large panels of odors and receptors, and so we leave the inclusion of timing effects for future work.
A challenge specific to the study of the olfactory system as compared to other senses is the limited knowledge we have of the space of odors. It is difficult to identify common features shared by odorants that activate a given receptor type (Rossiter, 1996; Malnic et al., 1999), while attempts at defining a notion of distance in olfactory space have had only partial success (Snitz et al., 2013), as have attempts to find reduceddimensionality representations of odor space (Zarzo and Stanton, 2006; Koulakov et al., 2011). In this work, we simply model the olfactory environment as a vector $\mathbf{\mathbf{c}}=\{{c}_{1},\mathrm{\dots},{c}_{N}\}$ of concentrations, where ${c}_{i}$ is the concentration of odorant $i$ in the environment (Figure 1a). We note, however, that the formalism we describe here is equally applicable for other parameterizations of odor space: the components ${c}_{i}$ of the environment vector $\mathbf{\mathbf{c}}$ could, for instance, indicate concentrations of entire classes of molecules clustered based on common chemical traits, or they might be abstract coordinates in a lowdimensional representation of olfactory space.
Once a parameterization for the odor environment is chosen, we model the statistics of natural scenes by the joint probability distribution $P({c}_{1},\mathrm{\dots},{c}_{N})$. We are neglecting temporal correlations in olfactory cues because we are focusing on odor identity rather than olfactory search where timing of cues will be especially important. This simplifies our model, and also reduces the number of olfactory scene parameters needed as inputs. Similar static approximations of natural images have been employed powerfully along with the efficient coding hypothesis to explain diverse aspects of early vision (e.g., in Laughlin, 1981; Atick and Redlich, 1990; Olshausen and Field, 1996; van Hateren and van der Schaaf, 1998; Ratliff et al., 2010; Hermundstad et al., 2014).
To construct a tractable model of the relation between natural odor statistics and olfactory receptor distributions, we describe the olfactory environment as a multivariate Gaussian with mean ${\mathbf{\mathbf{c}}}_{0}$ and covariance matrix $\mathrm{\Gamma}$,
This can be thought of as a maximumentropy approximation of the true distribution of odorant concentrations, constrained by the environmental means and covariances. This simple environmental model misses some sparse structure that is typical in olfactory scenes (Yu et al., 2015; Krishnamurthy et al., 2017). Nevertheless, approximating natural distributions with Gaussians is common in the efficientcoding literature, and often captures enough detail to be predictive (van Hateren, 1992a; van Hateren, 1992b; Van Hateren, 1993; Hermundstad et al., 2014). This may be because early sensory systems in animals are able to adapt more effectively to loworder statistics which are easily represented by neurons in their mean activity and pairwise correlations.
The number $N$ of odorants that we use to represent an environment need not be as large as the total number of possible volatile molecules. We can instead focus on only those odorants that are likely to be encountered at meaningful concentrations by the organism that we study, leading to a much smaller value for $N$. In practice, however, we are limited by the available receptor affinity data. Our quantitative analyses are generally based on data measured using panels of 110 odorants in fly (Hallem and Carlson, 2006) and 63 in mammals (Saito et al., 2009).
We next build a model for how the activity at the glomeruli depends on the olfactory environment. We work in an approximation in which the responses depend linearly on the concentration values:
where ${r}_{a}$ is the response of the glomerulus indexed by $a$, ${S}_{ai}$ is the expected response of a single sensory neuron expressing receptor type $a$ to a unit concentration of odorant $i$, and ${K}_{a}$ is the number of neurons of type $a$. The second term describes noise, with ${\eta}_{a}$, the noise for a single OSN, modeled as a Gaussian with mean 0 and standard deviation ${\sigma}_{a}$, ${\eta}_{a}\sim \mathcal{\mathcal{N}}(0,{\sigma}_{a}^{2})$.
The approximation we are using can be seen as linearizing the responses of olfactory sensory neurons around an operating point. This has been shown to accurately capture the response of olfactory receptors to odor mixtures in certain concentration ranges (Singh et al., 2018). While odor concentrations in natural scenes span many orders of magnitude and are unlikely to always stay within the linear regime, the effect of the nonlinearities on the information maximization procedure that we implement below is less strong (see Appendix 3 for a comparison between our linear approximation and a nonlinear, competitive binding model in a toy example). One advantage of employing the linear approximation is that it requires a minimal set of parameters (the sensing matrix coefficients $S}_{ai$), while nonlinear models in general require additional information (such as a Hill coefficient and a maximum activation for each receptorodorant pair for a competitive binding model; see Appendix 3).
Information maximization
We quantify the information that responses, $\mathbf{r}=({r}_{1},\dots ,{r}_{M})$, contain about the environment vector, $\mathbf{c}=({c}_{1},\dots ,{c}_{N})$, using the mutual information $I(\mathbf{r},\mathbf{c})$:
where $P(\mathbf{\mathbf{r}},\mathbf{\mathbf{c}})$ is the joint probability distribution over response and concentration vectors, $P(\mathbf{\mathbf{r}}\mathbf{\mathbf{c}})$ is the distribution of responses conditioned on the environment, and $P(\mathbf{\mathbf{r}})$ is the marginal distribution of the responses alone. Given our assumptions, all these distributions are Gaussian, and the integral can be evaluated analytically (see Appendix 2). The result is
where the overlap matrix $Q$ is related to the covariance matrix $\mathrm{\Gamma}$ of odorant concentrations (from Equation (1)),
and $\mathbb{K}$ and $\mathrm{\Sigma}$ are diagonal matrices of OSN abundances ${K}_{a}$ and noise variances ${\sigma}_{a}^{2}$, respectively:
The overlap matrix $Q$ is equal to the covariance matrix of OSN responses in the absence of noise (${\sigma}_{a}=0$; see Appendix 2). Thus, it is a measure of the strength of the usable olfactory signal. In contrast, the quantity $\mathrm{\Sigma}{\mathbb{K}}^{1}$ is a measure of the amount of noise in the responses, where the term $\mathbb{K}}^{1$ corresponds to the effect of averaging over OSNs of the same type. This implies that the quantity $\mathbb{K}{\mathrm{\Sigma}}^{1}Q$ is a measure of the signaltonoise ratio (SNR) in the system (more precisely, its square), so that Equation (4) represents a generalization to multiple, correlated channels of the classical result for a single Gaussian channel, $I=\frac{1}{2}\mathrm{log}\left(1+{\mathrm{S}\mathrm{N}\mathrm{R}}^{2}\right)$ (Shannon, 1948; van Hateren, 1992a; van Hateren, 1992b). In the linear approximation that we are using, the information transmitted through the system is the same whether all OSNs with the same receptor type converge to one or multiple glomeruli (see Appendix 2). Because of this, for convenience we take all neurons of a given type to converge onto a single glomerulus (Figure 1a).
The OSN numbers ${K}_{a}$ cannot grow without bound; they are constrained by the total number of neurons in the olfactory epithelium. Thus, to find the optimal distribution of receptor types, we maximize $I(\mathbf{\mathbf{r}},\mathbf{\mathbf{c}})$ with respect to $\{{K}_{a}\}$, subject to the constraints that: (1) the total number of receptor neurons is fixed (${\sum}_{a}{K}_{a}={K}_{\text{tot}}$); and (2) all neuron numbers are nonnegative:
Throughout the paper, we treat the OSN abundances ${K}_{a}$ as real numbers instead of integers, which is a good approximation as long as they are not very small. The optimization can be performed analytically using the KarushKuhnTucker (KKT) conditions (Boyd and Vandenberghe, 2004) (see Appendix 2), but in practice it is more convenient to use numerical optimization.
Note that in contrast to other work that has used information maximization to study the olfactory system (e.g. Zwicker et al., 2016), here we optimize over the OSN numbers ${K}_{a}$, while keeping the affinity profiles of the receptors (given by the sensing matrix elements ${S}_{ia}$) constant. Below we analyze how the optimal distribution of receptor types depends on receptor affinities, odor statistics, and the size of the olfactory epithelium.
Receptor diversity grows with OSN population size
Large OSN populations
In our model, receptor noise is reduced by averaging over the responses from many sensory neurons. As the number of neurons increases, ${K}_{\text{tot}}\to \mathrm{\infty}$, the signaltonoise ratio (SNR) becomes very large (see Equation (2)). When this happens, the optimization with respect to OSN numbers ${K}_{a}$ can be solved analytically (see Appendix 2), and we find that the optimal receptor distribution is given by
where $A$ is the inverse of the overlap matrix $Q$ from Equation (5), $A={Q}^{1}$, $\sigma}_{a}^{2$ are the receptor noise variances (Equation (6)), and $\overline{{\sigma}^{2}A}=\sum {\sigma}_{a}^{2}{A}_{aa}/M$ is a constant enforcing the constraint $\sum {K}_{a}={K}_{\mathrm{t}\mathrm{o}\mathrm{t}}$. When ${K}_{\text{tot}}$ is sufficiently large, the constant first term dominates, meaning that the receptor distribution is essentially uniform, with each receptor type being expressed in a roughly equal fraction of the total population of sensory neurons. In this limit, the receptor distribution is as even and as diverse as possible given the genetically encoded receptor types. The small differences in abundance are related to the diagonal elements of the inverse overlap matrix $A$, modulated by the noise variances ${\sigma}_{a}^{2}$ (Figure 2a). The information maximum in this regime is shallow because only a change in OSN numbers of order ${K}_{\text{tot}}/M$ can have a significant effect on the noise level for the activity of each glomerulus. Put another way, when the OSN numbers ${K}_{a}$ are very large, the glomerular responses are effectively noiseless, and the number of receptors of each type has little effect on the reliability of the responses. This scenario applies as long as the OSN abundances ${K}_{a}$ are much larger than the elements of the inverse overlap matrix $A$.
Small and intermediatesized OSN populations
When the number of neurons is very small, receptor noise can overwhelm the response to the environment. In this case, the best strategy is to focus all the available neurons on a single receptor type, thus reducing noise by summation as much as possible (Figure 2b). The receptor type that yields the most information will be the one whose response is most variable in natural scenes as compared to the amount of receptor noise; that is, the one that corresponds to the largest value of ${Q}_{aa}/{\sigma}_{a}^{2}$—see Appendix 2 for a derivation. This is reminiscent of a result in vision where the variance of a stimulus predicted its perceptual salience (Hermundstad et al., 2014).
As the total number of neurons increases, the added benefit of summing to lower noise for a single receptor type diminishes, and at some critical value it is more useful to populate a second receptor type that provides unique information not available in responses of the first type (Figure 2b). This process continues as the number of neurons increases, so that in an intermediate SNR range, where noise is significant but does not overwhelm the olfactory signal, our model leads to a highly nonuniform distribution of receptor types (see the trend in Figure 2b as the number of OSNs increases). Indeed, an inhomogeneous distribution of this kind is seen in mammals (IbarraSoria et al., 2017). Broadly, this is consistent with the idea that living systems conserve resources to the extent possible, and thus the number of OSNs (and therefore the SNR) will be selected to be in an intermediate range in which there are just enough to make all the available receptors useful.
Increasing OSN population size
Our model predicts that, all else being equal, the number of receptor types that are expressed should increase monotonically with the total number of sensory neurons, in a series of step transitions (see Figure 2c). Strictly speaking, this is a prediction that applies in a constant olfactory environment and with a fixed receptor repertoire; in terms of the parameters in our model, the total number of neurons ${K}_{\text{tot}}$ is varied while the sensing matrix $S$ and environmental statistics $\mathrm{\Gamma}$ stay the same. Keeping in mind that these conditions are not usually met by distinct species, we can nevertheless ask whether, broadly speaking, there is a relation between the number of functional receptor genes and the size of the olfactory epithelium in various species.
To this end, we looked at several mammals for which the number of OR genes and the size of the olfactory epithelium were measured (Figure 2f). We focused on the intact OR genes (Niimura et al., 2014), based on the expectation that receptor genes that tend to not be used are more likely to undergo deleterious mutations. We have not found many direct measurements of the number of neurons in the epithelium for different species, so we estimated this based on the area of the olfactory epithelium (Moulton, 1967; Pihlström et al., 2005; Gross et al., 1982; Smith et al., 2014). There is a known allometric scaling relation stating that the number of neurons per unit mass for a species decreases as the 0.3 power of the typical body mass (HerculanoHouzel et al., 2015). Assuming a fixed number of layers in the olfactory epithelial sheet, this implies that the number of neurons in the epithelium should scale as $N}_{\text{OSN}}\propto (\text{epithelial area})/(\text{body mass}{)}^{\frac{2}{3}\cdot 0.3$. We applied this relation to epithelial areas using the typical mass of several species (Rousseeuw and Leroy, 1987; FCI, 2018; Gross et al., 1982; Smith et al., 2014). The trend is consistent with expectations from our model (Figure 2f), keeping in mind uncertainties due to species differences in olfactory environments, receptor affinities, and behavior (e.g. consider marmoset vs. rat). A direct comparison is more complicated in insects, where even closely related species can vary widely in degree of specialization and thus can experience very different olfactory environments (Dekker et al., 2006). As we discuss below, our model’s detailed predictions can be more specifically tested in controlled experiments that measure the effect of a known change in odor environment on the olfactory receptor distributions of individual mammals, as in IbarraSoria et al. (2017).
Optimal OSN abundances are contextdependent
We can predict the optimal distribution of receptor types given the sensing matrix $S$ and the statistics of odors by maximizing the mutual information in Equation (4) while keeping the total number of neurons ${K}_{\text{tot}}={\sum}_{a}{K}_{a}$ constant. We tested the effect of changing the variance of a single odorant, and found that the effect on the optimal receptor abundances depends on the context of the background olfactory environment. Increased exposure to a particular ligand can lead to increased abundance of a given receptor type in one context, but to decreased abundance in another (Figure 3). In fact, patterns of this kind have been reported in recent experiments (Santoro and Dulac, 2012; Zhao et al., 2013; Cadiou et al., 2014; IbarraSoria et al., 2017). To understand this contextdependence better, we analyzed the predictions of our model in various signal and noise scenarios.
One factor that does not affect the optimal receptor distribution in our model is the average concentration vector ${\mathbf{\mathbf{c}}}_{0}$. This is because it corresponds to odors that are always present and therefore offer no new information about the environment. This is consistent with experiment (IbarraSoria et al., 2017), where it was observed that chronic odor exposure does not affect receptor abundances in the epithelium. In the rest of the paper, we thus restrict our attention to the covariance matrix of odorant concentrations, $\mathrm{\Gamma}$.
The problem of maximizing the amount of information that OSN responses convey about the odor environment simplifies considerably if these responses are weakly correlated. In this case, standard efficient coding theory says that receptors whose activities fluctuate more extensively in response to the olfactory environment provide more information to brain, while receptors that are active at a constant rate or are very noisy provide less information. In this circumstance, neurons expressing receptors with large signaltonoise ratio (SNR, i.e. signal variance as compared to noise variance) should increase in proportion relative to neurons with low signaltonoise ratio (see Appendix 2 for a derivation). In terms of our model, the signal variance of glomerular responses is given by diagonal elements of the overlap matrix $Q$ (Equation 5), while the noise variance is ${\sigma}_{a}^{2}$; so we expect ${K}_{a}$, the number of OSNs of type $a$, to increase with ${Q}_{aa}/{\sigma}_{a}^{2}$. Responses are less correlated if receptors are narrowly tuned, and we find indeed that if each receptor type responds to only a small number of odorants, the abundances of OSNs of each type correlate well with their variability in the environment (narrowtuning side of Figure 2d). This is also consistent with the results at high SNR: we saw above that in that case ${K}_{a}\approx C{\sigma}_{a}^{2}{({Q}^{1})}_{aa}$, and when response correlations are weak, $Q$ is approximately diagonal, and thus ${({Q}^{1})}_{aa}\approx 1/{Q}_{aa}$.
The biological setting is better described in terms of widely tuned sensing matrices (Hallem and Carlson, 2006), and an intermediate SNR level in which noise is important, but does not dominate the responses of most receptors. We therefore generated sensing matrices with varying tuning width by changing the number of odorants that elicit strong activity in each receptor (as detailed in Appendix 1). We found that as receptors begin responding to a greater diversity of odorants, the correlation structure of their activity becomes important in determining the optimal receptor distribution; it is no longer sufficient to just examine the signal to noise ratios of each receptor type separately as a conventional theory suggests (widetuning side of Figure 2d). In other words, the optimal abundance of a receptor type depends not just on its activity level, but also on the context of the correlated activity levels of all the other receptor types. These correlations are determined by the covariance structures of the environment and of the sensing matrix.
In fact, across the range of tuning widths the optimal receptor abundances ${K}_{a}$ are correlated with the inverse of the overlap matrix, $A={Q}^{1}$ (Figure 2e). For narrow tuning widths, the overlap matrix $Q$ is approximately diagonal (because correlations between receptors are weak) and so ${Q}^{1}$ is simply the matrix of the inverse diagonal elements of $Q$. Thus, in this limit, the correlation with ${Q}^{1}$ simply follows from the correlation with $Q$ that we discussed above. As the tuning width increases keeping the total number of OSNs ${K}_{\text{tot}}$ constant, the responses from each receptor grow stronger, increasing the SNR, even as the offdiagonal elements of the overlap matrix $Q$ become significant. In the limit of high SNR, the analytical formula ${K}_{a}\approx C{\sigma}_{a}^{2}{Q}_{aa}^{1}$ (Equation 8) ensures that the OSN numbers ${K}_{a}$ are still correlated with the diagonal elements of ${Q}^{1}$, despite the presence of large offdiagonal components. Because of the matrix inversion in ${Q}^{1}$, the optimal abundance for each receptor type is affected in this case by the full covariance structure of all the responses and not just by the variance ${Q}_{aa}$ of the receptor itself. Mathematically, this is because the diagonal elements of ${Q}^{1}$ are functions of all the variances and covariances in the overlap matrix $Q$. This dependence of each abundance on the full covariance translates to a complex contextdependence whereby changing the same ligand in different background environments can lead to very different adapted distributions of receptors. In Appendix 6 we show that the correlation with the inverse overlap matrix has an intuitive interpretation: receptors which either do not fluctuate much or whose values can be guessed based on the responses of other receptors should have low abundances.
Environmental changes lead to complex patterns of OSN abundance changes
To investigate how the structure of the optimal receptor repertoire varies with the olfactory environment, we first constructed a background in which the concentrations of 110 odorants were distributed according to a Gaussian with a randomly chosen covariance matrix (e.g., Figure 4a; see Appendix 4 for details). From this base, we generated two different environments by adding a large variance to 10 odorants in environment 1, and to 10 different odorants in environment 2 (Figure 4b). We then considered the optimal distribution in these environments for a repertoire of 24 receptor types with odor affinities inferred from (Hallem and Carlson, 2006). We found that when the number of olfactory sensory neurons ${K}_{\text{tot}}$ is large, and thus the signaltonoise ratio is high, the change in odor statistics has little effect on the distribution of receptors (Figure 4c). This is because at high SNR, all the receptors are expressed nearly uniformly as discussed above, and this is true in any environment. When the number of neurons is smaller (or, equivalently, the signaltonoise ratio is in a low or intermediate regime), the change in environment has a significant effect on the receptor distribution, with some receptor types becoming more abundant, others becoming less abundant, and yet others not changing much between the environments (see Figure 4d). This mimics the kinds of complex effects seen in experiments in mammals (Schwob et al., 1992; Santoro and Dulac, 2012; Zhao et al., 2013; Dias and Ressler, 2014; Cadiou et al., 2014; IbarraSoria et al., 2017).
Changing odor identities has more extreme effects on receptor distributions than changing concentrations
In the comparison above, the two environment covariance matrices differed by a large amount for a small number of odors. We next compared environments with two different randomly generated covariance matrices, each generated in the same way as the background environment in Figure 4a. The resulting covariance matrices (Figure 5a, top) are very different in detail (the correlation coefficient between their entries is close to zero; distribution of changes in Figure 5b, red line), although they look similar by eye. Despite the large change in the detailed structure of the olfactory environment, the corresponding change in optimal receptor distribution is typically small, with a small fraction of receptor types experiencing large changes in abundance (red curve in Figure 5c). The average abundance of each receptor in these simulations was about 1000, and about 90% of all the abundance change values $\left\mathrm{\Delta}{K}_{i}\right$ were below 20% of this, which is the range shown on the plot in Figure 5c. Larger changes also occurred, but very rarely: about 0.1% of the abundance changes were over 800.
If we instead engineer two environments that are almost nonoverlapping so that each odorant is either common in environment 1, or in environment 2, but not in both (Figure 5a, bottom; see Appendix 4 for how this was done), the changes in optimal receptor abundances between environments shift away from midrange values towards higher values (blue curve in Figure 5c). For instance, 40% of abundance changes lie in the range $\left\mathrm{\Delta}K\right\phantom{\rule{thinmathspace}{0ex}}>\phantom{\rule{thinmathspace}{0ex}}50$ in the nonoverlapping case, while the proportion is 28% in the generic case.
It seems intuitive that animals that experience very different kinds of odors should have more striking differences in their receptor repertoires than those that merely experience the same odors with different frequencies. Intriguingly, however, our simulations suggest that the situation may be reversed at the very low end: the fraction of receptors for which the predicted abundance change is below 0.1, $\left\mathrm{\Delta}K\right\phantom{\rule{thinmathspace}{0ex}}<\phantom{\rule{thinmathspace}{0ex}}0.1$, is about 2% in the generic case but over 9% for nonoverlapping environment pairs. Thus, changing between nonoverlapping environments emphasizes the more extreme changes in receptor abundances, either the ones that are close to zero or the ones that are large. In contrast, a generic change in the environment leads to a more uniform distribution of abundance changes. Put differently, the particular way in which the environment changes, and not only the magnitude of the change, can affect the receptor distribution in unexpected ways.
The magnitude of the effect of environmental changes on the optimal olfactory receptor distribution is partly controlled by the tuning of the olfactory receptors (Figure 5d). If receptors are narrowly tuned, with each type responding to a small number of odorants, changes in the environment tend to have more drastic effects on the receptor distribution than when the receptors are broadly tuned (Figure 5d), an effect that could be experimentally tested.
Model predictions qualitatively match experiments
Our study opens the exciting possibility of a causal test of the hypothesis of efficient coding in sensory systems, where a perturbation in the odor environment could lead to predictable adaptations of the olfactory receptor distribution during the lifetime of an individual. This does not happen in insects, but it can happen in mammals, since their receptor neurons regularly undergo apoptosis and are replaced.
A recent study demonstrated reproducible changes in olfactory receptor distributions of the sort that we predict in mice (IbarraSoria et al., 2017). These authors raised two groups of mice in similar conditions, exposing one group to a mixture of four odorants (acetophenone, eugenol, heptanal, and Rcarvone) either continuously or intermittently (by adding the mixture to their water supply). Continuous exposure to the odorants had no effect on the receptor distribution, in agreement with the predictions of our model. In contrast, intermittent exposure did lead to systematic changes (Figure 6a).
We used our model to run an experiment similar to that of IbarraSoria et al. (2017) in silico (Figure 6b). Using a sensing matrix based on odor response curves for mouse and human receptors (data for 59 receptors from Saito et al. (2009)), we calculated the predicted change in OSN abundances between two different environments with random covariance matrices constructed as described above. We ran the simulations 24 times, modifying the odor environments each time by adding a small amount of Gaussian random noise to the square roots of these covariance matrices to model small perturbations (details in Appendix 4; range bars in Figure 6b). The results show that the abundances of already numerous receptors do not change much, while there is more change for less numerous receptors. The frequencies of rare receptors can change dramatically, but are also more sensitive to perturbations of the environment (large range bars in Figure 6b).
These results qualitatively match experiment (Figure 6a), where we see the same pattern of the largest reproducible changes occurring for receptors with intermediate abundances. The experimental data is based on receptor abundance measured by RNAseq which is a proxy for counting OSN numbers (IbarraSoria et al., 2017). In our model, the distinction between receptor numbers and OSN numbers is immaterial because a change in the number of receptors expressed per neuron has the same effect as a change in neuron numbers. In general, additional experiments are needed to measure both the number of receptors per neuron and the number of neurons for each receptor type.
A framework for a quantitative test
Given detailed information regarding the affinities of olfactory receptors, the statistics of the odor environment, and the size of the olfactory epithelium (through the total number of neurons ${K}_{\text{tot}}$), our model makes fully quantitative predictions for the abundances of each OSN type. Existing experiments (e.g. IbarraSoria et al., 2017) do not record necessary details regarding the odor environment of the control group and the magnitude of the perturbation experienced by the exposed group. However, such data can be collected using available experimental techniques. Anticipating future experiments, we provide a Matlab (RRID:SCR_001622) script on GitHub (RRID:SCR_002630) to calculate predicted OSN numbers from our model given experimentallymeasured sensing parameters and environment covariance matrix elements (https://github.com/ttesileanu/OlfactoryReceptorDistribution).
Given the huge number of possible odorants (Yu et al., 2015), the sensing matrix of affinities between all receptor types in a species and all environmentally relevant odorants is difficult to measure. One might worry that this poses a challenge for our modeling framework. One approach might be to use lowdimensional representations of olfactory space (e.g. Koulakov et al., 2011; Snitz et al., 2013), but there is not yet a consensus on the sufficiency of such representations. For now, we can ask how the predictions of our model change upon subsampling: if we only know the responses of a subset of receptors to a subset of odorants, can we still accurately predict the OSN numbers for the receptor types that we do have data for? Figure 7a and b show that such partial data do lead to robust statistical predictions of overall receptor abundances.
First steps toward a dynamical model in mammals
We have explored the structure of olfactory receptor distributions that code odors efficiently, that is are adapted to maximize the amount of information that the brain gets about odors. The full solution to the optimization problem, Equation (7), depends in a complicated nonlinear way on the receptor affinities $S$ and covariance of odorant concentrations $\mathrm{\Gamma}$. The distribution of olfactory receptors in the mammalian epithelium, however, must arise dynamically from the pattern of apoptosis and neurogenesis (Calof et al., 1996). At a qualitative level, in the efficient coding paradigm that we propose, the receptor distribution is related to the statistics of natural odors, so that the life cycle of neurons would have to depend dynamically on olfactory experience. Such modulation of OSN lifetime by exposure to odors has been observed experimentally (Santoro and Dulac, 2012; Zhao et al., 2013) and could, for example, be mediated by feedback from the bulb (Schwob et al., 1992).
To obtain a dynamical model, we started with a gradient ascent algorithm for changing receptor numbers, and modified it slightly to impose the constraints that OSN numbers are nonnegative, ${K}_{a}\ge 0$, and their sum ${K}_{\text{tot}}={\sum}_{a}{K}_{a}$ is bounded (details in Appendix 5). This gives
where $\alpha $ is a learning rate, ${\sigma}_{a}^{2}$ is the noise variance for receptor type $a$, and $R$ is the covariance matrix of glomerular responses,
with the angle brackets denoting ensemble averaging over both odors and receptor noise. In the absence of the experiencerelated term ${({R}^{1})}_{aa}$, the dynamics from Equation (9) would be simply logistic growth: the population of OSNs of type $a$ would initially grow at a rate $\alpha $, but would saturate when ${K}_{a}=1/\lambda $ because of the populationdependent death rate $\lambda {K}_{a}$. In other words, the quantity $M/\lambda $ sets the asymptotic value for the total population of sensory neurons, ${K}_{\text{tot}}\to M/\lambda $, with $M$ being the number of receptor types.
Because of the last term in Equation (9), the death rate in our model is influenced by olfactory experience in a receptordependent way. In contrast, the birth rate is not experiencedependent and is the same for all OSN types. Indeed, in experiments, the odor environment is seen to have little effect on receptor choice, but does modulate the rate of apoptosis in the olfactory epithelium (Santoro and Dulac, 2012). Our results suggest that, if olfactory sensory neuron lifetimes are appropriately anticorrelated with the inverse response covariance matrix, then the receptor distribution in the epithelium can converge to achieve optimal information transfer to the brain.
The elements of the response covariance matrix ${R}_{ab}$ could be estimated by temporal averaging of cooccurring glomerular activations via lateral connections between glomeruli (Mori et al., 1999). Performing the inverse necessary for our model is more intricate. The computations could perhaps be done by circuits in the bulb and then fed back to the epithelium through known mechanisms (Schwob et al., 1992),
Within our model, Figure 8a shows an example of receptor numbers converging to the optimum from random initial values. The sensing matrix used here is based on mammalian data (Saito et al., 2009) and we set the total OSN number to ${K}_{\text{tot}}=2000$. The environment covariance matrix is generated using the random procedure described earlier (details in Appendix 4). We see that some receptor types take longer than others to converge (the time axis is logarithmic, which helps visualize the whole range of convergence behaviors). Roughly speaking, convergence is slower when the final OSN abundance is small, which is related to the fact that the rate of change $d{K}_{a}/dt$ in Equation (9) vanishes in the limit ${K}_{a}\to 0$. For the same reason, OSN populations that start at a very low level also take a long time to converge.
In Figure 8b, we show convergence to the same final state, but this time starting from a distribution that is not random but was optimized for a different environment. The initial and final environments are the same as the two environments used in the previous section to compare the simulations to experimental findings (Figure 6b). Interestingly, many receptor types actually take longer to converge in this case compared to the random starting point, perhaps because there are local optima in the landscape of receptor distributions. Given such local minima, stochastic fluctuations will allow the dynamics to reach the global optimum more easily. In realistic situations, there are many sources of such variability, for example, sampling noise due to the fact that the response covariance matrix $R$ must be estimated through stochastic odor encounters and noisy receptor readings. In fact, in Figure 8b, we added a small amount of noise (corresponding to $\pm 0.05{K}_{\text{tot}}/M$) to the initial distribution of receptors to improve convergence rates.
Discussion
We built a model for the distribution of receptor types in the olfactory epithelium that is based on efficient coding, and assumes that the abundances of different receptor types are adapted to the statistics of natural odors in a way that maximizes the amount of information conveyed to the brain by glomerular responses. This model predicts a nonuniform distribution of receptor types in the olfactory epithelium, as well as reproducible changes in the receptor distribution after perturbations to the odor environment. In contrast to other applications of efficient coding, our model operates in a regime in which there are significant correlations between sensors because the adaptation of OSN abundances occurs upstream of the brain circuitry that can decorrelate olfactory responses. In this regime, OSN abundances depend on the full correlation structure of the inputs, leading to predictions that are contextdependent in the sense that whether the abundance of a specific receptor type goes up or down due to a shift in the environment depends on the global context of the responses of all the other receptors. All these striking phenomena have been observed in recent experiments and had not been explained prior to this study.
In our framework, the sensitivity of the receptor distribution to changes in odor statistics is affected by the tuning of the olfactory receptors, with narrowly tuned receptors being more readily affected by such changes than broadly tuned ones. The model also predicts that environments that differ in the identity of the odors that are present will lead to greater deviations in the optimal receptor distribution than environments that differ only in the statistics with which these odors are encountered. Likewise, the model broadly predicts a monotonic relationship between the number of receptor types found in the epithelium and the total number of olfactory sensory neurons, all else being equal.
A detailed test of our model requires more comprehensive measurements of olfactory environments than are currently available. Our hope is that studies such as ours will spur interest in measuring the natural statistics of odors, opening the door for a variety of theoretical advances in olfaction, similar to what was done for vision and audition. Such measurements could for instance be performed by using mass spectrometry to measure the chemical composition of typical odor scenes. Given such data, and a library of receptor affinities, our GitHub (RRID:SCR_002630) online repository provides an easytouse script that uses our model to predict OSN abundances. For mammals, controlled changes in environments similar to those in IbarraSoria et al. (2017) could provide an even more stringent test for our framework.
To our knowledge, this is the first time that efficient coding ideas have been used to explain the pattern of usage of receptors in the olfactory epithelium. Our work can be extended in several ways. OSN responses can manifest complex, nonlinear responses to odor mixtures. Accurate models for how neurons in the olfactory epithelium respond to complex mixtures of odorants are just starting to be developed (e.g. Singh et al., 2018), and these can in principle be incorporated in an informationmaximization procedure similar to ours. More realistic descriptions of natural odor environments can also be added, as they amount to changing the environmental distribution $P(\mathbf{\mathbf{c}})$. For example, the distribution of odorants could be modeled using a Gaussian mixture, rather than the normal distribution used in this paper to enable analytic calculations. Each Gaussian in the mixture would model a different odor object in the environment, more closely approximating the sparse nature of olfactory scenes discussed in, for example, Krishnamurthy et al. (2017).
Of course, the goal of the olfactory system is not simply to encode odors in a way that is optimal for decoding the concentrations of volatile molecules in the environment, but rather to provide an encoding that is most useful for guiding future behavior. This means that the value of different odors might be an important component shaping the neural circuits of the olfactory system. In applications of efficient coding to vision and audition, maximizing mutual information, as we did, has proved effective even in the absence of a treatment of value (Laughlin, 1981; Atick and Redlich, 1990; van Hateren, 1992a; Olshausen and Field, 1996; Simoncelli and Olshausen, 2001; Fairhall et al., 2001; Lewicki, 2002; Ratliff et al., 2010; Garrigan et al., 2010; Tkacik et al., 2010; Hermundstad et al., 2014; Palmer et al., 2015; Salisbury and Palmer, 2016). However, in general, understanding the role of value in shaping neural circuits is an important experimental and theoretical problem. To extend our model in this direction, we would replace the mutual information between odorant concentrations and glomerular responses by a different function that takes into account value assignments (see, e.g. Rivoire and Leibler, 2011). It could be argued, though, that such specialization to the most behaviorally relevant stimuli might be unnecessary or even counterproductive close to the sensory periphery. Indeed, a highly specialized olfactory system might be better at reacting to known stimuli, but would be vulnerable to adversarial attacks in which other organisms take advantage of blind spots in coverage. Because of this, and because precise information regarding how different animals assign value to different odors is scarce, we leave these considerations for future work.
One exciting possibility suggested by our model is a way to perform a first causal test of the efficient coding hypothesis for sensory coding. Given sufficiently detailed information regarding receptor affinities and natural odor statistics, experiments could be designed that perturb the environment in specified ways, and then measure the change in olfactory receptor distributions. Comparing the results to the changes predicted by our theory would provide a strong test of efficient coding by early sensory systems in the brain.
Materials and methods
Software and data
Request a detailed protocolThe code (written in Matlab, RRID:SCR_001622) and data that we used to generate all the results and figures in the paper is available on GitHub (RRID:SCR_002630), at https://github.com/ttesileanu/OlfactoryReceptorDistribution (Teşileanu, 2019; copy archived at https://github.com/elifesciencespublications/OlfactoryReceptorDistribution).
Appendix 1
Choice of sensing matrices and receptor noise variances
We used three types of sensing matrices in this study. Two were based on experimental data, one using fly receptors (Hallem and Carlson, 2006), and one using mouse and human receptors (Saito et al., 2009); and another type of sensing matrix was based on randomlygenerated receptor affinity profiles. These can all be either directly downloaded from our repository on GitHub (RRID:SCR_002630), https://github.com/ttesileanu/OlfactoryReceptorDistribution, or generated using the code available there.
Fly sensing matrix
Some of our simulations used a sensing matrix based on Drosophila receptor affinities, as measured by Hallem and Carlson (Hallem and Carlson, 2006). This includes the responses of 24 of the 60 receptor types in the fly against a panel of 110 odorants, measured using singleunit electrophysiology in a mutant antennal neuron. We used the values from Table S1 in (Hallem and Carlson, 2006) for the sensing matrix elements. To estimate receptor noise, we used the standard deviation measured for the background firing rates for each receptor (data obtained from the authors). The fly data has the advantage of being more complete than equivalent datasets in mammals.
Mammalian sensing matrix
When comparing our model to experimental findings from (IbarraSoria et al., 2017), we used a sensing matrix based on mouse and human receptor affinity data from (Saito et al., 2009). This was measured using heterologous expression of olfactory genes, and tested in total 219 mouse and 245 human receptor types against 93 different odorants. However, only 49 mouse receptors and 10 human receptors exhibited detectable responses against any of the odorants, while only 63 odorants activated any receptors. From the remaining 59 × 63 = 3717 receptor–odorant pairs, only 335 (about 9%) showed a response, and were assayed at 11 different concentration points. In this paper, we used the values obtained for the highest concentration (3 mM).
Random sensing matrices
The random sensing matrices matrices used in the main text (and referred to as ‘tuning’ in some of the figures in this Appendix) were generated as follows. We started by treating the column (i.e. odorant) index as a onedimensional odor coordinate with periodic boundary conditions. We normalized the index to a coordinate $x$ running from 0 to 1. For each receptor, we then chose a center ${x}_{0}$ along this line, corresponding to the odorant to which the receptor has maximum affinity, and a standard deviation $\sigma $, corresponding to the tuning width of the receptor. Note that both ${x}_{0}$ and $\sigma $ are allowed to be real numbers, so that the maximum affinity can occur at a position that does not correspond to any particular odorant from the sensing matrix.
To obtain a belllike response profile for the receptors while preserving the periodicity of the odor coordinate we chose, we defined the response affinity to odorant $x$ by
This expression can be obtained by imagining odorant space as a circle embedded in a twodimensional plane, with odorant $x$ mapped to an angle $\theta =2\pi x$ on this circle, and considering a Gaussian response profile in this twodimensional embedding space. This is simply a convenient choice for treating odor space in a way that eliminates artifacts at the edges of the sensing matrix, and we do not assign any significance to the particular coordinate system that we used.
The centers ${x}_{0}$ for the Gaussian profiles for each of the receptors were chosen uniformly at random, and the tuning width $\sigma $ was either a fixed parameter for the entire sensing matrix, or was uniformly sampled from an interval. Before using the matrices we randomly shuffled the columns to remove the dependencies between neighboring odorants, and finally added some amount of random Gaussian noise (mean centered and with standard deviation 1/200). The overall scale of the sensing matrices was set by multiplying all the affinities by 100, which yielded values comparable to the measured firing rates in fly olfactory neurons (Hallem and Carlson, 2006).
For the robustness results below we also generated random matrices in additional ways: (1) ‘gaussian’: drawing the affinities from a Gaussian distribution (with zero mean and standard deviation 2), (2) ‘bernoulli’: drawing from a Bernoulli distribution (with elements equal to 5 with probability 30%, and 0 with probability 70%), (3) ‘signed’: drawing from a Bernoulli distribution followed by choosing the sign (so that elements are 5 with probability 15%, –5 with probability 15%, and 0 with probability 70%); and (4, 5) ‘fly scrambled’ and ‘mammal scrambled’: scrambling the elements in the fly and mammalian datasets (across both odorants and receptors).
Robustness of results to changing the sensing matrix
Our qualitative results are robust across a variety of different choices for the sensing matrix (Appendix 1—figure 1). For instance, the optimal number of receptor types expressed in a fraction of the OSN population larger than 1% grows monotonically with the total number of neurons (Appendix 1—figure 2). Similarly, the general effect that environment change has on optimal OSN numbers, with less abundant receptor types changing more than more abundant ones, is generic across different choices of sensing matrices (Appendix 1—figure 3).
Appendix 2
Mathematical derivations
Deriving the expression for the mutual information
In the main text we assume a Gaussian distribution for odorant concentrations and approximate receptor responses as linear with additive Gaussian noise, Equation (2). Thus it follows that the marginal distribution of receptor responses is also Gaussian. Taking averages of the responses, $\u27e8{r}_{a}\u27e9$, and of products of responses, $\u27e8{r}_{a}{r}_{b}\u27e9$, over both the noise distribution and the odorant distribution, and using Equation (2) from the main text, we get a normal distribution of responses:
where the mean response vector ${\mathbf{\mathbf{r}}}_{0}$ and the response covariance matrix $R$ are given by
where $S$ is the sensing matrix, $\mathbb{K}$ is a diagonal matrix of OSN abundances, and $\mathrm{\Sigma}$ is the covariance matrix of receptor noises, $\mathrm{\Sigma}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}({\sigma}_{1}^{2},\dots ,{\sigma}_{M}^{2})$ (also see the main text). Here, as in Equation (1) in the main text, ${\mathbf{\mathbf{c}}}_{0}$ is the mean concentration vector, $\mathrm{\Gamma}$ is the covariance matrix of odorant concentrations, and we use the overlap matrix from Equation (5) in the main text, $Q=S\mathrm{\Gamma}{S}^{T}$. Note that in the absence of noise ($\mathrm{\Sigma}=0$), the response matrix is simply the overlap matrix $Q$ modulated by the number of OSNs of each type, $R}_{\text{noiseless}}=\mathbb{K}Q\mathbb{K$.
The joint probability distribution over responses and concentrations, $P(\mathbf{\mathbf{r}},\mathbf{\mathbf{c}})$, is itself Gaussian. To calculate the corresponding covariance matrix, we need the covariances between responses, $\u27e8{r}_{a}{r}_{b}\u27e9\u27e8{r}_{a}\u27e9\u27e8{r}_{b}\u27e9$, which are just the elements of the response matrix $R$ from Equation (13) above; and between concentrations, $\u27e8{c}_{i}{c}_{j}\u27e9\u27e8{c}_{i}\u27e9\u27e8{c}_{j}\u27e9$, which are the elements of the environment covariance matrix $\mathrm{\Gamma}$, Equation (1) in the main text. In addition, we need the covariances between responses and concentrations, $\u27e8{r}_{a}{c}_{i}\u27e9\u27e8{r}_{a}\u27e9\u27e8{c}_{i}\u27e9$, which can be calculated using Equation (2) from the main text. We get:
with
The mutual information between responses and odors is then given by (see below for a derivation):
From Equation (13) we have
and from Equation (15),
where we used Equation (13) again, and employed Schur’s determinant identity (derived below). Thus,
This recovers the result quoted in the main text, Equation (4).
By using the fact that the diagonal matrices $\mathbb{K}$ and ${\mathrm{\Sigma}}^{1}$ commute, we can also write:
This shows that the mutual information can be written in terms of a symmetric ‘SNR matrix’ $\stackrel{~}{Q}={\mathrm{\Sigma}}^{1/2}Q{\mathrm{\Sigma}}^{1/2}$. This is simply the covariance matrix of responses in which each response was normalized by the noise variance of the corresponding receptor.
Schur’s determinant identity
The identity for the determinant of a 2 × 2 block matrix that we used in Equation (18) above can be derived in the following way. First, note that
Now, from the definition of the determinant it can be seen that
since all the products involving elements from the offdiagonal blocks must necessarily also involve elements from the 0 matrix. Thus, taking the determinant of Equation (21), we get the desired identity
Mutual information for Gaussian distributions
The expression from Equation (16) for the mutual information $I(\mathbf{\mathbf{r}},\mathbf{\mathbf{c}})$ can be derived by starting with the fact that $I$ is equal to the KullbackLeibler (KL) divergence from the joint distribution $P(\mathbf{\mathbf{r}},\mathbf{\mathbf{c}})$ to the product distribution $P(\mathbf{\mathbf{r}})P(\mathbf{\mathbf{c}})$. As a first step, let us calculate the KL divergence between two multivariate normals in $n$ dimensions:
where
Plugging the distribution functions into the logarithm, we have
where the normalization property of $p(\mathbf{\mathbf{x}})$ was used. Using also the definition of the mean and of the covariance matrix, we have
which implies
for any vector $\mu $ and matrix $C$. Plugging this into Equation (26), we get
We can now return to calculating the KL divergence from $P(\mathbf{\mathbf{r}},\mathbf{\mathbf{c}})$ to $P(\mathbf{\mathbf{r}})P(\mathbf{\mathbf{c}})$. Note that, since $P(\mathbf{\mathbf{r}})$ and $P(\mathbf{\mathbf{c}})$ are just the marginals of the joint distribution, the means of the variables are the same in the joint and in the product, so that the last term in the KL divergence vanishes. The covariance matrix for the product distribution is
so the product inside the trace becomes
where the entries replaced by '$\mathrm{\dots}$' need not be calculated because they drop out when the trace is taken. The sum of the dimensions of $R$ and $\mathrm{\Gamma}$ is equal to the dimension, $n$, of $\mathrm{\Lambda}$, so that the term involving the trace from Equation (29) also drops out, leaving us with the final result:
which is the same as Equation (16) that was used in the previous section.
Deriving the KKT conditions for the information optimum
In order to find the optimal distribution of olfactory receptors, we must maximize the mutual information from Equation (4) in the main text, subject to constraints. Let us first calculate the gradient of the mutual information with respect to the receptor numbers:
The cyclic property of the trace allows us to use the usual rules to differentiate under the trace operator, so we get
We now have to address the constraints. We have two kinds of constraints: an equality constraint that sets the total number of neurons, $\sum {K}_{a}={K}_{\text{tot}}$; and inequality constraints that ensure that all receptor abundances are nonnegative, ${K}_{a}\ge 0$. This can be done using the KarushKuhnTucker (KKT) conditions, which require the introduction of Lagrange multipliers: $\lambda $ for the equality constraint, and ${\mu}_{a}$ for the inequality constraints. At the optimum, we must have:
where the Lagrange multipliers for the inequality constraints, ${\mu}_{a}$, must be nonnegative, and must vanish unless the inequality is saturated:
Put differently, if ${K}_{a}\phantom{\rule{thinmathspace}{0ex}}>\phantom{\rule{thinmathspace}{0ex}}0$, then ${\mu}_{a}=0$ and $\partial I/\partial {K}_{a}=\lambda /2$; while if ${K}_{a}=0$, then $\partial I/\partial {K}_{a}=\lambda /2{\mu}_{a}\le \lambda /2$. Combined with Equation (34), this yields
The magnitude of $\lambda $ is set by imposing the normalization condition $\sum {K}_{a}={K}_{\text{tot}}$.
The manyneuron approximation
Suppose we are in the regime in which the total number of neurons is large, and in particular, each of the abundances ${K}_{a}$ is large. Then we can perform an expansion of the expression appearing in the KKT equations from Equation (37):
whose $aa$ component is
where we used $\stackrel{~}{Q}={\mathrm{\Sigma}}^{1/2}Q{\mathrm{\Sigma}}^{1/2}$. With the notation
we can plug into Equation (37) and get
This quadratic equation has only one large solution, and it is given approximately by
Combined with the normalization constraint, ${\sum}_{a}{K}_{a}={K}_{\text{tot}}$, this recovers Equation (8) from the main text.
Optimal distribution for uncorrelated responses
When the overlap matrix $Q=S\mathrm{\Gamma}{S}^{T}$ is diagonal, the optimization problem simplifies considerably. By plugging $Q=\mathrm{diag}({Q}_{aa})$ into Equation (4) in the main text, we find
We can again use the KKT approach and add Lagrange multipliers $\lambda $ and ${\mu}_{a}$ for enforcing the equality and inequality constraints, respectively,
and take derivatives with respect to ${K}_{a}$ to find the optimum,
with the condition that ${\mu}_{a}\ge 0$ and either ${\mu}_{a}$ or ${K}_{a}$ must vanish, ${\mu}_{a}{K}_{a}=0$. This leads to
showing that receptor abundances grow monotonically with ${Q}_{aa}/{\sigma}_{a}^{2}$. This explains the correlation between OSN abundances ${K}_{a}$ and receptor SNRs ${Q}_{aa}/{\sigma}_{a}^{2}$ when the responses are uncorrelated or weakly correlated.
First receptor type to be activated
When there is only one active receptor, ${K}_{x}={K}_{\text{tot}}$, ${K}_{a\ne x}=0$, the KKT conditions from Equation (37) are automatically satisfied. The receptor that is activated first can be found in this case by calculating the information $I(\mathbf{\mathbf{r}},\mathbf{\mathbf{c}})$ using Equation (4) from the main text while assuming an arbitrary index $x$ for the active receptor, and then finding $x={x}^{*}$ that yields the maximum value. Without loss of generality, we can permute the receptor indices such that $x=1$. Using Equation (19) and setting ${K}_{1}={K}_{\text{tot}}$, we have:
Thus, in general, the information when only receptor type $x$ is activated is given by
which implies that information is maximized when $x$ matches the receptor corresponding to the highest ratio between the diagonal value of the overlap matrix $Q$ and the receptor variance in that channel ${\sigma}_{x}^{2}$; that is the receptor that maximizes the signaltonoise ratio:
Another way to think of this result is by employing the usual expression for the capacity of a single Gaussian channel, and then finding the channel that maximizes this capacity.
Invariance of mutual information under invertible and differentiable transformations
Consider the mutual information between two variables $\mathbf{r}\in {\mathbb{R}}^{M}$ and $\mathbf{c}\in {\mathbb{R}}^{N}$:
Let us now define two different variables that depend on $\mathbf{\mathbf{r}}$ and $\mathbf{\mathbf{c}}$ in an invertible and continuouslydifferentiable (but in general nonlinear) way,
The joint probability distribution for the new variables is related to the joint distribution of the original variables through the Jacobian determinants,
where
For the marginals, we have
where we used the standard substitution formula for multiple integrals. We can now calculate the mutual information between the new variables:
Thus, invertible and continuouslydifferentiable transformations of either the response variables $\mathbf{\mathbf{r}}$ or the concentration variables $\mathbf{\mathbf{c}}$ in our model leave the mutual information unchanged.
Multiple glomeruli with the same affinity profile
In mammals, the axons from neurons expressing a given receptor type can project to anywhere from 2 to 16 different glomeruli. Here we show that in our setup, information transfer only depends on the total number of neurons of a given type, and not on the number of glomeruli to which they project.
The key observation is that mutual information, Equation (3) in the main text, is unchanged when the responses and/or concentrations are modified by invertible transformations (see previous section). In particular, linear transformations of the responses do not affect the information values. Suppose that we have a case in which two receptors $p$ and $q$ have identical affinities, so that ${S}_{pi}={S}_{qi}$ for all odorants $i$. We can then form linear combinations of the corresponding glomerular responses,
and consider a transformation that replaces $({r}_{p},{r}_{q})$ with $({r}_{+},{r}_{})$. Since ${r}_{}$ is pure noise, that is it does not depend on the concentration vector $\mathbf{\mathbf{c}}$ in any way, it has no effect on the mutual information.
We have thus shown that the amount of information that $M$ receptor types contain about the environment when two of the receptors have identical affinity profiles is the same as if there were only $M1$ receptor types. The two redundant receptors can be replaced by a single one with an abundance equal to the sum of the abundances of the two original receptors. The sum of two Gaussian variables with the same mean is Gaussian itself and has a variance equal to the sum of the variances of the two variables, meaning that the noise term ${\eta}_{+}$ in the ${r}_{+}$ response has variance $\frac{{K}_{p}{\sigma}_{p}^{2}+{K}_{q}{\sigma}_{q}^{2}}{{K}_{p}+{K}_{q}}$.
Appendix 3
A nonlinear response example
Estimating the mutual information numerically
Consider an extension of our model in which the responses depend in a nonlinear way on concentrations, but are still subject to pure Gaussian noise:
Note that here we are calculating the average OSN response ${\overline{r}}_{a}={r}_{a}/{K}_{a}$, while in the main text we used the total response ${r}_{a}$. As far as mutual information calculations are concerned, the difference between ${\overline{r}}_{a}$ and ${r}_{a}$ does not matter, as they are related by an invertible transformation.
Unless the functions ${f}_{a}$ are linear, a closedform solution for the mutual information between concentrations and responses cannot be found. It is thus necessary to calculate the mutual information integral numerically. We can still do part of the calculation analytically, though:
In our case, $P(\overline{\mathbf{r}}\mathbf{c})$ is a multivariate Gaussian distribution whose covariance matrix is $\mathrm{\Sigma}{\mathbb{K}}^{1}$ and does not depend on the concentrations. This means that the $\mathbf{\mathbf{c}}$ integral in the second term can be performed independently of the $\overline{\mathbf{\mathbf{r}}}$ integral, in which case it drops out of the calculation, as it is equal to 1. The $\overline{\mathbf{\mathbf{r}}}$ integral is simply the negative entropy of a multivariate Gaussian distribution, and is thus equal to
The first term in Equation (58) is the entropy of the responses, which needs to be calculated numerically. We use a histogram method, in which we split the space of possible responses along each dimension into bins of equal size $\mathrm{\Delta}$. We then estimate the probability in each bin. If ${i}_{1}\mathrm{\dots}{i}_{M}$ indexes the bins, we can then think of the response distribution as a discrete PDF ${P}_{{i}_{1}\mathrm{\dots}{i}_{M}}$, and we can estimate the entropy using
In this approach, the challenge remains to estimate the PDF of the responses,
where $\mathbf{\mathbf{f}}$ is the vector of response functions $\mathbf{\mathbf{f}}=({f}_{1},\mathrm{\dots},{f}_{M})$. We do this using a sampling technique based on the law of large numbers. Given $n$ sample concentration vectors ${c}_{i}$ drawn from the probability distribution $P(\mathbf{\mathbf{c}})$, we have
where $\mathbb{E}}_{P(\mathbf{c})}\{\cdots \$ denotes the expected value under the distribution of concentrations. We use this formula to estimate the histogram elements ${P}_{{i}_{1}\mathrm{\dots}{i}_{M}}$ and then use Equation (60) to estimate the response entropy $H(\overline{\mathbf{\mathbf{r}}})$. We then plug $H(\overline{\mathbf{\mathbf{r}}})$ and Equation (59) into Equation (58) to find the mutual information. Note that we have not assumed anything about the natural distribution of odor concentrations, $P(\mathbf{\mathbf{c}})$, so that we are not restricted to Gaussian environments with this method.
Competitive binding model
The way in which olfactory neurons respond to arbitrary mixtures of odorants is not completely understood. However, simple kinetic models in which different odorant molecules compete for the same receptor binding site have been shown to capture much of the observed behavior (Singh et al., 2018). In such models, the activation of an OSN of type $a$ in response to a set of odorants with concentrations ${c}_{i}$ is given by
where $\mathsf{E}\mathsf{C}\mathsf{50}}_{ai$ is the concentration of odorant $i$ for which the response for the OSN of type $a$ reaches half its maximum, and ${e}_{ai}$ is the maximum response elicited by odorant $i$ in an OSN of type $a$.
Results from a toy problem
The computation time from the method outlined above for calculating mutual information grows exponentially with the dimensionality $M$ of the response space. Additionally, it grows linearly with the number $n$ of samples drawn from the odor distribution, which in turn needs to grow exponentially with the number $N$ of odorants we are considering in order to sample concentration space sufficiently well. For this reason, largescale simulations involving this method are infeasible.
Thus we focused on a simple example with $M=3$ receptors and $N=15$ odorants. We used an arbitrary subset of elements from the fly sensing matrix and a pair of randomlygenerated nonoverlapping environments (Appendix 3—figure 1) to first calculate the optimal receptor distribution using the linear method described in the main text (Appendix 3—figure 2, top). We chose the scale of the environment covariance matrices to get a variability in the responses of around 1, large enough to enter the nonlinear regime when using the nonlinear response function (described below). We then set the total neuron population to ${K}_{\text{tot}}=200$, which put us in an intermediate SNR regime in which all the receptor types were used in the optimal distribution, but their abundances were different (Appendix 3—figure 2, top).
In the linear approximation, we found that receptor 1 is underrepresented in environment 1, while in environment 2 receptor 3 has very low abundance. We wanted to see how much this result is affected by a nonlinear response function. We used a competitive binding model as described above in which the matrix of EC50 values was taken equal to the sensing matrix used in the linear case, and the efficacies ${e}_{ai}$ were all set to 1:
To calculate the mutual information between responses and concentrations for a fixed choice of neuron abundances ${K}_{a}$, we used the procedure outlined above with 20 bins between –0.75 and 1.5 for each of the response dimensions. We sampled $n={10}^{4}$ concentration vectors to build the response histogram. We calculated the information values in both environments at a 10 × 10 grid of OSN abundances (Appendix 3—figure 2, middle row), and found the cell which maximized the information. The OSN abundances at this maximum (Appendix 3—figure 2, bottom) show the same pattern of change as we found in the linear approximation, with receptors 1 and 3 exchanging places as least abundant in the OSN population.
Appendix 4
Random environment matrices
Generating random covariance matrices
Generating plausible olfactory environments is difficult because so little is known about natural odor scenes. However, it is reasonable to expect that there will be some strong correlations. This could, for instance, be due to the fact that an animal’s odor is composed of several different odorants in fixed proportions, and thus the concentrations with which these odorants are encountered will be correlated.
The most straightforward way to generate a random covariance matrix would be to take the product of a random matrix with its transpose, $\mathrm{\Gamma}=M{M}^{T}$. This automatically ensures that the result is positive (semi)definite. The downside of this method is that the resulting correlation matrices tend to cluster close to the identity (assuming that the entries of $M$ are chosen i.i.d.). One way to avoid this would be to use matrices $M$ that have fewer columns than rows, which indeed leads to nontrivial correlations in $\mathrm{\Gamma}$. However, this only generates rankdeficient covariance matrices which means that odorant concentrations are constrained to live on a lowerdimensional hyperplane. This is too strong a constraint from a biological standpoint.
To avoid these shortcomings, we used a different approach for generating random covariance matrices. We split the process into two parts: we first generated a random correlation matrix by the method described below, in which all the variances (i.e. the diagonal elements) were equal to 1; next we multiplied each row and corresponding column by a standard deviation drawn from a lognormal distribution.
In order to generate random correlation matrices, we used a modified form of an algorithm based on partial correlations (Lewandowski et al., 2009). The partial correlation between two variables ${X}_{i}$ and ${X}_{j}$ conditioned on a set of variables $L$ is the correlation coefficient between the residuals ${R}_{i}$ and ${R}_{j}$ obtained by subtracting the best linear fit for ${X}_{i}$ and ${X}_{j}$ using all the variables in $L$. In other words, the partial correlation between ${X}_{i}$ and ${X}_{j}$ is equal to that part of the correlation coefficient that is not explained by the two variables depending on a common set of explanatory variables, $L$. In our case the ${X}_{i}$ are the concentrations of different odorants in the environment and the partial correlations in question are, for example, the correlation between any pair of the odorants conditioned on the remaining ones. We want to construct the unconditioned correlation matrix between the odor concentrations vectors of the environment. There is an algorithm to construct this matrix that starts by randomly drawing the partial correlation between the first two odorants ${X}_{1}$ and ${X}_{2}$ conditioned on the rest, and then recursively reducing the size of the conditioning set while generating more random partial correlations until the unconditioned correlation values are obtained. For details, see Lewandowski et al. (2009).
The specific procedure used in Lewandowski et al. (2009) draws the partial correlation values from beta distributions with parameters depending on the number of elements in the conditioning set $L$. This is done in order to ensure a uniform sampling of correlation matrices. This, however, is not ideal for our purposes because these samples again tend to cluster close to the identity matrix. A simple modification of the algorithm that provides a tunable amount of correlations is to keep the order of the beta distribution fixed $\alpha =\beta =\text{const}$ (see Stack Exchange, at https://stats.stackexchange.com/q/125020). When the parameter $\beta $ is large we obtain environments with little correlation structure, while small $\beta $ values lead to stronger correlations between odorant concentrations. The functions implementing the generation of random environments are available on our GitHub (RRID:SCR_002630) repository at https://github.com/ttesileanu/OlfactoryReceptorDistribution (see environment/generate_random_environment.m and utils/randcorr.m).
Perturbing covariance matrices
When comparing the qualitative results from our model against experiments in which the odor environment changes (IbarraSoria et al., 2017), we used small perturbations of the initial and final environments to estimate error bars on receptor abundances. To generate a perturbed covariance matrix, $\stackrel{~}{\mathrm{\Gamma}}$, from a starting matrix $\mathrm{\Gamma}$, we first took the matrix square root: a symmetric matrix $M$, which obeys
We then perturbed $M$ by adding normallydistributed i.i.d. values to its elements,
and recreated a covariance matrix by multiplying the perturbed square root with its transpose,
This approach ensures that the perturbed matrix $\stackrel{~}{\mathrm{\Gamma}}$ remains a valid covariance matrix—symmetric and positivedefinite—which would not be guaranteed if the random perturbation was added directly to $\mathrm{\Gamma}$. We chose the magnitude $\sigma $ of the perturbation so that the error bars in our simulations are of comparable magnitude to those in the experiments.
We used a similar method for generating the results from Figure 3, where we needed to apply the same perturbation to two different environments. Given the environment covariance matrices ${\mathrm{\Gamma}}_{k}$, with $k\in \{1,2\}$, we took the matrix square root of each environment matrix, ${M}_{k}={\mathrm{\Gamma}}_{k}^{1/2}$. We then added the same perturbation to both, ${\stackrel{~}{M}}_{k}={M}_{k}+P$, then recovered covariance matrices for the perturbed environments by squaring ${\stackrel{~}{M}}_{k}$, ${\stackrel{~}{\mathrm{\Gamma}}}_{k}={\stackrel{~}{M}}_{k}{\stackrel{~}{M}}_{k}^{T}$. In the examples used in the main text, the perturbation $P$ was a matrix in which only one column was nonzero. The elements in this column were chosen from a Gaussian distribution with zero mean and a standard deviation five times larger than the square root of the median element of ${\mathrm{\Gamma}}_{1}$. This choice was arbitrary and was made to obtain a visible change in the optimal receptor abundances between the ‘control’ and ‘exposed’ environments.
Finally, we employed this approach also for generating nonoverlapping environments. Given two environments ${\mathrm{\Gamma}}_{1}$ and ${\mathrm{\Gamma}}_{2}$ and their matrix square roots ${M}_{1}$ and ${M}_{2}$, we reduced the amount of variance in the first half of ${M}_{1}$’s columns and in the second half of ${M}_{2}$’s. We did this by dividing those columns by a constant factor $f$, which in this case we chose to be $f=4$. We then used the resulting matrices ${\stackrel{~}{M}}_{k}$ to generate covariance matrices $\stackrel{~}{\mathrm{\Gamma}}}_{k}={\stackrel{~}{M}}_{k}{\stackrel{~}{M}}_{k}^{T$ with largely nonoverlapping odors.
Appendix 5
Deriving the dynamical model
To turn the maximization requirement into a dynamical model, we employ a gradient ascent argument. Given the current abundances ${K}_{a}$, we demand that they change in proportion to the corresponding components of the information gradient, plus a Lagrange multiplier to impose the constraint on the total number of neurons:
The brain does not have direct access to the overlap matrix $Q$, but it could measure the response covariance matrix $R$ from Equation (13). Thus, we can write the dynamics as
where we used the fact that ${\mathrm{\Sigma}}^{1/2}$ and $\mathbb{K}$ are diagonal and thus commute. These equations do not yet obey the nonnegativity constraint on the receptor abundances. The divergence in the ${K}_{a}^{1}$ term would superficially appear to ensure that positive abundances stay positive, but there is a hidden quadratic divergence in the response covariance term, ${R}_{aa}^{1}$; see Equation (13). To ensure that all constraints are satisfied while avoiding divergences, we multiply the righthandside of Equation (69) by ${K}_{a}^{2}$, yielding
which is the same as Equation (9) from the main text.
If we keep the Lagrange multiplier $\lambda $ constant, the asymptotic value for the total number of neurons ${K}_{\text{tot}}$ will depend on the statistical structure of olfactory scenes. If instead we want to enforce the constraint $\sum {K}_{a}={K}_{\text{tot}}$ for a predetermined ${K}_{\text{tot}}$, we can promote $\lambda $ itself to a dynamical variable,
where $\beta $ is another learning rate. Provided that the dynamics of $\lambda $ is sufficiently slow compared to that of the neuronal populations ${K}_{a}$, this will tune the experienceindependent component of the neuronal death rate until the total population stabilizes at ${K}_{\text{tot}}$.
Appendix 6
Interpretation of diagonal elements of the inverse overlap matrix
In the main text we saw that the diagonal elements of the inverse overlap matrix ${Q}_{aa}^{1}$ were related to the abundances of OSNs ${K}_{a}$. Specifically,
where $\lambda $ is a Lagrange multiplier imposing the constraint on the total number of neurons. As noted around Equation (13) above, the overlap matrix $Q$ is related to the response covariance matrix $R$: in particular, $Q$ is equal to $R$ when there is a single receptor of each type (${K}_{a}=1$) and there is no noise (${\sigma}_{a}=0$). That is, the overlap matrix measures the covariances between responses in the absence of noise. This means that its inverse $A={Q}^{1}$ is effectively a socalled ‘precision matrix’. Diagonal elements of a precision matrix are inversely related to corresponding diagonal elements of the covariance matrix (i.e. the variances), but, as we will see below, they are also monotonically related to parameters that measure how well each receptor response can be linearly predicted from all the others. Since receptor responses that either do not fluctuate much or whose values can be guessed based on the responses of other receptors are not very informative, we would expect that abundances ${K}_{a}$ are low when the corresponding diagonal elements of the inverse overlap matrix ${A}_{aa}$ are high, which is what we see. In the following we give a short derivation of the connection between the diagonal elements of precision matrices and linear prediction of receptor responses.
Let us work in the particular case in which there is one copy of each receptor and where there is no noise, so that $Q=R$, that is ${Q}_{ij}=\u27e8{r}_{i}{r}_{j}\u27e9\u27e8{r}_{i}\u27e9\u27e8{r}_{j}\u27e9$. Without loss of generality, we focus on calculating the first diagonal element of the inverse overlap matrix, ${A}_{11}$, where $A={Q}^{1}$. For notational convenience, we will also denote the meancentered first response variable by $y\equiv {r}_{1}\u27e8{r}_{1}\u27e9$, and the subsequent ones by ${x}_{a}\equiv {r}_{a+1}\u27e8{r}_{a+1}\u27e9$. Then the covariance matrix $Q$ can be written in block form
where $M$ is
and $\mathbf{\mathbf{x}}$ is a column vector containing the ${x}_{a}$ variables. Using the definition of the inverse together with Laplace’s formula for determinants, we get
Using the Schur determinant identity (derived above) on the block form (Equation (73)) of the matrix $Q$,
where we used the fact that the argument of the second determinant is a scalar.
Now, consider approximating the first response variable $y$ by a linear function of all the others:
where $q$ is the residual. Note that we do not need an intercept term because we meancentered our variables, $\u27e8y\u27e9=\u27e8x\u27e9=0$. Finding the coefficients $\mathbf{\mathbf{a}}$ that lead to a best fit (in the leastsquares sense) requires minimizing the variance of the residual, and a short calculation yields
where $M$ is the same as the matrix defined in Equation (74).
The coefficient of determination ${\rho}^{2}$ is defined as the ratio of explained variance to total variance of the variable $y$,
Comparing this to Equation (76), we see that
showing that the diagonal elements of the precision matrix are monotonically related to the goodnessoffit parameter ${\rho}^{2}$ that indicates how well the corresponding variable can be linearly predicted by all the other variables. In addition, the inverse dependence on the variance of the response ${\u27e8y\u27e9}^{2}$ shows that variables that do not fluctuate much (low ${\u27e8y\u27e9}^{2}$) lead to high diagonal values of the precision matrix . From Equation (72), we see that these variances should be considered ‘large’ or ”small’ in comparison with the noise level in each receptor (${\sigma}_{a}$). Since receptor responses that either do not fluctuate much or whose values can be guessed based on the responses of other receptors are not very informative, we should find that receptor abundances ${K}_{a}$ are low when the corresponding diagonal elements of the inverse overlap matrix ${A}_{aa}={Q}_{aa}^{1}$ are high.
Data availability
All the code necessary to reproduce our results and the figures from the paper is available on GitHub, at https://github.com/ttesileanu/OlfactoryReceptorDistribution (copy archived at https://github.com/elifesciencespublications/OlfactoryReceptorDistribution). The olfactory receptor affinity data were originally published in Hallem et al. (2006) and Saito et al. (2009), and the olfactory receptor expression levels in mouse were originally published in IbarraSoria et al. (2017).
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Decision letter

Upinder Singh BhallaReviewing Editor; Tata Institute of Fundamental Research, India

Catherine DulacSenior Editor; Harvard University, United States

David ZwickerReviewer
In the interests of transparency, eLife includes the editorial decision letter and accompanying author responses. A lightly edited version of the letter sent to the authors after peer review is shown, indicating the most substantive concerns; minor comments are not usually included.
Thank you for submitting your article "Adaptation of olfactory receptor abundances for efficient coding" for consideration by eLife. Your article has been reviewed by 3 peer reviewers, one of whom is a member of our Board of Reviewing Editors, and the evaluation has been overseen by a Senior Editor. The following individual involved in review of your submission has agreed to reveal his identity: David Zwicker (Reviewer #3).
The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.
Summary:
In this manuscript Tesileanu and colleagues present a theoretical analysis of optimal coding in olfactory systems. They derive analytical results and use simulations to ask how receptor distributions depend on the number of neurons, the tuning width of receptors, and environment, with a core assumption of efficient coding. The study leads to the interesting prediction of strikingly changed receptor distribution following olfactory experience.
Essential revisions:
The current paper lays out a good framework but would be much stronger if some essential ramifications of the core idea were to be addressed.
1) The model must make predictions that can be falsified by experimental or evolutionary data.
2) The authors should incorporate more biological activation functions and receptor sensitivity distributions and examine how these affect the conclusions of the model.
3) The authors should comment on the diversity of olfactory systems across evolution and note how their model does or does not account for this diversity.
4) The authors should address the question of what happens when the number of receptors changes (as opposed to the number of neurons), as this is one of the main variables that seems to differ across evolution.
Reviewer #1:
This study builds its analysis on the idea that olfactory coding lies in a regime where sensor responses are correlated, hence efficient coding leads to divergent receptor abundance.
In addition to the assumption of efficient coding, the manuscript also assumes that olfactory receptor populations adapt to achieve such coding, within the timeframe of receptor turnover. This assumption leads to the interesting prediction of strikingly changed receptor distribution following olfactory experience, a phenomenon that has been observed experimentally.
I find the analysis interesting and potentially insightful, but it misses out on a few key biological points, that I feel really should be taken on board if the analysis is to be biologically relevant. I'll enumerate three of these, in increasing order of concern.
1) The authors explicitly ignore temporal correlations in olfactory cues, with a brief line in the introduction to their model that states that spike timing could be incorporated into the model. I do not see how this will work for respiratory phase tuning of odor responses, and would be interested to see what the authors had in mind for this.
2) The authors choose an operating point where they can apply a linear model for glomerular responses. In the animal, the operating range of different receptors for different odors is rather diverse, with the halfmax varying substantially and the slope also varies. Thus a subset of odors will be saturating for some receptors, but linear or even subthreshold for other receptors. I suspect that this will affect the analysis of the responses.
My view is that any coding theory has to account for the very wide range of odor concentrations encountered in nature. One could possibly add this to the analysis reported in Equations 4 to 6, by summing the mutual information over a set of odor ranges, in which different but overlapping subsets of receptors are involved. I would be interested to see if this alters the conclusions.
3) A major point of concern with the whole analysis is of salience. The obvious outlier here is pheromones. Enormous resources are allocated to pheromone detection, and clearly this doesn't seem to fall within the framework presented in the paper. Even with the general olfactory system, the assumption of efficient coding needs to be further mapped to the distribution of odor salience, that is, relevance for animal survival. There seems to be a subtle nod to this point in the thirdlast paragraph of the Discussion, where 'value of detecting different odorants' is mentioned. I feel that the point is central enough that it needs to be fully addressed.
The constraint is not just to efficiently code the environment, it is to efficiently code those aspects of the environment which matter for survival. This seems to give rise to a fundamental challenge to this model, as follows: Assume a rare predator with a characteristic odor. Even if the predator is absent from the odor scene for long periods, it would be fatal to the prey species to underexpress receptors sensitive to the predator. One can come up with numerous other examples on these lines where selection pressures necessitate receptor expression for reasons other than efficient coding. There may be a couple of ways to go about incorporating this into the model: an evolutionarily determined 'prior' that weights salience of receptors, or a more general rule that tries to ensure a certain degree of broad coverage even at the expense of efficient coding. I suspect both may be relevant.
In summary, I think that the current paper lays out a good framework but would be much stronger if some essential ramifications of the core idea were to be addressed.
Reviewer #2:
In this manuscript Tesileanu and colleagues present a theoretical analysis of optimal coding in olfactory systems. The goal is to find the distribution of olfactory receptor abundances that maximizes the information an olfactory system can gain about odors in its environment, and to predict how receptor abundances should change when the environment changes. Given a set of assumptions about how odors are encoded by a population of receptors, they derive an expression for the mutual information between the response of a receptor population and a vector of environmental odors. They then evaluate this expression and show that the information depends on an overlap matrix, related to the covariance of the environmental odor vector. Based on these analytical results, they use simulations to ask how receptor distributions depend on the number of neurons, and the tuning width of receptors. They then ask how receptor abundances should change when the environment changes. They report a number of findings: (1) Receptor abundances are more sensitive to environmental perturbations when the number of neurons is small or intermediate, (2) Receptor abundances are more sensitive to environmental perturbations when they are narrowly tuned, (3) changes in optimal receptor abundances cannot be simply predicted from changes in odor abundances or variances.
At an abstract level, olfactory systems can be thought of as arrays of receptors, which have evolved from distinct receptor families many times over the course of evolution. Olfactory receptor genes are among the largest and most rapidly evolving gene families. Therefore I highly support the goals of this study to provide a theoretical understanding of how receptor arrays should change in response to changes in odor environment. In general, the level of abstraction adopted in this study is appropriate, and some of the findings are interesting. However, I have a number of questions about the analyses performed and conclusions reached, particularly concerning how the results might be related to biologically testable phenomena.
1) The conclusions concerning how receptor abundances should change following a change in environment are disappointing. While their model recapitulates IbarraSoira's result which predicts that the distribution of high abundance receptors is likely to remain unchanged, they do not provide any concrete predictions on the receptors which change their abundance in either direction of change or magnitude. As currently stated, the central predictions of the model – that optimal receptor abundances can increase or decrease or stay the same following a change in environment – seems to be unfalsifiable.
The manuscript could be strengthened by making more concrete predictions about how receptor abundances should change, at least in particular regimes. For example, the authors note that for intermediate numbers of neurons, optimal receptor distributions are anticorrelated with the inverse of the overlap matrix Q^{1}. They expand on this to say that receptors with high Q^{1} can be uninformative because they do not fluctuate or because they provide redundant information. Although I did not fully follow the arguments here, it seemed like this was saying that abundance is inversely related to information, and there are two ways to be uninformative, one by having low variance, and two by being highly correlated with other receptors. Could this be used to make more concrete predictions about predicted changes in receptor abundance, at least for a given number of neurons? In addition, the authors also provide model evidence for predicting the magnitude of the change based on the change in olfactory environment, but it is unclear the characteristics which group types of changes together.
2) Some of the conclusions seem odd when considered in the context of olfactory evolution. For example, the authors conclude that if the number of neurons is large, then the optimal receptor distribution is approximately uniform. Olfactory systems differ greatly in magnitude across organisms. In particular, two of the moststudied models, fly and mouse, differ by an order of magnitude in the number of receptors (~60 for fly, ~1000 for mouse), as well as the total number of neurons. The finding that total neuron number determines receptor distribution should be tied numerically to the olfactory systems of flies and mice, if not also for other organisms. It is unclear, for example, whether the olfactory receptor number of mice is considered large, or whether it would fall in the intermediate signal to noise regime. Does the model predict that mouse receptor distributions are uniform while fly distributions are highly skewed? Why then is any adaptation observed in mouse receptor abundances as has been observed experimentally?
Given the results presented here one might imagine that the optimal strategy would be to make a very large number of broadly tuned receptors. Instead, what we observe across evolution are olfactory systems of various sizes, with various widths of odor tuning, all constantly evolving. The number of receptors in particular seems to be under strong evolutionary pressure, with new gene families expanding (as in ant ORs) or collapsing (as in humans). This discrepancy, or the other constraints that might lead to the biological situation, should be commented on.
The authors state that receptor abundances do not change in insects and therefore focus on a mammalian example to test their hypothesis. However, insect olfactory systems evolve quite rapidly between closely related species, and there is a large literature on this, especially from the Hansson group (e.g. Dekker…Hansson, 2006). Can these studies be used to test any of the hypotheses here? Or can the authors propose comparative studies that would test their hypotheses?
3) Several concepts used in the text are a bit unclear, at least to a biological reader:
Could the authors provide some intuition for what is meant (biologically) by the inverse of the overlap matrix?
Could the authors please unpack the following sentence:
The quantity KQ thus behaves as a signaltonoise ratio (SNR), so that Equation 4 is essentially a generalization to multiple, correlated channels of the standard result for a single Gaussian channel, I = 1 log(1 + SNR^{2}).
Could the authors please clarify in the discussion of Equation 7 whether K_{tot} represents the total number of neurons, the number of receptors, or the number of receptor types? Is the total number of neurons the most sensible thing to vary or would it be interesting to look at olfactory systems with different numbers of receptor types? This seems related to the question of where noise arises in the system, and what other constraints, besides information as quantified here, an animal might have on the design of its olfactory system.
4) The investigation of how optimal coding changes with broad versus narrow tuned receptors was interesting. However, real receptor arrays, at least as seen in the Hallem data, contain a mix of broadly and narrowlytuned receptors, and receptor tuning width depends on odor intensity, with many receptor showing narrowly tuned response at low concentrations and wider tuning at high concentrations. Could the authors explore what happens in this regime, and provide any explanation for why animals might have both broad and narrowly tuned receptors? This finding could be further explored by making predictions for olfactory systems with receptors of mixed tuning widths, as is generally accepted to be the case in most organisms. This would provide a more concrete prediction for future experiments.
5) The authors claim that their model is robust to nonlinearities and as well as their choice to represent the olfactory environment as a vector of concentrations. These ideas should be tested and demonstrated within the paper. For example, the nonlinearities involved in receptor encoding are well known: receptor responses can be expressed as a Hill function of odor concentration:
r = (c^n)/(c^n+Kd)
In many olfactory systems n=1, further simplifying this equation. The authors should explicitly show that the model generalizes when this nonlinearity is included. In addition, the main sources of noise in receptor encoding are likely to be (1) difference in receptor abundance across neurons that express the same receptor, (2) stochasticity in receptor binding and activation. The authors might consider incorporating these sources noise and showing that the model extends in this case.
The first section of the Results is difficult to read because it contained a number of statements justifying elements of the model and claiming that these do not affect the conclusions. This section would be easier to read if these points were saved for later in the manuscript where they could be explicitly demonstrated.
6) The section on dynamical optimization at the end seemed least wellconstrained by data, and also (as noted) somewhat preliminary. The authors might consider reserving this material for a future manuscript that explores dynamics and tests them more thoroughly, and instead using this space to show that the model still holds when certain assumptions in the first version of the model are relaxed.
7) The authors should consider including graphical representations, similar to those provided in Figure 1, for concepts such as the mutual information measure, the covariance matrix, the overlap matrix, and the inverse overlap matrix. This would help provide insight for readers with less mathematical background, who may nonetheless be interested in the predictions of the models.
Reviewer #3:
The paper investigates theoretically how changing copy numbers of olfactory sensory neurons affects the coding properties of the olfactory system. The authors introduce a simple model based on the maximization of mutual information, which they analyze analytically and numerically using both artificial and measured values for the receptor sensitivities. Their analysis reveals a complex dependence of the optimal copy numbers of expressed receptors on the correlation structure of the receptor sensitivities and the odor environment. Since qualitatively similar dependencies have been observed in experiments, the model is very valuable for understanding the dynamics of copy number adaptation in the olfactory system. More generally, the presented model of the olfactory system is helpful for discussing how sensory systems adapt to changes in the environment and whether the aim for efficient coding is the driving mechanism.
The manuscript is well written and the arguments are clearly presented for the most part. My main concerns with the manuscript are that some limitations are not spelled out explicitly and that the theoretical analysis could have been more comprehensive. In particular, the authors do not investigate how their model would fair in the realistic case where odors are sparse and they do not discuss how the results depend on the number of different receptor types and the number of different odor molecules. The latter might be important to assess how relevant the results would be for realistic situations, since the current analysis is necessarily restricted to smaller numbers for the lack of adequate experimental data.
Taken together, I believe that the manuscript provides a substantial advance of our understanding of the olfactory system and of the adaptation of sensory systems to changing environments in general. I can therefore recommend publication of the manuscript in eLife once my comments have been taken into account.
https://doi.org/10.7554/eLife.39279.024Author response
Essential revisions:
The current paper lays out a good framework but would be much stronger if some essential ramifications of the core idea were to be addressed.
1) The model must make predictions that can be falsified by experimental or evolutionary data.
Our model makes both qualitative and quantitative predictions, which we have now highlighted in the paper, along with strategies for testing them.
First, our model makes qualitative predictions. For example, the number of receptor types should grow with the number of neurons in the olfactory epithelium, all else being equal, at least for closelyrelated species in similar ecological niches. Testing this prediction requires surveys of the number of receptor types and OSNs in different species, data which is currently fragmentary. As a step towards a test we plotted the number of intact OR genes in several mammalian species against an estimate of the number of OSNs derived from measures of the area of the olfactory epithelium and an allometric scaling law relating neural density and body mass (new Figure 2F). The trend is consistent with our predictions; a precise match is not expected since the species for which we found data live in different ecological niches and have presumably evolved distinct receptor repertoires.
Second, our model makes fully quantitative predictions for the abundances of olfactory neurons of different types, given receptor affinities and the statistics of the odor environment. Likewise, the model makes detailed predictions for how the abundances should change when the olfactory environment is modified. The predictions can be experimentally checked, for example using a protocol like that from IbarraSoria et al., 2017. We now describe this in detail in the section “A framework for a quantitative test”, which describes a procedure for working out the predictions in a given setting. This section is paired with a Matlab script that allows an interested researcher to plug measured affinity data and environmental statistics into our model and obtain numeric predictions for OSN abundances. We also applied our procedure to an insilicoexperiment imitating IbarraSoria et al., 2017 with an available panel of 59 mouse and human receptors responding to 63 odorants. The results (Figure 6) qualitatively resemble the outcome of in vivoexperiments in mouse (~1000 receptor types responding to a complete olfactory environment). Finally, we investigated the robustness of our model’s predictions to subsampling of the receptors and the odor environment (new Figure 7, and accompanying discussion). We find that the predictions of absolute receptor abundances in an environment are robust to subsampling. Of course, more complete measurements will be required for predictions of smaller differences in receptor abundances between different environments.
Olfaction is a complex sense with many receptors sensing diverse odorants. Because of this, olfactory neuroscience lags behind visual neuroscience in the characterization of complete receptor repertoires and of natural olfactory scenes. However, large scale surveys of such data have begun, sponsored partly by the BRAIN program in the USA and by the NSF Olfaction Ideas Lab. Techniques are certainly available – e.g., mass spectrometry of volatile molecules harvested in a given environment. Our theoretical work motivates such largescale surveys, and, given the data, will make precise predictions for new experiments.
Finally, in addition to these avenues for new experimental tests, our work is, to our knowledge, the first to propose a normative explanation for the observed qualitative behavior of receptor abundances in the olfactory epithelium including: (1) the inhomogeneous receptor distribution in the OSN population, and (2) the reproducible but apparently sporadic patterns of adaptation in receptor abundances following olfactory experience in mammals.
2) The authors should incorporate more biological activation functions and receptor sensitivity distributions and examine how these affect the conclusions of the model.
A challenge here is that the experimental data on receptor sensitivity distributions and biological activation functions is limited. To answer this question, we have leveraged available datasets surveying receptor responses to panels of odorants, as well as existing studies of response nonlinearities.
We are using receptor sensitivity values from fly (Drosophila) and from mammals (mouse and human) in the figures of the main text in addition to artificial sensitivity distributions with scalable tuning widths. In the updated Appendix (Appendix 1, Figures 1, 2, 3) we also include scrambled versions of these sensitivity distributions and additional artificial sensing matrices to show that the qualitative conclusions are robust to the details.
Fully including nonlinear effects in OSN responses requires data from new experiments. Indeed, doseresponse curves for neurons responding to single odorants are only available in a small number of cases, and nonlinearities in mixture responses that we would need in general are only beginning to be understood. There is some evidence (e.g., Singh et al., 2018, now cited in the paper) that a simple competitive binding model might give a reasonably good description of mixture responses in many cases. Following the reviewer’s suggestion, we used this framework as the starting point for the nonlinear results that we added to the Appendix (Appendix 3, A nonlinear response example). Such a model needs data on Hill coefficients and maximum activation values for every pair of receptor and odorant, and we estimated these from data in the fly. In the nonlinear case the mutual information must be numerically estimated, and, doing this, we found that in a simple example the qualitative structure of the results was the same as in a linear sensing model based on the same receptor data.
It is worth adding that typical neural nonlinearities show an approximately linear regime between the activation threshold and saturation. Our model should be regarded as a linearized approximation of this regime. Also, the mutual information that we optimize is invariant under invertible, smooth nonlinearities (Appendix 2, section “Invariance of mutual information under invertible and differentiable transformations”). For these reasons we expect our linear sensing model to provide a reasonable approximation which can be numerically extended to a fully nonlinear model when such data become broadly available for OSNs.
At a technical level, calculating the mutual information outside the linear and Gaussian idealization that we used is much more difficult because the required integrals must be calculated numerically, as we now describe in Appendix 3. The runtime for the simple code that we used in this case is orders of magnitude slower than that for the linear case and, worse, it grows exponentially with the number of receptor types and the number of odorants used in the problem. There are more advanced methods for estimating mutual information numerically, and there may be new approximation schemes that are better suited for our problem, but these are entire research projects in their own right and are beyond the scope of the present work. A related point is that entropy estimation and maximization are inherently difficult computationally, and so neural circuits might have no choice but to only approximately adapt to natural statistics.
On a broader methodological level, we feel that an important role of theory is to try to find aspects of biological systems that are “universal”, in the sense that the behavior of the system is roughly the same independent of microscopic details. We implicitly make use of such universality when we study olfaction without explicitly modeling the interaction between every molecule in the nasal epithelium and every volatile molecule that reaches it. Our premise is that capturing just the rough aspects of receptor responses as we do in our model might be enough to get a first approximation of the receptor abundances. The theoretical model can then be improved upon comparing the results from this simplified analysis to experiment.
3) The authors should comment on the diversity of olfactory systems across evolution and note how their model does or does not account for this diversity.
Our study focuses on the question of how to optimally use an available repertoire of olfactory receptors. We therefore take the set of available receptors, as well as their affinities to odorants, to be fixed. We do not seek to say anything about how these evolve.
Our model does suggest that larger olfactory systems (more OSNs in the epithelium), should support a greater diversity of receptor types. Strictly speaking the prediction is that, given a fixed repertoire of receptor types and olfactory environment, the number of types that are expressed should increase with the number of OSNs. Of course, even related animal species can typically have different genetically encoded receptor types and occupy different environmental niches. Unfortunately, information about the receptor repertoires and olfactory environments of different species is fragmentary and sometimes nonexistent. Nevertheless, our theory leads to expect a general trend of receptor diversity increasing with OSN numbers. As a preliminary study, we illustrate this trend for some mammalian species in the new Figure 2F.
Receptor abundances can change faster than affinity profiles – in mammals, this even happens during the lifetime of an individual. Thus, by focusing on understanding the receptor abundances, we in effect focus on questions of adaptation on shorter timescales. There are other recent studies that approach the question of the evolution of receptor genes (e.g., Zwicker et al., 2016), but we are considering a different, complementary question here.
We edited the text to make these points clearer.
4) The authors should address the question of what happens when the number of receptors changes (as opposed to the number of neurons), as this is one of the main variables that seems to differ across evolution.
It is indeed interesting to see how our results change when we change the number of receptors (and also the number of odorants). We have now added results that show how the optimal abundances of the remaining receptors change when a fraction of the receptors is removed (new Figure 7).
The same results can also be interpreted in terms of the robustness of our results to incomplete sampling of the receptors. We find that even without having measurements for the affinity profiles for all receptor types, we can still get reasonable estimates for the optimal abundances of the receptors we do have data for. We also showed that a similar robustness holds for subsampling of odorants. This suggests that we can obtain reasonable results even without recording the affinity profile against every odorant in an environment, which would be difficult to achieve.
That said, we wish to emphasize again that we are not trying to build an evolutionary model of the olfactory periphery. We are mainly interested in changes that occur either during the lifetime of an individual, or on short evolutionary time periods, during which it may be easier to alter the abundances of receptors rather than their affinities. It thus seems reasonable to assume that the receptor types are fixed while optimizing the mutual information. Despite this, as we show in Figure 2, our model predicts that in certain regimes the receptor distribution will be inhomogeneous, so that some receptor types will be used in very small numbers. We would predict that the corresponding receptor genes will be more likely to undergo lossoffunction mutations.
Reviewer #1:
[…] I find the analysis interesting and potentially insightful, but it misses out on a few key biological points, that I feel really should be taken on board if the analysis is to be biologically relevant. I'll enumerate three of these, in increasing order of concern.
1) The authors explicitly ignore temporal correlations in olfactory cues, with a brief line in the introduction to their model that states that spike timing could be incorporated into the model. I do not see how this will work for respiratory phase tuning of odor responses, and would be interested to see what the authors had in mind for this.
One way in which temporal correlations could be implemented in our model would be to consider the timing of the first spike for each receptor type/glomerulus, with respect to the onset of respiration, as part of the response. This time is shorter for higher concentrations, and thus a linear expansion around an operating point like the one we use for firing rates could be used for timing, as well. The timing variables could be used instead of, or in addition to, the rate variables.
We now explain this in more detail in the text.
2) The authors choose an operating point where they can apply a linear model for glomerular responses. In the animal, the operating range of different receptors for different odors is rather diverse, with the halfmax varying substantially and the slope also varies. Thus a subset of odors will be saturating for some receptors, but linear or even subthreshold for other receptors. I suspect that this will affect the analysis of the responses.
My view is that any coding theory has to account for the very wide range ofodor concentrations encountered in nature. One could possibly add this to theanalysis reported in Equations 4 to 6, by summing the mutual information overa set of odor ranges, in which different but overlapping subsets of receptors are involved. I would be interested to see if this alters the conclusions.
There are three points here: (1) different receptors have different operating ranges for different odors, (2) odors can occur in diverse concentrations, (3) receptors have nonlinear response functions with a threshold and saturation. To address these points in a fully naturalistic setting we need precise measurements of the natural olfactory environment and a complete set of doseresponse curves; comprehensive data of this kind is not available.
So, as a first step we have used available data from Hallem and Carlson, 2006 and from Saito et al.,2009. These works survey responses of a subset of receptors in fly, mouse and human to a panel of odorants. In these studies, a given receptor may respond strongly to some odorants and weakly (or not at all) to others; we use these experimentally measured receptor sensitivities. Also, in our model the threshold for informative response is effectively set by the noise level in the receptor, which was also taken from data (see Appendix 1). So, in effect, different overlapping sets of receptors respond to different odors, as the reviewer would like to see. Similarly, if the intensity of a particular odor mixture is lowered (i.e., if all the component concentrations are scaled down) then some of the receptors will stop responding informatively to some of the components of the mixture. Thus, diverse response thresholds and response gains have effectively been included in our study.
We also wanted to model olfactory environments. We are not aware of any dataset describing the actual variances and covariances that are typically observed. So, for convenience we chose a Gaussian distribution because this permits parametric variations and analytic calculations. Since we pick the odor covariance matrix randomly, some of the odorants will have a large variance and some have a small one. Thus, diverse concentration ranges have been included in our study.
Including nonlinearities is more challenging because we must posit a functional form for mixture responses, and because computing mutual information with such nonlinear sensing requires new computational innovations that are out of the scope of this paper. Nevertheless, we checked, using experimentally measured nonlinearities, that the broad predictions of our model will be robust. This is now reported in Appendix 3. Please also see the response above to Essential Revision #2 in the editor’s summary.
3) A major point of concern with the whole analysis is of salience. The obvious outlier here is pheromones. Enormous resources are allocated to pheromone detection, and clearly this doesn't seem to fall within the framework presented in the paper. Even with the general olfactory system, the assumption of efficient coding needs to be further mapped to the distribution of odor salience, that is, relevance for animal survival. There seems to be a subtle nod to this point in the thirdlast paragraph of the Discussion, where 'value of detecting different odorants' is mentioned. I feel that the point is central enough that it needs to be fully addressed. […]
In summary, I think that the current paper lays out a good framework but would be much stronger if some essential ramifications of the core idea were to be addressed.
It is of course true that some odors are more meaningful to the animal than others. However, it is not clear to what extent this kind of distinction is already implemented at the level of the sensory periphery. As an example, many efficientcoding studies in early vision and audition rely on approximations in which only information transfer is taken into account, without reference to meaning or value. The predictions nevertheless yield very good agreement with experiment. This might in fact be a result of the kind of prior on broad coverage that the reviewer is suggesting – in order to achieve breadth, the system would not adapt to extreme variations in value. This would ensure that stimuli that are not presently valuable (but might turn out to be at some point) are not ignored in favor of the ones to which high value is currently assigned. The idea here is that filtering for value and salience should occur deeper in the brain, perhaps in the olfactory cortex which has extensive projections to and from areas associated to meaning and value. From this perspective the sensory periphery should focus on simply taking in informative signals broadly. In addition, arguments based on compressed sensing suggest that, by focusing on preserving information, the olfactory system might in fact be able to sense any odor that is sufficiently sparse (e.g., Krishnamurthy et al., 2017). In this case, emphasizing salience in the periphery might actually be counterproductive, leading to a narrower distribution of receptors than desirable. We have added some of these remarks to the Discussion.
That said, the reviewer’s suggestion of an innate, evolutionarily determined prior is very interesting. From our perspective, this prior would represent the olfactory environment that the species has been subject to over generations, and could be “deformed” relative to the statistics of actual odor occurrence to account for the special importance of some odors. Concretely, suppose a particular odorant occurs only rarely but is associated to a predator and so is disproportionately important. Then, artificially inflating the variance of that odorant in the effective background environment effectively increases its importance to the optimization. In this way, the “background” olfactory environments that we start from can be regarded as incorporating the priors suggested by the referee. In fact, the data from IbarraSoria et al. supports this sort of picture, showing that genetically different strains of mice have somewhat different receptor distributions even when reared in the same environment. The picture we have in mind is that the changes reported in that paper due to environmental factors can be seen as a perturbation on an innate prior which incorporates an effective longterm olfactory environment, perhaps discounted for salience, that a species has been subject to.
Finally, a fully grounded approach requires new experimental and theoretical quantification of the notion of “value”. This is a major goal of neuroscience and of the study of behavior, but the field is far from achieving this. Thus, any treatment of value in our paper can at best be a preliminary step and a detailed investigation of value lies out of the scope of this work. In effect, we are hypothesizing, like in the literature on vision and audition, that the value of signals in the early olfactory system (which does not have access to cognitive portions of the brain) is dependent largely on the information content of the signal. There may of course be some odors with a special valence, and there is some data suggesting that even the main olfactory system can adapt to these. Extensions along the lines suggested by the reviewer would incorporate such specific effects. But, since the necessary experiments are sparse at present, we believe it is best to postpone the incorporation of value until more data are available to fix the parameters.
Reviewer #2:
[…] 1) The conclusions concerning how receptor abundances should change following a change in environment are disappointing. While their model recapitulates IbarraSoira's result which predicts that the distribution of high abundance receptors is likely to remain unchanged, they do not provide any concrete predictions on the receptors which change their abundance in either direction of change or magnitude. As currently stated, the central predictions of the model – that optimal receptor abundances can increase or decrease or stay the same following a change in environment – seems to be unfalsifiable.
As we discussed above in answer to the first point on the list of essential revisions, our model certainly makes quantitative predictions for the abundances of OSNs in the olfactory epithelium. These are of course falsifiable. Given specified changes in an olfactory environment and receptor affinities the theory makes specific predictions for which receptors will change in abundance and in which direction. We have highlighted this in the section “A framework for a quantitative test”. An important point of our paper is that the predicted changes are not readily summarized in terms of a simple catchphrase like “receptors with greater response variance should increase in number”. This is because, as we discuss in detail in the paper, in the presence of widespread correlations in the responses, the optimal abundance of one receptor depends on the context of the responses of all the others. Nevertheless, given receptor affinities and a characterization of the odor environment the model predicts changes in abundances precisely.
The doubt here may have arisen because of the way we phrased the prediction in the original text (“…receptor abundances can increase, decrease or the stay the same following a change in the environment…”). This statement may have been misread to mean that a given receptor will sometimes increase, decrease, or stay the same in replicates of the same experiment. We intended to say that receptors may increase or decrease in number after increased exposure to a particular ligand in different contexts, and that the effect will be reproducible, although the specific change in a receptor will depend on the context of all the others. We have edited the text throughout to state this better in order to avoid confusion. For example, we have edited the Abstract to read “Experimentally, increased exposure to odorants leads variously, but reproducibly, to increased, decreased, or unchanged abundances of different activated receptors. We demonstrate that this diversity of effects is required for efficient coding when sensors are broadly correlated, and provide an algorithm for predicting which olfactory receptors should increase or decrease in abundance following specific environmental changes.”
Incidentally, the qualitative predictions of our model are themselves falsifiable. It is a nontrivial observation that increasing exposure to an odorant does not necessarily lead to an increase in the abundance of the receptor types that respond to it, as we might naively expect if the lifetime of OSNs was simply tied to their activity. The fact that this counterintuitive effect of exposure to ligands is seen experimentally is corroborative evidence for the framework that we are proposing.
The manuscript could be strengthened by making more concrete predictions about how receptor abundances should change, at least in particular regimes. For example, the authors note that for intermediate numbers of neurons, optimal receptor distributions are anticorrelated with the inverse of the overlap matrix Q^{1}. They expand on this to say that receptors with high Q1 can be uninformative because they do not fluctuate or because they provide redundant information. Although I did not fully follow the arguments here, it seemed like this was saying that abundance is inversely related to information, and there are two ways to be uninformative, one by having low variance, and two by being highly correlated with other receptors. Could this be used to make more concrete predictions about predicted changes in receptor abundance, at least for a given number of neurons?
In some limits, the structure of the maximally informative receptor distribution can be given a simple intuitive description. We explain these limits in some detail now in the revamped section “Optimal OSN abundances are contextdependent”. However, in general the abundances are dependent on the context of all the receptor responses. For instance, in certain regimes, they are related to elements of the inverse overlap matrix, which depends on the full covariance matrix of responses. The conceptual meaning of the inverse overlap matrix in terms of response variance and predictability from other responses is now discussed in Appendix 6. To summarize again, running the optimization from Equation 7 in our model makes fully concrete predictions about receptor abundances, given the required parameters. The discussion involving the inverse overlap matrix provides an intuition explaining the results of this optimization, but it is not meant to replace it.
In addition, the authors also provide model evidence for predicting the magnitude of the change based on the change in olfactory environment, but it is unclear the characteristics which group types of changes together.
A key point of our paper is that there is not a simple characterization of receptors that increase vs. decrease in number after particular environmental changes. This is because, as we show in the paper, there is a global dependence on the context of the responses of the rest of the population of receptors. The specific, quantitative changes that should occur for a given environmental change can, however, be predicted using the full optimization framework we describe (the new section “A framework for a quantitative test”describes how to do this). We discuss this, and simple intuitions that apply in special limits (e.g., high/low SNR), in the section “Optimal OSN abundances are context dependent”.Indeed, the complex context dependence is necessary to understand the apparently sporadic patterns of change seen in experiments, and is predictable given a fuller characterization of the receptor affinities and odor environment.
To further clarify the complex patterns of change, we updated Figures 4 and 5 regarding differences in the optimal receptor distribution for different sorts of changes in the environment. Figure 4 shows results for a pair of environments that differ only in the variance of a few odorants. Figure 5 compares results for a pair of randomly differing environments and two largely nonoverlapping environments. The results are discussed in sections entitled “Environmental change leads to complex patterns of OSN abundance changes” and “Changing odor identities has more extreme effects on receptor distributions than changing concentrations”.
2) Some of the conclusions seem odd when considered in the context of olfactory evolution. For example, the authors conclude that if the number of neurons is large, then the optimal receptor distribution is approximately uniform. Olfactory systems differ greatly in magnitude across organisms. In particular, two of the moststudied models, fly and mouse, differ by an order of magnitude in the number of receptors (~60 for fly, ~1000 for mouse), as well as the total number of neurons. The finding that total neuron number determines receptor distribution should be tied numerically to the olfactory systems of flies and mice, if not also for other organisms. It is unclear, for example, whether the olfactory receptor number of mice is considered large, or whether it would fall in the intermediate signal to noise regime. Does the model predict that mouse receptor distributions are uniform while fly distributions are highly skewed? Why then is any adaptation observed in mouse receptor abundances as has been observed experimentally?
We were using the terms “large” and “small” in a limiting sense, as the numbers went to infinity or one. In these limits, the effective SNR becomes either very large or very small, driving the receptor distribution to either high diversity and uniformity or low diversity and inhomogeneity (section entitled “Receptor diversity grows with OSN population size”). Such limits are useful to analyze because they give a sense of the factors and considerations that are influencing the results. The degree of diversity and inhomogeneity seen in olfactory receptor distributions in animals suggests that they are effectively in an intermediate regime between the “large” and “small” population sizes.
In the intermediate regime, the number of receptor types used and their relative abundances are determined by the interplay between receptor affinities, noise levels, and environmental odor statistics. One way of thinking about this is that, in our model, the total number of neurons K_{tot} is a constraint, reflecting the limited resources that can be allocated to the olfactory epithelium. Given a fixed bank of receptor types, some of these will be more useful for transmitting information compared to others. Thus, not all of them will be used when the number of neurons is small, and allowing more neurons in the system allows more of the receptor types to be used (see Figures 2A, B, C). At some level of the neuron number, K^{*}_{tot}, all receptor types will be used. We are definitively in the largeneuronnumber regime if K_{tot} is much larger than K^{*}_{tot}. Since this is effectively defined by using all the receptor types available, it makes sense that it increases with the size of the receptor bank.
Our assumption is that the biologicallyrelevant regime is typically one where K_{tot} is intermediate, comparable to K^{*}_{tot}. This is because for smaller numbers of neurons, some receptor types would not be used, and thus we would expect these to mutate into nonfunctional forms. And for much larger numbers of neurons, the improvement in information transmission would no longer be significant. Thus, we would expect that the OSN population size will be selected over time such that the functional receptors are all useful, and such that there is not much information benefit to having more neurons. This is the “intermediate” regime of our analysis, and both fly and mouse should be in it. We would thus expect the number of receptor types in mouse to be larger than in fly, given the increased size of the epithelium. If the receptor pool that both animals used was the same, and if the odor environments they experienced were the same, then Figure 2C would provide a quantitative prediction for exactly how their number of receptor types and number of OSNs are related. Of course, insect olfactory receptors are evolutionarily distinct from mammalian ones, and the environments that flies and mice inhabit may have very different odor statistics. Thus, we cannot directly compare their receptor repertoires although it is indeed true that mice have more OSNs and more receptor types. Such a comparison can perhaps more meaningfully be done for different species of mammals, and we now include some results in Figure 2F.
Given the results presented here one might imagine that the optimal strategy would be to make a very large number of broadly tuned receptors. Instead, what we observe across evolution are olfactory systems of various sizes, with various widths of odor tuning, all constantly evolving.
As we explained in answer to point 3) of the editor's summary, we feel that our goals in relation to the evolution of olfactory systems have been misunderstood. In our model, the tuning of olfactory receptors is taken as given. Our model does not say anything about what the tuning should be. In any case, it could be that the characteristics of receptors depend on biochemical properties that do not allow them to all be similarly broadlytuned, even if it turned out that this was optimal from an informationtransmission viewpoint. Actually, it is not even obvious that broad tuning is necessary given the presumed combinatorial nature of the odor code. For example, suppose any given odor elicits responses in just 10 out of 100 receptors. There are O(10^{13}) such patterns, more than enough to encode the possible species of volatile molecules an animal is likely to encounter. That said, given a certain number of noisy receptor types, it will still be useful to redistribute them to best represent the particular odor scenes that an animal encounters. It is also worth noting that the notion of tuning width is always dependent on (a) which odorants we test the receptor with, and (b) an arbitrary threshold separating what we call an active receptor vs. an inactive one. Thus, the same data can look broadlytuned to some researchers and sparse to others. Studies that attempt to answer the question of how an optimal olfactory receptor repertoire should be built exist (e.g., Zwicker et al. 2016), but we stress again that their goals are complementary to ours.
The number of receptors in particular seems to be under strong evolutionary pressure, with new gene families expanding (as in ant ORs) or collapsing (as in humans). This discrepancy, or the other constraints that might lead to the biological situation, should be commented on.
Again, we do not attempt to model the evolution of olfactory receptor genes, or to find an optimal set of receptor types. For work along these lines, see Zwicker et al., 2016. Our model instead focuses on a complementary question that takes the available receptor types for granted and asks how these should be used, i.e., how many receptors of each type should an animal have. For a related perspective in early vision see Ratliff et al., 2010, where the tuning of ON and OFF cells in the retina is assumed, and the relative fractions of these types is predicted. That said, there is a possible connection with the collapse of the OR gene family in humans, in that the optimal receptor repertoire for the typical odor environment of interest to humans might have included vanishing or negligible amounts of some of the available receptor types. If this were the case, we would expect these ORs to mutate to nonfunctional forms due to genetic drift. To test this hypothesis, we would need a good grasp on the way in which human environments and olfactory behaviors differ from those of our remote ancestors. We do not have such data, but new Figure 2F and the associated discussion in the section “Increasing OSN population size” bear broadly upon these points.
The authors state that receptor abundances do not change in insects and therefore focus on a mammalian example to test their hypothesis. However, insect olfactory systems evolve quite rapidly between closely related species, and there is a large literature on this, especially from the Hansson group (e.g. Dekker…Hansson, 2006). Can these studies be used to test any of the hypotheses here?
The prediction of the way in which the number of receptor types grows with the number of OSNs is contingent on the receptor repertoire and the environment being similar between the species we are comparing. This makes the rapid evolution of insect olfactory systems a hurdle, rather than an advantage, for detailed comparison. For instance, in Dekker et al., 2006, the generalist D. melanogaster is compared to the highly specialized D. sechellia. It is clear that the typical olfactory environments for the two species are very different, and it would thus be difficult to say to what extent our prediction should hold without having measurements of these environments.
We have added a plot (Figure 2F) showing how the number of intact OR genes scales with a measure of the size of the olfactory epithelium across several species of mammals. While the trend in these data are in agreement with our model, we stress that there are many caveats about this comparison, as described in the text.
Insects could perhaps be used in experiments in which the olfactory environment is tightly controlled. This would be interesting to do, and our model could be tested in this context.
Or can the authors propose comparative studies that would test their hypotheses?
Studies very similar to that in IbarraSoria et al., where mice were raised in two different olfactory environments, would be ideal for testing our model. This would involve measuring the statistics of a few dozen odorants in the environment of control mice, and the same statistics for the exposed group, combined with the response profiles of a set of mouse ORs to those same odorants. Using these parameters, our model would give precise quantitative predictions (including signs) regarding the amounts by which the abundances of different ORs should change. Given the approximations we make, we would not expect these to be in exact agreement with the experimental values, but we would expect a significant correlation. This would be a strong test of our hypothesis. We now explain this in, e.g., the section “A framework for a quantitative test”.
3) Several concepts used in the text are a bit unclear, at least to a biological reader:
Could the authors provide some intuition for what is meant (biologically) by the inverse of the overlap matrix?
Thank you for this question. Interestingly, the elements of the inverse overlap matrix characterize how much the responses of one receptor type can be predicted if we know the responses of the others. This predictability might happen for various reasons – for instance, some receptor responses might not vary much, and then they can be easily predicted. Note that this depends on the environmental statistics of odors – receptors that do not vary much in one environment might well vary a lot in another environment. Another reason for a receptor response to be predictable would be if its affinity profile is similar to that of other receptors. Finally, it could be that, due to properties of the odor environment, certain odorants that activate receptor a are always accompanied by odorants that activate receptor b; in this case, one receptor type's response would be predictable given the others, even though their affinity profiles could be completely different.
To be a little more precise, the offdiagonal elements of the inverse overlap matrix, A_{ab} are related to the correlation coefficients between the responses in two glomeruli, a and b, while controlling for the responses of all the others. They are also inversely proportional to the product of the standard deviations of the two responses. The diagonal elements A_{aa} depend inversely on the variance of the response of the a^{th} glomerulus, and are also related to a parameter which measures how well the response in the a^{th} glomerulus can be linearly predicted from responses in all the others. In this way the correlation of optimal receptor distribution with the inverse overlap matrix has an intuitive interpretation: receptors which either do not fluctuate much or whose values can be guessed based on the responses of other receptors should have low abundances. We now state this at the end of the section “Optimal OSN abundances are context dependent” and develop the details further in Appendix 6.
Could the authors please unpack the following sentence:
The quantity KQ thus behaves as a signaltonoise ratio (SNR), so that Equation 4 is essentially a generalization to multiple, correlated channels of the standard result for a single Gaussian channel, I = 1 log(1 + SNR^{2}).
This sentence was indeed difficult to follow in part because noise in the receptors had been absorbed into various expressions as an effective normalization. To increase clarity, we decided to reinstate the noise standard deviations for each receptor type, Equation 2, so that both responses and concentrations can be measured in natural units (e.g., firing rate and molarity, respectively). Now the quantity that used to be KQ is seen to actually be K*𝛴^{1}*Q, which is, in matrix form, the ratio between the covariance matrix of glomerular responses (Q) and the covariance matrix for the noise (𝛴 K^{1}), where the K^{1} term corresponds to the decrease in noise variance due to averaging over OSNs with the same receptor. When glomerular responses are uncorrelated (i.e., Q is diagonal), the determinant in Equation 4 is easily calculated, and we obtain I = sum over all receptor types of 1/2*log(1 + SNR_{i}^{2}), where SNR_{i} is the signaltonoise ratio in channel i, SNR_{i}^{2} = Q_{ii} / (𝜎_{i}^{2} / K_{i}). The result 1/2*log(1+SNR^{2}) for the mutual information in a Gaussian channel has been known since the work of Shannon (1948), and so we wanted to emphasize the connection to this classic result. We have tried to clarify these points in the section “Information Maximization”. More technical details are also presented in Appendix 2.
Could the authors please clarify in the discussion of Equation 7 whether K_{tot} represents the total number of neurons, the number of receptors, or the number of receptor types?
K_{tot} refers to the total number of neurons. We updated the text.
Is the total number of neurons the most sensible thing to vary or would it be interesting to look at olfactory systems with different numbers of receptor types? This seems related to the question of where noise arises in the system, and what other constraints, besides information as quantified here, an animal might have on the design of its olfactory system.
Our model takes the odorant affinities for each receptor as an input. This means that in order to increase the number of receptor types, we need to fix the affinities of the added receptors. There are many ambiguities in doing this. Decreasing the number of receptor types is, however, more straightforward – we can simply remove some receptors from the analysis. We now do this in Figure 7. This analysis has an auxiliary benefit: it provides a test of how robust the receptor distribution is to changes in the repertoire of available receptor types.
4) The investigation of how optimal coding changes with broad versus narrow tuned receptors was interesting. However, real receptor arrays, at least as seen in the Hallem data, contain a mix of broadly and narrowlytuned receptors, and receptor tuning width depends on odor intensity, with many receptor showing narrowly tuned response at low concentrations and wider tuning at high concentrations. Could the authors explore what happens in this regime, and provide any explanation for why animals might have both broad and narrowly tuned receptors? This finding could be further explored by making predictions for olfactory systems with receptors of mixed tuning widths, as is generally accepted to be the case in most organisms. This would provide a more concrete prediction for future experiments.
In various of our results (Figure 2A, B, C, Figures 3, 6 and 7) we are indeed using data from Hallem, Carlson, 2006 and also from Saito et al., 2009. These data include both broadly and narrowly tuned receptors. In the original submission we had also examined situations where all the tuning widths were narrow or wide (Figure 2D, E). We now also present additional results where the artificial receptor arrays are made up of heterogeneous receptors with varying tuning widths (Figure 5).
Regarding the question of why animals have both broadlytuned and narrowlytuned receptors, we stress again that in our study the affinity profile of the available receptor genes is considered as given. As such, our model is not trying to address the optimal way to build the receptor repertoire, but simply the optimal way of using this repertoire (i.e., relative proportions of different receptor types).
We also emphasize again that our model makes fully precise, quantitative predictions once the affinity profile for olfactory receptors is known. Thus, if we are interested in making testable predictions, the best approach is to use measured affinity profiles. In this case, we do not need to worry about how to choose the tuning widths for the receptors since nature has already chosen them for us.
5) The authors claim that their model is robust to nonlinearities and as well as their choice to represent the olfactory environment as a vector of concentrations. These ideas should be tested and demonstrated within the paper.
We have made these points more precise in the main text and added two sections in the Appendices to explicitly address them. The section entitled “Invariance of mutual information under invertible and differentiable transformation”in Appendix 2, is a mathematical explanation of the statement in the section title. For example, consider a linearnonlinear model,
r_{a} = g_{a}(K_{a} S_{ai} c_{i} + 1/sqrt{K_{a}} 𝜂_{a}), with g_{a} a set of invertible functions. It is a mathematical identity that the mutual information is invariant under such transformations. Of course, some nonlinear transformations may not preserve information if they do not satisfy the stated conditions. From a biological standpoint, it is most interesting to study nonlinearities like those in competitive binding models like the ones suggested by the reviewer. We have now examined these in Appendix 3,as explained in the response to the next question, as well as in answer to point 2) from the editor's summary.
Regarding the way in which the olfactory environment is represented, we were trying to say that as far as our model is concerned, it does not matter what the numbers c_{i} represent: the same kind of model with the same generic mathematical results would apply if c_{i} were concentrations of distinct odorants or if they were, for instance, aggregates over several chemical species related by some property. Of course, the parameters feeding into the model – the sensing matrix S and the environment covariant matrix 𝛤 – depend on the meaning of the environment vector c_{i}, and thus the specific results would change. However, the analysis itself wouldn't. We have tried to make this clearer in the text in the section “Olfactory response model”.
For example, the nonlinearities involved in receptor encoding are well known: receptor responses can be expressed as a Hill function of odor concentration:
r = (c^n)/(c^n+Kd)
In many olfactory systems n=1, further simplifying this equation. The authors should explicitly show that the model generalizes when this nonlinearity is included.
To our knowledge, there is still debate regarding models for how olfactory receptors respond to odors. While the response to single odorants can be wellapproximated by Hill functions, the response to mixtures is harder to describe. Competitive binding models do perform relatively well (Singh et al., 2018), as in fact do linear approximations for small numbers of mixture components and in the regime between the response threshold and saturation. We used a linear approximation in the main text because it uses fewer parameters and is analytically tractable. Furthermore, a complete competitive binding model of the sort reported in Singh et al., 2018 requires measurements of doseresponse curves of all the receptors being studied against all odorants of interest. We do not have such data. Therefore, we used existing data to create a simple competitive binding model for a few receptors and compared the results (obtained through a numerical analysis) with those obtained analytically from the linear approximation. The new Appendix 3 shows that the results are broadly similar between the two models. As we explain in more detail above in the response to point 2) in the Essential revisions, our linear sensing model provides a reasonable and tractable approximation which can be numerically extended to a fully nonlinear model when such data become broadly available for more OSNs.
In addition, the main sources of noise in receptor encoding are likely to be (1) difference in receptor abundance across neurons that express the same receptor, (2) stochasticity in receptor binding and activation. The authors might consider incorporating these sources noise and showing that the model extends in this case.
We agree that these separate sources of noise may be present in the olfactory periphery, but we are not aware of specific, quantitative noise models for which the relevant parameters have been measured. As above, we think that in the absence of these data, it makes more sense to start with a simple, analyticallytractable noise model, and to leave more complex descriptions for future work when experimental guidance is available. Also, the qualitative structure of our results is not going to depend on the source of noise. For example, even after including the two sources of independent noise the referee mentions separately, the optimal distribution of receptors is still going to be contextdependent.
The first section of the Results is difficult to read because it contained a number of statements justifying elements of the model and claiming that these do not affect the conclusions. This section would be easier to read if these points were saved for later in the manuscript where they could be explicitly demonstrated.
We went through the section and tried to improve the presentation in the manner suggested by the reviewer.
6) The section on dynamical optimization at the end seemed least wellconstrained by data, and also (as noted) somewhat preliminary. The authors might consider reserving this material for a future manuscript that explores dynamics and tests them more thoroughly.
The dynamical model part of the paper is intended as an indication that simple birth and death processes modulated by experience can achieve the sorts of optima that our model describes. We feel that it is useful to see this, and have retained the section.
Efficientcoding arguments of the sort that we used in the paper are normative: we derived optimal rules for how the olfactory periphery should be organized, given a simplified model of receptor responses, and we argued that organisms would benefit from approaching these optima. But a fundamental issue with such approaches is that there is no guarantee that the optimum is actually reachable using the resources available to an organism. This is why we believe it is important to show that the optima can be dynamically reached. It is interesting that our dynamical model requires that the death rate of the neurons, but not their birth rate, should depend on olfactory experience, as experimentally observed.
And instead using this space to show that the model still holds when certain assumptions in the first version of the model are relaxed.
We have made extensive additions to paper, both in the main text and in the Appendices to relax various assumptions. Nonlinearities, diversity of tuning widths, realistic receptor affinity distributions, changing the number of receptor types, and changing the tuning widths are all now addressed as described above.
7) The authors should consider including graphical representations, similar to those provided in Figure 1, for concepts such as the mutual information measure, the covariance matrix, the overlap matrix, and the inverse overlap matrix. This would help provide insight for readers with less mathematical background, who may nonetheless be interested in the predictions of the models.
We have revamped all of our figures to aid the reader. We decided not to give an introduction to information theory (e.g., the notions of mutual information, covariance matrix) because there are many standard textbooks and also review articles in the field of neuroscience itself. However, we did include a more extensive discussion of the overlap matrix, as described above. In addition, we have now included a section in the paper (A framework for a quantitative test) and a Matlab script that makes it easy to plug measurements of receptor affinities and natural odor statistics into our model and obtain the predicted optimal OSN numbers. This allows people who are not expert in the mathematics behind the model to still use it or test it.
Reviewer #3:
[…] The manuscript is well written and the arguments are clearly presented for the most part. My main concerns with the manuscript are that some limitations are not spelled out explicitly and that the theoretical analysis could have been more comprehensive. In particular, the authors do not investigate how their model would fair in the realistic case where odors are sparse and they do not discuss how the results depend on the number of different receptor types and the number of different odor molecules.
We have added an analysis of the latter: Figure 7 now shows the amount by which the results change when receptors and/or odorants are subsampled. This also allows us to see that our overall results are robust to such subsampling.
Natural odors like foods are often sparse, containing maybe 4050 components that are important for perception. Recent work has suggested that this sparsity may influence the structure of the sensing matrix implemented by the receptor repertoire (e.g., Zwicker et al., 2016, Krishnamurthy et al., 2017). However, an odor environment typically contains a mixture of such odors, and many odorants can activate multiple receptors. In this context it is unlikely that the odor response is sparse even if many individual natural odors only contain a few tens of components. The correlated structure of the resulting responses is the key factor driving our results. Thus, we should expect the results to remain qualitatively the same for environments consisting of combinations of odors that are sparse in chemical space.
An attempt to directly model odor sparsity runs into the same lack of data about the structure of natural odor environments that we discussed in our response to the overall Essential revisions. We have some idea that individual natural odors (e.g., strawberry) typically contain a few tens of components (perhaps 4050) but we do not have a detailed survey over odors. We also do not know how many of these odors cooccur in natural settings and with what frequencies, variances and covariances. Right now we are in a setting where knowing even just the mean and covariance matrix of a set of odorant concentrations would be an important advance. In this context, modeling olfactory environments requires exploring many arbitrary choices each of which would require separate justification. We could vary over all of these choices, but that would require a study in itself. In fact, the random covariance matrices that we generate have a structure reminiscent of sparse odors when the 𝛽 parameter is small (see Appendix 4 for a discussion of how these matrices are generated) and are thus a good starting model of a sparse odor environment (i.e., where each odor is composed of a small fraction of all odorants). The next step might be to treat the odor environment as a mixture of such Gaussians in which each term of the mixture represents an odor object, itself a sparse odor. But many choices are necessary – how many odor objects to include, how many odorants are present in each one, variance and covariance of the objects, etc. In addition, calculating mutual information from such distributions is numerically challenging if we include many odors and odorants as necessary to be realistic (this is discussed further above and in Appendix 3). Meeting that challenge would be worthwhile if we knew enough about the odor environment to make it useful, and the effort would make an interesting computational paper in its own right, but is out of the scope of the present work.
https://doi.org/10.7554/eLife.39279.025Article and author information
Author details
Funding
Simons Foundation (400425)
 Vijay Balasubramanian
Aspen Center for Physics (PHY160761)
 Vijay Balasubramanian
Swartz Foundation
 Tiberiu Teşileanu
National Science Foundation (PHY1734030)
 Tiberiu Teşileanu
 Vijay Balasubramanian
United States  Israel Binational Science Foundation (2011058)
 Vijay Balasubramanian
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Acknowledgements
We thank Joel Mainland and David Zwicker for helpful discussions, and Elissa Hallem, Joel Mainland, and Darren Logan for olfactory receptor affinity data. This work was supported by a grant from the Simons Foundation/SFARI Mathematical Modeling in Living Systems program (400425, VB). VB was also supported by Aspen Center for Physics NSF grant PHY160761 and US–Israel Binational Science Foundation grant 2011058. TT was supported by the Swartz Foundation. This work was also supported by NSF grant PHY1734030 (Center for the Physics of Biological Function).
Senior Editor
 Catherine Dulac, Harvard University, United States
Reviewing Editor
 Upinder Singh Bhalla, Tata Institute of Fundamental Research, India
Reviewer
 David Zwicker
Publication history
 Received: July 3, 2018
 Accepted: February 13, 2019
 Accepted Manuscript published: February 26, 2019 (version 1)
 Version of Record published: March 4, 2019 (version 2)
Copyright
© 2019, Teşileanu et al.
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
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Funding:
This research received no specific grant from any funding agency in the public, commercial, or notforprofit sectors.

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