Abstract
The olfactory system faces the difficult task of identifying an enormous variety of odors independent of their intensity. Primacy coding, where the odor identity is encoded by the receptor types that respond earliest, is one possible representation that can facilitate this task. So far, it is unclear whether primacy coding facilitates typical olfactory tasks and what constraints it implies for the olfactory system. In this paper, we develop a simple model of primacy coding, which we simulate numerically and analyze using a statistical description. We show that the encoded information depends strongly on the number of receptor types included in the primacy representation, but only weakly on the size of the receptor repertoire. The representation is independent of the odor intensity and the transmitted information is useful to perform typical olfactory tasks, like detecting a target odor or discriminating similar mixtures, with close to experimentally measured performance. Interestingly, we find situations in which a smaller receptor repertoire is advantageous for identifying a target odor. The model also suggests that overly sensitive receptor types could dominate the entire response and make the whole array useless, which allows us to predict how receptor arrays need to adapt to stay useful during environmental changes. By quantifying the information transmitted using primacy coding, we can thus connect microscopic characteristics of the olfactory system to its overall performance.
Author summary Humans can identify odors independent of their intensity. Experimental data suggest that this is accomplished by representing the odor identity by the earliest responding receptor types. Using theoretical modeling, we here show that such a primacy code allows discriminating odors with close to experimentally measured performance. This performance depends strongly on the number of receptors considered in the primacy code, but the receptor repertoire size is less important. The model also suggests a strong evolutionary pressure on the receptor sensitivities, which could explain observed receptor copy number adaptations. Taken together, the model connects detailed molecular measurements to large-scale psycho-physical measurements, which will contribute to our understanding of the olfactory system.
Introduction
The olfactory system identifies and discriminates odors for solving vital tasks like navigating the environment, identifying food, and engaging in social interactions. These tasks are complicated by the enormous variety of odors, which vary in composition and in the concentrations of their individual molecules. In particular, the olfactory system needs to separately recognize the odor identity (what is there?) and the odor intensity (how much is there?). For instance, the identity is required to decide whether to approach or avoid an odor source, whereas the intensity information is important for localizing it. It is unclear how these two odor properties are separated.
Odors are sensed by olfactory receptors that have distinct responses to different odor molecules. Generally, each receptor responds to a wide range of odors and each odor activates many receptor types. The resulting combinatorial code allows to distinguish odor identities [1–3], but also depends on the odor intensity, since receptors respond stronger to more concentrated molecules [4]. To obtain an intensity-invariant code in the olfactory cortex [5, 6], the neural information is processed in the olfactory bulb in mammals and the antenna lobe in insects [7–9]. For instance, inhibiting neurons in the olfactory bulb [10, 11] affect the neurons processing the receptor activities globally [12–18], which could result in a concentration-invariant representation of the odor identity [19, 20]. However, we showed that such a normalized representation still depends strongly on the number of ligands in a mixture and might thus not be optimal for solving olfactory tasks [21]. An alternative to these normalized representations is rank coding, where the order in which the receptors are excited is used to encode the odor identity robustly and independently of the odor intensity [22]. Indeed, experiments suggests that odors are encoded robustly by the receptor types that respond within a given time window after sniff onset [23, 24]. In particular, the odor identity could be robustly encoded by a fixed number of the receptors that respond first, which is known as primacy coding [23, 25]. So far, it is unclear whether this simple coding scheme is sufficient to explain the remarkable discriminatory capability of the olfactory system.
In this paper, we consider a simple model of primacy coding and investigate how well it represents the identity of complex odors. In particular, we identify how much information is transmitted and how well this information can be used to perform typical olfactory tasks, like identifying a target odor in a background or discriminating odor mixtures. Our model thus links parameters of the primacy code with results from typical psychophysical experiments. We show that primacy coding provides a robust and compact representation of the odor identity over a wide range of odors, independent of the odor intensity. However, this good performance of the olfactory system hinges on tuned receptor sensitivities, which suggests that there is a strong selective pressure to adjust the sensitivities on evolutionary and shorter timescales.
Results
We describe odors by concentration vectors c = (c1, c2, …,cNL), which determine the concentrations ci > 0 of all detectable ligands to the olfactory receptors. The number NL of possible ligands is at least NL = 2300 [26] although the realistic number is likely much larger [27]. Typical odors contain only tens to hundreds of ligands, implying that most ci are zero. The statistics of natural odors are difficult to measure [28]. We thus consider a broad class of odor distributions, where each ligand i has a probability pi to appear in an odor. For simplicity, we neglect correlations in their appearance, so the mean number s of ligands in an odor is s = ∑i pi. To model the broad distribution of ligand concentrations, we choose the concentration ci of ligand i from a log-normal distribution with mean μi and standard deviation σi if the ligand is present. Consequently, the mean concentration of a ligand in any odor reads ⟨ci ⟩= piμi and the associated variance is . For simplicity, we consider ligands with equal statistics in this paper, so the distribution Penv(c) of odors is characterized by the three parameters pi = p, μi = μ, and σi = σ.
Simple model of primacy coding
Odors are detected by an array of receptors in the nasal cavity in mammals and on the antenna in insects. The receptor array consists of NR different receptor types, which each are expressed many times. Typical numbers are NR ≈ 50 in flies [7], NR ≈ 300 in humans [29], and NR 1000 in mice [30]. The excitations of all receptors of the same type are accumulated in an associated glomerulus in the olfactory bulb in mammals and the antennal lobe in insects [31]. Since this convergence of the neural information mainly improves the signal-to-noise ratio, we here capture the excitation of the receptors on the level of glomeruli; see Fig. 1A. The excitation en of glomerulus n can be approximated by a linear map of the odor c [4, 32], where Sni denotes the effective sensitivity of glomerulus n to ligand i. Note that Sni is proportional to the copy number of receptor type n if the response from all individual receptors is summed [33].
The sensitivity matrix Sni could in principle be determined by measuring the response of each glomerulus to each possible ligand. However, because the numbers of receptor and ligand types are large, this is challenging and only parts of the sensitivity matrix have been measured, e. g., in humans [34] and flies [35]. We showed that the measured matrix elements are well described by a log-normal distribution with a standard deviation λ ≈ 1 of the underlying normal distribution [33]. Motivated by these observations, we here consider random sensitivity matrices, where each element Sni is chosen independently from the same log-normal distribution, which is parameterized by its mean ⟨Sni⟩ = S̄ and variance . Since these receptor sensitivities are broadly distributed, they might not include specific receptors related to innate behavior [36], but they can collectively discriminate concentration differences of several orders of magnitude [33].
The odor representation on the level of glomeruli excitations en depends strongly on the odor intensity ctot = ∑i ci, which complicates the extraction of the odor identity determined by the relative concentrations. A concentration-invariant representation could be achieved by normalizing the excitations by the mean excitation [16], which leads to an efficient neural representation on the level of projection neurons [21]. However, recent experimental data suggest an alternative encoding based on the timing of the glomeruli excitation [23]. The key idea of this primacy coding is that the set of receptor types that are excited first is independent of the total concentration ctot and thus provides a concentration-invariant representation. In the simple situation where bound ligands only affect the strength of the receptor output, but not the signaling dynamics, the receptors that first cross a threshold are the ones with the largest excitation. For simplicity, we also neglect the order in which excitations cross the threshold, in contrast to rank coding. Taken together, the primacy code is then given by the identity of the NC glomeruli with the largest excitation, which is known as the primacy set [37].
The primacy set can be represented by a binary vector , where an = 1 implies that glomerulus n belongs to the primacy set and is active, while an = 0 denotes an inactive glomerulus not belonging to the primacy set. Since the active glomeruli have the highest excitation, they can be identified using an excitation threshold γ; see Fig. 1B. Consequently, the activities are given by
Physiologically, the activities an could be encoded by projection neurons in insects and mitral and tufted cells in mammals. These neurons receive excitatory input from one glomerulus [38] and are inhibited by a local network of granule cells [20, 31]. These granule cells basically integrate the activity of all glomeruli [39] and could inhibit the glomeruli once a threshold is reached. Taken together, this would imply primacy coding since only the glomeruli that respond earliest would be activated. For simplicity, we consider the case where the number NC of active glomeruli is fixed and does not depend on the odor c. The associated constraint determines the threshold γ. The activity pattern a is sparse since only a fraction NC/NR of all glomeruli are activated. Moreover, a is concentration-invariant, since the odor intensity ctot does not affect a. This is because multiplying the concentration vector c by a constant factor changes both the excitations en and the threshold γ by the same factor, so that a given by Eq. (2) is unaffected. In essence, only relative excitations are relevant for our model of primacy coding.
The amount of information that can be learned about the odor c by observing the activity pattern a is quantified by the mutual information I given by where the probability P (a) of observing an output a depends on the odor environment Penv(c) as well as the properties of the olfactory system, which in our model are quantified by NC, NR, and λ.
In an optimal receptor array, each output a occurs with equal probability when encountering odors distributed according to Penv(c) [33]. In the case of primacy coding, only outputs with exactly NC active receptor types are permissible. Consequently, in the optimal representation each receptor type would be activated with a probability ⟨an⟩ = NC/NR and all types would be uncorrelated, cov(an, am) = 0 for n ≠ m. The associated information provides an upper bound for I given by Eq. (4). Here, the approximation on the right hand side is obtained using Stirling’s formula for large receptor repertoires (NR ≫ NC). Note that primacy coding contains much less information than simple binary coding (where all glomeruli are considered [33, 40]) and rank coding (where the order of activation of the first NC glomeruli is also included [22]); see Fig. 2A. Nonetheless, we will show below that primacy coding provides useful information for solving typical olfactory tasks and can even outperform alternatives encoding more information.
Transmitted information depends weakly on receptor repertoire
We start by analyzing the information I transmitted by the primacy code using numerical ensemble averages of Eqs. (1)–(4); see Methods and Models. Fig. 2B shows that I is very close to the maximal information Imax given by Eq. (5), which is obtained when all receptor types have equal activity and are uncorrelated [33]. This indicates that the primacy code uses the different receptor types with similar frequency and that correlations between them are negligible. The expression for Imax implies that the information grows linearly with the primacy dimension NC, but only logarithmically with the number NR of receptor types. Consequently, the number of distinguishable signals, given by 2I, grows strongly with NC, but the dependence on the repertoire size is weaker. Given equal NC, our model thus predicts that the transmitted information in mice is only twice that of flies, although mice possess about 20 times as many receptor types; see Fig. 2C. However, the number of discriminable signals changes by many orders of magnitudes because of the exponential scaling with I.
The logarithmic scaling of the transmitted information I with the receptor repertoire size NR could explain why the ability of rats to discriminate odors is not significantly affected when half the olfactory bulb is removed in lesion experiments [41, 42]. If this operation removes half the receptor types, our model implies that the transmitted information I is lowered by NC bits; see Eq. (5). This corresponds to a reduction of I by about 10 % in rats where NR ≈ 1000; see Fig. 2D. Conversely, the transmitted information can be reduced by almost 50 % in flies, which have a much smaller receptor repertoire of NR ≈ 50. Our model thus predicts that lesion experiments have a much more severe affect on the performance of animals with smaller receptor repertoires.
Taken together, this first analysis already suggests that the primacy code provides a robust odor representation, which is sparse, concentration-invariant, and depends only weakly on the details of the receptor array. However, for this representation to be useful to the animal, it needs to allow solving typical olfactory tasks.
Primacy coding discriminates odors efficiently
Typical olfactory tasks involve detecting a ligand in a background, detecting the addition of a ligand to a mixture, and discriminating similar mixtures. All these tasks involve discriminating odors with common ligands, implying that the associated primacy sets are correlated. In particular, discriminating similar odors will be impossible if their primacy sets are identical. To see when discrimination is possible, we quantify the distance d between two primacy sets by simply counting the number of glomeruli with different activities.
Discriminating uncorrelated odors
To build an intuition for the distance d between primacy sets, we start by considering two uncorrelated odors. In this case, each receptor type has an expected activity of ⟨an⟩ = NC/NR and the resulting distance reads d∗ = 2NC(1 − NCNR−1), which implies that two uncorrelated odors will typically be distinguishable (⟨d⟩ ≥ 2), even for very small primacy dimension NC. Moreover, this expression implies that odors become more easily discriminable when NC is increased, whereas increasing the receptor repertoire size NR has a negligible effect in the typical case NC ≪ NR. This is similar to the scaling of the transmitted information I discussed above.
Detecting the presence of a target odor in a background
One simple task where correlated primacy sets matter is the detection of a target odor in a distracting background. To understand when a target can be detected, we analyze how the primacy set a changes when a single ligand at concentration ct is added to a background ligand at concentration cb. Because of concentration-invariance, only the relative target concentration ct/cb matters and we expect that the target is easier to detect when it is more concentrated (larger ct/cb). Fig. 3A shows that this is indeed the case, since the mean change ⟨d⟩ in the primacy set a increase with ct/cb. Moreover, ⟨d⟩ increases with the primacy dimension NC in the same way as the distance d∗ of uncorrelated odors (see inset). In fact, ⟨d⟩ must approach d∗ when the target dominates the background (ct/cb → ∞). This scaling implies that the receptor repertoire size NR only has a weak effect on ⟨d⟩, which is confirmed by Fig. 3B. Surprisingly, the dependence on NR is not monotonic and very dilute odors (small ct/cb) are actually more difficult to discriminate with larger receptor repertoires.
The fact that increasing the receptor repertoire size NR can impede the detection of the target odor can be understood in a simplified statistical model, where we calculate the expected distance ⟨d⟩ using ensemble averages over sensitivity matrices; see Methods and Models. Since the primacy set a corresponds to the NC receptor types with the largest excitations, a will only change when adding the target odor brings the excitation of an inactive receptor type above the excitation of a previously active one. Intuitively, this is more likely when the difference ∆e between the excitation of the weakest active receptor type and the strongest inactive one is small. Fig. 3C shows that large ∆e are more likely for larger NR, essentially because the distribution of the glomeruli excitation en has a heavy tail, so that sampling more excitations leads to larger gaps between the largest excitations. In this case, it is less likely that perturbing the odor changes the order of the excitations and thus the primacy set. Consequently, the maximal concentration ct/cb at which a target can still be detected increases with the receptor repertoire size NR; see Fig. 3B. In contrast, increasing the primacy dimension NC always improves the detection limit.
Detecting the addition of a ligand to a mixture
So far, we considered simple odors consisting of single ligands. However, realistic odors are comprised of many different ligands and a more realistic olfactory task is thus the detection of a target in a background of many distracting ligands. For simplicity, we first consider the case where all ligands have the same concentration and we only vary the number s of ligands in the background odor. Using ensemble averages over sensitivity matrices, we show in Fig. 4A that the discriminability ⟨d⟩ decreases both with larger mixture sizes s and smaller primacy dimension NC. In experiments, humans can detect the presence or absence of a ligand for mixtures of up to about 16 ligands [43] and mice perform even better [44]. To see whether this performance is achievable with primacy coding, we use our statistical model to determine the maximal mixture size s∗ at which the addition of the target odor can still be detected (i. e. when ⟨d⟩ ≥ 2). The solid lines in Fig. 4B show that s∗ = 16 is feasible for NC ≈ 7 in humans if all ligands have the same concentration . However, if the concentration of the individual ligands is drawn from a distribution with significant variance, a much larger primacy dimension of NC ≈ 15 is necessary to still detect the absence or presence of an additional ligand for s = 16 (dashed lines in Fig. 4B, ).
Interestingly, we find that target odors can be detected more reliably when the background at a given total concentration cb consists of more ligands. This can be seen by comparing single-ligand backgrounds (Fig. 3A) with multi-ligand backgrounds (Fig. 4A), where the effective target concentration is ct/cb = 1/s. Considering NC = 8, the target can only be detected for ct/cb ≲ 1/3 in the single-ligand case, while the ratio can be much smaller (ct/cb ≲ 1/20) for multiple ligands. This puzzling result can again be understood in the simplified statistical model, which predicts that the variance of the excitations associated with the background odor is smaller if this odor is comprised of many ligands; see Eq. (8) in Methods and Models. This smaller variance implies smaller ∆e, so that adding the target has a higher chance of shuffling the order of the excitations to change the primacy set. The same logic implies that the target is easier to detect when the concentrations of the background ligands vary less, which is confirmed by Fig. 4B. Taken together, numerical results and the statistical model suggest that a target odor is easier to notice if the background odor contains many ligands and small concentration variations.
Primacy coding permits the detection of the addition of ligands to mixtures more efficiently than alternative simple encodings. To show this, we also calculate the mean change ⟨d⟩ of the activity when a ligand is added to a mixture in two alternative models that have been discussed in the literature; see Fig. 4C. First, we consider a normalized code where glomeruli are active when their excitation normalized to the mean excitations exceeds a threshold value α. We showed in [21] that the encoded information and the discriminability strongly depends on the mixtures size s in this model. Consequently, a normalized code cannot detect the addition of a ligand at large s while at the same time providing a sparse response for individual ligands (small NC/NR); see Fig. 4C. In an even simpler model of the olfactory system, glomeruli do not interact at all and are simply activated when their excitation exceeds a threshold [40]. This binary code is not sparse and is strongly affected by the odor intensity, implying that mixtures cannot be discriminated over any significant concentration range [33]. Fig. 4C shows that the discriminability measured by ⟨d⟩ decreases much more slowly with the mixture size s in primacy coding compared to the alternatives. Taken together, primacy coding provides odor discriminability on physiologically relevant levels using a sparse code for all mixtures sizes.
Discriminating similar mixtures
To consider the discrimination of similar odors that have common ligands, we next consider odors that each contain s ligands, sharing sB of them. Such odors are uncorrelated (⟨d⟩ = d∗) when they do not share any ligands (sB = 0) and they are identical (⟨d⟩ = 0) when they share all ligands (sB = s). Between these two extremes, the expected distance ⟨d⟩ of the primacy sets of the two odors can be determined by a numerical ensemble average over sensitivities and by the statistical model; see Methods and Models. Fig. 5 shows that both methods predict that more similar odors are harder to discriminate. However, the discriminability of odors only depend on their relative similarly (the fraction of shared ligands) and is independent of the total number of ligands in the odor, consistent with psychophysical experiments [45]. Our model predicts that odors should be distinguishable even if they differ by only about 10 % for NC ≳ 4.
Overly sensitive receptors degrade the coding efficiency
So far, we calculated the transmitted information and tested the discrimination performance of primacy coding under the assumption that all receptor types behave similarly. In fact, we established that the maximal information is achieved when all receptor types are activated with equal probability NC/NR. However, neither the receptor sensitivities nor the odors themselves are distributed equally in realistic situations. Variations in these quantities affect the transmitted information and thus the usefulness of the primacy code. For instance, the transmitted information decreases if a single receptor is activated less often than all the others; see Fig. 6A. This effect is small, since in the worst case the receptor is never active and the transmitted information thus corresponds to an array with the receptor removed. Conversely, having a receptor that is active more often than all others can have a much more severe effect; see Fig. 6A. In fact, if the receptor type is more than three times as active, the transmitted information I is lower than if the receptor type was remove completely; see Methods and Models. This indicates that receptors can shadow the response of other receptors and thus be detrimental to the overall array when they are overly sensitive.
The effect of varying receptor sensitivities can be studied in our model of primacy coding by discussing more general sensitivities matrices. We consider , where each receptor type can have a different sensitivity factor ξn, which modulates the uniform sensitivity matrix where each entry is independently chosen from the same log-normal distribution. The case of homogeneous sensitivities that we discusses so far thus corresponds to ξn = 1.
To investigate the effect of heterogeneous sensitivities, we start by varying the sensitivity factor of one receptor type while keeping all others untouched, i. e., we change ξ1 while keeping ξn = 1 for n ≥ 2. There are three simple limits that we can discuss immediately. For ξ1 = 0, the first receptor type will never become active, the array behaves as if this type was not present, and the transmitted information is approximately Imax(NC, NR − 1). This value is lower than the maximally transmitted information Imax(NC, NR) reached for the symmetric case ξ1 = 1. However, the associated information loss ∆I = Imax(NC, NR) − Imax(NC, NR − 1) ≈ NC/(NR ln 2) is relatively small in large receptor arrays (NR ≫ NC); see Fig. 6B. Conversely, the transmitted information can be affected much more severely if the sensitivity of the first receptor type is increased beyond ξ1 = 1 and the receptors will thus be active more often than the others. In the extreme case of ξ1 → ∞, the first receptor type will always be active and thus not contribute any information. Since this receptor type would always be part of the primacy set, the information transmitted by the remaining receptor types is approximately Imax(NC − 1, NR − 1), which is smaller than Imax(NC, NR 1) in the typical case NR ≫ NC. Consequently, an overly active receptor type can be worse than not having this type at all under primacy coding.
The fact that overly sensitive receptors are detrimental to the transmitted information is also visible in numerical simulations. Fig. 6B shows ensemble averages of the information I transmitted by receptor arrays as a function of the sensitivity factor ξ1. As qualitatively argued above, I is maximal for ξ1 = 1 and it is slightly lower for smaller ξ1 since the receptor type is active less often. In contrast, for ξ1 > 1, I decreases dramatically and falls below the value of ξ1 = 0 for ξ1 ≳ 1.5. These data suggest that it would be better to remove receptor types that exhibit a 50 % higher sensitivity than the other types.
To see whether overly sensitive receptor types are also detrimental when all types have varying sensitivities, we next considering sensitivity factors ξn distributed according to a lognormal distribution. Numerical results shown in Fig. 6C indicate that the transmitted information indeed decreases with increasing variance var(ξn) of the sensitivity factors. In fact, a variation of var(ξn)/ ⟨ξn⟩2 = 0.5 already implies a reduction of the transmitted information by almost 50 % for small concentration variations σ/μ = 1. If the odor concentrations vary more, the information degradation is less severe, but the same trend is visible. Interestingly, rescaling the information by the maximal information Imax given in Eq. (5) collapses the curves for all dimensions NC and NR, suggesting that this analysis also holds for realistic receptor repertoire sizes. Note that the reduced transmitted information also implies poorer odor discrimination performance; see Fig. 6D. Taken together, this provides a strong selective pressure to limit the variability of the receptor sensitivities so overly sensitive receptors do not dominate the whole array.
Discussion
We analyzed a simple model of primacy coding, where odors are identified by the NC strongest responding receptor types. This primacy coding provides a sparse representation of the odor identity that is independent of the odor intensity. We showed using numerical simulations and a statistical model that the primacy dimension NC strongly affects the transmitted information and the discriminability of odors. However, we showed that typical olfactory discrimination tasks can be carried out with performances close to experimentally measured ones for small NC ≲ 10 already. Conversely, the number NR of receptor types does not strongly affect the coding capacity and the discriminability of similar odors, in accordance with lesion experiments. Interestingly, our model even indicates that lowering NR can improve the identification of a target ligand in a background.
Our model predicts that receptors need to respond with similar frequencies to incoming odors to be useful. This is because receptor types that are overly sensitive and respond strongly to many odors could dominate the response of other types and thus degrade the total information. In fact, having a receptor type that is 50 % more sensitive than others, and thus responds about three times as often, can lead to less transmitted information than when this type is absent. This observation is related to the primacy hull discussed in [37], which also predicts strong restrictions on the receptor sensitivities stemming from primacy coding. Various strategies could play a role in keeping the activity of the receptor types similar [46]: On timescales as short as a single sniff, the inhibition strength could be adjusted to regulate the relative importance of receptor excitations [47]. On longer timescales of several weeks, there are changes of the receptor copy number that directly affect the sensitivity of the glomeruli [48–50] and the processing neurons in the olfactory bulb [51, 52]. Receptor copy number adaptations influence the signal-to-noise ratio at the receptor level, so the copy number could be increased to improve the detection of frequently appearing odors [53]. In contrast, we predict a decrease of the copy number of overly sensitive receptor types that respond often. Combining the two alternatives, receptor copy numbers could be controlled such that noise is suppressed sufficiently while ensuring that single receptor types do not dominate the array. Finally, receptor sensitivities can also be adjusted by genetic modifications on evolutionary timescales [54, 55]. Moreover, direct feedback from higher regions of the brain could modify the processing of olfactory signals, e. g., in response to the behavioral state [7]. Although our work shows that the activities of the receptors need to be balanced, the actual distribution of the sensitivities matters much less. For instance, log-uniform distributions, which have been suggested to describe realistic receptor arrays [40, 56], lead to similar odor discriminability as log-normally distributed sensitivities; see Fig. S1.
Our results raise the question why mice have 20 times as many receptor types than flies, although the transmitted information under primacy coding is only increased by a factor of 2; see Eq. (5). The apparent usefulness of large receptor repertoires hints at roles of the olfactory system beyond transmitting the maximal information and discriminating average odors. For instance, having many receptor types might help to hardwire innate olfactory behavior when receptors are narrowly tuned to odors. In this case, our model would only apply to the fraction of the receptor types that are broadly tuned and are not connected to innate behavior. Alternatively, having many receptor types might be advantageous to discriminate very similar odor mixtures, to cover a larger dynamic range in concentrations of individual ligands, or to allow for a larger variation in average sensitivities, enabling quick adaptation to new environments. Finally, biophysical constraints of the receptor structure might imply that many receptors are required to cover a large part of chemical space.
Our model of primacy coding is very similar to our previous model of normalized receptor responses [21], which also exhibits concentration-invariant representations and predicts similar evolutionary pressure on the receptor sensitivities. In that case however, the mean activity decreases with larger mixture sizes, leading to diminishing discriminability of large mixtures caused by the constant inhibition strength [21]. Conversely, primacy coding can be interpreted as normalization with an inhibition strength that depends on the non-dimensional width of the concentration distribution; see Methods and Models. Primacy coding is thus an example for global inhibition with instantaneous adaptation, which displays better performance than a simple fixed threshold. Note that both models can detect targets in background odors, while this task is almost impossible without concentration invariance; see Fig. S2. Concentration invariance, here achieved by global inhibition, is therefore paramount for discriminating odors at various intensities. Taken together, our model suggests that primacy coding is superior at discriminating odors (Fig. 4C) while at the same time transmitting less information (Fig. 2A) compared to alternative models [21, 33, 40]. This implies that the information is more useful, which potentially allows for simpler processing downstream.
We discussed the simplest version of primacy coding with a minimal receptor model and a constant primacy dimension NC implemented by a hard threshold. This model neglects the complex interactions of ligands at the olfactory receptors, which can affect perception [57]. In particular, antagonistic effects can already provide some normalization at the level of receptors [58]. Generally, it is likely that many mechanisms contribute to the overall normalization of the receptor response [59]. A more realistic model of primacy coding might also consider a softer threshold, where receptor types with larger excitation are given higher weight in the downstream interpretation, which is related to rank coding [22]. In this case, information from fewer glomeruli might be sufficient to identify odors, since the rank carries additional information; see Fig. 2A. Realistic olfactory systems could also use a timing code, taking into account more and more receptor types (with decreasing excitation) until an odor is identified confidently. Such a system could explain that the response dynamics in experiment depend on the task [60, 61]. Generally, a better understanding of the temporal structure of the olfactory code [8, 62–66] might allow to derive more detailed models. These could rely on attractor dynamics that are guided by the excitations and thus respond stronger to the early and large excitations [67, 68].
Methods and Models
Numerical simulations
All numerical simulations are based on ensemble averages over odors c and sensitivity matrices Sni. The elements of Sni are drawn independently from a log-normal distribution with var(Sni)/S̄2 = 1.72 corresponding to λ = 1. Odors c are chosen by first determining which of the NL ligands are present using a Bernoulli distribution with probability p = s/NL and then drawing their concentration from a log-normal distribution with mean μ and standard deviation σ. The primacy set a corresponding to c is given by the NC receptors with the highest excitation calculated from Eq. (1). Statistics of a and the transmitted information I given by Eq. (4) are determined by repeating this procedure 105 and 107 times, respectively.
Statistical model
The statistics of the output a given by Eqs. (1)–(3) can be estimated using ensemble averages of sensitivity matrices for different odors c, similar to our treatment presented in [21] and [33]. In particular, Eq. (1) implies that the excitations en are well approximated by a log-normal distribution with mean and variance and ) whereas correlations are negligible [21]. The probability that the excitation en exceeds the threshold γ and the associated receptor type is part of the primacy set reads with being the cumulative density function of a log-normal distribution with ⟨x⟩ = 1 and var(x) = exp(2ζ) − 1. The width of the distribution is determined by the positive parameter ), which reads for an ensemble average over sensitivities. Note that ζ is concentration-invariant, since it does not change when the concentration vector c is multiplied by a constant factor. In the simple case of ligands that are distributed according to Penv(c), we find . Consequently, the distribution width ζ is large for broadly distributed sensitivities (large λ), few ligands in an odor (small s), and wide concentration distributions (large σ/μ).
The constraint Eq. (3) implies ⟨an⟩ = NC/NR, so that the mean threshold reads where G−1 is the inverse function of G defined in Eq. (7). Using this expression as an estimate for γ in Eq. (6) results in concentration-invariant activities an, since ⟨γ⟩ is proportional to the excitation ⟨en⟩. This situation is comparable to simple normalized representations resulting from the threshold γ = α ⟨en⟩, where α is a constant inhibition strength [21]. In fact, primacy coding can be interpreted as global inhibition with an inhibition threshold depending on the width of the excitation distribution, α = G−1(1 − NCNR−1; ζ).
Inter-excitation intervals
The expected difference between excitations corresponding to a given odor c can be studied using order statistics, where excitations are re-indexed such that they are ordered, . For simplicity, we consider the case where the excitations en are distributed identically when considering all odors according to Penv(c). Denoting the cumulative distribution function of the excitations by and the associated probability density function by f (e), the probability density function associated with the excitation e(n) at rank n reads [70]
The joint distribution of E(n) and E(m), 1 ≤ n < m ≤ N, reads [70]
Consequently, the distribution of the difference ∆e = e(n) e(n−1) of consecutive excitations is
Hence, the expected difference ⟨∆e⟩ = ∫ x f∆E(x; NR NC 1) dx between the strongest excited inactive receptor type and the weakest active receptor type can be evaluated.
Distances between primacy set
The expected number ⟨d⟩ of changes in the primacy set a when a target odor ct is added to some background cb reads where pon is the probability that a receptor type that was inactive for cb is turned on by the perturbation ct and poff is the probability that a receptor type that was active is turned off. Both probabilities depend on the excitation thresholds γ(1) and γ(2) associated with the odors cb and cb + ct, respectively, which can be estimated from Eq. (9) using the respective excitation statistics. With this, pon follows from the probability that the excitation was at the value x below γ(1) and the additional excitation by the target brings the total excitation above γ(2), where g(e;ζ) is the probability density function associated with G(e;ζ) given in Eq. (7). Here, and ζj describe the excitation statistics of the target (j = t) and the background (j = b). Similarly, we obtain so we can use Eq. (13) to calculate the expected Hamming distance ⟨d⟩. Note that γ(1) and γ(2) depend on NR, so the distance ⟨d⟩ does thus not scale trivially with NR, in contrast to the case of normalized representations [21].
We use Eqs. (13)–(15) to calculated ⟨d⟩ when a target ligand with concentration ct is added to a background ligand at concentration cb. The associated statistics of the excitations obey and follows from chosen values of σ/μ and λ. Similarly, when a ligand with concentration c is added to a mixture of s ligands, all at concentration c, we have
The third case of correlated odors that we discuss in the main text concerns two odor mixtures of equal size s sharing sB of the ligands. In this case, the excitation threshold γ is the same for both odors and we can express the probability pxor that a receptor type is excited by one mixture but not the other as where the statistics and ζj need to be evaluated for the excitations associated with the sB ligands that are the same (j = B) and the s sB ligands that are different (j = D) between the two mixtures. Taken together, the expected distance reads ⟨d⟩ = 2NRpxor and we recover ⟨d⟩= d∗ for unrelated mixtures (sB = 0) and ⟨d⟩= 0 for identical mixtures (sB = s).
Information transmitted by diverse receptors
In the case where the primacy sets a can be partitioned into NM groups with all elements within a group appearing with the same probability, we can write the information I given by Eq. (4) as where Mm is the number of elements within group m and pm is the probability that group m appears in the output, such that and . In the simple case of one receptor type with deviating statistics, we have NM = 2 with while the remaining activities are ⟨an⟩= (NC − ⟨a1⟩)/(NR − 1) for n ≥ 2 to obey Eq. (3). For p1 = 0, Eq. (19) reduces to I = Imax(NC, NR − 1), whereas the maximum I = Imax(NC, NR) is reached for p1 = NC/NR. The information decreases for larger p1 and eventually reaches values lower than Imax(NC, NR 1) when . For , it would thus be advantageous to remove this receptor type. Using and expanding Eq. (19) around p1 = eNR/(NR − 1), we find in the limit NR ≫ NC of large repertoires, so .
Acknowledgments
This work was funded by the Simons Foundation and the German Science Foundation through ZW 222/1-1.
Footnotes
↵* david.zwicker{at}ds.mpg.de