Normalized Neural Representations of Complex Odors

The olfactory system removes correlations in natural odors using a network of inhibitory neurons in the olfactory bulb. It has been proposed that this network integrates the response from all olfactory receptors and inhibits them equally. However, how such global inhibition influences the neural representations of odors is unclear. Here, we study a simple statistical model of the processing in the olfactory bulb, which leads to concentration-invariant, sparse representations of the odor composition. We show that the inhibition strength can be tuned to obtain sparse representations that are still useful to discriminate odors that vary in relative concentration, size, and composition. The model reveals two generic consequences of global inhibition: (i) odors with many molecular species are more difficult to discriminate and (ii) receptor arrays with heterogeneous sensitivities perform badly. Comparing these predictions to experiments will help us to understand the role of global inhibition in shaping normalized odor representations in the olfactory bulb.

We numerically calculated ensemble averages over odors c and sensitivity matrices S ni . Here, we first choose S ni by drawing all entries independently from a log-normal distribution with meanS = 1 and variance var(S ni ) = e λ 2 − 1. We then draw an odor c using the following procedure: First, we determine which of the N L ligands are present according to their probabilities p i . Second, we draw the concentrations c i for each ligand i that is present from a log-normal distribution with mean µ i and standard deviation σ i . We then use Eqs. 1-3 given in the main text to map the odor c to a binary activity vector a, from which we can for instance calculate the number of active channels. We obtain ensemble averages of such quantities by repeating these steps 10 5 times. This allows us to calculate the mean activities a n , the covariances cov(a n , a m ), and the Pearson correlation coefficient ρ, which is defined as cov(a n , a m ) var(a n ) var(a m ) We also estimate P (a) from an ensemble average to calculate the information I from its definition given in Eq. 4 in the main text. The resulting statistics only weakly depend on the dimensions N L and N R of the input and the output space, respectively, see Fig. A. This is because we consider uncorrelated odors and uncorrelated sensitivities. Conversely, Fig. B shows that these quantities significantly depend on the width λ of the sensitivity distribution, but we only consider the experimentally motivated value λ = 1 in the main text. * dzwicker@seas.harvard.edu; http://www.david-zwicker.de

B. STATISTICS OF NORMALIZED CONCENTRATIONS AND EXCITATIONS
Let p i be the probability that ligand i is present in an odor. If it is present, its concentration c i is drawn from a log-normal distribution with mean µ i and standard deviation σ i , while c i = 0 if the ligand is not present. Hence, while the covariances cov(c i , c j ) = c i c j − c i c j vanish for i = j since the ligands are independent. The statistics of the total concentration c tot = i c i read The excitations e n are given by e n = i S ni c i , where the sensitivities S ni are log-normally distributed with mean S ni =S and variance var(S ni ) =S 2 (e λ 2 − 1). Hence, and cov(e n , e m ) = 0 for n = m. We next determine the statistics of the normalized concentrationsĉ i = c i /c tot . For simplicity, we consider large odors, i p i 1, where c tot can be considered as an independent random variable. Since c tot is the sum of (a variable) number of log-normally distributed random variables, its distribution can be approximated by another log-normal distribution [2], which we parameterize by its mean µ tot and variance σ 2 tot . We consider the simple approximation where these parameters are directly given by Eq. B.3 [1]. This choice approximates the tail of the distribution well, but leads to errors in the vicinity of the mean [2].
Since both c tot and c i are log-normally distributed when ligand i is present in an odor (c i > 0),ĉ i is also log-normally distributed in this case and Note that the covariance cov(ĉ i ,ĉ j ) does not vanish since theĉ i are not independent. In particular, var( iĉ i ) = 0, since iĉ i = 1 by definition. This condition is only consistent with Eq. B.6a if χ ≈ 1, which implies that c tot must not vary much, σtot In the simple case where all ligands are drawn from the same distribution (p i = p, µ i = µ, σ i = σ), we obtain which is equivalent to Eq. 5 in the main text.

C. APPROXIMATE CHANNEL ACTIVITY
We estimate the expected activity a n by the probability that the normalized excitationsê n exceed the expected normalized threshold α. Since both the sensitivities S ni and the normalized concentrationsĉ i are approximately log-normally distributed,ê n can also be approximated by a log-normal distribution [1]. The associated probability distribution function reads and the cumulative distribution function is The parameters M n and S n can be determined from the mean and variance ê n = exp M n + where ζ = 1 2 ln(1 + var(ê n ) ê n −2 ). Eq. 6 of the main text follows from this and Eq. 5. For small a n we have a n ≈ 2 ζ/π ln(α) + ζ exp − (ln(α) + ζ) 2 4ζ , (C.14) which follows from erfc(x) ≈ e −x 2 /(x √ π), valid for x 1. For small ζ, we obtain the approximate scaling ln a n ∼ −(ln α) 2 /(4ζ), where ζ ∼ s −1 for s 1. Fig. 2C of the main text shows that Eq. C.14 approximates the channel activity a n very well. Moreover, this approximation together with Eq. (7) of the main text can be used to estimate the transmitted information I.

D. ODOR DISCRIMINABILITY
We quantify the discriminability of two odors by the Hamming distance d of their respective representations a for several different cases: a. Uncorrelated odors The expected distance d between the activity patterns a (1) and a (2) of two independent odors is where a (1) n and a (2) n denote the expected activities of the two odors, averaged over sensitivity matrices, and we neglect correlations cov(a n , a m ) for simplicity.
b. Adding target to background We calculate the expected change d of the representation when a target odor c t is added to a background odor c b . Because the odor concentrations are specified, we consider the actual excitations e n instead of the normalized quantitiesê n . Taking an ensemble average over sensitivity matrices, the excitations associated with the two odors are characterized by probability distribution functions f t E (e t ) and f b E (e b ) for the target and the background, respectively. We here consider log-normally distributed e n , which are parameterized by their mean and variance, where var(S ni ) =S 2 (e λ 2 − 1). When the target is added to the background, the expected threshold γ increases from γ b = α e b to γ s = α( e b + e t ), where e κ denotes the mean excitation e κ = z f κ E (z) dz for κ = t, b. This increase in the threshold can deactivate a channel if it was previously active, i. e. if its excitation was larger than the threshold associated with the background, e b > γ b . For such e b , the probability that the receptor gets deactivated by adding the target is P (e b + e t < γ s |e b ). Integrating over all possible e b , we thus get the probability p off that a channel becomes inactive, where F t E (e t ) is the cumulative distribution function associated with f t E (e t ). Conversely, a channel becomes active when the additional excitation by the target odor brings it above the threshold γ s . The associated probability p on reads Taken together, the expected number d of channels that change their state reads There are three simple limits that we can solve analytically: If there is no target, e t = 0, the activation pattern does not change and we have d = 0. In the opposing limit of a dominant target, e t → ∞, the activation patterns are independent and we recover the distance d max for uncorrelated odors, which is given by Eq. D.15. Lastly, in the case where the target and the background are identically distributed, e b = e t and var(e b ) = var(e t ), we have d = 1 2 d max . c. Discriminating two odors of equal size We consider the simple case of two odors that each contain s ligands at equal concentration, sharing s b of them, such that the expected threshold γ is the same for both odors. Similar to the derivation above, we here calculate the probability p that a channel is active for one odor, but not for the other. The s b ligands that are present in both odors cause a baseline excitation e b , which is distributed according to f b E (e b ). A channel is inactive for an odor with probability F d E ( γ − e b ), where F d E (e d ) is the cumulative distribution function of the excitation caused by the s d = s − s b different ligands. Hence, where z = γ − e b . Note that the upper bound of the integral is γ since channels will be active for both odors if e b ≥ γ . The associated Hamming distance d between the two odors is then given by d = pN R .

E. RECEPTOR BINDING MODEL
We consider a simple model where receptors R n get activated when they bind ligands L i . This binding