Generating sparse and selective third-order responses in the olfactory system of the fly

Transformation from ORN to PN Firing Rates.

The transformation from ORN to PN responses consists of two functional components: a feedforward nonlinearity and lateral suppression. The relationship between the firing rate of an ORN and that of a PN is nonlinear with strong saturation at high ORN rates (21), a result due at least in part to short-term depression at ORN-PN synapses (24). We refer to this transformation as a feedforward nonlinearity, although lateral circuitry may also contribute to this effect. This nonlinearity was described using an exponential function, but it can also be fit by an equation of the form , where r_PN and r_ORN are the firing rates of a corresponding PN and ORN pair, and R_max and σ are constants. The advantage of using this latter form is that it fits in well with more recent work in which the effects of lateral suppression have been incorporated (22). Lateral suppression, which is the dominant effect of ORNs that do not drive a given PN directly, can be described by adding a term proportional to the sum of the firing rates of all of the ORNs to the denominator of the expression for r_PN. Defining s_ORN as the sum of all ORN firing rates (independent of the glomerulus to which they project), the equation

where m is another constant (see Methods for parameter values), provides an accurate description of both the feedforward nonlinearity and lateral suppression generated by the antennal lobe circuitry (22). Eq. 1 refers to average responses, but trial-to-trial variability is included in the PN responses we use (Fig. 1B) by adding randomly generated fluctuations modeled after the data (21) (Methods).

Before modeling LHNs and KCs, it is instructive to examine how the transformation of Eq. 1 modifies olfactory responses. The sum of all ORN firing rates, s_ORN, introduced above and the analogous sum over all PN firing rates, s_PN, are measures of the total magnitude of the response to a given odor, which has a strong impact on LHN and KC performance. A third-order neuron tuned to respond to an odor that elicits a weak overall PN response (small s_PN) will tend to respond nonselectively to odors that elicit stronger total responses (large s_PN). This problem can be avoided if the magnitudes of PN responses are equalized, meaning that the total activity across the population of PNs (i.e., s_PN) is roughly the same for each odor. We use the term equalization, rather than normalization, because normalization has been used to refer to the entire transformation of Eq. 1 in the visual system literature where such a transformation was first proposed and has been used extensively (25, 26). The transformation of Eq. 1 has a strong magnitude-equalizing effect; the sum of ORN rates (s_ORN, Fig. 1C) is much more variable across odors than the sum over PN rates (s_PN, Fig. 1E). To isolate the origin of this equalization of the response magnitude, we also computed s_PN when lateral suppression was eliminated by setting the parameter m in Eq. 1 to zero. The results indicate that much, but not all, of the response magnitude equalization arises from the feedforward nonlinearity (Fig. 1D).

Selectivity is also difficult to achieve if the patterns of PN activation across odors are too similar, that is, too highly correlated. We performed a principal component analysis (PCA) of the ORN and computed PN firing rates to compare their degree of correlation. For totally correlated data, a single principal component accounts for 100% of the data variance, whereas for completely uncorrelated data, all of the components account for an equal percentage of the variance. For the olfactory responses, >40% of the ORN response variance is accounted for by a single principal component projection, indicating a high degree of correlation (Fig. 1F). The maximal principal component projection for the PN responses accounts for only 15% of the total variance (Fig. 1H), so PNs are significantly decorrelated by the transformation of Eq. 1. This decorrelation is a consequence of lateral suppression in the antennal lobe, because it is almost entirely eliminated if we set the parameter m in Eq. 1 to zero (Fig. 1G). A previous mathematical characterization of PN responses did not show decorrelation (21, 27) because it did not include lateral suppression. In agreement with more recent data (22), the more accurate characterization we use here does decorrelate.

Discrimination from Odor-Specific Weights (Lateral Horn).

We hypothesize that neurons in the lateral horn are connected in a manner that generates highly selective responses to specific odors, allowing them to evoke innate behaviors. To study selectivity in the lateral horn from the data we are using, we constructed 110 LHNs, each selective for a different one of the 110 odorants in the dataset. In other words, we assigned a single target odor from this set to each LHN and attempted to make it respond exclusively to this target odor. We are not suggesting that each of these odors generates an innate behavior. Instead, the model LHNs are used as a proof of principle to demonstrate how highly specific odor-selective LHNs can be constructed.

We assign synaptic weights for the LHNs using Fisher’s linear discriminant, a standard technique (Methods). Importantly, this approach uses only PN responses to the targeted odor and general statistical properties of the PN responses across odors such as the mean and covariance to determine the weights, and avoids overfitting. The weights of inputs onto LHNs produced in this way are both positive and negative. We interpret negative weights as arising from inhibitory input to the LHN from unmodeled interneurons driven by the corresponding PN. Using these weights, we can generate LHNs tuned to respond almost exclusively to a single odorant for all 110 odors in the dataset (Fig. 2).

LHNs will respond exclusively to their target odor if the input for this odor is consistently higher than the inputs for any other odors. Histograms of the sum of 20 weighted PN inputs (Fig. 2A) for targeted (red) and nontargeted (blue) odors for all 110 LHNs over repeated trials reveal two roughly Gaussian distributions that are well separated. This allows us to divide the inputs into those producing and those not producing a response with a fixed threshold (dashed vertical line in Fig. 2A). Using this threshold, we computed response probabilities across trials (PN responses are different in each trial due to the response variability built into our PN model) for 110 differently tuned LHNs, each labeled by the odor to which it is targeted (LHN index in Fig. 2B). The diagonal elements of the matrix of response probabilities (Fig. 2B) correspond to each LHN responding to its targeted odor, and the nondiagonal entries represent responses to nontargeted odors. For perfect performance of the LHNs, response probabilities would be 100% along the diagonal and zero off the diagonal, meaning that the LHNs respond every time to the odor for which they are targeted and never to any other odor. The selectivity of the LHNs is quite close to being perfect in this sense (Fig. 2B). This excellent performance is in agreement with a theoretical analysis of the capacity of a system of this type (28).

Performance of the LHNs can also be characterized by the fraction of untargeted odors that generate a response (the false-positive rate) and the fraction of targeted odors that fail to generate a response (the false-negative rate). These quantities depend on the threshold, but if we adjust this so that the false-positive and false-negative rates are equal, the value they take is 0.4%. The area under a receiver operating characteristic (ROC) curve provides a threshold-independent measure of discrimination performance, and this area is 0.999 for model LHNs driven by 20 PN inputs.

We also considered what happens if the number of PN inputs to the model LHNs is reduced to five, a number suggested from analysis of KCs (15). In this case, the weights are constructed again using Fisher’s linear discriminant, but for a reduced set of inputs (Methods). With only five PN connections per LHN, the input distributions for targeted and nontargeted odors are slightly more overlapping (Fig. 2C), and there are more nonzero response probabilities for nontargeted odors (Fig. 2D). Nevertheless, the overall performance is still good. The equal false-positive and false-negative probabilities are 2%, and the area under the ROC curve is 0.987. Thus, acceptable discrimination can be obtained with as few as five PN inputs per LHN.

Sparse KC Responses from Random Weights (Mushroom Body).

Our modeling work for KCs is based on the assumption that they provide a sparse representation across a wide variety of odorants. Of course the LHNs, being highly selective, also provide a sparse representation, but not across nearly as broad a range of odors as the KCs do. Our goal is to construct a population of 2,500 model KCs, the number estimated in the mushroom body (29), such that on average 5% of the KCs respond to each odor, and each KC responds to ∼5% of the odorants presented, corresponding to experimental estimates (14, 15). We allow fluctuations around this 5% value from cell to cell or from odor to odor, but we associate large fluctuations with a low-quality sparse representation.

Our procedure for constructing a KC response is to randomly choose PNs from n glomeruli and to set the strengths of the synapses connecting them to a KC randomly and independently from a uniform distribution over positive values (Methods). Good performance can be achieved with such random connections (Fig. 3A) even though we used the same threshold for all of the KCs and did not individually adjust the threshold to achieve a precise 5% sparseness in each KC. The scattering of color shows that the sparse KC responses are well distributed across all 2,500 cells and 110 odorants. Fig. 3 shows KCs receiving excitatory input from PNs connected to five different glomeruli, along with global inhibition (30) (Methods). In this case, all 110 odors elicit responses above the 95% threshold in, on average, 125 cells and, at a minimum, at least two KCs.

One measure of the quality of the sparse representation for different numbers of PN inputs is the number of missed odors (Fig. 3B Upper). Such an odor would not be represented in the mushroom body and, presumably, could not be detected by the fly unless it was represented separately in the lateral horn. We define a neuron as responding if it receives an above-threshold input in at least 50% of trials. Even with this fairly stringent criterion, none of the 110 odors is missed for most values of n, the number of PNs connected to each KC (Fig. 3B Upper). The best performance is for n < 15. A high-quality sparse representation should also efficiently use its neuronal resources, which means there should not be a large number of KCs that never respond to any odor. Whereas 1 or 2 PN inputs are sufficient to obtain a good sparse representation, this leads to a large number of silent KCs (Fig. 3B Lower). With only 1 PN input, about 800 of the 2,500 KCs do not respond to any odor, according to the 50% criterion. The number of nonresponding KCs drops progressively as the number of PN to KC connections increases. Combining the results of Fig. 3B Upper and Lower, the best performance seems to arise for between 5 and 15 PN inputs per KC, numbers consistent with estimates obtained from electrophysiology (15).

The synaptic weights connecting PNs to our model KCs are all positive, representing pure excitation, but the model KCs also receive global inhibition (Methods). Inhibition is useful because it allows the KCs to take advantage of the decorrelation of the PN inputs. However, we also examined the performance of model KCs when all inhibition was removed and they were driven solely by PN input (Fig. S2). Performance is still good with regard to missed odors (Fig. S2 Upper), but there are significantly more silent KCs when inhibition is eliminated (Fig. S2 Lower).

Concentration Dependence.

Changing odor concentration is a particularly stringent test of a model like ours in which responses are determined by a fixed threshold. Without response magnitude equalization across concentrations, selectivity that is present at low concentrations will be lost at high concentrations or, alternatively, neurons that are selective at high concentrations will fail to respond at low odor concentrations. Imaging and electrophysiological experiments (14, 15) indicate that at least some of the KCs retain their selectivity to the same odors across a range of different concentrations. It is reasonable to assume that at least some LHNs share this feature, as this would have obvious behavioral advantages. Hallem and Carlson measured ORN firing rates at four different concentration levels for 10 odorants (20). We studied the effects of varying concentration by replacing the ORN data used in the previous sections with this collection of 40 response sets, representing 10 odors presented at four different concentrations, 10⁻², 10⁻⁴, 10⁻⁶, and 10⁻⁸ dilutions.

For the model LHNs, we generalized Fisher’s linear discriminant for multiple stimuli across different concentrations (Methods) and used five PN inputs per LHN. The performance of these model LHNs is shown as four response probability matrices (Fig. 4A), one for each concentration. In this case, the bright diagonal is recapitulated four times, illustrating that these cells respond reliably to their target odors at all of the different concentrations. The off-diagonal response probabilities, representing responses to nontargeted odors, remain small, indicating that the false-positive rate is low across concentrations. The main error comes, not surprisingly, from discrimination at low concentration, where the false-positive rate increases somewhat. Over all odors and concentrations, the false-positive and false-negative rates are 7% and the area under the ROC curve is 0.92.

To construct KCs, we again used random positive synaptic weights with global inhibition and adjusted the single threshold used for all KCs so that 5% of the neurons respond, averaged across concentrations. Note that in contrast to previous models of concentration invariance (31), the KCs in our model do not receive compensatory inputs or change their thresholds adaptively for different concentrations. These fixed-threshold KCs respond to odors sparsely at both low (10⁻⁸ SV) and high (10⁻² SV) concentrations, and an analysis of the results shown in Fig. 4B indicates that, collectively, the population responds to every odor in the dataset regardless of its concentration. In other words, none of the odors is missed across this wide range of concentrations, so a full representation of the odors is retained.

Role of the ORN-to-PN Response Transformation.

To study the effect of the nonlinear transformation that takes place between the ORN and PN responses, we repeated our analysis of model LHNs and KCs but driven either by direct ORN input or by PN input with lateral suppression removed (m = 0 in Eq. 1).

The results are summarized in Fig. 5, and further details can be found in Figs. S3–S5.

For LHNs driven by either 20 (Fig. 5A) or 5 (Fig. 5B) inputs, the false-positive and false-negative error rates (when the threshold is adjusted to make them equal) rise only slightly when the effect of lateral suppression is removed (bars marked “NL” in Fig. 5 A and B), but there is a larger increase in error percentage when the feedforward nonlinearity is also eliminated (equivalent to using ORN rather than PN input, bars marked “ORN” in Fig. 5 A and B; also Fig. S3). KCs driven by PN input lacking lateral suppression perform basically as well as those using full PN input, as in neither case are any odors missed from the KC representations with 5 inputs per KC (Fig. 5C). Performance degrades significantly in the absence of the feedforward nonlinearity when ORN input is used (Fig. 5C and Fig. S4). In this case, ∼30 odors are missing from the representation.

Lateral suppression, and hence decorrelation, plays a larger role in buffering the concentration dependence of LHN responses. The equal false-positive and false-negative error percentages rise more significantly when lateral interactions are eliminated, although most of the decrement in performance associated with using ORN input still arises from the absence of the feedforward nonlinearity (Fig. 5D and Fig. S5A). KCs driven by ORN rather than PN input (removing the effects of Eq. 1) fail to respond to many of the tested odors at low concentrations (Fig. S5B) and thus lose the ability to represent odors across the wide range of concentrations seen in Fig. 4A.

In summary, the most important factor in generating selective (LHNs) and sparse (KCs) third-order responses is the response magnitude equalization provided by the feedforward nonlinearity of the antennal lobe. The effects of lateral suppression, which are responsible for response decorrelation, are also important, although smaller than those of the feedforward nonlinearity, for retaining selectivity over a range of odor concentrations.