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. Author manuscript; available in PMC: 2018 Jul 1.
Published in final edited form as: Neuron. 2018 Apr 26;98(4):736–742.e3. doi: 10.1016/j.neuron.2018.04.004

Odor Perception on the Two Sides of the Brain: Consistency Despite Randomness

Evan S Schaffer 1,7,*, Dan D Stettler 1,5, Daniel Kato 1, Gloria B Choi 1,6, Richard Axel 1,2,3, LF Abbott 1,4
PMCID: PMC6026547  NIHMSID: NIHMS976460  PMID: 29706585

SUMMARY

Neurons in piriform cortex receive input from a random collection of glomeruli, resulting in odor representations that lack the stereotypic organization of the olfactory bulb. We have performed in vivo optical imaging and mathematical modeling to demonstrate that correlations are retained in the transformation from bulb to piriform cortex, a feature essential for generalization across odors. Random connectivity also implies that the piriform representation of a given odor will differ among different individuals and across brain hemispheres in a single individual. We show that these different representations can nevertheless support consistent agreement about odor quality across a range of odors. Our model also demonstrates that, whereas odor discrimination and categorization require far fewer neurons than reside in piriform cortex, consistent generalization may require the full complement of piriform neurons.

In Brief

Random connectivity from the olfactory bulb to piriform implies that the representation of odors will differ in different cortices. Schaffer et al. demonstrate in a model that consistent agreement about odor quality occurs but requires the full complement of piriform neurons.

INTRODUCTION

Olfactory perception involves both odor discrimination and generalization. Discrimination relies on the detection of differences between odors, whereas generalization requires the identification of similarities and dissimilarities across odors. The ability to generalize is vital to performance in any olfactory task because natural variability implies that the same stimulus is never experienced twice. Perceptual consistency across odors requires that generalization be similar among individuals. We thus explore whether the unstructured representation of odors in piriform cortex can support consistent generalization.

Individual olfactory sensory neurons in the mouse express one of 1,500 receptor genes (Buck and Axel, 1991; Godfrey et al., 2004; Zhang and Firestein, 2002). Neurons expressing a given receptor project with precision to two spatially invariant glomeruli within the olfactory bulb (Mombaerts et al., 1996; Ressler et al., 1993, 1994; Vassar et al., 1993, 1994). Odors elicit distributed neural activity in the sensory epithelium that is transformed in the olfactory bulb, where each odor evokes a distinct spatial pattern of glomerular activity (Davison and Katz, 2007; Igarashi and Mori, 2005; Ma et al., 2012; Meister and Bonhoeffer, 2001; Soucy et al., 2009; Takahashi et al., 2004). A second transformation occurs in the piriform cortex, where individual odors activate unique ensembles of neurons that lack discernable spatial patterning (Illig and Haberly, 2003; Iurilli and Datta, 2017; Poo and Isaacson, 2009; Rennaker et al., 2007; Stettler and Axel, 2009; Sugai et al., 2005; Zhan and Luo, 2010). Representations of individual odors are distributed across the entire piriform with no apparent large-scale (Stettler and Axel, 2009) or local spatial structure (Figure S1). One model consistent with both the anatomy and the physiology assumes that each piriform neuron samples a random combination of glomerular inputs (Choi et al., 2011; Davison and Ehlers, 2011; Ghosh et al., 2011; Miyamichi et al., 2011; Sosulski et al., 2011). Random connectivity implies that the piriform representation of a given odor will differ among different individuals and across brain hemispheres in a single individual. Does random connectivity from the olfactory bulb to piriform cortex support generalization and allow shared experiences to “align” the perception of odor quality in different individuals across a wide range of odors?

We have combined mathematical modeling with analysis of in vivo optical imaging of odor responses in piriform cortex and find that many properties of the piriform odor representation are in good accord with predictions of a model with random input. Consistent with previous models (Babadi and Sompolinsky, 2014; Barak et al., 2013; Cho and Saul, 2009; Litwin-Kumar et al., 2017), we observe that piriform representations simulated with random input are less correlated than the bulb representations but maintain sufficient correlation between odors to support generalization. We consider how individuals with different piriform odor representations can agree consistently about odor quality across a range of odors and how conflicts are avoided between the two sides of a single individual’s brain. Our model demonstrates that whereas odor discrimination and categorization require far fewer neurons than reside in piriform cortex (Babadi and Sompolinsky, 2014), consistent generalization requires the full complement of piriform neurons.

RESULTS

Piriform Responses to Odor Pairs Exhibit Correlations Predicted by a Randomly Connected Model

Similar odors evoke correlated patterns of activity in the olfactory bulb (Davison and Katz, 2007; Igarashi and Mori, 2005; Ma et al., 2012; Meister and Bonhoeffer, 2001; Soucy et al., 2009; Takahashi et al., 2004). Random wiring from the bulb to piriform might suggest that these correlations would be absent in cortex. However, a number of theoretical studies have shown that correlations in a set of inputs are reduced, but not eliminated, when activity is transmitted through random synaptic connections (Babadi and Sompolinsky, 2014; Barak et al., 2013; Cho and Saul, 2009). Indeed, piriform responses to pairs of odors activating similar glomeruli can show considerable overlap (Figure 1D; octanal and hexanal, 24% overlap in piriform [Stettler and Axel, 2009]; 60%–70% overlap in bulb [Meister and Bonhoeffer, 2001; Igarashi and Mori 2005]; see also 26% in bulb [Ma et al., 2012]).

Figure 1. The Transformation from Model Bulb to Model Piriform Partially Preserves Stimulus Similarity.

Figure 1

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(A) The representation of a model odor in the bulb is transformed through random connections to a representation in the piriform. Example model odors in the bulb (top) and piriform (bottom), with three different values of the excess overlap parameter f: nonclass odors (f = 0, black), slightly overlapping odors (f = 0.3, blue), and very overlapping odors (f = 0.7, green). For visual clarity, just 50 glomeruli and 50 piriform neurons are shown. Color scale is normalized; white, max response; red, weak response; black, no response.

(B) Correlation in piriform representation versus correlation in glomerulus representation. Each dot corresponds to the correlation coefficient between two odors with the same level of overlap (f = 0.3 or f = 0.7), color-coded according to the conventions in (A). For reference, the full spectrum of values of f ranging from 0 to 1 is shown in gray. Distributions of correlation in the glomeruli and in piriform are shown along their respective axes.

(C) Observed probability of a piriform cell responding to two odors in the model versus the expected probability if odor representations were statistically independent (f = 0.0, f = 0.3, and f = 0.7 shown in black, blue, and green, respectively). In order to recapitulate the variability in representation sparseness observed in vivo, model odors were generated with a range of sparseness values from Sx = 0.05 to Sx = 0.2.

(D) Same as (C), but for in vivo odor responses (average ± SEM).

(E) Class selectivity of a neuron is quantified as the difference in that neuron’s mean response to class odors versus nonclass odors. Shown is the probability distribution over the fraction of class (green) and nonclass (black) odors eliciting a response for model neurons in the 4th percentile for class selectivity.

(F) Same as (E), but for a class defined by an arbitrary collection of odors not necessarily activating similar glomeruli. Selected class members in gray and nonclass members in black.

To explore the implications of random connectivity, we constructed a model in which the connectivity between the olfactory bulb and piriform cortex is random (STAR Methods). The model has an input layer of Nx = 1,000 glomeruli and a cortical layer with Ny neurons. Model odors activate a sparse ensemble of SxNx glomeruli, with sparseness Sx = 0.1, unless otherwise noted. Bulb activity both excites and inhibits the cortical layer in a balanced manner, so that the average bulb-derived input across the cortex is zero. Connectivity from bulb to piriform is sparse (Sc = 0.2), so each piriform neuron receives input from ScNx = 200 random glomeruli, and approximately 20 of these are activated by a given odor. Model piriform neuron responses are a threshold-linear function of the sum of their inputs with the threshold chosen so that, on average, SyNy cortical neurons respond to an odor with Sy = 0.06, matching the sparseness of responses observed in vivo. Firing rates of the active glomeruli are chosen from a lognormal distribution. We defined classes of model odors with a mean shared fraction f of active glomeruli (STAR Methods), and we focus our analysis on three sets of odors (Figure 1A), defined by f = 0 (nonclass), f = 0.3 (weak class), and f = 0.7 (strong class).

Correlations between simulated piriform activities for odor pairs are related to but smaller than correlations of glomerular activities (Figure 1B), in agreement with previous results (Babadi and Sompolinsky, 2014; Cho and Saul, 2009). The shape of this curve follows from the observation that sparse activity patterns can more faithfully preserve positive correlations than negative correlations. Whereas input correlations close to 1 tend to be largely preserved, negative input correlations are transformed into piriform correlations near zero because uncorrelated sparse representations have very little overlap, leaving minimal dynamic range for further anticorrelation. Consequently, the curve relating input correlation to output correlation has a shallow slope for low input correlation.

Comparison of model results with data is facilitated by considering the fraction of neurons responding to pairs of odors, rather than the correlation. The fraction of neurons responding to pairs of simulated odors, drawn from all three classes, exceeds the value expected from an uncorrelated odor representation (Figure 1C). For independent representations, the fraction of cells responding to both odor A and odor B would be pApB, where pA and pB are the probabilities for responses to A or B alone. For simulated odors that activate shared sets of glomeruli (Figure 1C), this is a direct result of input-derived correlations maintained despite random connectivity (Babadi and Sompolinsky, 2014; Cho and Saul, 2009). In addition, more neurons than expected for independent representations respond to odor pairs that do not share active glomeruli (Figure 1C). This correlation arises from common glomerular inputs that occur by chance despite random connectivity (see also Litwin-Kumar et al., 2017).

The model results make two predictions about correlations in the piriform representations of odor pairs. First, the average correlation between molecularly unrelated odor pairs should be positive, not zero. Second, odors that activate similar sets of glomeruli should evoke correlated representations in piriform, but these correlations should be smaller than in the bulb. Two-photon calcium imaging of layer 2 of piriform cortex (STAR Methods) reveals that the fractions of neurons responding to odor pairs (Figure 1D) are consistent with the model (Figure 1C). Most pairs produce larger fractional responses than would be expected for statistically independent activity (p < 0.02, Wilcoxon signed-rank test). In our model, correlations in the bulb are the major source of correlations in piriform. Adequate data from imaging of the response to the same odor pairs in both bulb and piriform are unavailable. Therefore, we cannot at present experimentally compare correlations across these two olfactory areas.

Random Connectivity Can Produce Odor-Class Selective Piriform Neurons

We next asked whether piriform neurons that respond selectively to a set of similar odors can arise from random input wiring. We define class selectivity as the difference between the mean responses of a neuron to class and nonclass odors and then ranked the neurons in our model according to this selectivity. We illustrate the results by showing the probability distribution over the fraction of odors eliciting a response for model neurons in the 4th percentile for class selectivity (Figures 1E and 1F). On average, these neurons respond near chance levels (Sy = 6%) to nonclass odors, but they respond to 40% of class odors (Figure 1E). We show that this selectivity reflects overlap in the bulb by constructing a nonsensical class defined as a random collection of 1,000 odors with f = 0. Neurons in the 4th percentile of selectivity to this artificial class respond near chance levels to both class and nonclass odors (Figure 1F). Thus, even a randomly connected piriform will have some cells with selectivity to a class of similar odors.

Learning Single-Odor Associations Can Align Population Readouts from Different Piriform Representations

Given the randomness of the input to piriform, the same odor will be represented differently in different brains. We now consider how these individuals can nevertheless align the qualities they assign across a wide range of odors. A similar problem arises in considering the alignment of odor qualities inferred from the two sides of a single individual’s brain. To define an odor quality in our model, we introduce a linear readout that characterizes the output of an entire population of piriform neurons. The readout is a weighted sum of piriform activities z = Σiwiyi, where yi is the activity of piriform neuron i and wi is its weight. Initially, the readout weights are chosen randomly, and we examine the resulting readout responses from two different randomly wired model piriform cortices (Figure 2A, top). Readout responses obtained from two different model piriform cortices using the same weights are uncorrelated (correlation coefficient = 0.03, −0.03, and 0.005, for odors with f = 0.7, 0.3, and 0, respectively) because the two piriform representations are completely different.

Figure 2. Agreement between Readout Units Requires a Large Piriform.

Figure 2

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(A) Response of a readout unit versus the response of a second readout connected to an independent set of piriform neurons, either before training (top) or after training to a single odor (bottom), for a panel of odors with 70%, with 30%, or at chance levels of overlap with the trained odor, color coded as in Figure 1. Gray boxes denote the regions in which the readouts do not agree for θ = 0.5 and f = 0.7.

(B) Scaling with Ny of readout agreement (with f = 0) versus other measures of readout performance: readout agreement with a threshold of θ = 0.5 (brown) or θ = 0.9 (orange), readout correlation (black), SNR (gray), and accuracy (magenta). All quantities except SNR are defined from 0 to 1 (Max[SNR] = 3.4). We normalize SNR such that the minimum value is 0 and the maximum value is 1 to enable comparison to other quantities.

(C) Comparison with experimental data from the Drosophila mushroom body. The correlation between two model MBONs as a function of the number of inputs they receive (black). Published data (Hige et al., 2015b) showing the correlation across odors in three conjugate pairs of MBONs versus the number of KCs that innervate each MBON (gray; mean ± SEM).

(D) Readout agreement is smaller than readout correlation (top), while readout accuracy is greater than normalized readout SNR (bottom).

(E) Intuition for why readout agreement is smaller than readout correlation. As the correlation between two readouts grows, their joint probability distribution becomes elongated (gray), but even when this distribution is very elongated, an appreciable portion remains in the off-diagonal quadrants (orange).

We next asked whether shared training to a single odor in two different individuals results in readouts that agree across a wide range of odors. In this simulation, we set the weights for each readout equal to the activity produced by the odor being learned in the corresponding piriform ( Wi=yi∗, where the piriform response to the trained odor is yi∗; STAR Methods). This implements a form of Hebbian learning (Oja, 1982). This single shared experience, corresponding to associative learning with a common odor, does an excellent job of aligning the two readouts in response to other odors (Figure 2A, bottom). Simulated odors sharing either 30% or 70% of activated glomeruli with the trained odor evoke significantly larger responses in the readouts than all nonclass odors. Two readouts not only distinguish class from nonclass, but can also rank the similarity of a panel of odors extremely well, even if the odors are not similar to one another or to the trained odor (the f = 0 case). Moreover, the responses of the two readouts across all simulated odors are well correlated (correlation coefficient = 0.98, 0.97, and 0.86, for f = 0.7, 0.3, and 0, respectively). It is worth emphasizing that the observation that this works for all odors implies that the identity of the training odor is inconsequential. Thus, after training with a single exemplar odor, two readouts can generalize in a similar way across other odors.

The degree of correlation between two readouts due to single-odor learning depends on the number of piriform neurons. This is of interest because of the observation that the number of piriform neurons exceeds the number required for efficient categorization (Babadi and Sompolinsky, 2014). The readout correlation increases and saturates as a function of the number of piriform neurons (Figures 2B and S2). The correlation coefficient between readouts modified by a shared training experience provides a measure of consistency. The ability of a single readout to categorize odors, on the other hand, is characterized by its signal-to-noise ratio (SNR; Babadi and Sompolinsky, 2014; Litwin-Kumar et al., 2017). As a function of the number of piriform neurons, the SNR saturates (Babadi and Sompolinsky, 2014; Litwin-Kumar et al., 2017), and in our model, this saturation occurs when roughly 100,000 piriform neurons provide input to the readout. It has been noted that this is an order of magnitude less than the actual number of piriform neurons (Babadi and Sompolinsky, 2014). The curve showing the correlation coefficient for two readouts (black curve in Figure 2B) overlaps with the curve for the SNR scaled so that its maximum value is 1 (gray curve in Figure 2B). We discuss the reasons for this overlap in the STAR Methods.

To provide a test of our calculation of readout alignment, we exploit experimental data from the Drosophila mushroom body. The fly affords an opportunity to examine genetically defined neurons that readout from the two different mushroom bodies in the two hemispheres of the fly brain. In Drosophila, projection neurons connect glomeruli in the antennal lobe (the insect analog of the olfactory bulb) to Kenyon cells (KCs) that form the mushroom body (the piriform analog) (Marin et al., 2002; Wong et al., 2002). As in the piriform, glomerular inputs to the KCs show no apparent structure (Caron et al., 2013; Gruntman and Turner, 2013; Murthy et al., 2008). Further support for the random nature of these connections is provided by electron microscopy (EM) data from a Drosophila larva (Eichler et al., 2017), indicating that input connections to the KCs are not only unstructured, but are also completely different on the two sides of the brain. KCs synapse onto mushroom body output neurons (MBONs) that are analogous to the model readout we have introduced. Learned associations are established by modifying the strength of the KC to MBON synapses (Aso and Rubin, 2016; Hige et al., 2015a; Séjourné et al., 2011). Thus, homologous MBONs on the two sides of the brain of a single fly should exhibit the response alignment that we have discussed. Correlations were determined for three MBON types that receive different numbers of KC inputs (Hige et al., 2015b). This allows us to test not only the values of the correlations we computed, but also their dependence on the number of neurons driving the readout.

We constructed a fly analog of our piriform model, adjusting the numbers of neurons and synapses appropriately (STAR Methods). The correlation coefficient between two model MBONs as a function of the number of inputs (Figure 2C) resembles the piriform correlation curve in Figure 2B. The fly model correlation saturates at a value of roughly 2,000 KC inputs, the actual number of KCs in the mushroom body. Hige et al. (2015b) have analyzed the responses of left-right pairs of MBONs that receive different numbers of KC inputs to a panel of odorants. The correlation coefficient between MBON pairs determined experimentally shows a striking match with the predictions of the model (Figure 2C).

Readout of a Sufficiently Large Piriform Population Supports Consistent Choice

Odor-guided behavior typically involves making a binary choice; for example, to act or not to act. We have described a readout that provides a continuous measure of the piriform response. We now model choices by comparing this readout value to a threshold. Readout values greater than the threshold are interpreted as a choice to act; those below threshold as a choice not to act. This allows us to compute the accuracy with which a trained readout can guide choice (Figure 2B; accuracy is defined as fraction correct). Accuracy is an increasing function of the SNR that saturates before the SNR reaches its maximum (Figure 2D, top). As a result, accuracy saturates at a lower number of piriform neurons than does the SNR (Figure 2B), further emphasizing the disparity between the actual number of piriform neurons and the number needed to support categorization and choice.

The threshold we have introduced allows us to investigate the consistency of choice across individuals or brain hemispheres. We trained two readouts, each connected to an independent model piriform cortex (106 neurons), on a single odor. We then compared the readouts to a threshold and examined the choice agreement across a large number of nonclass odors. Good agreement between readouts is substantially more difficult to obtain than is accuracy in a typical categorization task such as distinguishing class from nonclass odors. We characterize the value of the threshold with a parameter θ that is equal to the fraction of odors that produce a subthreshold readout response. At a threshold for which odors are equally divided between supra-and subthreshold (θ = 0.5), 95% of the choices are the same for two trained readouts, but agreement is at chance levels (~50%) for two untrained readouts (Figure 2A). Note that this consistency is observed for odors that are both similar to and different from the trained odor. Thus, readouts receiving input from different piriform representations can support consistent choices after a single shared training experience.

We next asked how the consistency of choices depends on the number of neurons in piriform. For this purpose, we introduce a quantity Aθ called the agreement. In defining the agreement, we correct for the bias introduced by thresholds different from 0.5 by subtracting the fraction of choices due to chance from the fraction of choices that agree. Normalizing the resulting quantity to a maximum value of 1 yields our measure of agreement Aθ (STAR Methods).

Whereas accuracy is determined by SNR, agreement is a function of the readout correlation coefficient and is always the smaller quantity (Figure 2D, bottom). Aθ saturates at higher numbers of piriform neurons than the accuracy, SNR, or readout correlation coefficient (Figures 2B). This is illustrated graphically in Figure 2E. As the number of piriform neurons grows, the values of the two readouts become more correlated and fall in an increasingly elongated elliptical distribution (Figure 2E, gray). For θ = 0.5, readout agreement corresponds to the fraction of the distribution not in the off-diagonal quadrants (Figure 2E, orange). Even when the distribution is very elongated (high correlation), an appreciable portion remains in these quadrants. The slow increase of Aθ as a function of the number of piriform neurons is a general property that is relatively insensitive to the threshold used because Aθ has been corrected for bias and normalized.

Our conclusions concerning the agreement between two readouts hold when we extend our analysis to multiple readouts. We introduce an additional parameter to specify the fraction of similarly responding readouts required to say that the ensemble “agrees” (STAR Methods). For small values of this parameter, the many readout case looks very similar to the two-readout case; for large values, maximal agreement requires at least an order of magnitude more inputs from piriform (Figure S2). Furthermore, the number of piriform inputs required for maximal readout agreement reduces only slightly for odors that more densely activate the olfactory bulb (Figure S2). Thus, a single piriform cortex can support accurate categorization and choice with fewer than 105 neurons, an order of magnitude less than the actual number. However, 106 piriforms are required for choice agreement between different individuals or across the two sides of the brain. This may provide a rationale for the large number of piriform neurons.

DISCUSSION

Neurons in piriform cortex receive input from a random collection of glomeruli, resulting in an odor representation that lacks the stereotypy of the olfactory bulb. Random connectivity implies that each piriform cortex is unique, posing the problem as to how different individuals generalize consistently across stimuli. Generalization implies that different individuals, after learning a conditioned response to odor A, will generate the learned behavior in response to odors with glomerular representations similar to A, but not to odors that activate different glomeruli. Consistent generalization is also necessary across the two differently wired piriform cortices of a single individual if inter-hemispheric conflicts are to be avoided. Our model demonstrates that readouts from two different randomly wired cortices can be highly correlated after a single shared experience. In this manner, perceptual consistency will not require individuals to have lived identical lives. The degree of correlation between model readouts depends on the number of randomly wired neurons, generating predictions that are in excellent agreement with data from the mushroom body output neurons of the fly. Moreover, the ability of the model piriform to support consistent choices across individuals requires that the piriform cortex contain a large number of neurons, a number in accord with that observed in the mouse. Modeling studies using more traditional performance measures such as dimensionality and readout SNR have suggested that piriform has an excess of neurons (Babadi and Sompolinsky, 2014). We find that consistent decision-making may require the full complement of piriform neurons.

The model we have considered is based on an assumption of random wiring. We investigated this assumption further by examining in vivo data for evidence of structure. We could detect no local spatial structure, no periodic structure, and no clustering across the piriform surface. We did find positive correlations in the responses of individual neurons to odor pairs, but these are consistent with expectations given random wiring from the bulb to piriform. The fact that piriform odor representations generated by random connectivity can “inherit” correlations from bulb representations is essential for generalization across odors.

A number of studies have examined the impact of random connectivity on the ability of a sensory representation to support categorization (Babadi and Sompolinsky, 2014; Barak et al., 2013; Cortes and Vapnik, 1995; Litwin-Kumar et al., 2017; Marr, 1969; Rigotti et al., 2013). In these studies, categorization consists of dividing stimuli (odors) into two classes based on arbitrarily assigned valences. This type of categorization task emphasizes the importance of decorrelation because correlations in piriform responses have a negative impact. In our model, associations are learned for a small subset of odors and behavioral decisions are made on the basis of generalization. In this task, correlations between piriform responses to odors are essential for assessing odor similarity, and similarity provides the basis for generalization. Correlations are also essential for the consistency of generalization across individuals and between brain hemispheres despite the presence of different representations in different cortices. Lastly, in tasks in which structure in the stimulus space can be exploited, plasticity on the random layer itself can further aid the ability to categorize (Babadi and Sompolinsky, 2014; Litwin-Kumar et al., 2017; Pehlevan and Chklovskii, 2014).

Correlations between odor representations in the bulb reflect correlations between the binding of odors to receptors in the nose. These correlations are retained in the piriform despite the random connectivity from the bulb. Since individuals in a species express similar repertoires of receptors, correlations will be preserved across neuronal populations in different piriform cortices, but the neurons responsible for these correlations will be different. In our model, individuals with different piriform cortices can nevertheless consistently assess the similarity of odors. In this manner, “a rose by any other name would smell as sweet” (Shakespeare, 2015).

STAR★METHODS

Detailed methods are provided in the online version of this paper and include the following:

KEY RESOURCES TABLE

REAGENT or RESOURCE SOURCE IDENTIFIER
Deposited Data
Calcium imaging data of odor responses in mouse piriform cortex Stettler and Axel, 2009 N/A
Calcium imaging and electrophysiological data of odor responses in Drosophila mushroom body output neurons Hige et al., 2015b N/A
Software and Algorithms
Algorithms for calculation of readout correlation and agreement This paper https://github.com/schafferEvan/coherentGeneralization
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CONTACT FOR REAGENT AND RESOURCE SHARING

Further requests for resources should be directed to and will be fulfilled by the Lead Contact, Evan S. Schaffer (ess2129@columbia.edu).

METHOD DETAILS

Piriform Model

Our network model has feedforward connectivity and consists of three layers that we refer to as the glomerular layer, cortical layer, and readout layer, respectively, as shown in Figure 1. A fraction Sx of the Nx neurons in the glomerular layer are activated by a given odor. Glomerular representations of class odors with excess overlap f are chosen such that all such odors activate a common set of fNxSx glomeruli and an additional randomly selected (1 − f) NxSx glomeruli. The case with no glomeruli designated as shared (f = 0) corresponds to independent or nonclass odors. Response magnitudes of glomeruli whose responses are nonzero are chosen from an No-dimensional multivariate lognormal distribution, exp (𝒩(μg, Σg)), where the diagonal elements of the covariance matrix Σg all equal σg2, and off-diagonal elements equal ln[f(exp[σg2]-1)+1], and No is the number of class odors. This yields response magnitudes whose average cross-odor correlation is f.

Each of Ny cortical neurons receives SceNx excitatory and SciNx inhibitory synaptic inputs from a random selection of glomeruli with weights, represented by matrix elements Jij, set to 1 and Sce/Sci, respectively. The response of each cortical neuron is a threshold-linear function of the sum of its inputs, yi = θ[ΣjJijxj − θ], where θ (x) = x if x > 0, and θ (x) = 0 otherwise. The threshold θ is chosen such that an average of 6.2% of piriform neurons are activated by each odor, matching our imaging data. For the parameters chosen, this results in an excess of 9.2 excitatory inputs being required, on average, for a piriform neuron to be activated, i.e., θ≈9.2∗exp[μg+(σg2/2)]. Finally, the readout unit receives a weighted input from every piriform neuron, z = ΣiWiyi, where Wi=yi∗ after training, where the piriform response to the trained odor is yi∗. Before training, Wi is chosen randomly with the same statistics as y. For all simulation results shown, Nx = 1000, μg = 0.1, σg = 0.5, Sx = 0.1, Sce = 0.2, Sci = 0.4, θ = 11.9.

In all quantities computed from the performance of one or multiple trained readout units, a single odor is trained, and the readout is tested with a panel of unrelated odors. Thus, readout accuracy is calculated as the fraction of novel odors correctly rejected as different from the trained odor. A more symmetric task in which the stimulus test set also includes noisy presentations of the trained odor, to which the readout should respond with the same output as the trained odor, gives qualitatively similar results.

Fly Model

Our fly model differs from our piriform model in the following parameters: Nx = 50, Sx = 0.2 (see Wang et al., 2003), Sce = 7/50 (see Caron et al., 2013), Sci = 1 (see Papadopoulou et al., 2011). The only other difference between these models is that in our fly model, the readout plasticity is anti-Hebbian (rather than Hebbian), so that synapses from KCs responsive to the trained odor are set to zero, and synapses from KCs not responsive to the trained odor remain strong. This feature is not essential for our results but is motivated by the experimental observation that KC-MBON synapses depress upon pairing of an odor with an unconditioned stimulus (Aso and Rubin, 2016; Hige et al., 2015a; Séjourné et al., 2011).

Similar Scaling of the SNR and Two-readout Correlation Coefficient as a Function of the Number of Piriform Neurons

As seen in Figure 2B, the signal-to-noise ratio (SNR) of a single readout and the correlation coefficient (CC) between two readouts, all trained by a Hebbian rule, have virtually identical scaling as a function of the number of piriform neurons supporting the readout. We denote the response of the single readout to odor a by za and the valence of this odor by va. Similarly, the responses of the two read-outs are denoted by za(1) and za(2). The signal-to-noise ratio for the single readout is given by

SNR=〈vaza〉a2Var(za),

where the average in the numerator is over odors and the denominator is the variance across odors. The correlation coefficient between two readouts is

CC=〈za(1)za(2)〉a(Var(za(1))Var(za(2)))12.

The numerators of these two expressions depend only weakly (as O(1)+O(Ny-1/2)) on the number of piriform neurons when this number is significantly greater than 1. The dependences of the denominators on this number thus determine the scaling we are discussing. Because the single readout za and the pair of readouts za(1) and za(2) all have the same statistics, Var(za(1))Var(za(2))=Var(za)2 and thus the denominators of SNR and CC are identical. These two facts explain their similar scaling.

Agreement for Two Readouts

Agreement Aθ is calculated by subtracting the chance probability of agreement β from the raw fraction of similarly-responding read-outs, α, and then normalizing to one, so that Aθ= (α − β)/(1 − β), where β = θ2 + (1 − θ)2. The relationship between Aθ and θ can be seen more easily in the Gaussian approximation of Aθ, which makes explicit the dependence on the correlation σ,

AθGauss=1-(1π1-σ2)∫∫x>θ,y<θdxdyexp[(12σ2-2)(x2-2σxy+y2)].

Agreement for a Population of Many Readouts

Although the case of two readouts is of particular interest because of the application to modeling opposing hemispheres of the same brain, the agreement between more than two readouts is also of interest, with applications to readouts in multiple brain areas or in multiple individuals. We define the agreement of a population of Nz readouts, Aθϕ,Nz, with a threshold criterion ϕ on the fraction of readouts giving the same response (either 1 or 0); in other words, a “yes” choice results when the fraction of readouts with supra-threshold values is greater than a number ϕ. Our original definition of Aθ in the two-readout case can be seen as population agreement with ϕ = 1. As in the two-readout case, we subtract the probability of agreement occurring by chance and normalize to a maximum of 1. More precisely, we subtract the chance probability of agreement β from the raw fraction of similarly-responding readouts, α, and then normalizing to one, so that Aθϕ,Nz=(α-β)/(1-β), where β is the sum of binomial cumulative density functions, β=∑i=0Nz(1-ϕ)(Nz/i)θi(1-θ)Nz-i+∑i=NzϕNz(Nz/i)θi(1-θ)Nz-i. For Nz = 2 and ϕ = 1, this reduces to β = θ2 + (1 − θ)2

To properly compare the required piriform resources for readout populations of varying size, we examine population agreement as a function of the total number of piriform neurons – the number of piriform inputs per readout, multiplied by the number of readout units. This is an assumption of nonoverlapping inputs to the readouts. Population agreement, Aθϕ,Nz, for a range of ϕ values with θ= 0.5 and a population of 10 readouts is qualitatively similar to the original Aθ curves (Figure S2A, left); independent of the value of the choice threshold ϕ, A0.5ϕ,10 requires in excess of 10 million piriform inputs for performance to saturate. With a larger readout population (Nz = 50), a dependence on the value of the choice threshold ϕ emerges, such that A0.5ϕ,50 looks very similar to Aθ when ϕ = 0.55 but requires several orders of magnitude more piriform neurons when ϕ = 0.95 (Figure S2A, right). Intuitively, the appearance of an effect of choice threshold when the readout population is large is a consequence of the ability of a large readout population to detect smaller deviations from chance: low values of the choice threshold ϕ make the population agree more often and therefore require less piriform inputs, but it also makes the chance probability of agreement higher. Only a sufficiently large readout population can distinguish between real and chance agreement and take advantage of this lower threshold. Most importantly, even in this case, performance does not saturate until the number of piriform inputs reaches ~1 million, suggesting that the importance of a large piriform cortex is general.

Piriform Data Analysis

Our analyses are based aggregated data from calcium imaging of layer 2 of piriform cortex expressing GCaMP3, GCaMP5, and Oregon Green 488 BAPTA-1 AM (Stettler and Axel, 2009). The responses of thousands of neurons to odor delivery in freely-breathing anesthetized animals were compared to assess response selectivity. Approximately 6% of piriform neurons respond selectively to odors at concentrations of 1–10 ppm. A small fraction of the neurons (less than 1%) respond to most or all presented odors. These cells may represent a separate functional class and have been excluded from further analyses. Cells responsive to a given odor are found across the piriform with no spatial preference, and the representations of different odors exhibit considerable spatial overlap, in agreement with prior work (Illig and Haberly, 2003; Iurilli and Datta, 2017; Stettler and Axel, 2009). Cells responsive to the ethologically relevant odors TMT and mouse urine were also found to be distributed without spatial preference (Illig and Haberly, 2003; Stettler and Axel, 2009). Cells and odor responses were identified and quantified using custom software written in MATLAB as previously described (Stettler and Axel, 2009).

Nearest-neighbor statistics were calculated using the distance from each cell responsive to a specific odor within a field to the nearest other cell responsive to that odor. We then compared the distribution of these values with the distribution derived from Monte Carlo simulations in which responsive cells were drawn randomly from all the cells in the field, and nearest-neighbor statistics for each cell were then computed in the same manner. In Figure S1F, 5th and 95th percentiles of the Monte Carlo-derived distribution were computed separately for each distance bin.

If responsive cells were clustered, they would tend to have smaller nearest-neighbor separations than randomly distributed cells. At a representative imaging site, the in vivo nearest neighbor distances closely match randomly generated distances (Figure S1F). The site-averaged nearest neighbor distances in observed and randomly shuffled data are closely matched across all imaging sites and odors (Wilcoxon signed-rank test, p > 0.3; Figure S1G), providing no evidence for clustering. We also inspected in vivo piriform odor representations for local patterning by analyzing their spatial wavelengths. The spatial power spectra of in vivo response patterns were compared to spectra from simulations for the same sites. No significant discrepancies between the in vivo and randomly generated spectra are apparent for any spatial period across the sites (Figure S1H).

QUANTIFICATION AND STATISTICAL ANALYSIS

Results for readout accuracy, SNR, correlation, and agreement are averaged over input patterns and over instantiations of the network architecture.

DATA AND SOFTWARE AVAILABILITY

Software was written in MATLAB (https://www.mathworks.com/). Code used to compute quantities presented in this study is available at: https://github.com/schafferEvan/coherentGeneralization.

Supplementary Material

Supplement

Highlights.

  • A random model predicts observed preservation of correlations in piriform responses

  • The model supports consistent agreement about odor quality among individuals

  • Consistent generalization may require the full complement of piriform neurons

Acknowledgments

We thank Mike Shadlen and Torsten Wiesel for constructive skepticism that motivated this work, Toshi Hige and Glenn Turner for generously sharing data, Ashok Litwin-Kumar for helpful comments and suggestions, Phyllis Kisloff and Clayton Eccard for assistance in preparation of the manuscript, and Miriam Gutierrez and Adriana Nemes for general laboratory support. This work was supported by the Howard Hughes Medical Institute (453), the Gatsby Charitable Foundation, NSF NeuroNex Award DBI-1707398, and the Simons Collaboration on the Global Brain (SCGB) Award 328055. E.S.S. was supported by SCGB postdoctoral fellowship 481778.

Footnotes

SUPPLEMENTAL INFORMATION

Supplemental Information includes two figures and can be found with this article online at https://doi.org/10.1016/j.neuron.2018.04.004.

AUTHOR CONTRIBUTIONS

Conceptualization, E.S.S., D.D.S., R.A., and L.F.A.; Software, E.S.S. and D.D.S.; Methodology, E.S.S., D.D.S., and G.B.C.; Formal Analysis, E.S.S., D.D.S., D.K., and L.F.A.; Investigation, E.S.S. and D.D.S.; Data Curation, D.D.S. and D.K.; Writing – Original Draft, E.S.S., D.D.S., R.A., and L.F.A.; Writing – Review & Editing, E.S.S., D.D.S., R.A., and L.F.A.

DECLARATION OF INTERESTS

The authors declare no competing interests.

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