Neural Circuits for Fast Poisson Compressed Sensing in the Olfactory Bulb
Authors: Jacob A Zavatone-Veth
Paul Masset
William Lingxiao Tong
Joseph Zak
Venkatesh N Murthy
Cengiz Pehlevan
Venue: NeurIPS 2023 poster (2023)
Keywords: Olfaction
Bayesian inference
neural circuits
normative models
population geometry
TL;DR: We map an algorithm performing Poisson compressed sensing onto the circuits of the olfactory bulb.

Abstract: Within a single sniff, the mammalian olfactory system can decode the identity and concentration of odorants wafted on turbulent plumes of air. Yet, it must do so given access only to the noisy, dimensionally-reduced representation of the odor world provided by olfactory receptor neurons. As a result, the olfactory system must solve a compressed sensing problem, relying on the fact that only a handful of the millions of possible odorants are present in a given scene. Inspired by this principle, past works have proposed normative compressed sensing models for olfactory decoding. However, these models have not captured the unique anatomy and physiology of the olfactory bulb, nor have they shown that sensing can be achieved within the 100-millisecond timescale of a single sniff. Here, we propose a rate-based Poisson compressed sensing circuit model for the olfactory bulb. This model maps onto the neuron classes of the olfactory bulb, and recapitulates salient features of their connectivity and physiology. For circuit sizes comparable to the human olfactory bulb, we show that this model can accurately detect tens of odors within the timescale of a single sniff. We also show that this model can perform Bayesian posterior sampling for accurate uncertainty estimation. Fast inference is possible only if the geometry of the neural code is chosen to match receptor properties, yielding a distributed neural code that is not axis-aligned to individual odor identities. Our results illustrate how normative modeling can help us map function onto specific neural circuits to generate new hypotheses.

Neural Circuits for Fast Poisson Compressed Sensing in the Olfactory Bulb
Jacob Zavatone-Vethh Paul Masset 1.3, William L. Tong 1,455
Joseph D. Zak
Venkatesh N. Murthy 1.3 Cengiz Pehlevan 1,45f CCenter for Brain Science, DDpartment of Physics, DDepartment of Molecular and Cellular Biology,
*John A. Paulson School of Engineering and Applied Sciences,
KKmmpne Institute for the Study of Natural and Artificial Intelligence,
Harvard University
Cambridge, MA 02138
Department of Biological Sciences,
University of Illinois at Chicago
Chicago, IL 60607
jzavatonevethhg harvard. edu,, paul_masset@fatt haaasatt@fas.hhaarrddd harvard, edu,
vmmurthyōfas harvard. edu, cpehlevan@seas harvard. edu
Abstract
Within a single sniff, the mammalian olfactory system can decode the identity and concentration of odorants wafted on turbulent plumes air. Yet. must do so given access only to the noisy, dimensionally-rrduued representation of the odor world provided by olfactory receptor neurons. As result, the olfactory system must solve compressed sensing problem, relying the fact that only handful of the millions of possible odorants are present in a given scene. Inspired by this principle, past works have proposed normative compressed sensing models for olfactory decoding. However, these models have not captured the unique anatomy and physiology of the olfactory bulb, nor have they shown that sensing can be achieved within the 100-millisecond timescale of a single sniff. Here, we propose rate-based Poisson compressed sensing circuit model for the olfactory bulb. This model maps onto the neuron classes of the olfactory bulb, and recapitulates salient features of their connectivity and physiology. For circuit sizes comparable to the human olfactory bulb, we show that this model can accurately detect tens of odors within the timescale of single sniff. We also show that this model can perform Bayesian posterior sampling for accurate uncertainty estimation. Fast inference is possible only if the geometry of the neural code is chosen match receptor properties, yielding a distributed neural code that is not axis-aligned to individual odor identities. Our results illustrate how normative modeling help map function onto specific neural circuits to generate new hypotheses.
1 Introduction
Sensory systems allow organisms to detect physical signals in their environments, enabling them to maximize fitness by acting adaptively. This experience of the physical environment, also known the Umwell, depends on the sensors and sensory organs each organism [1, 2]. Throughout evolution, organisms have developed specialized sensory mechanisms extract specific information about the
"JAZ-V and PM contributed equally to this work.
t VNM and CP jointly supervised this work.
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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physical world. In vision and audition the most studied sensory modalities in neuroscience-stinn are characterized by intuitive metrics such as orientation or frequency. which have been shown to map onto neural representations from the earliest stages of the sensory systems [3-5]. One can continuously vary the orientation of an object or the pitch of tone and quantify resulting changes in perception and neural representations. From computational point of view, this structure in the representations can be viewed as optimizing the information transfer in the network [6-16].
In contrast, the geometrie structure of the olfactory world is far less clear: How can one 'rotate' a smell? Despite significant effort, attempts to find such structure in olfactory stimuli and link that geometry to maps in olfactory areas have succeeded only in identifying coarse principles for high- level organization, far from the precision of orientation columns or tonotopy in visual and auditory cortices [17-19]. In the absence of geometric intuitions, the principles of compressed sensing (CS) have emerged as an alternative paradigm for understanding olfactory coding [20-27]. This framework provides partial answer to the question of how an organism could identify which of millions of possible odorants are present given the activity of only a few hundred receptor types [28-33]. However, existing CS circuit models do not admit convincing perform fast inference at scale. Indeed, many proposals assume that the presence of each odorant is represented by single, specialized neuron, which is inconsistent with the distributed odor coding observed in vivo [22, 34-36]. This aligned coding does not leverage the geometric structure of the sensory space, which can be defined even in the absence of interpretable dimensions [37-41].
implementations that can
In this paper, we propose Poisson CS model for the mammalian olfactory bulb. Our primary contributions are as follows:
We derive a normative CS circuit model which can be mapped onto the circuits of the bulb ($3). Importantly, this mapping goes beyond basic counting of cell types; it includes detailed biological features like symmetric coupling and state-dependent inhibition ($4).
We show that this model enables fast, accurate inference of odor identity in a biologically reasonable regime where tens of odorants are present in a given scene ($5). This fast inference is enabled by considering the geometry of the olfactory receptor code. This consideration leads to distributed odor coding, resolving a major tension between previous CS circuit models and neural data.
We extend our circuit model to allow Bayesian inference of uncertainty in odor concentrations ($6).
In total, our results demonstrate the importance of considering representational geometry when trying to understand neural coding in the olfactory bulb (OB). Importantly, we show that it is the geometry in the space defined by the receptor affinity (or OSN activation) that controls the speed of inference. This view is distinct from previous geometric theories of olfaction, which have focused on the space of odorants [42, 43]. We propose that thinking in terms of the geometry of OSN coding will allow for deeper understanding of early olfactory processing.
2 Related work and review of the olfactory sensing problem
We begin with a review of the principles of olfactory coding, and of previous CS models for olfactory circuits. The structural logic of early olfactory processing is broadly conserved across the animal kingdom (Fig. IA) [4447], and this distinctive circuit structure is thought to play a key role in the computational function of the olfactory bulb [48-50]. In mammals, volatile odorants are first detected by olfactory receptor (ORs) proteins expressed on the surface of olfactory sensory neurons (OSNs). Each OSN expresses only a single OR type; in humans there are around 300 distinct ORs, in mice around 1000 [45]. Importantly, most ORs have broad affinity profiles, and the OSN code for odor identity is combinatorial [5]]. In contrast to the immune system's 's highly adaptable chemical recognition capabilities arising from somatic recombination [52]. ORs are hard coded into the genome as single genes [53], and therefore can only change over evolutionary timescales [54, 55]. Some adaptation of expression levels across the receptors is possible [56], but the chemical affinity of the receptor array is fixed. OSNs expressing the same OR then converge onto the same glomerulus, synapsing onto the principal projection cells of the olfactory bulb (OB), the mitral and tufted cells. These in turn send signals to olfactory cortical areas. The OB contains several types of inhibitory interneurons, whose computational role remains to be clarified [49]. Importantly, the excitatory mitral and tufted cells are not reciprocally connected across glomeruli. Instead, they are connected through network of inhibitory granule cells, the most numerous cell type in the OB [57].

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Figure 1: Circuit architecture of the mammalian olfactory bulb. A. Outline of the anatomy of OB circuits. B. Fit of the responses of 228 OSN glomerular responses to a panel of 32 odorants. (Gray line, response of single glomeruli, Blue line, Fitted average response.) See Appendix F for details of how these data were collected. C. Affinity matrix generated from the data-driven model ensemble. For illustrative purposes, we use only 30 receptors and 100 odorants; in simulations we use 300 receptors to roughly match humans [45], and 1000 or more odorants.
In our model, we will focus on a particular task: odor component identification within single sniff [58, 59]. This computational problem differs from many experimental tasks focusing on discrimination between two odorants [60], which underlie most scientific work on rodent olfactory decision making [60-62]. Here, the goal is to identify the components of complex olfactory scene [63-66]. Importantly, the limits human performance in this setting remain to our knowledge unknown [67, 68]. To render the problem more tractable, we will make number of simplifications of the anatomy and physiology of the OB. We will not distinguish between mitral and tufted cells, the two classes of projection neurons. Recent works have shed some light the distinct computational roles of these cell types; these distinctions are likely to become important when considering richer environmental dynamics than we do here [69, 70J. We will also ignore the fact that odorant concentrations can vary over many orders of magnitude, and OSN responses to large changes in concentration strongly nonlinear. Here, will focus on concentration changes of one or two orders magnitude. Within such ranges, the responses of OB neurons are well characterized by linear models [71].
the theoretical side, is widely recognized that the tremendous compression of dimensionality inherent in the transformation from odorant molecules to receptor activity means that the olfactory decoding problem is analogous to the one faced compressed sensing (CS) [21-27, 72-74). Classical CS theory shows that sparse high-dimensional signals can be recovered from a small number of random projections [28-33]. Inspired by these results, previous works have used the principles to build circuit models for olfactory coding [21-27, 72-74J. However, these model circuits do not map cleanly onto the neural circuits the OB and the olfactory cortical areas, particularly because they often assume each granule cell encodes exactly one odorant. Moreover, these works usually assume Gaussian noise model for OSNs, which is biologically unrealistic (but see [21, 22]). Instead, OSN activity is better captured by Poisson noise model [75, 76). Some theoretical guarantees for Poisson CS are known, but the situation is less well-understood than the Gaussian case [32, 33, 77, 78).
3 A neural circuit architecture for Poisson compressed sensing
We now derive a normative, rate-based neural circuit model that performs Poisson CS, which in subsequent sections we will map onto the circuitry of the OB. The design of this model will follow general biological principles, without initially drawing on specific knowledge of the OB. With the goal of sensing within single sniff in mind, the circuit's objective is to rapidly infer the odorant concentrations с E Riodor underlying a single, static sample of OSN activity E RRoss For simplicity, we model the mean activity of OSNs linear function of the concentration, with receptor affinity matrix A e noss Xioddo and baseline rate RRoSN, As motivated above,

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we use Poisson noise model for OSN activity given the underlying concentration signal c, and correspondingly a Gamma prior over concentrations with shape a Rodud and scale X Rfoddor
s Poisson(ro + Ac), C Gamma(a,
(I)
Given this likelihood and prior, we construct a neural circuit to compute the maximum a posteriori (MAP) estimate of the concentration using gradient ascent on the log-posterior probability. Here, we sketch the derivation, deferring some details to Appendix B. Our starting point is the gradient ascent equation
c(t) = Ve logp(c|s) s) [ (ru + Ac) (a-1))cc--
where 2 denotes elementwise division and é(!) dt. Here, the estimate e is formally constrained to IR Radda in numerical simulations we will sometimes ignore this constraint. Circuit algorithms of this form were studied in previous work by
et al. [21].
However, in this most basic setup there is a one-to-one mapping between neurons and odorants, which is at variance with our knowledge of biological olfaction ($2). To distribute the code, we instead use a projected setup where the firing rates g R of the neurons are mapped to the concentration estimate e through a matrix r E I Rodde xx;; c(t) = [t) Even if T is non-square in particular, if ng > Rodor so long as IT is desitive-definte and the rates follow the dynamics (t)(() t) = (Ar) [s (ro + A[') 1] + [(a - 1) @ (I'g) ^] for some time constant Tg. the concentration estimates will still converge the MAP. This corresponds preconditioned gradient ascent [39, 79J. Again, we formally require the constraint that c E IR which translates into a constraint on g. Most simply, we may take g E R2 and choose I to be positivity-preservii
These dynamics include two divisive non-linearities, which can be challenging implement in biophysical models of single neurons [80]. Using the approach proposed by Chalk et al. [81], we can linearize the inference by introducing two additional cell types that have as their fixed points the elementwise divisions s 2 (To + Al'g and (a - 1) Ø (Tg). Concretely, we introduce cell types with rates p E ROOSs and I Roddr such that their fixed-point rates for fixed g are p* = s0 (ro+ + Arg)
and z = (a - 1) @ (Tg), respectively. This yields the coupled circuit dynamics
c(t) Tg((), (b)(() = p& (ro + ATg),
(tÉ(() (Ar) (p-1) (t, = a 2
(z A)
(3)
for cell-typ--peecific time constants Tg. Tp. and 72. where O denotes elementwise multiplication. In the limit Tp. Tu < Tg. this circuit will recover the MAP gradient ascent. If Tp and Tz are not infinitely fast relative to Tg. we expect these dynamics approximate the desired dynamics [81] (see Appendix C for a preliminary analysis of the linear stability of the MAP fixed-point). We will test the accuracy of this approximation for biologically-resooabbl time constants using numerical experiments. Moreover, p should formally he constrained to RT"", such that the non-negativity of the target ratio respected. Finally, we note that in the special case a 1 in which the Gamma prior reduces to an exponential prior, the introduction of the cell type z is no longer required.
4 Biological interpretation and predictions of the circuit model
We now argue that the normative model derived in the preceding section can be mapped onto the circuitry of the OB. In particular, though the model was derived based only on general biological principles, its specific features are biologically implementable based on the detailed anatomy and physiology of the OB. In terms of the levels of understanding of neural circuits proposed by David Marr, this is an example of how normative modeling can bridge the gap between algorithmic and mechanistic understanding [82, 83].
As foreshadowed by our notation, we interpret the cell type g as the granule cells of the OB, and the cell type p as the mitral cells, which are projection neurons. Provided that the elements of the matrix AT are non-negative, this interpretation is justified at the coarsest level by the signs with which the two cell types appear in the dynamics: the p neurons excite the g neurons, which in turn inhibit the p neurons. Finally, we interpret the cell type z as representing a form of cortical feedback. In the remainder of this section, we will justify this mapping in detail.
See Appendix A for detailed description of our notational conventions.

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Figure State-dependent inhibition of mitral cells. Upon stimulation, mitral cells exhibit transient burst of activity followed by relaxation to plateau. Darker color indicates stronger stimulation. B. In the circuit for Poisson activation of second mitral cell (MCB))s is gated by the activity of the cell we are recording from (MCA). as observed experimentally by Arevian et al. [95] in their Fig. 2D. This state-dependent gating does not occur in circuit derived from Gaussian noise model (orange line). Rates normalized to the maximal stimulation of the principal cell. Dashed line indicates the unity line. C. Strength of inhibition function of time and of stimulation intensity; c.f. [95] Fig. 3B.
we propose (blue line), the inhibition due to the
4.1
Circuit anatomy: weight transport, dendro-dendritic coupling, and cortical feedback
The first salient feature of the dynamics (3) is that the cell types g and p do not make direct lateral connections amongst themselves. Rather, they connect only indirectly through the neurons of the opposing cell type, matching the connectivity structure of mitral and granule cells (Fig. IA). Moreover, the synaptic weights AT of the connections from g p neurons mirror exactly the weights (AT)" of the connections from to g neurons. Naïvely, this creates a weight transport problem of the form that renders backpropagation biologically implausible: how should the exact transpose of matrix be copied to another synapse [84-86]? However, in the unique case of the OB this does not pose substantial obstacle, as the mitral and granule cells are coupled by dendro-dendritic synapses, meaning that bi-directional connectivity occurs at a single physical locus (Fig. 1) [49, 57, 87].
This interpretation accounts for the interactions between the cell types g and p in the dynamics (3), but what about the cell type z? These cells give direct excitatory input to the granule cells g with contribution to the log-prior gradient. We
weights P and represent the pnucentraaion-depeene can therefore interpret these cells as representing feedback from olfactory cortical areas to the OB, which arrives at the granule cells [88-90]. Though our model can thus flexibly incorporate cortical feedback, for our subsequent simulations we will focus on the simple case which the prior is static and has 1, in which case reduces to exponential and feedback is not explicitly required.
Given this mapping of the cell types of our model to the cell types the bulb, we will henceforth choose the membrane time constants to match experiment, taking R = 20 ms [91] and Ta 30 ms [92]. This matching of timescales is required for our comparison of model inference timescales to the timescale of single sniff to be meaningful.
4.2 Choosing the affinity matrix
In theories of Poisson CS, the optimal sensing matrix A one that has columns that are in some precise sense as orthogonal as possible, so that acts as an approximate isometry [27-33]. However, biologically, the olfactory system is not free to choose optimal sensors. Rather, the affinity profiles of each receptor are dictated by biophysics and by evolutionary history [53, 54, 93]. To build a more realistic model for OSN sensing, we therefore turn to biological data. Using two-phor two-photon calcium imaging, we recorded the responses of 228 mouse OSNs glomeruli to 32 odorants (see Appendix for details; these data were previously published in [94]). In Fig. IB, we show that the distribution of responses well-fit by Gamma distribution with shape 0.37 and scale 0.36 (Appendix F). We then define our ensemble of sensing matrices A by drawing their elements as independently and identically distributed Gammaa0.37 0.36) random variables. example matrix drawn from this ensemble is shown in Fig. IC.
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4.3 Divisive predictive coding by mitral cells
In our model (3), odorant concentration estimates are decoded from granule cell activity. This feature is shared with previous Gaussian CS models of olfactory coding [20, 24, 25J. Our model. however, matches biology more closely than these previous works because it allows for a distributed code rather than assuming that each granule cell codes for single odorant.
What, then, is the functional role of the mitral cells in our model? We can interpret their dynamies as implementing form of predictive coding, in which they are trying to cancel their input by the current prediction. Because the mitral cell activity converges to the ratio of their input to the prediction, this a divisive form of predictive coding [81], In contrast, Gaussian noise model gives a subtractive form of predictive coding which the activity converges to the difference between input and prediction (Appendix D) [15]. In Fig. 2A, we show example timecourses of model mitral cell activity following the onset of stimulus. Consistent with experimentally measurrd responses [96, 97], a sharp transient response at the onset of stimulation is followed by decay to low level of tonic activity (Fig, 2A).
4.4 State-dependent inhibition of mitral cells
A salient feature of our circuit model is that the inhibition from the granule cells onto mitral cell is gated by the activity of the mitral cell itself (3). This dependepnnd inhibition is reminiscent of in vitro experiments showing that granule cell mediated inhibition is activity dependent [95]. In these experiments, Arevian et al. measured the activity of a primary mitral cell (MCa) while increasing its level of stimulation under two conditions. In the first condition, other cells in the eircuit are being stimulated. In the second condition, they also activate another mitral cell (MCB). The activation
of the second mitral cell leads to the activation of granule cells connected to both mitral cells and a reduction in the firing evoked by stimulation of the primary cell alone. Strikingly, these authors showed that this inhibition dependent on the activity of the primary cell [95].
Here, we show that our proposed circuit reproduces these observations of state-dependent inhibition, and that they do not arise in
circuit for Gaussian noise model. To model Arevian et al. [95]'s in vitro experiments, we simulated a reduced circuit with 2 mitral cells and 10 granule cells. We stimulated MCA with input $A E 400] while toggling stimulation s = 80 of the second mitral cell MC As observed experimentally, when MCB B is activated, the inhibition of the primary mitral cell MCA is state dependent (Fig. 2B). To show that this effect arises from our Poisson circuit, we build a similar inference circuit with a Gaussian noise model (Appendix D). Under similar conditions, the inhibition in the Gaussian circuit is independent of the activity of the primary mitral cell MCA (Fig. 2B). Furthermore, the dynamics of the relative inhibition qualitatively recapitulate those observed experimentally, with sustained inhibition throughout the stimulation period at the stimulation level of MCA leading to maximal inhibition and inhibition followed by relaxation for stronger stimulation levels of MCA (Fig. 2C and see [95] Fig. 3b).
off the
5 Geometry, speed, and capacity
We have argued that the model introduced in $3 could be implemented biologically, but can it perform at scale? Concretely, can a circuit of this architecture with neuron counts comparable to the human OB correctly identify which among a large set of odorants are present in given scene? This is precisely the question of the capacity of the CS algorithm [98, 99]. In Fig. 3A, we present our algorithm with scenes composed of varying numbers of randomly-selccte panel of 1000, which for simplicity we take be at the same concentration. We first ask how many odorants can be reliably detected within a single sniff, i.e., 200 ms. To convert MAP concentration estimates into presence estimates, we simply binarize the estimated concentrations based on whether they are larger than half of the true odorant concentration. The ability of the one-to-one code to successfully detect odorants falls off rapidly, with the detection fraction falling below one-half even if only handful of odorants are present (Fig. 3A).1
odorants out of a
The limited capacity of the one-to-one code can be overcome by distributing the code in way that takes into account the geometry of the sensing problem. Here, the information geometry of the
We remark that, with a one-to-one one code, our model is identical to that proposed by Grabska- Barwińiska et al. [21] except for the introduction of the granule cells.

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Figure 3: Fast detection of many odorants. A. Fraction of odorants correctly detected within 100 ms (left), 200 ms (center), and s (right) after odorant onset as a function the number of odorants present, for models with one-to-one, naively distributed, and geometry-aware codes. Here, consider a repertoire of 1000 possible odorants. B. Heatmap with overlaid smoothed contours of correct detection fraction a function of number of present odorants and time window for models with one-to-one (left), naïvely distributed (center), and geometry-aware (right) codes. As in A, repertoire of 1000 possible odorants is used. C. Threshold number of odorants for which half can be correctly detected as a function of total repertoire size within 100 ms (left), 200 ms (center), and 1 (right) after odorant onset. See also Supp. Figs. G.1 and G.2 for versions of panels A and B with varying repertoire sizes, from which the capacities shown here are derived. See Appendix G for details of numerical methods. Shaded patches show 1.99 SEM over realizations throughout.
problem is governed by the sensing matrix A, which introduces correlations in the input signals to mitral cells because the off-diagonal components of A A are non-negligible. To counteract these detrimental correlations, we can choose a readout matrix such that ITT Ps (ATA) A)* up to constants of proportionality. where we must take a pseudoinverse because T A is highly rank- deficient (see Appendix G for details). Geometrically, this corresponds an approximation of natural gradient descent, in which we use the Gauss-Newton Newton matrix instead of the Fisher information matrix because the latter is state-dependent for Poisson likelihoods [37-40]. As control, also consider naively distributed code with IT R Inddar Distributing the code, even without accounting for the geometry of inference, markedly improves the single-sniff capacity to around 10-20, and taking into account the geometry produces a further improvement (Fig. 3A).
To gain more granular view of how tuned geometry enables faster inference, in Fig. 3B we test the three models' detection capabilities at sub-sniff resolution. We can see that at long times-around second after odorant onset the naïvely distributed and geometry-aware codes achieve similar capacities of around 50-60 odorants. However, the geometry-aware code reaches this detection capacity within single sniff, whereas the naïvely distributed code requires the full second of

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processing time. The detection capacity of the one-to-one code reaches only around 20 odorants after 1 second, and even then does not appear to have reached its asymptote. These results illustrate two important conceptual points: First, when the strength of individual synapses is bounded, distributed coding can speed up dynamies by increasing the effective total input to a given neuron. Second, given sensing matrix that induces strong correlations, geometry-aware distributed coding can accelerate inference by counteracting that detrimental coupling.
These tests show show that our model can reliably detect tens of odorants from a panel of thousands of possible odorants, but do not probe how the model's capacity scales to larger odor spaces. While the true dimensionality of odor space remains unknown [100, 101]. there may be orders of magnitude more than thousands of possible odorants. As an upper bound, there are on the order of millions of known volatile compounds that are plausibly odorous [102]. Thus. it is important to determine how odor spaces. From the literature on compressed sensing logarithmically with increasing Modor [28-33]. As a first step, Supp. Figs. G. and G.2 we reproduce Fig. 3A-B for between 500 and 8000 possible odorants, showing that performance does indeed drop off slowly with increasing odor space dimension. To get a more precise estimate of how performance scales with repertoirse size, in Fig. 3C we plot the threshold number of odorants for which half can be reliably detected as a function of the repertoire size, showing that for the geometry soometry-aware aware code decays only bit faster than logarithmically. Our ability to simulate larger systems was limited by computational resources. This limitation is present in previous works, and the repertoires tested here comparable to or substantially larger
our model scales to more realistically-sizze performance bounds, we expect the threshold sparsity to decay slowly-rough
used in past studies [21, 22, 24, 26, 72].
Fast sampling for uncertainty estimation
Thus far, we have focused on circuit that performs MAP estimation of odorant concentrations. However, to successfully navigate a dynamic, noisy world, animals must estimate sensory uncertainty at the timescale of perception [38, 103-107]. Fortunately, our circuit model can be easily extended to perform Langevin sampling of the full posterior distribution, allowing for uncertainty estimation while maintaining its attractive structural features. Appendix B, we provide a detailed derivation of model that implements Langevin sampling through the granule cells for a Poisson likelihood and Gamma prior. This yields circuit that identical to the MAP estimation eircuit introduced in $3 up to the addition of Gaussian noise to the granule cell dynamics:
c(t) = [g(() (tt(t) = - - P (TO + Arg)
Ty&(()= (AT) T(p 1)++ (z- X) + E(t) (t,((t) a - 2
Here, E(t) is a vector of ng independent zero-mean Gaussian noise processes with covariance BiC, (t)) = 6 and once again the rates and should in principle be constrained to be non-negative. In this case, the readout matrix I both preconditions the effective gradient force and shapes the structure of the effective noise I'g, allowing us to mold the geometry of the sampling manifold [37, 40]. By using a projected readout, we maintain the independence of the noise processes for different neurons. This both allows us to have many independent samplers if fgg > Modor and important for biological realism if we interpret the sampling noise as resulting from fluctuations in membrane potential due to synaptic noise [108].
As test of how this sampling circuit performs, we consider simple setup in which one set of odorants appears at a low concentration, and then a second set of odorants appear at a higher concentration while the low odorants are still present. This setup tests both the circuit's ability to converge rapidly enough to give accurate posterior samples within a single sniff, and its ability to correctly infer odorant concentrations even in the presence of distractors [63, 66]. In Fig. 4, we show that circuits with one-to-one or naively distributed codes do not give accurate estimates of the concentration mean within 200 ms, while tuning the geometry to match the receptor affinities enables fast convergence. All three circuits overestimate the posterior variance at short times, consistent with what one would expect for unadjusted Langevin samplers [109], but the geometry-aware model's estimate decays most rapidly towards the target. Therefore, when the synaptic weights are tuned, our circuit model can enable fast, robust estimation of concentration statistics, even in the presence of distractors.
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Figure 4: Fast uncertainty estimation using Langevin sampling of the posterior. Here, use simple concentration estimation task in which 5 randomly-selected 'low' odorants out of a panel of 1000 appear at concentration 10 at time s. and then a further 5 randomly-selected 'high' odorants appear at concentration 40 at time 1 s. A. Smoothed timeseries of instantaneous concentration estimates for low, high, and background odorants, for models with one-to-one (top), naïvely distributed (middle), and geometry-ave aware (bottom) codes. Background odorant estimates are shown as mean + standard deviation over odorants. Dashed lines show true concentrations over time. B. Cumulative estimates of concentration mean for low (top) and high (bottom) odorants after the onset of the low odorants for codes. Black lines indicate baseline estimates
one-to-one, naively distributed, and geometry-awe of the posterior mean. Thick colored lines indicate means over odorants. C. As in B, but for the estimated variance. D. As in B, but after the onset of high odorants. E. As in C, but after the onset of high odorants. See Appendix G for details of our numerical methods, and for individual-odor traces.
7 Discussion
In this paper, we have derived a novel, minimal rate model for fast Bayesian inference early olfaction. Unlike peviously-proposee algorithms for CS in olfaction, this model has a clear mapping onto the circuits of the mammalian OB. We showed that this model successfully performs odorant identification across biologically-relevar therefore exemplifies how normative approaches can blur the lines between algorithmic and mecha- nistic understanding of neural circuits [82, 83]. We now conclude by discussing possible avenues for future inquiry, as well as some of the limitations of our work.
range of scene sparsities and circuit sizes. This model
One limitation of our simulations is that we have chosen to distribute the neural code randomly, and have allowed for negative entries in the mitral- granule synaptic weight matrix AT (see methods in Appendix G). These features are not entirely biologically satisfactory. However, our model only captures the mitral cells and granule cells, overlooking a number of other inhibitory cell types that could contribute to solving this problem, e.g., through feedforward inhibition onto mitral cells or lateral inhibition across granule cells [48, 49]. Fundamentally, negative values in the geometry-aware decoding matrix I' arise as although the affinity matrix is positive, its inverse will contain negative elements. moologiallly-pausiiie
realization of the geometry-aware code through the introduction of additional cell types could be achieved by decomposing the inverse into several components, yielding sparse, consistently-ste signed connectivity [110] As a first step, in Supp. Fig. G.4 we show that similar performance to Fig. 4 can be achieved using sparse non-negative randomly distributed code. As result, one objective for future work will be to develop better models for the decoding matrix that result in more realistic connectivity.
Though our model captures two of the interesting features of the anatomy and physiology of the OB symmetric dendrodendritic coupling and state-dependent inhibition of mitral thlse

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many biological details which we have not addressed. First, our linear model for OSN mean firing
neglects receptor antagonism. gain control, and other nonlinear effects [94, 111, 112], which are known to affect the performance of Gaussian CS models for olfaction [26, 113]. Second, our models are rate-based, while neurons in the OB spike. In spiking implementations of sampling networks, the noise is not uncorrelated across neurons, complicating their biological interpretation [38, 1 114]. Constructing models that capture these richly nonlinear effects will be an important objective for future work. A first step towards such nonlinear model would be to build a spiking network that approximates the rate-based models considered here, which could be accomplished using the efficient balanced network formalism for distributed spiking networks [38]. Another step would be to add a Hill function nonlinearity to the OSN model to approximate competitive binding, as studied for Gaussian compressed sensing by Qin et al. [26]. One challenge in constructing models that incorporate additional nonlinearity is that the simple linear strategy for distributing the code used here may no longer be directly applicable. In Appendix E,, we illustrate this obstacle for the relatively simple case of a model with the same linear OSNs and Poisson likelihood but an Lo instead of Gamma prior, building on recent work on circuits for Gaussian CS with Lo priors [115].
A closely related point is that we model the weights of the synapses between mitral and granule cells as fixed, and do not consider synaptic plasticity. In particular, we assume that they are tuned to the statistics of the receptor affinities without specifying a mechanism by which this tuning could take place. In biology, receptor abundances and other OSN properties display setivity-dependeed adaptation over long timescales, meaning that the optimal tuning is unlikely to be static [56, 1 116]. Some past works have sought to incorporate plasticity of the mitral-gramiun cell synapses into decoding models [24, 1 17], tying into a larger body of research on how plasticity can enable flexible feature extraction in olfaction [1 (118, 119]. This learning should lead measurable changes in the population response to odorant panels, which our model predicts should be linked in a precise way to account for receptor-induced correlations in the responses to the most frequently present odorants. Experimentally characterizing and carefully modeling these changes in responses across timescales will be an interesting avenue for future work [120]. Experimental techniques to probe these ideas at the neural and behavioral level have recently been proposed [121, 122] which allow more precise control of stimulus and subjective geometry.
Though the circuit model derived in $3 incorporates a general Gamma prior represented by cortical feedback, our simulations focus on the special case in which the prior reduces to an exponential, in which the feedback neurons are not needed. Future work will therefore be required to carefully probe the effect of incorporating Gamma prior with non-unit shape and to dissect the structure of the resulting modeled cortical feedback. More generally, it will be interesting to extend our framework to incorporate data-adaptive priors. Importantly, the stimuli used in this paper constitute an extremely impoverished model for the richness of the true odor world; we do not account for its rich dynamical structure and co-occurence statistics. Adaptive priors as encoded by cortical feedback would allow circuits to leverage this structure, enabling faster and more accurate inference [88-90, 123, 124].
We conclude by noting that our work provides example of how distributed coding can lead to faster inference than axis-aligned disentangled coding. In recent years, the question of when axis- aligned coding is optimal has attracted significant attention in machine learning and neuroscience [125-135]. Much of this work focuses on the question of when axis-aligned codes are optimal for energy efficiency or for generalization, whereas here we focus on the question of which code yields the fastest inference dynamics. These ideas are one example of the broader question of how agents and algorithms should leverage the rich geometry of the natural world to enable fast, robust learning and inference [136]. We believe that investigating how task demands and biological constraints affect the optimal representational geometry for that task a promising avenue for illuminating neural information processing in brains and machines [41, 136, 137].
10

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Acknowledgments and Disclosure of Funding
We thank Naoki Hiratani, Shanshan Qin, and Vikrant Kapoor for useful discussions. This work was supported by NSF grants DMMS-1141577 and CAREER IIS-2239780 to CP. NIH grants ROIDC017311 and ROI DC016288 to VNM, and NTT Research award A47994 to VNM. support from Sloan Research Fellowship. PM was partially supported by grant from the Harvard Mind Brain Behavior Interfaculty Initiative. JDZ was partially supported by NIH grant K99DC017754. This work has been made possible in part by gift from the Chan Zuckerberg Initiative Foundation to establish the Kempner Institute for the Study of Natural and Artificial Intelligence. A subset of the the FASRC cluster supported by the FAS Division of Science
received additional
computations in this paper were Research Computing Group at Harvard University.
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A Notational conventions
In this Appendix, we define the notational conventions used throughout the paper. We write R {x E R" ........ Fn 2 0f for the non-negative reals, and N {0, .....) . for the natural numbers. For vector X E IR,, we write
Poisson X)
(A.1)
if the probability mass function of x E N is
As_ p(x)
(A.2)
For scalars a > and > O, we write that a vector x E Ry is distributed as
x ~ Gamma(a, X)
(A.3)
its probability density function
zj
p(x)
Au
(A.4)
I(a)
Similarly, for vectors a; RT, we write
Gamma X)
(A.5)
if
7 0--1 p(x) I(ay)
A52s
(A.6)
For vectors x, y E RR, we write x @ y for their Hadamard (elementwise) product
② y); x;yi
(A.7)
and x 2 y for their elementwise ratio
E2
(x 2 y);
(A.8)
y/1
B Detailed derivation of circuit model
In this Appendix, we give a detailed derivation of the circuit models introduced in Sections 3 and of the main text. Here, focus on the setting the sampling circuit; the circuit algorithm to compute the MAP introduced in Section 3 of the main text can be recovered at each step by dropping the additive noise terms from the dynamics.
We recall that our goal is to sample the posterior p(c s) over concentrations c given OSN activity s, for Poisson likelihood
c ~ Poisson(ro + Ac)
(B.1)
and a Gamma prior
Gamma(a: A).
(B.2)
Here, we use the conventions of Appendix A for Poisson and Gamma random vectors. Discarding normalization constants that do not depend on c, the density of the posterior is then
p(c s) p(s c)p(c)
(B.3)
Pader II
(ro + Ac) Are)
cj - Aye
(B.4)
S1

---

We will sample from this distribution using Langevin dynamics, without explicitly constraining the concentration estimate be non non-negative [21. 22]. This corresponds to the stochastic dynamics
c(t) = V. p(c s) + (7),,
(B.5)
where mtt) is an dimenssonal zero-mean Gaussian noise process with (t)n (t)) 26zz.0(t Ó(t t'). The gradient of the log- pog-posterior is
Ø Ologp(cls)
NOEN
Σ
Sk
-1) 1) Akj + 0j ~A3.
(B.6)
dey
(ro + Ac)
or, in vector notation,
Ve log s) A I + 1a-
(B.7)
The Langevin dynamics then become
c(t) = A Is @ (ro + Ac) - (a- + (t)).
(B88)
This is closely related to the Langevin sampling algorithm introduced in equation (3.9) of Grabska- Barwinska et al. [21], but here we are not scparately inferring odor concentration and odor presence. We note that, for oj > the term (a 1) gives force that diverges for small cj-
However, this setup single neurons encode single odors. We instead want allow for distributed code, where the population of neurons responsible for sampling may not directly encode single odors. To distribute the code, we follow previous work by Masset et al. [38] in using the "complete recipe" for stochastic gradient MCMC [37]. From the "complete recipe", we know that the dynamics
Tg Ttt(t) r T V. log s) + re(t),
(B.9)
for (expectantly named) time constant Tx and any (potentially non-square) matrix Moder Xng such that rT is positive-definite, will have as their stationary distribution the posterior p(c s). here, we have replaced the Yodor -dimensional noise process ()) with an ny-dimensional noise process E(t) with zero mean and covariance
(t)E = 2Tz t¹).
(B.10)
Then, write the concentration estimate
c(t) = rg(t)
(B.11)
where the activity of the neurons g 2 follows the dynamics
(t) = r logg s)
+ E(t).
(B.12)
c=
Using the gradient of the pog-posterior computed above, we then have
(tg(t) = (Ar) 2 (ro + Ar') - -1]+1 [(a - 1) Ø (Tg) - A E(t).
(B.13)
These dynamics include two divisive non- non-lincarities, which can be complex to implement in biophysi- cal models of single neurons [80]. Using the approach proposed in Chalk et al. [81], we can linearize the inference by introducing two additional cell types that have as their fixed points the elementwise divisions s @ (ro + ATg) and (a 1) Ø (Tg). We then introduce a population p of nOSN neurons, with dynamics
Pt s p (ro + Arg),
(B.14)
such that the fixed point is
p' = (ro ATg). We finally introduce a third population z of nodor neurons, with dynamics
(B.15)
T2i(t) = a 1- (Tg),
(B.16)
such that the fixed point is
z' (a= (Tg).
(B.17)
S2

---

To be more precise, the cell types p and z compute the desired elementwise divisions in the pseudo-
steady-state regime Tp. Tx + 0.
Putting everything together, this gives the circuit dynamics
c(t) [t(()
(B.18)
(2- [t) - (Ar) ( E(t)
(B.19)
Ptp(t)
② (ro A[g)
(B.20)
Tz²(() = a c,
(B.21)
where we recall that the covariance of the zero-mean Gaussian noise process E(t) is
(t),(t)) = 2Tg 6t - t').
(B.22)
We note that the two constant terms in the dynamics of g can be grouped into overall leak term -PT (A A + A). As detailed in the main text, we interpret p as M/T cells, g as granule cells, and z cortical feedback.
C Stability of the MAP fixed point
In this Appendix, we aim to get some understanding for how the introduction of the cell type p to represent the elementwise division affects the inference, particularly when Tp is comparable to Tg- For simplicity, we specialize to the case of an exponential prior (i.e., we set 1). in which case the general circuit (3) simplifies
c(t) rg(()
(tg(() (AT) (p- 1) A
(C.1)
P(t() = s - p @ (ro + ATg).
C.1 Analysis of a two-cell circuit
To build intuition, we first consider circuit with Aodor NOSN ng 1:
c(t) = 7g(()
dg a> (p 1) - 7d
Tx dl
(C.2)
Tp dp p(ro + arg).
In this case, it is useful to non-dimenssioaalize the system. Re-scale time
(C.3)
Tk
and define the dimensionless parameters and input
f Tp
(C.4)
Tg
70 =
(C.5)
Ā
(C.6)
(C.7)
Then, the dynamics
dg
p -
(C.8)
di
dp
3 p(ño + g).
(C.9)
di
S3

---

These dynamics have single fixed point at
8
g+
50
(C.10)
1 +
p-
+ 2.
(C.11)
Linearizing about this fixed point, we have
d
8g 6g
6g
M
(C.12)
di
6g
for
(
0 x) -(1+
F
M
-(1 + (1+1))115 2)-18)
(C.13)
The cigenvalues of M are easily computed as
A+ (M) -3 + 82 - 45(1 + 2)3
(C.14)
2F + X)
For any positive 7 and non-negative A, we have
-3 + Re
52 - 4F 2)3
Re ReA1(M)
(C.15)
27(1 + X)
v 52 4F + 2)3
8² -F
2F(1 + X)
(C.16)
otherwise,
27(1 + X)
hence it is easy to that the real parts of both of these eigenvalues strictly negative so long as
3 > 0, meaning that the system is stable. If 3 = O-which should be exceedingly rare if the OSNs
have a baseline rate-then
+ Á
A (M)
(M)
+28
(C.17)
6=0
and there can be oscillations. Therefore, a linear stability analysis suggests that the MAP fixed point of this two-cell circuit should be stable even for large F in the presence of non-zero input, though we expect the relaxation timescales to grow as F becomes larger.
C.2 Analysis of the full circuit
We now consider the full MAP circuit
c(t) Tg(()
Ttt(() (AT) T (p- 1) -
(C.18)
p() - p @ (ro + Arg).
We assume that nOSN < Rodoo < ng and that AT is of full row rank, i.c., it has rank nOSN- We recall that E IR*s and p E RT Therefore, the dynamics of span only an Rodor bodor-dimensiona subspace, and, in particular, the p-dependent term affects only an noSN dimeeniinsii subspace.
We start by observing that the dynamics of p depend on g only through
ATg.
(C.
Importantly, if AT is
then q E gRoSN Then, we have the closed dynamics
ワ = (ATT A (p - 1) A
(C.20)
T = O (ro 9)
S4

---

for (q.p) E R The fixed point of this system is of course determined by the conditions
d(q.p)/d p// O, which gives
ArT A (p ATT X p (ro +-" =s
(C.21)
(C.22)
subject to the non-negativity constraints. By our assumptions on the rank AT, the symmetric matrix AIT A is positive-definit and thus can be inverted to solve the first condition for p":
p' = 1+ (ATT A A -AA-AAAT' X.
(C.23)
For the second condition to be satisfied, we can that the elements of p must be strictly positive at the fixed point, which gives a self-connistency condition on A, T, and X. Assuming that this holds, then have
q' 2 p" ro.
(C.24)
which again gives a self-consistency condition as non-negativity is required.
Assuming these conditions hold, we can then linearize the dynamics about the fixed point. For convenience, we non-dim non-dimensionalize time through
(C.25)
g
and
F = Tp
(C.26)
TR
which yields the linearized dynamics
d
pq 6p/
(óq)
M
(C.27)
di
6p
for
0
(ATT AT)
M
(C.28)
diag(p")
7 diag(ro + q*))
0
(APTT AT)
(C.29)
diag(p")
diag(s p²))
Using the fact that the diagonal matrices diag(p") and diag(s 2 p*) commute, the characteristic polynomial of M is
(APITA) det(MM (Al2nos)) det 7 diag(p") #- diag(s 2 p²) JIROSN
InoS
(C.30)
det(jt Imoax + μē diag(s @ p²) + (ATTT A diag(("))) (C.31)
One case that is particularly easy to solve is when the symmetric positive-defintte matrix ATT A in fact diagonal, with positive diagonal entries aj. Then, we have
BOBN
8j
asp; 7
dct(M - -JIZnoss
1 +
+
(C.32)
Tpj
meaning that the eigenvalues of M are
2
1
8j
8j
a
+
(C.33)
Fp'
7
which under the given assumptions always have strictly positive real part for non-negative inputs.
S5

---

Now, more generally, assume that M is diagonalizable, and let m be un-normalized eigenvector of M with eigenvalue p. Then, writing
m
u
(C.34)
for ChoaN, the eigenvector condition
Mm pm
(C.35)
implies that u and v satisfy
(ATT A JJv== pm
(C.36)
Tg
1 diag(p u diag(s @ p')v pvv
1
(C.37)
Tp
Tp
Assuming that μ is non-zero, we can solve the first equation for u and then substitute the result into the second to obtain a quadratic eigenproblem for j and v:
1 diag(s 2 p')v+ Tp
1
+
AAAATTATYe Avv= 0.
(C.38)
TpTs
We will not attempt to solve this cigenproblem, but will instead attempt to extract information about possible values of μ for a fixed v. Suppose (without loss of generality given the assumption that is nonzero, as otherwise we may divide by its norm) that v is a unit vector. vector.Then, Then, acting with vf from the left, we have
1²2 +
+ = 0. Tp
(C.39)
TpTs
where we define the coefficients
a vl diag(s 2 p')vv
(C.40)
and
= v diag(p (iiag(""((AA)))) (APT) A J)..
(C.41)
So long and p' are positive, a is real and positive. Let
== (APT AT) Arr X
(C.42)
such that
=1=1+À
(C.43)
By assumption, the elements of À are strictly greater than -1 Then,
Re(b) = yt diag(p") AT) + (AITTA AT) diag(p")
(C.44)
2
- AT) vt diag (ATTT AT) (ATTTA AT) diag( X)
V.
(C.45)
while
Im(b) = diag( (p' (ATT" A - (ATTT A diag(p"))
(C.46)
2i
diag (AATT A
(AITTA A diag(A) X)
(C.47)
2i
Then, solving the quadratic equation for μ4, we have
a + Va
47b
(C.48)
Tp
2
S6

---

for
Tp 7g
(C.49)
This gives
Re(n)) =
a
+
Re V a² - 4Fb
(C.50)
2Tp
2Tp
a -
Via² - 48 Ro(2))² Ro( + 16 (m) +
47 Re(b)
(C.51)
2Tp
2Tp
2
If [m) 0, then we are the same situation as we were for the two-neuron cireuit, and the fixed
point is thus stable.
More generally, for the system to be stable we want to have
v aa2 47 Re(b)) + 167 Im (m)² +
4F Re(b)
a>>
(C.52)
For either sign of a² 4F Re(b), one can show that this holds provided that Re(b)
(C.53)
and
Iml < Re(b).
(C.54)
This amounts to a combined condition on the effective prior strength À and the relative time constants. Therefore, for sufficiently small X, we have stability for broad range of Tp-
How can we go from stability of the (q, p) circuit stability of the full circuit? Heuristically, this follows by decomposing the dynamics of g in terms of the different subspaces. We will not attempt do this rigorously, as our goal only get a rough sense of how the system should behave. By the fact that their dimensions are strictly ordered, we know that span(I D span[(Ar) We can first exclude components outside Sp as they will be unchanged by the dynamics and do not affect the concentration estimate. Recalling the non-negativity constraint, components in TT \ span (ppr)(AA) J will decay linearly to zero. Then, the components in span[(A]) should be controllable using the argument for the (q. p) space. Giving a fully rigorous analysis of the conditions under which the MAP fixed point is stable will be an interesting avenue for future investigation.
D Derivation for the Gaussian circuit model
To compare the result of the state dependent inhibition due estimating the MAP for Poisson noise model with the fixed inhibition resulting from Gaussian noise we build an alternative circuit model. This circuit still shares the separation into two cell types, but the granule cells g converge on the solution for a Gaussian noise model.
Our starting point is isotropic Gaussian likelihood
s s/c ~ N(r, + Ac, o d'Inoss)
(D.1)
variance o², and an exponential prior
Exp(A).
(D.2)
Gradient ascent on the resulting log- log-posterior over e leads to the dynamics:
é(t)
A A (ro Ac)] X.
(D.3)
Distributing the code such that c(t) Ig and splitting the inference as we did in the Poisson case gives the circuit dynamics
c(() = rg(()
Ttt(() o2 (AT) p T
(D.4)
P() - (ro Arg).
Unlike in the Poisson case, the inhibition onto the projection cells (mitral cells) is not gated by their activity, leading to the fixed offset shown in Fig, 2B.
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---

E Extending the sampling circuit to incorporate an Lo prior
In this Appendix, consider the possibility of extending our circuit model to incorporate Lo spike-and-slad prior
p(c) = we Ar; + (1 (1- (1-w))((a),
(E.1)
where is the probability that the odor is present and À is the rate of the exponential prior on concentrations given that the odor is present. To sample from the resulting posterior using Langevin dynamics, will follow the approach of Fang et al. [115], who developed Langevin algorithm to perform sparse coding with this prior given a Gaussian likelihood. This approach only works in the sampling setting; it cannot be applied to MAP estimation because the Dirac mass at C: 0 means that the MAP estimate will always vanish.
In this approach, we define an auxiliary variable u that is mapped to concentration estimates c via element-wise soft thresholding:
= f(u),
(E.2)
where
u < 4g
[(u)
(E.3)
too
u 2 Ug
is the soft-thresholding function for threshold
log w
(E.4)
We then posit the following Langevin dynamics for u, given an observation S:
ill)= V log c)
O([ul un) À siggn()) + (()
(E.5)
e=f([m))
with no constraint on the sign of u, where (7( is white Gaussian noise process. Here, all nonlinearities are applied element-wise, and uo = ug vector with all elements equal to the threshold wp. Fang et al. [115] argue that the stationary distribution induced on с = f(u) by these dynamics should be the desired posterior with Lo prior.
Using the Poisson likelihood gradient as computed before, we have
il)) (A T [s @ (ro ((ul))) - 1]} @ O([u] un) X sign(u) + (()
(E.6)
At this stage, we can see that the soft thresholding has picked out preferred basis. If we apply the complete recipe in way analogous what did in Appendix B by writing
u u= rg.
(E.7)
we will have
Tyg(()= -T (A (s (ru + +f((gg)) 11} o o([ggl - sign + +
(E.8)
where E(t) is zero-mean Gaussian noise process with covariance
(t), = 5t - t').
(E.9)
Here, we cannot simply regroup terms; if we introduce an additional cell type p before compute the division we will have a sort of effective weight matrix
IIaAA - uo)
(E.10)
for the p-to-g connections, and an entirely different coupling Ag(( for the g-to-p connections. Concretely, introducing p as before, we have the system
c(t) f(Tg(t))
(E.11)
(t))) T A (p 1)] O ( uo) Asign(Tgg dt + E(I)
(E.
(tt(t = s p 3 Fo + Ag()
(F.13)
This issue illustrates the limitations of the simple, linear approach to distributing the neural code used in Appendix In particular, the nonlinearity some sense picks out a preferred basis, meaning that we can no longer perform simple linear change of coordinates to distribute the code.
S8

---

F
Experimental details and affinity matrix fitting
Here we briefty describe the experimental procedures used to collect the data used to fit the parameters of the affinity matrix A in Fig. 1. The full experimental methods are described in the paper that first presented the data, Zak et al. [94]. All the experiments were performed in accordance with the guidelines set by the National Institutes of Health and approved by the Institutional Animal Care and Use Committee at Harvard University.
F.1 In vivo recordings from mouse OB
Adult (> 8 weeks) OMP-GCaMP3 mice of both sexes were used in this study, A craniotomy was performed to provide optical access to olfactory sensory neuron axon terminals in both olfactory bulbs. A custom-built two-photon microscope was used for in vivo imaging. Images were acquired 16-bit resolution and 4-8 frames/s. The pixel size was 0.6 pm and the fields of view were 720 X 720 μim. Monomolecular odorants (Allyl butyrate, Ethyl valerate, Methyl tiglate, and Isobutyl propionate) were used as stimuli and delivered by 16-channel olfactometer controlled by custom- written software in LabWiew. For the odorant concentration series, the initial odorant concentration was between 0.08% 30%(v/v) in mineral oil and further diluted times with air. The relative odorant concentrations were measured by photoionization detector, then normalized to the largest detected signal for each odorant. For all experiments, the airflow to the animal was held constant at 100mLL min, and odorants delivered 2-6 times in pscudorandom order.
injected into carrier stream. Each odorant concentration was
Images were processed using both custom and available MATLAB scripts. Motion artifact compen- sation and denoising were done using NoRMcorre [138]. The AF/F signal was calculated by finding the peak signal following odorant onset and averaging with the two adjacent points. To account for changes respiration and anesthesia depth, correlated variability was corrected for [139]. Thresholds for classifying responding ROIs were determined from noise distribution of blank (no odorant) trials from which three standard deviations were used for responses.
F.2 Gamma distribution fitting
In order to fit the affinity matrix A to the experimentally recorded data, we normalized the response of each glomerulus to its maximum response across the panel of 32 odorants. We then vectorized the resulting matrix and fitted a Gamma distribution using the gamfit it function in MATLAB. As mentioned in the main text, this resulted in Gamma(0.37, 0.36) distribution.
G Numerical methods and supplemental figures
All numerical simulations were performed using MATLAB 9.13 (R2022b, The Math Works, Natick, MA, USA) either on desktop workstations (CPU: Intel i9. 99-99 Xeon W-2145, 64GB RAM) or on the Harvard University FASRC Cannon HPC cluster (https: : / //www rc. fas hhwvaard.d.............. edu). computationally intensive, and required around that 6000 CPU-hours of
Our simulations total compute time.
G.1 State-dependent inhibition simulations
For the simulations to highlight the signatures of state-dependent inhibition of the Poisson network and compare its behavior with the experimental observation by Arevian al. [95] we use reduced network with nOSN = 2 and n, = 10. We chose this reduced circuit both to reduce computational costs and match the experimental parameters of the in-vitro experiment. The other parameters of the simulation where as follows: Tp = 0.02, Tg 0.03, = 10, X = 2, dt 14 The simulation ran for 600 mss stimulation was applied starting at tstart 100 ms and ended at fend = 500 ms. We stimulated the principal mitral cell MCA with 80 equally spaced values in $ E [1. 400). When the second mitral cell was active, its stimulation was set at 8 80. The entries of the matrix were sampled from a normal distribution were non-zero. We sampled 32 "pairs' of cells by resampling the matrix for each 'pair' of cells. The simulation for the Gaussian circuit used the same parameters except the dynamics followed those
which we applied mask such that only 25% of its entries
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---

from equation D.4. For plotting purposes, we normalized the range of firing in Figure 2B to the maximum firing rate for each circuit model.
G.2 Capacity simulations
In our capacity simulations, we use BOSN 300 and Rodor 500, 1000, or 2000. We take the odor stimulus to rectangular pulse, with varying numbers randomly-selected odors appearing at concentration Cj = 40. As elsewhere, we take Tp 20 ms [91] and Ty 30 ms [92] to match experiment. We set the baseline rate be ro and the prior mean to be À 1; based on some experimentation our results appear relatively insensitive to small variations in these choices. We integrate the MAP circuit dynamics (3) using the forward Euler method with timestep Át 10-4 s. To determine which odors the model estimates as being present at given timepoint, we simply check which concentration estimates exceed 20 at that time.
As discussed in the main text, we consider three variants of the MAP circuit, defined by different choices of the matrix r:
For the one-to-one code, we let fgg fodorr and simply set r = / /N(ii)) (Aiz).
For the naïvely distributed code, we let fig 5nodor and choose as follows: We sample a random matrix Q IR "oder x*x with QQ T Innder by drawing a Gaussian matrix and orthogonalizing its rows. Then, we define
Q Q/N Q/N()
For the geometry-awart code, we let fgg 5roooo and choose I as follows: We sample a random matrix Q IR. RRodd Xng with QQ I by drawing Gaussian matrix and orthogonalizing
rows. Then, we compute an approximate inverse square root of the low-rank matrix A A constant a. In our simulations, we set 0.5; we find empirically that our results are not substantially sensitive small variations in a. Finally, we let
(A A + )
for small positive regularizing
where N is normalization function defined as
((A) max(A) Vng
C
with C = 50 identified reasonable choice. Our normalization convention for T is motivated by the idea that, in biology, the strength of individual synapses should be bounded. Moreover, changing the overall scale of the synaptic weights in our model corresponds to changing the effective time constant of the dynamics, which can produce a trivial speedup.
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---

after 100
after 200 ms
after
A
odor 500
1-1 dist.
proportion correct
geom.
0.5
50
100
odors present
B
Oaor 1000
C
ootor 2000
n 4000
2our 8000
Figure G.1: A. Fraction of odors correctly detected within 100 ms (left), 200 ms (center), and (right) after odor onset as function of the number of odors present, with nodor 500 possible odors, for models with one-to-one, naively distributed, and geometry-aware codes. B. As in A, but for nndor 1000. This matches Fig. C. As in but for Moder 2000, 4000, or 8000 possible odorants, respectively. Shaded patches show + 1.96 SEM over realizations throughout.
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---

one-to-one
naively distributed
geometry-aware
A
odor
500
100
# of odoors present
proportion correct
50% correct
0.5
time after odor onset (s)
B
1000
C
2000
D
alo 4000
E
n abs 8000
Figure G.2: A. Heatmap with overlaid smoothed contours of the fraction of odors correctly detected as
function of number of present odors and time window out of panel of nodor 500 possible odors
for models with one-to-one (left), naively distributed (center), and geometry-aware (right) codes. B. As in A, but for Rader 1000. This matches Fig. 3B. C. D, E. As in A, but for Bador 2000, 4000, or 8000 possible odorants, respectively.
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G.3 Sampling simulations
In our sampling simulations, we use nOSN 300, Rodor 500 or 1000, and ng 5Rodor. Here, the odor stimulus was composed of two rectangular pulses: At time flow 0 s, five "low" odors appear at concentration 10, and remain present at that concentration until time = 2 s. Then, at time (high = 1 s, five "high" odors appear at concentration 40, and remain present until time loff 2 s. As in our capacity simulations, we take Tp = 20 ms [91] and Ty = 30 ms [92] to match experiment, and we set the baseline rate to be = 1 and the prior mean to be À 1. As for our capacity results, our results appear relatively insensitive to small variations in these choices. We integrate the sampling circuit dynamics (4) using the forward Euler-Maruyama method with timestep At 10-$s. s.
As in our capacity simulations, in Fig. 4 we choose I follows:
For the one-to-one code, we let fgg = Rodor and simply set r Ln max (Aig).
For the naïvely distributed code, we let 7g = 5Rodor and choose I as follows: We sample a random matrix Q E IR (RRndorXii with QQ T Inwdor by drawing a Gaussian matrix and Q/ max (1))
orthogonalizing its rows. Then, we define
For the geometry-ave aware code, we let ngg = 5nodor and choose I as follows: We sample a random matrix Q E "odeer X with QQ T Indder by drawing a Gaussian matrix and orthogonalizing its rows. Then, we compute an approximate inverse square root of the low-rank matrix A A B A + ( )-1/2 for small positive regularizing constant a. In our simulations, we set a 0.5; we find empirically that our results are not substantially sensitive small variations in a. Finally, we let r BQ/ max[](ABQQ)
In Fig. 4A, we smooth the concentration estimate timeseries using a 100 ms moving average. In Fig. 4B-C, we show cumulative estimates of the mean and variance. Concretely, given a concentration timeseries (t). we estimate the mean and variance as
At T
Py
Σ
(1(()
(G.1)
Hoo S flow +
and
(T) At Σ -(t)2-1((1))
(G.2)
fJom StStio +T
respectively, where we assume T and are integer multiples of At. In Fig. 4D-E, we do the same except for times after hhghh In both cases, baselines were obtained by running the naïve sampling algorithm for 10 steps with a burn-in period of 10? steps. In Supp. Fig. G.3, we reproduce Fig. showing estimates for individual odorants as well as the mean across odorants.
In Supp. Fig. G.4, we show a preliminary experiment with an alternative, mostly non-negative choice for T. This gives the following three models:
For the one-to-one code, we let fig = Rodor. and simply set r - max (Aij)-
For the naïvely distributed code, we let nig 5todorr and choose as follows: We sample sparse, non-negative random matrix Q E P X with entries that are non-zero with density 0.15, and the non-zero entries are drawn uniformly on [0, 1]. Then, we define r Q/ max[[(AQ)))
For the geometry- code, we let fgg = 5modor. and choose T as follows: We sample a sparse, non-negative random matrix Q E Ro XRg with entries that are non-zero with density 0.15, and the non-zero entries are drawn uniformly on [0. 1]. Then, we compute an approximate inverse square root of the low-rank matrix A T A as B = (A A A Al ) 1/2 for small positive regularizing constant a. In our simulations, we set a 0.5; we find empirically that our results are not substantially sensitive to small variations in Finally, we let BQ/ max (I (| (Q)
This yields naively distributed synaptic weight matrix AT that is entirely non-negative, while the geometry-aware weight matrix has a small number of negative elements due to the fact that the inverse of non-negative matrix need not be neg-tegaeive We see that the behavior of this circuit is
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---

A
B
D
50
onn--t-onee
10
low odors
high
est. mean (arb. units)
low odors
est. mean (arb. units)
smoothed concentration estimate (arb. units)
odors
low odors
10
40
high odors
one-to-one naively dist.
naively ddistribueed
geometry
high odors
0.5
0.5
time since low odor onset (s)
time since high odor onset (s)
20
20
est. variance (arb, units)
low odors
est. variance (arb. units)
low odors
geomelry-aware
20
1150
high odors
high odors
2
time (s)
1
0.5 time since high odor onset (s)
time since low odor onset (s)
Figure G.3: Fast uncertainty estimation using Langevin sampling of the posterior, showing estimates for individual odorants. Here, we use a simple concentration estimation task in which randomly-
selected 'low' odorants out of a panel of 1000 appear concentration 10 at time O and then further 5 randomly-selected 'high' odorants appear at concentration 40 at time I s. A. Smoothed timeseries of instantancous concentration estimates for low, high, and background odorants, for models with one-to-one (top), naīvely distributed (middle)) and geometry- (bottom) codes. Background odorant estimates are shown as mean standard deviation over odorants. Dashed lines show true concentrations over time. B. Cumulative estimates of concentration mean for low (top) and high (bottom) odorants after the onset of the low odorants for one-to-one, naively distributed, and seometry-awart codes. Black lines indicate baseline estimates of the posterior mean. Thick colored lines indicate means over odorants, while thin lines show traces for individual odorants. C. As B, but for the estimated variance. D. As in B, but after the onset of high odorants. E. As in C, but after the onset of high odorants. See Appendix G for details of our numerical methods.
similar to that observed in Fig. 4. As mentioned in the Discussion, an important topic for future work will be to devise method to choose I that yields fully non-negative, sparse synaptic weight matrix A..
H
Code and data availability
All code and data required to reproduce the figures presented is available under an MIT License at https: //gi tigghhu..... www/peheeeaaaaaeeeeeeeeaooaaaaaa wtt.eeeeaanrroooooaattioooooo http://ooo.ccciooooeeetr///////
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A
B
D
50
10
10
one-to-one
low odors
high
est. mean (arb. units)
est. mean (arb. units)
smoothed concentration estimate (arb. units)
odors
low odors
low odors
10-
high odors
40-
one-to-one
naively dist.
naively distributed
geometry
high odors
0.5
0.5
time since low odor onset (s)
time since high odor onset (s)
est. variange (arb. units m
20
20p
est. variance (arb. units)
low odors
low odors
[geometry-aww aware
$100
20
high odors
high odors
2
time (s)
0.5
0.5 time since high odor onset (s)
time since low odor onset (s)
Figure G.4: Langevin sampling of the posterior using sparse non-negative distributed codes. Here, simple concentration estimation task in which 5 mandomly-selcctee 'low' odors out of panel of 500 appear at concentration 10 at time s, and then a further 5 randomly-selected 'high' odors appear concentration 40 at time s. This figure replicates Fig. 4, except that the code is distributed using sparse non-negative random matrix rather than an orthogonal matrix. Smoothed timeseries of instantaneous concentration estimates for low, high, and background odors, for models with one-to-one (top), naively distributed (middle), and geometry- aware (bottom) codes. Background odor estimates shown mean + standard deviation concentrations over time. B. Cumulative estimates of odor concentration mean for low (top) and high (bottom) odors after the onset of the low odors for one-to-one, naïvely distributed, and geometry- aware codes. Black lines indicate baseline estimates of the posterior mean. Thick lines indicate means over odors, and thin lines individual odors, C. As in B, but for the estimated variance. D. As in B. but after the onset of high odors. E. As in C, but after the onset of high odors. See Appendix G for details of our numerical methods.
odors. Dashed lines show true odor
S15


Reviews:
[{"review_id": "qd8hacVwSp", "reviewer": "Reviewer_Wphi", "summary": "The study introduces a novel compressed sensing (CS) rate-based circuit model\nrelying on Poisson noise models instead of Gaussian noise models. A central\nclaim is that the model can be mapped onto the neural circuits of the olfactory\nbulb while considering neurophysiological features and a biologically plausible\nconnectivity scheme. Importantly, compared to previous Gaussian CS models, the\nPoisson noise model enables distributed odor coding instead of indicating odor\npresence via the activity of a single neuron.", "questions": "1. What are the simulation parameters for the capacity simulations and the\n   sampling simulations? Some parameter settings seem to be missing in the\n   supplement.\n2. If in-vitro recordings are available, wouldn't it be possible to estimate\n   more model parameters from data directly?\n3. What do you want to show using Figure 1C that could not be shown using\n   summary statistics?\n", "limitations": "Limitations of the model are adequately addressed in the discussion section.", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "**Originality:** The authors propose a novel extension of prior work on\ncompressed sensing capturing properties of the the olfactory bulb using a\nPoisson noise model. They convincingly show that their model can explain\nobservations found in in vitro recordings while outperforming previous CS\nmodels.\n\n**Quality:** The experiments are well designed and performed convincingly. The\ninterpretations and intuitions in this work are carefully constructed and\neloquently presented. The figures complement the main text well. The discussion\nsection is one of the highlights of the paper.\n\n**Clarity:** Exceptionally clear writing and presentation. Sections are\nstructured well and link together to form a coherent picture.\n\n**Significance:** Excellent starting poiont for spike-based implementations of\nolfactory microcircuits and how the framework of compressed sensing can be used\nto deduce the functionality of circuits under investigation.", "weaknesses": "Minor remarks:\n\n1. Panels B to E in Figure 4 appear crammed, chaotic and could be plotted\n   differently. Maybe figures with individual odorants could be provided in the\n   supplement instead and the main figures just show the thicker lines?\n   Alternatively, the transparency of the lines could be reduced?\n2. Figure 1C is unreadable after printing on paper. Even on screen, the non-zero\n   entries are hard to spot.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the referee for their positive assessment of our work. We hope that our revisions will resolve their concerns about the clarity of the figures and of simulation parameters. \n\n### Weaknesses:\n\nMinor remarks:\n\n1. *Panels B to E in Figure 4 appear crammed, chaotic and could be plotted differently. Maybe figures with individual odorants could be provided in the supplement instead and the main figures just show the thicker lines? Alternatively, the transparency of the lines could be reduced?*\n\nThank you for this comment; we will follow your suggestion of providing a figure with individual odorants only in the supplement. We include the updated main-text figure showing only the thicker lines representing the averages in the PDF attached to the common response. \n\n2. *Figure 1C is unreadable after printing on paper. Even on screen, the non-zero entries are hard to spot.*\n\nThank you for noticing this legibility issue; after printing we agree that it is difficult to spot the non-zero entries. As we do not want to show much about the resulting sensing matrix other than the distribution of individual entries and the fact that it will have a few large entries, which Figure 1B is adequate to show (c.f. your 3rd question), we will remove it from the paper. \n\n### Questions:\n1. *What are the simulation parameters for the capacity simulations and the sampling simulations? Some parameter settings seem to be missing in the supplement.*\n\nThank you for noticing this omission. For the capacity and sampling simulations, the time constants used were $\\tau_{p} = 0.020\\ \\mathrm{s}$ and $\\tau_{g} = 0.030\\ \\mathrm{s}$, matching the state-dependent inhibition simulations and experimentally measured values for mitral/tufted and granule cells respectively (see below). We set the baseline rate to $r_{0} = 1$, and the prior strength to be $\\lambda = 1$; based on some experimentation our results appear relatively insensitive to small variations in these choices. We will make sure to state these parameter settings more clearly in the updated manuscript. \n\n2. *If in-vitro recordings are available, wouldn't it be possible to estimate more model parameters from data directly?*\n\nThank you for this suggestion. We already chose the time constants to match experimental measurements. For mitral/tufted cells we used $\\tau_p = 20ms$ (see Burton, S.D. and Urban, N.N., 2014. Greater excitability and firing irregularity of tufted cells underlies distinct afferent‐evoked activity of olfactory bulb mitral and tufted cells. The Journal of Physiology, 592(10), pp.2097-2118.) For granule cells, we used $\\tau_g = 30 ms$ (see Burton, S.D. and Urban, N.N., 2015. Rapid feedforward inhibition and asynchronous excitation regulate granule cell activity in the mammalian main olfactory bulb. Journal of Neuroscience, 35(42), pp.14103-14122.) We will clearly mention this in the updated version of our manuscript. We agree that it would be interesting to constrain the values of other parameters based on experiment; we will comment on this in our revised manuscript. \n\n3. *What do you want to show using Figure 1C that could not be shown using summary statistics?*\n\nAs noted above, we will remove Figure 1C, as the distributions shown in Figure 1B are sufficient to communicate our point.  \n"}, {"review_id": "alVfyvmfcj", "reviewer": "Reviewer_xx4u", "summary": "The authors develop a normative model for olfactory bulb using compressed sensing theory. They map the derived circuit model to known properties of cell types and  further hypothesize their roles.", "questions": "* Are the exact time-scales of dynamics biologically correct?\n", "limitations": "Limitations have been appropriately discussed.", "rating": 6, "confidence": 1, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "I am not an expert in olfaction. However, I enjoyed reading this paper as it is very well written - introduces the basic neuroscience of olfaction, the state of the art and the contribution of the paper. \n\nThe rationale, and analyses seem correct, but I cannot comment on the novelty.", "weaknesses": "there is no connection to real data (except defining the distribution of A).\n\nSince the goal of this paper is to build a normative circuit for olfaction based on biological principles, and since I am not an expert on olfaction, I cannot verify the design choices of the model. \n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their careful assessment of our work, and are pleased to hear that they found it enjoyable to read. We’d like to note that beyond using OSN activity measurement to define the affinity matrix $A$, we also reproduce the state-dependent inhibition effect observed in Arevian et al., Nature Neurosci, 2008, and this effect does not occur in a circuit model derived for Gaussian noise in the OSNs. Moreover, we tune the timescales of the neural dynamics to match experimentally measured membrane time constants of mitral/tufted and granule cells (see below).\n\n### Questions:\n\n*Are the exact time-scales of dynamics biologically correct?*\n\nYes, the values for the membrane time constant of the neurons were chosen to match values measured experimentally and published in the literature. For mitral/tufted cells we used $\\tau_p = 20ms$ (see Burton, S.D. and Urban, N.N., 2014. Greater excitability and firing irregularity of tufted cells underlies distinct afferent‐evoked activity of olfactory bulb mitral and tufted cells. The Journal of Physiology, 592(10), pp.2097-2118.) For granule cells, we used $\\tau_g = 30 ms$ (see Burton, S.D. and Urban, N.N., 2015. Rapid feedforward inhibition and asynchronous excitation regulate granule cell activity in the mammalian main olfactory bulb. Journal of Neuroscience, 35(42), pp.14103-14122.) We will make this point more clearly and cite the relevant papers in the updated version of the manuscript."}, {"review_id": "2fM1j8ibbU", "reviewer": "Reviewer_DAo3", "summary": "The paper proposes a normative model for odor processing and representation in the olfactory bulb (of mammals). I did not track the conclusions, since I got distracted by the choices required to build a normative (mathematically tractable) model.", "questions": "Context: I am not the right audience for this paper, since I do not grok the normative modeling approach. So the comments below can perhaps best be viewed not comments to be acted on, or even as criticisms, but as a window on how non-normative biological modelers might see the paper... for whatever that is worth, which may be not much :)\n\n-----------\n\nI don't get how this is a compressed sensing problem: the ORNs' chemical sensitivities correspond to our visual spectrum - they are simply the subset of chemicals that are measurable. Non-measurable chemicals effectively don't exist, and the sensing of detectable odors is not, to my understanding, random projection (see eg lines 171 - 172). I believe this is different from compressed sensing. I thought compressed sensing tied into olfaction further downstream, at the much larger (50x), sparsely firing layer (Mushroom Body in insects; I think there is a mammal equivalent), and in particular in the way the MB projects onto readout neurons (extrinsic neurons in insects) via random projections.(see Ganguli 2012).\n\nFor clarity, the biological interpretation/mapping (section 4) would go better much earlier, so the reader can map the mathematical elements to the OB as they show up. As it is, the math model is developed without context that would be highly useful.\n\nI wonder if the model for odor concentrations (gamma prior with poisson noise at the ORNs) corresponds to actual odor plumes, or if it was chosen for mathematical convenience.\n\nEquation 2: words to motivate what this equation is doing would be helpful (in general I prefer math motivated by words).\n\nline 118: why is c (the odor) defined in terms of g (the projection neuron firing rates)? This seems backwards. Is it a convenience to make the math work, and if so is it stepping on any biological realities?\n\nline 125: is it cricket to linearize the system? I thought the OB (AL) is strongly nonlinear. See for example Rachel Wilson's papers on divisive normalization in the AL. Here again (my refrain) I worry about biological sacrifices to make the math nice.\n\nline 148: It sounds like g represent the lateral inhibition neurons. Is it the case that the mathematics accurately reflects what is concretely known about lateral inhibition in the OBL The symmetry of weights (expressed in the A-Gamma matrix) look highly unrealistic. Even having the projection neurons excite the inhibitory neurons is not accurate - both g and p are excited by the ORNs (caveat: my understanding is based in the insect AL, not the OB).\n\nsection 4.4: The inhibitory effects of the model may be reminiscent of experimental findings, but I believe those same findings can be more directly explained by known structure and dynamics of lateral inhibition within the OB. I would argue for subordinating the math to fit the known structure.\n\nline 285: I don't see how negative synaptic weights can be justified (assuming that inhibition is expressed by a separate minus sign in the equations).\n\nline 278- 279: I regret that I do not agree that the math clearly maps onto the OB", "limitations": "NA", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The paper is well written and the model is developed soundly (within the normative style) and with good detail.\n\nIf one accepts the sacrifices of biological detail needed to make the math clean, that is, if one supports the idea of normative modeling, the model may be a good one.  \n", "weaknesses": "Note to Area Chair and authors:\n\nI am not well suited to this paper, because I do not understand the motives behind the normative program, which prioritizes clean math over known biological information.  I see normative modeling as a Procrustes bed, and how far to distort biology for the sake of math is a matter that admits wide differences of opinion. I am uncomfortable with several of the choices (details below). \n\nHowever, I realize that those sympathetic to the normative program will feel differently. Since they, and not I, are the audience for this paper, I'm happy to defer to their preferences.\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their careful and fair assessment of our manuscript, and appreciate the importance of their perspective given that we respectfully disagree about the fundamental utility of normative modeling. We emphasize that our normative model is not being used as a Procrustean bed, since we are not imposing exact conformity to a certain arbitrary standard. We are saying that if we posit certain reasonable axioms, non-trivial outcomes arise that agree with data. We are not saying anywhere that other explanations are not possible.  We respond to each of their concerns in turn below; we hope that this helps address some of their concerns regarding the biological relevance of our findings. \n\n1. *I don't get how this is a compressed sensing problem [...]*\n\nIn the OB (and across brain areas), compression and expansion occur at several stages of processing. In the mammalian olfactory system, there is indeed an expansion from the mitral/tufted cells to their projections in olfactory cortical areas (resembling the expansion of the projection into the MB in insects). But at the sensing stage, there is a compression from the space of possible odors to the space of OSN (and then M/T cell) responses that also maps onto a compressed sensing problem. Importantly, only a few out of the many detectable odorants are present in a given olfactory scene (see Krishnamurthy et al, 2022 Front. Comp. Neuro., or Koulakov and Rinberg 2011, Neuron) \n\n2. *For clarity, the biological interpretation/mapping (section 4) would go better much earlier [...]*\n\nAs with all of our paper, this organization is motivated by the normative framework - we aim to fully derive the mathematical model and only then connect it to biology. We appreciate that the reviewer will disagree with our preferences.\n\n3. *I wonder if the model for odor concentrations (gamma prior with poisson noise at the ORNs) corresponds to actual odor plumes [...].*\n\nModeling the odor concentration as gamma-distributed is a common approximation to the distribution of concentrations wafted on turbulent plumes, as it offers a useful balance of mathematical tractability with physical realism. Indeed, it can be physically motivated (see Celani, Villermaux, and Vergassola, 2014, PRX). A more realistic prior would include a spike at zero in addition to the approximately-Gamma slab of positive concentrations, but as we mention in the Discussion this is more challenging to sample and is left for future work. Similarly, the linear-mean OSNs with a Poisson noise model represents a compromise between detailed realism and mathematical tractability. As we mention below, a more detailed model would account for sensing nonlinearities due to receptor antagonism, but would be considerably more complicated. Therefore, these design choices reflect a balance between realism and conceptual simplicity. We will add a sentence to comment on the motivation for the Gamma prior to our updated manuscript. \n\n4. *Words to motivate what this equation is doing would be helpful [...].*\n\nAs noted in Line 111 above eq. (2), this is simply gradient ascent on the (logarithm of the) posterior probability resulting from (1). To make the motivation clearer, we propose to replace the sentence on Lines 109-110 starting with “Given this…” with “Given this likelihood and prior, we construct a neural circuit to compute the maximum a posteriori (MAP) estimate of the concentration $\\mathbf{c}$ using gradient ascent on the log-posterior probability.”\n\n5. *Why is c (the odor) defined in terms of g (the projection neuron firing rates)? [...]*\n\nIt is not the underlying concentration of the odor that is defined in terms of the model granule cell firing rates, but rather the system’s estimate of the concentration. If the reviewer thinks it would be helpful to adopt notation to explicitly disambiguate the concentration estimate from the underlying concentration—perhaps by writing $\\hat{\\mathbf{c}}$ rather than just $\\mathbf{c}$, we would be happy to do so. \n\n6. *Is it cricket to linearize the system? [...]*\n\nAs we noted in Lines 83-87, mammalian OSNs do indeed display nonlinear (saturating) responses when exposed to wide ranges of odor concentration. However, within a moderate concentration range as we use here, linear models provide an accurate description; see Gupta, Albeanu, and Bhalla, 2015 Nat Neurosci. 2015 (ref. [66]). \n\n7. *[...] Is it the case that the mathematics accurately reflects what is concretely known about lateral inhibition in the OB?*\n\nIn mammals, the granule cells receive excitatory input from OSNs only indirectly via the M/T cells, not directly (this is diagrammed in Figure 1A). This is an anatomical difference between OB and insect AL. Similarly, as we discussed in Lines 152-160, due to the fact that granule and M/T cells are coupled by dendrodendritic synapses, including a large number of reciprocal synapses, it is not implausible to assume approximately symmetric coupling.  \n\n8. *[...] I would argue for subordinating the math to fit the known structure.*\n\nWe hope that the above clarification regarding the anatomy of the OB helps address some of your concerns regarding the biological plausibility of our model. We respectfully disagree with the reviewer that it is always beneficial to subordinate math to known biological structure; we would argue that depending on the level of understanding desired it can be useful to make simplifications for the sake of conceptual clarity. \n\n9. *I don't see how negative synaptic weights can be justified [...]*\n\nAs we discuss in the paragraph starting on Line 284, we agree that this feature is not entirely biologically satisfactory; our model is of course not perfect, but it is useful. In that paragraph, we offer potential accounts for a few negative weights, but this will be an important topic for future work. \n\n10. *I regret that I do not agree that the math clearly maps onto the OB*\n\nWe appreciate the reviewer’s reasons for disagreement.\n"}, {"review_id": "hEgLUVSVel", "reviewer": "Reviewer_cVJu", "summary": "This work presents a compressed sensing circuit model of the olfactory bulb. This circuit model performs the task of computing an estimate of odorant concentrations, given receptor activity. Notably, the proposed circuit takes great care to maintain biological plausibility. Linearized workarounds are found for non-linear circuit components. And the receptor activity is subject to Poisson noise, which is more biologically plausible than the typically considered Gaussian noise.  ", "questions": "- Line 42: What does \"distributed coding\" mean in this context? Distributed across receptor types? Distributed across receptors? \n- Could you briefly speculate what modifications might be made to the circuit to reproduce the nonlinear responses described in lines 84-85?\n- Line 119: I'm missing something basic here. As far as I can tell, $n_g$ describes the number of receptor types and $n_{odor}$ is the number of odors. So, why is the case of $n_{g} > n_{odor}$ being considered? Isn't this at odds with the fact that there should be more odors than receptor types?\n- Section 4.2: Do I understand correctly that the affinity matrix is fixed? Are there any pieces of the circuit that are \"learned\" or \"fitted\"?\n- Could you help me understand why [Grabska-Barwinska et al. 2013](https://proceedings.neurips.cc/paper_files/paper/2013/file/2bcab9d935d219641434683dd9d18a03-Paper.pdf) refer to Poisson noise as an \"unrealistic\" assumption (page 8 under \"future directions\").", "limitations": "Yes, limitations are addressed, and initial solutions are sketched in the supplementary material, although I did not closely read these.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The proposed circuit model is more biologically plausible than existing circuit models. And proposing a  circuit model in the first place avoids the weight transport problem of neural networks, or at least confines the problem to a regime where it could be plausibly explained.\n- The proposed model recapitulates some biological phenomena that a model with a Gaussian noise assumption does not\n- The proposed model performs better in the presence of distractors than the naive model\n- Overall, the paper is well written, and seems like an improvement in explanatory power over existing work that the community follows", "weaknesses": "- Limitations are noted by the authors, namely the dimensionality of the odor space considered, but possible solutions, and an analysis of how the system would degrade are presented as well.\n- There are simplifications, and points at which the model breaks with biology. For example, the roles of mitral and tufted cells are not distinguished. But for an understanding at the level of concepts, I believe this is alright.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their careful assessment of our work and its limitations. We agree that building a model with distinguished roles for mitral and tufted cells will be an important objective for future studies. We respond to each of their questions below. \n\n# Questions:\n\n1. *Line 42: What does \"distributed coding\" mean in this context? Distributed across receptor types? Distributed across receptors?*\n\nThank you for this question. Here, “distributed coding” means that the presence and concentration of a specific odorant is not signaled by the activity of a specialized granule cell; rather, a response across many neurons in the population is triggered. We will expand our discussion of this point to avoid any loss of clarity. \n\n2. *Could you briefly speculate what modifications might be made to the circuit to reproduce the nonlinear responses described in lines 84-85?*\n\nTo model the nonlinear responses, a simple approach would be to apply a saturating nonlinearity to the linearly sensed concentrations. This is the approach that was taken by Qin, S., Li, Q., Tang, C. and Tu, Y., 2019. Optimal compressed sensing strategies for an array of nonlinear olfactory receptor neurons with and without spontaneous activity. Proceedings of the National Academy of Sciences, 116(41), pp.20286-20295.: if one assumes a simple competitive binding model for receptor activity, the resulting nonlinearity is $x/(1+x)$. We will add a sentence to the paper to comment on this point. \n\n3. *Line 119: I'm missing something basic here. As far as I can tell, n_g describes the number of  receptor types and n_{odor} is the number of odors. So, why is the case of n_{g} > n_{odor} being considered? Isn't this at odds with the fact that there should be more odors than receptor types?*\n\nThe number of receptor types in $n_{\\mathrm{OSN}}$; $n_{\\mathrm{g}}$ is the number of inhibitory granule cells. We keep $n_{\\mathrm{OSN}} \\ll n_{\\mathrm{odor}}$ throughout, but allow $n_{\\mathrm{g}} \\geq n_{\\mathrm{odor}}$. We will add a sentence to clarify that the number of receptor types is $n_{\\mathrm{OSN}}$. \n\n4. *Section 4.2: Do I understand correctly that the affinity matrix is fixed? Are there any pieces of the circuit that are \"learned\" or \"fitted\"?*\n\nYour understanding is correct: the affinity matrix is fixed. At the moment, the decoding matrix $\\mathbf{\\Gamma}$ is tuned by hand in the geometry-aware model to match the properties of the affinity matrix, such that $\\mathbf{\\Gamma} \\mathbf{\\Gamma}^{\\top} \\simeq (\\mathbf{A}^{\\top} \\mathbf{A})^{+}$. We focus here on the problem of inference (and which can be studied by experimentally measuring the firing rates of neurons). There is no dynamic learning in the current version of our model. As noted in lines 312-323 of the Discussion, extending this model to incorporate synaptic plasticity will be an interesting topic for future work. \n\n5. *Could you help me understand why Grabska-Barwinska et al. 2013 refer to Poisson noise as an \"unrealistic\" assumption (page 8 under \"future directions\").*\n\nThank you for this question. For completeness, we reproduce the relevant sentences from Grabska-Barwinska et al:\n\n> We have made several unrealistic assumptions in this analysis. For instance, the generative model was very simple: we assumed that concentrations added linearly, that weights were binary (so that each odor activated a subset of the olfactory receptor neurons at a finite value, and did not activate the rest at all), and that noise was Poisson. None of these are likely to be exactly true. \n\nTo the best of our understanding, the point being made here is that the true noise distribution is not perfectly Poisson. Of course, no choice of noise distribution is perfect, but Poisson noise is a better model for the statistics of OSN responses than Gaussian noise."}, {"review_id": "PYAN1OzSb5", "reviewer": "Reviewer_Qhvj", "summary": "The authors derive a normative model to describe neural computations performed by the olfactory bulb. They describe in detail the biological interpretation of their model and show that the model makes realistic prediction about the biology of the olfactory bulb, matching the experimental observations. The authors validate the model on synthetic data, presenting different implementations of the model and evaluating its performance.", "questions": "Can the authors quantify better the performance gain and or distill better the conceptual advance made by their study with respect to current computational models of olfactory discrimination? Maybe with direct application of competing models to the same simulated tasks? (this would be the key point  to increase my enthusiasm for the paper)\n\nCan the authors distill the bibliography, only keeping the most relevant references?\n\nHow does the model perform in the fast identification of several odors, compared to previously existing (e.g. Gaussian CS) models? \nCan the authors provide an estimate of the human performance in their task, as a benchmark for the model performance?\nBased on Appendix G, it appears that naively distributed and the geometry-aware codes simulations ran using 5 times the number of granule cells compared to the one-to-one code. How much of the increase in model performance is due to the larger number of neurons and how much is due to a better coding scheme? Wouldn't it have been fairer to also set n_g = n_odor for the naively distributed and the geometry-aware codes? Additionally, from Fig. 3A it appears that the naively distributed code performance is quite low – also compared to the one-to-one code – for less than 10 odors. Authors should comment on why this is the case.\n\n", "limitations": "Yes, the authors describe well and fairly the  limitations in the Discussion section.", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "The paper is well written with excellent language and clear logic. The authors made a major effort in relating the equations they obtained to the biology of the network. The maths is rigorous and sound and is also well explained. The prediction of state-dependent inhibition obtained using the normative approach is elegant. ", "weaknesses": "The bibliography is far too long and it makes it difficult for the reader to focus on references that are most relevant. While I like a well referenced paper, this is too much. I struggle to understand which papers I should read to evaluate better the present contribution. \n\nMajor - but hopefully addressable - concern. It is not fully clear whether the new model outperforms the existing ones. It is nice to see that the Poissonian model does better than the Gaussian one in predicting the effect of state-dependent inhibition. However, it would have been more relevant to compare the performances of the model with existing ones (Figures 3 and 4). \n\nIt is not easy to understand how far (or different) the model performance is from human performance in olfactory discrimination. Clarifying this would make it easier to evaluate the overall performance of the model.\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the referee for their careful reading of our work, and are glad to hear that they found it elegant, rigorous, and well-written. We reply to each of their concerns in turn below; we hope that the proposed changes strengthen their view that our paper should be presented at NeurIPS.  \n\n### Questions:\n\n1. *Can the authors quantify better the performance gain and or distill better the conceptual advance made by their study with respect to current computational models of olfactory discrimination? Maybe with direct application of competing models to the same simulated tasks? (this would be the key point to increase my enthusiasm for the paper)*\n\nPlease see our response to Point 3 below.\n\n2. *Can the authors distill the bibliography, only keeping the most relevant references?*\n\nThank you for this suggestion. We acknowledge that our paper is somewhat heavily referenced, which reflects the substantial body of work on the olfactory system. We will endeavor to distill each block of references down to the most relevant few.\n\n3. *How does the model perform in the fast identification of several odors, compared to previously existing (e.g. Gaussian CS) models? Can the authors provide an estimate of the human performance in their task, as a benchmark for the model performance? Based on Appendix G, it appears that naively distributed and the geometry-aware codes simulations ran using 5 times the number of granule cells compared to the one-to-one code. How much of the increase in model performance is due to the larger number of neurons and how much is due to a better coding scheme? Wouldn't it have been fairer to also set n_g = n_odor for the naively distributed and the geometry-aware codes? Additionally, from Fig. 3A it appears that the naively distributed code performance is quite low – also compared to the one-to-one code – for less than 10 odors. Authors should comment on why this is the case.*\n\nThank you for these questions. We respond to each of your points in turn:\n- Based on the extensive theoretical literature on compressed sensing in Poisson noise, it is difficult to make a meaningful, controlled comparison between a Gaussian CS model and a Poisson model if the data is generated using Poisson noise, as Gaussian algorithms will perform poorly even in the moderate-intensity regime. See, for example, Raginsky et al., “Compressed sensing performance bounds under Poisson noise,” IEEE Trans. Sig. Proc., 2010, and references therein. More biologically, it is not always clear how to make a fair comparison in terms of matching single-neuron time constants. \n- Second, the 1-to-1 code model is closely related to the Poisson CS model introduced by Grabska-Barwinska et al. 2013. Concretely, they differ only in that we split up the circuit to linearize the division, which allows an interpretation in terms of cell types in olfactory bulb circuits; the idea of performing gradient ascent on the log-posterior is otherwise identical. We alluded to this relationship in Line 113-114, but will add a sentence to further reify this point.\n- To the best of our knowledge, human psychophysics data for complex olfactory mixture tasks is not available. More precisely, we are not aware of experiments testing humans’ ability to detect single mixture components. Work from Liang and colleagues (Jinks, A. and Laing, D.G., 2001. The analysis of odor mixtures by humans: evidence for a configurational process. Physiology & behavior, 72(1-2), pp.51-63. or Laing, D.G. and Francis, G.W., 1989. The capacity of humans to identify odors in mixtures. Physiology & behavior, 46(5)) has investigated humans’ ability to identify all components of a mixture; humans perform poorly on this task.\n- It is of course true that the number of granule cells affects the speed of convergence and the accuracy of sampling. This is, however, precisely the reason we include the naively-distributed code, as it provides an intermediate point of comparison: comparing the naively-distributed code shows that the geometry-aware code’s performance gain relative to the 1-to-1 code is not due purely to the increased number of granule cells. We will elaborate on this point in our updated manuscript. \n"}]

Author rebuttal:
We thank the reviewers for their favorable comments on our manuscript. We outline the primary changes we propose to make to the manuscript here, and respond to the reviewer’s detailed comments point-by-point individually. The largest changes to our manuscript will be as follows: 
- Following Reviewer Wphi’s suggestion, we will simplify Figure 4 by removing the lines for individual odors, and include a version with those lines only in the supplement. We include this revised figure in the PDF attachment. 
- We will remove Figure 1C, which Reviewer Wphi found to be illegible. 
- We will add a new figure showing how the capacity of the different algorithms tested in Figure 3 scales with the number of odors. This panel is included in the PDF. 
- We will elaborate on the motivation underlying our choice of a Gamma prior, as requested by Reviewer DAo3.
- We will mention that we chose the model membrane time constants to match experimental measurements, addressing a question raised by Reviewers Wphi and xx4u.

Meta-review: A new model for olfaction which performs compress sensing in a biologically plausible way with an extension for Bayesian inference to measure uncertainty. The tools developed here could see application to other domains where uncertainty is important.

There is great interest in the community in biologically plausible ML. This is a fairly unique arena in which to consider biological plausibility.

Multiple reviewers noted that contextualizing this work better would be beneficial. A large part of the value of this publication in NeurIPS is explaining the problems in this domain clearly where members of the community can engage with both the problem and the models offered here. As came up in the rebuttals, a precise discussion about what aspects are or are not biologically plausible, what problems are being solved and what problems are left, would both be important for a final submission. The more clearly the authors articulate this, the more engagement with their work there will be.
Decision: Accept (poster)