SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Jacob A. Zavatone-Veth $^{1,2}$ ; Paul Masset $^{1,3}$ ; William L. Tong $^{1,4,5}$ ; Joseph D. Zak $^{6}$ , Venkatesh N. Murthy $^{1,3\dagger}$ ; Cengiz Pehlevan $^{1,4,5\dagger}$
$^{1}$ Center for Brain Science,  $^{2}$ Department of Physics,  $^{3}$ Department of Molecular and Cellular Biology,
$^{4}$ John A. Paulson School of Engineering and Applied Sciences, Empner Institute for the Study of Natural and Artificial Intelligence
$^{6}$ Department of Biological Sciences,
Chicago, IL 60607
jzavatoneveth@g.harvard.edu, paul_masset@fas.harvard.edu, vnmurthy@fas.harvard.edu, cpehlevan@seas.harvard.edu
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.
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 as the Umwelt, depends on the sensors and sensory organs of each organism [1, 2]. Throughout evolution, organisms have developed specialized sensory mechanisms to extract specific information about the
*JAZ-V and PM contributed equally to this work.
$\dagger$  VNM and CP jointly supervised this work.
physical world. In vision and audition—the most studied sensory modalities in neuroscience—stimuli 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 a tone and quantify resulting changes in perception and neural representations. From a 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 geometric 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 a 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 biologically-plausible implementations that can perform fast inference at scale. Indeed, many proposals assume that the presence of each odorant is represented by a single, specialized neuron, which is inconsistent with the distributed odor coding observed in vivo [22, 34-36]. This axis-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].
In this paper, we propose a 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. 1A) [44-47], 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 [51]. In contrast to the immune system'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 a network of inhibitory granule cells, the most numerous cell type in the OB [57].
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 a 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 a complex olfactory scene [63-66]. Importantly, the limits of human performance in this setting remain to our knowledge unknown [67, 68]. To render the problem more tractable, we will make a 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 on 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, 70]. We will also ignore the fact that odorant concentrations can vary over many orders of magnitude, and OSN responses to large changes in concentration are strongly nonlinear. Here, we will focus on concentration changes of one or two orders of magnitude. Within such ranges, the responses of OB neurons are well characterized by linear models [71].
On the theoretical side, it 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 in 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 of CS to build circuit models for olfactory coding [21-27, 72-74]. However, these model circuits do not map cleanly onto the neural circuits of the OB and the olfactory cortical areas, particularly because they often assume each granule cell encodes exactly one odorant. Moreover, these works usually assume a Gaussian noise model for OSNs, which is biologically unrealistic (but see [21, 22]). Instead, OSN activity is better captured by a Poisson noise model [75, 76]. Some theoretical guarantees for Poisson CS are known, but the situation is less well-understood than in 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 a single sniff in mind, the circuit's objective is to rapidly infer the odorant concentrations  $\mathbf{c} \in \mathbb{R}_{+}^{n_{\mathrm{odor}}}$  underlying a single, static sample of OSN activity  $\mathbf{s} \in \mathbb{R}_{+}^{n_{\mathrm{OSN}}}$ . For simplicity, we model the mean activity of OSNs as a linear function of the concentration, with a receptor affinity matrix  $\mathbf{A} \in \mathbb{R}_{+}^{n_{\mathrm{OSN}} \times n_{\mathrm{odor}}}$  and a baseline rate  $\mathbf{r}_0 \in \mathbb{R}_{+}^{n_{\mathrm{OSN}}}$ . As motivated above,
we use a Poisson noise model for OSN activity given the underlying concentration signal  $\mathbf{c}$ , and correspondingly a Gamma prior over concentrations with shape  $\alpha \in \mathbb{R}_{+}^{n_{\mathrm{odor}}}$  and scale  $\lambda \in \mathbb{R}_{+}^{n_{\mathrm{odor}}}$ :<sup>3</sup>
$$
\mathbf {s} \mid \mathbf {c} \sim \operatorname {P o i s s o n} \left(\mathbf {r} _ {0} + \mathbf {A c}\right), \quad \mathbf {c} \sim \operatorname {G a m m a} (\boldsymbol {\alpha}, \boldsymbol {\lambda}). \tag {1}
$$
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. Here, we sketch the derivation, deferring some details to Appendix B. Our starting point is the gradient ascent equation
$$
\dot {\mathbf {c}} (t) = \boldsymbol {\nabla} _ {\mathbf {c}} \log p (\mathbf {c} \mid \mathbf {s}) = \mathbf {A} ^ {\top} [ \mathbf {s} \oslash (\mathbf {r} _ {0} + \mathbf {A c}) - \mathbf {1} ] + (\boldsymbol {\alpha} - \mathbf {1}) \oslash \mathbf {c} - \boldsymbol {\lambda}, \tag {2}
$$
where  $\varnothing$  denotes elementwise division and  $\dot{\mathbf{c}}(t) = d\mathbf{c} / dt$ . Here, the estimate  $\mathbf{c}$  is formally constrained to  $\mathbb{R}_+^{n_{\mathrm{odor}}}$ ; in numerical simulations we will sometimes ignore this constraint. Circuit algorithms of this form were studied in previous work by Grabska-Barwińska 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  $\mathbf{g} \in \mathbb{R}^{n_{\mathrm{g}}}$  of the neurons are mapped to the concentration estimate  $\mathbf{c}$  through a matrix  $\Gamma \in \mathbb{R}^{n_{\mathrm{odor}} \times n_{\mathrm{g}}}$ :  $\mathbf{c}(t) = \Gamma \mathbf{g}(t)$ . Even if  $\Gamma$  is non-square—in particular, if  $n_{\mathrm{g}} > n_{\mathrm{odor}}$ —so long as  $\Gamma \Gamma^{\top}$  is positive-definite and the rates follow the dynamics  $\tau_{\mathrm{g}} \dot{\mathbf{g}}(t) = (\mathbf{A} \Gamma)^{\top}[\mathbf{s} \oslash (\mathbf{r}_0 + \mathbf{A} \Gamma \mathbf{g}) - \mathbf{1}] + \Gamma^{\top}[(\boldsymbol{\alpha} - \mathbf{1}) \oslash (\Gamma \mathbf{g}) - \boldsymbol{\lambda}]$  for some time constant  $\tau_{\mathrm{g}}$ , the concentration estimates will still converge to the MAP. This corresponds to preconditioned gradient ascent [39, 79]. Again, we formally require the constraint that  $\mathbf{c} \in \mathbb{R}_{+}^{n_{\mathrm{odor}}}$ , which translates into a constraint on  $\mathbf{g}$ . Most simply, we may take  $\mathbf{g} \in \mathbb{R}_{+}^{n_{\mathrm{g}}}$  and choose  $\Gamma$  to be positivity-preserving.
These dynamics include two divisive non-linearities, which can be challenging to 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  $\mathbf{s} \oslash (\mathbf{r}_0 + \mathbf{A}\Gamma\mathbf{g})$  and  $(\alpha - 1) \oslash (\Gamma\mathbf{g})$ . Concretely, we introduce cell types with rates  $\mathbf{p} \in \mathbb{R}^{n_{\mathrm{OSN}}}$  and  $\mathbf{z} \in \mathbb{R}^{n_{\mathrm{odor}}}$  such that their fixed-point rates for fixed  $\mathbf{g}$  are  $\mathbf{p}^* = \mathbf{s} \oslash (\mathbf{r}_0 + \mathbf{A}\Gamma\mathbf{g})$  and  $\mathbf{z}^* = (\alpha - 1) \oslash (\Gamma\mathbf{g})$ , respectively. This yields the coupled circuit dynamics
$$
\mathbf {c} (t) = \boldsymbol {\Gamma} \mathbf {g} (t), \quad \tau_ {\mathrm {g}} \dot {\mathbf {g}} (t) = (\mathbf {A} \boldsymbol {\Gamma}) ^ {\top} (\mathbf {p} - \mathbf {1}) + \boldsymbol {\Gamma} ^ {\top} (\mathbf {z} - \boldsymbol {\lambda}),
$$
$$
\tau_ {\mathrm {p}} \dot {\mathbf {p}} (t) = \mathbf {s} - \mathbf {p} \odot \left(\mathbf {r} _ {0} + \mathbf {A} \boldsymbol {\Gamma} \mathbf {g}\right), \quad \tau_ {\mathrm {z}} \dot {\mathbf {z}} (t) = \boldsymbol {\alpha} - \mathbf {1} - \mathbf {z} \odot \mathbf {c}, \tag {3}
$$
for cell-type-specific time constants  $\tau_{\mathrm{g}}$ ,  $\tau_{\mathrm{p}}$ , and  $\tau_{\mathrm{z}}$ , where  $\odot$  denotes elementwise multiplication. In the limit  $\tau_{\mathrm{p}}, \tau_{\mathrm{z}} \ll \tau_{\mathrm{g}}$ , this circuit will recover the MAP gradient ascent. If  $\tau_{\mathrm{p}}$  and  $\tau_{\mathrm{z}}$  are not infinitely fast relative to  $\tau_{\mathrm{g}}$ , we expect these dynamics to 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-reasonable time constants using numerical experiments. Moreover,  $\mathbf{p}$  should formally be constrained to  $\mathbb{R}_+^{n_{\mathrm{p}}}$ , such that the non-negativity of the target ratio is respected. Finally, we note that in the special case  $\alpha = 1$  in which the Gamma prior reduces to an exponential prior, the introduction of the cell type  $\mathbf{z}$  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  $\mathbf{g}$  as the granule cells of the OB, and the cell type  $\mathbf{p}$  as the mitral cells, which are projection neurons. Provided that the elements of the matrix  $\mathbf{A}\Gamma$  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  $\mathbf{p}$  neurons excite the  $\mathbf{g}$  neurons, which in turn inhibit the  $\mathbf{p}$  neurons. Finally, we interpret the cell type  $\mathbf{z}$  as representing a form of cortical feedback. In the remainder of this section, we will justify this mapping in detail.
See Appendix A for a detailed description of our notational conventions.
C
Figure 2: State-dependent inhibition of mitral cells. A. Upon stimulation, mitral cells exhibit a transient burst of activity followed by relaxation to a plateau. Darker color indicates stronger stimulation. B. In the circuit for Poisson CS we propose (blue line), the inhibition due to the activation of a second mitral cell  $(MC_{B})$  is gated by the activity of the cell we are recording from  $(MC_{A})$ , as observed experimentally by Arevian et al. [95] in their Fig. 2D. This state-dependent gating does not occur in a circuit derived from a Gaussian noise model (orange line). Rates are normalized to the maximal stimulation of the principal cell. Dashed line indicates the unity line. C. Strength of inhibition as a function of time and of stimulation intensity; c.f. [95] Fig. 3B.
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  $\mathbf{g}$  and  $\mathbf{p}$  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. 1A). Moreover, the synaptic weights  $\mathbf{A}\Gamma$  of the connections from  $\mathbf{g}$  to  $\mathbf{p}$  neurons mirror exactly the weights  $(\mathbf{A}\Gamma)^{\top}$  of the connections from  $\mathbf{p}$  to  $\mathbf{g}$  neurons. Naively, this creates a weight transport problem of the form that renders backpropagation biologically implausible: how should the exact transpose of a matrix be copied to another synapse [84-86]? However, in the unique case of the OB this does not pose a 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  $\mathbf{g}$  and  $\mathbf{p}$  in the dynamics (3), but what about the cell type  $\mathbf{z}$ ? These cells give direct excitatory input to the granule cells  $\mathbf{g}$  with weights  $\Gamma^{\top}$ , and represent the concentration-dependent contribution to the log-prior gradient. We 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 in which the prior is static and has  $\alpha = 1$ , in which case it reduces to an exponential and feedback is not explicitly required.
Given this mapping of the cell types of our model to the cell types of the bulb, we will henceforth choose the membrane time constants to match experiment, taking  $\tau_{p} = 20$  ms [91] and  $\tau_{g} = 30$  ms [92]. This matching of timescales is required for our comparison of model inference timescales to the timescale of a single sniff to be meaningful.
4.2 Choosing the affinity matrix
In theories of Poisson CS, the optimal sensing matrix  $\mathbf{A}$  is one that has columns that are in some precise sense as orthogonal as possible, so that it 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-photon calcium imaging, we recorded the responses of 228 mouse OSNs glomeruli to 32 odorants (see Appendix F for details; these data were previously published in [94]). In Fig. 1B, we show that the distribution of responses is well-fit by a Gamma distribution with shape 0.37 and scale 0.36 (Appendix F). We then define our ensemble of sensing matrices  $\mathbf{A}$  by drawing their elements as independently and identically distributed Gamma(0.37, 0.36) random variables. An example matrix drawn from this ensemble is shown in Fig. 1C.
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, 25]. 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 a single odorant.
What, then, is the functional role of the mitral cells in our model? We can interpret their dynamics as implementing a 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, a Gaussian noise model gives a subtractive form of predictive coding in 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 a stimulus. Consistent with experimentally-measured responses [96, 97], a sharp transient response at the onset of stimulation is followed by decay to a low level of tonic activity (Fig. 2A).
4.4 State-dependent inhibition of mitral cells