Simultaneous detection and estimation in olfactory sensing

Simultaneous detection and estimation in olfactory sensing | bioRxiv /* */ /* */ <!-- <!-- /*! * yepnope1.5.4 * (c) WTFPL, GPLv2 */ (function(a,b,c){function d(a){return"[object Function]"==o.call(a)}function e(a){return"string"==typeof a}function f(){}function g(a){return!a||"loaded"==a||"complete"==a||"uninitialized"==a}function h(){var a=p.shift();q=1,a?a.t?m(function(){("c"==a.t?B.injectCss:B.injectJs)(a.s,0,a.a,a.x,a.e,1)},0):(a(),h()):q=0}function i(a,c,d,e,f,i,j){function k(b){if(!o&&g(l.readyState)&&(u.r=o=1,!q&&h(),l.onload=l.onreadystatechange=null,b)){"img"!=a&&m(function(){t.removeChild(l)},50);for(var d in y[c])y[c].hasOwnProperty(d)&&y[c][d].onload()}}var j=j||B.errorTimeout,l=b.createElement(a),o=0,r=0,u={t:d,s:c,e:f,a:i,x:j};1===y[c]&&(r=1,y[c]=[]),"object"==a?l.data=c:(l.src=c,l.type=a),l.width=l.height="0",l.onerror=l.onload=l.onreadystatechange=function(){k.call(this,r)},p.splice(e,0,u),"img"!=a&&(r||2===y[c]?(t.insertBefore(l,s?null:n),m(k,j)):y[c].push(l))}function j(a,b,c,d,f){return q=0,b=b||"j",e(a)?i("c"==b?v:u,a,b,this.i++,c,d,f):(p.splice(this.i++,0,a),1==p.length&&h()),this}function k(){var a=B;return a.loader={load:j,i:0},a}var l=b.documentElement,m=a.setTimeout,n=b.getElementsByTagName("script")[0],o={}.toString,p=[],q=0,r="MozAppearance"in l.style,s=r&&!!b.createRange().compareNode,t=s?l:n.parentNode,l=a.opera&&"[object Opera]"==o.call(a.opera),l=!!b.attachEvent&&!l,u=r?"object":l?"script":"img",v=l?"script":u,w=Array.isArray||function(a){return"[object Array]"==o.call(a)},x=[],y={},z={timeout:function(a,b){return b.length&&(a.timeout=b[0]),a}},A,B;B=function(a){function b(a){var a=a.split("!"),b=x.length,c=a.pop(),d=a.length,c={url:c,origUrl:c,prefixes:a},e,f,g;for(f=0;f<d;f++)g=a[f].split("="),(e=z[g.shift()])&&(c=e(c,g));for(f=0;f<b;f++)c=x[f](c);return c}function g(a,e,f,g,h){var i=b(a),j=i.autoCallback;i.url.split(".").pop().split("?").shift(),i.bypass||(e&&(e=d(e)?e:e[a]||e[g]||e[a.split("/").pop().split("?")[0]]),i.instead?i.instead(a,e,f,g,h):(y[i.url]?i.noexec=!0:y[i.url]=1,f.load(i.url,i.forceCSS||!i.forceJS&&"css"==i.url.split(".").pop().split("?").shift()?"c":c,i.noexec,i.attrs,i.timeout),(d(e)||d(j))&&f.load(function(){k(),e&&e(i.origUrl,h,g),j&&j(i.origUrl,h,g),y[i.url]=2})))}function h(a,b){function c(a,c){if(a){if(e(a))c||(j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}),g(a,j,b,0,h);else if(Object(a)===a)for(n in m=function(){var b=0,c;for(c in a)a.hasOwnProperty(c)&&b++;return b}(),a)a.hasOwnProperty(n)&&(!c&&!--m&&(d(j)?j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}:j[n]=function(a){return function(){var b=[].slice.call(arguments);a&&a.apply(this,b),l()}}(k[n])),g(a[n],j,b,n,h))}else!c&&l()}var h=!!a.test,i=a.load||a.both,j=a.callback||f,k=j,l=a.complete||f,m,n;c(h?a.yep:a.nope,!!i),i&&c(i)}var i,j,l=this.yepnope.loader;if(e(a))g(a,0,l,0);else if(w(a))for(i=0;i (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0];var j=d.createElement(s);var dl=l!='dataLayer'?'&l='+l:'';j.src='//www.googletagmanager.com/gtm.js?id='+i+dl;j.type='text/javascript';j.async=true;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-M677548'); Skip to main content Home About Submit ALERTS / RSS Search for this keyword Advanced Search New Results Simultaneous detection and estimation in olfactory sensing View ORCID Profile Chen Jiang , View ORCID Profile Matthew Y. He , View ORCID Profile Venkatesh N. Murthy , View ORCID Profile Cengiz Pehlevan , View ORCID Profile Jacob A. Zavatone-Veth , View ORCID Profile Paul Masset doi: https://doi.org/10.1101/2025.11.01.686013 Chen Jiang 1 Department of Psychology, McGill University , Montréal, QC, H3A 1G1, Canada 2 Quantitative Life Sciences, McGill University , Montréal, QC, H3A 1G1, Canada Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Chen Jiang Matthew Y. He 1 Department of Psychology, McGill University , Montréal, QC, H3A 1G1, Canada Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Matthew Y. He Venkatesh N. Murthy 3 Center for Brain Science, Harvard University , Cambridge, MA, 02138, USA 4 Department of Molecular and Cellular Biology, Harvard University , Cambridge, MA, 02138, USA 5 Kempner Institute for the Study of Natural and Artificial Intelligence, Harvard University , Cambridge, MA, 02138, USA Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Venkatesh N. Murthy Cengiz Pehlevan 3 Center for Brain Science, Harvard University , Cambridge, MA, 02138, USA 5 Kempner Institute for the Study of Natural and Artificial Intelligence, Harvard University , Cambridge, MA, 02138, USA 6 John A. Paulson School of Engineering and Applied Sciences , Cambridge, MA, 02138, USA Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Cengiz Pehlevan Jacob A. Zavatone-Veth 3 Center for Brain Science, Harvard University , Cambridge, MA, 02138, USA 7 Society of Fellows, Harvard University , Cambridge, MA, 02138, USA Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Jacob A. Zavatone-Veth For correspondence: jzavatoneveth{at}fas.harvard.edu paul.masset{at}mcgill.ca Paul Masset 1 Department of Psychology, McGill University , Montréal, QC, H3A 1G1, Canada 8 Mila - Québec AI Institute , Montréal, QC, H2S 3H1, Canada Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Paul Masset For correspondence: jzavatoneveth{at}fas.harvard.edu paul.masset{at}mcgill.ca Abstract Full Text Info/History Metrics Preview PDF Abstract The mammalian olfactory system shows an exceptional ability for rapid and accurate decoding of both the identity and concentration of odorants. Previous works have used the theory of compressed sensing to elucidate the algorithmic basis for this capability: decoding odor information from the responses of a restricted repertoire of receptors is possible because only a few relevant odorants are present in any given sensory scene. However, existing circuit models for olfactory decoding still cannot contend with the complexity of naturalistic olfactory scenes; they are limited to detection of a handful of odorants. Here, we propose a model for olfactory compressed sensing inspired by simultaneous localization and mapping algorithms in navigation: the set of odors that are present in a given scene, and the concentration of those present odors, are inferred separately. To enable rapid inference of odor presence in a biologically-plausible recurrent circuit, our model leverages the framework of Mirrored Langevin Dynamics, which gives a general recipe for sampling from constrained distributions using rate-based dynamics. This results in a recurrent circuit model that can accurately infer presence and concentration at scale and can be mapped onto the primary cell types of the olfactory bulb. This frame-work offers a path towards circuit models—for olfactory sensing and beyond—that both perform well in naturalistic environments and make experimentally-testable predictions for neural response dynamics. PARP positron emission tomography radioligand breast glioblastoma prostate pancreas 1 Introduction Animals sense the physical properties of their external world using specialized neural circuitry [ 1 ]. To enable rapid adaptation of behavior to the demands of a changing world, early stages of sensory processing leverage the statistical properties of sensed signals [ 2 – 5 ]. This evolutionary adaptation leaves an imprint on the structure of neural circuits [ 5 – 7 ]. The most important statistical analysis problem faced by many mammals is that of odorant detection, of analyzing the composition of a given mixture of scents. Mammals—particularly rodents—rely on reliable olfactory sensing to perform tasks essential for survival, such as detecting predators [ 8 ], recognizing conspecifics [ 9 , 10 ], and locating food sources [ 11 , 12 ]. However, olfactory sensing is a highly challenging computational task, as an animal has access only to intermittent samples of odorants wafted on turbulent plumes of air, from which it must determine which among a myriad of possible sources is present [ 11 , 13 – 15 ]. This challenge is exacerbated by the fact that olfactory sensing is bottlenecked by the limited repertoire of olfactory receptor proteins, which range in number from ∼ 300 in humans to ∼ 1000 in mice ( Figure 1a-b ) [ 16 ] Download figure Open in new tab Figure 1. The structure of olfactory sensing. a . Outline of the anatomy of the mammalian olfactory bulb (OB). Volatile odorant molecules first bind to olfactory receptors on the surface of the olfactory sensory neurons (OSNs) that tile the olfactory epithelium. Each OSN expresses a single receptor type; humans have a repertoire of about 300 receptor types, while mice have about 1000 [ 9 , 16 , 43 ] OSNs synapse onto the primary excitatory projection neuron types of the OB—mitral and tufted cells—in clustered olfactory glomeruli containing axons of OSNs expressing the same, single receptor type [ 44 , 45 ]. Mitral and tufted cells then project to higher areas, notably including the posterior piriform (primary olfactory) cortex and the anterior olfactory nucleus (AON), respectively [ 22 , 25 , 46 , 47 ]. Recurrent connectivity within the mitral and tufted cell layers is mediated via inhibitory granule cells, which make dendro-dendritic synapses with their excitatory partners. Feedback from higher areas to the bulb comes in the form of synapses onto the granule cells. b . Dimensionality of the sensing problem. The compressed OSN representation s of the olfactory world ( left ) is given by a (stochastic) function f (·) of the product of the matrix A of the affinities of each OSN to each odorant ( center ) with the sparse, high-dimensional odor scene vector giving the concentrations of which out of millions of possible odorants are present in a given scene ( right ). For illustrative purposes, we show 10 receptors and 15 odorants. Despite this challenge, rodents can perform many olfactory sensing tasks with exquisite speed and accuracy, often within the hundred-millisecond timescale of a single sniff [ 12 – 14 , 17 – 19 ]. Due to the difficulties inherent in generating tunable yet naturalistic olfactory stimuli in the laboratory, detailed probes of these capabilities have only become possible in recent years [ 12 , 14 , 18 – 20 ]. This growing body of work has begun to elucidate the key role of the statistics of olfactory landscapes in perception and detection. Moreover, it has revealed that the different cell types of the olfactory bulb (OB)—the locus of early olfactory processing in mammals ( Figure 1a )—display strikingly diverse tuning and dynamics in response to odor stimuli [ 21 – 25 ]. How might these remarkable abilities arise? A growing body of theoretical work suggests that the answer may lie in the theory of compressed sensing [ 26 , 27 ]: accurate analysis is possible because only a few odorants—tens or hundreds out of a million or so possibilities—are present in a given scene [ 3 , 28 – 40 ]. Given a compressed sensing framework, the uniquely-structured recurrent circuits of the OB offer a potential substrate for the implementation of simple algorithms to solve the resulting problem of inferring what odorants are present given the noisy activity of olfactory sensory neurons (OSNs) ( Figure 1a ). However, the algorithms proposed by these past works—including our own [ 39 ]—have not shown how to simultaneously overcome two key challenges: First, to generate experimentally-testable predictions, it is useful to show that the algorithm can be implemented as a biologically-plausible recurrent neural network (RNN). Second, the algorithm should be able to scale up to dynamic olfactory scenes with a naturalistically large odorant repertoire. In this work, we propose a framework for olfactory sensing that allows us to achieve high detection capacity in a recurrent circuit model. Our model combines two central ideas: First, odorant presence and concentration are inferred separately. This split inference procedure is inspired by simultaneous localization and mapping (SLAM) algorithms in vision-guided navigation [ 41 ], and builds on previous works that have used split inference in olfaction to specify more flexible priors on presence [ 29 , 36 ]. However, inference of odor presence is challenging to implement in a rate-based recurrent circuit because presence is binary; past works did not overcome this obstacle. This leads to the second main idea of our model: we leverage the methodology of Mirrored Langevin Dynamics [ 42 ] to design a recurrent circuit sampler that samples a softened presence estimate while remaining biologically plausible. This approach to designing simple recurrent circuits that can solve constrained inference problems should be applicable to sensory neuroscience beyond the context of olfaction. We show that the resulting model enables fast and robust inference of more odorants than structurally-similar recurrent circuit models that do not split the inference problem, accurately analyzing static scenes with hundreds of odorants present out of a repertoire of tens of thousands within the timescale of a single sniff. This significantly exceeds the detection capacity demonstrated by previous olfactory compressed sensing models. At the same time, our model is itself a recurrent circuit that demonstrates rich cell-type-specific neural dynamics in response to odor stimuli. These diverse dynamics qualitatively resemble those that have been measured in the mitral and tufted cells of the OB [ 21 – 25 ]. Thus, our work lays the groundwork for the development of performant circuit models that can generate detailed, experimentally-testable predictions about response dynamics in the olfactory bulb. 2 Simultaneous detection and estimation in olfactory scenes Following the principle of analysis-by-synthesis, the starting point for our odorant recognition model is a generative model for olfactory scenes. At the most basic level, an odor landscape is determined by which sources are present; in the Montréal cityscape one might imagine coffee alongside bagels pulled freshly from a wood-fired oven. The presence of these sources then implies the presence of a characteristic set of volatile odorants which they emit. These odorants are then transported by turbulent airflow to the nose, where they can at last bind to olfactory receptors, driving activity in OSNs and thus generating a perceivable smell. Importantly, there is a separation of the timescales at which odorant presence and measured concentration fluctuate. The underlying presence of an odorant varies relatively slowly, as sources appear and disappear. In contrast, the measured concentration at the olfactory sensory epithelium displays millisecond-timescale fluctuations over many orders of magnitude. These concentration fluctuations are driven both by the physical nature of turbulent transport in air before an odorant molecule arrives at the nose [ 14 , 15 , 48 ], and by the process of sniffing as a mammal actively inhales air into the nasal cavity [ 48 ]. This means that apparent concentrations of present odors will fluctuate wildly—and can indeed appear to be zero, as in some regimes of turbulence an odorant will appear only in whiffs [ 15 ]—but non-present odorants will remain absent until the sources change. These physical and statistical distinctions suggest that it should be computationally advantageous to separately infer which odorants are present in a given scene, and the underlying concentration of the present odorants. This separation is reminiscent of the principles underlying simultaneous localization and mapping (SLAM) algorithms for visually-guided navigation in robots, which separate the inference of an agent’s rapidly-changing position from that of a map of slowly-changing landmarks [ 41 ]. From a probabilistic perspective, it corresponds to treating separately the distribution over presence p and the conditional distribution of concentration c given p : Here, we focus on inference from a single, static snapshot of the olfactory world, modeling the decoding of odorant presence from a single sniff [ 13 ]. As we will contemplate at some length in the Discussion, this means that our inference model cannot take advantage of concentration measurements across multiple sniffs, which would allow it to directly leverage the separation of timescales in presence and concentration dynamics. However, focusing on static snapshots makes it easier to develop and test the formalism for olfactory SLAM we introduce here. This lays required groundwork for future incorporation of dynamical priors. With this setup, we can now state the statistical model for odor scenes that we use in this work. We assume that there are n odor potential odorants, and n OSN types of olfactory sensory neurons, with n odor ≫ n OSN . Thus, , while . We first specify a model for OSN firing rates as a function of apparent concentration, which gives the likelihood in our statistical model. Then, in keeping with the discussion above, we separately specify priors on the presence of odorants and of the concentration of present odorants. For simplicity, we assume that different odorants are independent and identically distributed under the prior. This could be relaxed, but as a starting point we focus on the i.i.d. case here (see Discussion). To determine the likelihood, we model the mean activity of OSNs as a linear function of the concentration, with a receptor affinity matrix and a baseline rate . This is an accurate model within moderate concentration ranges [ 49 ], but neglects nonlinear effects which are known to be important in olfactory sensing; see the Discussion for details. Following past works [ 29 , 39 ], we then use a Poisson noise model for OSN activity . The OSNs fire independently, so the overall likelihood is for where we assume that the OSNs have identical baseline firing rates r 0 , and use in the likelihood the presence-masked concentration p i c i . We will compare several toy models for the distribution of affinities. In defining the affinity matrix A , we make the simplifying assumption that each OSN expresses a distinct receptor type. This disregards the fact that although each OSN expresses a single receptor type, there are multiple OSNs that express the same receptor type at different levels that converge on the same glomerulus [ 9 , 16 , 43 , 44 ]. We now specify a realistic ‘spike and slab’ prior on the presence-masked concentration c i p i , where the ‘spike’ assigns a large prior probability to the measured concentration being zero, and the ‘slab’ captures the broad distribution of concentrations of present odorants ( Figure 2a ). Using masking makes this easy, as by specifying a prior on presence that favors p i = 0 we immediately obtain the desired ‘spike’ at zero in the distribution of p i c i . For simplicity, we use a Bernoulli distribution for ϖ small. Thus, under the prior, the average number of present odors is . Now we must specify the prior on c p . In the final estimated concentration c i p i , the value of c i is unconstrained when p i = 0. Thus, for simplicity we use the same prior as when p i = 1 (see Appendix D and Figure S3 for a discussion of presence-dependent priors). To mimic the broad distribution of odorant concentrations, we use as our prior on c i a Gamma distribution with parameters α and β : Download figure Open in new tab Figure 2. Prior and Mirrored Langevin dynamics. a . Spike and slab prior over concentration, which is a mixture of a Dirac delta function (orange curve) and a Gamma distribution (blue curve). b . Mirror map illustrating the nonlinear transformation between dual and primal spaces with the mirror map σ and its inverse σ − 1 . Here, we set the gain of the sigmoidal mirror map σ to γ = 3. This transformation projects a bounded interval of primal space to an unconstrained dual space. The sampling is performed in the dual space and projected back to the primal space. c . Example dynamics of the dual variable u ( t ) and the corresponding primal variable p ( t ). The mirror map σ is a sigmoid with gain γ = 5. The distribution of samples in the primal space is concentrated near the boundaries at 0 and 1. This gives us the desired ‘slab’ for present odors ( Figure 2a ). This probabilistic model for olfactory scenes was used in previous work by Grabska-Barwińska et al . [ 29 , 50 ]. However, they did not provide a circuit implementation, limiting the ability of their model to generate testable predictions for experiment. Moreover, they did not show that their model could scale to scenes with more than a handful of odorants. In [ 39 ], we provided a circuit algorithm for non-separated inference of presence and concentration, but likewise found a limited capacity of only tens of odorants out of a repertoire of up to about 8000. It is these two limitations which we seek to overcome in this work. 3 Estimating presence through mirrored Langevin dynamics The key obstacle to a recurrent circuit sampler is the fact that the presence is binary. This means that one cannot use the standard approach of writing down the Langevin dynamics that sample the posterior [ 29 , 39 , 51 ]. Grabska-Barwińska et al . [ 29 ] used Gibbs sampling to estimate odor presence, which maintains the exact binary nature of presence estimates, but cannot be readily mapped onto a rate-based recurrent neural network. To resolve this issue, we consider a continuous relaxation of the presence variables from { 0, 1 } to [0, 1]. Then, we can leverage the framework of Mirrored Langevin Dynamics (MLD), which allows Langevin sampling to be applied to constrained problems [ 42 ]. We emphasize that we will not endeavor to provide mathematically rigorous guarantees for the efficacy of our models—indeed, not all of the relevant distributions are log-concave, see Appendix B for details—and will rely on simulations. Building on the framework of mirror descent [ 52 , 53 ], the core idea of MLD is to express the constrained variable of interest in terms of an unconstrained “dual space” variable, and to sample using unconstrained Langevin dynamics in the dual space [ 42 ] ( Figure 2b ). Importantly, this mapping between the constrained “primal space” and the dual space is defined using a “mirror map”, so that it is reversible. This framework provides a powerful way to sample from constrained distributions using recurrent dynamics, but to our knowledge has not seen previous application in neuroscience. We therefore give a general, informal overview of the MLD framework in the Methods. In the present case, we express in terms of a dual variable where the mirror map is the sigmoid with gain γ , and where we use the notation [ n odor ] = { 1, …, n odor }. With this choice of mirror map, we can follow the MLD recipe to obtain a rate-based circuit that samples the posterior of interest. We give a detailed derivation in Appendix B, and state the final circuit model here. Let be the ratio between the observed snapshot of OSN activity and the mean prediction from the likelihood based on the current concentration estimate c i ( t ) p i ( t ), that is, a ratiometric prediction error [ 39 ]: In terms of h ( t ), the dynamics of the dual presence variable u ( t ) and the latent concentration variable c ( t ) are: where d B u and d B c are independent n odor -dimensional isotropic Gaussian noises, and the presence prior parameter is packaged into ρ = − log[ ϖ/ (1 − ϖ )]. We define ρ such that it is positive when ϖ < 1 / 2, i . e ., when odors are more often not present under the prior. Here, we have also introduced time constants τ u and τ c to set the timescale of the inference dynamics; these should be similar to ensure reliable sampling. Because this model performs s imultaneous d etection and estimation in o lfactory scenes, we will refer to it by the acronym “SDEO”. If we simulate the dynamics (8) with modestly large gain values, the mirror map translates a relatively smoothly-varying dual space signal u ( t ) into a neatly nearly-binarized presence estimate p ( t ) ( Figure 2c ). Therefore, at the most basic level, the continuous relaxation yields an interpretable estimate of presence. The joint presence-concentration dynamics involve multiple forms of gating; the dynamics for u are in particular gated both by presence and concentration through p i (1 − p i ) c i . If α > 1, the divisive 1 /c i term from the concentration prior acts as a repulsive barrier preventing concentration estimates from going to zero. As shown in Figure 3c , our SDEO model accurately tracks the presence and concentration of changing odorants in a simple scene. Here, we simply model the sensitivity matrix A as a sparse binary matrix; see Figure S2 for a similar test with a different model for the affinity matrix. Fixing p = 1 and running Langevin sampling to infer c , we recover the model studied in our previous work [ 39 ] ( Figure 3c ). Returning to Figure 3 and comparing this non-separated model to the SDEO model developed above, we see that that the new model converges far more rapidly to a more accurate estimate of presence and concentration upon changes in the olfactory scene. This performance improvement—even in a relatively simple scene—is consistent with our conceptual motivations for considering split inference of presence and concentration. Download figure Open in new tab Figure 3. Dynamics of non-separated and SDEO models during estimation of odorants present in a slowly-changing scene. a. A static sensor downstream of two odor sources will measure an intermittent, variable mixture of the emitted odorants as they are transported by turbulent airflow. We illustrate this with a sketch inspired by the plume measurements of Nowotny and Szyszka [ 48 ]. b . In the laboratory, it is challenging to mimic the fast-timescale dynamics of a turbulent plume, but one can generate changing steps of concentration [ 14 , 18 , 19 ]. We model this scenario by selecting two groups of three random odorants each. Each group—intended to model the odor emitted by one source—has common concentration fluctuations, but remains present throughout the entire in silico experiment. Within each group, the concentrations of each odorant are identical, but the concentrations for the two groups change independently. We present this simulated odor scene to three models: non-separated ( left ; as in [ 39 ]), SDEO ( center )), and SDEO with circuit implementation ( right ). In each plot, the colored lines denote the estimated concentration for the presented odorants, while the gray lines represent those for the background (non-presented) odorants. The dashed line traces true concentration over time. In these simulations, we use a sparse binary affinity matrix. For details of our numerical methods and the corresponding plot using a dense gamma-distributed affinity matrix, see Appendix F.4.2 and Figures S1 and S2 . 4 Cell-type-specific computations and dynamics To this point, we have shown how MLD allows us to build a recurrent circuit model for olfactory SLAM, and demonstrated that it can enable faster inference relative to an analogous model that simultaneously infers presence and concentration. We now seek a biological interpretation for this model, in particular one that attributes distinct functional roles to distinct cell types. 4.1 Removing divisive nonlinearities by introducing auxiliary neurons As in practice we see that the presence estimate p is essentially binary ( Figure 4a ), we hard gate it in the dynamics of c while keeping the continuous relaxation in the dynamics of u . We set when p i ≥ p th and when p i 0 is a given threshold parameter. Replacing p with in the dynamics of c while keeping the dynamics of u unchanged, we have the dynamics of the hard-gated version of the model: where . Download figure Open in new tab Figure 4. Internal dynamics of SDEO models during estimation of odorants present in the slowly-changing scene shown in Figure 3 . In a and b , we show the basic SEDO model and its circuit implementation, respectively. The first row shows the estimated dual-space presence variable u , the second row shows the estimated presence in the [0, 1 ]-bounded primal space p (with a dashed line marking the threshold used to binarize presence during inference), the third row shows the latent concentration estimate c (which does not return to zero for non-present odors because of our choice of prior), and the fourth row shows the output concentration estimate c i p i . The panels in the fourth row are identical to the corresponding panels in Figure S2c . In each plot, the colored lines denote the estimated concentration for the presented odorants, while the gray lines represent those for the background (non-presented) odorants. Fluctuations visible in the primal space presence for non-present odors occur for non-present odors with low estimated concentrations, for which the presence can fluctuate without substantially changing the masked concentration c i p i . This is visible in the fact that the output concentrations for non-present odors remain near zero in the fourth row. In these simulations, we use a sparse binary affinity matrix. For details of our numerical methods and the corresponding plot using a dense gamma-distributed affinity matrix, see Appendix F.4.2 and Figures S1 and S2 . These dynamics include divisive non-linearities, which make it hard to implement in a biologically plausible way [ 54 ]. Following our previous work [ 39 ], we linearize the dynamics using the method proposed in Chalk et al . [ 55 ]. Concretely, we introduce three additional cell types with rates and , with dynamics chosen such that they have as their fixed points the required divisions that define h , , and the divisive prior term ( α − 1) /c i = z i : Substituting the instantaneous rates for the divisions in the dynamics for c and p , we obtain a circuit model with five cell types. If the time constants , and τ g are small relative to the timescale of the dynamics of p and c , we expect this expanded circuit to perform comparably well to the model where the divisions are computed exactly [ 39 , 55 ]. We can map these dynamics on the circuit architecture of the olfactory bulb [ 7 , 56 ]. As they are excited by the OSN input, we interpret h and as the two classes of projection neurons in the OB (mitral and tufted cells). Then, the concentration estimate c and presence estimate p are encoded by local interneurons (granule cells), which inhibit the projection neurons and gate each other’s dynamics. The z neurons required to linearize the prior can then be interpreted as a form of cortical feedback onto the granule cells. With this coarse mapping in mind, we use time constants that are comparable to those measured in experiment for the major cell types of the olfactory bulb: we set ms to roughly match the primary projection neurons of the bulb, the mitral and tufted cells [ 57 ], and we set τ u = τ c = 30 ms to roughly match the primary local interneurons of the bulb, the granule cells [ 58 ]. Though the resulting timescales are clearly not substantially separated, reliable inference is possible despite the resulting approximation to the divisions. In particular, the dynamics of this elaborated circuit model closely match that for the algorithmic split inference model introduced before ( Figure 4a-b ), with performance remaining far superior to the non-split model ( Figure 3c ). With this performance check in hand, we now turn to a more careful study of the dynamics of the different cell types in our model. 4.2 Dynamics and tuning of the hard- and soft-gated projection neurons Although computational models of the olfactory bulb have provided great insights into the computational role of different cell types, they have usually ignored the differences between the two distinct dominant classes of projection neurons, the mitral and tufted cells [ 28 , 29 , 39 , 50 ]—but see Tootoonian and Schaefer [ 59 ]. Suggestively, our elaborated model as introduced above has two classes of projection neurons—hard and soft threshold, respectively—which raises the question of how their responses compare to known properties of mitral and tufted cells. In simple scenes modeling the static odorant mixtures presented in experiments, we find that our model recapitulates the differences in response amplitude, concentration dependence, and duration that are characteristic of mitral and tufted cells. At the population level, both classes exhibit a transient burst of activity at the time of odor onset ( Figure 5 ). Softgated neurons then relax to a plateau above the baseline whose early amplitude and duration increase monotonically with concentration ( Figure 5a,b,g ). This is consistent with the response dynamics of the single class of projection neurons in our previous work [ 39 ] and with what has been observed experimentally in tufted cells [ 21 , 22 , 25 ]. By contrast, hard-gated neurons show a biphasic pattern: a steep suppression below baseline after the burst and then raising slowly before settling into baseline activity. Higher concentrations evoke deeper initial drops followed by a slower but slightly higher rise. ( Figure 5a,c,h ), consistent with the more complex dynamics of mitral cells [ 21 , 22 , 25 ]. Figure 5d further quantifies how stimulus intensity shapes the response dynamics by plotting the time-window–averaged firing rate (0–100 ms after onset) as a function of concentration, which suggests that soft-gated neurons (putative tufted cells) have a clearer monotonic response to increases in concentration that hard-gated neurons (putative mitral cells) as observed experimentally [ 25 ]. Download figure Open in new tab Figure 5. Comparison between hard gated projection neuron (labeled as hard threshold) and soft gated projection neuron h (labeled as soft threshold). A set of fixed odor stimuli is present at 0s and withdrawn at 0.6s. We set the Gamma distribution as the concentration prior with shape parameter α = 5 and rate parameter β = 0.1, and the Bernoulli prior for presence with ϖ = 0.01. a Mean firing rate over time of two thresholding projection neurons. Upon stimulation, both the hard-gated neuron and the soft-gated neuron exhibit a transient burst of activity. After the burst, soft-gated neurons reduce to a plateau higher than baseline, while hard-gated neurons drop immediately, then rise again before settling into baseline activity. When the stimulation disappears, soft-gated neurons drop to baseline instantly, whereas hard-gated neurons exhibit a negative peak and then converge to baseline smoothly. b. Mean firing rate of soft-gated neurons under varying odor concentrations. Darker color indicates stronger stimulation. At odor onset, higher concentrations evoke longer-lasting responses, yet all activities converge to the same plateau. c . Mean firing rate of hard-gated neurons under varying odor concentrations. At odor onset, higher concentrations evoke deeper initial drops followed by a slower but higher rise. At odor offset, higher concentrations evoke lower negative peak, after which all responses converge back to baseline d . Mean firing rate as a function of odor concentration, averaged over the first 100 ms after odor onset. Error bars indicate standard error across trials. The soft-gated neurons respond monotonically to concentration. e . Example dynamics of a single soft-gated neuron under varying concentration. f . Example dynamics of a single hard-gated neuron under varying concentration. g, h . Baseline-subtracted activity of 300 soft-gated neurons and 300 hard-gated neurons, ordered by the response magnitude at concentration 50 (averaged over 50 ms after odor onset). A subset of soft-gated neurons and hard-gated neurons exhibits a stronger and loner peak at odor onset when concentration increases. The rest soft-gated neurons are stable, remain baseline or slightly below, whereas the remaining hard-gated neurons exhibit a negative peak, after which some recover to baseline and others rise above baseline. For additional single-neuron responses, see Figure S4 . Examining baseline-subtracted single neuron responses, we can see that there are activated and suppressed neurons in both classes ( Figure 5e-h ). The activated soft-gated neurons exhibit sustained activities during stimulation. The suppressed soft-gated neurons are stable, remaining at or slightly below baseline. In contrast, activated hard-gated neurons show transient bursts, which are strengthened and prolonged as concentration increases. The other hard-gated neurons show a negative peak, after which followed by one of three outcomes: persistent suppression, return to baseline, or raise above baseline. These contrasting motifs identified here yield experimentally-testable predictions for the diversity of neural response dynamics. However, the model’s response dynamics still deviate from experimental observations, particular in onset latency, sparsity, and the response threshold [ 21 , 22 , 25 , 60 ]. We will return to this issue, and more broadly to the limitations of how well our model can be mapped to the circuitry of the olfactory bulb, in the Discussion. These limitations notwithstanding, these results illustrate the potential of our framework to generate testable predictions for cell-type-specific dynamics. 5 Scaling of olfactory inference Thus far, we have shown that our model yields biologically-plausible computational dynamics. This fulfills one important desideratum of models for olfactory sensing. However, an algorithm must also be performant. Therefore, we developed two scaling simulations to evaluate our model under large network size and high dimensionality. The first quantifies the model’s ability to estimate odorants on a timescale of hundreds of milliseconds, while the second examines how the required number of OSNs scales with the size of potential odorant dictionary. As a baseline, we use the non-split inference model we proposed in [ 39 ]. For the non-separated model which lacks explicit presence estimation, we approximate presence by binarizing the concentration estimates using half of the true concentration as the threshold. In general, to show the higher performance of the SDEO model, our evaluation criteria are much stricter than those used in [ 39 ]. We give a comprehensive discussion of our rationale for the choice of metrics in Appendix F.3. 5.1 Speed of inference We first evaluate our model’s ability to infer odor presence rapidly. With a potential odorant dictionary of size 1000 and 300 OSNs, the SDEO models can detect the concentration of ∼ 40 simultaneously presented odorants with at least 80% accuracy and mean absolute error ≤ 25% · c True ( Figure 6b,d ; see also Figure S5 ). For presence estimates we can also see a similar but slightly improved limits of detection at ∼ 50 simultaneously presented odorants. On the other hand, the non-separated model can only detect concentration of less than 10 odorants reliably, and detect presence reasonably accurate for at most 25 odorants. Moreover, we can see that SDEO models have significantly faster convergence rate where they reach steady states after ∼ 100 milliseconds, while the non-separated model needs seconds to converge. In both concentration and presence estimation, the SDEO models demonstrate considerable improvement in both estimation speed and maximum number detection capacity compared to non-separated model under the same conditions. Download figure Open in new tab Figure 6. Improvement in fast detection of multiple odorants through separation of inference. We evaluate the same three models as in Figure 3 in a series of simulation where increasing numbers of odorants are simultaneously presented. In each run, in a set of 1000 odorants, a number of them are randomly selected and presented to the model for a duration of 0.75 s. We increase number of presented odors from 1 to 100, while repeat each setting for 40 times, compute the metrics and then take the average as final results. The shaded areas in a and c show ± 1.96 SEM (representing 95% C.I) over realizations throughout. The first and second rows assess the models’ performance in odorant concentration and presence estimation, respectively. a. Mean absolute error of estimated concentration as a function of the number of odorants present at two timepoints after odor onset. b . Heatmap of mean absolute error over inference time and number of presented odorants, with smoothed contours of correct detection fraction overlaid. c . AUROC score as a function of the number of odors present at two timepoints after odor onset. d . Heatmap of AUROC score over inference time and number of presented odors, with smoothed contours overlaid. For details of implementations and corresponding figures using dense Gamma sensing matrices, see Appendix F.4.3 and Figure S5 . Noticeably, we have also found the SDEO models exhibit a clear phase transition. Although having higher capacity compared to the non-separated model, the SDEO models collapse dramatically once the number of presented odorants exceeds their capacity, indicating qualitative changes in the network dynamics ( Figure 6b,d ; see also Figure S5 ). In contrast, the performance of the non-separated model deteriorates more gradually as the number of presented odorants increases. We also note that the phase transition for concentration and presence estimation occurs simultaneously at 50 presented odorants. This reflects the strong correlation between concentration and presence estimation in the SDEO models and suggests that the models indeed leverage the coupled dynamics to achieve better sampling rather than merely carrying out two independent tasks. 5.2 Detection capacity We now consider detection capacity, which we define as the detection capacity as the maximum number of simultaneously presented odorants that a model can detect with desired accuracy (see Appendix F.3.4 and the figure captions for details). From a compressed sensing perspective, the crucial question is how the number of sensors required to achieve a particular detection capacity scales with the size of the dictionary of potential odorants [ 26 , 61 ]. In Figures 7 and S6 , we examine the scaling of detection capacity with dictionary size and sensor repertoire across different combinations of models and sensing matrix. In the first two columns, we can see the SDEO model significantly out-performs the non-separated model in the range of entire heatmap. Thus, splitting the inference of presence and concentration improves the speed and accuracy of inference, yielding higher capacity. Upon closer inspection of the SDEO model’s dynamics, we find that the ultimate bottleneck of capacity arises from presence estimation, which drives the sampling of concentration. Download figure Open in new tab Figure 7. Scaling of detection capacity with dictionary size and sensor repertoire for different priors and sensing matrix models. The three columns correspond to three models—non-separated, SDEO and SDEO with circuit implementation—and the three rows correspond to three types of affinity matrices. These are: dense gamma, whose entries are i.i.d. random variable following Gamma(0.37, 0.36); sparse gamma, obtained by applying a 0.1 sparsity mask to a dense gamma matrix; and sparse binary, whose entries are i.i.d. random variable following Bernoulli(0.1). Each heatmap shows the maximum detection capacity assessed by presence estimates for combinations of sensors counts (from 100 to 800, equally spaced linearly) and dictionary size (1000 to 16000 equally spaced on a log scale). The maximum detection capacity κ AUROC is defined as the largest number of simultaneously presented number of odorants that the model can detect with a AUROC score ≥ 0.85. Smoothed contours are overlaid and can be interpreted as the required number of sensors to maintain a certain capacity as a function of dictionary size. The total inference time duration is 0.2 s for all runs, and the value in each cell of the heatmap is the average of three independent runs. For further discussion on affinity matrix and scaling capacity, see Appendix E. For additional details of implementation and the corresponding figures using mean absolute error to compute maximum detection capacity, see Appendix F.4.4 and Figure S6 . 5.3 Improving capacity by modifying the presence prior As detection capacity appeared to be bottlenecked by presence estimation, we therefore sought to improve presence estimation by modifying the corresponding prior. Up to this point, we have treated the presence of odorants as a binary variable, and correspondingly specified for it a Bernoulli prior. However, this is an imperfect choice under the continuous relaxation for presence estimation. The natural distribution of presence probability is bimodal on [0, 1 ] with density clustering near the endpoints. However, the density of the continuous Bernoulli prior decreases monotonically on [0, 1 ] with our chosen parameter ( Figure S7 ). To overcome this issue in the continuous relaxation of presence estimation, we want a naturalistic bimodal distribution on [0, 1 ] that can capture the natural distribution of presence probability. Ideally, such a distribution should also admit a tractable density and score. The obvious candidate for such a prior would be the Beta distribution, but its density and score are non-trivial to analyze and costly to compute. Past works have proposed the use of the Kumaraswamy (KS) distribution as an alternative to the Beta distribution in similar contexts [ 62 – 64 ]. Unlike the Beta distribution, the KS distribution involves solely elementary functions, and is therefore straightforward to incorporate into our model. However, the vanilla KS distribution suffers from instability issues due to exploding gradients near boundaries of its open support (0, 1). We overcome this by mapping a truncated KS distribution on the closed interval [0 + ε , 1 − ε ] to the full interval [0, 1 ] for a small cutoff ε ∼ O (10 − 5 ) (see Appendix C for details). This truncated KS prior allows for efficient and numerically-stable computation. As shown in Figures 7 and S6 , the SDEO model with truncated Kumaraswamy distribution prior on presence empirically outperforms the model with a continuous Bernoulli prior. Given the same odorant dictionary size, fewer sensors are required to reach a given capacity. Moreover, the limiting dimensionality is also increased, depicted by delayed explosion in the slope of the contours. This improvement is perhaps because the bimodal Kumaraswamy distribution better accounts for the underlying structure of presence, thus facilitating efficient sampling. Just as altering the prior improves the capacity, similar improvements result from sparsifying the affinity matrix: dense Gamma-distributed affinity matrices yield the lowest capacity, followed by sparse Gamma, and then by sparse binary. As we analyze and discuss in Appendix E and Figure S8 , this improvement likely results from the decreased mutual coherence—the maximum overlap between the affinity profiles of two sensors—of the sparse ensembles relative to the dense one, consistent with the existing compressed sensing literature [ 61 ]. In sum, particularly with an appropriate choice of prior and of affinity matrix, our SDEO model can detect and estimate the concentration of up to a hundred or so odorants out of a repertoire of tens of thousands ( Figures 7 and S6 ). This significantly exceeds the capacity of the model studied in our previous work, and of other models for olfactory compressed sensing. 6 Discussion In this work, we have combined insights from simultaneous localization and mapping in robotics [ 41 ], from compressed sensing [ 26 ], and from mirrored Langevin dynamics sampling algorithms [ 42 ] to propose a biologically-constrained model for odorant sensing ( Figure 1 - 3 ). Our algorithm recapitulates the experimentally observed distinct properties of the two main projection neuron classes in the bulb, mitral and tufted cells ( Figure 5 ). We further show that our approach scales to large sensory scenes, successfully detecting the presence and estimating the concentration of tens to hundreds of odorants amongst thousands of potential ones ( Figure 6 - 7 ). Our model suggests a map between anatomy and computational function, with the tufted cells contributing to the odor presence detection and the mitral cells to the concentration estimation computation. The dynamics of each cell type is broadly consistent with experimental observations [ 21 , 22 , 25 ] ( Figure 5 ) but our model leaves open how the information is used by cortical areas. The major cortical projection targets are distinct across the two cell types—with, roughly speaking, tufted cells projecting predominantly to anterior cortical regions like the AON and mitral cells projecting to more posterior regions like the piriform cortex—but there are secondary projections with substantial projection from mitral cells into the AON and some projection from tufted cells into piriform cortex [ 22 , 46 , 47 ]. This implies that the computational function of cortical areas is not readily mapped onto the presence and concentration axes. Representations in piriform cortex have been shown to carry both concentration and presence information [ 65 , 66 ], consistent with inputs from both mitral and tufted cells. Piriform also carries contextual information such as location [ 67 ], which could play a role in adapting the odor-processing prior [ 68 , 69 ]. Less is known about tuning to concentration in the AON but experimental data is consistent with a role in detection of odor identity which would be mediated by the tufted cell inputs in our model [ 70 – 73 ]. The circuit implementation of our model could also be extended by introducing a more detailed model of the granule cells and other cell types of the bulb [ 39 , 74 ]. In contrast to previous accounts of the M/T computational split [ 59 ], our model proposes that these two cell types interact through indirect, reciprocal multiplicative gating rather than additive excitation. This gating effect could be probed in future experiments. As in our previous work [ 39 ], the M/T cells encode a ratiometric prediction error between the current OSN activity and the prediction thereof based on the current estimate of concentration encoded in the granule cells. However, by introducing hard and soft gating, we form two predictions, and correspondingly two types of prediction error neurons. Existing models for predictive coding in other sensory systems that incorporate multiple cell types generally assume Gaussian noise models, and correspondingly subtractive prediction errors [ 75 ]. It will be interesting to contrast predictions of models encoding subtractive versus ratiometric errors in future work. Attempts to experimentally probe cell-type-specific response dynamics in the OB have traditionally been limited by the challenges inherent in controlling an odor stimulus. Unlike for vision or audition, it is challenging to deliver tunable olfactory stimuli that mimic the statistics of those encountered in the natural world. Typical experiments in psychology and neuroscience have delivered static odor flows at high concentrations over timescales of hundreds of milliseconds. It is only recently that experimental paradigms have allowed to combine recordings of neural activity with rapid control of the physical stimulus, through rapidly switching air flows for odor delivery [ 14 , 18 , 76 , 77 ], invasive approaches controlling breathing [ 49 ], or direct optogenetic activation of olfactory sensory neurons [ 78 – 81 ]. Thus, it is now possible to test predictions from a model like ours about the neural dynamics in response to temporally-structured stimuli. Despite the richness of its predictions, our model only provides at best a partial picture of the computational architecture of the olfactory bulb. As mentioned before, our work does not take full advantage of the olfactory SLAM framework, as we do not model the dynamics of the odor environment. A more complete circuit would incorporate into its prior a motion model for the slow dynamics of odor presence, as well as a prior for fast fluctuations in concentration due to sniffing [ 18 ] and turbulent transport [ 15 , 48 ]. Here, we have neglected odor dynamics to focus on the minimal setting in which we can test implementations of olfactory SLAM using MLD. However, dynamical priors could be incorporated into this framework without much difficulty. This would in turn allow us to make richer predictions about cell-type-specific dynamics in the olfactory system [ 21 , 24 , 55 ], and improve inference from multiple samples of more complex scenes. With the fact that one would eventually like to incorporate different dynamical priors on presence and concentration in mind, it is important to note that splitting the inference is not the only way to implement inference using a static spike-and-slab prior in a rate network. In recent work, Fang et al . [ 82 ] have proposed a Langevin sampling network for a spike-and-slab prior based on thresholding. As we detail in Appendix G, applying this methodology to the olfactory sensing problem leads to a gated RNN with a single population of neurons that directly encodes the sparse vector of concentration elements. It is not immediately clear how one could implement priors over concentration and presence with distinct timescales within this framework. Moreover, it cannot instantiate the separation of computational labor that is at the heart of the SLAM approach. Additionally, we used a simplified model for OSN responses that assumes that receptor affinities are independent and identically distributed (iid), and that the mean firing rate of each OSN is linear in concentration. The assumption of iid receptor affinities makes it easier to systematically probe scaling properties of the model, but is an idealization. Using these iid affinities, we saw that improved scaling capacity was obtained for sparse random affinities, which as mentioned before is qualitatively consistent with ideas from the theory of compressed sensing with Gaussian noise. It is also consistent with previous works that have shown that optimizing affinities to maximize the mutual information between odor concentrations and OSN activities (assuming either linear or simple non-linear models for the firing rate, and Gaussian noise) leads to sparse affinity matrices [ 3 , 4 , 32 ]. In these optimized models, the non-zero affinities are broadly-distributed—which our sparse Gamma model aims to roughly capture—but the distributions of individual elements contain structure beyond long tails [ 4 ]. Investigating how our results change given optimized sensitivity matrices and non-linear OSNs could be an interesting avenue for future investigation, but the optimization procedures used in past works are too computationally expensive to enable the investigations of scaling we perform here. Another level of structure which we have not yet incorporated into our model is correlations between the presence of different odorants that arise from their generation by a common source. Recent work has shown that such environmental correlations can shape the distribution of optimal receptor affinities [ 4 ]. Closer to our work here, Tootoonian and Schaefer [ 40 ] have recently proposed a circuit basis for the encoding of correlated odorant priors in the olfactory bulb. Their construction leverages sister mitral cells, which receive input from the same receptors but are connected to different granule cells [ 78 ]. Our model does not leverage this structure, and incorporating structure and hierarchies into the prior will be important as we move towards testing its ability to parse realistically-structured scenes. Another substantial limitation of the present work is that we consider a non-distributed coding scheme, in which each granule cell represents the estimate of the presence or concentration of a specific odorant. This appears inconsistent with physiology [ 7 , 70 ]. It also stands in contrast to our previous work on the non-split version of this inference model in [ 39 ]. There we showed that linearly distributing the odorant code— i . e ., writing the estimated concentration vector as c = Γg for model granule cell activities g and a decoding matrix Γ —allowed for significant improvements in performance if one chose Γ to cancel correlations in OSN responses induced by a dense sensing matrix. It is therefore possible that taking advantage of this geometry would close some of the gaps between different sensing matrix ensembles which we observed in Figure 7 . However, the simple linear mixing we used in [ 39 ] will not suffice here, as the additional nonlinearities in the model dynamics fix a preferred basis. Naïvely following the linear mixing approach leads to a model without a clear circuit interpretation. Determining how to construct a split inference model with fully distributed coding will be an important task for future work. This will also be an important prerequisite to more direct comparison of cell-type-specific dynamics in models with those measured in the OB. More generally, our work points to compositionality of neural representations, as has been studied across other modalities and neural circuits [ 83 – 85 ]. Decoupling concentration estimating and odor detection, would allow extensions of our model to learn complex priors over these two domains separately. For example the dependence on context of the structure of odor presence and of the dynamics of concentration fluctuations due to air flow to are likely to be only loosely correlated (presence of odor emitting objects vs properties of air flow in a given environment). In closing, we emphasize that the framework we develop here can be adapted to a broad array of inference problems that arise in other sensory modalities, beyond olfaction alone. In particular, MLD provides a general recipe for the design of recurrent circuits that solve constrained probabilistic inference problems. Recent work has applied mirror descent—which is to optimization as MLD is to sampling—to study learning in the presence of biological constraints on synaptic weights [ 86 ]. However, the application of this toolkit to neural dynamics is to our knowledge novel. At the same time, MLD has recently gained popularity in machine learning as a method to enable learning how to sample from constrained distributions using diffusion generative models [ 87 , 88 ]. It therefore holds much promise as a foundation for building circuit models that can process naturalistic, complex stimuli. 7 Methods 7.1 The Mirrored Langevin Dynamics recipe Here, we give a brief introduction to the Mirrored Langevin Dynamics (MLD) framework introduced by Hsieh et al . [ 42 ], and summarize its application to presence estimation. Abstractly, suppose that we wanted to sample a distribution P ( x ) over a variable x that is subject to some constraint. MLD prescribes that we should define a convex function ϕ ( x ) that transforms the constrained variable x into an unconstrained dual-space variable y via the “mirror map” and maps the un-constrained variable y back to the constrained variable x via Here, is the Fenchel dual—that is, the Legendre transform—of the convex function ϕ . With appropriate choices of mirror map, this framework can handle many classes of constraints, far beyond the simple hypercube constraint considered here [ 42 ]. Then, since y ( t ) is unconstrained, we can run naïve Langevin dynamics on y , for Gaussian noise d B u ( t ), where the stationary distribution Q ( y ) is chosen such that x ( y ) = y ϕ ∗ ( y ) has the desired distribution P ( x ). For x ( y ) to have density P ( x ), the density of y should be as the definition of the Fenchel conjugate implies that the Jacobian of the change-of-variables x ( y ) = y ϕ ∗ ( y ) is just inverse of the the Hessian of ϕ : where we abbreviate the Hessian as . By the chain rule, the score appearing in the Langevin dynamics for y is then where the right-hand-side of the equation should be viewed as a function of y through x ( y ). This informal sketch recovers the MLD prescription of Hsieh et al . [ 42 ]. For the case of presence estimation, where the constraint is just that p should lie within the hypercube , we can apply this recipe with for any γ > 0, which has Fenchel dual and thus yields as the mirror map the desired sigmoid With this choice, for the SDEO posterior the MLD recipe as sketched above leads to the dynamics (8); we defer the full details of this derivation to Appendix B. 7.2 Summary of numerical methods We now briefly summarize our numerical methods, deferring a detailed description to Appendix F. Our model is defined as a system of stochastic differential equations, which we integrate numerically using the Euler-Maruyama method with a fixed timestep of Δ t = 10 − 5 seconds. To enable efficient simulation in high dimensions—as is required to study capacity scaling in Figure 7 —we implement our simulations in PyTorch [ 89 ], which allows efficient, parallelizable multiplication of large matrices. We evaluate concentration estimation using the mean absolute error between estimated and true concentrations of present odorants. As a complementary measure, we consider the proportion of correct concentration estimates, which we define as those for which the estimated concentration is within a ± 25% interval of the true value. To assess the quality of presence estimation, we adopt the AUROC score, because inferring presence can be framed as a binary classification problem. For the non-separated model which lacks explicit presence estimation, we approximate presence by binarizing the concentration estimates using half of the true concentration as the threshold. We give a comprehensive discussion of our rationale for the choice of metrics in Appendix F.3. Author contributions PM and JAZ-V conceptualized this work, with input from VNM and CP. CJ and MYH contributed equally to coding, investigation, and figure preparation, with input from JAZ-V and PM. CJ, MYH, JAZ-V, and PM wrote an initial draft of the paper. All authors contributed to review and editing of the manuscript. JAZ-V and PM contributed equally to supervision and funding acquisition. Data and code availability Code to reproduce all figures is available on GitHub at: https://github.com/labmasset/Simultaneous-detection-and-estimation-in-olfactory-sensing . Supplemental Information A Notational conventions B Derivation of the full model In this Appendix, we give a detailed derivation of the inference model introduced in the main text. Before launching into the derivation, we note that we will use vector notation, with ⊙ representing elementwise multiplication and ⊘ representing element-wise division, i . e ., for two vectors a and b we write ( a ⊙ b ) i = a i b i and ( a ⊘ b ) i = a i /b i . We recall that our goal is to sample the Bayes posterior over odor presence and concentration given a snapshot of olfactory sensory neuron (OSN) activity. Following the simultaneous localization and mapping (SLAM) idea, we first specify a prior P ( p ) on presence, and then a prior P ( c | p ) on the concentration of the present odorants. As introduced in the main text, we aim to approximately sample the posterior by applying the framework of Mirrored Langevin Dynamics (MLD) [ 42 ] to a continuous relaxation of the presence variable to . The posterior enters these dynamics only through the gradients of the potential with respect to p and c , meaning that the marginal likelihood P ( s ) does not enter the subsequent calculations. Throughout, we use a Poisson likelihood as in prior works [ 29 , 39 ]. We leave the priors unspecified for now, as we will consider several different choices. B.1 Deriving the Mirrored Langevin Dynamics for presence estimation We now give a detailed derivation of the dynamics that result from applying the MLD recipe to joint inference of presence and concentration. As mentioned in the main text, we consider the mirror map where for any γ > 0, is the logistic sigmoid with gain γ , taken to act elementwise. The gain factor γ allows us to control the steepness of the sigmoid. To apply the MLD recipe, our goal is to construct a convex function with Fenchel conjugate such that which means that we should have The natural choice—as discussed by Hsieh et al . [ 42 ]—is then of course to take which has Fenchel conjugate Here, we use the usual information-theoretic convention that 0 log 0 = 0. As ϕ is additively separable, its Hessian is diagonal: This allows us to easily obtain hence For a potential V = − log P ( p, c | s ), the MLD recipe [ 42 ] then gives the dynamics Substituting in the properties of our choice of ϕ , we have or, in vector form We can then jointly sample c and u by simultaneously running the dynamics for an independent Brownian motion B c . To be extremely formal, we can also introduce a trivial mirror map for c , and consider the combined mirror map induced by which handles the constraint on p while leaving c intact. We remark in passing that not all of the distributions of interest are log-concave in the dual space (which is the relevant notion for MLD [ 42 ]). Guarantees for the convergence of stochastic gradient Langevin dynamics given certain sufficient conditions when sampling on non-log-concave target distributions are weaker in general than those known in the log-concave case, but theoretical progress has in recent years been rapid [ 90 , 91 ]. Since we are anyway relying on numerics rather than rigorous proofs for fast convergence, we will not dwell on this issue further. B.2 Gradients of the combined presence-concentration posterior We now explicitly compute the energy gradients that appear in the MLD. From the definition of the posterior (B.1), we have: so the required gradients are Since we are using a Poisson likelihood (B.3), denoting , then we have: where λ s , e − λ are all element-wise operations. Notably, although the Poisson PMF is discontinuous with respect to s , it is continuous and hence differentiable with respect to c and p . Therefore, we proceed with: Denote h = s ⊘ λ = s ⊘ ( r 0 + A ( c ⊙ p ), then we have Following the same procedure, we have the gradient with respect to p : Substituting (B.26) and (B.27) back to (B.20), (B.21), we have We leave the prior terms unspecified as we will subsequently explore different choices of prior. B.3 Model equations for a continuous Bernoulli prior on presence and a Gamma prior on concentration We now write down the version of our model that follows the setup in Section 2 , i . e ., using a continuous Bernoulli prior on presence and a Gamma prior on concentration. As introduced in the main text, we assume that different odorants are independent and identically distributed under the prior. This could be relaxed, but would require more notation. With the continuous relaxation of the presence variables, we choose as our prior a continuous Bernoulli distribution where ϖ ∈ (0, 1) is a parameter, and Z ( ϖ ) is a normalization constant. If we were using binary presence variables, we would simplify have Z ( ϖ ) = 1, but in the continuous case we have As only the score appears in the dynamics, this normalization constant is for us largely irrelevant. As introduced in Section 2 of the main text, we choose the prior on c i | p i to be the same for p i = 0 and p i = 1, as the final estimate c i p i is independent of the value of c i when p i = 0. This choice was also made by Grabska-Barwińska et al . [ 29 ]. Following those authors, we use a Gamma distribution with parameters α and β : Now we consider the two components of the prior. Clearly, we have ∇ c log P ( p ) = 0 , while The log-prior on concentration is Because of our simplifying choice that the concentration prior does not depend on presence, we have while the derivative with respect to c yields Substituting (B.33), (B.35) and (B.36) into (B.28) and (B.29), we then have Substituting (B.37) and (B.38) into the derived mirrored Langevin dynamics (B.16) and (B.17) respectively, we obtain where h = s ⊘ ( r 0 + A ( c ⊙ p )) and p = σ γ ( u ). This is the soft-gated version of the model. As the presence p is essentially binary, we hard gate the dynamics of c . We set when p ≥ p th and when p < p th . This gating can be encoded by . Replacing p with in (B.40), while keeping (B.39) unchanged, we have the dynamics of c with hard gated presence: where and . The SDEs (B.39) and (B.41) represent the full dynamics of our model when using a continuous Bernoulli prior on presence and a Gamma prior on concentration with hard-gating. This variation of our model is the basic version, while further variations are introduced in the following sections. B.4 Circuit implementation of the model To implement the coupled dynamics in a biologically plausible way, we first linearize the dynamics using the method proposed in Chalk et al [ 55 ] following our previous work [ 39 ]. We first introduce a population h of n OSN neurons with the following dynamics with a fixed point that is and a population of n OSN neurons, whose dynamics are with a fixed point that is Then we introduce another population z of n odor neurons, whose dynamics are with a fixed point that is Putting everything together, we have the full dynamics. We can map these dynamics on the circuit architecture of the olfactory bulb [ 7 , 56 ]. As they are excited by the OSN input, we interpret h and as the two classes of projection neurons in teh OB (mitral and tufted cells). Then, the concentration estimate c and presence estimate p are encoded by local interneurons (granules cells), which inhibit the projection neurons and gate each other’s dynamics. The z neurons required to linearize the prior can then be interpreted as a form of cortical feedback onto the granule cells. C Kumaraswamy Prior on Presence In our olfactory sensing model, a prior distribution that reflects the underlying structure of the natural odorants landscape yields a more informative posterior and is expected to promote more efficient inference. Earlier, we used a unimodal continuous Bernoulli (CB) distribution as the prior on presence. However, the nature of the presence variable is bimodal, where p i tends to be either 0 or 1. Concretely, the natural density of p i should not be monotonically decreasing near the upper boundary 1, whereas in a Bernoulli prior the density taper off as p i → 1 ( Figure S7a ). Therefore, compared to a unimodal prior, a bimodal prior is more realistic, and this motivates us to adopt a bimodal prior distribution. A commonly used bimodal distribution is the Beta distribution with a correct choice of parameters. However, practically it is hard to adopt the Beta distribution because its density and cumulative distribution function involve non-elementary functions. Fortunately, there is a Beta-type distribution that is easier to work with: the Kumaraswamy (KS) distribution, originally proposed by Kumaraswamy in 1980 [ 92 ]. The original KS distribution A random variable X has the KS distribution if X has density function and cumulative distribution function (CDF) where a, b > 0 are shape parameters. The KS distribution, similar to the Beta distribution, can be bimodal, when a < 1 and b < 1, as visualized in Figure S7a . On the other hand, the KS distribution is also much easier to work with than the Beta distribution for several reasons. One of them is that it only involves simple functions in its density function [ 93 ]. Truncation of the KS distribution The support of the original KS distribution is (0, 1); our presence variable p i of each odorant, however, is defined on [0, 1 ]. Hence, we must transform variables to ensure that the support of the prior distribution matches the domain of p i . To eliminate the asymptotic behavior of the KS distribution near 0 and 1, we first restrict the support of KS distribution to the close interval C = [0 + ϵ , 1 − ϵ ] for a small ϵ > 0. The restricted KS distribution has compact (closed and bounded) support [0 + ϵ , 1 − ϵ ] and Lipschitz smooth density function. We then remap the restricted KS distribution to the full interval [0, 1 ]. We define a function where ϵ ∈ (0, 0.5). Given X is a random variable with density function , we define a new random variable Y as Y = g ( X ). Since the function g is injective, differentiable and has positive derivative, we can use the change of variables formula to get the probability density function of . Since we have where y ∈ [0, 1 ]. Denoting the normalization factor , we have: The density function f Y ( y ) defines a new probability distribution that is well-defined on [0, 1 ]. and we call this distribution the transformed-KS (TKS) distribution. It follows that if we let z = g − 1 ( y ) = y (1 − 2 ϵ ) + ϵ , the first and second derivative of the log distribution (C.8) are: The first and second derivative of the log distribution are visualized in the third and fourth columns (left-to-right) in Figure S7a . We now choose a particular TKS distribution with the set of parameters a, b and ϵ to be the prior on presence. As before, we denote the prior distribution density function as P ( p ) in vector form. The gradient of the log-prior with respect to c is 0, while the derivative with respect to p is where g ( p ) = ( p − ϵ ) ⊘ (1 − 2 ϵ ) is the remapping function. The above equation (C.11) directly follows from (C.8). Using (C.9), the gradient of the energy function with respect to p is while the gradient with respect to c is unchanged: Thus, the full dynamics yields and We now try to gain some more intuition for the behavior of the prior over u resulting from the truncated KS prior over p . Importantly, it is clear that it is not log-concave. As the prior is factorized over odorants, consider the j -th odorant, for which we have: where C is a normalizing constant. We can also derive W ′ ( u j ) and W ′′ ( u j ) as following: where we from (C.9). We can then substitute (C.9) and (C.10) in and get the second derivative w.r.t. u j . For any positive small 1 ≫ ϵ > 0, W ( u j ) is roughly constant for any u j of even modestly large absolute value, with a transition region between. In particular, for u j → −∞ we have while for u j → ∞ we have In between, we note that What remains is to figure out the behavior of the function in the transition region, as well as the width of that region. By direct computation, we find that the only stationary point of W ( u j ) (that is, the solution to is Using Mathematica, we can verify that the Hessian at this point, meaning that it is a local minimum. Moreover, we can see that W ′ ( u j ) > 0 for , while W ′ ( u j ) < 0 for . Thus, the two plateaus are separated by a low-probability well. By inspection, and from Figure S7c , we can see also that the Hessian W ′′ ( u j ) does not have definite sign, meaning that the prior is not log-concave. In addition, the necessity to truncate the original distribution becomes obvious as we compare between Figure S7c &d. The log prior distribution as a function of u j is unbounded on ℝ, causing numerical instability in implementation. Re-parametrization of the original KS distribution To select the desired parameters a, b , we introduce a reparameterization of the original KS distribution PDF. Specifically, we re-parameterize the PDF w.r.t a, r instead of the original a, b . We start from the CDF of the original KS distribution (C.2) and set F X (0.5) = P ( X ≤ 0.5) = r : Hence we have . Under this parameterization, we interpret a as a shape parameter and r as a bias parameter the balance between the density near the two end points. This parameterization is much easier to interpret and useful when finding a desired distribution. Although we lose the exact relation that P ( X ≤ 0.5) = r after the truncation process, the qualitative properties of the distribution is mostly unchanged. Hence, this reparameterization is still helpful for selecting parameter for truncated KS distribution. Particularly, in the simulations, we let a = 0.055 and r = 0.75, which yields ( a, b ) = (0.055, 0.422) in the original expression. D Presence-dependent concentration priors For most of this paper, we have imposed the simplifying assumption that the conditional prior on concentration is the same irrespective of whether or not an odorant is present. However, such a formalism yields an artifact that our latent concentration estimates c i does not goes to 0 when p i = 0. To address this, we introduce a presence-dependent concentration prior that favors having c i = 0 when p i = 0. We design a differentiable prior composed of a sparsity-encouraging exponential distribution (that is, an L 1 penalty) with rate λ when p i = 0, and a Gamma distribution with parameters α and β otherwise. This is defined as: It follows that the log prior is Its derivatives with respect to p and c are and respectively. With the same definition of the energy function as before, we have and where we now let Thus, the full SDEO model dynamics with this prior are and We show an example simulation of these dynamics in Figure S3 . In contrast to Figures 3 , S1, and S2, we see that with this choice of prior the latent concentration estimates for non-present odors decay to zero once the presence estimate indicates that the odor is absent. Moreover, in the dynamics above, we see that for α > 1 the update to the presence estimate coming from the conditional prior on c | p has an interesting effect: it will tend to push low- c odors towards being recognized as absent, as ( α − 1) log c will become strongly negative. E Mutual coherence of affinity matrix and scaling capacity In this section we will discuss our model from a pure compressive sensing perspective, focusing on how the properties of the random sensing matrix—in our case, the affinity matrix—affect the scaling capacity. The key property of interest is the mutual coherence of the sensing matrix, which is one measure of how closely the measurement approximates an isometry. It is important to note that compressed sensing with Poisson noise is not as well-understood as the standard case of additive Gaussian noise [ 27 , 94 ]. However, many of the same desiderata for the sensing matrix carry over. Namely, the mutual coherence should ideally be small. The mutual coherence is a measure of the worst-case similarity between the columns of a projection matrix A . It is defined as: where a i is the i -th column of A . Equivalently, the mutual coherence is the maximum absolute value of the off-diagonal elements of the normalized Gram matrix i . e ., µ { A} = max i ≠ j | G ij |. To achieve good compressed sensing performance, the mutual coherence should be small. However, the mutual coherence is a worst-case measure, and thus can give extremely pessimistic predictions relative to the performance of a particular sensing matrix in practice. On these grounds, Elad [ 61 ] argued that the t-averaged mutual coherence can provide a more informative measure. Here, 1 (·) is an indicator function which is equal to one when the predicate in its argument is true, and zero otherwise. Using these two measures, we can evaluate the quality of an affinity matrix A . Particularly, we will use the t-averaged mutual coherence , where the t value is set to be the 80% quantile of the Gram matrix, which gives us the average of top 20% worst cases. We are interested in three ensembles of random affinity matrices: 1. Dense gamma affinity matrix each element of the affinity matrix follows a Gamma distribution, and the matrix is normalized by its largest element; 2. Sparse binary affinity matrix each element of the affinity matrix follows a Bernoulli distribution, being 1 with a probability p and 0 otherwise; 3. Sparse gamma affinity matrix each element of the affinity matrix follows Gamma distribution with probability p and 0 otherwise. We plot examples of these three types of sensing matrices in Figure S8a . In Figures 7 and S6 , we saw that sparse binary affinity matrices seem in general to achieve the highest capacity, followed by sparse gamma and then dense gamma. Examining the mutual coherence distributions for each of these ensembles, we see that a moderate system size (dictionary of 5000 odorants, 600 sensors), we notice that the sparse Gamma sensing matrix has the lowest average mutual coherence µ t but the sparse binary sensing matrix has the lowest mutual coherence µ ( Figure S8b ). The average mutual coherence varies with the dictionary size and with the number of sensors, but this ordering remains roughly consistent across scales ( Figure S8c ). In the sparse sensing ensembles, a crucial parameter governing the mutual coherence is the mean sparsity (average fraction of non-zero elements) p [ 27 ]; sparser matrices have lower average mutual coherence ( Figure S8d ). As we would expect, this results in a scaling capacity that decreases with increasing p across a variety of dictionary and sensor repertoire sizes ( Figure S8e-f ). Therefore, our empirical results are consistent with the broad conclusion that lower average mutual coherence enables higher compressive sensing capacity. F Numerical methods and additional scaling results F.1 General numerical method pipeline This section illustrates the general numerical analysis pipeline used in this paper. As an example, we showed how we numerically simulate the SDEs (B.39), (B.41) and (C.14). These equations describe the full model dynamics under the assumption of 1-on-1 coding in granule cells. They use either the Bernoulli prior (B.39) or the Kumaraswamy prior (C.14) on the presence. We first discretized the time domain by considering a step size of Δ t = 10 − 5 , resulting in a total of timesteps. Afterward, applying the Euler-Maruyama method, along with additional treatments to stabilize the numerical computation, we obtained a discrete-time Markov chain ( X n : 0 ≤ n ≤ N ) representing the evolution of the estimated c and u over time. Since the subscript n is already used to denote the time step, we use p n [ i ] to denote the i -th element of the vector p n in this section. In particular, we have with its transition rules given by: where and In particular, we used the following hyperparameter settings: and ε = 10 − 5 . For the Bernoulli prior, we set ω = 0.1 and p th = 0.2; for the Kumaraswamy prior, we set a = 0.1, b = 0.5 and p th = 0.5. The affinity matrix A is one of the three types of random matrix discussed in Appendix E. Since the Markov chain produced by the Euler-Maruyama method approximates both the trajectory and stationary distribution of the SDEs, its behavior—particularly empirical convergence and convergence rate—reflects the model’s capacity to infer the presence and concentration of odorants. In general, analyzing the behavior of this Markov chain suffices as a numerical approach for evaluating the model’s system of SDEs. Unless otherwise specified, the same numerical procedure was applied to variants of the basic model. Modifications specific to each variant are discussed in the corresponding sections. We summarize all parameter used in the scaling simulation in Table 2 . View this table: View inline View popup Download powerpoint Table 1: Variable and parameter names View this table: View inline View popup Download powerpoint Table 2: Hyperparameter used in simulations F.2 Further computational implementation and computing resources All scaling simulations were implemented in Python 3.11 using the PyTorch framework (version 2.6.0+computecanada) and executed on AMD EPYC 9655 CPU nodes of the Digital Research Alliance of Canada Fir cluster. Several runtime optimizations were applied to improve performance and ensure stability across different CPU types and under varying levels of cluster load. Because iterated Euler integration involves repeated multiplication of large matrices—an increasingly expensive operation as dimensionality grows—single-simulation runtime optimization focused on efficient matrix computation. We used PyTorch tensors instead of NumPy arrays for most matrix operations to leverage the multi-processing in PyTorch. Further speed-up was obtained by using PyTorch’s Just-In-Time (JIT) compilation for the core Euler-forward SDE integration kernels. In addition, we parallelize matrix computations across 4 CPU cores for the baseline model with OpenMP. All of these optimizations yield a 2-3x speedup over vanilla NumPy implementations in high-dimensional settings. At the experiment level, each sweep experiment was partitioned into multiple jobs and dispatched as independent tasks using SLURM’s array feature. File I/O was handled via the h5py library for efficient data storage and convenient retrieval. Additional algorithmic optimizations for scaling capacity experiments are discussed in Appendix F.4.4. Specific runtime varies across platform. In our case, one simulation with 5 × 10 4 iteration steps at the highest dimensionality (16K possible odorants) took at most 1200s for SDEO model, and up to ∼ 7000s for the baseline model. Thus, a significant proportion of the CPU hours was spent running the baseline comparison in Figures 7 and S6 . Total compute time required to generate all the figures is ∼12000 CPU hours. F.3 Baseline model and metrics for evaluation F.3.1 Baseline model To demonstrate the improved capacity of our model through comparison, we use a model we proposed in [ 39 ] as a baseline. Specifically, we use the proposed model with one-to-one coding between neurons and odorants. We confirm that the performance of the baseline model we obtain here is consistent with the results in previous work. For validation of consistency, Figures S5 and S6 in this work can be indirectly compared with Figures 3 in [ 39 ]. We note that the displayed performance appear worse because we adopt stricter evaluation criteria and shorten available time for inference to ensure comparability with the high performance of the SDEO model. For example, we define correct estimate as one in which the estimated concentration falls within ±25% × True Concentration, while the previous work use ±50%. F.3.2. Metrics for concentration estimation We introduce the two metrics for evaluating the accuracy of the concentration inference. Recall that we denote the number of presented odors as n present , for any concentration estimates ĉ given the model, we have: 1. Mean absolute error We compute the mean absolute error (MAE) between inferred and true concentrations for the presented odors: . In Figures 6 and S5 , MAE is plotted as curves in panel a and as heatmaps in panel b . 2. Correct proportion We calculate the proportion of presented odors whose predicted concentrations fall within a ± δ neighborhood of the true value . The correct proportion computed using δ = 10 is shown as contours in panel b of Figures 6 and S5 . We note that since we set the true concentration value to be 40, the correct proportion with δ = 10 corresponds of the proportion of correct concentration estimates within a tolerance of ±25%×True concentration. F.3.3 Metrics for presence estimation To introduce metrics for the quality of the presence inference, we begin by noting that the detection of odorants is inherently a binary classification problem. We therefore adopt a metric, AUROC score, from receiver operating characteristic (ROC) analysis to quantify the probability that the presence of odorants can be decoded from based on the output of the model. In ROC analysis, the ROC curve is the function of the true positive rate (TPR) over the false positive rate (FPR), which can be plotting by interpolating the TPR versus FPR at all possible thresholds. The AUROC score ( AUC ∈ [0, 1 ]) is defined as the area under the ROC curve. It can be computed given estimated value vector and a ground truth label vector p True . Importantly, its value is exactly the probability of responding correctly in the two-alternative forced-choice test based on a given data [ 95 ], and it reflects the discrimination between the representation of present and absent odorants in . AUROC score can also be interpreted as the integration of the accuracy over all possible classifying thresholds. We briefly showed that this interpretation aligns with the standard definition of AUROC score. Let z denote the threshold that downstream neurons use to classify p . Define: the true positive rate as β ( z ) = P ( p ≥ z |present) the probability density function of presence p given the odorant is absent as where F abs ( z ) = P ( p ≤ z |absent). We then have the probability of decoding correctly as: Now consider the false positive rate α ( z ) = P ( p ≥ z |absent), notice that: Substitute (F.12) into (F.11) we have: Given that the ROC curve is β = ψ ( α ), where ψ : [0, 1 ] → [0, 1 ] maps the FPR α to the TPR β , the right hand side of (F.13) corresponds to the area under the ROC curve. Therefore, the AUROC score is indeed the probability of correct classification in a 2FAC task. As a direct consequence of its definition, AUROC score is threshold-independent. It also inherently incorporates both TPR and FPR in a single scaler value and reflects the discrimination between the two classes in the data we are performing the classification on. In comparison, most other metrics cannot account for TPR and FPR simultaneously, requiring additional metric as complementary indicator. Therefore, AUROC score provides a more comprehensive assessment of a classifier’s performance by considering the probability of correct classification than traditional threshold-based binary classification metrics. A AUROC score of 0.85 indicates that, for any randomly chosen pair consisting of a present odor and an absent odor, the model will correctly classify the present odor with a probability of 85%. Effectively, this means that, on average, 85% of all odors can be correctly classified, and in particular, 85% of the presented odors can be correctly classified as present. We can see that although the AUROC score captures more information about the model’s presence estimation performance, the two metrics have some extent of equivalency in their interpretation, and this allows a clear and direct comparison between the two models. Hence we use the AUROC score as the primary metric for evaluating the presence estimation performance of our model. In Figures 6 and S5 , we plotted the AUROC score as curves in panel c and as heatmaps with contours overlaid in panel d . F.3.4 Maximum detection capacity Lastly, to investigate the scaling properties of the SDEO model, we introduce the maximumdetection capacity κ to quantify the largest number of simultaneously presented odorants that the model can reliably detect given a certain number of sensors and dictionary size. For a fixed n OSN and n odor , we first define maximum detection capacity assessed by presence estimates: κ AUROC . Specifically, we let it to be the largest n present such that the AUROC score on model’s presence estimates remains above a threshold ϵ t : where S ⊂ ℕ denotes the search space. In practice, the search space is chosen to balance computational cost and the resolution. Similarly, we define its counterpart that evaluates the concentration estimates as κ MAE . Notably, since larger MAE implies worse performance, κ MAE is defined by MAE remaining less than some threshold ϵ t : In Figures 7 and S8 (panel f ), we plot the maximum detection capacity evaluated by the AUROC score as heatmaps with contours overlaid. The threshold used is the AUROC score = 0.85. In Figure S6 , we plot the maximum detection capacity evaluated by the MAE as heatmaps with contours overlaid. The threshold used is 10 concentration unit. This represents a 25% relative error tolerance as the true concentration is 40. F.4 Experimental setting and supplemental figures F.4.1 Single simulation Here we explain how a single simulation—the building block of all scaling experiment—is executed. The first step of each simulation is to generate sensory scene, which includes generating ground truth time series, affinity matrix for OSNs, and multiplying it with ground truth time series to get OSNs responses. We considered the OSN responses s to be static throughout the inference, hence the values of s are drawn once from the Poisson distribution in (F.7). Since the affinity matrix is randomly generated in each trial, without loss of generality, we always pick the first n present in the dictionary to be the presented odors in the implementation level. This is followed by the initialization of network. Each components of the concentration vector c 0 is independently drawn from a Gamma distribution: c 0 [ i ] ∼ Gamma(6, 4) for i = 1, …, n odor , while each components of the presence vector p 0 was set such that . Consequently, the initial concentration states are randomly assigned, while the initial presence values correspond to their prior probability ϖ . In our simulations, the prior probability ϖ was set to 0.1 for all odorants. However, these prior need not be identical—they can be assigned heterogeneously or updated adaptively based through a learning process. In fact, non-uniform and adaptive prior may better reflect the implementation in biological system. Further exploration on the dynamical prior and OSN responses were left for the future work. After initialization, we run the RNN through the forward integration method discussed in Appendix F.1, and record data at some chosen sample rate into a tensor, where the further analysis and metrics are computed on. F.4.2 Dynamics demonstration In Figures S1 and S2 , we show the dynamics of the model during the process of estimating a single group of odorants, under sparse binary and dense gamma sensing matrices respectively. We set the the number of presented odors n present = 5, the dictionary size n odor = 500, and the number of sensor n OSN = 300. Other parameters remain the same as in Table 2 except we use γ = 1 in dynamics simulation. We set the appearance of odorants to be 0.25 sec after the network starts running. Then the odorants are appears with concentration 40 for a duration of 0.5 seconds, while the network runs for another 0.25 sec after odorants disappear. This design allows us to illustrate the baseline steady state of the network right after initialization and after odorants disappearing. We record the activities of neurons in the network with a sample rate of 1000 Hz. In Figure 3 , we show the dynamics of the model during the process of estimation two groups of odorants, under sparse binary sensing matrices. The simulation procedure remains the same, except that we generate a sensory scene that involves independent groups of odorants with slow-changing concentration. We plot the output concentration estimate c ⊙ p to illustrate the final output of our model. We set the number of presented odors n present = 6, arranged in two groups of three odorants. We maintain all other setting the same as in the single group dynamics simulation above. F.4.3 Simultaneous and rapid detection capacity In Figures 6 and S5 , we assess the model’s ability to infer the presence and concentration of multiple odorants simultaneously presented. We vary the number of presented odors n present from 1 to 100, while fixing the total number of sensors n OSN = 300 and the dictionary size. For simulation using the sparse binary sensing matrix, we let n odor = 1000 ( Figures 6 and S1 ). For simulation using the dense Gamma sensing matrix, we set n odor = 500 ( Figure S5 ). This is because using a sparse binary matrix improves the capacity of the model. For each 1 ≤ n present ≤ 100, we run 40 independent simulations. In each simulation, we present n present odorants to the model for an interval of length T = 0.75 s and record the models’ estimates with a sample rate of 100 Hz. Under this setting, a slice of the recorded tensor at some timepoint t represent the estimates produced by the model when given an available time window of t s. To assess the quality of the estimates, we compute three metrics score from these slices: mean absolute error (MAE), correct proportion with tolerance δ =10, and AUROC score (defined in Appendix F.3.2). We then take the averaged scores throughout the 40 trials as the final results. Results of this simulation are shown in Figures 6 and S5 . We first focus on evaluating the concentration estimates. The averaged MAE for each 1 ≤ n present ≤ 100 over the time course of T = 0.75 s is shown as heatmaps in panel b . As a complement, smoothed contours of equal correct proportion are overlaid on top of the heatmaps. While MAE shows the quantitative measure of the accuracy, the correct proportion scores give more qualitative indication of the model performance. For visualization purposes, we smoothed the contours with a Gaussian filter of standard deviation σ = 1. Across heatmaps, we ensured the color axis are the same so the fair visual comparison across different conditions can be made. Furthermore, we plot vertical slices of the heatmaps at t 1 = 100ms and t 2 = 600ms separately in panel a . They shows as MAE as a function of n present at that two timepoint. The shaded area around the curves represents ± 1.96 × SEM , indicating 95% confidence interval of the mean. When evaluating the presence estimates, similar procedure is repeated. The averaged AUROC score for each 1≤ n present ≤ 100 over the time course of T = 0.75 s is shown as heatmaps in panel d . Contours of equal AUROC score are overlaid. Vertical slices of the AUROC score heatmaps at t 1 = 100ms and t 2 = 600 are also plotted as curves with shaded region indicating 95%C.I. in panel c . F.4.4 Scaling capacity In Figures 7 and S6 , we aim to investigate how the number of receptor types n OSN required to effectively detect a fix number of presented odors n present scales with the dictionary size n odor . Thus, we vary n OSN linearly from 100 to 800 with an increment of 50 and sampled n odor over 16 evenly spaced points in log scale ranging from 2000 to 16000. This created a 2D grid of ( ñ OSN , ñ odor ) pairs. For each pair of ( ñ OSN , ñ odor ), we perform three independent runs of binary search to find the maximumsimultaneous detection capacity κ defined in Appendix F.3.4, where we set the search space S = { n | n = 5 i, i = 1, …, 20 }. Particularly, in Figure 7 , we use maximum detection capacity assessed by presence estimates κ AUROC with a threshold ϵ t = 0.85; in Figure S6 , we use its concentration counterpart κ MAE with a threshold of 10, while the true concentration is 40. In each binary search, we ran simulations under the condition n OSN = ñ OSN , n odor = ñ odor , and n present = j , where j ∈ S is taken from the sequence (search path) produced by the binary search algorithm. The available time for estimation T is 0.2 sec in Figure 7 and 0. 5 sec in Figure S6 . The concentration is harder to estimate so we increase the time available. We then used the averaged results of the three independent run, , as the final estimate of the maximum simultaneous detection capacity. We repeated this process across the 15 × 16 pairs of ( ñ OSN , ñ odor ), and visualized the resulting grid of as a heatmap. Contours of equal AUROC score or MAE is further overlaid onto the heatmaps. All heatmaps within the same figures share the same color axis. The use of binary search design significantly reduces the computational cost, without whom the experiment would be computationally infeasible, as binary search guarantees we can find the κ using at most ⌈log 2 ( | S |) ⌉ = 5 simulations, whereas a naive linear search would easily require more than 10 simulations. Using this efficient pipeline, we are not only able to directly compare the scaling capacity of our SDEO model with the non-separated baseline model, but also investigate the effect of the affinity matrix and presence prior on the scaling capacity. We tested 9 combinations between three types of affinity matrix (Appendix E) and three types of model variants (non-separated, SDEO with Bernoulli prior and SDEO with Kumaraswamy prior). G Fang et al.’s approach to sampling with L 0 priors in rate networks As noted in the Discussion, Rao-Blackwellization is not the only way one can design a rate network to sample with an L 0 prior. In particular, Fang et al . [ 82 ] proposed a Langevin sampling algorithm for sparse coding with an L 0 prior. Here, we detail this model as applied to the olfactory sensing problem, following the discussion in Appendix E of our previous work [ 39 ]. Our starting point is a spike-and-slab prior on odor concentrations: where for simplicity we will assume that a priori each odor is present with the same probability ϖ , and given that it is present its concentration is drawn from an exponential distribution of rate λ . Now, define an auxiliary variable u that is mapped to concentration estimates c via element-wise soft thresholding: where is the soft-thresholding function for threshold Given an observation s , we then run the following unconstrained Langevin dynamics for u : where u 0 = u 0 1 . For a Poisson likelihood as used elsewhere, we have Like our model, this is a gated RNN, though of a different form. Download figure Open in new tab Figure S1: Dynamics of non-separated and SDEO models during the estimation process under sparse binary sensing matrices. We test three models— non-separated (as in [ 39 ]), SDEO, and SDEO with circuit implementation—on a simple estimation task to give qualitative illustration of the model dynamics during olfactory sensing. In the task, a randomly selected set of 5 out of 500 odorants appears at concentration 40 for a duration of 0.5 s. The three columns illustrate the dynamics of three models respectively, and the three rows show different quantities estimated. In each plot, the colored lines denote the values for the presented odorants, while the gray lines represent those for the background (non-presented) odorants. a . Estimated concentration. The dashed line traces true concentration over time. b . Estimated presence in the dual space. c . Estimated presence in the [0, 1 ]-bounded primal space. The dashed line marks the threshold used to binarize the presence variable during inference. Here we used sparse binary sensing matrices with sparsity 0.1 (defined in Appendix E). We ran the same simulation using dense Gamma sensing matrices and showed the results in Figure S2 . For details of implementations, see Appendix F.4.2. Download figure Open in new tab Figure S2: Dynamics of non-separated and SDEO models during the estimation process under dense Gamma sensing matrices. Here we re-ran the same simulation as in Figure S1 but used dense gamma affinity matrices instead. Hence, for details of the experiment, see the caption under Figure S1 . a . Estimated concentration. The dashed line traces true concentration over time. b . Estimated presence in the mirror space. c . Estimated presence in the [0, 1 ]-bounded space. The dashed line marks the threshold used to binarize the presence variable during inference. For details of implementations, see Appendix F.4.2. Download figure Open in new tab Figure S3: Dynamics of SDEO models using presence-dependent concentration priors with sparse binary and dense gamma affinity matrices. We set the rate of the exponential prior λ as 2.5. Once the odors disappear at 0.5s, the presence estimate falls below the threshold, and the exponential prior gradually suppresses the concentration estimate toward 0. Download figure Open in new tab Figure S4: Responses of single projection neurons under different thresholding schemes. Each pair of panels depicts the corresponding neurons simulated with soft and hard gating, allowing direct comparison of their effects on firing dynamics. A set of fixed odor stimuli is present at 0s and withdrawn at 0.6s. Darker color indicates stronger stimulation. Download figure Open in new tab Figure S5: Improvement in fast detection of multiple odorants through separation of inference. We repeat the same simulation as in Figure 6 but using dense Gamma affinity matrices instead of sparse binary affinity matrices. We evaluate the same three models as in Figure 3 in a series of simulation where increasing numbers of odorants are simultaneously presented. In each run, in a set of 1000 odorants, a number of them are randomly selected and presented to the model for a duration of 0.75 s. We increase number of presented odors from 1 to 100, while repeat each setting for 40 times, compute the metrics and then take the average as final results. The shaded areas in a and c show ± 1.96 · SEM (representing 95% C.I) over realizations throughout. Row 1 assesses the models’ performance in odorants concentration estimation using mean absolute error. a . Mean absolute error of estimated concentration as a function of the number of odorants present at two timepoints after odor onset. b . Heatmap of mean absolute error over inference time and number of presented odorants, with smoothed contours of correct detection fraction overlaid. Row 2 assesses the models’ performance in odorants presence estimation. For the non-separated model, we convert the concentration estimation into presence estimation by binarizing the estimated concentrations based on whether they exceeds half of the true odorant concentration. Since presence estimation is a binary classification task, we use AUROC score as the performance metric. c . AUROC score as a function of the number of odors present at two timepoints after odor onset. d . Heatmap of AUROC score over inference time and number of presented odors, with smoothed contours overlaid. For details of implementation, see Appendix F.4.3. Download figure Open in new tab Figure S6: Scaling of detection capacity with dictionary size and sensor repertoire for different priors and sensing matrix models. We run the same simulation as in Figure 7 , but showing maximum detection capacity assessed by concentration instead of presence estimates. The three columns correspond to three models—non-separated, SDEO, and SDEO with circuit implementation—and the three rows correspond to three types of affinity matrices. These are: dense Gamma, whose entries are i.i.d. random variable following Gamma(0.37, 0.36); sparse Gamma, obtained by applying a 0.1 sparsity mask to a dense Gamma matrix; and sparse binary, whose entries are i.i.d. random variable following Bernoulli(0.1). Each heatmap shows the maximum detection capacity assessed by concentration estimates for combinations of sensors counts (from 100 to 800, equally spaced linearly) and dictionary size (1000 to 16000 equally spaced on a log scale). The maximum detection capacity κ MAE is defined as the largest number of simultaneously presented number of odorants that the model can detect with a mean absolute error ≤ 10, while c True = 40. Smoothed contours are overlaid and can be interpreted as the required number of sensors to maintain a certain capacity as a function of dictionary size. The total inference time duration is 0.5 s for all runs, and the value in each cell of the heatmap is the average of three independent runs. For details of implementation, see Appendix F.4.4. Download figure Open in new tab Figure S7: In a . and b ., we plot the Truncated Kumaraswamy distribution and continuous Bernoulli distribution on [0, 1 ]. a . Truncated Kumaraswamy distribution with parameters a = 0.055, b = 0.422, ε = 1 e − 5. The four columns from left to right respectively are: probability density function (PDF), log PDF, the first derivative of log PDF, and the second derivative of log PDF. b . Continuous Bernoulli distribution with ϖ = 0.01. The three columns from left to right respectively are: probability density function (PDF), log PDF, and the first derivative of log PDF. We didn’t show the second derivative of log PDF because it vanishes. In c ., d . and e . we plot three distributions with respect to u ∈ ℝ such that . This is to illustrate the score function in the dual space. The three columns from left to right respectively are the log PDF, the first derivative of the log PDF, and the second derivative of the PDF. c . Truncated Kumaraswamy distribution with the same parameters as in a. d . Un-truncated Kumaraswamy distribution with parameters a = 0.055, b = 0.422, ε = 0. e . Continuous Bernoulli distribution with ϖ = 0.01. Download figure Open in new tab Figure S8: a . Zoomed in example of the three types of sensing matrix—dense Gamma with Gamma(0.36, 0.37), sparse Gamma with Gamma(0.36, 0.37) and sparsity 0.1, and sparse Binary with sparsity 0.1. b . Histograms showing the distribution of absolute off-diagonal entries of the Gram matrix of the sensing matrices, which is equivalent to column-wise correlation in the sensing matrix. Three shows distinct sensing matrix types matching the column titles. Here we use a system with 600 sensors and 5000 dictionary size. The worst case correlation µ , namely the mutual coherence, is indicated by the green vertical dashed line overlaid to the histogram, and also printed out in the legend. The average of the top 20% largest correlation µ t , is indicated by the orange vertical dashed line overlaid to the histogram, and also printed out in the legend. c . Mutual coherence of the sensing matrices under different dimensionality, averaged across 5 random trials. d . Histogram showing the probability density of absolute off-diagonal entries of sparse binary sensing matrices with different sparsity. e . Maximum detection capacity as a function of sparsity of the sparse binary sensing matrix, under different combination of dictionary size and sensor repertoire. The dimensionality is indicated by dots with corresponding color in panel f. f . Heatmap of maximum detection capacity under different dictionary size and sensor repertoire when using sparse binary sensing matrices with increasing sparsity (s=0.1 to 0.5) Acknowledgements We thank Dmitri Chklovskii, Farhad Pashakhanloo, and Sina Tootoonian for inspiring conversations. Moreover, we thank Juan Carlos Fernández del Castillo, Siddharth Jayakumar, and Farhad Pashakhanloo for helpful comments on a previous version of this manuscript. JAZV was supported in part by the Office of the Director of the National Institutes of Health under Award Number DP5OD037354. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. JAZV is further supported by a Junior Fellowship from the Harvard Society of Fellows. This research was carried out in part thanks to funding from the Canada First Research Excellence Fund, awarded to PM through the Healthy Brains, Healthy Lives initiative at McGill University. PM acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) (grant numbers RGPIN-2025-05676 and DGECR-2025-00255), and of the Fonds de Recherche du Québec (grant number CB-365865). PM was further supported by a Sloan Research Fellowship. This research was enabled in part by support provided by Calcul Québec and the Digital Research Alliance of Canada. Funder Information Declared Office of the Director of the National Institutes of Health , DP5OD037354 Harvard Society of Fellows , Junior Fellowship Natural Sciences and Engineering Research Council , RGPIN-2025-05676 , DGECR-2025-00255 Fonds de recherche du Québec , CB-365865 Alfred P. Sloan Foundation , Sloan Research Fellowship Canada First Research Excellence Fund , Healthy Brains, Healthy Lives References [1]. ↵ Jakob Johann von Uexküll . Umwelt und Innenwelt der Tiere . Springer , 1909 . doi: 10.1007/978-3-662-24819-5 . OpenUrl CrossRef [2]. ↵ Eero P Simoncelli and Bruno A Olshausen . Natural image statistics and neural representation . Annual Review of Neuroscience , 24 (Volume 24, 2001): 1193 – 1216 , 2001 . ISSN 1545-4126 . doi: 10.1146/annurev.neuro.24.1.1193 . URL https://www.annualreviews.org/content/journals/10.1146/annurev.neuro.24.1.1193 . OpenUrl CrossRef PubMed Web of Science [3]. ↵ Shanshan Qin , Qianyi Li , Chao Tang , and Yuhai Tu . 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 ): 20286 – 20295 , September 2019 . ISSN 1091-6490 . doi: 10.1073/pnas.1906571116 . URL http://dx.doi.org/10.1073/pnas.1906571116. OpenUrl Abstract / FREE Full Text [4]. ↵ Juan Carlos Fernández del Castillo , Farhad Pashakhanloo , Venkatesh N Murthy , and Jacob A Zavatone-Veth . Convergent motifs of early olfactory processing are recapitulated by layer-wise efficient coding . bioRxiv , 2025 . doi: 10.1101/2025.09.03.673748 . URL https://www.biorxiv.org/content/early/2025/09/03/2025.09.03.673748 . OpenUrl Abstract / FREE Full Text [5]. ↵ Nikolai M. Chapochnikov , Cengiz Pehlevan , and Dmitri B. Chklovskii . Normative and mechanistic model of an adaptive circuit for efficient encoding and feature extraction . Proceedings of the National Academy of Sciences , 120 ( 29 ): e2117484120 , 2023 . doi: 10.1073/pnas.2117484120 . URL https://www.pnas.org/doi/abs/10.1073/pnas.2117484120 . OpenUrl CrossRef PubMed [6]. Paul Cisek and Benjamin Y. Hayden . Neuroscience needs evolution . Philosophical Transactions of the Royal Society B: Biological Sciences , 377 ( 1844 ): 20200518, 2022 . doi: 10.1098/rstb.2020.0518 . URL https://royalsocietypublishing.org/doi/abs/10.1098/rstb.2020.0518 . OpenUrl CrossRef PubMed [7]. ↵ Kara A Fulton , David Zimmerman , Aravi Samuel , Katrin Vogt , and Sandeep Robert Datta . Common principles for odour coding across vertebrates and invertebrates . Nature Reviews Neuroscience , 25 ( 7 ): 453 – 472 , 2024 . OpenUrl CrossRef PubMed [8]. ↵ David M. Ferrero , Jamie K. Lemon , Daniela Fluegge , Stan L. Pashkovski , Wayne J. Korzan , Sandeep Robert Datta , Marc Spehr , Markus Fendt , and Stephen D. Liberles . Detection and avoidance of a carnivore odor by prey . Proceedings of the National Academy of Sciences , 108 ( 27 ): 11235 – 11240 , 2011 . doi: 10.1073/pnas.1103317108 . URL https://www.pnas.org/doi/abs/10.1073/pnas.1103317108 . OpenUrl Abstract / FREE Full Text [9]. ↵ Bogdan Bintu , Yoh Isogai , Xiaowei Zhuang , and Catherine Dulac . Molecular and spatial organization of the primary olfactory system and its responses to social odors . bioRxiv , 2025 . doi: 10.1101/2025.05.02.651832 . URL http://dx.doi.org/10.1101/2025.05.02.651832. OpenUrl Abstract / FREE Full Text [10]. ↵ Adam C. Nelson , Vikrant Kapoor , Eric Vaughn , Jeshurun A. Gnanasegaram , Nimrod D. Rubinstein , Mustafa Talay , Venkatesh N. Murthy , and Catherine Dulac . Molecular and neural control of social hierarchy by a forebrain-thalamocortical circuit . Cell , 2025/09/03 2025 . ISSN 0092-8674 . doi: 10.1016/j.cell.2025.07.024 . URL https://doi.org/10.1016/j.cell.2025.07.024. OpenUrl CrossRef [11]. ↵ Neil J. Vickers . Mechanisms of animal navigation in odor plumes . The Biological Bulletin , 198 ( 2 ): 203 – 212 , 04 2000 . doi: 10.2307/1542524 . URL https://doi.org/10.2307/1542524. OpenUrl CrossRef PubMed Web of Science [12]. ↵ Siddharth Jayakumar , Nicola Rigolli , Mackenzie W Mathis , Massimo Vergassola , Alexander Mathis , and Venkatesh N Murthy . Mice navigate scent trails using predictive policies . bioRxiv , 2025 . doi: 10.1101/2025.08.27.672631 . URL https://www.biorxiv.org/content/early/2025/09/01/2025.08.27.672631 . OpenUrl Abstract / FREE Full Text [13]. ↵ Adam Kepecs , Naoshige Uchida , and Zachary F. Mainen . The Sniff as a Unit of Olfactory Processing . Chemical Senses , 31 ( 2 ): 167 – 179 , 12 2005 . ISSN 0379-864X . doi: 10.1093/chemse/bjj016 . URL https://doi.org/10.1093/chemse/bjj016. OpenUrl CrossRef PubMed Web of Science [14]. ↵ Tobias Ackels , Andrew Erskine , Debanjan Dasgupta , Alina Cristina Marin , Tom PA Warner , Sina Tootoonian , Izumi Fukunaga , Julia J Harris , and Andreas T Schaefer . Fast odour dynamics are encoded in the olfactory system and guide behaviour . Nature , 593 ( 7860 ): 558 – 563 , 2021 . OpenUrl CrossRef PubMed [15]. ↵ Antonio Celani , Emmanuel Villermaux , and Massimo Vergassola . Odor Landscapes in Turbulent Environments . Physical Review X , 4 ( 4 ): 041015 , October 2014 . doi: 10.1103/PhysRevX.4.041015 . URL https://link.aps.org/doi/10.1103/PhysRevX.4.041015 . Publisher: American Physical Society . OpenUrl CrossRef [16]. ↵ Barry W. Ache and Janet M. Young . Olfaction: Diverse species, conserved principles . Neuron , 48 ( 3 ): 417 – 430 , 2005 . ISSN 0896-6273 . doi: 10.1016/j.neuron.2005.10.022 . URL https://www.sciencedirect.com/science/article/pii/S0896627305008949 . OpenUrl CrossRef PubMed Web of Science [17]. ↵ Naoshige Uchida and Zachary F Mainen . Speed and accuracy of olfactory discrimination in the rat . Nature Neuroscience , 6 ( 11 ): 1224 – 1229 , October 2003 . ISSN 1546-1726 . doi: 10.1038/nn1142 . URL http://dx.doi.org/10.1038/nn1142. OpenUrl CrossRef PubMed Web of Science [18]. ↵ Luis Boero , Hao Wu , Joseph D Zak , Paul Masset , Farhad Pashakhanloo , Siddharth Jayakumar , Bahareh Tolooshams , Demba Ba , and Venkatesh N Murthy . Perception and neural representation of intermittent odor stimuli in mice . bioRxiv , pages 2025 – 02 , 2025 . [19]. ↵ Dan Rokni , Vivian Hemmelder , Vikrant Kapoor , and Venkatesh N Murthy . An olfactory cocktail party: figure-ground segregation of odorants in rodents . Nature Neuroscience , 17 ( 9 ): 1225 – 1232 , August 2014 . ISSN 1546-1726 . doi: 10.1038/nn.3775 . URL http://dx.doi.org/10.1038/nn.3775. OpenUrl CrossRef PubMed [20]. ↵ Matt Wachowiak , Adam Dewan , Thomas Bozza , Tom F O’Connell , and Elizabeth J Hong . Recalibrating olfactory neuroscience to the range of naturally occurring odor concentrations . Journal of Neuroscience , 45 ( 10 ), 2025 . [21]. ↵ Izumi Fukunaga , Manuel Berning , Mihaly Kollo , Anja Schmaltz , and Andreas T Schaefer . Two distinct channels of olfactory bulb output . Neuron , 75 ( 2 ): 320 – 329 , 2012 . OpenUrl CrossRef PubMed Web of Science [22]. ↵ Kei M Igarashi , Nao Ieki , Myungho An , Yukie Yamaguchi , Shin Nagayama , Ko Kobayakawa , Reiko Kobayakawa , Manabu Tanifuji , Hitoshi Sakano , Wei R Chen , et al. Parallel mitral and tufted cell pathways route distinct odor information to different targets in the olfactory cortex . Journal of Neuroscience , 32 ( 23 ): 7970 – 7985 , 2012 . OpenUrl Abstract / FREE Full Text [23]. Kevin A Bolding and Kevin M Franks . Recurrent cortical circuits implement concentration-invariant odor coding . Science , 361 ( 6407 ): eaat6904 , 2018 . OpenUrl Abstract / FREE Full Text [24]. ↵ Honggoo Chae , Arkarup Banerjee , and Dinu F Albeanu . A non-canonical feedforward pathway for computing odor identity . bioRxiv , 2020 . [25]. ↵ Honggoo Chae , Arkarup Banerjee , Marie Dussauze , and Dinu F. Albeanu . Long-range functional loops in the mouse olfactory system and their roles in computing odor identity . Neuron , 110 ( 23 ): 3970 – 3985 .e7, 2022 . ISSN 0896-6273 . doi: 10.1016/j.neuron.2022.09.005 . OpenUrl CrossRef PubMed [26]. ↵ Richard G Baraniuk . Compressive sensing [lecture notes] . IEEE signal processing magazine , 24 ( 4 ): 118 – 121 , 2007 . OpenUrl [27]. ↵ Maxim Raginsky , Rebecca M. Willett , Zachary T. Harmany , and Roummel F. Marcia . Compressed sensing performance bounds under Poisson noise . IEEE Transactions on Signal Processing , 58 ( 8 ): 3990 – 4002 , 2010 . doi: 10.1109/TSP.2010.2049997 . OpenUrl CrossRef [28]. ↵ Alexei A Koulakov and Dmitry Rinberg . Sparse incomplete representations: A potential role of olfactory granule cells . Neuron , 72 ( 1 ): 124 – 136 , 2011 . OpenUrl CrossRef PubMed Web of Science [29]. ↵ C.J. Burges , L. Bottou , M. Welling , Z. Ghahramani , and K.Q. Weinberger Agnieszka Grabska-Barwińska , Jeff Beck , Alexandre Pouget , and Peter Latham . Demixing odors -fast inference in olfaction . In C.J. Burges , L. Bottou , M. Welling , Z. Ghahramani , and K.Q. Weinberger , editors, Advances in Neural Information Processing Systems , volume 26 . Curran Associates, Inc ., 2013 . URL https://proceedings.neurips.cc/paper_files/paper/2013/file/2bcab9d935d219641434683dd9d18a03-Paper.pdf . [30]. Z. Ghahramani , M. Welling , C. Cortes , N. Lawrence , and K.Q. Weinberger Sina Tootoonian and Mate Lengyel . A dual algorithm for olfactory computation in the locust brain . In Z. Ghahramani , M. Welling , C. Cortes , N. Lawrence , and K.Q. Weinberger , editors, Advances in Neural Information Processing Systems , volume 27 . Curran Associates, Inc ., 2014 . URL https://proceedings.neurips.cc/paper_files/paper/2014/file/c8ba76c279269b1c6bc8a07e38e78fa4-Paper.pdf . [31]. Yilun Zhang and Tatyana O. Sharpee . A robust feedforward model of the olfactory system . PLOS Computational Biology , 12 ( 4 ): 1 – 15 , 04 2016 . doi: 10.1371/journal.pcbi.1004850 . URL https://doi.org/10.1371/journal.pcbi.1004850. OpenUrl CrossRef [32]. ↵ David Zwicker , Arvind Murugan , and Michael P. Brenner . Receptor arrays optimized for natural odor statistics . Proceedings of the National Academy of Sciences , 113 ( 20 ): 5570 – 5575 , April 2016 . ISSN 1091-6490 . doi: 10.1073/pnas.1600357113 . URL http://dx.doi.org/10.1073/pnas.1600357113. OpenUrl Abstract / FREE Full Text [33]. Daniel Kepple , Brittany N. Cazakoff , Heike S. Demmer , Dennis Eckmeier , Stephen D. Shea , and Alexei A. Koulakov . Computational algorithms and neural circuitry for compressed sensing in the mammalian main olfactory bulb . bioRxiv , 2018 . doi: 10.1101/339689 . URL https://www.biorxiv.org/content/early/2018/06/05/339689 . OpenUrl Abstract / FREE Full Text [34]. Daniel R Kepple , Hamza Giaffar , Dmitry Rinberg , and Alexei A Koulakov . Deconstructing odorant identity via primacy in dual networks . Neural Computation , 31 ( 4 ): 710 – 737 , 2019 . doi: 10.1162/neco_a_01175 . OpenUrl CrossRef PubMed [35]. Naoki Hiratani and Peter E. Latham . Rapid Bayesian learning in the mammalian olfactory system . Nature Communications , 11 ( 1 ): 3845 , Jul 2020 . ISSN 2041-1723 . doi: 10.1038/s41467-020-17490-0 . URL https://doi.org/10.1038/s41467-020-17490-0. OpenUrl CrossRef [36]. ↵ Vijay Singh , Martin Tchernookov , and Vijay Balasubramanian . What the odor is not: Estimation by elimination . Phys. Rev. E , 104 : 024415 , Aug 2021 . doi: 10.1103/PhysRevE.104.024415 . URL https://link.aps.org/doi/10.1103/PhysRevE.104.024415 . OpenUrl CrossRef [37]. Kamesh Krishnamurthy , Ann M Hermundstad , Thierry Mora , Aleksandra M Walczak , and Vijay Balasubramanian . Disorder and the neural representation of complex odors . Frontiers in Computational Neuroscience , 16 , 2022 . doi: 10.3389/fncom.2022.917786 . OpenUrl CrossRef [38]. Sina Tootoonian , Andreas T. Schaefer , and Peter E. Latham . Sparse connectivity for map inference in linear models using sister mitral cells . PLOS Computational Biology , 18 ( 1 ): 1 – 40 , 01 2022 . doi: 10.1371/journal.pcbi.1009808 . URL https://doi.org/10.1371/journal.pcbi.1009808. OpenUrl CrossRef [39]. ↵ Jacob Zavatone-Veth , Paul Masset , William Tong , Joseph D. Zak , Venkatesh Murthy , and Cengiz Pehlevan . Neural circuits for fast Poisson compressed sensing in the olfactory bulb . In Advances in Neural Information Processing Systems , volume 36 , pages 64793 – 64828 , 2023 . URL https://proceedings.neurips.cc/paper_files/paper/2023/file/cc8638553a347b1834d98be7613fa3f0-Paper-Conference.pdf . OpenUrl [40]. ↵ Sina Tootoonian and Andreas T. Schaefer . Inference with correlated priors using sisters cells . arXiv , 2025 . URL https://arxiv.org/abs/2505.14579 . [41]. ↵ Hugh Durrant-Whyte and Tim Bailey . Simultaneous localization and mapping: part i . IEEE robotics & automation magazine , 13 ( 2 ): 99 – 110 , 2006 . OpenUrl Web of Science [42]. ↵ S. Bengio , H. Wallach , H. Larochelle , K. Grauman , N. Cesa-Bianchi , and R. Garnett Ya-Ping Hsieh , Ali Kavis , Paul Rolland , and Volkan Cevher . Mirrored Langevin dynamics . In S. Bengio , H. Wallach , H. Larochelle , K. Grauman , N. Cesa-Bianchi , and R. Garnett , editors, Advances in Neural Information Processing Systems , volume 31 . Curran Associates, Inc ., 2018 . URL https://proceedings.neurips.cc/paper_files/paper/2018/file/6490791e7abf6b29a381288cc23a8223-Paper.pdf . [43]. ↵ David H. Brann , Tatsuya Tsukahara , Cyrus Tau , Dennis Kalloor , Rylin Lubash , Lakshanyaa Thamarai Kannan , Nell Klimpert , Mihaly Kollo , Martín Escamilla-Del-Arenal , Bogdan Bintu , Thomas Bozza , and Sandeep R. Datta . A spatial code governs olfactory receptor choice and aligns sensory maps in the nose and brain . bioRxiv , 2025 . doi: 10.1101/2025.05.02.651738 . URL http://dx.doi.org/10.1101/2025.05.02.651738. OpenUrl Abstract / FREE Full Text [44]. ↵ Venkatesh N. Murthy . Olfactory maps in the brain . Annual Review of Neuroscience , 34 ( 1 ): 233 – 258 , 2011 . doi: 10.1146/annurev-neuro-061010-113738 . URL https://doi.org/10.1146/annurev-neuro-061010-113738. OpenUrl CrossRef PubMed Web of Science [45]. ↵ Naoshige Uchida , Cindy Poo , and Rafi Haddad . Coding and transformations in the olfactory system . Annual review of neuroscience , 37 : 363 – 385 , 2014 . OpenUrl CrossRef PubMed [46]. ↵ Yushu Chen , Xiaoyin Chen , Batuhan Baserdem , Huiqing Zhan , Yan Li , Martin B. Davis , Justus M. Kebschull , Anthony M. Zador , Alexei A. Koulakov , and Dinu F. Albeanu . High-throughput sequencing of single neuron projections reveals spatial organization in the olfactory cortex . Cell , 185 ( 22 ): 4117 – 4134 .e28, Oct 2022 . ISSN 0092-8674 . doi: 10.1016/j.cell.2022.09.038 . URL https://doi.org/10.1016/j.cell.2022.09.038. OpenUrl CrossRef PubMed [47]. ↵ Shin Nagayama , Allicia Enerva , Max L. Fletcher , Arjun V. Masurkar , Kei M. Igarashi , Kensaku Mori , and Wei R. Chen . Differential axonal projection of mitral and tufted cells in the mouse main olfactory system . Frontiers in Neural Circuits , 4 , 2010 . ISSN 1662-5110 . doi: 10.3389/fncir.2010.00120 . URL https://www.frontiersin.org/journals/neural-circuits/articles/10.3389/fncir.2010.00120 . OpenUrl CrossRef PubMed [48]. ↵ Thomas Nowotny and Paul Szyszka . Dynamics of Odor-Evoked Activity Patterns in the Olfactory System, page 243–261 . Springer International Publishing , 2017 . ISBN 9783319536736 . doi: 10.1007/978-3-319-53673-6_15 . URL http://dx.doi.org/10.1007/978-3-319-53673-6_15. OpenUrl CrossRef [49]. ↵ Priyanka Gupta , Dinu F Albeanu , and Upinder S Bhalla . Olfactory bulb coding of odors, mixtures and sniffs is a linear sum of odor time profiles . Nature Neuroscience , 18 ( 2 ): 272 – 281 , 2015 . doi: 10.1038/nn.3913 . OpenUrl CrossRef PubMed [50]. ↵ Agnieszka Grabska-Barwińska , Simon Barthelmé, Jeff Beck , Zachary F. Mainen , Alexandre Pouget , and Peter E. Latham . A probabilistic approach to demixing odors . Nature Neuroscience , 20 ( 1 ): 98 – 106 , Jan 2017 . ISSN 1546-1726 . doi: 10.1038/nn.4444 . URL https://doi.org/10.1038/nn.4444. OpenUrl CrossRef PubMed [51]. ↵ Guillaume Hennequin , Laurence Aitchison , and Máté Lengyel . Fast sampling-based inference in balanced neuronal networks . Advances in Neural Information Processing Systems , 27 : 2240 – 2248 , 2014 . OpenUrl [52]. ↵ Amir Beck and Marc Teboulle . Mirror descent and nonlinear projected subgradient methods for convex optimization . Operations Research Letters , 31 ( 3 ): 167 – 175 , 2003 . OpenUrl [53]. ↵ Garvesh Raskutti and Sayan Mukherjee . The information geometry of mirror descent . IEEETransactionson Information Theory , 61 ( 3 ): 1451 – 1457 , 2015 . OpenUrl [54]. ↵ Gary R Holt and Christof Koch . Shunting inhibition does not have a divisive effect on firing rates . Neural computation , 9 ( 5 ): 1001 – 1013 , 1997 . OpenUrl CrossRef PubMed Web of Science [55]. ↵ Matthew Chalk , Paul Masset , Sophie Deneve , and Boris Gutkin . Sensory noise predicts divisive reshaping of receptive fields . PLoS computational biology , 13 ( 6 ): e1005582 , 2017 . OpenUrl PubMed [56]. ↵ Shin Nagayama , Ryota Homma , and Fumiaki Imamura . Neuronal organization of olfactory bulb circuits . Frontiers in neural circuits , 8 : 98 , 2014 . OpenUrl PubMed [57]. ↵ Shawn D Burton and Nathaniel N Urban . 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 ): 2097 – 2118 , 2014 . OpenUrl CrossRef PubMed [58]. ↵ Shawn D Burton and Nathaniel N Urban . Rapid feedforward inhibition and asynchronous excitation regulate granule cell activity in the mammalian main olfactory bulb . Journal of Neuroscience , 35 ( 42 ): 14103 – 14122 , 2015 . OpenUrl Abstract / FREE Full Text [59]. ↵ Sina Tootoonian and Andreas T. Schaefer . If mitral cells are the answer, what is the question? In Cosyne Abstracts , 2023 . [60]. ↵ Matthew Geramita and Nathan N Urban . Differences in glomerular-layer-mediated feedforward inhibition onto mitral and tufted cells lead to distinct modes of intensity coding . Journal of Neuroscience , 37 ( 6 ): 1428 – 1438 , 2017 . OpenUrl Abstract / FREE Full Text [61]. ↵ Michael Elad . Optimized projections for compressed sensing . IEEE Transactions on Signal Processing , 55 ( 12 ): 5695 – 5702 , 2007 . doi: 10.1109/TSP.2007.900760 . OpenUrl CrossRef [62]. ↵ M.C. Jones . Kumaraswamy’s distribution: A beta-type distribution with some tractability advantages . Statistical Methodology , 6 ( 1 ): 70 – 81 , 2009 . ISSN 1572-3127 . doi: 10.1016/j.stamet.2008.04.001 . URL https://www.sciencedirect.com/science/article/pii/S1572312708000282 . OpenUrl CrossRef [63]. P. Kumaraswamy . A generalized probability density function for double-bounded random processes . Journal of Hydrology , 46 ( 1 ): 79 – 88 , 1980 . ISSN 0022-1694 . doi: 10.1016/0022-1694(80)90036-0 . URL https://www.sciencedirect.com/science/article/pii/0022169480900360 . OpenUrl CrossRef GeoRef Web of Science [64]. ↵ Max Wasserman and Gonzalo Mateos . Stabilizing the kumaraswamy distribution . Transactions on Machine Learning Research , 2025 . ISSN 2835-8856 . URL https://openreview.net/forum?id=baZLwdphqw . [65]. ↵ Kevin A Bolding , Shivathmihai Nagappan , Bao-xia Han , Fan Wang , and Kevin M Franks . Robust odor coding across states in piriform cortex requires recurrent circuitry : evidence for pattern completion in an associative network . bioRxiv , 2019 . [66]. ↵ Kevin A Bolding and Kevin M Franks . Complementary codes for odor identity and intensity in olfactory cortex . eLife , 6 : e22630 , April 2017 . ISSN 2050-084X . doi: 10.7554/eLife.22630 . URL https://doi.org/10.7554/eLife.22630. Publisher: eLife Sciences Publications, Ltd . OpenUrl CrossRef PubMed [67]. ↵ Cindy Poo , Gautam Agarwal , Niccolò Bonacchi and Zachary F. Mainen . Spatial maps in piriform cortex during olfactory navigation . Nature , 601 ( 7894 ): 595 – 599 , January 2022 . ISSN 1476-4687 . doi: 10.1038/s41586-021-04242-3 . URL https://www.nature.com/articles/s41586-021-04242-3 . OpenUrl CrossRef PubMed [68]. ↵ Alison M. Boyd , James F. Sturgill , Cindy Poo , and Jeffry S. Isaacson . Cortical Feedback Control of Olfactory Bulb Circuits . Neuron , 76 ( 6 ): 1161 – 1174 , 2012 . ISSN 08966273 . doi: 10.1016/j.neuron.2012.10.020 . URL http://dx.doi.org/10.1016/j.neuron.2012.10.020. arXiv: NIHMS150003 ISBN: 1097-4199 (Electronic)\r0896-6273 (Linking). OpenUrl CrossRef PubMed Web of Science [69]. ↵ GH Otazu , P Masset, and DF Albeanu . A computational role for cortical feedback in odor detection in natural scenes . Chemical Senses , 41 ( 7 ): E101 – E102 , 2016 . OpenUrl [70]. ↵ David Wolf , Lars-Lennart Oettl , Laurens Winkelmeier , Christiane Linster , and Wolfgang Kelsch . Anterior Olfactory Cortices Differentially Transform Bottom-Up Odor Signals to Produce Inverse Top-Down Outputs . Journal of Neuroscience , 44 ( 44 ), October 2024 . ISSN 0270-6474, 1529-2401 . doi: 10.1523/JNEUROSCI.0231-24.2024 . URL https://www.jneurosci.org/content/44/44/e0231242024 . OpenUrl Abstract / FREE Full Text [71]. Renata Medinaceli Quintela , Daniela Brunert , and Markus Rothermel . Functional role of the anterior olfactory nucleus in sensory information processing . Neuroforum , 28 ( 3 ): 169 – 175 , August 2022 . ISSN 2363-7013 . doi: 10.1515/nf-2022-0008 . URL https://www.degruyterbrill.com/document/doi/10.1515/nf-2022-0008/html . Publisher: De Gruyter . OpenUrl CrossRef [72]. Max Levinson , Jacob P. Kolenda , Gabriella J. Alexandrou , Olga Escanilla , Thomas A. Cleland , David M. Smith , and Christiane Linster . Context-dependent odor learning requires the anterior olfactory nucleus . Behavioral Neuroscience , 134 ( 4 ): 332 – 343 , 2020 . ISSN 1939-0084 . doi: 10.1037/bne0000371 . Place: US Publisher: American Psychological Association . OpenUrl CrossRef PubMed [73]. ↵ Shu Kikuta , Kenichiro Sato , Hideki Kashiwadani , Koichi Tsunoda , Tatsuya Yamasoba , and Kensaku Mori . Neurons in the anterior olfactory nucleus pars externa detect right or left localization of odor sources . Proceedings of the National Academy of Sciences of the United States of America , 107 ( 27 ): 12363 – 12368 , 2010 . ISSN 00278424 . doi: 10.1073/pnas.1003999107 . ISBN: 1003999107 . OpenUrl Abstract / FREE Full Text [74]. ↵ Shawn D Burton . Inhibitory circuits of the mammalian main olfactory bulb . Journal of neurophysiology , 118 ( 4 ): 2034 – 2051 , 2017 . OpenUrl CrossRef PubMed [75]. ↵ Abdelrahman Sharafeldin and Hannah Choi . Cell-type-specific encoding of prediction and reward in cortical microcircuits during novelty detection . bioRxiv , 2025 . doi: 10.1101/2025.05.13.653877 . URL https://www.biorxiv.org/content/early/2025/10/08/2025.05.13.653877 . OpenUrl Abstract / FREE Full Text [76]. ↵ Ofer Mazor and Gilles Laurent . Transient dynamics versus fixed points in odor representations by locust antennal lobe projection neurons . Neuron , 48 ( 4 ): 661 – 673 , 2005 . ISSN 0896-6273 . doi: 10.1016/j.neuron.2005.09.032 . URL https://www.sciencedirect.com/science/article/pii/S0896627305008925 . OpenUrl CrossRef PubMed Web of Science [77]. ↵ Srinivas Gorur-Shandilya , Carlotta Martelli , Mahmut Demir , and Thierry Emonet . Controlling and measuring dynamic odorant stimuli in the laboratory . Journal of Experimental Biology , 222 ( 23 ): jeb207787 , 11 2019 . ISSN 0022-0949 . doi: 10.1242/jeb.207787 . URL https://doi.org/10.1242/jeb.207787. OpenUrl Abstract / FREE Full Text [78]. ↵ Ashesh K. Dhawale , Akari Hagiwara , Upinder S. Bhalla , Venkatesh N. Murthy , and Dinu F. Albeanu . Non-redundant odor coding by sister mitral cells revealed by light addressable glomeruli in the mouse . Nature Neuroscience , 13 ( 11 ): 1404 – 1412 , Nov 2010 . ISSN 1546-1726 . doi: 10.1038/nn.2673 . URL https://doi.org/10.1038/nn.2673. OpenUrl CrossRef PubMed [79]. Jonathan V Gill , Gilad M Lerman , Hetince Zhao , Benjamin J Stetler , Dmitry Rinberg , and Shy Shoham . Precise holographic manipulation of olfactory circuits reveals coding features determining perceptual detection . Neuron , 108 ( 2 ): 382 – 393 , 2020 . OpenUrl PubMed [80]. Edmund Chong , Monica Moroni , Christopher Wilson , Shy Shoham , Stefano Panzeri , and Dmitry Rinberg . Manipulating synthetic optogenetic odors reveals the coding logic of olfactory perception . Science , 368 ( 6497 ): eaba2357 , 2020 . OpenUrl Abstract / FREE Full Text [81]. ↵ Nirag Kadakia , Mahmut Demir , Brenden T Michaelis , Brian D DeAngelis , Matthew A Reidenbach , Damon A Clark , and Thierry Emonet . Odour motion sensing enhances navigation of complex plumes . Nature , 611 ( 7937 ): 754 – 761 , 2022 . OpenUrl CrossRef PubMed [82]. ↵ Michael Y.-S. Fang , Mayur Mudigonda , Ryan Zarcone , Amir Khosrowshahi , and Bruno A. Olshausen . Learning and Inference in Sparse Coding Models With Langevin Dynamics . Neural Computation , 34 ( 8 ): 1676 – 1700 , 07 2022 . ISSN 0899-7667 . doi: 10.1162/neco_a_01505 . URL https://doi.org/10.1162/neco_a_01505. OpenUrl CrossRef [83]. ↵ Gonzalo H Otazu and Christian Leibold . A corticothalamic circuit model for sound identification in complex scenes . PLoS One , 6 ( 9 ): e24270 , 2011 . OpenUrl CrossRef PubMed [84]. Jeff Hawkins , Marcus Lewis , Mirko Klukas , Scott Purdy , and Subutai Ahmad . A framework for intelligence and cortical function based on grid cells in the neocortex . Frontiers in neural circuits , 12 : 121 , 2019 . OpenUrl PubMed [85]. ↵ Philipp Schwartenbeck , Alon Baram , Yunzhe Liu , Shirley Mark , Timothy Muller , Raymond Dolan , Matthew Botvinick , Zeb Kurth-Nelson , and Timothy Behrens . Generative replay underlies compositional inference in the hippocampal-prefrontal circuit . Cell , 186 ( 22 ): 4885 – 4897 .e14, October 2023 . ISSN 0092-8674, 1097-4172 . doi: 10.1016/j.cell.2023.09.004 . URL https://www.cell.com/cell/abstract/S0092-8674(23)01025-5 . Publisher: Elsevier . OpenUrl CrossRef PubMed [86]. ↵ Jonathan Cornford , Roman Pogodin , Arna Ghosh , Kaiwen Sheng , Brendan A. Bicknell , Olivier Codol , Beverley A. Clark , Guillaume Lajoie , and Blake A. Richards . Brain-like learning with exponentiated gradients . bioRxiv , 2024 . doi: 10.1101/2024.10.25.620272 . URL https://www.biorxiv.org/content/early/2024/10/26/2024.10.25.620272 . OpenUrl Abstract / FREE Full Text [87]. ↵ Guan-Horng Liu , Tianrong Chen , Evangelos Theodorou , and Molei Tao . Mirror diffusion models for constrained and watermarked generation . Advances in Neural Information Processing Systems , 36 : 42898 – 42917 , 2023 . OpenUrl [88]. ↵ Berthy Feng , Ricardo Baptista , and Katherine Bouman . Neural approximate mirror maps for constrained diffusion models . In The Thirteenth International Conference on Learning Representations , 2025 . URL https://openreview.net/forum?id=vgZDcUetWS . [89]. ↵ H. Wallach , H. Larochelle , A. Beygelzimer , F. d’Alché-Buc , E. Fox , and R. Garnett Adam Paszke , Sam Gross , Francisco Massa , Adam Lerer , James Bradbury , Gregory Chanan , Trevor Killeen , Zeming Lin , Natalia Gimelshein , Luca Antiga , Alban Desmaison , Andreas Kopf , Edward Yang , Zachary DeVito , Martin Raison , Alykhan Tejani , Sasank Chilamkurthy , Benoit Steiner , Lu Fang , Junjie Bai , and Soumith Chintala . PyTorch: An imperative style, high-performance deep learning library . In H. Wallach , H. Larochelle , A. Beygelzimer , F. d’Alché-Buc , E. Fox , and R. Garnett , editors, Advances in Neural Information Processing Systems , volume 32 . Curran Associates, Inc ., 2019 . URL https://proceedings.neurips.cc/paper_files/paper/2019/file/bdbca288fee7f92f2bfa9f7012727740-Paper.pdf . [90]. ↵ H. Wallach , H. Larochelle , A. Beygelzimer , F. d’Alché-Buc , E. Fox , and R. Garnett Santosh Vempala and Andre Wibisono . Rapid convergence of the unadjusted Langevin algorithm: Isoperimetry suffices . In H. Wallach , H. Larochelle , A. Beygelzimer , F. d’Alché-Buc , E. Fox , and R. Garnett , editors, Advances in Neural Information Processing Systems , volume 32 . Curran Associates, Inc ., 2019 . URL https://proceedings.neurips.cc/paper_files/paper/2019/file/65a99bb7a3115fdede20da98b08a370f-Paper.pdf . [91]. ↵ Po-Ling Loh and Maxim Raginsky Krishna Balasubramanian , Sinho Chewi , Murat A Erdogdu , Adil Salim , and Shunshi Zhang . Towards a theory of non-log-concave sampling: First-order stationarity guarantees for Langevin Monte Carlo . In Po-Ling Loh and Maxim Raginsky , editors, Proceedings of Thirty Fifth Conference on Learning Theory, volume 178 of Proceedings of Machine Learning Research, pages 2896–2923 . PMLR , 02–05 Jul 2022 . URL https://proceedings.mlr.press/v178/balasubramanian22a.html . [92]. ↵ P. Kumaraswamy . A generalized probability density function for double-bounded random processes . Journal of Hydrology , 46 ( 1 ): 79 – 88 , March 1980 . ISSN 0022-1694 . doi: 10.1016/0022-1694(80)90036-0 . URL https://www.sciencedirect.com/science/article/pii/0022169480900360 . OpenUrl CrossRef GeoRef Web of Science [93]. ↵ M. C. Jones . Kumaraswamy’s distribution: A beta-type distribution with some tractability advantages . Statistical Methodology , 6 ( 1 ): 70 – 81 , January 2009 . ISSN 1572-3127 . doi: 10.1016/j.stamet.2008.04.001 . URL https://www.sciencedirect.com/science/article/pii/S1572312708000282 . OpenUrl CrossRef [94]. ↵ Alex Dytso , Michael Fauß , and H. Vincent Poor . The vector Poisson channel: On the linearity of the conditional mean estimator . IEEE Transactions on Signal Processing , 68 : 5894 – 5903 , 2020 . doi: 10.1109/TSP.2020.3025525 . OpenUrl CrossRef [95]. ↵ P. Dayan and L.F. Abbott . Theoretical Neuroscience . MIT Press , Cambridge, MA , 2001 . View the discussion thread. Back to top Previous Next Posted November 03, 2025. Download PDF Email Thank you for your interest in spreading the word about bioRxiv. NOTE: Your email address is requested solely to identify you as the sender of this article. Your Email * Your Name * Send To * Enter multiple addresses on separate lines or separate them with commas. 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