Inference with correlated priors using sisters cells
Authors: Sina Tootoonian
Andreas T. Schaefer
Venue: NeurIPS 2025 poster ()
Keywords: bayesian inference
olfaction
architecture
latent variable models
structured priors
Primary area: neuroscience_and_cognitive_science
TL;DR: A novel network architecture that uses sister cells - same inputs, different connectivity - to perform inference using correlated priors without direct interaction between latents.

Abstract: A common view of sensory processing is as probabilistic inference of latent causes from receptor activations. Standard approaches often assume these causes are \textit{a priori} independent, yet real-world generative factors are typically correlated. Representing such structured priors in neural systems poses architectural challenges, particularly when direct interactions between units representing latent causes are biologically implausible or computationally expensive. Inspired by the architecture of the olfactory bulb, we propose a novel circuit motif that enables inference with correlated priors without requiring direct interactions among latent cause units. The key insight lies in using \textit{sister cells}: neurons receiving shared receptor input but connected differently to local interneurons. The required interactions among latent units are implemented indirectly through their connections to the sister cells, such that correlated connectivity implies anti-correlation in the prior and vice versa. We use geometric arguments to construct connectivity that implements a given prior and to bound the number of causes for which such priors can be constructed. Using simulations, we demonstrate the efficacy of such priors for inference in noisy environments and compare the inference dynamics to those experimentally observed. Finally, we show how, under certain assumptions on latent representations, the prior used can be inferred from sister cell activations. While biologically grounded in the olfactory system, our mechanism generalises to other natural and artificial sensory systems and may inform the design of architectures for efficient inference under correlated latent structure.

Inference with correlated priors using sisters cells
Sina Tootoonian and Andreas T. Schaefer
Sensory Circuits and Neurotechnology Laboratory
The Francis Crick Institute
London, UK andreas schaefer] Scrick. ac. uk
Abstract
A common view of sensory processing is as probabilistic inference of latent causes from receptor activations. Standard approaches often assume these causes are a priori independent, yet real-world generative factors are typically correlated. Representing such structured priors in neural systems poses architectural challenges, particularly when direct interactions between units representing latent causes are biologically implausible or computationally expensive. Inspired by the architecture of the olfactory bulb, proposc novel circuit motif that enables inference with correlated priors without requiring direct interactions among latent cause units. The key insight lies in using sister cells: neurons receiving shared receptor input but connected differently to local interneurons. The required interactions among latent units implemented indirectly through their connections the sister cells, such that correlated connectivity implies anti correlaorrtion in the prior and vice versa. We use geometric arguments to construct connectivity that implements a given prior and to bound the number of causes for which such priors can be constructed. Using simulations, demonstrate the efficacy of such priors for inference noisy environments and compare the inference dynamics to those experimentally observed. Finally, show how, under certain assumptions latent representations, the prior used can be inferred from sister cell activations. While biologically grounded in the olfactory system, our mechanism generalises other natural and artificial sensory systems and may inform the design of architectures for efficient inference under correlated latent structure.
Introduction
common view of sensory processing is probabilistic inference of latent causes,
[r [ NNN)
from receptor inputs [l] Causes are often assumed to be a priori independent (2), so that
p(x) p( ..))
II (w:))
This assumption is not only mathematically convenient, but is also appropriate in some settings. For example, in the celebrated 'cocktail party problem," signals from multiple microphones must be demixed into as many simultaneous conversations, and is reasonable to assume that the audio waveforms from different conversations will be independent.
In other situations, however, the independence assumption may not be appropriate. For example, some notable models 33 of the early visual system describe it as explaining retinal input in terms of simple features like oriented Gabors. Although they arrive at such features by searching for
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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independent causes that explain the visual input, natural scenes are likely to impose correlations on the presence of such features due to large-scale visual structures.
Including correlations is principle easy to do by incorporating corresponding terms in the prior. For example, pairwise correlations, the focus of our present work, can be incorporated by augmenting the prior with quadratic terms,
p( T2, In) x exp E di((.)) ! Cijuiay 15
Non-zero off-diagonal elements of Ct capture correlations among the corresponding features. A prior of this form, which sets (x;²) proportional to 2i, the Gaussian Markov random field (5).
Although analytically simple to incorporate, neural implementations such priors can pose architectural challenges. To illustrate this, we will use simple model of the mammalian olfactory bulb, from which we take inspiration. In this model, receptors yi are linearly excited by latent features 2, (c.g. molecular species comprising odour), and corrupted by Gaussian noise, so that p[)[x) x) N (E, Aujz,, Combining this with the prior above yields the postenior distribution, whose logarithm
inference circuits that use
log (II,,..,,IIII,......... ..M...........)........). = Σ وزرر) Σ Czazzan - jk
Σ
20 (y Σ An
(I)
log
log likelihood
can be maximized over features to yield the maximum posteriori (MAP) estimate of the combination of features (e.g. odour) most likely to have produced the observed receptor activations. A circuit that performs this maximization (see e.g. [6]) contains 'mitral cells' X; one per input channel, that compare actual receptor inputs yi with the system's 's current estimate, Σ; Asz
πA = - + Σ AijTy.
(2)
interacting with 'granule cells' representing the latent causes 2y.. Granule cells are driven by the mitral cells, and are subject to a prior that includes both the individual terms (x,) & (1,((;))/dz and the pairwise terms Cyk,
Tad -0)(0,)) - CHAT + Σ Aigdz.
&
is easy to see that the fixed point of the coupled dynamics Eqn. (2) and Eqn. (3) maximizes the log posterior in Eqn. ① When the dynamics converge, the activity of the granule cells represents the inferred concentrations of the molecules represented by each granule cell.
The problem with this formulation is that each latent cause must interact with many others, as determined by the elements Cjk- Such connectivity may be difficult implement when the number of causes N is very large. For example, in our model system of the olfactory bulb we assume that latent causes are represented by the millions of granule cells, and no direct connections have been observed between them 77. These points suggest looking elsewhere to implement the pairwise interactions the prior.
2 Encoding correlated priors using sister cells
To encode correlated priors without direct interaction between granule cells representing latent causes, note that in the olfactory bulb, each mitral cell has many sisters other mitral cells that receive the same receptor input but connect differently to granule cells 779. We now show how correlated priors can be encoded indirectly by the way granule cells connect the sister cells.
In the dynamics of Eqn. 2 and Eqn. 3 a mitral cell X indexed by the receptor input yi receives. To extend these dynamics to sister cells, we simply add an index to identify individual sisters, and endow each sister Ais with its own connectivity A; to the granule cells:
TAXrs -02 22xi + Vi Σ Aig

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The granule cell dynamics in Eqn. 33 are correspondingly changed to drop the latent interaction term and instead distinguish sister cells
Tad (0 + ΣΣ Aigdin-
(5)
Although we derived these dynamics heuristically by analogy to the case of a single mitral cell per input channel, show in Sec. [ST that their fixed points minimize the loss
[(x) Σ (z ΣΣ
20 yys Σ
(6)
We can give a probabilistic interpretation to this loss by first defining few connectivity statistics. Letting S; indicate the number of sister cells receiving receptor input yi, we define
As
S. Σ A.
Cj.
1 S. E
A A.jA
Aij
Cjs
Σ S.CC
(7)
as the average synaptic weight from sisters to granule cell z., the covariance the weights which connect two granule cells T; and zę to of sisters, and the weighted sum of these covariances across input channels, respectively. Expressing the loss in Eqn. (6) using these terms and completing the square (see Sec.
(S) we arrive at
[(x) = Eg((o) + 20
ΣΣ k
Caagarr +
20 Σ Si(:: Σ Aigag))
(8)
log prior
log likelibood
This has the same form as the loss in Eqn. demonstrating that sister cell dynamics perform MAP inference under a correlated prior, as desired. Note that the connectivity statistics play two roles: first, the mean strength A: of the connections between the unit representing feature j and the sisters sampling receptor encodes the affinity of the receptor for that feature. Second, the prior correlation Cjk relating two features is determined by the covariance of the connectivity between the sisters and the units representing those features. We show this schematically in Fig. K
3 Connectivity for correlated priors
How do we encode a desired affinity and correlated prior into the connectivity? As we show below, we may not be able to encode correlations among all N features. To achieve the desired affinities Ay and correlated priors Cjk- n of the N features, will first assume that the affinities zero, because once we have weights with the desired covariance, we can add the required affinities as offset to all the weights, without affecting their covariances. The covariance of the weights that connect sisters sampling receptor to the units representing latent features j and is then
1 Σ Al S,
C
Our correlated priors set the weighted sum of covariances across input channels. So we must have
Cjk = E -. - Ai
We see that correlated priors are encoded by the scalar product of the weights, when indexed by feature. To make this explicit we reshape the weights into S-element vectors, ay, one per feature making correlated priors scalar products of these vectors.
ax) [ayl:> A A = C5k- = (ay. ag) = Ckr as) = CCr.Cuppjk
where Pjk is the scalar product of the unit vectors, and we've 've used that ||ay| /Cii (see Fig. IC)
The decomposition indicates that we can determine the weight vectors two steps: First, we adjust their orientations to achieve the angles required by the scalar product. We do this by first ordering

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V
B
Ba
0-
02
X
-11AA -Xa)
Eli
Elil
Leteent
Altritty
Latent
Fiii
Latent
Mitiit
Laberi
Gii
Giil
Latent
Aititt
Latent
Figure 1: Encoding correlated priors with sister cells. (A) Schematic of the connectivity between sister cells and units representing latent features and k. The mean Aij of the weights Ai, connecting sister cells A sampling receptor y, to units zj and my reflects the affinity of that receptor for those features. (B) The weighted sum of the covariances of the synaptic strengths with which units I, and z& connect to sisters sampling each input channel encodes the prior correlation between those features. (C) Desired correlations can be constructed by first assuming affinities are zero and adjusting the angles between vectorization a, of the weights Aie- After this adjustment the arfinnffinity condition will likely be violated, but can be satisfied by (D) rotating the weights into the null space of the constraint matrix B. Weights are then rescaled by the desired standard deviations, and the required affinities are added. (Ei) Example weight matrix connecting mitral cells (columns) and latent feature units (granule cells, rows). Neighbouring mitral cells are sisters except across input channel boundaries, indicated by vertical dotted lines. Weights are coloured by their deviation from the affinity of a given channel for a given latent. (Eii) Correlated prior achieved by the weight matrix in panel Ei, showing only the first 10 latents for clarity. (Eiii) Affinity of the first (blue) and second (orange) input channels for the latents, ordered by the former. (Fi-Gili) Many weight matrices can achieve the same correlated prior and affinity, for example the ones shown in panels Eii and Eiii The remaining panels are in Ei-Eiii but for a weight matrix that was optimized for sparsity i.e. changing as few synapses as possible from the value dictated by the affinity (Fi-Fiii), and one that optimized for weighted sparsity, where some input channels had more changes than others (Gi-Giii). See Sec. I details.
the unit vectors arbitrarily, then taking the first and assigning be the first standard unit vector the second vector to be [P12, /I - P12, O,... and so on. Continuing in this way we
e = [1, [1,0,...
J. .... define unit vectors for all features since for the k'th weight vector, we use the first A elements to achieve the correlations with the previously considered features, the k'th element to achieve unit length, setting the remaining n elements to zero. At the end of this procedure, the feature vectors are at the desired orientations relative to each other. We then simply scale each unit vector to its desired length, as specified by the elements /Czj-
It is clear from this geometric formulation of the problem that we have rotational degree of freedom: any fixed rotation applied to all of the feature vectors will retain their lengths and relative orientations, and therefore the desired co-occurrence data. We will use this rotational degree of freedom to achieve the desired zero-affinity condition. This condition states that E, AG = 0 for all input channels and latent features j. To relate this sum to our weight-vectors we convert channels by defining a binary indicator.
sum over sisters and input
{1 if i, indexes sister that samples input channel B
0 otherwise.

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The zero arro-affinity condition then becomes
EΣ Bi for all input channels and latent features
By converting each iim. into an S-element vector bn like we did the weights, this condition becomes
(bm a₂) 0 for all input channels m and features j.
We can specify these MN conditions single matrix equation by stacking the M vectors by into
the M x S matrix
B a [b1, be,....bal .bsl".
and the N vectors ay into the S x N matrix
W a a2 aNl
whereby our zero-affinity condition becomes
BW 0.
To that solutions exist, notice first that has an M dimensional nullspace. Notice also that W has an n-dimensional column space, because we only specified weight-vectors for features, and left the rest at zero. Therefore, as long as
S S -M M
always rotate W into the null null-space of B, and achieve the zero-affinity condition (see the schematic Fig. [TD) This bound says, first, that we can only specify covariance priors if we have more sisters than imput channels. Second, it says that the number of features for which correlated priors grows with the number sister cells. The value sisters we have than input channels. If this number is smaller than the number of latent features N of interest to the system, it has to choose the most important n for which to encode correlations.
encode M counts how many more
To determine the set of all possible solutions, we decompose W into Uw Vir using singular value decomposition (SVD), where we've ve taken Uw to be x n. Rotating this column space leaves the correlations W W unchanged. To find the subset of rotations that satisfy the affinity condition BW = O, we apply SVD to B and get a basis for its row space in the columns of the S x M matrix VB. Letting the S x S M matrix V 3 be arbitrary orthogonal completion of this basis, that the affinity condition is met if and only if V Uw 0, i.e. if the span of Uw is in the span of V We can therefore choose the first column of Uw to be any weighted combination of the degrees of freedom. The second
columns of V that has unit norm, giving column of Uw can be selected in the same way, but orthogonal to the first, giving m - degrees of freedom. Continuing in this way until we have selected all n columns of Uw. Em - i) = nm 1 nn + +) degrees of freedom when determining Uw.
S M
that we have
In practice, we find solutions by first applying an m-dimenniionll rotation to Uw. then picking its first n-columns. We can pick rotations randomly, or to optimize certain properties of the resulting weight matrices. In Fig. E-G we show three different weight matrices, all producing the same affinity and correlated prior, but with different sparsity properties. We investigate the effects of these difference on sister cell responses in Section below.
4 Inference with correlated priors
latent features are correlated, then inference that incorporates this information will outperform that which does not. To verify this, considered the receptor input produced by the simultancous presence of five latent features at high concentration, and corrupted by noise of a fixed variance. We compared the performance of two inference circuits. The first treated all latent features as independent. For the second, we used the approach Sec. 3 to construct connectivity that encoded prior correlations just the five features present. In Fig. 2 compare the inferred feature concentrations at convergence for both circuits. It's clear that the circuit that incorporates the correlated prior outperforms the vanilla circuit.
In Fig- B we show the time course of activity in the granule cells encoding latent features for the circuit that uses the correlated prior, when presented with the input in panel A. The dynamics, which
5

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A
C
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0.2
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0.0
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Time (s)
On
Figure 2: Inference with correlated priors. We simulated network with 50 input channels, each with a uniform random number 4-9 sisters, and 200 latent units (see Supplementary Information for additional details.) (A) Results of inferring the true feature values (gray) when using correlated prior (orange), or assuming feature independence (bluc), with receptor noise s.d. of 0.5. Only the first 10 of 200 features are shown, for clarity.
(B) Time course of inference readout in the latent feature units, the first five of which (orange) correspond to the features actually present, for the correlated prior setting of panel A. Red triangles indicate the exact solution to the problem for those features as determined by convex optimization.
(C) Mean (dots) +/- s.d. (bars) of the error (Euclidean norm) between the true and inferred feature vectors computed over random noisy corruptions of the same receptor input, for different receptor noise standard deviations. See Sec. S2.2 for details.
use biologically realistic time constants, show that the inference result is achieved within few 100 ms, consistent with the time-course of respiration.
Finally, compared the performance of the two circuits over range of receptor noise settings. An important parameter of the inference circuity is the assumed variance of the receptor noise, which can differ from the true noise level and can be adjusted to improve MAP inference. Therefore, for each level of input noise we reported the performance gave the lowest error. The results for both circuits plotted in Fig. 2C and reveal that the circuit using correlated priors outperforms the vanilla circuit at all but the highest noise levels, where both circuits perform equally poorly.
the circuit using the inference noise that
Effect of different connectivity solutions
Next, we examine the effect of connectivity solutions on response heterogeneity. We saw Section that many connectivity solutions exist that produce a desired correlated prior and affinity. Different connectivity solutions will produce different levels of heterogeneity in sister cell responses. For example, sparse solutions like those in Fig. D in which differences in the connectivity of sisters are limited a few sites, will result in more homogeneous responses, while dense solutions, like those in Fig. [ where sister cell connectivity varies widely, will produce more heterogeneous responses. The heterogeneity of responses can therefore inform about the connectivity structure of the circuit, as we demonstrate by comparing to experimentally recordings from the olfactory system.
In Fig. Ai we show the responses of the first (green) and second (olive) sister cells sampling the first input channel to three different stimuli presented at fixed concentration for one second starting at t (gray), for the system using the random connectivity of Fig. E. We see that the responses to some stimuli e.g. stimulus 2, are quite similar, while those to others, e.g stimulus 1, can be different. In Fig.3/Air ii we summarize this response heterogeneity by computing the Pearson correlation of responses over the t = 0-1.5 see time window and averaging over all pairs of sisters in each input channel, yielding one value per channel and stimulus. We took values less than 0.3 (brown) as indicating that the sisters in an input channel had diverse responses to that stimulus, while values above 0.7 (bronze) we took to indicate stereotyped responses. In Fig. A we show how response heterogeneity was distributed per odour channel, ordered by the average similarity of responses. Since

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the random connectivity used was distributed evenly among input channels, we see that response heterogeneity is similar across all channels.
In 3 Bi-Bii we perform the same procedure as in panels Ai-Aii,, but for a system that used the uniform sparse connectivity of Fig. I Because differences in the connectivity of sisters cells to latent feature units occurred at fewer sites, sister cell responses are more homogeneous, and similar across all input channels. Finally, in Fig. Cii-Cii we do the same in the previous panels, but now using the weighted sparse connectivity of Fig. DG. In this setting, input channels with lower indices receive more connectivity changes. This results in high diversity of responses in those channels, as seen in the individual responses of panel Ci and the summary statistics of panel Ciii. The higher density of changes for sisters sampling some input channels, compared the low density in others also results in larger fractions of both diverse, and stereotyped, responses, as revealed by panel Cii.
The patterns of response heterogeneity can suggest the connectivity of given circuit. We demonstrate this by comparing our simulated responses in panels A-C, to those recorded in the olfactory system by [9]. In Fig. Di we show experimentally recorded odour responses of pairs sisters sampling three different input channels, revealing that in all three cases, sisters can respond similarly to some odours, but differently others, like we saw in simulations. In Fig. D we show the distribution of response heterogeneity per olfactory input channel. The prominent presence of both diverse and stereotyped responses, and their uneven distribution among input channels, is similar to our simulations using weighted sparse connectivity in panels Cii, suggesting that similar connectivity in the recorded olfactory system. However, the overall distribution of responses for the experimental data, shown in Fig. 3 Diii shows more diverse responses, and different trend in the distribution than for our simulated data of Fig. 3Cii. We address possible reasons for this in the Discussion.
Estimating correlated priors from responses
Inference using priors that reflect natural feature statistics would improve inferential accuracy. Therefore, natural and artificial systems performing sensory inference would be predicted to use such priors. In our theory, these priors are encoded in the strengths of connections between sister cells and latent feature units (granule cells). Directly measuring these strengths is difficult, while sister cell responses are much easier to measure. Can we infer the priors from sister cell responses alone?
The principal difficulty in determining connectivity from sister cell responses is that multiple latent feature units may respond given stimulus. Therefore, the steady-state response of sister cell reflects the influence of a corresponding number of weights, making difficult to determine the strength of any particular one.
To avoid this problem, we appeal to the assumed sparsity of feature representations and assume a 'one-hot', or "grandmother cell' model of latent feature unit responses in which only one is activated per stimulus. Thus when stimulus j is presented, the feature unit activation vector at the end of inference is approximately,
x = [0, 0, o..., 0, c, 0,0,0,...0)
where ej is the inferred value of the stimulus, and we 'vve used [j]] to emphasize that this is the response to stimulus From Eqn. (4) the corresponding steady-state activity of sisters cells simplifies to
we've
0² Aa 5 v Aijc.
We still need the input yı to this sister cell, but this is common to all sister cells sampling input channel i, and eliminated in the covariance computation (see below).
We can rearrange the steady-state response and express the sister weights in terms of the activity as
Ar,- = (wisl - o² > l/)/c..
To compute the covariance we also need the average value of the weights. Since all sisters receive the same input y. this average is simply
Au = (Hibi) oxx(ii))oo
so the difference need is
A A.j -
- -
7

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Ai
Stimulus 0
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Figure 3: Effect of different connectivity solutions on inference. (Ai) Activity of the first (green) and second (olive) sister cells in the first input channel to three different stimuli, presented at fixed concentration during the gray time window, when using the random connectivity of Fig. D (Aii) Distribution of sister cell response heterogeneity, computed the Pearson correlation of responses of two sisters from t=0 to 1=1.5 seconds, averaged over all pairs of sisters in each input channel, yielding value for each input channel-odour pair. Values below 0.3 (brown) above 0.7 (bronze) stereotyped. (Aiii) Distribution of response types per input channel, sorted by prevalence of stereotyped responses. (Bi-iii.) As in panels Ai-ii but for a system using the sparse connectivity of Fig. F (Ci-iit). As in panels Ai-iii but for a system using the weighted sparse connectivity of Fig. GG. (Di) Calcium responses (traces) from pairs of experimentally recorded sisters cells from three different olfactory input channels to two different odours per channel [9]. (Dii, As in Aii-ii but for the experimentally recorded responses. See Sec. S2.3 for details.
termed diverse, those
In other words, under our sparsity assumption about latent features, the deviation of the weights from their average is proportional to the deviation of the sister cell activities from their average.
We can now use sister cell responses compute the contribution of the i'th input channel to the prior,
Cj
Σ (Ai
Aij) Asx))
S
0²
Aisli] - Aali]
dia [k] [.(k]
Cj
Ck
0²
cov,
(Ai)()))
cjck The correlation prior is the weighted sum of these
Cju = E
cjck Σ S; cov covv,oo)) (Ai)je
(10)
Therefore, to determine the correlated priors from the sister cell activations, we also need to know the inferred values of each stimulus. We address this requirement in several ways.
One possibility is to assume that all the inferred stimulus values are the same, c. In that case.
C/ - cov (Xi)
8

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so, the correlated prior is proportional to the variance of the sister cell activations. In Fig. A we have plotted the correlated priors estimated in this way from the experimentally recorded sister cell responses [9] (see Sec. S2.4). The results suggest a broad trend of mild anticorrelation. There positive correlation of methyi valerate, which has a fruity odour [Q. with, for example with 2-heptanone, which has 'banana-like, fruity odour' [.. This can suggest a prior modelling the co-occurrence of fruity odours. Nevertheless, the most prominent feature of the panel is the block of strong negative correlations, which include that of methyl varelate with valeraldehyde, which has 'fruity, nutty, berry' odour 12] contradicting a simple association of fruity odours.
The counterintuitive priors in Fig. 4/ were computed assuming constant inferred odour concentrations. Another possibility is that these concentrations are related to the vapour pressure (see Table (ST). Because the vapour pressures varied widely, using them directly resulted in large fluctuations in the estimated priors. Instead, we selected a function that produced visually smoother estimates, whereby
= 2 2(0; + 0.1)
where 2/ is the vapour pressure of that odour. In Fig. 4B shows this approach has reduced the block of anticorrelation around methyl valerate. This procedure suggests how concentration functions can be fit to natural statistics. In fact, when sufficient data is present, the inferred concentrations can be treated as free parameters and individually to any stimulus correlation data. Note, however, that because concentrations are positive, adjusting them can only change the magnitude of an inferred prior correlation, not its sign - see Eqn. (IO).
A final possibility that we considered was to assume that because correlated priors can only be specified between a subset all possible pairs of odours, the priors that are encoded are likely to be strong. We can then find inferred concentrations that maximize the magnitude |Czs] of the encoded priors. We first defined the weighted sum of activity covariances,
2ik- A
Σ S; cov
The magnitude of the correlated prior can then be written
o² 1Qjal- -11Qm
1C,ul = EsiCcal
Σ covs(Aii)
Cjck
cyck
then searched for the inferred concentrations that maximized the summed magnitude of the correlation priors. Because inferred concentrations appear in the denominator, we needed to avoid degenerate solutions that drive them to zero. So rather than working with concentrations, we used inverse concentrations rj e,"
We
1 We then found the vector & [ri...., .n]] of inverse inferred concentrations that maximized the summed magnitude of the correlated priors. That is, we solved
ΣIC₁al = argmax ||||S| 3.k
Σ 0 rrrr (2jal:
argmax
||||
J.A:
where the length constraint on avoids degenerate maximization by scaling of the inverse concen- trations. Given this formulation, the solution to the above problem was the principal eigenvector of [Q In Fig. 4c we have plotted the correlated prior estimated in this way. By construction, many more terms have elevated values, maximizing the sum of absolute correlations.
To quantitatively evaluate the quality of these estimates, consulted a publicly available database of 214 essential oils and their monomolecular components [13]. A subset of these monomolecular odours were also used in the experiments of g these are indicated by the dotted lines the pannels of Fig. 4 We then searched the essential oils database and marked any pairs that co-occurred in at least essential oil with black rectangle. Assuming that the essential oils, which are extracted from plants, are ethologically relevant enough for the animals to encode co-occurrence information about them, we would expect the correlated priors to reflect this, and the corresponding pairs to be postive (green) in the panels of Fig, 4 What we we actually observe is that the estimated terms are all near zero, suggesting lack of correlation. We comment on this observation in the Discussion.
7 Discussion
In this work we have taken inspiration from the olfactory system to show how natural and artificial systems performing inference can use sister cells units that receive the same input but connect

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A
B
C
-20
-20 20
Figure 4: Estimating correlated priors from the experimentally recorded sister cell responses in 9 The values shown are -Caa- positive (green) means prior promoting correlation, negative (red) is anti-correlation. Estimates when assuming inferred concentrations are (A) are constant, (B) fixed function of the vapour pressure of each odour, or (C) determined by the eigenvectors covariance. Dotted lines mark odours present in the essential oils database of [13]. squares indicate
the response
odour pairs that co-occur in at least one essential oil that database. See Sec. S2.4 for details.
differently to units representing latent variables to incorporate correlated priors on the latents, without requiring direct interactions between the latent units. We used geometric arguments to show how the connectivity between sister cells and the latent units can be constructed, and verified improved inference performance when latents were correlated. We demonstrated how different connectivity solutions can affect the heterogeneity of sister cell responses, providing clues about connectivity from responses alone. Finally, we showed how under certain assumptions about latent representations, the correlated priors used by system can be estimated from the sister cell responses alone. Although our approach is derived from the olfactory system, the ideas involved are general and should be applicable to other natural and artificial systems that perform inference in environments with correlated latents.
Limitations. Our work is simple proof-of-concee and has number of limitations. A key aspect of our approach the linear, isotropic Gaussian observation model of receptor responses, which lends itself to completing the square and from which the correlated prior emerges. In many systems, including the olfactory system from which we take inspiration, such a model may be inappropriate or too simplistic, and it is unclear whether extending more realistic models to use sisters cells would readily yield correlated priors. Further, more testing than the single stimulus corrupted by range of noise that used in Section Sec. 4 would be needed to robustly establish the performance of the model. Scction 3 we used geometric arguments to demonstrate how to find connectivity that achieves desired stimulus affinity and correlated prior. An important extension of this work would be to show how natural and artificial systems can Iearn such connectivity from natural stimulus statisties. In that section we also showed how variety of connectivity solutions exist and in Section 5) we explored the effects of different solutions on the heterogeneity of sister cell responses, and compared them those observed in the olfactory system. The weighted sparse connectivity, in which sisters in some channels had more heterogeneous synaptic strengths than others, qualitatively matched the per-channel heterogeneity statistics (compare Fig. Dii. but fell short the pooled statistics (Fig. [ Diii)..ii... However, we did not directly optimize the sparsity weighting to match these statistics, and doing so may improve the match further, and suggest a similar connectivity the olfactory bulb. In Section we showed how priors can be estimated from sister cell responses alone. This was only possible because of our assumption of "grandmother-cell latent feature representations. We have not explored whether estimation is possible when this assumption is relaxed.
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Acknowledgements
This work was supported by the Francis Crick Institute, which receives its core funding from Cancer Research UK (CC2036 to A.T.S.), the UK Medical Research Council (CC2036 to A.T.S.), and the Wellcome Trust (CC2036 to A.T.S.). This work was also supported by the National Science Foundation / Canadian Institutes of Health Research / German Research Foundation Fonds de Recherche du Quebec/ UK Research and Innovation-Medica Research Council Next Generation Networks for Neuroscience Program (Award No. 2014217 to A.T.S.). We are grateful to the members of the Schaefer lab at the Francis Crick Institute and the Latham lab the UCL Gatsby Computational Neuroscience Unit for useful discussions.

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References
[1] Peter Dayan and Larry F. Abbott. Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems. MIT Press, Cambridge, Mass., new ed edition edition, September 2005. ISBN 7778--262--51885-
[2] A. Hyvärinen and F. Oja. Independent component analysis: algorithms apppiictoonn,,, Neural Neural Networks, 13(4):411-430 June 2000. ISSN 0893-6080. doi: 10.0016550022--60000000000060010
URL hRLtt::/://www www.wscceinncnerec........m httpi/reetcccceseeeeeeeeeeeeeeee wtt.:ccooccccieeee htt.://hc.........coc..oooooooan
[3] Bruno A. Olshausen and David J. Field. Emergence of simple-cell receptive field properties by learning sparse code for natural images. Nature, (686))))60007600 June 1996. ISSN 1476-
4687. doi: 10. (038/3816077a))
[4] Anthony J. Bell and Terrence J. Sejnowski. The "independent componetts" of natural scenes are edge filters. Vision Research, 37(23):3327-3333 December 1997. ISSN 0042-6989. doi:
articllepp11/00029999770022
[5] Havard Rue and Leonhard Held. Gaussian Markov Random Fields; Theory and Applications. Chapman and Hall/CRC, New York, February 2005. ISBN 978-0-429-20882-9, doi: 10.1201/ 9780203492024.
[6] Sina Tootoonian, Andreas T. Schaefer, and Peter E. Latham. Sparse connectivity for MAP infer- ence in linear models using sister mitral cells. PLOS Computational Biology. 1801)..0009808 January 2022. ISSN 1558-7358, doi: 10. 1 I.jura//oor 1009808. URLhhttps://jounna 13710/00rn3/1//r pcbi 1009808
plos.oor/////oeplloppppoolpl org/
Publisher:
Public Library of Science.
[7] Gordon M. Shepherd. The Synaptic Organization of the Brain. Oxford University Press, Oxford; New York, 5th edition, 2004. ISBN 977-0-19-511955--4 978-0-19-93 19-15566-1.
[8] 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): (((11):14444 444--4412, November 2010. ISSN 1097-6256. doi: 10.108//n.2677. URL http::tt::/ste nature...... www.ta...o/eeeeeeei http////./..//ooo//o///aaa////// m11/full/nn nn. 2673.66777 html
[9] Yuxin Zhang, Carles Bosch, Tobias Ackels, Alfred Laugros, Anne Bonnin, Jayde Livingstone, Carsten Waltenberg, Manuel Berning, Sina Tootoonian, Mihaly Kollo, Andrea Nathansen, Nor- man Rzepka, Peter Cloetens, Alexandra Pacureanu, and Andreas T. Schaefer. Cruucture-Functioo Mapping of Olfactory Bulb Circuits with Synchrotron X-ray Nanotomography, April 2025.
URL https: www. biorxiv. org/ wttp://ntttt...1//////..0/1 11011102111222255 .4. 650439vv
Pages:
2025.04..4.650433 Section: New Results.
[10]
June 2022. URL URL /ttps://e wwwp://wwwiiiiiiiiirreeeeaa. phpPtitle
Page Version ID: 1094960830.
[11] 2- 2-Heptanone, March 2025.
URL http:://ww...kipiiaaaa..///////p
2 --Heppaanoeekoodddd=-222233311-- Page Version ID: 1282131562.
[12] Pentanal, April 2025.
URL https::/een.wikjjeeaaoooog// http:/peedaa..gg////eee...... phpptittteeer...ptiteeee
Pentamalkollddd-184467777 Page Version ID: 1284463797.
[13] Wesley Qian, Jennifer Wei, Benjamin Lanchez-Lengeling. Brian K Lee, Yunan Luo, Marnix Vlot, Koen Dechering, Jian Peng, Richard C Gerkin, and Alexander B Wiltschko. Metabolic activity organizes olfactory representations. eLife, 12:e82502, May 2023. ISSN 2050-084X. doi: 777554/eLifc.885000 URL https::/doii ieciiiorg//i 10. (1554/eLifee eL24f Publisher: eLife Sciences Publications, Ltd.
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Supplementary Information
S1 Derivations
To show that the fixed points the dynamics Equ. (4) and Eqn. (5) set the temporal derivatives to zero. Doing for Eqn. 44 gives
minimize the loss in Equ. (6)
a² Ais V/ Σ A.g
Doing for Eqn. (5) and using the above expression for Aia gives
,イ(ي))= - 1 DE A4 -
Σ Aljzy).
Setting the partial derivative of the loss in Eqn. (6) with respect æy gives the equation, proving that the fixed points of the dynamics are the same as those the loss. The latter will be minima when the (1y)) are the convex functions typically used in the literature.
To show that the loss Eqn. (8) is equivalent to that Eqn. (6), let's focus on the contributions to the former loss from single input channel. Expanding it out and combining terms over sisters,
Ev,, -
2y Σ
I3 + ,
Σ
Σ
A TT
The first term is S; copies of the squared input ve" since each sister will receive the same input. The second term involves term summing the weights Ai, connecting sister cells sampling input channel i to granule cell j. We can represent this sum as the average of the weights Aij. scaled by the number of sisters S; sampling that input channel. We then have
E(u - A1,,) S
v² - Σ A.j23
+
+Σ
Ai,44 2YZk.
The first term in brackets is the contribution to the loss from input channel with single mitral cell but is missing the pairwise interaction terms between the granule cells. We can add this missing term, but must also subtract it to leave the overall sum unchanged. *Completing the square' in this way, we arrive at
Sr 21 Σ
+ ΣΣ
ΣΣ &
Ang AnD = - A00z) -E
Ary AuRsEr
3 &
&
Substituting this into our expression above,
Σ (14 Σ A1,;)) S: Σ 1 ΣΣ
Σ +
1y0k S Σ Σ A Au.
Factoring out S, from the middle term and combining interaction terms, we get
Σ (1n Σ ~ =S Ag + 5; ΣΣ
S Σ
A1y - Aij Are
We recognize the last term in brackets as the covariance C22, of the weights with which sister cells sampling input channel connect to granule cell j and granule cell k. We can finally express the contribution to the loss from input channel as
Σ( Σ A² = 5.(v: -
A + S. ΣΣO CLITAA
k
Returning to the loss in Eqn. (6) we now see that we can write it as
[(x) Σ b(aa)) + Σ 20 (U: Σイ + 20
ΣΣΣ
S.C myaat
which after pooling covariance terms across input channels, yields the loss in Eqn. 8
13

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S2 Simulation Details
The standard deviation of the receptor noise and that of inference were on 0.5 and ont 20, respectively, unless otherwise noted. In all simulations the individual prior on the latents was the elastic net
74 2 24
(x,2,) Br;
where the & parameter 3 = 0.1, unless otherwise noted, and the l parameter "y was set per unit so that its sum with the corresponding diagonal term coming from the correlated prior was 0.1. These values for the parameters of the loss function were selected because they gave good inference performance for the example inputs used in the text. The integration time constants of the mitral cells and latent feature units were Time = 50 msec. and Tge = 100 msec., respectively. These were selected because they were biologically realistic and gave smooth dynamics that converged within respiration time. We used first-order Euler integration with a step size of 200μssec to integrate the dynamics. All simulations were carried out in python version 2 3.9 running mid-2015 2.8 GHz Intel Core i7 MacBook Pro, and all individual simulations ran in about one minute or less.
S2.1
Figure 1
To generate the connectivity matrices in Fig. 1 E-G first generated our desired affinity and correlated prior. We simulated a system with M = 20 input channels and N = 50 latents. The affinities of the input channels for the latents were selected independently from the uniform distribution over [0- 3]. The number of sister cells sampling each input channel was selected independently and uniformly at random from the integer range 10 20. Given this large number of sister cells per input channel, we could use a correlated prior that involved all N = 50 of the latents. We generated such prior by setting a random 10% of the upper triangular elements of N x N matrix 0.1, adding the result to its transpose, and finally setting the diagonal elements to 1. The sparsity and strength of the correlations were set ensure that the resulting matrix remained positive definite. We then scaled the result by 7 so that when scaled down by ain (see Eqn. (8) the diagonal elements would equal the l prior 7.
With the affinity and correlated prior selected in this way, we generated connectivity to achieve using the geometric approach described in Sec. 3 We first generated the S x N matrix W with unit norm columns and angles set by the desired correlations, and performed SVD on it, keeping the product of the singular values and right eigenvectors Sw V We then constructed the constraint matrix B, computed its right eigenvectors V n and an orthonormal complement V We replaced the column space of W by applying a random N x N orthonormal matrix R to this complement, so (updating W).
W V Vir
The matrix in Fig. Ti used such random matrix. For the weight matrix in Fig. Fi optimized over R to find sparse solutions by minimizing the sum 3w |W| with 3w = 10, doing so times with random initializations and keeping the sparsest result. For the weight matrix in Fig. [ used weighted penalty based on input channel, where Sw for weights for sister cells sampling the first channel was 0, those sampling the second was 1, and on. We again repeated this procedure 5 times with random initialization, and kept the best result. Once W was determined in this was, we scaled the columns of W their required lengths. Optimizations were performed using the pymanopt package using its SteepestDDescet optimizer. The resulting matrices are what are plotted in Fig. [ In the final step, we added the affinities of the corresponding channel for the corresponding latent to every weight, to achieve the affinity condition.
S2.2 Figure? 2
To generate the inference results in Fig. 2 we simulated a system with M 50 input channels, N = 200 latents, and a random number between 4 to 9 sisters per channel. Our correlated prior promoted coactivation of the first identity diagonal, and -0.24 off-diagonal for the first nzn block. This was then scaled by o (see previous section). Other parameters were as Figure 1. In panel A we were only interested in the end result of inference so instead of simulating the dynamics we simply computed their solution by solving the convex optimization directly, using the cvxpy python package with the SCS solver. Fig. 2 we were interested in the dynamics so simulated them, with the same parameters. In Fig. 2 we were
latents. We achieved this with a correlation matrix with
14

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again only interested in the end result so we minimized the loss directly using convex optimization.
We computed the results for different settings of receptor and inference noise standard deviation, testing every combination of
{0.1, 0.2, 0.5, 1, 2,5, 5, 10) and 0if {1, 2, 5, 10, 20, 50, 100). The receptor standard deviation was used to generating random noise on each of 5 trials to corrupt the receptor input generated by the presence of the first features at unit concentration. For each circuit, we found the setting of dif that gave the best trial-averaged error at each noise level, and reported the results for each on using that setting.
S2.3 Figure 3
The simulations Fig. Bi, we used the connectivity and parameters described in the details of Figure 1, and ran the dynamics of these three circuits for three stimuli, each consisting of a single latent at unit concentration. Temporal similarity indices for comparing sister cell responses were defined as the Pearson correlation of responses over the first 1.5 seconds following stimulus onset. In Fig. Bii we ran the three circuits for 50 stimuli, each consisting of one the latent features at unit concentration. For each input channel and stimulus similarity among all pairs of sister cells, and plotted these per channel, labeling similarities below 0.3 diverse (orange), those above 0.7 stereotyped (bronze). In Fig. 3 Biii,,iii,CCii Ciii,Cii plotted the cumulative distribution of response similarities pooled over all input channels and stimuli.
then computed the average response
S2.4
Figure 4
estimated correlation priors from the experimentally recorded responses of (9] by assuming 'grandmother cell' feature representations to relate responses directly to connectivity. The only unknowns were then the inferred concentration for each odour. In Fig. [A assumed all these concentrations were the same constant value, in Fig. 4 we used an ad-hoc function of the vapour pressures listed in Table ST and in Fig. C we used concentrations whose inverses, the principal eigenvector of the matrix of absolute value of covariances. Since the experimental data were pooled across multiple experiments, we first normalized the odour responses of each sister cell by its standard deviation across stimuli. We then scaled the odour responses by the inverse of the assumed inferred concentration for each odour. Next, for each input channel, we computed the biased covariance of the odour responses across sister cells. We used biased covariance so that the required division was by the number of sister cells, not this number mminus 1. We then weighted the resulting odour odour covariance by the number of sisters in glomeruls, and summed the result to produce overall correlation prior (see Eqn. 100).
In Fig.
vector, was
15

---

S3
Miscellancous
Table S1: Vapour pressures for odours used in 9
Odour
Vapour Pressure (mmHg)
Nonanoic Acid
9
e-yyproxyaccoopheeone
I-Nonanol
41
PPeenylproopoonalellyyd
294
1 D--Dimethoxyybenzenee
0.47
Ethyl Valerate
4.745
Arethoee
69
2-Nonaone
645
2- 2-Methyl2--Butnne
16.8
Benzaldehyde
1.27
Hexanoic Acid
158
Methyl Valerate
11.043
Benzyl Acetate
177
I-Heptanol
325
alpha-Tesr Terpinene
1.64
Acetophenone
397
Valeraldehyde
31.792
Geranyl Acetate
256
(+)-eench
463
Ethyl Heptanoate
0.68
4- 4-Allylanisole
0.21
Cyclohexanol
975
Dodecanal
Propyl Acetate
35.223
1.4-Cineole
1.93
Guaiacol
78
Butanoic Acid
1.65
Methyl Salicylate
343
2- 2-Methoxy-4-MMetyylhheee
2-Heptanone
4.732
Nonanal
532
cis-3-Hexenyl Tiglate
306
Ethyl Caproate
1.665
Eugenol
104
S-(+)-Carvone
66
Methyl Benzoate
0.38
Octanal
2.068
Valeric Acid
452
Mineral Oil
0
LacDipennyyleettoohennne
63
Eucalyptol
1.9
Ethyl Tiglate
4.269
Undecanal
83
2-Citronellic Acid
Methyl Butyrate
35.9
R- R-()--iimonene
198
Ethyl Butyrate
12.8
4- 4-Methyloctanoic Acid
Ethyl Acetate
111.716
leMehhhlacetooheeonee
187
16

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NeurIPS Paper Checklist
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Answer: [Yes]
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Reviews:
[{"review_id": "H7uJ51glJ3", "reviewer": "Reviewer_W7wp", "summary": "This paper presents a mathematical formulation inspired by sister cells, demonstrating how the connectivity between these and the latent units can enhance inference performance when latent variables are correlated. \n\nI have minimal knowledge on this topic and am not up to date with the current state of this research. So, please read my criticism below with a grain of salt.", "questions": "Q1)  Can authors show how much boost they get by optimizing the sparse connectivities? \nQ2) Can authors sketch the inference steps if the receptor responses were not modelled as linear/isotropic Gaussian observations? What are the challenges and what solutions are there to overcome those?", "limitations": "Yes.", "rating": 5, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n1. Extensive experimental study\n2. Solid mathematical modelling\n3. Well-explained, very clearly written paper\n\nWeaknesses: \n1. As mentioned by the authors, the receptor responses were modelled as linear/isotropic Gaussian observations, which can be viewed too simplistic. \n2. Sparse connectivity can also be optimized by introducing hyperparameters (as mentioned in the limitation section).", "was_revised": true, "final_justification": "The authors' responses addressed my concerns during the rebuttal period. I keep my original score, as it reflects the state of the paper post-rebuttal as well.", "rebuttal": "We thank the reviewer for their comments. Below we provide answers to the two questions they raised.\n\nQ1) The reviewer raises an important point about the role of sparsity in our model. As we show in the main text, many connectivities encode the same receptor affinities and molecular coactivation statistics, spanning the spectrum from sparse to dense. Therefore, the computational boost when using sparse connectivity is not in the final result of inference, as that would be the same for all such connectivities. The advantages gained from sparsity would principally be in the sparsity itself: fewer synapses would need to be adjusted to encode the same statistics. There may be dynamical advantages wherein sparser connectivities arrive at the solution faster than denser ones, but we have not explored this important question. Our main motivation for exploring the role of sparsity was the empirical observation in the olfactory system that sister cells can have similar responses to some odours, and different responses to others. We achieved a comparable level of heterogeneity using a weighted sparsity solution, where sisters from some glomeruli had denser connectivity to the granule cells than others, and therefore suggest that the olfactory system may operate in this regime. This sparsity may be a result of learning - work by others has shown that learning in neural systems using 'exponentiated gradients' produces synapses that obey Dale's law, so don't change sign during the learning process, is competitive or superior to standard gradient descent on many tasks, and produces sparse, log-normally distributed synaptic weights. Such learning in the olfactory system could produce the sparse connectivity that appears to be necessary to reproduce the response heterogeneity observed empirically. We plan to address this point in future work on how connectivity encoding odour statistics, which we constructed by hand in this manuscript, can be learned. \n\nQ2) The reviewer raises an important point that we also mention in our \"Limitations\" section. The way we encode second-order odour statistics relies on the linear Gaussian aspect of our model, which allows 'completing the square' and an interpretation of the sister cell loss as encoding receptor affinities and molecular coactivation statistics. We have not yet explored whether deviations from such a simple model would readily yield a similar encoding of odour statistics. We speculate that non-isotropy would be easy to incorporate, as differences in variance among the glomeruli could be analytically equalised by absorption into synaptic weights and the strengths of glomerular inputs, perhaps at the cost of some degrees of freedom. Deviations from the linear-Gaussian model, however, seem much more difficult to accommodate. For example, a more accurate model of receptor responses would use Poisson statistics for the spike counts. However, at least in our back-of-the-envelope derivations, which used a linear-Poisson model, we were unable to readily encode higher-order statistics in the connectivity, though it's possible that with further work, we could do so. Therefore, at this point, our results should be interpreted as limited to linear Gaussian models."}, {"review_id": "kJngNngtjJ", "reviewer": "Reviewer_2zE9", "summary": "This paper develops a neural circuit model of the olfactory system that performs probabilistic inference using priors that capture correlations between latent generative factors of stimuli. To overcome the architectural challenges associated with modeling such correlations as direct connections between units representing latent variables, the authors leverage \"sister cells\" — a type of cell that receives copies of the receptor input — to bake this prior indirectly into their model. In this framework, the requisite coupling between latent units are captured by the strength of the synaptic connections between sister cells and latent units. The authors provide geometric intuition and a bound on the number of correlated latent causes that can be modeled. The model outperforms models that assume priors with independent latents for stimuli when inferring stimuli generated from correlated latent factors. The authors also show that the connectivity structure for a desired correlation structure exhibits degeneracy, and that different connectivity patterns provide different levels of response heterogeneity. They demonstrate that this link may be used to infer the connectivity structure from experimentally recorded responses. Finally, the authors investigate whether the responses can be used to infer the priors themselves. Under assumptions of one-hot feature representation and given a choice of inferred stimulus values, they can calculate a prior correlation matrix, which is then compared against latent factor correlations present in an ethologically relevant dataset.", "questions": "1. Since the objective in eq 8 uses the synaptic weights that average across sister cells (from eq 7), am I right in saying that the dynamics implied here are specified for the average sister cell activity?\n2. To verify the bound derived in section 3, can you show via simulations what would happen if there are fewer sisters than input channels, and how the number of features grows with the number of sister cells? \n3. If the stimuli were composed of features where only a portion of the features are correlated (with the rest being independent), how would inference with correlated priors look like? Similarly, it would be useful to show the converse of section 4, that if the stimuli were composed of fully independent features, performing inference using correlated priors would still recover the independent factors (i.e. match the performance of a standard model with independent priors). \n4. In section 6, it is surprising that there is a mismatch between correlated prior estimates and the empirical observation of correlations in monomolecular components. Is this purely a function of assuming grandmother cells, or are there other assumptions that lead to this mismatch? Conversely, under which regimes would I recover the observations even under a grandmother cell assumption?", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths \n\n[+] Quality: The model successfully overcomes the problem initially proposed at the onset of the paper, as to whether correlations can be included in the prior without invoking direct interactions between latent cause units, which is both empirically unsupported and computationally impractical. The mathematical derivations are sound and well reasoned. I also appreciated the attempts to form a tighter connection to empirical data, both by comparing model predictions with experimentally measured unit responses, and by comparing the model correlation matrix against the empirical correlations extracted from the ethologically relevant odor dataset. \n\n[+] Clarity: It was clear which parts were assumptions, which were novel contributions, and which were consequences of the data/model. The investigatory questions following the model formulation were thoughtful and flowed logically, including a sanity check to see if this model outperforms the vanilla model with a prior of independent latent factors. \n\n[+] Significance: This model proposes a novel way to incorporate correlations in the latent factors in the prior of a neural circuit model of the olfactory system, and provides what are the necessary conditions for using sister cells to encode n correlated features in the prior. \n\n[+] Originality: while encoding priors and performing inference via fixed points of a dynamical system is not new, this work provides a novel way to induce implicit correlated priors in a circuit via the connectivity structure between latent units and neurons that receive shared receptor input. \n\nWeaknesses\n\n[-] It would be useful to have a little bit more context to connect Eq 1 to Eq 2 and 3, otherwise ref 6 must be read to understand the model setup. \n\n[-] Some parts of section 3 (finding the bound on the number of features, and finding the set of all possible solutions to W and B) were overly dense, especially since not all of the conclusions were necessary for the rest of the paper and/or verified via simulations. I believe it could be tightened by keeping only the essentials and punchlines of the derivations and moving the rest to the appendix. \n\n[-] Clarity: Some of the wording explaining why it is reasonable to first zero out the affinities is unclear. There are also a couple of typos; line 218: \"this the\" → \"this is the\"; line 223: \"compute to\" → \"compute the\"; line 266: \"quantitavely\" → \"quantitatively\"; line 273: \"esimated\" → \"estimated\"; line 303: \"statistcs\" → \"statistics\".", "was_revised": true, "final_justification": "The authors have provided thorough responses, both to the questions that I raised and to the identified weaknesses. I have also read the other reviews and in my opinion further benchmarking is not necessary (aGi5). The authors have already demonstrated that their method can infer correlated features in synthetic data (in Fig 2) and compared their estimate of the correlated priors to a naturalistic database of odor latents (in Fig 4). While the correlations in the model's priors do not match those observed in data, in the rebuttal the authors have expounded on the reasons as to why this might be the case. Overall, I am satisfied with the responses provided by the authors, and maintain my decision to accept the paper, with the same level of confidence as before.", "rebuttal": "We thank the reviewer for their detailed and thoughtful comments on our submission, and apologise for any lack of clarity in our presentation. We will first address the weaknesses the reviewer raised, and then their questions.\n\n# Weaknesses\n- We regret that the connection between the log posterior in Eq 1, and the dynamics in Eq 2 and 3 wasn't clearer. Although ref 6 does indeed provide more context, the fundamental connection is simply gradient ascent: the 'granule cell' dynamics in Eq 3 ascend the gradient of the log posterior in Eq 1 relative to the latents, employing the 'mitral cells' in Eq 2 as auxiliary variables to compute a component of the gradient. \n\n- Density of parts of section 3: We apologise for the compressed presentation in this section - it was an issue raised by another reviewer as well. Next time we will take the reviewer's suggestion and leave only the essentials in the main text, moving the rest, decompressed, to the appendix.\n\n- Zeroing out affinities: We apologise that the reasoning behind this subtle point was not completely clear. In that section, we are determining how to set connectivities to achieve a particular covariance of the weights. These covariances are computed over sisters, and measure covariability relative to the mean value of the weights computed over sisters. Thus, changing this mean value, which equals the affinities, does not affect the covariances. Therefore, to simplify the derivation, we can first pretend the affinities are zero and compute the weights that achieve the desired covariance. We can then add the correct affinities to all the weights, without changing the covariances achieved by the weights. \n\n- Spelling mistakes: We apologise for these and thank the reviewer for pointing them out.\n\n# Questions\nWe first would like to say that we are grateful for the reviewer's thoughtful questions, which reveal their engagement with our submission.\n\n## Q1 \nThe objective in eq 8 indeed uses synaptic weights that are averaged across sister cells. And indeed, if one averages the sister cell dynamics in eq 4, one see that the dynamics of the resulting average sister cell activity use the weights averaged across sister cells. However, a key aspect of those dynamics is how they are influenced by the granule cells (x_j). If we replace the sister cells with their average, then the information about correlations in the prior can no longer be represented in the different activity levels among sisters, since these have been replaced by an average. Instead, it must be supplied by direct interactions between the granule cells, similar to eqn 3. Therefore, the loss in eq. 8 can correspond to dynamics using average sister cells, using the average weights, interacting with granule cells that interact directly in order to implement correlations in the prior. \n\n## Q2\nVerifying the bound would be an important addition to the manuscript. The number of sister cells S is always at least equal to the number of input channels M, since there is at least one mitral cell per input channel (otherwise the information from that channel would be unavailable to the system), and we call all mitral cells sampling a channel sisters, even, perhaps confusingly, when there is only one such cell. One way to verify the bound would be to directly optimise the weights to achieve a desired set of affinities and covariances. We currently construct such weights using the geometric approach that we used to derive the bound. However, the optimisation approach could be made agnostic to our derivation, and to merely search for sister cell weights that have the relationships we require. If our bound is correct, then we should find that when the dependencies we require are among more latents than our sister cell budget allows, the optimisation fails because the problem is not feasible. Therefore, plotting the frequency of such failures as we increase the number of dependencies encoded could serve as a partial verification of our bound.\n\n##  Q3\nThe configuration of correlations the reviewer wonders about is, in fact, already the case for the simulation results in Figure 2, for which only 5 of the features (the first 5) are correlated in the prior, and the rest are independent.\n\nShowing that the model using correlated priors can still perform inference when the latents are, in fact, independent, is a good suggestion, and we could certainly include such results. However, we would expect that performance in that setting would be inferior to that using a prior that assumed independence, since the correlations in our prior would bias inference to reflect such correlations, when in fact none would exist. This would essentially mirror the results shown in Figure 2C, and would reflect the reduction in inferential accuracy when the assumed prior does not match the statistics of the data.\n\n## Q4   \nThere could be a number of reasons for the mismatches we observed in Figure 4, relating both to our model and to the available data on the frequencies with which pairs of molecules are present in natural odours. First, the predictions in those panels require assuming that our model of sister cell dynamics in equations 4 and 5 are correct, and that the grandmother cell assumption is correct. Both of these are at best approximations.\n\nEven under the grandmother cell assumption, the predictions require knowing the inferred concentration of each odour (c_j and c_k in Eq 10). A given data point on the observed correlation of two monomolecular odours would provide a value for Cjk in Eq 10, and thereby constrain the product of concentrations on the right-hand side of that equation. We only had values for 6 pairs of odours, so rather than fitting the concentration values to optimise the fit, we opted to make three different assumptions about the inferred concentrations and show the results in the panels of Figure 4. If we had at least as many pairwise datapoints as we have monomolecular odours, then we would have enough data to fit all the concentrations, up to an overall scaling factor, and would then hope to improve our match to the observations. \n\nFinally, assuming our model is correct and we had sufficient data on odour co-occurrence statistics, we would only expect a match if those statistics are those of the environment in which the animals whose bulbar response data lived. Thus, the ideal experiment would be to raise animals in an environment in which the co-occurrence statistics are known, and then to use our approach to estimate these co-occurence statistics from the sister cell responses."}, {"review_id": "ya5dhJeIsy", "reviewer": "Reviewer_j9ec", "summary": "The paper demonstrates a way to implement correlated priors over latent features in a dynamical systems model of Bayesian inference, inspired by the rodent olfactory bulb. Experimental results show that allowing nonzero correlation in the feature priors can better capture true feature distributions than assuming independence in the prior. Under the assumptions that the \"sister cell\" model holds, the paper shows how to estimate priors empirically from the responses of those sister cells, which could hypothetically be recorded.", "questions": "Can the authors provide citations to computational papers on the mammalian olfactory bulb demonstrating that its architecture as a latent-factor inference circuit generalizes across species?\n\nIn what sense does the paper's non-spiking, real-valued model of neurons incorporate interneurons vs pyramidal cells?  Interneurons would typically (admittedly, this is in cortex) refer to inhibitory cells with many-to-one connectivity, or (in the spinal cord or in very simple model organisms) simply to any neuron that is neither a sensory nor a motor neuron.", "limitations": "The paper's model, as stated by the authors under Limitations, can only enable estimation of neural priors under the \"grandmother cell\" assumption, an assumption that we have fairly strong reasons not to believe in the form it is stated here: https://www.tandfonline.com/doi/full/10.1080/23273798.2016.1235279 .", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strength: mathematical fluency\n\nWeakness: stating log-probability densities algebraically instead of as members of exponential families, parameterized in a standardized way, robs this paper of much of its clarity.  Definitions are not given in the manuscript and the primary theorem appears to be in the supplementary material.  Lines 129-137 on page 5 basically hand-waves a sizeable amount of linear algebra that should have been stated in Proposition form and then had reasonable-length proofs given in the Supplemental Material for checking.\n\nStrength: good experimental results\nStrength: despite use of the problematic \"grandmother cell\" assumption, decoding of priors from network activity is an interesting demonstration", "was_revised": true, "final_justification": "Upon reading the Author Response I must grant that spike-rate encoding is standard in this field, while also reducing my confidence in my final score. I hope that with the other reviewers and the area chair we can come to a more confident assessment of the paper.", "rebuttal": "We appreciate the reviewer's feedback, which raises a number of important points, both about the biological realism and applicability of our model and its mathematical presentation. We will address the points relating to biology first.\n\n# Relation to Biology\nWe would like to begin by stating that our primary aim was not to propose a new model of an olfactory bulb in a particular species, but to take inspiration from its architecture, particularly in rodents, to propose an algorithm that we believe applies more generally to other inference tasks that require correlated priors, including those beyond olfaction, encompassing natural as well as artificial systems, which we believe is of interest to the machine learning community. Hence, we avoided detailed explication of olfactory bulb anatomy and instead focused on the elements most relevant to our algorithm. However, we do believe that our algorithm provides some insight into the structure and function of the rodent olfactory bulb, hence our comparisons to experimental data from that system. Should we decide to apply our work more directly to understanding mammalian olfactory bulbs, we will be sure to take the reviewer's comments on board and highlight the connections in more biological detail.\n\n## Q1\nThe reviewer is right in their summary to point out that we speak of \"the\" olfactory bulb - implying that all mammalian olfactory bulbs are similar - and asks in their first question for citations of computational papers demonstrating its architecture as a latent-factor inference circuit generalises across species. We are unaware of computational papers demonstrating such cross-species generalisation. However, we point out that it is common in normative modelling of olfaction to speak of \"the\" mammalian olfactory bulb, rather than specialising to a particular species. For example, see the following papers, which all treat \"the\" olfactory bulb within the latent-factor inference framework, and none of which are from us: Koulakov, Rinberg 2011, Grabska-Barwinska et al. 2016, Kepple et al. 2019, Zavatone-Veth, Masset et al. 2023. This is because such models at present, including ours, aim primarily to explain qualitative aspects of olfactory system dynamics that depend on anatomical properties shared by many mammalian olfactory systems, such as ORNs projecting by receptor type to glomeruli, mitral/tufted cells sampling these glomeruli and projecting to the cortex, while interacting indirectly via inhibition by granule cells. As such models increase in sophistication and include additional biological details, we expect that they will correspondingly specialise their results to specific olfactory bulbs. We thank the reviewer for raising the issue of cross-species comparison, and believe that it will be fruitful to look, in future work, at how different sister cell architectures in different species may impact inference.\n\n## Q2\nThe reviewer is correct that our model incorporates non-spiking neurons, although the real-valued responses are meant to represent spike rate, in absolute terms for the 'granule cells', and relative to a baseline for the 'mitral cells'. Modeling spike rates rather than spikes is common in the field; see e.g. the first three papers cited in our response to the previous question. Our 'mitral' cells interact with inhibitory 'granule' cells. The latter are interneurons in the sense the reviewer describes: they are neither sensory nor motor neurons. They, in fact, lack axons, so cannot themselves report the results of inference beyond the bulb. This is OK when our model is viewed, as we intend, as an inference algorithm _inspired_ by the olfactory bulb, since the results of inference can be read out from the 'granule cells'. It may be problematic, however, if our model is interpreted as a description of the operation of the olfactory system, since the bulb would be unable to directly communicate the results of inference, as represented by granule cell activity, to downstream cortex - although these results could be read out indirectly through the influence of granule cells on mitral cells, which _do_ project to the cortex. Other works, e.g. that of Koulakov and Rinberg cited above, also focus on how odours can be represented by the granule cells. Such issues are perhaps why the reviewer asks whether we model interneurons or pyramidal cells. One way to resolve these issues is to think of our 'granule cells' as encapsulating a disynaptic circuit in which mitral cells project to pyramidal neurons in olfactory cortex, which (as is known anatomically) project back onto granule cells, which in turn inhibit the mitral cells. We did not delve into these important nuances because our model is an algorithm inspired by the olfactory bulb, but is not meant to describe it in detail. Should we decide to apply our model to explicating olfactory bulb structure, we will be sure to explicitly address the different roles of inhibitory interneurons in the bulb vs pyramidal cells in the cortex.\n\n# Mathematical presentation\nWe now address the reviewers' points about our mathematical explication.\n\nThe reviewer regretted our choice of stating log probability densities algebraically rather than as members of exponential families. We made this choice mainly to clarify the link between the log posterior being maximised (eqn 1) and the dynamics of the maximisation (eqn 2 and 3), hoping that it would illustrate how the latter equations extremize the former. We apologise for any confusion or lack of clarity that this may have caused.\n\nThe reviewer is also correct that our primary Theorem, showing how the loss function extremized by the sister cell dynamics, eqn 6, can be expressed as a loss incorporating correlated priors, eqn 8. Due to space constraints and because the derivation is purely algebraic, we limited our description in the main text to mentioning that the most important step is the standard procedure of completing the square and relegated the detailed step-by-step derivation to the supplement. We again apologise for any confusion this may have caused.\n\nThe reviewer also points to the \"hand-waving\" of a significant quantity of linear algebra in lines 129-137. Though we don't consider our description \"hand-waving\" as we believe the lines mentioned constitute a detailed proof, including such detail in less than 10 lines due to space constraints certainly produced an overly-compressed presentation, and we apologise for the resulting difficulty in parsing the dense text. In fact, this issue was also raised by another reviewer. In the future, we will take the reviewer's advice and provide the statement of the result as a Proposition in the main text, and relocate the proof to the supplementary material."}, {"review_id": "5oMksC52Gb", "reviewer": "Reviewer_aGi5", "summary": "The paper proposes a biologically-plausible formulation that lets a neural network perform infenence with correlated latent variables, even when the neurons representing those latents are not connected directly to one another.\n\nOn simulated data, the correlated prior performed better than it's uncorrelated counterpart.\n\nSome theoretical arguments where made about dense random weights vs. weighted sparse patterns, which best matched experimental calcium responses, and serve to act as a connection between the method and empirical neurophysiological results.", "questions": "- What's the notable take-home message from this paper?\n- Is the novelty here trying to test a neuroscientific theory in-silico, or is it to use a biologically-inspired mechanism to introduce a new algorithm?", "limitations": "I appreciate the authors acknowledgment of weaknesses, which seems to echo my point about the findings in this paper being unclear and potentially not self-contained in-terms of a final message/insight. Given the above, what's the real take-home message of this paper?", "rating": 3, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n- Creative method, with a relatively thorough attempt to mathematically motivate the work.\n- Attempt to close the loop between empirical observations and algorithmic inspiration. I appreciate the attempt to motivate the method through a neuroscientific lens, and then subsequently attempt to link the results back to the observed data.\n\n**Weaknesses**\n- Only applied to hand-designed, simple data. The significance of the results is very unclear to me. Especially as it pertains to modern ML methods.\n- The connection between the neuroscientific insight and algorithm is weak\n- The significance of the results in general are unclear and open to interpretation.", "was_revised": true, "final_justification": "Work has potential, but to make the right impact, some utility on real data needs to be illustrated. For example a neuroscience dataset.", "rebuttal": "We thank the reviewer for their comments and apologise for any lack of clarity in our presentation.\n\nThe reviewer's first question, and their comment in the Limitations section, asks what the main message of our paper is. It is this: Many natural and artificial systems must perform inference about latents that are a priori correlated. The 'natural' way to incorporate such correlations requires direct interactions among the latents. This can be computationally or architecturally expensive when there are many latents. Our main contribution takes inspiration from the mammalian olfactory system to show an alternative approach to incorporating these correlations, by introducing 'sister cells' which receive the same input but connect differently to the latent units. This both provides a hypothesis as to why there are sister cells, and provides a new way of efficiently implementing inference for more realistic (correlated) priors\n\nThe reviewer's second question is whether the novelty is testing a neuroscientific theory or proposing a new biologically-inspired algorithm. Our main aim is the latter: to propose a biologically inspired algorithm. Although we believe our algorithm sheds light on the structure and dynamics of the mammalian olfactory system, we believe it applies to other natural and artificial systems that perform inference of high-dimensional, a priori correlated latents."}]

Author rebuttal:

Meta-review: This paper introduces a novel circuit model for performing probabilistic inference with correlated priors using a framework called "sister cells". It shows both theoretically and empirically that this architecture outperforms models with independent priors, and links connectivity structure to observed neural response heterogeneity. Three of the four reviewers felt this work was above threshold for acceptance, and I ultimately concur with their assessment that the paper makes a worthwhile contribution to the literature and should be accepted.  Congratulations! Please be sure to address all reviewer comments and criticisms in the final manuscript.
Decision: Accept (poster)