The Bayesian sampling in a canonical recurrent circuit with a diversity of inhibitory interneurons
Authors: Eryn Sale
Wenhao Zhang
Venue: NeurIPS 2024 poster (2024)
Keywords: Sampling-based Bayesian Inference
recurrent circuit model
ring attractor network
inhibitory interneurons
Primary area: neuroscience_and_cognitive_science

Abstract: Accumulating evidence suggests stochastic cortical circuits can perform sampling-based Bayesian inference to compute the latent stimulus posterior. Canonical cortical circuits consist of excitatory (E) neurons and types of inhibitory (I) interneurons. Nevertheless, nearly no sampling neural circuit models consider the diversity of interneurons, and thus how interneurons contribute to sampling remains poorly understood. To provide theoretical insight, we build a nonlinear canonical circuit model consisting of recurrently connected E neurons and two types of I neurons including Parvalbumin (PV) and Somatostatin (SOM) neurons. The E neurons are modeled as a canonical ring (attractor) model, receiving global inhibition from PV neurons, and locally tuning-dependent inhibition from SOM neurons.
We theoretically analyze the nonlinear circuit dynamics and analytically identify the Bayesian sampling algorithm performed by the circuit dynamics. We found a reduced circuit with only E and PV neurons performs Langevin sampling, and the inclusion of SOM neurons with tuning-dependent inhibition speeds up the sampling via upgrading the Langevin into Hamiltonian sampling. Moreover, the Hamiltonian framework requires SOM neurons to receive no direct feedforward connections, consistent with neuroanatomy. Our work provides overarching connections between nonlinear circuits with various types of interneurons and sampling algorithms, deepening our understanding of circuit implementation of Bayesian inference.

The Bayesian sampling in a canonical recurrent circuit with a diversity of inhibitory interneurons
Eryn Sale
Wen-Hao Zhang wenhao v thang@utsoutthhesseen. edu
eryn. waw.eatssssuuuueeeeemn edu
Lyda Hill Department of Bioinformatics, UT Southwestern Medical Center "o'Dnnllll Brain Institute, UT Southwestern Medical Center
Abstract
Accumulating evidence suggests stochastic cortical circuits can perform sampling- based Bayesian inference compute the latent stimulus posterior. Canonical cortical circuits consist of excitatory (E) neurons and types of inhibitory (I) in- terneurons. Nevertheless, nearly no sampling neural circuit models consider the diversity of interneurons, and thus how interneurons contribute to sampling re- mains poorly understood. provide theoretical insight, we build nonlinear canonical circuit model consisting of recurrently connected E neurons and two types of neurons including Parvalbumin (PV) and Somatostatin (SOM) neurons. The E neurons are modeled canonical ring (attractor) model, receiving global inhibition from PV neurons, and locally tuning-dependent inhibition from SOM neurons. We theoretically analyze the nonlinear circuit dynamics and analytically identify the Bayesian sampling algorithm performed by the circuit dynamics. We found a reduced circuit with only E and PV neurons performs Langevin sampling, inhibition speeds up
and the inclusion of SOM neurons with the sampling via upgrading the Langevin into Hamiltonian sampling. Moreover, the Hamiltonian framework requires SOM neurons to receive no direct feedfor- ward connections, consistent with neuroanatomy. Our work provides overarching connections between nonlinear circuits with various types of interneurons and sampling algorithms, deepening understanding of cireuit implementation of Bayesian inference.
Introduction
The brain lives in a world of uncertainty and ambiguity and thus has to infer unobserved world states. The Bayesian inference is normative framework to implement inference, and extensive studies have suggested that the brain's 's perception is consistent with the Bayesian inference, forming the concept of Bayesian brain [I], including, e.g., visual processing [2]. multisensory integration [3], decision-making [ sensorimotor learning [5], etc. Studying how neural circuits in the brain realize Bayesian inference has been active topic in neuroscience 6 Many neural circuit models of Bayesian inference have been developed with distinct representational and algorithmic mechanisms, e.g., parametrie-based representation [ 1 and sampling- based representation [9
Despite a large body of neural circuit models of Bayesian inference, there are still gaps between our current Bayesian circuit models and canonical recurrent circuits in the cortex. One obvious distinction previous Bayesian circuit models haven't considered the rich diversity neuronal types in the cortex, especially inhibitory interneurons. The canonical cortical microcircuit contains
Cooresponding author.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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three major types of inhibitory (1) interneurons 20 24], (SOM), and Vasoactive Intestinal Peptide (VIP) neurons (Fig. [TA) These interneurons have different electrical properties, stimulus-tuning profiles, distinct connectivity, and synaptic modulations with other neurons. For example, PV neurons are weakly tuned to stimulus [20]. and multiplicatively modulate E. neurons in way called divisive normalization by sending axons to E neurons' cell body [2 25 26]. In contrast, SOM neurons are tuned to the stimulus, and module E neurons in an additive way via sending axons to the distal dendrites of E neurons [22] 25 26). Another gap comes from the nonlinearity of cortical circuit dynamics, which impedes the analytical understanding of circuits' Bayesian inference algorithms. Most earlier studies relied on numerical methods to analyze the algorithm in nonlinear circuits (e.g.. g.. [7 19), ), or considered linear neural dynamics to obtain We still lack a comprehensive understanding about how the nonlinear recurrent circuit dynamics with diversity of interneurons implement Bayesian inference.
including Parvalbumin (PV), Somatostatin
analytical solutions (e.g., [2
8
27
28
To gain insight into Bayesian computation in a nonlinear recurrent circuit with types of interneurons, we build canonical recurrent eircuit model consisting of excitatory (E) and two types of interneurons, including PV and SOM neurons (Fig. [B), and investigate how the model implements sampling- -bannd Bayesian inference. The E neurons are modeled as a rate-based ring (attractor) model that emerges the tunings over ID stimulus such as orientation. The E neurons receive internal Poisson-like variability mimicking stochastic spike generation, which provides a variability source to drive sampling [18]. For simplicity, the PV neurons in the model are not tuned to the stimulus as a limiting case of their weak tuning found in experiments [20], and provides global inhibition to E neurons via divisive normalization to ensure stability [21 255 In contrast, the SOM neurons have stimulus tunings and provide locally tuned inhibitory feedhack to E neurons in an additive way (20) (9]. The circuit model
with the above connectivity successfully reproduces multiplicative and additive modulations E
neurons' tunings from PV and SOM neurons respectively (Fig. [G-H) (26].
We perform theoretical analysis on the nonlinear recurrent circuit dynamics, and analytically identify the sampling algorithm adopted in the circuit. We find the reduced circuit with only E and PV neurons can implement Langevin sampling in the stimulus feature manifold. The tuning-dependent inhibitory feedback from SOM speeds up the sampling by upgrading the Langevin sampling into Hamiltonian sampling. And the two types of interneurons have different effects sampling speed. Moreover, we find that Hamiltonian sampling requires SOM neurons not to receive feedforward sensory inputs, consistent with neuroanatomy with few feedforward synapses targeting SOM neurons (2 24 The nonlinear circuit model with fixed weights can flexibly sample posteriors with different uncertainties, if located in the linear input-output regime. At last, the circuits can be extended to sample multivariate stimulus posteriors and bimodal posteriors.
2 The recurrent neural circuit with various types of interneurons
The cerebral cortex is repetition of the canonical neural circuit composed of multiple types of neurons (Fig. [A), including excitatory (E) neurons and three major types of inhibitory (I) interneurons (PV, SOM, and VIP: classified via biomarkers (30).. To study how sampling-based Bayesian inference is implemented by the canonical neural circuit composed of various types of neurons, build recurrent neural circuit model consisting of E neurons and two types of interneurons of PV and SOM neurons (Fig. (B). The model doesn't 't include VIP neurons, which will form our future research (see Discussion). The basic wiring diagram of the proposed circuit model is consistent with the structure of the canonical cortical circuit (Fig. [T) In the model, the E neurons are selective for a ID periodic stimulus feature 2 E (-T, x. e.g., the orientation moving direction. Denote 0, as the preferred stimulus feature of the j-th E neuron, and the preferred stimulus features of all Ng E neurons, Y uniformly cover the whole range of feature space (Fig. 1 and F). This setting is the same as the canonical ring network model that has been widely used in modeling cortical circuits (e.g-- [7 (1) Mathematically, in the continuum limit (0, 0) corresponding to infinite number of neurons, the dynamics of the E neurons is 34 371.
@ug(0.t) 01
-u((0,t)+ +) t
Σ. (WEX (WEx* rx)(0,,) ( (F (I)
X=E,P.
where 10, and (0, () are the synaptic input and firing rate respectively of the E neuron preferring = 0. X denotes neuronal types with F, and S representing E neurons, sensory feedforward inputs, and the SOM neurons respectively. is the time constant of synaptic input,
2

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Figure 1: The recurrent circuit model. (A) The canonical cortical circuit consists of E neurons and three types of inhibitory interneurons. (B-C) The recurrent circuit model with two types of interneurons (B) and detailed recurrent circuit structure (C). (D) The Gaussian recurrent connection kernels in the circuit model. (E) An example of population responses of E (top) and SOM (bottom) neurons over time. (F) The tuning curves of E neurons in the model. (G-H) The tuning curve of an example E neuron in control state compared with enhancing PV neurons (G) and SOM neurons (H). The model result is qualitatively similar to the experimental data from 26
and = NE /T is the neuronal density covering the space of stimulus feature 2. E neurons receive internal Poisson variability mimicking stochastic spike generation (Eq. last term), and FE the Fano factor of the injected variability whose value will be adjusted to make the Fano factor of E neurons' activities
the order of {(0, is standard Gaussian white noise, i.e., (E(O, (i(e,t)((',,, (')) = 8(0 - 0 (') t¹) with ö(t - t') being the Dirac delta function. The internal Poisson variability provides the variability for the circuit draw random samples from the posterior 18], which may arise from the E and I balance in the cortex as a complex network effect 38 40
Recurrent connection kernels. Wy (0) denotes the recurrent connection kernel from neurons with type X to those with type Y, which are modeled as Gaussian functions in the model (Fig. DD)
Wy x (0) wy (errx(v2nx)) (V2maxy) exp(-0" )2xxxr))
where x controls the peak strength of the recurrent weight, and the connection width across the stimulus feature space. The kernels Wy (0) connecting different neuronal types can have different peak weights wyx (wy > 0 or < 0 regards to E or synapses respectively). Moreover, different Wyx (0) may have different connection widths yxy, although most of them have the same width axy = unless noted otherwise (see Supplementary Information (SI.) Sec. 6.1). In Eq. ( the symbol denotes the spatial convolution, i.e., (W * (0) = (0 - 0' (r((')()))) which implies the manslation-invariance of the connection weight between neurons in the stimulus feature space.
Sensory feedforward inputs. The recurrent circuit model receives sensory feedforward input (0, t) (Eq. [ randomly evoked from a stimulus feature in the world. Given a stimulus feature 2, we assume the feedforward inputs rF are conditionally independent Poisson spikes with Gaussian tuning (Fig. \-)), which has been widely used before (e.g., probabilistic population code 4 g )
rF (0|2) ~ Poisson[Ar(0]::
>=(0 = R expl - z)² /2a"),
where (0)=)) is the mean firing rate. In simulating our rate-based model, the ry is approximated as a continuous Gaussian random variable with multiplicative noise to mimic the Poisson statistics.
2.1 Inhibitory interneurons in the circuit model
hence, for simplicity, we consider PV neurons the model are not tuned for stimulus features and only provide global unstructured inhibition to E neurons to keep stability. Moreover, it was suggested PV neurons provide divisive normalization (DN) to modulate E neurons' responses via shunting inhibition 21 |25 42. Hence the proposed circuit model absorbs PV neurons' effects in the divisive normalization E neurons which has been widely used circuit models 21
PV neurons. The stimulus orientation weakly modulates the PV neurons 20
26 41
44
(u t)
rg (0,1) = 1 pw [u (0', (e..))).3 d0
(4)
3

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where the DN acts as an activation function transferring the instantaneous synaptic input ug(0, t) of E neurons into their firing rate (r((. t). [x]+ = max( 0) denotes negative rectification. The integral [[[ [ (0, 4) d0' =\t rrv, reflects PV neurons globally summing all E neurons' activities. wer is the global inhibition strength characterizing the inhibitory weight from PV to E. neurons.
SOM neurons. It is suggested that SOM neurons linearly modulate E neurons' responses, in contrast multiplicative modulation from PV to E neurons [26]. Therefore the model considers the E neurons receive additive synaptic inputs from SOM neurons ((W ES rs) (0, t), Eq. D. Furthermore, SOM neurons are tuned to stimulus feature with a strength comparable to E neurons 20) unlike the weak tunings of PV neurons. Thus, the dynamics of SOM neurons are governed by,
aus(a, t) Ot = us(, +P
Σ (Wax (0, rs(0, t) = gs [us(r,)))+ (5) X=E,F
where the Is (scalar) controls the "gain" of SOM neurons and is set as a fixed value across the study. Two features about SOM neurons' connectivity are worth noting (Eq, [5) First, the model doesn't include recurrent inhibitory connections between SOM neurons, due to few mutual inhibitions between SOM neurons 222 24 This simplification will only change the effective gain Is of SOM neurons without affecting our conclusions of the circuit algorithm. Second, the proposed circuit model allows the existence of the feedforward connections to SOM neurons (W SF in Eq. 55) even they are rare in reality [22 24]. The reason for allowing this rare connection is we want to test whether the Bayesian sampling theory has the power to constrain it.
Overall, the proposed circuit model is consistent with most canonical recurrent circuit models in the field (e.g.- (3))))))) The simplifications considered above reserve the main characteristics of neuronal connectivity and response properties observed in experiments, especially interactions between E neurons and interneurons. For example, it was found enhancing PV neurons will multiplicatively modulate E neurons' tuning curves (Fig. [G, right), whereas SOM neurons modulate neurons' tuning additively (Fig. [H, right) [26). Both effects are successfully reproduced in the proposed circuit model (Fig. TG-H, left; see details in SI. Sec. 6.5).
3 From recurrent circuit dynamics to Bayesian inference
The proposed recurrent circuit dynamics (Eqs. D and 5 is supposed to implement Bayesian inference by computing the stimulus posterior based on a received feedforward input,
p(z(rF) x p(rrl=)p(z).
(6)
regards the stage from external stimulus to the feedforward input ry as the probabilistic generative process (Fig. [2). Implementing Bayesian inference requires the recurrent circuit to store the genera- tive model. Rather than defining a generative model and deriving its neural circuit implementation as in many previous studies, here we ask the question the other way around: if the proposed recurrent circuit model based on neurophysiology could do Bayesian inference (Eq. [A], what generative model is stored in the circuit and what stimulus posteriors are computed by the circuit? Furthermore, what is the Bayesian inference algorithm adopted by the circuit dynamics?
ry
Stimulus likelihood. The stochastic feedforward input from the stimulus feature (Eq. 3 naturally specifies the stimulus likelihood that can be calculated Gaussian likelihood (see SI. Sec. 2.T
p(rr|z) II Poisson[A (f]==) x (Z(lHz.
where the Gaussian stimulus likelihood function comes from the the Gaussian profile of feedforward
input tuning A=(0 z) (Eq. 3 [9 The mean 14= and the precision A of the stimulus likelihood can be
read out from rF via linear decoder called population vector 45 46).
# Σ, (r0,)), 2, r=;)),, A ~ rF)))
(8)
Geometrically, 22 is regarded as the location of FF in the stimulus feature space, and A is proportional to the input spike count (Fig. B In this way, single snapshot of Fr parametrically conveys the whole stimulus likelihood function p(r (2) (
Subjective prior. We suppose the recurrent circuit utilizes its stored subjective prior to compute the subjective stimulus posteriors, with a mild assumption that the subjective prior in the circuit matches the objective prior in the outside world. Nevertheless, the subjective circuit prior remains unknown at this point. Next, we theoretically analyze the circuit dynamics to identify the stored subjective prior in the circuit and find out the circuit algorithm of Bayesian sampling.

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Figure 2: (A) A reduced circuit after blocking SOM neurons. (B-C) A schematic of the feedforward (spiking) input (B) and a linear read out of the stimulus likelihood conveyed by the feedforward input (C). Geometrically, the position and spike count of feedforward input determine the mean and variance of likelihood respectively. (D-E) E neurons' population responses (D) and the location of instantaneous E population response the y-axis is regarded as stimulus sample 2E generated by the network (E) and can be read out via a linear decoder called population factor. (F) The cross- coosscoorrlation function of stimulus samples generated by the circuit. (G) The circuit with fixed parameters flexibly
samples posteriors with various uncertainties (controlled by feedforward input intensity).
The Bayesian sampling in the stochastic circuit dynamics
4.1 Theoretical analysis of the neural dynamics
We theoretically analyze the nonlinear eircuit dynamics investigate how it can implement Bayesian sampling. We perform perturbative analysis of the nonlinear circuit dynamics, identify the low- dimensional stimulus feature manifold (subspace) the high-dimensional population response space, and eventually study the circuit dynamics on the stimulus feature manifold. Given a feedforward input rF (Eq. 3). it can be checked that the synaptic input (0) and the firing rate rx xx of neurons X in the equilibrium attractor states are both Gaussian profiles (Fig. D see SI. Sec. 3.1.
(ux(0)) = Ux expl-(0 - 2x)² /x/)/ (rx(0)
Rx expl-(0- 2x)²/2n") (X E, (9)
where (-) denotes the average over different realizations. Ux and Rx denote the height of the population synaptic input and firing rate respectively, and can be analytically computed (Eq. (S37). The position of population activity on the stimulus feature space is ãx Ms is the same as the location μr= of the feedforward inputs (SI. Sec. B.T). Intuitively, this is because the recurrent circuit is homogeneous along the stimulus feature space, i.c., all neurons are uniformly distributed on the stimulus feature space and the recurrent connections are translational invariance.
With the corruption of sensory noises and the internal Poisson variability, the instantaneous neural re- sponses will deviate from the equilibrium attractor state (Eq. 9.. We treat each instantaneous response (0)) + dux (0, t), (X = E, S),
as a perturbation from its equilibrium state, i.e., ux (0, t) and the relaxation dynamics of the perturbation dux (0, t) can be derived [47]. Then performing eigen-analysis of the perturbation dynamics we analytically find out the stimulus feature manifold (subspace), which specified by its (unnormalized) eigenvector
35 48).
(0(=x) & (0- 2x) xx) exp( (0 2x /4/a), (X=
(10)
Previous studies have shown the stimulus feature eigenvector has the largest eigenvalue in the perturbation dynamics [47]. Therefore, we project the dynamics of E and SOM neurons (Eqs. aad 5) onto their respective stimulus feature eigenvectors, where the projection is computing the inner product between the neuronal responses and the eigenvector, i.e., [ = J (())((0) with f()) representing the left or right handed side of Eqs. [5) The projection yields the dynamics of the E and SOM neurons on the stimulus feature manifold (see details in SI. Sec. 333
&E =s ges(2s - =g) + gEr (p= 2g) + GEVTEEL
(11) (12)
Tsăg R gs =s) + ger -2s)
The approximation comes from ignorance of some negligible nonlinear terms. Fæ and zş denote the
instantaneous positions of neural activities at the stimulus feature manifold at time (Fig. ()) Mz is
the observed stimulus feature conveyed by the feedforward input (Eq. 8). 7x 70x (X = E.
5

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is the time constant of the circuit dynamics on the stimulus feature manifold, where Ux is the peak value of population synaptic input (Eq. 19. The coefficients 9xy in the above equation denote the coupling strength between neural response positions, where 9xY x wxy Ry with Ry the peak firing rate of the pre-synaptic neural population (Eq. S48). The 0f 8aF (333) is the variance of the internal variability on the stimulus feature manifold coming from the internal Poisson variability (Eq. last term). gg is a constant value irrelevant with feedforward inputs and network responses.
D
4.2 Langevin sampling in the reduced circuit (E and PV neurons)
Since the circuit contains multiple types of neurons, we first analyze reduced circuit dynamics on the stimulus feature manifold without SOM neuron (Fig. 22, setting JES to zero in Eq. [1] and then study how the SOM neurons affect the sampling dynamics. Without SOM neurons (only E and PV neurons), the E neurons' dynamics on the stimulus feature manifold is,
E = TF gEF(Uz - -ZE) + GETE Ee,
(13)
which is first-order Langevin dynamies. This motivates the possibility that the E and PV neurons can implement Langevin sampling in the stimulus feature manifold. If true, the instantancous position of E activity, eg (Fig. 2D). can be regarded as stimulus feature sample generated by the circuit. In theory, the Langevin sampling of posterior p( p(z(rr) ) xr p(rr(2))((())) corresponds to performing
stochastic ascent on the log posterior surface 49 50
1 = Tx 1 V p(rr(>) + Inp(=)] + (T=/2) 1/2
(V = d/dz)
(14)
T. [A(Ms 2) V [nn((=)] + (Tx/2) 1//
where is the sampling time constant controlling the sampling speed. The 2nd row is obtained by substituting the Gaussian likelihood (Eq.
Uniform subjective prior. Comparing Eqs. (14 and 13) we can identify a stored uniform stimulus prior p(=) in the reduced circuit model stores This is because the gradient of uniform prior is Vn p(2) = O, and then the Langevin sampling dynamics (Eq. 14) reduces to a form similar to the circuit dynamics on the stimulus manifold (Eq. (3]. The uniform stimulus prior comes from homogeneous neurons in the circuit, i.e., neurons are uniformly distributed along the stimulus feature space, and the translation-invariann connection profile (Eq. 2. It implies that the circuit needs to break the symmetry of homogeneous neurons to store a non- non-uniform prior (see Discussion).
Condition for realizing Langevin sampling. Utilizing Langevin dynamics to sample the posterior requires the drift and diffusion coefficients to share the same time constant T= (Eq. [4]. To satisfy this requirement, the E dynamies the stimulus feature manifold (Fq. 13) should have g A/2. With the expressions of gEr and oZ (Eq. 548). the feedforward connection weight wgr should be,
VT OF
2 V3
3
WEF
FE-
(15)
Intuitively, larger internal variability Fg increases the sampling variance, and it requires a larger feedforward weigh wgr to compensate for sampling variance increase to match with the posterior variance. To verify our theoretical derivation (Eq. [5]. we search whether there is an optimal value of the feedforward weight weF allowing the circuit without SOM neurons to sample the posterior (likelihood). Indeed, we find once the recurrent circuit model is set with that optimal WEF, the reduced circuit model (consist of E and PV neurons) with all parameters fixed can flexibly sample the likelihood with various uncertainties (Fig. 2G). A characteristic Langevin sampling the cross- correlation function of samples (Fq. T4p exponentially decays with time which can be calculated as p(At) = exp( - 9 AtteAAA/TT (SI. Eqs. S7). To verify whether the reduced circuit performs sampling with the Langevin dynamics as suggested by Eq. (13), we estimate the noss-correeation function of stimulus sample eæ generated by the network, which indeed cxhibits an exponential form, and our theoretical calculation pl Af) predicts the actual cross-correlation function well (Fig. 2F).
4.3
SOM neurons accelerate Bayesian sampling in E neurons
The SOM neurons augment the dimensionality of the circuit dynamics on the stimulus feature manifold from the first order to the second order (Eqs. and 12). In principle, the second-order

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Figure 3: The Bayesian sampling in the full circuit model with PV and SOM neurons. (A) The population responses of E and SOM neurons. (B) The network's sampling distribution read out from E and SOM neurons in a way similar with Fig. 2D-E. The E neuron's position regarded as stimulus feature sample 2g. while the sample of SOM neurons 2g contribute the auxiliary variable in Hamiltonian sampling. The distribution of ng (right marginal) will be used to approximate the posterior. (C-D) The cross-correlation function of stimulus sample ag (C) and the decaying of KL divergence from posterior and the sampling distribution over time (D). (D) A linear manifold as the combination of feedforward weight WEF and the inhibitory weight from SOM to E neurons wEs allowing the full circuit to sample the posterior correctly. (E) The circuit with fixed weights (Fig. 3D,
orange dot) can flexibly sample posteriors with various uncertainties.
Hamiltonian sampling dynamics motivates us to explore whether and how the circuit can implement Hamiltonian-like sampling [50] [1]. The computational benefit from the increased complexity of Hamiltonian sampling compared with Langevin sampling is accelerating the sampling speed by
generating less correlated samples over time [50]. To investigate such a possibility, we write one
commonly used Hamiltonian sampling dynamics 50 52 53).
T = y; T = - By + V In ((2/rr) + (2Br.)
(16)
y is the auxiliary variable representing momentum in the Hamiltonian sampling. It can speed up sampling by increasing sampling step size when close to the posterior center, helping the sampling trajectories move toward the other side of the posterior (Fig. 3B; details SI. Sec. 2.1). is the friction strength determining how fast the momentum will decay to zero. We next bridge the circuit dynamics (Eqs. 11 and T2) with the Hamiltonian sampling dynamies (Eq. T6).
Intuitively, the sample from SOM neurons zs (Eq. 12 resemble the auxiliary variable y in the Hamiltonian dynamics (Eq. [6]. Nevertheless, there are several gaps between the two: First, in the Hamiltonian sampling, the stimulus sample = is purely driven by the auxiliary variable (Eq. 16). whereas in the circuit model 2g receives both 2g and μ1; (Eq, [T]. Second, Hamiltonian sampling injects variability into the auxiliary dynamics y (Eq. [6]. while the variability (from stochastic spike generation) is injected into the dynamics of zg. To bridge the gap between the circuit model and sampling dynamics, we assume that the circuit is conducting a mixture of Langevin (Eq. [4] and Hamiltonian sampling (Eq. 16), and thus we split the neurons' sampling dynamics into two parts,
TBêg = [gEs(Es zσ) + (1 - (OL)gFF(As) zE)] + [aLgEF(U zg) + TBEI,
(17)
Us
+ [aLgEr(M: - zg) + &EETT
where ar. E [0, denotes the proportion of feedforward input contributed by Langevin sampling.
We see the ys resembles the auxiliary variable in Hamiltonian sampling (Eq. which implies the auxiliary variable in the circuit is a mixture of samples from neurons zg and SOM neurons zs. Given the definition of ys, the stochastic dynamics of ys can be derived as form similar to the one in the Hamiltonian sampling (Eq. [6] details in SI. Sec. 4.Ip.
ús = Byys + BE(Mz - =g) + 8s(uz - 25) + oy
(18)
By. SE, B8 and By are functions of the coefficients in Eq. (17) (detailed expressions at Eq. S53
can be checked that equilibrium distribution of the mixed dynamics is the posterior (details of Fokker-Planck approach in SI. Sec. 2.5
Conditions for realizing mixed sampling. To utilize the full circuit model with SOM neurons (Eqs. 17 18 to implement Bayesian sampling, the coefficients in the circuit model should satisfy the
7

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relations of coefficients required in the Langevin sampling (Eq. 14) and the Hamiltonian sampling (Eq. [6.. First, setting up the Langevin sampling part in the circuit model (Eq. 17 blue) requires 01ger g = A/2, which finally leads to,
WEF = / (uarr))
(19)
whose value is 1//g times the feedforward weight in the Bayesian sampling circuit without SOM neurons (Eq. [5). Second, realizing the Hamiltonian sampling part in the circuit model (Eq. 17
orange; and Eq. 18 yields three conditions shown below.
Tv
3 B
A
(28Tz)
(a). Tz = TB: (b). Ag = 0: (c).
(20)
Bg
By
The conditions in Eqs. (9) and (0)) combined enable the full circuit model with SOM neurons to implement Bayesian sampling. An important insight from Eg. (20p) is that the SOM neurons should not receive feedforward inputs directly (wsF = in Eq. 5 Fig. IA, dashed line removed). This thcoretical result is consistent with the anatomy that SOM neurons receive much fewer feedforward synapses than other types of neurons 23 24]. In addition, substituting the detailed expression of coefficients into Eq. (20), we find there is low-dimensional manifold in the circuit model's connection weight space for the circuit to sample the posterior p(2(rz) (SI. Sec. 4.2).
Rs) WES [(1 az)= RF] WEF [G(ar)Ug Rg] USE.
(21)
G(az) is a nonlinear function of a which specifies the proportion of Langevin sampling in the circuit dynamics, which remains invariant with feedforward input and network activities (SL. Eq. [565) Ux and Rx are the height of the population synaptic input and firing rate of neuronal populations X (Eq. 9.. Eq. (2T) implies that the sampling in the full circuit model is robust, without the need to fine-tune recurrent weights. To verify the theoretical result (Eq. 2T). we fix the weight from E to SOM neurons, WSE, and search whether there is line manifold in the two-dimensional parameter space of Wgs and WgF for the circuit correctly sample the stimulus posterior. Indeed, Fig. 3E numerically confirms the line manifold of weights under which the sampling distribution matches the posterior. Moreover, the introduction of SOM interneruons makes the cross-correlation of sample zg decay faster (Fig. 3D), suggesting speeding up sampling (Fig. B3)).
Flexible sampling posteriors in the linear regime. Moreover, the circuit model with fixed weights should be able to sample the posteriors with different uncertainties. In the circuit model, the posterior uncertainty is determined by the feedforward input rate Rt (Eq. 3.. where a larger input rate leading to smaller posterior uncertainty (Eq. (8) We find that when the nonlinear circuit model is located the linear regime, it can flexibly sample posteriors with different uncertainties. To see this effect, we can change the feedforward input rate by multiplying gain factor g. i.c., RF + gRp that change the likelihood precision A + 9A (Eq. 8 S5). If the circuit is at the linear regime, the peak value of synaptic input, Ux + gUx, and the population firing rate, Rx + gRx, are both multiplied with the gain factor being applied to the feedforward input. And then it can be checked Eq. (2I) is still satisfied. This theoretical result is confirmed by numerical simulation (Fig. (3F) which shows the precision of circuit's 's stimulus samples increases with feedforward rate Rp and aligns well with the likelihood. Here we adjust the recurrent E-to-E weight WEF to set the circuit has an approximately linear response at the range of the feedforward input rate. Fig. [S] shows if the network deviates from the linear regime, the circuit's sampling distribution will deviate from the likelihood.
4.4 The Bayesian sampling performance from interneurons
We further investigate how quantitative measures of sampling, e.g., sampling speed and temporal correlation of samples, will be affected by interneurons such as the inhibitory feedback weight. In principle, both sampling speed and temporal correlation can be revealed by the eigenvalues of the circuit sampling dynamics (Fqs. 12). When organizing the circuit sampling dynamics into a matrix form (Eqs. 11 12) = Mz + μ. with z (=g. =s)" and μ lumping terms exclusive 2g or zg in Eqs.
the cigenvalues of the circuit sampling dynamics are (SI. Eq. S69).
A (r(M) + / /t(M - 4 de) 2 tr(M) + VA,
(22)
where tr(M) = TF (GEs + 9EF) + T ISE, dct(M)- = TE Tg 9EFOSE
-1
8

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A
E PV (orhanced) 1
B
1.2
1.0
D
10
LFP
Cross carr. (normalized)
F PV- SOM
Eigenvalues i.
0.
Roal Imaginary
Eigenvalues
Intensity
0.04
Real Imaginary
-0.2
10
Time
Glob.
0.02
Inh, Waglfmr
Freq. (Hz) 100
Figure 4: Interneurons' effects on sampling- (A) The cross-correlation of samples for perturbing PV and SOM. (B-C) The smallest eigenvalue of sampling dynamics when changing PV (A) and SOM (B) inhibitions. (A): obtained from the circuit without SOM (Fig. 2 (D) The local field potential (LFP), defined by the sum of synaptic inputs of E and SOM neurons (color: same A).
The sampling speed is limited by the smallest real part of eigenvalues, i.e., Re(A_) (Eq. 22 in that the KL. divergence from the distribution of samples up to time to the stationary distribution (Fig. 3D) decays exponentially as sxp[-Re(A). )t] [54].
PV neurons. PV's inhibition weight WEP modulates the cigenvalues by decreasing the common factor Tø (wE( (Eq. 22). where UE(WEr). the peak value of E population synaptic input
(Eq. 9T decreases with Wgp. Hence, stronger PV inhibition increases the slowest eigenvalue x_ (Fig. A and leads to faster sampling, exhibited by the faster decay of the temporal correlation between samples (Fig. 4A, blue and red). Moreover, the multiplicative modulation from PV neurons will not induce the imaginary part of eigenvalues (Fig. 4B, bottom), i.e., no oscillation between samples.
SOM neurons. The SOM inhibitory weight WEs will have non-monotonic effects on sampling speed measured by the real part of the slowest eigenvalue. There is value of wgs to maximize the sampling speed (Re(X_) (Fig. |4C, top). Moreover, SOM's inhibition will induce temporal oscillations between samples, i.e., emerging imaginary part of eigenvalue X_ (Fig. [ bottom). This oscillation is confirmed by the cross- cross-correlation of samples (Fig. AA, purple). Moreover, to mimic neural experimental data analysis, the oscillation induced by SOM neurons be revealed by the power spectrum analysis of the local field potential (LFP) (SL. Eq. (90).
Sampling complex posteriors in canonical recurrent circuits
High-dimensional stimulus posteriors. As a proof of concept example, we consider sampling bivariate stimulus posteriors by coupled circuits (Fig. (5A) with each circuit the same Fig. [B. And only neurons across circuits are coupled. Each circuit m (m = 1,2) receives a feedforward input generated by a latent stimulus feature 2m and will sample 2m. Hence, the number of coupled circuits equals to the stimulus feature dimension. We found the coupled circuits store an associative (subjective) prior, i.e., p(21, =2) expl-A, increasing with inter-circuit coupling weights (Fig. 5): Eq. S85). Math analysis is presented in SI. Sec. Concatenating the samples generated by E neurons two circuits (with the same readout as Fig. 3 AAB), we can obtain the 2D posterior sampled from the coupled circuit (Fig. 5B.. Similarly, the SOM interneurons speed up sampling, reflected by the sampling trajectories with SOM transverse over wider posterior region than the one without SOM in the same period (Fig. 5B))
)² / (Fig. [C), with the prior precision A,
Bimodal stimulus distributions. We also use the circuit (Fig. [C) to sample uni-variate bimodal posteriors (Fig. S2). in response to superpositions of two feedforward inputs with each generated sample bimodal distribution, where samples jump between
by latent stimulus. The circuit two modes alternatively, due to the bi-stability in the circuit dynamics (Fig. S2C and E). In contrast, without SOM neurons (Fig. (2)), the circuit only sample uni-modal distribution, as uni-modal approximation of the bimodal one (Fig. F2 Due to the space limit, we haven't I comprchensively linked the circuit' bimodal sampling distribution with posteriors, which will form our future work.
Conclusion and Discussion
The present study investigates how canonical recurrent circuits with diverse inhibitory interneurons implement Bayesian sampling. The nonlinear circuit model consists neurons, and two types of interneurons neurons (PV and SOM neurons). PV and SOM neurons have distinct tuning properties and modulations on E neurons: PV neurons have weak stimulus tuning and multiplicatively modulate E neurons, while SOM neurons have stimulus tuning and send additive inputs to E neurons. Through

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A
Coupled
B
sempllng
Al coupled
ed droult
Ganoaloenoods Sampleg
180
D.G
Price Preckion
Stimiux
Ragin SAE. Ne
SCM
-180
180
somple /Nee 1)
Etmukas 4.
Coupled weight (2//0)
Figure 5: Sampling 2D posteriors in coupled circuits. (A) Each circuit (net) is the same as the one in Fig. B (PV neurons are not shown). (B) The 2D sampling distribution generated by the coupled circuit shown in (A). (C-D) The coupled circuits store an associative (subjective) prior of two stimuli (C). The prior correlation increases with inter-cireuit coupling weights (D).
theoretical analysis and numerical simulations, we find the reduced circuit with only E and PV neurons implements Langevin sampling in the stimulus manifold (subspace) to compute the stimulus posterior. The SOM neurons accelerate the circuit sampling speed by upgrading the Langevin sampling into Hamiltonian sampling. We further find Hamiltonian sampling requires the SOM neurons to receive no feedforward connections, consistent with neuroanatomy. We also investigate how inhibitory strength from two types of interneurons affects the sampling speed. Our work is one of the earliest studies investigating the Bayesian sampling algorithm in canonical nonlinear recurrent circuits with diverse interneurons, and provides new insight into interneurons in Bayesian computation.
Comparison with other work. The computational mechanism of accelerated sampling by SOM neurons is similar to previous studies considering structured inhibitory feedback to E neurons (e.g..
14 17 53 55)), while some notable differences exist. Earlier studies considered the sampling in neural response space where the posterior dimension is the same as the number of neurons 17). whereas the current circuit samples in the stimulus feature manifold (subspace) in the neural response space. On the other hand, although other previous studies considered sampling in the stimulus feature manifold similar to the current model [53 55), their structured inhibitory feedback comes from the biophysical mechanism within single neurons, e.g., spike frequency adaptation [53 and potassium channels [55). From a neurobiology perspective, SOM neurons can be modulated by VIP neurons (Fig. [T],, whereas intracellular mechanisms are hardly modulated [53] 55 suggesting accelerated sampling from SOM neurons might be more flexible than single neuron mechanism in reality. Moreover, the proposed circuit model is similar to recent one [88 in terms of utilizing internal Possion variability to draw samples. However, the current circuit model nonlinear while the other study considered linear circuit model [18] Lastly, the coupled circuit sampling of bivariate posteriors was also considered in 34], but which didn't 't figure out the circuit's 's sampling algorithms.
Limitations and extensions of the model. The proposed circuit model doesn't 't include VIP neurons that exclusively target SOM neurons (Fig. [T]. Our future work will incorporate them and study their effects on circuit sampling. Based on our current conclusion, it is likely that VIP neurons act as "knob" modulate the sampling speed, depending on task needs, by changing the activation level ("gain") of SOM neurons. Moreover, we find the proposed canonical circuit stores uniform stimulus prior, due to the homogeneity of neurons distributing on the stimulus manifold. The homogeneous neuron simplification has been widely used in neural coding and continuous attractor networks (e.g., [3D 34 35] (56),, which simplifies the math analysis without altering results substantially. Nevertheless, the circuit has to break the neuronal homogeneity to store non- non-unifome uniform prior [57]. and certainly, cortical neurons are heterogeneous. The neuronal heterogeneity can be realized by manipulating the translation-invarian recurrent connection matrix (Eq. - c.g., introducing randomness (zero mean with certain variance) on recurrent weight which has also been widely used in (chaotic) Excitation and Inhibition (E/I) balanced networks ( (38 40 58) ). A potential function of
heterogeneity from random recurrent weights is that this puts the spiking networks into the chaotic regime where the network internally generates Poisson variability, which is statistically equivalent to the injected multiplicative variability in our rate-based network (Eq. [ last term). Moreover, the proposed circuit model only infers static stimulus, and will extend infer a dynamic stimulus described by a hidden Markov model in the future. All of these form our future research.
Acknowledgments
W.H.Z. is supported by the UT Southwestern Endowed Scholars program. The authors thank Chengcheng Huang for fruitful comments.
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Reviews:
[{"review_id": "x87DGW5Oq2", "reviewer": "Reviewer_6oXx", "summary": "This paper studied how the introduction of two inhibitory neuron population affects Bayesian inference in firing rate models with additive noise terms. Analytical derivation and simulations are done with circular 1D input variable (orientation). They found the inclusion of SOM leads to faster inference.", "questions": "Have the authors considered having a mixture of two orientations as input and study how the introduction of I-diversity resolves the potential interference? (mentioning it not to ask for more experiments for this submission but just as a general future direction)", "limitations": "Limitations are adequately addressed.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1) I would like to thank the authors for their clear writing and detailed technical reports; \n2) there are many decision points in building up the circuit model, e.g. firing rate vs. spiking model, choice of weight structure etc., all decisions are carefully thought of and reasoned in the paper, leading to a very technically solid paper.", "weaknesses": "My main reservation with the paper is on its significance - I think the paper (at least in the way it is written) interests people within the specific subfield of Bayesian sampling in simplified circuit models for its technical merits; the result of SOM leads to faster inference for simple 1D input may interest SOM researchers in neuroscience. I'm just debating how much of these merit a NIPS publication.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the reviewer's positive feedback on our writing and math analysis! \n\n> My main reservation with the paper is on its significance - I think the paper (at least in the way it is written) interests people within the specific subfield of Bayesian sampling in simplified circuit models for its technical merits; the result of SOM leads to faster inference for simple 1D input may interest SOM researchers in neuroscience. I'm just debating how much of these merit a NIPS publication.\n\nUnderstanding how neural circuits in the brain do computation is one of NeurIPS's interests, which is also acknowledged by the Reviewer K6Di (strengths). We believe this adventure would bring mutual benefit to both neuroscience and machine learning (ML) communities, in that it not only helps us understand the brain, but also has the potential to be a new building block for ML tasks (Of course, our current model is still a \"baby\" and is not able to deal with real tasks). \nThere are lots of famous examples of brain-inspired computations, e.g., Neocognition by Fukushima, Conv Net by Yan LeCun, etc. \n\nEven if the manuscript only presents a 1D posterior sampling, its analytical methodology and research philosophy (analytically identified circuit representation and algorithm) further our understanding of the __uncertainty representation__ in neural networks, and how different types of interneurons contribute to sampling, which, we believe, will have long-term impacts on ML. In contrast, conventional perceptron-based neural network models ignored neural fluctuations when they were developed decades ago, which causes __uncertainty representation__ in artificial neural networks, a fundamental question in ML. \nMoreover, the diversity of interneuron types may provide new possibilities to implement/link with the complicated components used in conventional RNNs used in ML such as LSTM.\nAt last, the 1D sampling is the preface of our series of work, and we have also considered extending the circuit model to deal with more complicated posteriors (please see our Global Rebuttal and Rebuttal Figure).\n\n> Have the authors considered having a mixture of two orientations as input and study how the introduction of I-diversity resolves the potential interference? (mentioning it not to ask for more experiments for this submission but just as a general future direction)\n\nYes, the proposed circuit with SOM neurons can sample multi-modal and high-dim posteriors. Please find the results in the rebuttal PDF and our response in Global Rebuttal. \nWe are happy to include the results to strengthen our paper in the revised version with one more page, if you encourage us to do so.\nThe reason for not including them is because of the page limit and the completeness of the manuscript, where we sacrificed the inference task complexity to present the reasoning of building the model and math analysis.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "E6IKXhDZmQ", "reviewer": "Reviewer_NhnW", "summary": "The paper proposes a dynamical model for how neurophysiologically realistic circuits of pyramidal excitatory neurons, with two types of inhibitory interneurons, could perform Bayesian inference in a generative model of a uniform prior (over a bounded domain such as orientation angles) and a Gaussian likelihood. Analyzing the proposed neuronal circuit, the paper finds that its dynamics implement Hamiltonian Monte Carlo, and that removing one type of inhibitory interneuron results in Langevin Monte Carlo.  The paper then claims to replicate certain neurophysiological findings, specifically regarding L2/3 of the laminar circuit but also, according to some experiments, of L5.", "questions": "Where do the authors propose that their circuit ought to be found, neuroanatomically?  How does it fit into the laminar structure proposed by existing cortical microcircuit theories?", "limitations": "See weaknesses.  The paper's chief limitation is considering an overly simplified generative model based on previous work on probabilistic population codes, neglecting the observation that priors can be decoded (under supervised learning settings and at above-chance rates, etc.) almost ubiquitously from cortical recordings and are modulated by top-down feedback activity.", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The paper's in-depth use of neurophysiology alongside dynamical systems is its greatest strength.", "weaknesses": "The paper is quite dense, often unclear for a machine-learning audience, and ultimately only addresses a uniform-Gaussian joint density that cannot capture natural stimuli or behavior.  However, the authors have addressed this weakness in their responses and demonstrated that their model can sample from multimodal distributions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for the reviewer's appreciation of our math analysis and the biological plausibility of the model.\n\n> The paper is quite dense, often unclear for a machine-learning audience, and ultimately only addresses a uniform-Gaussian joint density that cannot capture natural stimuli or behavior.\n\nWe will improve our manuscript to make it more friendly to machine learning audiences. \nWe are sorry that the detailed justification of the biological plausibility of the circuit model and the theoretical analysis might appear less interesting to machine learning audiences, whereas they are probably necessary for a complete theoretical neuroscience study.\n\nThe proposed circuit with SOM neurons can sample multi-modal and high-dim posteriors. Please find the results in the rebuttal PDF and our response in Global Rebuttal. \nWe are happy to include the results to strengthen our paper in the revised version with one more page, if you encourage us to do that.\nThe reason for not including them is because of the page limit and the completeness of the manuscript, where we sacrificed the inference task complexity to present the reasoning of building the model and math analysis. \n\n> Where do the authors propose that their circuit ought to be found, neuroanatomically? How does it fit into the laminar structure proposed by existing cortical microcircuit theories?\n\nThe micro-circuit structure (Fig. 2A) is canonical and ubiquitous in all cortical brain regions in mammals. In recent years, most experiments utilized mice's visual cortex to investigate this canonical micro-circuit, because we have high-throughput recording and perturbing tools on mice.\n\nIn terms of laminar layers, the canonical micro-circuit (Fig. 2A) typically exists in both layers 2/3 (superficial layer) and layer 5 (deep layer), although the connection strength among neurons varies across laminar layers and across brain regions. The Fig. 7 in Ref. [24] in the submitted manuscript is a good summary of this micro-circuit structure in both superficial and deep layers.\n\n> See weaknesses. The paper's chief limitation is considering an overly simplified generative model based on previous work on probabilistic population codes, neglecting the observation that priors can be decoded (under supervised learning settings and at above-chance rates, etc.) almost ubiquitously from cortical recordings and are modulated by top-down feedback activity.\n\nWe agree that the __derived__ generative model considered in our theoretical neuroscience study is simple compared with machine learning studies. \nIt's worth noting the difference in research philosophy of the presented study is that we didn't use a __top-down approach__ where we \"assign\" a simple 1D generative model and \"design\" a circuit model to implement the inference.\nInstead, we ask the question the other way around from a __bottom-up view__. That is, given the recurrent circuit models with types of interneurons that have been extensively studied in neuroscience in recent years (acknowledged by the Reviewer RifV by saying this is \"the norm of papers in neural sampling theory\"), what kind of generative model and probabilistic inference can be implemented by the biological circuit. This philosophy was detailed in Lines 139-143 and Lines 152-156 of the submitted manuscript. \n\nIn this way, even if the discovered 1D sampling in the proposed circuit model is simple, we feel the result still has its merit to be reported, because \n1) no previous study has proposed what kind of inference can be achieved in the circuit with types of interneurons; \n2) the simple 1D inference will encourage people to further investigate how the canonical circuit could do more complex inference. \n\nFor the results uploaded in the Global Rebuttal PDF, we found the circuit with SOM neurons could sample multi-modal stimulus. However, due to the page limit, we didn't include them in the submitted manuscript. And we are happy to include them in the revised manuscript if you would prefer.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "g1w92H2qTE", "reviewer": "Reviewer_K6Di", "summary": "The paper is build on previous work on Bayesian sampling to provide incremental but novel analytical insights about a neuronal circuit implementing bayesian inference. The circuit comprises excitatory neurons and two types of inhibitory neurons, PV and SOM, and uses the ring architecture. Authors find that the dynamics of such a circuit performs a Bayesian sampling algorithm and analyse analytically the properties of such sampling. They find that the circuit with E neurons and PV interneurons performs Langevin sampling while including SOM interneurons, the circuit performs Hamiltonian sampling.", "questions": "Unless I misunderstood something, the model is by design limited to encoding of a single stimulus feature. An increasing amount of literature in neuroscience shows that biological neural networks (in particular in the cortex), typically represent several stimulus- and behaviour-related variables (Or at least, such variables can be reliably decoded from neural activity in vivo; see for example Stringer et al. Nature 2019, Stronger et al. Science 2019, Mussall et al. Nature 2019). These observations include the same neuron being responsive to multiple features of the stimulus or of the behaviour. Can authors comment on that? \n\nDoes Bayesian sampling implemented by the model imply a static stimulus and a convergence of network activity to such stimulus? How does the network deal with a stimulus that has rapid shifts in dynamics or that changes the mean?\n\nCan the model deal with non-Gaussian stimuli?\n\nTuning curves of neurons of the same cell type are homogeneous  across neurons (have the same shape). Is this assumption required for analytical computations? Could there be a computational reason that would instead allow heterogeneity of tuning across neurons of the same cell type?", "limitations": "Limitations are adequately addressed.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The paper is build on previous work on bayesian sampling to provide incremental but novel analytical insights about a neuronal circuit implementing bayesian inference. The topic of Bayesian inference is in general of interest to the NeurIPS community and the paper builds on latest results in the field. While I have not checked the Equations in detail, the paper seems technically sound.", "weaknesses": "The biggest weakness of the paper is its convoluted style of presentation - the paper is hard to read. Authors could dedicate more space to give intuitive explanations about the meaning of the main results. For example, authors could give some intuition about Langevin and Hamiltonian sampling and why having different types of interneurons gives different types of sampling. The meaning of the main results of the paper remains largely unclear. \n\nThe authors emphasise that their results are relevant to a canonical circuit and describe the canonical circuit comprising three types of interneurons, however, their model only incorporates two types of interneurons. I understand this might be a work in progress. Nevertheless, I find it would be better to avoid mentioning the \"canonical circuit\" if the theory does not actually describe the canonical circuit.\n\nFurther, authors use the assumption that PV neurons are not tuned to the stimulus and justify this with findings of one empirical study. It seems rather that tuning of PV neurons is an open question, with some studies reporting relatively strong tuning of PV interneurons in the cortex (see e.g. Runyan et al., Neuron 2010,  Moore and Wehr, J. Neurosci. 2013). While it is justifiable to use the assumption of untuned PV interneurons, I find it better to present it as an assumption supported by some of the empirical literature.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the reviewer's feedback and positive comments about math analysis, and the insight of SOM neuron's role in inference presented in our manuscript\n\n> The biggest weakness of the paper...\n\nThanks for your suggestions; we will revise our manuscript accordingly by adding more content about intuitive descriptions. For example, we can add text related to Fig. 3C to explain why the structured inhibition from SOM neurons could speed up sampling by decreasing the temporal correlations of samples. Intuitively, Hamiltonian sampling comes with oscillation, and meanwhile, the structured inhibition from SOM will bring oscillations of stimulus samples computed by E neurons.\n\n> The authors emphasis that their results...\n\nThank you for your suggestion. We will weaken our wording in the revision. We will just use the wording \"canonical circuit\" in describing the Fig. 1A and call our proposed model a circuit with diverse types of interneurons. In addition, we can also remove the \"canonical\" in the title.\n\n> Further, authors use the assumption that PV neurons...\n\nThanks for the suggestion and we agree that the tuning strength of PV is under debate.  We will revise our text by emphasizing the untuned PV interneurons in the model is an __assumption__. This simplifies our theoretical analysis, while reserving a certain amount of biological plausibility, via multiplicative modulation on E tunings (Fig. 1G). Specifically, we will weaken our text in lines 104-105 into \\\n\"Some experiments found the stimulus orientation weakly modulates the PV neurons, hence, for simplicity, we assume PV neurons in the model are not tuned to stimulus features, providing only global unstructured inhibition to E neurons for stability (see Discussion for the complexity of PV neurons' tunings)\" \n\\\nIn Discussion we will cite Runyan 2010 and Moore 2013 to discuss the experimental discovery of some PV neurons with sharp tuning, remind readers that the tuning of PV neurons is debatable, and clearly state that global inhibition from PV is an assumption considered in the study.\n\n> Unless I misunderstood something....\n\nDue to the page limit, our submitted manuscript only presents the 1D posterior sampling.  The proposed circuit with SOM neurons can sample multi-modal and high-dim posteriors. Please find the results in the rebuttal PDF and our response in Global Rebuttal. \nWe are happy to include the results to strengthen our paper in the revised version with one more page.\n\n> Does Bayesian sampling...\n\nYes, the current manuscript only considers a static stimulus and thus the stimulus samples embedded in the spatiotemporal neuronal activity will converge to the 1D stimulus posterior density, rather the just the point value of the stimulus. \n\nThe proposed circuit model, with or without SOM neurons, can be used to infer the changing stimulus $z_t$, i.e., inferring the instantaneous posterior $p(z_t| r_t ... r_1)$  based on all history of neural inputs $(r_t, ..., r_1)$ in a hidden Markov model. \nThen the variance of the transition prior probability $p(z_t|z_{t-1})$ will be stored in the peak recurrent weight between E neurons, $w_{EE}$ (Eq. 2), decreasing the variance of $p(z_t|z_{t-1})$ (we have analytical results about this). This is because a larger $w_{EE}$ will incur a stronger bump response of E neurons (dark blue in Fig. 1E) which moves slower over the stimulus space, corresponding to a transition probability in MCMC with smaller variance.\n\n> Can the model deal with non-Gaussian stimuli?\n\nYes, please see the attached Rebuttal PDF where the circuit can sample bimodal posteriors.\nIn principle, the proposed framework can sample exponential family distributions (with a similar mechanism to Ma et al., Nat. Neurosci. 2006 and Zhang et al., Nat. Comms. 2023) or a mixture of exponential family distributions, where the __spatial profile__ of recurrent weights (Eq. 2) acts as the natural parameter of distributions and determines the type of sampling distribution in the circuit. The Gaussian distribution comes from the Gaussian profile of recurrent weights (Eq. 2).\nIf we change the recurrent weight profile into other shapes, e.g., von Mises function, the circuit model can sample von Mises distributions.\n\nTo sample distributions out of the exponential family, we may consider passing the responses of the current circuit model into a feedforward network which can rescale the Gaussian distribution into arbitrary distributions. This mechanism was used in a recent study, Chen et al., NeurIPS 2023 mentioned by Reviewer RifV.\n\n> Tuning curves of neurons of the same cell type are homogeneous...\n\nYes, the homogeneity simplifies the math analysis significantly, without altering our conclusion.  Real cortical circuits have heterogeneity among neurons, and the homogeneity assumption has been widely used in many computational neuroscience models as a justifiable simplification, especially in the work of continuous attractor networks, e.g., Ben-Yi Shai 1995; Wu, Neural Computation 2008; Khona, Nat. Rev. Neurosci., 2022, etc.\n\nHeterogeneous neuronal tunings could be realized by introducing __randomness (zero mean with certain variance)__ into the recurrent connection matrix (Eq. 2), which has also been widely used in (chaotic) Excitation and Inhibition (E/I) balanced networks (Vreeswijk, Science 1996; Neural Computation 1998; Litwin-Kumar, Nat. Neurosci., 2012; Rosenbaum, Nat. Neurosci., 2017, etc). \n\nA potential __function of heterogeneity__ from random recurrent weights is that this puts the spiking networks into the chaotic regime where the network __internally generates Poisson variability__ (last term in Eq. 1). We think the injected multiplicative variability in our rate-based network (Eq. 1) captures the chaotic Poisson variability from spiking network dynamics. Then, stronger random recurrent weights will induce large heterogeneity, corresponding to a larger Fano factor $F_E$ of the injected variability in Eq. 1.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "cNUIZGfbj1", "reviewer": "Reviewer_RifV", "summary": "This paper investigates how canonical recurrent circuits in the cortex can implement sampling algorithms. The authors show that circuits with only E and PV neurons implement Langevin dynamics, while including SOM neurons enables a mixture of Langevin and Hamiltonian sampling.", "questions": "I am curious about how the authors would construct a distributed version of the proposed circuit that can sample from high-dimensional stimulus posterior.", "limitations": "See weaknesses", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "The paper demonstrates its due diligence to connect canonical circuits and Bayesian sampling rigorously. Overall, the paper is well-written, and the experimental results are well-illustrated.", "weaknesses": "Two major limitations of the sampling circuit proposed in the paper are its low-dimensionality and uniform prior. Overall, the authors only demonstrate sampling of 1-D Gaussian distributions, due to the linear drift. While I understand that it is the norm of papers in neural sampling theory, recent works have shown that recurrent circuits are capable of sampling from much more complicated distributions [1,2]. The authors should discuss how the proposed circuits can scale in more detail.   \n\n[1] Lyo, Benjamin and Cristina Savin. “Complex priors and flexible inference in recurrent circuits with dendritic nonlinearities.” bioRxiv (2023): n. pag.\n[2] Chen, Shirui, et al. \"Expressive probabilistic sampling in recurrent neural networks.\" Advances in Neural Information Processing Systems 36 (2024).\n\nMinor:\n1. Figure 3 caption does not include panel F.\n2. Although it is straightforward for the linear case (S36), for the mixture of general Langevin and Hamiltonian dynamics (S34 & S35), it is worthwhile to include references or show that the marginal distribution of z is indeed p(z) (is that true in the general case?).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the reviewer's feedback and positive comments about the math rigor and biological plausibility of our work.\n\n> Two major limitations of the sampling circuit proposed in the paper are its low-dimensionality and uniform prior. Overall, the authors only demonstrate sampling of 1-D Gaussian distributions, due to the linear drift. While I understand that it is the norm of papers in neural sampling theory, recent works have shown that recurrent circuits are capable of sampling from much more complicated distributions [1,2]. The authors should discuss how the proposed circuits can scale in more detail.\n> \n> [1] Lyo, Benjamin and Cristina Savin. “Complex priors and flexible inference in recurrent circuits with dendritic nonlinearities.” bioRxiv (2023): n. pag. [2] Chen, Shirui, et al. \"Expressive probabilistic sampling in recurrent neural networks.\" Advances in Neural Information Processing Systems 36 (2024).\n\nThank you, we very much appreciate you bringing this issue.\nWe will include the discussion of recent papers that sampled more complicated posteriors in the revised manuscript, and do some comparison.\n\nIn addition, the proposed circuit with SOM neurons can sample multi-modal and high-dim posteriors. Please see the results in the rebuttal PDF and our response in Global Rebuttal. \nWe are happy to include the results to strengthen our paper in the revised version with one more page, if you encourage us to do that.\nThe reason for not including them is because of the page limit and the completeness of the manuscript, where we sacrificed the inference task complexity to present the reasoning of building the model and math analysis. \n\nCompared with the two papers mentioned by the reviewers, our study presented a simpler inference task, compensated by the analytical results of identifying sampling algorithms in the circuit, and biological plausibility (diverse interneurons, reproducing experimentally observed tuning curve). Our rebuttal PDF shows the proposed circuit model can also sample multi-modal and high-dim posteriors. We wish the reviewer could re-evaluate the significance of our model framework and analytical methodology.\n\n\n> Minor: Figure 3 caption does not include panel F.\n\nApologies for the typo. We will add it to the revised manuscript.\n\n> Minor: Although it is straightforward for the linear case (S36), for the mixture of general Langevin and Hamiltonian dynamics (S34 & S35), it is worthwhile to include references or show that the marginal distribution of z is indeed p(z) (is that true in the general case?).\n\nThe mixture of Langevin and Hamiltonian (Eqs. S34 & S35) was used by Ref. 14 and 50, and it is guaranteed to sample the correct $p(z)$.\nWe will emphasize this property in the revised manuscript.\n\n> I am curious about how the authors would construct a distributed version of the proposed circuit that can sample from high-dimensional stimulus posterior.\n\nPlease see our results of extending the proposed circuit model to sample high-dim posteriors in the Global Rebuttal and attached Rebuttal PDF.\n\nTo sample high-dim posteriors, we can extend the current model into many coupled networks with each the same as the one in the current manuscript. Each network receives a neural input generated from a latent 1D stimulus, and samples the corresponding 1D latent stimulus. As a whole, the coupled networks distributivity sample high-dim posteriors, where the dimension of the posterior is determined by the number of networks in the coupled circuits. The recurrent weights between subnetworks store a high-dim correlation prior of stimuli. In this way, the coupled networks are similar to the network presented in Zhang et al., Nat. Comms. 2023 (see its Fig. 6), but each subnetwork in Zhang's study didn't consider SOM neurons.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
## Global rebuttal
We thank all reviewers' efforts in reading our manuscript! 

### Neural circuit sampling of complex posteriors
Three Reviewers (6xXx, K6Di, and RifV) wondered whether the circuit model can sample more complex posteriors, e.g., multi-modal and/or high-dim posteriors. Yes, the proposed circuit model framework can sample these complex posteriors (see the attached rebuttal PDF file), including
- Multi-modal posteriors (Fig. 5). When simultaneously presenting two stimuli to the proposed circuit model (Fig. 5B), the circuit with SOM neurons can sample bimodal posteriors. In contrast, without SOM, the circuit only samples an unimodal distribution corresponding to the Gaussian approximation of the bimodal posterior.
- High-dim posteriors (Fig. 6). To sample high-dim posteriors, we can extend the current model into many coupled networks with each the same as the current manuscript. Each network receives a neural input generated from a latent 1D stimulus, and samples the corresponding 1D latent stimulus. Combining the samples generated by all networks, the coupled circuit as a whole samples high-dim posteriors with the dimension determined by the number of networks. The coupling weights between networks store a high-dim correlation prior of stimuli (Fig. 6C). In this way, the coupled networks are similar to the network presented in Zhang et al., Nat. Comms. 2023 (see its Fig. 6), but each subnetwork in Zhang's study didn't consider SOM neurons.

__We are happy to include the new results in the final version as long as reviewers think they can strengthen our manuscript__. 

Due to the page limit, we didn't include them in the submitted manuscript. Even for the 1D posterior sampling, the Reviewer Nhnw feels the manuscript is __dense__.  To fit the page limit, the submitted manuscript sacrificed the complexity of inference tasks for presenting our reasonings for crafting a circuit model with sufficient biological plausibility (lines 67-133, Fig. 1G-H and Fig 4C-D; acknowledged by the Reviewer 6xXx), and rigorously analytical results of the nonlinear neural dynamics (Eqs. 9-21; acknowledged by all Reviewers). We feel the __biological plausiblity__ and the __analytical results__ are just as important as the complexity of inference tasks, especially since no previous work investigated inference in circuit models with different types of interneurons.
Certainly, we are happy to revise it based on reviewers' suggestions.

### The simplicity of 1D posterior sampling in the circuit model
We agree with reviewers that 1D sampling is computationally simple, however, its implementation in __nonlinear__ neural circuits with types of __interneurons__ remains unclear. 
In addition, it's worth noting that the difference in the research philosophy of the present study is that we didn't use a __top-down approach__ where we "assign" this simple 1D generative model and "design" a circuit model implementation.
Instead, we ask the question the other way around from a __bottom-up view__: given the circuit models with types of interneurons that have been extensively studied in neuroscience (acknowledged by the Reviewer RifV, "the norm of papers in neural sampling theory"), what kind of generative model and probabilistic inference can be implemented by the biological circuit (detailed in Lines 139-143 and Lines 152-156 in submitted manuscript). 
In line with this philosophy, we feel the current result still has its merit to be reported, because 
1) no previous study has hypothesized what kind of inference can be achieved in the circuit with types of interneurons; 
2) the simple 1D inference will stimulate people to investigate more complex inferences in the canonical circuit (shown in the Rebuttal PDF). 

### The significance of the present study, and comparison with earlier studies
- The present study is probably one of the first one linking 1) a canonical nonlinear circuit model with __interneurons__, 2) __analytical__ results of circuit dynamics (Eqs. 11-12), and 3) Bayesian __inference algorithms__ (Eqs. 13 and 17) together. 
\
__Comparison__: Although some earlier work studied more complicated inferences than ours, they __sacrificed the complexity/biological plausibility/theoretical analyses__ of the circuit model. For example, they either considered a linear/simplified neural model sacrificing the biological plausibility (e.g., Hoyer, NIPS 2003; Savin, NIPS 2014; Aitchison, Plos Comp. Bio. 2016), or studied a nonlinear, biologically plausible dynamics but without analytical results of the nonlinear dynamics (e.g., Benjamin, ICLR 2024; Chen, NeurIPS 2023; Echeveste, Nat. Neurosci., 2020). In comparison, our work analyzes the nonlinear circuit dynamics and analytically identifies the circuit algorithm, which is also acknowledged by all Reviewers.

- The circuit model with __fixed__ weights can flexibly sample posteriors with __different uncertainties__ (Fig. 2G & 3F). 
\
This is a very important requirement because real neural circuits sample different posteriors within the time scale of hundreds of milliseconds which are too short to change synaptic weights. 
\
__Comparison with earlier work__: 
This flexible computation requirement has not been paid sufficient attention in many earlier studies, and thus we are unsure whether they could achieve this or not, including, e.g., Chen NeurIPS 2023 and Benjamin, ICLR 2024 (mentioned by Reviewer RifV); Dong, NeurIPS 2022; Savin NIPS 2014; Aitchison, Plos Comp. Bio. 2016; Hoyer, NIPS 2003.

- The injected __multiplicative__ variability is biologically plausible in that the neuronal responses are (Poisson-like)__, i.e., the variance of neuron's responses is proportional to its mean firing rate (Fig. 1G-H, the shared region increases with mean firing rate). The multiplicative noise is necessary for the circuit model with fixed weights to sample posteriors with different uncertainties.

We hope our reply will address the reviewers' concerns and reassess our work.

Meta-review: This paper considers how canonical cortical circuits with multiple interneuron types can implement sampling algorithms. The reviewers broadly agreed a strength of this paper is in its tight connection between the physiology and the sampling algorithm, which will be of interest to the Bayesian inference community. In light of this, I recommend acceptance.

There are some concerns that the paper only presents simple posteriors. I recommend the authors utilize the extra page allotted for accepted papers to include results with more complex priors.
Decision: Accept (poster)