Natural gradient enables fast sampling in spiking neural networks
Authors: Paul Masset
Jacob A Zavatone-Veth
J. Patrick Connor
Venkatesh N Murthy
Cengiz Pehlevan
Venue: NeurIPS 2022 Accept (2022)
Keywords: spiking neural networks
sampling
Bayesian inference
information geometry
neural population geometry

Abstract: For animals to navigate an uncertain world, their brains need to estimate uncertainty at the timescales of sensations and actions. Sampling-based algorithms afford a theoretically-grounded framework for probabilistic inference in neural circuits, but it remains unknown how one can implement fast sampling algorithms in biologically-plausible spiking networks. Here, we propose to leverage the population geometry, controlled by the neural code and the neural dynamics, to implement fast samplers in spiking neural networks. We first show that two classes of spiking samplers---efficient balanced spiking networks that simulate Langevin sampling, and networks with probabilistic spike rules that implement Metropolis-Hastings sampling---can be unified within a common framework. We then show that careful choice of population geometry, corresponding to the natural space of parameters, enables rapid inference of parameters drawn from strongly-correlated high-dimensional distributions in both networks. Our results suggest design principles for algorithms for sampling-based probabilistic inference in spiking neural networks, yielding potential inspiration for neuromorphic computing and testable predictions for neurobiology.

Natural gradient enables fast sampling in spiking neural networks
Paul Masset 2, Jacob Vavatone-Veth 1.3 J. Patrick Connor
Venkatesh N. Murthy 1.2f Cengiz Pehlevan 1,4f
CCenter for Brain Science, Department of Molecular and Cellular Biology, Department of Physics, *John A. Paulson School of Engineering and Applied Sciences, Harvard University
Cambridge, MA 02138
ma harvard edu, jzavatonevethhg, hww.ghaavadddee.. edu, cpehlevan@seas. harvard.edu
Abstract
For animals to navigate uncertain world, their brains need to estimate uncer- tainty at the timescales of sensations and actions. Sampling-based algorithms afford theoretically-groue grooudedd framework for probabilistic inference in neural circuits, but remains unknown how one can implement fast sampling algorithms in
spiking networks. Here, we propose to leverage the population geometry, controlled by the neural code and the neural dynamics, to implement fast samplers in spiking neural networks. We first show that two classes spiking cumplers-efficien efficient balanced spiking networks that simulate Langevin sampling, and networks with probabilistic spike rules that implement Metropolis- Hastings sampling can unified within a common framework. We then show that careful choice of population geometry, corresponding the natural space of parameters, enables rapid inference of parameters drawn from strongly-correlat digh-dimensional distributions in both networks. Our results suggest design prin- ciples for algorithms for sampling-bai -bated probabilistic inference in spiking neural networks, yielding potential inspiration for neuromorphic computing and testable predictions for neurobiology.
Introduction
Neural circuits perform probabilistic computations at the sensory, motor and cognitive levels [1-4]. From abstract representations of decision confidence [5] to estimates of sensory uncertainty in visual cortex [6, 7], evidence of probabilistic representations can found at all levels of the cortical processing hierarchy [8]. To be behaviorally useful, these probabilistic computations must occur at the speed of perception [9]. However, how neuronal dynamics allow brain circuits to represent uncertainty in high-dimensional spaces perceptual timescales remains unknown 10-12].
Several neural architectures probabilistic computation have been proposed, including: probabilistic population codes [13], which in certain cases allow direct readout of uncertainty; direct encoding of metacognitive variables, such as decision confidence [4, 5, 8]: doubly distributional codes [14, 15] which distinguish uncertainty from multiplicity; and sampling based codes [9, 16-25], where the variability in neural dynamics corresponds to signature of exploration of the posterior probability. Most experiments quantifying uncertainty representations in single biological neurons have only
PM and JAZ-V contributed equally to this work.
VVN and CP jointly supervised this work.
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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imput
appking
prediction
Euclidean space
Natural space
a
Alyetbre
Sytrent
Lataria sargliig
Rochark
gits
decoding msights
Sectard:
Figure 1: Probabilistic inference in spiking neural networks. a. Circuit diagram. b. Langevin sampling Euclidean space and natural space. c. Geometry of the inference space in (b). Arrows indicate the principal directions of the sampling noise covariance. d. Comparison of Langevin and Metropolis- -asttings sampling algorithms.
varied parameters along one or two dimensions, such as Bayesian cue combination [2, 10, 26]. In these conditions, many algorithms can perform adequately. However, probabilistic inference becomes more challenging as the entropy of the posterior distribution-whiee often scales with dimensionality-incceee increases [27, 28]. Some algorithms that work well in low dimensions, such as probabilistic population codes, may scale poorly to high-dimensional settings [3].
Of the proposed approaches to probabilistic computation in neural networks, sampling- based codes are grounded in the strongest theoretical framework [27, 29-33], and have been used to perform inference at scale [34-36]. Moreover, they predict specific properties of neural responses in visual cortex, including changes in Fano factor and frequency of oscillations with tuning and stimulus intensity [23]. Previous works have proposed several approaches to accelerate sampling in biologically-inspiee algorithms. Hennequín al. [9] showed that adding non-reversible dynamics to rate networks can reduce the sample autocorrelation time. However, they did not study convergence of the sampling distribution, and did not consider the biologically-relevan setting of spiking networks. Savin and Denève [20] used distributed code to parallelize sampling in spiking networks, but only considered two-dimensional distributions. Therefore, remains unclear how accurate sampling from high- timescales can be achieved using spiking networks.
dimensional distributions at bchaviorally-releee
In this paper, we show how the choice of the geometry of neural representations at the population level [37, 38], set by the neural code and the neural dynamics, can accelerate sampling-basee based inference in spiking neural networks. Ideas from information geometry allow us to perform inference in the natural space of parameters, which is manifold with distances measured by the Fisher-Rao metric (Figure I.c) [39-42]. Concretely, we leverage
methods for accelerating sampling from the machine learning literature [28, 42-44] to design novel efficient samplers in spiking neural networks. The structure and major contributions of this paper are divided as follows:
In $2, we construct from first principles novel spiking neural network model for sampling from multivariate Gaussian distributions. This model is based probabilistic spike rule that implements approximate Metropolis- Hagtinns sampling. We show that efficient balanced networks (EBNs) [20, 45-49] emerges limit of this model which spiking becomes deterministic.
In $3, we show that population geometry enables rapid sampling in spiking networks. Leveraging the 'ccoplltte recipe" for stochastic gradient MCMC [43], we establish principles for the design of efficient samplers in spiking neural networks. Then, we show how neural population geometry enables fast sampling-on -on the timescale of tens of milliseconds-ing two limits the model introduced in $2: EBNs in which sampling is driven by stochastic Langevin dynamics [20], and networks in which sampling is driven purely by a Metropolis- Hastings probabilistic spiking rule.
Finally, in $4 conclude by discussing the implications of our results in the context of prior works, and highlight their limitations as well as remaining open questions. In particular, we comment on possibilities for future experimental studies of sampling in biological spiking networks, and applications to neuromorphic computing.
Spiking networks for sampling-based probabilistic inference
We begin by proposing framework for probabilistic inference in spiking neural networks in which the spiking rule implements a Metropolis-Hastinn Hastingg step. We show that EBNs [20, 45-49] can be recovered as a limiting case of this more general framework.

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In this work, we will keep our discussion quite general, and state our results for sampling from a generic Gaussian distribution. However, the problem we aim to solve can be given concrete interpretation in a neuroscience context, and could also be extended to non- non-Gaussian distributions. The goal of neural network performing probabilistic inference is to estimate a posterior distribution P(0]x) over Np latent variables (or parameters) 0 given an input x (Figure 1). The input could correspond to the activity of sensory neurons in early sensory processing (e.g. input onto ganglion cells in the retina or onto mitral cells in the olfactory bulb) or inputs into a cortical column that linearly sense features in the environment through an affinity matrix. We provide a detailed discussion of this linear Gaussian model in Appendix C. In the rest of the paper, we will usually abbreviate the distribution from which we want to sample as P(0). rather than writing P(0 x).
2.1 Deriving an approximate Metropolis- Hasting spiking sampler
We will build a network of n, spiking neurons to approximately sample an nup -p-dimensiooal Gaussian distribution of time-varying mean {(t) and fixed covariance V. We first consider the case in which is constant, and then generalize the resulting algorithm to the case in which it is slowly time-varying.
As in prior work probabilistic inference using spiking networks, we take the samples z to be linearly decoded from the filtered spike trains r of nh, neurons [20, 45-49]. Working in discrete time for convenience and clarity (as in [48]), we let
Z Tre. where Fi 111-)))r11++ +o
the low-pass filtered history of spikes o, E 10,11 for some decay constant 0 S 1) S 1.
Metropolis-Hastine sampling constructs a Markov chain by drawing a proposed next state from some distribution, and then deciding whether accept or reject that proposal based on a probabilistic rule [27, 50J. The acceptance ratio is given in terms of the relative posterior probability of the proposed and current states. Here, we will randomly propose which neuron spikes given timestep.
In trying to build a Metropolis- Saaeliss-- sampler [27, 50] using a probabilistic spiking rule, we are immediately faced with two problems. First, the spikes are sign-constrained and discrete. Second, the dynamics of the filtered spike history incorporate a decay term, hence the readout z will change even if no spikes are emitted. These conditions mean that the proposal density over will not be symmetric, and that the Markov process will not satisfy the condition of detailed balance [27, 50]. We note that previous work has shown that violations of detailed balance can accelerate sampling [17, 18, 51]. but we will not carefully explore this possibility in the present work. To obtain a symmetric proposal distribution with sign-constrained spikes, we assume that the network is divided into two equally-sized populations with equal and opposite readout weights, i.e. that the readout matrix is of the form r = [+M,- MM for some matrix M E R This could be accomplished by dividing the total population of neurons into excitatory and inhibitory populations with weights that are fine-tuned to be equal and opposite. The second problem can be solved by assuming that n i.c., that we have access a perfect integrator of the spike trains.
At the t-th timestep, we choose one neuron j uniformly at random, and let the spike proposal be o' = ej, where e, is the j-th standard Euclidean basis vector (i.e., (e,)x= 6jk). This yields candidate readout
z' =1 -
(1-1)))-- 7) + Tej.
(2)
If = 0 and the balance assumption on r is satisfied, then the proposal distribution is exactly symmetric, in the sense that the probabilities of reaching z' from z and of reaching Z;-1 from z are equal. Then, if we accept the proposed spike with probability
A = min {1, P(x')
(3)
P(z-1))
we obtain Metropolis- Hastingss sampling algorithm for the discretized Gaussian [27,
We now relax the assumption of the perfect integration, and assume only that 7 < 1. Introducing decay smooths the readout, which in the perfect integrator is discretized. With the same proposal distribution as before, we take the acceptance ratio of the accept-reject step to be
A = min {1, P((1- 11);,-1 + Te,]
(4)
P[( 7)22-11
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This choice has two important features. First, the decay means that the proposal distribution will be asymmetric, and the Markov chain will no longer satisfy the condition of detailed balance [27. 50]. However, the resulting error will be small if < 1. Second, by comparing the likelihood of the proposal, P[(1 7)) + Tey], to the likelihond of the next state without the proposed spike but with the decay P((1 - 2)1 (instead of the likelihood of the current state P(z,-1) as in the Metropolis-Hastie
algorithm), this choice implements sort of look-ahead step that should allow the algorithm to partially compensate for the decay in the rate.
With the choices above, we show in Appendix B that one can write the acceptance ratio (4) as
A = min (1, exp (V2- -T))).
(5)
where V -(1 m) r T- has the interpretation of membrane potential,
(1-1))10rr
T 2 Rz
(6)
has the interpretation of a spiking threshold, and the the recurrent weight matrix is defined as
Tr
Thus far, have assumed that the mean signal is constant. The natural generalization of this algorithm to a time-varying mean signal 0, to take the membrane potential to be V, = S),,, + T yp 10. This leads to the voltage dynamics
V - (1 V = -(1 Ro; + +r-
v0 - (1- 0)),,1l.
(8)
which, when combined with the probabilistic spiking rule with uniform proposals and acceptance ratio (5), yields our final algorithm. This will not be an exact Metropolis-Hastings sampler unless the mean is constant, the decay term vanishes (77 0), and the readout matrix T satisfies an exact balancing condition. In particular, if these conditions are violated, the Markov chain will not satisfy detailed balance. However, if they are violated only weakly, this algorithm can still be a reasonable approximation to a true sampler. We will provide empirical evidence for this intuition in $3.
2.2 The continuous-timg limit
We now consider the continuous -imeeeeeuuueeeeeeeee limit of the model introduced above. This limit corresponds to taking the limit in which spike proposals are made infinitely often, and regarding the dynamics written down previously For a timestep A between spike proposals, we let the discrete-time decay rate be r = A A/Tm for time constant Tmm thusly named because has the interpretation of membrane time constant. Then, we show in Appendix that the A & 0 limit of the discretized rate dynamics (1) yields the familiar continuous-tima dynamics
forward Fuler discretization of an underlying continuous-time system.
dr(t) dt
1 -r()
r(t) + ott).
(9)
Tm
In continuous time, the spike train o(t) is now composed of Dirac delta functions, as the discretized spikes rectangular pulses of width A in time and height 1/2. We next consider the voltage dynamics of the leaky integrator for a varying mean signal (8), which have a similar continuum limit:
dv=(1)=_
V(t) Ro(() + + -T
(10(() + Fm e(t)) B(t)
(d0(t) dt
1
(10)
Ti
In this limit, the rate will decay by only infinitesimal amount between rejected spike proposal and the next proposal, meaning that the error incurred by neglecting the asymmetry in the proposal distribution due to the decay should be negligible. We also note that, though the probabilistic spike rule (5) does not explicitly include a reset step, the dynamics (10) prescribe that the j-th neuron's ss membrane voltage should be decremented by 2T, after it spikes.
2.3 Efficient balanced networks limit of the spiking Metropolis- -astopiniss-sssings sampler
This spiking network samples the posterior distribution by emitting spikes probabilistically, but we can use the same architecture to re-derive EBNs, which approximate continuous dynamical systems

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using spiking networks [45-49]. If we take V x In (for Inp the Ro x np identity matrix) the voltage dynamics (10) are identical to those of the EBN. The greedy spiking rule of the EBN can be recovered in this framework by taking a limit in which the variance of the Gaussian target distribution vanishes. Concretely, we let x0 = Lhyn and define re-scaled variables (() = ve (L). Q = 1-22, and T = T that will remain (11) even we take & 4 O. In terms of these new variables, the acceptance ratio is A = min [1 exp(((,()) /1], which tends to A = @ This explicitly recovers the greedy spike rule used in EBNs. In Appendix B, we further analyze how the overall scales of L and I affect the probabilistic spike rule.
T2) ) as v + O.
The network with voltage dynamics (10) samples a distribution with mean @ and covariance V. Instead of sampling using the structured proposal distribution on spikes, this network can implement sampling through slowly varying Langevin dynamics on 0. the limit where the spike rule becomes greedy, this recovers the spiking sampler studied by Savin and Denève [20]. We will discuss this model further in $3.
3 Population geometry for fast sampling
3.1 Leveraging the geometry of inference to accelerate sampling
We first review recent work from the machine learning literature for how population geometry can be chosen to accelerate simple Langevin sampling, which establishes principles for the design of fast samplers. For probability distributions belonging to the exponential family, including the Gaussian distributions on which we focus in this work, one can write the density P(z) in terms of an energy function U (z) such that P(z) x exp[-U(())] The classic algorithm to sample such a distribution is the discretization of the naive Langevin dynamies
dz(() VU (z) di v2dW(!),
(11)
where W(t) is a standard Brownian motion [29, 30, 33, 52]. By simply following these dynamics, one can obtain samples from the target distribution and therefore an estimate of the uncertainty at the timescale taken by the network to sufficiently explore the target distribution.
The Langevin dynamics (11) can be directly implemented in a rate network [9, 19] or approximately implemented in a spiking network [20], but their convergence properties for high-dimensional distributions have not been investigated in a neuroscience context. It is well known in statistics that, as the dimensionality of the target distribution increases, convergence of Langevin sampling to the target distribution slows dramatically. Furthermore, the discretization step can induce errors that cause the variance estimated from sampling to exceed the target variance [29, 30, 33, 52].
To overcome these issues, prior work in statistics and machine learning has proposed algorithms that can accelerate the sampling [28, 31, 42-44, 53-55] which we leverage here to propose our fast samplers in spiking neural networks. These ideas were unified into common framework by Ma et al. [43] (and see also [44, 56]) who proposed a "complete recipe" for stochastic gradient MCMC:
dz(t) = (z)(z) + S(z) + +b()) dt + Bv)d
(12)
with BBT = D. The matrix fields D(z) and S(z) modify the dynamics but keep the target distribution unchanged. D is positive semi- definie and defines the local geometry of the space in which the inference is occurring, while S is skew-symmetric and adds non-reversible dynamics. When D and S are state dependent, the correction term div(D + S) must be included [43].
The "complete recipe" provides a general framework to design samplers based on Langevin dynamics. Samplers based Riemannian geometry can be designed by choosing D to be the inverse of the Fisher information matrix G (or an approximation thereof), yielding the natural gradient V v = G- (Figure 1) [39-42, 57, 58]. Samplers incorporating dummy variables can be designed by expanding the parameter space and using the matrices D and S to obtain the desired dynamics [31, 43, 44, 53-55, 59]. Similarly, prior works have proposed methods accelerate sampling
'In Appendix B. we show how elastic net prior on firing rates used by Boerlin et al. [45] can be incorporated into this model. As it only modifies the definitions of the recurrent weights and spiking threshold, this extension does not add new conceptual difficulties,, hence we do not discuss it further in the main text. Additionally, we provide pedagogical introduction to the dynamics of the EBN in Appendix C.
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in biologically inspired neural networks by parallelizing the inference [20], using Hamiltonian dynamies [21] or by adding non-reversible dynamics [9, 17, 18]. The "complete recipe" provides general framework that encompasses all these examples, allowing for the principled design of bealogically-plausibbe samplers.
In the simple setting of linear rate networks sampling from a Gaussian distribution with constant mean ia and covariance Σ, the complete recipe for state-independent D and S takes the form
dz(t) - (D - S)Σ (z 1s) dt + / (W(
(13)
In Appendix D, we show the zero-ment case μ that large eigenvalues of the covariance matrix Σ introduce long timescales in the sample
for naive Langevin dynamics (D B In S 0), which slow convergence to the target distribution. Concretely, the 2- 2- distance between the ensemble sampling distribution at time / and the target is
ny
Wmative (1)
Σ
a. [1
(1 e 2//0))1/333
(14)
where o; are the eigenvalues of the covariance matrix. Networks performing inference in the natural
space (D = E. B = E1/2) are insensitive to these large cigenvalues, with 2- 2-Wasserstein distance
W Wnnatural (t)
Σ
(1-
(15)
For this analytical result, we consider the distribution of samples time over realizations of the noise process, which differs from the distribution of samples over time within a single realization, as used elsewhere the paper. This choice is made because the joint distribution of samples within single realization is challenging to characterize [60]. These results show how natural gradient can accelerate inference linear rate networks, and complement Hennequin et al. [9]'s 's study of how adding irreversible dynamics can reduce the sample autocorrelation time (i.e., D = B Ry o 0).
3.2 Fast sampling through population geometry in efficient balanced networks
We first consider a sampler based on efficient neural networks [20, 45-47] that leverages the geometry of the inference implement efficient sampling at the level of the population. In previous work, Savin and Denève [20] derived a sampler implementing naïve Langevin dynamics (we provide a full derivation using the notation from the present paper in Appendix C). Although they proposed to accelerate sampling by implementing parallel inference loops, they do not leverage the geometry of the inference nor do they test the convergence high (np > 2) dimensions.
Here, we approximate the "complete recipe" dynamics (13) using an EBN, and show that performing inference using natural gradients helps with speed and accuracy in high dimensions. As $3, we consider sampling from Gaussian distribution of possibly time-varying mean and covariance E.4 As we principally study the effect of the geometry, we henceforth set O. Note that even though the underlying dynamics of (13) are reversible if S = 0, the non- non-lineane lincarities introduced by spiking lead to a non-reversible sampler. To embed these dynamics in the EBN framework, we take -hleebbeh process (13), which we generalize to include an intrinsic timescale T., Following [20], we approximate the true state of the sampling dynamics by the instantaneous estimate from the filtered spike history. Then, as detailed in Appendix C, we obtain an EBN with the voltage dynamies
the target mean in the voltage dynamics (10) to evolve according to the
dV =
1 V - No +
1 I Tm
Ing
Tin
DE DE
1 Ir
T Ts
H dt +
VV r
Tm
To
Te
(16)
where the recurrent weight matrix is Ω = rT, and the spiking rule is greedy, with thresholds
T3 = r)
'In Appendix C. we give concrete interpretation of this task in terms of the Gaussian linear models used in previous work by Hennequin et al. [9] and Savin and Denève [20].

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Moving average of
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navv natural
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Figure Accelerating inference through population geometry in EBNs simulating Langevin sampling: a. 50 ms moving averages parameter estimates 0; over time, for naive and natural geometry in network sampling from "p 20-dimensional equicorrelated Gaussian distribution of correlation 0.75 using a network of nt, 200 neurons. At / = 0.5 s, the target mean shifts from zero to six, indicated by the black dashed line. b. The distribution of ISIs across neurons and trials is approximately exponential. See Supplemental Figure E.2 for more statisties of the resulting spike trains. c. Comparison of performance in the 50 ms following stimulus onset (after t 0.5 s) between naïve and natural geometry for varying p c.i. The estimate of the mean collapses towards zero with increasing p for naïve geometry, c.ii. The estimated variance increases catastrophically with for naïve geometry, but only mildly for natural geometry. c.iii. Inference accuracy measured by the mean marginal 2- 2-Wasserstein distance across dimensions decreases for naive. d. Similar analysis when varying n, for the d.i. Mean, d.ii. Variance and d.iii. mean 2- Wasserstein distance. In panels c,d shaded error patches show 95% confidence intervals computed via bootstrapping over 100 realizations in all panels. In Supplemental Figure E. we show the full time course of the inferred statistics for one set of network parameters and in Supplemental Figure E.3 we show the statistics after the full 1.5s, which are qualitatively similar. See Appendix E for detailed numerical methods.
To illustrate how correlations between parameters affect sampling in high dimensions, we will focus on equicorrelated multivariate distributions. The covariance matrix Σ of such distribution parameterized by an overall variance a > 0 and a correlation coefficient -1 < p < 1 such that Enj a 6. + p(1 5. We will explore the performance of the sampling algorithms across values of for different dimensions of the parameter (1pp) and neuron (n,, spaces. For a multivariate Gaussian distribution N (H, Σ), the Fisher information matrix G Σ-1 and we will therefore use D Σ (and B = vD) in our geometry aware implementation (see Appendix C). In our simulations, we compare the accuracy of sampling of the naïve implementation (D B with the geometry-aware version over a 50 millisecond window, which is roughly twice the membrane time constant Tm. 20 ms, well as steady-state. In Figure 2 and in Supplemental Figures E.1-E.3, we show that the geometry-aware sampler is more robust to increasing the correlation of the parameters and the dimensionality, allowing inference at behavioral timescales. As in non-spiking Langevin sampling. discretization leads to an overestimation of variance [27], but this effect, although still present, is strongly reduced in the geometry- implementation. Note here, that the geometry is imposed through the dynamics of the membrane potentials via the recurrent connectivity in the network.
3.3
Fast sampling through population geometry using probabilistic spike rules
In the preceding subsection, we have shown how neural population geometry, implemented through neural dynamics, can accelerate the speed of approximate sampling an efficient balanced network. However, this approach suffers from a fundamental conceptual gap: the firing rates of the spiking network are being used to simulate non-spiking Langevin dynamics; the spiking network itself has not been designed to sample. Specifically, the discretization introduced by the spiking exacerbates
7

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0.6
moving average of
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celabvee frequency
neurons neurons
0.3
-2
b
III
10%
10
relative frequency
relative freguency
relative frequency
104
0.5
10
10-
10
500
firing rate (Hz)
(ms)
Figure 3: Sampling using probabilistic spiking rules. a. Sampling from a np = 10-dimensional equicorrelated Gaussian distribution of correlation p 0.75 using network of n,, = 100 neurons.
a.i. 100 moving averages of parameter estimates O; over time, for naive and natural geometry. / 0.5 s, the target mean shifts from zero to one, as indicated by the black dashed line. a.ii. Marginal distributions of 0g after stimulus appearance. a.iii Spike rasters Variability in spiking across 1000 trials. b.i. Spike rate distributions across neurons and trials for stimulus distribution as in (a). b.i. The distribution of ISIs for an example neuron across trials is approximately exponential. b.iii. The distribution of coefficients of variation of ISIs across neurons. See Appendix for detailed numerical methods.
100 ms window. b.
the errors introduced by the discrete time implementation of the sampling dynamics, leading to overestimation of the stimulus variance. In this subsection, we take a first step towards bridging this gap by considering an alternative limit of the general model introduced in $2: the case in which sampling is performed leveraging only the stochasticity in the spike rule. Our objective here is not to demonstrate a fully biologically plauusile or practically useful sampling algorithm; rather, it is to illustrate the importance of population geometry in minimal model for probabilistic spiking.
As in 3.2, we focus on sampling from equicorrelated Gaussian distributions. N (1u, E). In this case, we set L Σ and 0 = in (10). Naïve sampling here corresponds to choosing I in some sense generically (see Appendix B), while geometry-aware sampling corresponds choosing I such that TT ~ Σ to overall constants of proportionality. In Figures 3 and 4, and in Supplemental Figure E.4, we show that naïve choices of I lead vanishing spike rates at strong correlations p and large parameter-space dimensionalities ny. This results in dramatic underestimation of the mean and variance of the target distribution, which is resolved by choosing the geometry appropriately, again allowing inference at behavioral time-scales. Moreover, these networks show Foisson-like variability in spiking statistics (Figure 3), consistent with cortical dynamics [61, 62].
In Appendix B, we provide a more careful analysis of the strongly-correlated limit f 1. Informally, we show that the probability of spiking should vanish I is chosen sufficiently naïvely and the mean of the target distribution is uniform across parameter dimensions, i.c., x Ing. This analysis is not specialized particular case, and holds generally for the model of $2. Therefore, careful choice of population geometry, as implemented by the neural code, is required for fast sampling in this model.
4 Discussion
In this paper, we have shown how careful choice of neural population geometry enables fast sampling in spiking neural networks. We presented a unified framework which EBN samplers approximating Langevin dynamics with greedy spiking and approximate Netropolis-Hastinne ministic voltage dynamics and probabilistic spiking can be unified. We then leveraged population geometry to perform rapid sampling at behaviorally-relevan timescales in these two disparate limits of our general model. We now discuss some of the limitations of our work, and highlight possible directions for future inquiry.
samplers with deter-
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Figure 4: Population geometry enables fast sampling from strongly-correlen high-dimensionn distributions networks with probabilistic spiking rules. a. Sampling from -eunglly-quicorrllttedd Gaussians in ny 10 dimensions using a network of n, = 100 neurons requires careful choice of geometry. The stimulus setup is in Figure 3. a.i. At strong correlations p. spiking is suppressed in networks with naive geometry, but not when the natural geometry is used. a.ii. Dimension -veeraeed estimate of the mean signal in the 50 ms after stimulus onset. See Supplemental Figure E.4 for steady-state statistics. a.iii. As in ii, but for the variance. a.iv. As in ii, but for the mean marginal 2- Wasserstein distance across dimensions. b. Sampling in high dimensions requires careful choice of geometry. b.i. At moderately strong correlations = 0.75 and high dimensions, spiking is suppressed networks with naïve geometry, but not when the natural geometry is used. Here, we use 10 neurons pnteension-averager estimate of the mean signal in the 50
per parameter, i.c., Ra 10mp b.ii. milliseconds after stimulus onset. See Supplemental Figure E.4 for steady-state statistics. b.iii. in ii, but for the variance. b.iv. As in ii, but for the mean 2- Wasserstein distance. Shaded error patches show 95% confidence intervals computed via bootstrapping over 100 realizations in all panels; see Appendix E for further details.
Like the original EBN model, the probabilistic spiking model introduced in $2 suffers from the limitations that requires instantaneous propagation of spike information and that only one neuron is allowed to spike at time [20, 45-49]. Moreover, the discretization timestep enforces a hard cutoff on maximum spike rate. Some these limitations could be partially circumvented by generalizing the spike proposal distribution to allow multiple spikes. However, such a model would still suffer from the issue that an accept/reject step that accounts for the effect of spikes from multiple neurons will require instantancous communication across the network. This limitation could possibly be overcome within the framework asynchronous Gibbs sampling [63, 64], which ignores the requirement that updates should be coordinated across the network.
The analysis of $2 shows that the models of $3 can be viewed as limiting cases of single framework. It is likely that the parallels between these limiting models could be further strengthened by viewing the Gaussian noise the voltage dynamics of the EBN sampler as an approximation to the effect of the stochastic spiking of other neurons in the Metropolis-HHan Hastings sampler. Such an approximation could be obtained the limit large network size given an appropriate treatment asynchronous updates, provided that one could neglect the coupling between spikes in different neurons induced by the proposal distribution [62, 65-67]. Careful analysis of the relationship between these two sources of stochasticity will be interesting subject for future investigation. Morcover, it will be interesting to investigate how they might be integrated in a single network, which would result in a spiking sampler somewhat reminiscent of the cestropolis-adjustee Langevin algorithms used in machine learning [68, 69].
Here, we have focused entirely on Gaussian target distributions. As we sketch in Appendices B and C, both the probabilistic spiking sampler and the EBN sampler could in principle extended to general exponential family targets. However, the most naïve extensions non-Gaussian distributions would involve non-linear and non-local interactions the voltage dynamics. In particular, one would have to account for state-dependence in the matrix fields ((z) and S(z), which would require the inclusion of

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the field $ (z)) in the complete recipe (12) (see ref. [43]). Mapping such interactions onto biological mechanisms requires more careful consideration. In recent work, Nardin al. [70] have proposed how dynamical systems with polynomial nonlinearities can be approximated by deterministic EBNs with multiplicative synapses. This approach could be extended to the stochastic setting of networks designed to sample, which will be interesting to test in future work.
In this work, we did not constrain neurons to follow Dale's law, and single neurons therefore have both excitatory and inhibitory effects on their neighbors. Many frameworks have been proposed to map the connectivity of unconstrained network algorithms onto distinct excitatory and inhibitory neuron types [9. 71, 72]. These refinements of the biological plausibility will not affect our key argument of accelerating inference through favorable population geometry. However, different possible implementations that comply with Dale's ' law will make different predictions for experimentally- measureable biophysical properties. For example, although less numerous, fast spiking inhibitory neurons have higher firing rates [71, 72], which could allow the approximate symmetry of the readout, as required by the construction of $2, to be maintained.
In biological spiking networks, probabilistic spike emission and probabilistic synaptic release are natural sources of stochasticity [73-76]. These two layers of probabilistic computation provide additional flexibility in processing beyond the simple accept/reject step considered here. As result, it is likely that one could construct sampling algorithms that are at once more biophysically detailed and more computationally efficient than the simple network constructed in $2. Further investigation of how violations of detailed balance through these mechanisms and the matrix S in the "complete recipe" framework could enable faster sampling will be a particularly interesting objective [17, 18, 51). Moreover, it will be interesting to investigate how natural gradients can accelerate inference in the presence of more complex synaptic dynamics, as studied in recent work by Kreutzer et al. [77].
The algorithms proposed in this work could also enable fast sampling in neuromorphic circuits [78-80]. As they require only the local membrane voltage to compute accept/reject steps, these algorithms could be implemented in a distributed neuromorphic architecture. They would then potentially be limited only by the timescale of individual computing units (which are much faster than biological neurons) rather than the dimensionality of the inference problem [81].
The sampling processes considered in this work focus on short-timescale perceptual inference. However, similar probabilistic inference can occur at longer timescales of learning [82-84]. These long-tin long-timescale sampling processes would allow networks to flexibly infer their synaptic weights. Importantly, learning the task-i.e., learning the matrix I-l_and learning the representational geometry-D, S. and r-aaee distinct processes that can occur in parallel. Once the task structure learned, adapting the geometry of the population code match changing stimulus geometry can be achieved through meta-learning [85]. Recent works have analytically studied the population geometry that results from this sampling procedure in rate-based networks [86], and proposed algorithms for efficient learning in EBNs [47, 87-89] and other classes of spiking networks [90-93]. However, the interactions between fast activity sampling and slow network parameter sampling in neural circuits- particularly spiking networks- rremain poorly understood [84]. Characterizing how adaptive population geometry accelerates perceptual inference in dynamic environments will be an important step towards more complete understanding of probabilistic inference in neural circuits.
Acknowledgments and Disclosure of Funding
PM was supported by the Harvard Mind Brain Behavior Interfaculty Initiative, JAZ-V and CP were supported by Google Faculty Research Award and NSF Award DMS-2134157. This work was also supported by a grant from the National Institute of Health (R01-DC-017311) to VNM. We thank Tony Zador for co-oraanizing the NeurIPS 1996 workshop on "Synaptic transmission: reliability and variability", whose follow-up discussions led to this paper.
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References
[1] David C Knill and Alexandre Pouget. The Bayesian brain: the role of uncertainty in neural coding and computation. Trends in Neurosciences, 27(12):712-71,, 2004. doi: 10.1016/j.tnns.20000 2004..iins.204...0000..
[2] Konrad Körding and Daniel M Wolpert. Bayesian integration in sensorimotor learning. Nature, 42776971) 2004. doi: 10. 10.1038/nature021669
[3] József Fiser, Pietro Berkes, Gergö Orbán, and Mâté Lengyel. Statistically optimal perception and learning: from behavior to neural representations. Trends in Cognitive Sciences, 14(3):
119 130, 2010, doi: 10.1016/j.ties.. 2010001..00...
[4] Torben Ott, Paul Masset, and Adam Kepecs. The neurobiology of confidence: From beliefs to neurons. In Cold Spring Harbor Symposia on Quantitative Biology. volume 83, pages 9-16. Cold Spring Harbor Laboratory Press, 2018. doi: 10.1 111100//sb.22188...3883388
[5] Paul Masset, Torben Ott, Armin Lak, Junya Hirokawa, and Adam Kepecs. Behavior-and mondality-general representation of confidence in orbitofrontal cortex. Cell, 182(1):112-126,
2020. doi: 10. 10.1016/j.cell... 20.2000.0.2200..022.
[6] Ruben S Van Bergen, Wei Ji Ma, Michael S Pratte, and Janneke FM Jehee. Sensory uncertainty decoded from visual cortex predicts behavior. Nuture Neuroscience, 18(12):122--1338 2015. doi: 10.1038/nn.4150
[7] Dylan Festa, Amir Aschner, Aida Davila, Adam Kohn, and Ruben Coen-Cagli. Neuronal vari- ability reflects probabilistic inference tuned to natural image statistics. Nature Communicutions, 12(1):1-11, 2021.
doi: 10.0038/44146--002228888
[8] Alexandre Pouget, Jan Drugowitsch, and Adam Kepecs. Confidence and certainty: distinct probabilistic quantities for different goals. Nature Neuroscience, (9(3):366-3744 2016. doi:
1..008/mm.4240.
[9] Guillaume Hennequin, Laurence Aitchison, and Mate Lengyel. Fast sampling-basee inference in balanced neuronal networks. In Z. Ghahramani, M. Welling, C. Cortes, N. Lawrence, and K.Q. Weinberger, editors, Advances in Neural Information Processing Systems, volume 27, pages 2240-2248. Curran Associates, Inc., 2014. URL https://papers: www.rrss..pps...... cttpps.ccpppppe.ee Taw-9944488---bbtaacccoo html.
2014/hash/
[10] Alexandre Pouget, Jeffrey M Beck, Wei Ji Ma, and Peter E Latham. Probabilistic brains: knowns and unknowns. Nature Neuroscience, 16(9):1 2013. doi: 10. 10.1038/nn.3495.
[11] Ádám Koblinger, József Fiser, and Máté Lengyel. Representations of uncertainty: where art thou? Current Opinion in Behavioral Sciences, 38:150-162, 2021. doi: 11111666.obehha.222.0..0000
[12] Hansem Sohn and Devika Narain. Neural implementations of Bayesian inference. Current Opinion in Neurobiology, 70:121-129, 2021. doi: 10.1016/j.conb.200 2tt.cc......2.00000..
[13] Wei Ji Ma, Jeffrey M Beck, Peter E Latham, and Alexandre Pouget. Bayesian inference with probabilistic population codes. Nature Neuroscience, 9(11):1432 1438, 2006. doi: 10. 10.1038/nn1191
[14] Maneesh Sahani and Peter Dayan. Doubly distributional population codes: simultaneous representation of uncertainty and multiplicity. Neural Computation, 15(10):225-222) 2003. doi: 10. 10.1162/899760033226655
[15] Eszter Vértes and Maneesh Sahani. Flexible and accurate inference and learning for deep generative models. In Bengio, Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018. URL haah1/85556676666s4440o222222...
wwwessnuuuip.... http:/e....ppp.pppp/////........ cctppper/2//1 Abstract. html.

---

[16] Patrik Hoyer and Aapo Hyvärinen. Interpreting neural response variability as Monte Carlo sampling of the posterior. In S. Becker, S. Thrun, and Ober- mayer, editors, Advances in Neural Information Processing Systems, volume 15. MIT Press, 2002. URL. https://proceeddnng neurips. wwc.iips.cpppppeee https/ccepppee///////seeee Abstract. html.
[17] Lars Buesing, Johannes Bill, Bernhard Nessler, and Wolfgang Maass. Neural dynamics as sampling: a model for stochastic computation recurrent networks of spiking neurons. PLoS Computational Biology, 7(1 (el1:e:0002111 2011 doi: 10.1371/journal.pe 1002211.
[18] Dejan Pecevski, Lars Buesing, and Wolfgang Maass. Probabilistic inference in general graphical models through sampling in stochastic networks of spiking neurons. PLOS Computational Biology, 7(12):1-25, 12 2011. 1. doi: 10. 1/337////ural..cc.. pcb....02...... 1002294. URL. httph://tp/o.oo org/ 10. 1t.1173/oouraal pebi. 1002294.
[19] Agnieszka Grabska-Barwinske Barwinska, Jeff Beck, Alexandre Pouget, and Peter Latham. Demixing odors- fast inference in olfaction. C.J. Burges, L. Bottou, M. Welling. Z. Ghahramani, and K.Q. Weinherger, editors, Advances in Neural Information Processing Systems, volume 26, pages 1968-1976. Curran Associates, Inc., 2013. URL httpshttp////ootoo htt.ttpps//proocedingg...o neurips. cc/ /appe///13/////aaa/pppppppaaa/a/ /ttaa//0000111111111111483388883 hostea/eeeaaaaaaaaaaaaaaattarrar Abstract.
[20] Cristina Savin and Sophie Denève. Spatio-temporal representations of uncertainty in spiking neural networks. In Z. Ghahramani, M. Welling, C. Cortes, N. Lawrence, and K.Q. Weinberger, editors, Advances in Neural Information Processing Systems, volume 27, pages 2024-2032. Curran Associates, Inc., 2014. URL https: h/tp://ee/o/o....../m/oe//s www.ddiinnnnn.nissne.eee. cc/pcccppp/111..//.... Abstract html.
1ass//422545/8116676061668
[21] Laurence Aitchison and Máté Lengyel. The Hamiltonian brain: Efficient probabilistic inference with excitatory-inhibioryy neural circuit dynamics. PLoS Computational Biology, 12(12): e1005186, 2016. doi: 10.1371 1 1005186.
[22] Wen-Hao Zhang, Tai Sing Lee, Brent Doiron, and Si Wu. Distributed sampling- Bayesian inference coupled neural circuits. bioRxiv, 2020. doi: 00.1001/2020.0.22..22222
[23] Rodrigo Echeveste, Laurence Aitchison, Guillaume Hemnequin, and Máté Lengyel. Cortical-like dynamics in recurrent circuits optimized for sampling- based probabilistic inference. Nature Neuroscience, 1119):1118-1144 2020. doi: 1110:///11133000-06666
[24] Wenhao Zhang, Si Wu, Kresimir Josic, and Brent Doiron. Sampling based Bayesian inference in recurrent circuits of stochastic spiking neurons. bioRxiv, 2022. doi: 10.
[25] Yang Qi and Pulin Gong. Fractional neural sampling as theory of spatiotemporal probabilistic computations in neural circuits. Nature Communications, 13(1):4572, Aug 2022. ISSN 2041-
1723. doi: 1038/s4 467-022-32279-2.
[26] Jeffrey M. Beck, Peter Latham, and Alexandre Pouget. Marginalization in neural circuits with divisive normalization. Journal of Neuroscience, (533))11110--3319 2011. ISSN 0270-6474, doi: 10. (0.1522//NEUROSCL.17000 31/43/15310.
110000--11..00 1.2011. URL https: :/ww:::///wwww// jnw.rosciiercccioo org/ cont/ntn/
[27] Radford M Neal. Prohabilistic inference using Markov chain Monte Carlo methods. Department of Computer Science, University of Toronto Toronto, ON, Canada, 1993. URL cs. toronto...... edu/"radf wrrrreoorrrriiiieerreeeeee abstract html.
[28] Christian P Robert, Victor Elvira, Nick Tawn, and Changye Wu. Accelerating MCMC algo- rithms. Wiley Interdisciplinary Reviews: Computational Statistics, 10(5):e1435, 2018. doi: 10.1002/wics. 1435..
[29] Crispin W Gardiner. Handbook of stochastic methods, volume 3. Springer Berlin, 1985,
[30] Max Welling and Yee W Teh. Bayesian learning via stochastic gradient Langevin dynamics. In Proceedings of the 28th international conference on machine learning (ICML-11), pages 681-688, 2011. URL https: https// http:////r/m//////e//eeee/////// pwppps/ppprs/ee/ee www.mmmppppp...mmmmmmmmmmmmmm.ep .df.
12

---

[31] Radford M Neal. MCMC using Hamiltomian dynamics. In Handbook of Markov Chain Monte Carlo, pages 139-188. Chapman and Hall/CRC, 2011. doi: 10.1201/b10905.
[32] Alain Durmus and Eric Moulines. Nonasymptotic convergence analysis for the unadjusted Langevin algorithm. The Annals of Applied Probability, 27(3):1551-1587, 2017. doi:
10.121411-AAP1238.
[33] Santosh Vempala and Andre Wibisono. Rapid convergence of the unadjusted Langevin algo- rithm: Isoperimetry suffices. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Bue, E. Fox, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019. URL https: https:///oocedd/eeeeeeeeeee www.eediiinnnnn..ii...... nww.ine.rcccccccoooo.. cttpppps/cc/pppp///1000//// (055a9996211155ddd22000000088800
Abstract. html.
[34] Sam Patterson and Yee Whye Teh. Stochastic gradient Riemannian Langevin dynamics on the probability simplex. In C.J. Burges, L. Bottou, M. Welling, Z. Ghahramani, and K.Q. Weinberger, editors, Advances in Neural Information Processing Systems, volume 26. Cur- ran Associates, Inc., 2013. URL https: http:::s//roceedinggs neurins..cccom wttpcc/ppppp///////eaaaaaa 30992844b100a5d7 www.aaffffaffffdaffafaaa www.ccdddbbbbbbrat...a..a.aaaa.. censtract.html.
[35] Pavel Izmailov, Sharad Vikram, Matthew D Hoffmann, and Andrew Gordon Gordon Wilson What are Bayesian neural network posteriors really like? In International Conference on Ma- chine Learning, pages 4629-4640. PMLR, 2021. URL htttps://proceedinng mlr www.mmlrreeeee press/
1.39/izmailov211 html.
[36] Florian Wenzel, Kevin Roth, Bastiaan Veeling, Jakub Swiatkowski, Linh Tran, Stephan Mandt, Jasper Snoek, Tim Salimans, Rodolphe Jenatton, and Sebastian Nowozin, How good is the Bayes posterior in deep neural networks really? In Hal Daumé III and Aarti Singh, editors, Proceedings of the 37th International Conference on Machine Learning, volume 119 of Proceedings of Machine Learning Research, pages 10248-1000 10259. PMLR, 13-18 Jul 2020. URL https: /prp://dig/ooeeiggsss mlr. press/v119//ee wwtp://wweeeaa....mm html.
[37] Nikolaus Kriegeskorte and Xue-Xin -in Wei. Neural tuning and representational geometry. Nature Reviews Neuroscience, 09 2021. ISSN 1471-0048. doi: 10. 10.1038/441583-001-0000
[38] SueYeon Yeon Chung and LF Abbott. Neural population geometry: An approach for understanding biological and artificial neural networks. Current Opinion in Neurobiology, 70:137-144, 2021. doi: cow....2..b...1..1..
[39] Shun-Ichi Amari. Natural gradient works efficiently in learning. Neural Computation, 10(2): 251-276, 1998. doi: 10.1 662089996698830007744
[40] Shun-Ichi Amari and Scott C Douglas. Why natural gradient? In Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 98 (Cat. No. 98CH36181), volume 2, pages 1213-1216. IEEE, 1998. doi: 10.1109/CAASPP1199... 6755499.
[41] Shun-Ichi Amari. Information geometry and its applications, volume 194, Springer, 2016. doi: 10.10077978-4441-55500
[42] Mark Girolami and Ben Calderhead. Riemann manifold Langevin and Hamiltonian Monte Carlo methods. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73
22::123-214, 2011.
doi: 10. 1011/11/j. 441444668.20011110 2010.000.....0.
[43] Yi-An Ma, Tianqi Chen, and Emily Fox. complete recipe for stochastic gradient MCMC, In C. Cortes, N. Lawrence, D. Lee, M. Sugiyama, and R. Gameet, editors, Advances in Neural Infor- mation Processing Systems, volume 28. Curran Associates, Inc., 2015. URL https: /papprs...... ccpps.cc//pper// pttpaperr/11////a/// 2t1///221///////// htt///aah//aaaaae
Abstract. html.
[44] Christopher Nemeth and Paul Fearnhead. Stochastic gradient Markov chain Monte Carlo. Journal of the American Statistical Association, 116(533):4334500 2021. doi: 10. 00.180000662145522000144777777
[45] Martin Boerlin, Christian K Machens, and Sophie Denève. Predictive coding of dynamical variables in balanced spiking networks. PLOS Computational Biology, ((1T)ee0032258 1003258, 2013. doi: 10.137 Lfournal. www.. 1003258.
13

---

[46] Sophie Denève and Christian K Machens. Efficient codes and balanced networks. Nature Neuroscience, 19(3):375-382, 2016. doi: 10.1038/nn.4243
[47] Sophie Denève, Alireza Alemi, and Ralph Bourdoukan. The brain as an efficient and robust adaptive learner. Neuron, 94(5):969 977, 2017. doi: 10.1016/j.neuron2001... 20...00..1...
[48] Camille E Rullán Buxó and Jonathan W Pillow. Poisson balanced spiking networks. PLoS Computational Biology, (e(11)xe108822) 2020. doi: 10.137//joumnal.pcie 100...006......... 1008261.
[49] Nuno Calaim, Florian A Dehmelt, Pedro J Gonçalves, and Christian K Machens. The geometry of robustness spiking neural networks. eLife, 11:e73276, may 2022. ISSN 2050-084X. doi: 10. 10.7554/eLife.725 73276. URL https: //toi. org/10. 7554/eLife e
[50] W. Keith Hastings. Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57(1):97-109 1970. doi: 10. 10.1099//iomec/57.1.9.7
[51] Radford M. Neal. Improving asymptotic variance of MCMC estimators: Non-reversible chains are better. arXiv, 2004. doi: T1A850/ARIIVMAAHH000225 URL http:/xxxvvvvrrr.//. org/ abs hts/math//407281.
[52] Bernt Øksendal. Stochastic differential equations. Springer, 2003, doi: 10.1007/978-3-642- 14394-6.
[53] Michael Betancourt. A conceptual introduction to Hamiltonian Monte Carlo. arXiv preprint arXiv:1701.02434, 2017. doi: 10.48550/arxiv.170..oom
[54] Zongchen Chen and Santosh S Vempala. Optimal convergence rate of Hamiltonian Monte Carlo for strongly logconcave distributions. Theory of Computing. 18(1):1-18, 2022. doi:
20188///oc.22..01188888
[55] Nan Ding, Youhan Fang, Ryan Babbush, Changyou Chen, Robert Skeel, and Hartmut Neven. Bayesian sampling using stochastic gradient thermostats. In Z. Ghahramani, M. Welling, C. Cortes, N. Lawrence, and K.Q Weinberger, edi- tors, Advances in Neural Information Processing Systems, volume 27. Curran As- sociates, Inc., 2014. URL 0a155baa775eeaec7744554845333333
wwwesnnnrrppp.. cc/ Abstract. html.
[56] Jianghong Shi, Tianqi Chen, Ruoshi Yuan, Bo Yuan, and Ping Ao. Relation new interpreta- tion of stochastic differential equations to Ito process. Journal of Statistical Physics, 148(3): 579- 590, 2012. doi: 10. 10.10077/
[57] Razvan Pascanu and Yoshua Bengio. Revisiting natural gradient for deep networks. arXiv preprint arXiv: 1301.3540 2013. doi: 10.48550/arXiv. 1301.3384.
[58] James Martens. New insights and perspectives on the natural gradient method. Journal of Machine Learning Research, 21:1-76, 2020. URL https://jmlr. http:///mmmoooooppprrreee wttppppprs//v//////
17-678. html.
[59] Jascha Sohl-Dickstein, Mayur Mudigonda, and Michael DeWeese. Hamiltonian Monte Carlo without detailed balance. In Eric P. Xing and Tony Jebara, editors, Proceedings of the 31st International Conference on Machine Learning. volume 32 of Proceedings of Ma- chine Learning Reseurch, pages 719-726, Bejing, China, 22-24 Jun 2014. PMLR. URL attps://proceeddngg: mlr. press//32s..3 hts//solllaas http://ool-diccsseeeeeee wial-ldcckteiii1ee html.
[60] Thomas Rippl, Axel Munk, and Anja Sturm. Limit laws of the empirical Wasserstein distance: Gaussian distributions. Journal of Multivariate Analysis, 151:90-109, 2016. ISSN 0047-259X. doi: scc/amcc/atttccep///////222....0
20w...0......6m...mmmm URL https: wttp://wciinnddreet..... com/
[61] William R Softky and Christof Koch. The highly irregular firing of cortical cells is inconsistent with temporal integration random epsps. Journal of Neuroscience, 13(1):334-350 1993. doi: 10. 10.15//NNEUROOCLL1--0000033
14

---

[62] Carl van Vreeswijk and Haim Sompolinsky. Chaotic balanced state in model of cortical circuits. Neural Computation, 10(6):1321-131, 1998. doi: 10.11220899766933011722
[63] Christopher De Sa, Chris Re, and Kunle Olukotun. Ensuring rapid mixing and low bias for asynchronous Gibbs sampling. In International Conference on Machine Learning, pages 1567-1576. PMLR, 2016. URL htt..httts://poceediiae
mlr www.:ppeesss////ee sa16. html.
[64] Alexander Terenin, Daniel Simpson, and David Draper. Asynchronous Gibbs sampling. In International Conference on Artificial Intelligence and Statistics, pages 144-154. PMLR, 2020. URL https:
htt.ittts:/prrceediigg http://procccee..cc/oe/
.ml.iinnnsmm...mmmmmmmm.m....m mlr. www..pressssv.... http////////eeeeeaaaaa.aa html.
[65] Nicolas Brunel. Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. Journal of Computational Neuroscience, 8(3):183-208, 2000. doi:
10.1023/a: 10te:/220000995000000
[66] Alfonso Renart, Nicolas Brunel, and Xiao-Jing Wang. Mcan-field theory of irregularly spiking neuronal populations and working memory recurrent cortical networks. Computational Neuroscience: A comprehensive approach, pages 431-490, 2004. doi: 10. 1211/9970000044944422 22.
[67] Alexander Lerehner, Cristina Ursta, John Hertz, Mandana Ahmadi, Pauline Ruffiot, and Søren Enemark. Response variability in balanced cortical networks. Neural Computation, 18(3): 634-659, 2006. doi: 10.1 10.166/09997666775993616
[68] Gareth O Roberts and Richard L Tweedie. Exponential convergence of Langevin distributions and their discrete approximations. Bernoulli, pages 341-363, 1996. doi: 10.2307/331 8418.
[69] Ruqi Zhang, A Feder Cooper, and Christopher De Sa. AMAGOLD: Amortized Metropolis adjustment for efficient stochastic gradient MCMC. In International Conference on Artificial Intelligence and Statistics, pages 2142-2152. PMLR, 2020. URL https::/ mlr. ppess/v108/ 2atp/////1//zaaa2eeeeeee html.
[70] Michele Nardin, James W Phillips, William F Podlaski, and Sander W Keemink. Nonlinear computations in spiking neural networks through multiplicative synapses. Peer Community Journal, 1, 2021. doi: 10.24072/pcjourmal.69
[71] Mengchen Zhu and Christopher Rozell. Modeling inhibitory interneurons in efficient sensory coding models. PLoS Computational Biology. 11(7)xe1004353, 2015. doi: 10.137 1/jourr nal.pebi. 1004353.
[72] Arish Alreja, Ilya Nemenman, and Christopher Rozell. Constrained brain volume in an efficient coding model explains the fraction of excitatory and inhibitory neurons in sensory cortices. PLOS Computational Biology. 18(1):e1009642, 2022. doi: 10. ho. 1009644.....
[73] Jose Del Castillo and Bernard Katz. Quantal components of the aud-plate potential. The Journal of Physiology. 124(3):560, 1954. doi: 10. 10.11 httpphysiol.1954.p0000
[74] Rob R de Ruyter van Steveninck, Geoffrey D Lewen, Steven P Strong, Roland Koberle, and William Bialek. Reproducibility and variability in neural spike trains. Science, 275(5307):
1 1805-1808, 1997. doi: 10.1 23f.science227.5337....
[75] Gianluigi Mongillo, Omri Barak, and Misha Tsodyks. Synaptic theory of working memory. Science, 31955566)11443115544 2008. doi: 10.1 1 16//science 150769.
[76] Dmitri A Rusakov, Leonid Savtehenko, and Peter Latham. Noisy synaptic conductance: Bug or feature? Trends in Neurosciences, 43(6):363-372, 2020. doi: 10.
[77] Elena Kreutzer, Walter Senn, and Mihai A Petrovici. Natural-gradient learning for spiking neurons. el.ife, 11:666526, 05 2022. ISSN 2050-084X. doi: 2f7.54/eL.fe.665266
[78] Kwabena Boahen, neuromorph's prospectus. Computing in Science & Engineering, 19(2): 14-28, 2017, doi: 10.1 10.1109/MCSE..01..3...
15

---

[79] Mike Davies, Narayan Srinivasa, Tsung-Han, Lin, Gautham Chinya, Yongqiang Cao, Sri Harsha Choday, Georgios Dimou, Prasad Joshi, Nabil Imam, Shweta Jain, al. Loihi: A neuro- morphic manycore processor with on-chip learning. IEFF Micro, 38(01):82 2018. doi:
10.1 10110MM.0008 12130359.
[80] Kaushik Roy, Akhilesh Jaiswal, and Priyadarshini Panda. Towards spike-based machine intelligence with neuromorphic computing. Nature, 575(7784):00--617, 2019. doi: https://oi..or//10.. 1ttp://..0088/415800111111111110
[81] Benjamin Cramer, Sebastian Billaudelle, Simeon Kanya, Aron Leibfried, Andreas Grübl, Vitali Karasenko, Christian Pehle, Korbinian Schreiber, Yannik Stradmann, Johannes Weis, et al. Surrogate gradients for analog neuromorphic computing. Proceedings of the National Academy doi: 10.10733pnas.2109111 21091991110199111199
of Sciences, 119(4). 2022.
[82] David Kappel, Stefan Habenschuss. Robert Legenstein, and Wolfgang Maass. Network plas- ticity as Bayesian inference. PLoS Computational Biology. 11(11)ee1004455, 2015. doi:
10. 10.1371/journal..pc pcbi.1004485.. 1004485.
[83] Laurence Aitchison, Jannes Jegminat, Jorge Aurelio Menendez, Jean-Pascal Pfister, Alexandre Pouget, and Peter E Latham. Synaptic plasticity as Bayesian inference. Nature Neuroscience, 24(4):565-571, 2021. doi: 10. 199--011-000000
[84] Paul Masset, Shanshan Qin, and Jacob Zavatone- Veth. Drifting neuronal representations: Bug or feature? Biological Cybernetics, pages 1-14, 2022. doi: 10. 10.1007//00422-021-00000
[85] Wenbo Gong, Yingzhen Li, and José Miguel Hernández-Lobato. Meta-learning for stochastic gradient MCMC. In 7th International Conference on Learning Representations, ICLR 2019, 2019. doi: 10.48550/arXiv. 180.00452.
[86] Jacob A. Zavatone-Vett Veth, Abdulkadir Canatar, Benjamin S. Ruben, and Cengiz Pehlevan. Asymptotics of representation learning in finite Bayesian neural networks. M. Ran- zato, A. Beygelzimer, Y. Dauphin, P.S. Liang, and J. Wortman Vaughan, editors, Ad- vances in Neural Information Processing Systems, volume 34, pages 14765-24777 Cur- Associates, Inc., 2021. URL https: ptsttps:/prroeeeeee /ttpc//eeccceeeeeeeeeeeeeee www.rpp..ccppppr////..... cc/paperr2001//aah) #fgd55544e144411111177444aa3a333 awt.reeccccae----ttaaaaaaa abstract.htmm
[87] Ralph Bourdoukan, David Barrett, Sophie Deneve, and Christian Machens. Learning op- timal spike-based representations. In F. Pereira, C.J. Burges, L. Bottou, and K.Q. Wein- berger, editors, Advances in Neural Information Processing Systems, volume 25. Curran Associates, Inc., 2012. URL https: https://proceedingg www..ps.cc/paper///0... ccccppppp/2202//ha// Ba1677/bberr33..3ffffffaaaaaa haw.aabbaaaaa--bbbaaraccac Abstract html.
[88] Alireza Alemi, Christian Machens, Sophie Denève, and Jean-Jacques Slotine. Learning nonlinear dynamies in efficient, balanced spiking networks using local plasticity rules. In Proceedings of the AAA! Conference on Artificial Intelligence, volume 32, 2018. doi: 10.1609/aaii v32il 13300.
[89] Wieland Brendel, Ralph Bourdoukan, Pietro Vertechi, Christian K Machens, and Sophie Denéve. Learning to represent signals spike by spike. PLoS Computational Biology, 16(3):c100792) 2020. doi: 10. 137 1/joum/l pcbi. 1007692.
[90] Dongsung Huh and Terrence J Sejnowski. Gradient descent for spiking neural networks. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Gar- nett, editors, Advances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018. URL https: http:://: httppp/ed///oooiiiigg.... nww.inuuun...e cttpripppccppppe//188////// h2ah//hhh//hh///////// Abstract....tt........... html.
[91] Emre O Neftci, Hesham Mostafa, and Friedemann Zenke. Surrogate gradient learning in spiking neural networks: Bringing the power of gradient-based optimization to spiking neural networks. IEEE Signal Processing Magazine, 36(6):51-6) 2019. doi: 10. 10.1009//..2099.999..1
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[92] Cengiz Pehlevan. A spiking neural network with local learning rules derived from non- negative similarity matching. In ICASSP 2019 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 7958-7962, 2019. doi: N0.1109/CASSP220199 8682999
[93] Nicolas Perez-Nieves, Vincent CH Leung, Pier Luigi Dragotti, and Dan FM Goodman. Neural heterogeneity promotes robust learning. Nature Communications, 12(1):1-9, 2021. doi: 10.108//41467--011-60223.3
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Reviews:
[{"review_id": "wPyhVXai08s", "reviewer": "Reviewer_TEk8", "summary": "New framework for constructing fast sampling dynamics in spiking neural networks, encompasses many of the previously proposed solutions as limit cases but also allows for a unified treatment and interesting variations.", "questions": "- decoding requires low pass filtering of spikes, which means that the filter width puts a hard limit on the autocorrelation function of the outputs, same as in distributed sampling a la savin et al. Why is then this faster? is it because the time constant is not tied to the membrane time constant the way it needs to be there?\n\n- explicit rejection step in MH seems to require fundamentally global knowledge which affects classic criteria for biological plausibility, is the discretization of time that assumes asynchronous updates making this local again?\n\n-  hard to intuitively get the source of the speedup, especially given the earlier proofs of hennequin that langevin is the slowest possible random walk with a given gaussian stationary distribution, how does the structure of the covariance play with the  sample autocorrelation function?\n\n- is there a way to make more formally precise statements about sampling speed in EBNs for the different variants or is this mainly relying on the ma work for generic speedup arguments?", "limitations": "no issues", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- clear biological and computational motivation \n- interesting knowledge transfer between machine learning and computational neuroscience\n- interesting mechanics: deterministic dynamics, stochastic spike generation process (usually stochastic dynamics, deterministic theshold)\n\nWeaknesses:\n- restricted to multivariate gaussian posteriors, although the time varying mean makes it somewhat more unusual/interesting as a setup\n- dependence of input (the actual inference part) missing in the construction (for simplicity i would agree this is fine for static posteriors but seems awkward once there is a time constant in the posterior changes themselves)\n- emphasis on the practically relevant scale of sampling being unexplored is an overstatement of fact, both the hennequin and savin type of dynamics can be used to extract moments of interest at time scale of hundreds of milliseconds (explicitly with spiking neurons at least for the second case), which many would argue is perceptually relevant enough, especially given the inherent tradeoff between precision and time for all sampling based codes; that is not to say that there is no room for alternative models of fast sampling but it's not nearly as bad as the introduction would make you believe\n- very precise/artificial architectural constraints on the solution (pair of readout pools tied weights etc)\n- in the neurally relevant regime the dynamics are not guaranteed to have the target posterior statistics. unclear why the series of approximatons should be expected to be constrained rather than lead to accumulation of errors and big deviations from the target\n\nMinor:\n- strictly speaking it is not the dimensionality of the latent space per se but rather posterior entropy that limits sampling speed, although admittedly the two tend to go together in simple cases\n- this is semantics i would say that typically the goal of bayesian perception is stated as computing a posterior over latent variables, not parameters\n- potentially relevant refs: radford neal tech report on speeding up sampling by dropping the detailed balance requirement, and the old lars buesing sampling paper on a biological realization of that idea (refractory period of sampling).\n", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "uP5wb03A8J", "reviewer": "Reviewer_jyfu", "summary": "The present paper studies sampling in spiking network models and tries to unify the Langevin sampling in the efficient balanced networks and Metropolis-Hastings sampling in networks with probabilistic spike rules. And it also studies how the geometric structure in the neural code speeds up the sampling.", "questions": "Please refer to my comments in the strengths and weaknesses.", "limitations": "The limitation of the current framework is the analog-digital conversion (DAC) in sampling a continuous distribution in discrete spiking networks, as mentioned in line 238. This makes people feel that the spiking network is not designed by implementing sampling although the error of DAC can be reduced by considering population geometry or in large number limit. Similar feelings were mentioned by the authors in the paper. I hope to see more discussion of alternative solutions to the DAC error. One possible solution is utilizing the math techniques in probabilistic population code to sample continuous distribution in discrete spikes, where the representation of a continuous distribution in spiking networks comes from the continuously smooth tuning curves of neurons. ", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The strength of this paper is its unifying nature. And its proposal of a probabilistic spike rule implementing Metropolis-Hastings sampling is novel. The present study did some theoretical derivations in linking the sampling dynamics with neural dynamics, but some of the presentations is not very clear.\n\n### Major\n\nI have a couple of major concerns about this present paper.\n-  A strong requirement of neural sampling is that the neural circuit dynamics with fixed parameters is able to sample posteriors with different uncertainties. It is not clear whether this can be achieved in the current framework. The proposed network model seems not able to achieve this if I understood correctly, in that the network parameters in Eq. 12 depend on the mean and the variance (explained by the text in line 160). This implies that to sample distribution with different parameters we need to adjust the network connections.\n\n\n- As mentioned in Eq. 14 that the D matrix only modifies the temporal dynamics but leaves the sampling distribution unchanged. That means sampling in the Euclidean space and the natural space (including the inverse Fisher information matrix, line 195) should not change the sampling distribution. However, from Fig. 2 and Fig. 3 the two sampling trajectories (blue vs. red) differ a lot and have different variances. I am wondering whether the sampling is correct or not.\n\n### Writing \nThe writing in sec 2.1 is not clear and I feel difficulty sometimes in following the flow. A lot of questions about the presentation remain.\n\n- It is not clear why the problem from discrete and sign-constrained spikes can be solved by imposing a fine-tuned balancing condition on the readout weights (lines 93-98). More explanations are needed. Also, it is not clear the motivation for choosing a neuron uniformly at random (line 104).\n\n- In Eq. 1, should I interpret the estimate $\\hat{\\theta}_t$ as a sample at time t? Is the uncertainty of the sampling distribution represented by the fluctuation of $\\hat{\\theta}_t$ over time? It is better to explicitly mention this in the beginning.\n\n- Eqs. 3 and 4: the argument in the distribution in the denominator of Eq. 4, i.e., $(1-\\eta)\\Gamma r_{t-1}$ is not consistent with $\\hat{\\theta}_{t-1}$ in the denominator of Eq. 3.\n\n- It is unclear how Eq. 6 and 10 are related to the undefined generative model which was mentioned in line 71. Is there an implicit assumption of the linear Gaussian generative model because the terms in Eq. 10 are the derivative of quadratic terms. Moreover, it is not clear how \\psi is defined at this moment until later I read the text in line 160. Also, in Eq. 6 the input to the neural dynamics is not the observation x defined in the generative model (line 71), but the latent variable $\\theta$ directly. More explanations are needed here.\n\n- Line 195: it is not clear what the matrix G is until I read line 232. Please define symbols when they first appear.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ["I don’t know"]}, "extra_text": {}}, {"review_id": "BlvE5lszsL", "reviewer": "Reviewer_Pjbs", "summary": "In this paper, the authors:\n- construct a spiking neural network that performs Metropolis-Hastings sampling of a target posterior distribution (for, e.g., estimating external features given noisy sensory input), and extend it to continuous time dynamics\n- demonstrate its relationship to efficient balanced networks\n- show how the network can be augmented (by imposing population geometry) to implement the \"generalized\" stochastic gradient MCMC\n- and demonstrate that geometry-aware networks are superior to geometry-naive networks, especially with increasing correlation and posterior dimension\n", "questions": "I'm somewhat confused about the central point of the paper, that population geometry improves sampling efficiency. If I understand correctly, population geometry, in the most important sense, means the way in which the sampling space can be projected (e.g., to the principal components, or some other non-Euclidean manifold) to essentially decorrelate the dimensions of the posterior (in the case of the correlated Gaussian). \n\nIf this is correct, then I would suggest for the authors to more clearly and explicitly mention this much earlier in the paper (apologies if I I had missed it). Figure 1C is helpful in this regard, but I'm still uncertain if I really got what \"population geometry\" means, since typically, in the neuroscience context, population geometry refers to a e.g., a lower dimensional manifold on which population activity resides, and I'm not sure if this idea is invoked at all in this paper.\n\nFurthermore, and this is probably due to my naivety still, I'm a bit confused as to how the network should have access to the natural geometry of the problem (in the form of the projection B, or D)? Similarly, it's a bit unclear how the weights of the network should be learned, and whether it is dependent on the problem. From the perspective of a naive reader unfamiliar with this subfield of comp neuro, it would be very helpful if the authors took a more descriptive approach to provide intuition for such basic high-level ideas.\n\nOverall, I believe my lack of expertise hinders my ability to understand this paper's contribution. However, I think I am not alone in that, unless one works exactly in the field of spiking network samplers, it is a bit difficult to understand the paper even at a high level.", "limitations": "Extensive discussion of the paper's limitations and assumptions, and no discussion of negative societal impact (though I don't foresee any immediate impact).", "rating": 6, "confidence": 1, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "I'm quite naive to the field of spiking network as samplers, so I cannot really evaluate the originality, significance, or the technical quality of the work. Broadly speaking, I believe the work is of interest to the computational neuroscience community, since \"the Bayesian brain\" is an active field of research, and therefore, a central question is how a biologically realistic network (i.e., spiking network) could implement such sampling dynamics. In addition, at a superficial level, the work appears to be carefully done and cites relevant literature, as well as carefully discussing its relations to previous work, and its limitations. Based on the figures, it seems that the central claim of the paper is supported, that access to the natural geometry of the sampling space enables more efficient and accurate sampling.\n\nThere are several assumptions that seem biologically implausible to me, many of which the authors touch on in the discussion, and I understand certain assumptions are required for such theoretical work and cannot comment on whether they are standard for the field. However, on the issue of long membrane time constant (nu<<1), it would be nice to see if/when the results break down, with a shorter integration window. Similarly, the fact that only one neuron is allowed to spike at a time significantly dampens my enthusiasm for the work, since at some point the actual model become so far away from the (very nicely stated) high level motivation.\n\nThere are some low-level editing mistakes that can improve the quality of the work, e.g., Figure 4 panels a and b are labeled as b and c.", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "7J0U3mg6Zbq", "reviewer": "Reviewer_96zE", "summary": "In this work, the authors demonstrate the possibility of implementing sampling algorithms through spiking neural networks. They provide details on how to realize the metropolis-hastings sampler using the probabilistic spike rules. Overall the work is interesting and extends the potential application of spiking neural networks. ", "questions": "1. What's the connection between the brain's capacity for probabilistic computation and the ability of SNN for sampling? What's the evidence for the brain to realize its probabilistic computation through the same sampling approach as we do in the Bayesian inference?\n\n2. The implementation needs to accept the spikes probabilistically. How to realize this mechanism in a spiking neural network? \n\n3. Why is it called fast sampling? How is it faster when compared to other methods? \n\n4. What's the infrastructure for the SNN used for sampling? \n\n5. If the probabilistic spiking mechanism is allowed, why do we need such a spiking neural network rather than using a regular artificial neural network for the computation in the sampling procedure? ", "limitations": "NA. ", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. The application is relatively new for SNN. \n2. The writing is clear. \nWeakness:\n1. The motivation for applying SNN for sampling is not convincing enough. \n2. The comparison to traditional sampling is not enough. ", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept