Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: In this paper, the authors:
- 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
- demonstrate its relationship to efficient balanced networks
- show how the network can be augmented (by imposing population geometry) to implement the "generalized" stochastic gradient MCMC
- and demonstrate that geometry-aware networks are superior to geometry-naive networks, especially with increasing correlation and posterior dimension

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.

There 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.

There 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.
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). 

If 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.

Furthermore, 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.

Overall, 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).
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: 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.
strengths_and_weaknesses: Strengths:
- clear biological and computational motivation 
- interesting knowledge transfer between machine learning and computational neuroscience
- interesting mechanics: deterministic dynamics, stochastic spike generation process (usually stochastic dynamics, deterministic theshold)

Weaknesses:
- restricted to multivariate gaussian posteriors, although the time varying mean makes it somewhat more unusual/interesting as a setup
- 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)
- 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
- very precise/artificial architectural constraints on the solution (pair of readout pools tied weights etc)
- 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

Minor:
- 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
- this is semantics i would say that typically the goal of bayesian perception is stated as computing a posterior over latent variables, not parameters
- 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).

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?

- 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?

-  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?

- 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
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: 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.
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.

### Major

I have a couple of major concerns about this present paper.
-  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.


- 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.

### Writing 
The 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.

- 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).

- 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.

- 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.

- 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.

- Line 195: it is not clear what the matrix G is until I read line 232. Please define symbols when they first appear.

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.
review_title: null

final_score: rating: 5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.
review_content: 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. 
strengths_and_weaknesses: Strengths:
1. The application is relatively new for SNN. 
2. The writing is clear. 
Weakness:
1. The motivation for applying SNN for sampling is not convincing enough. 
2. The comparison to traditional sampling is not enough. 
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?

2. The implementation needs to accept the spikes probabilistically. How to realize this mechanism in a spiking neural network? 

3. Why is it called fast sampling? How is it faster when compared to other methods? 

4. What's the infrastructure for the SNN used for sampling? 

5. 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.
review_title: null

Meta-review: metareview: Although some reviewers have reservations about strong modelling assumptions, the main contribution of the paper is clearly presented and technically sound.
confidence: Less certain
Decision: Accept