Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: In this work, the authors:
- review Bayesian inference for linear gaussian models, as well as HMC
- derive how the equations of a CANN with noisy adaptation is identical to that of HMC, with the appropriate substitutions, and how parameter regimes translates to sampling behavior
- demonstrate how CANN with noisy adaptation converges to the true posterior (in mean, variance, and DKL) more quickly than without, in single and coupled networks
strengths_and_weaknesses: originality: spiking networks have been shown to approximate HMC sampling (in ref 16 & 18). The current authors extend this investigation  to rate-based bump attractor networks, which is commonly used in computational neuroscience. I don't know of other works on this particular topic of adaptation, though I'm not well read in this literature (but see https://www.biorxiv.org/content/10.1101/2021.03.15.434192v2)

quality: at a superficial level and without re-deriving the proofs, I believe the work is of high quality, and that the results fully support the central claim of efficiency (but not necessarily the claim of biologically relevant speed, expanded below)

clarity: overall, I find this paper to be of high quality, and very clearly presented, even for a relative outsider of the literature. The brief but concise review of Bayesian inference, HMC and CANNs touched on all the relevant background without unnecessary length, and I commend the authors for writing such an enjoyable paper and without assuming basic knowledge from the reader, since I learned much from reading it. The figures are also well-presented and with readable font sizes.

significance: I believe this paper links two very interesting ideas: adaptation and neural implementation of probabilistic sampling, and will be of significance for a segment of the computational neuroscience community
questions: - the work is motivated by the claim that naive Langevin sampling cannot work at perceptually relevant speeds due to its slow dynamics around the posterior mode. This is not strictly true, since speed can come at a cost of certainty, and it's unclear if FLD performs unreasonably (and unrealistically) poorly compared to human performance. While it's convincingly shown that higher adaptation (up to the derived max speed) increases sampling efficiency, I'm curious how the performance of both algorithms are when measured against absolute time, instead of in units of synaptic time constant tau_s (e.g., in Figure 2B). In other words, if one plugs in a reasonable value for tau_s (e.g., 5-10ms for AMPA receptors), would HDF actually achieve reasonable performance in reasonable time?

- a critical component of the model seems to be the stochasticity in the adaptation dynamics (Eq13). To my limited knowledge, this is not a common assumption for CANNs, and is typically not included for adaptation dynamics in other models (such as spiking neuron). I would ask the authors to provide some evidence for why this assumption is justified (other than the currently cited "biological systems is noisy"). While SI3 provides some evidence that small deviations from the exact value of sigma_V is fine, perhaps a broader treatment is necessary, since one typically assumes the cell has near-perfect access to its own spike history (through calcium dynamics, etc)

- my only concern wrt to originality / significance is, how different is this work from ref 16 & 18? In other words, does CANN need an entirely different treatment than from spiking networks, which is arguably a more difficult problem? This question stems from my naivety, but it would be good if the authors further discuss why CANN should be significantly different, perhaps from the perspective of adaptation (or something else)
limitations: very brief discussion of limitation is included, and while the robustness of model parameters (in relation to deriving an exact match to HMC) is discussed in the SI, it's worth mentioning again in the limitations, as well as other strong assumptions of the model. No discussion of societal impact (and none from my perspective)
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: In this paper, the authors address the question of how sampling-based probabilistic inference could be implemented in the brain. An important concern about sampling-based theories is that the sampling speed is too slow to be compatible with brain function. 

One potential approach that circumvents the speed issue is Hamiltonian Monte Carlo (HMC) sampling. Previous works have shown that HMC can be implemented by a biologically plausible neural network with balanced excitation-inhibition among neurons. 
Here, they present an alternate neural implementation of HMC using a canonical network model called continuous attractor neural networks (CANNs). 
 
Using a linear Gaussian generative model, the authors analytically show how the dynamics of a CANN are equivalent to HMC. Further, they show that adding a noisy adaptation term in the CANN dynamics accelerates the sampling process. They also show how this model could be extended to sampling from a multivariate Gaussian posterior. Finally, they present simulation results that confirm their theoretical analyses. 
strengths_and_weaknesses: This work demonstrates how continuous attractor neural networks (CANNs) could implement Hamiltonian Monte Carlo (HMC) sampling. The authors show that (i) when the dynamics of the adaptation current in CANNs take the form of spike frequency adaptation, and (ii) CANN dynamics are projected onto the first two dominating motion modes, the resulting dynamics of the latent features are equivalent to HMC.
In figure 2 they also clearly show how varying the adaption strength results in different sampling regimes. In particular, the simulation results show how the sampling is sped up as the adaptation strength increases in the HMC regime. Overall, this is an interesting implementation of HMC with friction using CANNs. 

As the authors mention in the discussion, the results presented here are for a linear Gaussian model. Real-world distributions are more complicated and often multimodal. It would be interesting to consider how the current approach could be extended to perform inference on more complicated distributions. 
It would also be useful to provide a sense of how the speedup obtained here compares with inference timescales observed in the brain. 

The paper is quite clear and the figures are very neat and well-annotated. 
However, there are a few minor typos and grammatical errors. 
questions: - There are certain assumptions placed on the form of the neuronal connection strengths, firing rate, adaptation, and external input for the CANN dynamics to be equivalent to HMC sampling for a linear Gaussian model. Are all of these biologically plausible? How would the dynamics change if any of these assumptions are relaxed? 
- Are there any thoughts about how this approach could be extended to multimodal, non-Gaussian distributions? 
- The speed-up observed here is relative to first-order Langevin dynamics. Could you provide a sense of how this compares with inference time scales observed in neuroscience experiments?
- Clarifying question: in eq 2, where the prior is assumed to be uniform, is the support of $s$ finite? 
- Are there any concrete experimental predictions that could arise from this work? 

limitations: There is no potential negative societal impact - the authors state that their study is fundamental in neuroscience and not tied to applications. 

The authors also discuss the limitations of their work - the analytical results derived here and the simulations correspond to a linear Gaussian generative model. It would be interesting to see how their work could be extended to performing inference on more complicated, multimodal distributions.
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 manuscript the authors describe how adaptation can speed up posterior sampling in continuous attractor neural networks, a neural population model known to be capable of posterior sampling. Concretely, they show that the dynamics of networks with adaptation is equivalent to Hamiltonian dynamics with friction. This equivalence gives a theoretical justification why the methods with adaptation perform better than the ones without adaptation, which effectively implement (first order) Langevin dynamics. Furthermore, their theoretical analysis allows the authors to compute good parameter settings to sample from simple one and two dimensional Gaussians, which are confirmed in simulations.
strengths_and_weaknesses: 
The observation that an adaptation mechanism that pushes the sampling chain away from previously visited states can improve convergence speed makes intuitive sense and the derivations in the paper seem correct. Also, the formal connection to hamiltonian sampling with friction are interesting. However, parts of the theory and all of the evaluation are specific to Gaussian probability distributions, for which sampling is not strictly necessary. Thus, this paper is an interesting step towards understanding the dynamics of sampling networks, but not a big break through.

Concrete points:
1) I am generally worried that all theory here is implemented only for Gaussians and some of the parameters like the optimal and threshold adaptation strength depend on the input distribution. Computing with Gaussians really does not require sampling or a whole population of neurons arranged along the feature space. To make sampling sensible at least arbitrary distributions should be covered I think.
2) The setup for the CANNs discussed here seems a bit odd, because the input current in itself already a representation of the distribution, which allows computation as discussed in the probabilistic population coding framework. It is unclear to me why one would want to sample from a distribution given in this format.
3) Without a deep dive into the CANN literature, some of the derivations are extremely hard to follow. In particular, which properties are first derived here, which ones are taken from which older literature and under which assumptions these results hold is impossible to take from the manuscript alone. Here, the authors could definitely communicate more clearly.


Small points:
In (14) the standard deviation in the exponential is a, but should be Lambda.

questions: If the authors can show robustness in some way, i.e. that this works for other distributions, this would go a long way towards convincing me that their sampling method could work for other distributions where sampling is necessary.
limitations: ---
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: The paper shows that a particular model of continuous attractors very relevant in neuroscience performs HMC sampling.
strengths_and_weaknesses: Strengths:
* It is actually surprisingly hard to make rigorous connections between HMC sampling and real neural circuits.  In the Aitchison and Lengyel, they were for instance forced to use linear networks, without the spiking nonlinearity.  But they managed to make such a connection here!  The key innovation here was to consider HMC sampling over higher-level variables such as the location of the bump of activity, rather than considering e.g. firing rates themselves as the variables that HMC samples.

The key weakness is the need for clarification of the actual derivation of the equivalence, which is currently quite unclear.  Here are some specific issues (but my comments are not limited to these issues):
* Eq. 15: no x on RHS. 
* Eq. 15/16 G() vs G[] (i.e. sometimes G has curly, sometimes square brackets)
* The projection seems like the key step.  Can we give some flavour of how that works in the main text, and whether any assumptions go in there (e.g. linearity).
* It would be worth exploring the biology.  Is there really this kind of adaptation present in e.g. head-direction cells?  That would seem strange, as it would destabilise the bump.  This mechanism would be a potential explanation.
questions: N/A
limitations: It isn't clear how far the basic ideas here will extend (e.g. to learned representations, visual system etc.)  But it is definitely an approach worth considering in future.
review_title: null

Meta-review: metareview: The reviewers agree that the paper makes an interesting contribution, connecting inference in probabilistic models with network models from computation neuroscience.
confidence: Certain
Decision: Accept