Venue: NeurIPS 2022 Conference

Reviews:

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:   The authors of the current submission set out to provide a biologically inspired "neural field" (herein a PDE model) model of the experimental data reported in Wang et al. (2020).
  The proposed model, unlike those previously proposed, accounts for "phase procession" (cf. phase precession), as reported in Wang et al.
  To incorporate the above phenomenon, the usual PDE equations are combined with "feedback inhibition".
  The resulting system is shown to undergo a Hopf bifurcation giving rise to "oscillatory tracking", which is proposed as an explanation for the coexistence of phase precession and procession.
  Additionally, neurons with "unimodal" and "bimodal" tuning functions (as measured wrt to the theta oscillation) are shown by varying model parameters thus giving a unified description of the two.
strengths_and_weaknesses: Admittedly I was not familiar with Wang et al. (2020), which attests to my limited ability to judge the significance of this submission.
That being said, I have since familiarized myself with the works citing Wang et al and I believe the contribution offered by the current submission to be novel.
Overall, I found the writing and derivations to be very clear. Passages and figures requiring further are mentioned in the `Questions` field below.

In sum, I believe this to be a technically sound, interesting contribution.
Nevertheless, I hesitate to enthusiastically recommend it for acceptance for the following reason:

1. As stated I cannot accurately gauge the significance of this submission, especially to the NeurIPS community. 
2. Because the submission is devoted to deriving a particular model of place cell function (ie. without relying on statistical or machine learning methods, nor without any immediate implications for those communities), it might be an equally good if not better fit in specialized journals like PlOS CB, Neural Computation, eLife etc. 
Naturally, this is no grounds for dismissal but I would like to allow fellow reviewers and the authors to discuss whether NeurIPS is the appropriate venue for this work.
questions: Regarding clarity:
1. Having read Wang et al. I found that their subfigure 1A (which is also reproduced in the appendix of the current submission Figure SI 1 ) to be a clearer explanation of phase precession vs procession. I believe a schematic in the vein of SI 1B would make for an easier explanation than the current Fig. 1B from the main text.

2. There is little information as to what the heatmap in Fig 5C represents, and a fortiori how this normalized histogram was computed.

Terminology:
The authors say the model is a "continuous attractor neural network". Please correct me if I am wrong, but the incorporation of feedback inhibition intentionally de-stabilizes the bump attractor; it appears that feedback inhibition was *specifically* added to allow for a Hopf bifurcation.
If I understood it correctly, I believe this ought to be clarified that though derived from a CANN, the model is not a CANN.

Misc:
$I^{ext}$  is modeled assumed to be linear function of time. I would find it interesting to compare numerically how the proposed model deals with stochastic $I^{ext}$ in comparison with previously proposed models.
Neutrally stable dynamics in conjunction with a white noise drive ought to give rise to a diffusing receptive field. I believe the proposed model will not be susceptible to such behavior.

limitations: Not applicable
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 authors propose an attractor network model which includes a “feedback inhibition” term that jointly exhibits phase precession and procession in hippocampal place cells.  The model is defined via standard neural network rate equations in one-dimension using recurrent connections with a Gaussian profile.  Feedback inhibition to a neuron encoding location x is modeled as a dynamic term that depends only on the activity of that neuron (it is a low pass filter of the synaptic input to neurons encoding location x).  This is described qualitatively as the “overall effect of the interplay between the hippocampus and the medial septum”.  The consequence second order dynamics gives rise to traveling wave solutions.  The competing effects of the external input and intrinsic traveling wave dynamics induces oscillatory behavior in a region of phase space that accounts for the jointly observed precession and procession.  The authors also show how this model can be tuned to account for both “unimodal” and “bimodal” cells.  
strengths_and_weaknesses: This papers presents a simple and elegant model that incorporates multiple observations into a single framework (unimodal and bimodal cells).  It is clearly presented and the figures illustrate well all main conceptual points (the example movies in the supplementary material are also excellent and helpful).   The structure of the model is sufficiently detailed with all important steps of the construction and derivation of results presented.  The paper appears to exhibit overall high quality and clarity (though there are a few places where writing can be cleaned up and polished).

The results also appear original (to me, though I am less familiar with this section of the literature on hippocampal place cells than I used to be) and significant (in that they demonstrate a model that jointly exhibits phase precession and procession - again, to me).  

 The primary weaknesses I see are addressed in “Questions” below.


questions: I have a couple of presentation related points and one main issue regarding the formulation of the model.  

Regarding the model:  Using a term like equation 3 brushes a lot under the rug in terms of how the system itself works.  The second order dynamics that arises seems to exhibit the desired effect, and it is great to have identified a dynamical system that shows the appropriate empirical effects, but this formulation leaves as a mystery exactly how the “interplay between the hippocampus and medial septum” generates this effective inhibition term.   As it is, the term essentially just introduces the simplest second order temporal dynamics into equation (1) that (along with the Gaussian profile weights) turns it into an effective wave equation.  This is all great in terms of identifying the possible mechanism, but it would be nice to have some idea of what model of such interactions would give rise to such a term.  (Note that as I write this, this criticism sounds more severe than I intend it.)

Regarding presentation:  I think lines 217-231 could be greatly improved so the reader has to do less work.  In particular, it could be more immediately clear how h(t) relates to the position relative to the probe neuron (you erase this by defining h(t) as an absolute value).  It was easier for me to think in terms of the actual distance from the probe, personally.  Putting the sign back and considering the separate cases was an easier way to think about it for me.

Finally, the language around “forward/reverse theta sequences” and “forward and backward” sweeps of decoded position is likely very confusing for readers who are not sufficiently familiar with the place cell literature.  (There is some probability that they have confused this reviewer and I have fundamentally misunderstood the paper.).
limitations: I agree with the authors that there are negligible societal impacts of this work.
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: This paper proposed a CANN model with feedback inhibition to elucidate the mechanism underlying phase precession and procession of the firing of hippocampal place cells. Compared to standard CANN models, a feedback inhibition mechanism is used to perturb the steady state of a standard CANN, so that the activity bump oscillates around the true location of a rat. This paper derived analytic formulas of the CANN under different input and feedback conditions. This paper derived the expression of the trajectory of the oscillatory state. This paper simulated the proposed model and observed that the activity bump sweeps back and forth across the true location, similar to what observed in previous experiments. The firing rate dynamics of the model show behaviors similar to phase precession and procession.
strengths_and_weaknesses: Strengths:
+ The presentation and writing of this paper is good.
+ The mechanism underlying phase precession and procession is still unknown. It is meaningful to propose computational models to understand the mechanism.
+ The theoretical results are sound and comprehensively show the steady state behaviors of the proposed model.
+ Both analysis and simulations show that the proposed model provides a reasonable and relatively abstract explanation of both the phase precession and procession phenomena.

Weaknesses:
- The CANN model with feedback inhibition is itself an existing model of oscillatory behaviors of activity bumps. The main novelty is the analysis and applications of such models to understand the phase precession and procession.
- The animal movement model is too simple (movement along a line) to account for many more realistic movement scenarios.
- The synaptic strength distribution of the model is assumed to be an idealized form without considering learning effects.
- Phase precession and procession are defined in terms of spiking times during consecutive  theta cycles. The previous model does not simulate the generation of individual spikes.

Update after rebuttal:
Thank you to authors for their responses. From their responses, the major difficulty of generalizing the current framework lies in the lack of experimental records. Regardless, it would be helpful to generalize their framework or experimental records to 2D or learning scenarios. Such generalization will increase the impact of this paper and might be very helpful to machine learning community.

questions: 1. 1D model can only explain the phase precession and procession of a rat running on a linear track. How the proposed model can be generalized to 2D cases where a rat can run in an open field? 
2. The synaptic strength of the CANN is preconfigured through an ideal Gaussian form. In reality, such synaptic strengths might fluctuate continuously due to plasticity. Does the main results still hold if synaptic strengths become somewhat not ideal? How the synaptic strengths are learned through experiences?
3. Is there a missing link between the firing rate model of a CANN and precise spike firing times observed during phase precession and procession?
limitations: Yes.
review_title: null

Meta-review: metareview: This paper presents non-trivial and novel theoretical and computational modeling that accounts for experimentally observed phenomena: the theta phase procession and precession. These phenomena are implicated in the neural representation and learning of neuronal networks involving hippocampus. Although the current manuscript does not address the learning and the presentations are limited to a linear track environment, it represents a clear advance in linking spatial and temporal information representations by extending the standard continuous attractor models that do not exhibit phase procession/precession. Furthermore, the model elegantly explains the biologically observed unimodal and bimodal cells. The authors are encouraged to improve the clarity of some parts of the writing.

confidence: Less certain
Decision: Accept