Venue: NeurIPS 2021 Conference

Reviews:

final_score: rating: 7: Good paper, accept
review_content: summary: The submission suggests a neural mechanism underlying Levy flights in continuous attractor neural networks (CANN). More specifically, the author(s) proposes that superdiffusion (as observed in free memory retrieval and foraging behavior) occurs in CANN with noisy neural adaptation. External input noise together with adaptation makes the CANN switch between two regimes, local random walks and long-jump motion, separated in the CANN by the traveling wave boundary. Together, these regimes generate superdiffusion. An analytical treatment in reduced dimensionality by a Fokker-Planck approach reveals the two control parameters controlling the (super)diffusion in the model. Implications for free memory retrieval are discussed in a toy model, and as expected, superdiffusion consistent with experimental evidence is observed.

In conclusion, the study proposes an exciting new link to generate in a widely used neural network architecture (CANN) experimentally observed Levy flights using a neuronally plausible mechanism and studies it analytically.
main_review: #postrebuttal:
All major points were addressed.

Originality:
The submission proposes a simple CANN model that is being reduced to two dynamic variables. I am not an expert in the field, but I have not seen that model before; it combines two common 'ingredients' (adaptation and continuous attractor models) in a mathematically tractable model.
(Here seems a related work, but I don't have access to it: https://ses.library.usyd.edu.au/handle/2123/18808
The abstract says: "we develop a new type of two-dimensional neural field model that incorporates refractoriness as nonlinear negative feedback. We construct explicit bump solutions and perform a linear stability analysis, which reveals the emergence of stable bump activity as well as a critical transition from the bump state into propagating waves. Using numerical simulation, we show that the neural field exhibits local propagating patterns with rich dynamics including periodic, rotating and chaotic dynamics. We then show that propagating local patterns undergoing Levy flight emerge from a realistic cortical circuit model and that they can account for a wide range of neural response properties during both spontaneous and stimulus-driven activities."

Even if this work would be similar, the abstract doesn't say anything about an analytical treatment, so I would consider the presented manuscript still to be sufficiently original. Still, the work should be cited, and similarities and differences should be explained.
While previous work on Levy flights and Levy walks (and anomalous diffusion in general) is cited in the discussion, it would help a lot if the mathematical differences would be explained (e.i. why is it not necessary to use a fractional Fokker-Planck equation and fractional calculus here). 

Quality:
The model has the appeal of simplicity, so it can be (in a reduced version) be analytically treated using a Fokker-Planck approach, yielding the stationary distributions of s (the separation between bump U and adaptation current V) assuming separation of time scales. Based on that, the phase diagram of 3 dynamics regimes (traveling waves, Brownian motion, and Levy flights) is constructed, and the Levy exponent of the bump movement is determined, which depends on two parameters, 'distance to boundary \mu and noise-to-strength ratio \gamma. Their effect is then compared both analytically and numerically, and numerics and analytical calculations approximately agree.

Major points:
* What is not so clear to me is to what extent the described model actually exhibits a Levy flight. In my (admittedly non-expert) understanding, in Levy flights, increments are drawn from a Levy distribution which asymptotically follows a power law. In the presented manuscript, I don't see any direct evidence that the increments follow a Levy distribution and/or power law. Figure 1B shows an example, but the y-axis is not log-scales, so it is now clear if the decay follows a power law. Moreover, based on the analytical derivation, it is also not fully transparent where such a power law would come from. In the supplement, where the power-law is derived, besides the assumption of a separation of timescales, the assumption is made that "gamma an infinitesimal" (line 121), which, I guess, should read "gamma is infinitesimal". But in the rest of the work, gamma is not infinitesimal -this might be a misunderstanding on my side, clarification would be desirable. I also ran the Matlab code in the supplementary material, but it didn't show how the Levy exponent \alpha reported in the figures was estimated. Also for this, clarification would be desirable. 
* Potentially related to the first point, the main model ( 
* There is no code provided to reproduce the results on free memory retrieval (Figure 4), which seem to be purely numerically in nature.
* An analysis of how the core characteristics of the network (e.g., the Levy exponent \alpha) depend on the model parameters is missing. Do you expect the same behavior as in the model corresponding to figure 3? Based on that, can you predict how the empirically observed \alpha =0.73 would change when you change your model parameters, e.g., input noise \sigma_U, time constants \tau and \tau_V, inhibitory strength k, and Recurrent connection strength J_0? Is it important to have a variance of the SFA strength (\sigma_m) to make the model work? If so, why? 
* In the analysis of the Hopfield-like network, it is not clear how \alpha is measured. The manuscript text says: "Further analysis shows that long retrieval intervals are interrupted by bursts of very short intervals and the length of long intervals increases exponentially over time (Fig. 4E). If the length of long intervals increases over time, does it mean the distribution of intervals changes over time, thus also \alpha changes over time? Does it mean there is an aging effect? How is this consistent with the Fokker-Planck analysis, where a stationary distribution of s and z is assumed? 


Minor points:
One minor comment on the comparison of analytical and numerical results in Figures 3D and 3F: Please clarify, what the error bars indicate. Are these standard errors of the mean or standard deviation or inter-quartile rate? Also, what ensemble are these error bars calculated over?
* Figure 3D and 3F report numerical results in orange, it is enough to should the orange data points, the lines between the orange points might create the illusion that there is data, but as I understand, there is none. This might also help to make the blue (analytical results) visible.
I'd be delighted to be convinced that the limitations mentioned above are just resulting from my lack of understanding and are not intrinsic to the work. 

* It would also be interesting to look at the search efficiency as a function of noise-to-strength \gamma.
* It would demonstrate that as claimed with a larger grid (when the distribution is less truncated), the numerical simulations approach the theoretical predictions.
* Could the spontaneous activity in the Hopfield-like attractor network also be generated from spontaneous recurrent activity instead of external input drive (e.g., chaos as in asymmetric Hopfield-type networks)?
* Figure 3A and Supplementary figure 2C seem to be smoothed, it would be desirable just to display the raw numerical results without any smoothing that might create the illusion of intermediate values where there are none.

Some typos:
* The label in 3E reads "effeciency" but should read "efficiency".
* "Noises in neural adaptation causes" should read "Noise in neural adaptation causes"
* "also termed as anomalous" should read "also termed anomalous"
* "agree with our theoretical analyses very well" should read "agree with our theoretical analysis very well"
* "The neuronal recurrent connections" should read "The recurrent neuronal connections"
* "represent the coordinates of s on the the axes" should read "represent the coordinates of s on the axes"
limitations_and_societal_impact: There is no societal impact to discuss.
Limitations:
The authors discuss an extension to STD (short-term depression), but a discussion of the limitations is largely missing.
Here are some potential limitations:
* The derivation in the supplementary material (4) seems to be rather heuristic:
- e.g., the variance of Levy flights is generally diverging; this seems not to be considered here (also not in the supplement).
- the origin of the power-law (scale-invariance) is unclear; I would assume that the traveling waves have a typical scale. 
- the switching statistics between traveling wave and Brownian motion are not characterized; it is unclear why the length by which the bump travels should be long-tailed.
- a formal Kramers-Moyal expansion to derive the Fokker-Planck equation is missing.
* If the authors claim their proposed mechanism to generate Levy flights is beyond the mathematical physics literature of anomalous diffusion, it should be made clear what is different from known mechanisms to generate Levy flights (e.g., fractional Fokker-Planck equation). In case the mechanism proposed here can actually be mapped back to a previously described mechanism (which would be totally fine), this should be clearly stated.
* An analysis of the dependence of the Levy-flight phenomenon on the parameters of the Hopfield type model (figure 4) is missing.
* A clear explanation of the link between the model described in figure 3 and the Hopfield-type model (figure 4) is missing. Specifically, while in the CANN, a stationary state was assumed, the Hopfield-type model (figure 4) seems not to be characterized in a stationary state (see above).
I'd be delighted to be convinced that the limitations mentioned above are just resulting from my lack of understanding and are not intrinsic to the work.
review_title: null

final_score: rating: 6: Marginally above the acceptance threshold
review_content: summary: The paper discusses the appearance of Levy flights in animal behaviour, and attempts to justify the presence of a "neural mechanism" for this behaviour. The authors begin with a continuous attractor neural network (CANN) model: a linear stochastic partial differential equation involving an "adaptive current" term V, here called "spike frequency adaptation" (SFA). This SFA is the noise adaptation component mentioned in the title, and naturally involves another white noise term multiplied by the activity level U. The authors then approximate the dynamics by condensing the SPDE into a system of SDEs through projection onto two modes. One of the resulting SDEs also contains a multiplicative noise term. This results in a stationary distribution with an asymptotic power law. The appearance of this power law in the motion of these modes is claimed to justify the presence of Levy flights in "neural circuits", and hence in the brain, supported experimentally by a Hebbian-learning attractor network. 
main_review: I do not believe this is the correct venue for this paper. The problems and motivations appear highly scientific, rooted in applied mathematics and neuroscience, and not related to computer science (or, indeed, to neural networks, despite the name of the model used).

As a mathematician, I recognize that I am not the target audience for this work. From my point of view, when stripped back to its most basic elements, the authors appear to have rediscovered the relationship between multiplicative noise and heavy tails in a particular SPDE model --- see, for example, [1]. Some discussion of this general phenomenon is probably warranted to put the results in context. There is an active area of research surrounding the consequences of this property in ML, including [2,3]. However, I do not see how this paper would be of interest to the ML/statistics community, and think it is better submitted to a neuroscience outlet. 

Nevertheless, the paper appears grammatically sound. A few minor issues: 
- The CANN SPDE is not ill-posed per se, but Ito/Stratonovich properties of the white noise are not specified
- The intuition and discussion surrounding Levy flights as combining Brownian motion with long jumps is incorrect. Levy flights generally do not exhibit properties analogous to Brownian motion (unless alpha=2). An alpha-stable Levy process is one example of a Levy flight and is purely discontinuous with no Brownian motion part. - "Fokker-Plank" should be written "Fokker-Planck"
- Figure 1 doesn't seem like an especially fair comparison. The variance of the Brownian motion should be adjusted to give a more appropriate match between the Levy flight and the Brownian motion. 
- In Figure 4, estimating power laws using curve-fitting to a log-log plot should be avoided, as this approach incurs tremendous bias. An MLE-based approach should be considered instead.


[1] Biró, T. S., & Jakovác, A. (2005). Power-law tails from multiplicative noise. Physical review letters, 94(13), 132302.

[2] Gurbuzbalaban, M., Simsekli, U., & Zhu, L. (2021). The heavy-tail phenomenon in SGD. In International Conference on Machine Learning (pp. 3964-3975). PMLR.

[3] Hodgkinson, L., & Mahoney, M. (2021, July). Multiplicative noise and heavy tails in stochastic optimization. In International Conference on Machine Learning (pp. 4262-4274). PMLR.
limitations_and_societal_impact: The conclusions are not justified up to the standards of mathematical or statistical rigor expected at this venue.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: Levy flights are a class of random walks that are efficient search strategies, and human cognitive functions have been shown to exhibit Levy flight characteristics. This work demonstrates that an adaptation of continuous attractor neural networks (CANNs) can generate Levy flight dynamics. Essentially, by setting the parameters of the network near a threshold where the network transitions from Brownian dynamics to traveling wave dynamics, and adding external noise, one can induce the dynamics to randomly switch between Brownian dynamics and traveling wave dynamics, thus mimicking Levy flight dynamics.
main_review: This work is the first to study how the brain can generate Levy flight patterns which have been observed experimentally. Some of the interesting theory unpinning this work was previously established in [30,31], where it was shown that depending on the parameters, the CANN will exhibit Brownian dynamics or traveling wave dynamics. The main novelty here is to show that when the parameters are near the threshold between the 2 modes and there is external noise, the network will switch between the 2 modes and they obtain an analytical expression for the Levy exponent. In the conclusion, they propose that short-term depression can destabilize attractor states 

Minor comments:

1. Following Eq. (2), should "Fig. 2B" be "Fig. 2A"?
2. Caption of Fig. 2, I believe the authors mean "for clarity" instead of "for clearance".
limitations_and_societal_impact: How would one test whether brains use "spike frequency adaption" to generate Levy flights?
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: The study describes a model of wave propagation in recurrent neural network displaced on a 2D plan. The neural activation is modeled with a simple divise inhibition model, and the lateral connectivity is given by a Gaussian kernel.  The paper reviews some works that have studied the stability of the activation pattern in this setting and the motion if the neuron model has spike frequency adaptation (SFA). It seems that this novelty here is to include noisy in the dynamics of adaptation. By solving this analytically, the authors shows Levy flight dynamics.

By studying a special case where the recurrent connectivity is derived from Hebbian learning, it is argued that it could model jumpy wandering thoughts.
main_review: Overall the paper is very well written. The review of previous works is very clear even for me although I did not know about CANNs.

Many of the intermediate steps of the mathematical demonstration are inherited from other papers and the rest is detailed in the appendix. I tried to review some of that and I did not find problems but I did not go through the entire supplementary materials.
limitations_and_societal_impact: One limitation is that the Levy-Flights seem to appear only in the model when the adaptation is noisy. This seems to be a strong requirement because the chosen model of adaptation is simplified and I am convinced whether it captures well the wide range of adaptation mechanisms and their variety. I am not sure for instance that this model of adaptation is as solidly grounded on data as divisive inhibition. So would Levy flights emerge if the adaptation model is slightly different? or noisy in a different way? Perhaps at least in simulation it is possible to explore whether it is sufficient that noisy connectivity pattern or neuron position is enough to produce the effect rather that this particular model of noisy adaptation? I also wonder whether other types of frequency adaptation (for instance non-linear "spike" frequency adaptation involving r rather than U) would still lead to this behavior.

A second limitation is that the authors make an implicit far-fetched connection between Levy flights in behavior from Drosophilia and humans and Levy-flights in neural activity (line 35 to 44). The geographical motion of human is certainly explained by a different phenomena than the behavior Drosophilia, is it therefore ok to suggest that your model is an answer to all this by mentioning it in the intro? I believe it would be more convincing  to minimize or clarify this paragraph, at least I do not think that the presented model explains all these behavior. I do find that the speculation that the model could explain mind wandering is charming, although quite hypothetical. Ideally, this third section should acknowledge that this connection is a speculation or justify it better.
review_title: null

Meta-review: All the reviewers have agreed that the submission is original and interesting with clear contributions to the domain. Authors’ responses also addressed most major concerns. Hence, I recommend an acceptance.

On another note, as reviewer JwRM partially pointed out, using power-laws and Levy flights for modeling stochastic optimization algorithms have attracted considerable attention in other parts of ML as well. In this respect, I suggest the authors to check the papers and discuss in their paper if they think relevant, that reviewer JwRM suggested, as well as the following one, which is directly based on a Levy-driven Langevin equation:

Simsekli, U., Sagun, L., & Gurbuzbalaban, M. (2019, May). A tail-index analysis of stochastic gradient noise in deep neural networks. In International Conference on Machine Learning (pp. 5827-5837). PMLR.

Zhou, P., Feng, J., Ma, C., Xiong, C., & Hoi, S. C. H. (2020). Towards Theoretically Understanding Why Sgd Generalizes Better Than Adam in Deep Learning. Advances in Neural Information Processing Systems, 33.

Decision: Accept (Poster)