The motion planning neural circuit in goal-directed navigation as Lie group operator search

Thank you for your positive review on our work! Following are detailed replies to your comments.

”First, it is very puzzling ... it cannot be fit into the framework.”

Thank you for mentioning PFL2 neurons, and we are happy to include more discussions in the revised manuscript. Let’s discuss some issues here first.

• The behavior of animals in the anti-aligned situation is still debated. While Westeinde et al, Nature, 2024 observed maximum speed near the anti-aligned direction, Green et al, Nature Neuroscience, 2019 reported minimal turning velocity, which agrees with our model’s predictions.
• We do acknowledge that a reasonable model should not have the ‘false nulling’ problem (settling at the opposite direction, Westeinde et al, Nature, 2024), and the necessity of introducing PFL2 neurons. Essentially, in Westeinde's model, PFL2 only amplifies the velocity near the opposite direction, and relies on noises to escape the exact opposite direction. Our model similarly allows random turning when the goal is directly behind, mimicking natural behavior.
• Our model can be revised easily to include PFL2 neurons. We only need to speed up the turning velocity near the opposite direction by using the PFL2 activities as a gain factor to modulate the P-EN to E-PG feedback. In principle, the inclusion of PFL2 will only change the $\lambda$ in our theoretically defined objective function.

” Second, I don't agree with ... some of the claims?”

Thank you for your deep thinking. We have carefully checked your example, and found that it is also consistent with our conclusion $u(x|s) = u(x-s)$. Based on your example, we define

The preferred directions of the two neurons correspond to $x_1=0$ and $x_2=\pi/2$. Then we define a $2 \times 2$ rotation matrix $R(\theta)$ with rotation angle $\theta$ as,

Applying the rotation matrix $R(\theta)$ to $u(x|s)$ yields,

Comparing $u(x|s) = (\cos s, a \sin s)^\top$ with $u(x|s+\theta) = [\cos (\theta + s), a\sin (\theta + s)]^\top$, it can be checked that $u(x|s) = u(x-s)$.

Moreover, it can be checked that the above rotation matrix $R(\theta)$ satisfies the definition of rotation group (Eq. 4 is an equivalent definition), including

We need to point out that our theory and model correspond to the special case that $a=1$, where the magnitude or sum of neural responses is invariant over stimulus direction $s$. Moreover, we feel the cases where $a\neq 1$ can be used to explain the heterogeneity of neural responses to stimulus feature $s$, e.g., the summed firing rate of V1 neurons depends on the stimulus orientation (the cardinal effect).

"Third, the contribution is, ... an elegant mathematical reframing of existing ideas. "

We agree with the reviewers that the recurrent circuit model is not brand new, and is based on circuit models developed via bottom-up approaches in recent Drosophila studies (Refs. 19-20). The contribution of this paper is taking a top-down approach and marrying the abstract operator search with the circuit model, and meanwhile we provide strong analytical results. On the dynamics side, we also mathematically explain why PFL3 neurons need to have nonlinear activation function, while such math understanding are absent in Refs. 19-20.

Yes, shifting the sensory representation to left/right directions followed by nonlinear activation function and pooling can be the basic circuit motif to implement a more complicated group operator search. The basic idea is that we need to shift the sensory representations along each dimension of the group space. Please see the example of the 2D case in Global Rebuttal.

” on figure 4A ... such an arrow be added?”

Yes, the E-PG neurons send their axons to both P-EN and PFL3 neurons (Eqs. S13a and S14a in Supplementary). And actually this is the original meaning we would like to convey by Figure 4A: the long arrow should be treated as a single axon emanating from E-PG neurons and targeting both PFL3 and P-EN neurons, rather than two axons with each from E-PG to P-EN and another from P-EN to PFL3 neurons. We will revise our figure 4A accordingly to avoid this confusion.

” There seemed to be a few bodies ... does this match any known cell types?”

Thank you for leading us to these valuable works of literature!

• Certainly, we will discuss and compare these models in the revised manuscript.
• Second, we go through the two papers you mentioned. Nanda et al. 2023 proved that an ANN performs modular addition tasks by projecting inputs into a circle and working in the Fourier space, which is somehow similar to our model. This could be a strong support for the application potential of our model. As for Zhong, et al, 2023, we are not sure if you are referring to ‘Goal Driven Discovery ...’. To our knowledge, it is not related to our model.
• Third, we are glad that you bring up grid cells. Very recently, Mosers’ lab posted a biorxiv paper Vollan et al, 2024, which demonstrated a 2D counterpart of our model in rodents' grid cell systems. Please see strategy 2 in the 2D example in Global Rebuttal.

” how much do these results hold for discrete groups?”

Unfortunately, our result does not apply to discrete groups. Our model relies on the continuous property of Lie groups (Eq17), which does not hold for discrete groups.

” a few typos: ...”

Thank you for pointing out the typos, we really admire your accuracy and meticulousness. We will correct them in our revised version.

Thank you for your review! We want to clarify about the writings first:

‘Suppose the motor system ...through translation and rotation of the sensory input?’

In the heading direction example, the fly’s brain could use its wings to rotate its heading (flying) directions. Here, the direction $s$ corresponds to the world state, and all rotations constitute a rotation group. Flapping wings to change the heading direction (world state) corresponds to the upper left green arrow shown in Fig. 1A. Meanwhile, the fly’s sensory system will keep monitoring its heading direction, and update instantaneously (bottom left green arrow, Fig. 1A), corresponding to the motor system transforming the sensory inputs. The sensory-action loop depicted in Fig. 1A is canonical for the brain and embodied robots.

A similar principle also works for other transformations, e.g., walking/running in a 2D plane constitutes a 2D translation group, the 3D rotations of our head constitute a 3D rotation group SO(3), and 3D translations and rotations of our elbow constitute a SE(3) group.

The present study only considers the simple 1D rotation group, and eventually lands on concrete neural circuits in flies.

” And the computational complexity ... these "certain conditions"?”

The common signal processing algorithm refers to Fourier transformation (Eqs. 13 and 15, detailed algorithmic steps in Table S2 of Supplementary Information, computational complexity in Table.1 and lines 138-149). We find the complexity of finding the group operator in the neural circuit ($\mathcal{O}(N log|h-s|)$) will be lower than the Fourier transformation ($\mathcal{O}(N log N)$) when the stimulus direction $s$ is close to the goal direction $h$ enough, i.e., $|h-s| < N$. This was explained by the text in lines 220-223.

Of course, we will revise our text accordingly to make it less vague.

The comparison between the complexity of the FFT-based algorithm with the neural circuit is a natural one between artificial algorithms and neural circuit algorithms. We hope this comparison could show the advantage of neural circuit computation, and its potential to be a new building block for goal-directed navigation tasks in the future.

”The group operator search formulation ...this generalization be done?”

We will revise our manuscript and discuss/explain with more details based on our current discussion in Lines 312-324. Specifically, the theoretical derivation presented in Sec. 2 is a systematical way that guarantees to find required operators in complex group transformations, e.g., in 3D rotation SO(3) group, or SE(2) group. In addition, our Global Rebuttal lists one example of 2D translation operators. We can incorporate this discussion in the revised manuscript.

Here are our answers to the questions:

1. ” This paper seems to propose ... generic trajectory planning?”

Due to the page limit and the amount of experimental support, we only showed a 1D rotation case that can link to the recently discovered Drosophila’s heading direction system. However, we provided a general framework in Section.2, which, quote reviewer wau7, ‘’helps the reader grasp the theoretical framework of group operator search’. Besides, a recent work (Vollan et al,2024 from Mosers’ lab) indicates that there exists a 2D version of our model in rodent brains, which endorses the generality of our model to high dimensional cases. Please see Global Rebuttal for its generalization to generic trajectories.

2. ” The authors mention that ... this work fit in the literature?”

To our best knowledge, we don’t know any previous literature that ever linked the motion planning computation (group operator search), concrete neural circuits, and analytically solved recurrent neural dynamics together, so we don’t insert a citation here. The text “One of the first studies …” is actually our modest rhetoric, and we are sorry that the wording confused you. We will revise the text accordingly.

In addition, the goal of the present study is to build a theoretical framework of goal-directed navigation with both mathematical and biological groundings, and we have provided a thorough comparison with other works in the ‘Comparison to other work’ (lines 300-311). Reviewer a5Tr acknowledges our paper as ‘an elegant mathematical reframing’ of other models, and we believe this mathematical reframing would endow it with potential to extend to more complicated situations. Regarding citations, we provided references in the ‘Comparison to other work’ paragraph (Ref 19-21), whereas all these cited works only built the circuit model but didn’t connect it to the group operator search. In addition, Reviewer a5Tr also provided a few references in his/her review (references 7-14 of Westeinde et al), which will be included in the revised manuscript.

3. ” What is the meaning ...”

We intend to use this sentence to motivate the challenge of developing a neural circuit searching the required operators, which is not as simple as it appears as computing the arithmetic subtraction of $\theta^*= h-s$. This is because the stimulus direction $s$ and goal direction $h$ are distributedly embedded in two neuronal responses $u(s)$ and $u(s)$ respectively, where each $u$ is a vector with each element denoting the firing rate of a neuron and dimension determined by the number of neurons in the population. The challenge is the subtraction of angles $\theta^*= h-s$ doesn’t imply we can find the operator by naively subtracting $u(h) – u(s)$, otherwise the problem will be trivial.

In principle, finding the neural operator involves below steps

1. Decode the embedded directions $s$ and $h$ from neuronal responses.
2. Compute the angular difference, i.e., $\theta^* = h-s$.
3. Substituting $\theta^*$ into the expression of the rotation operator (Eq. 8).

The above steps are much more than calculating a subtraction $h-s$. And the present work linked the biological neural circuitry to the abstract group operator problem.

We hope our reply can help answer your questions. We wish the contribution of our work can be acknowledged and you could reconsider the rating.

Thank you for your positive comments on the strengths of our work, especially on how it links the theoretical framework to concrete neural circuits. Please check our Global Rebuttal for our replies to your concerns about the generalization of the proposed framework, and the equivariant assumption.

Reply to review’s questions:

1. “Given the current framing ... the authors touch on this aspect in their limitations section as well.”

We really appreciate the reviewer’s suggestion to improve our writing. Section.2 appears to be unnecessary or even overkilling in the current case because the 1D case is intuitive (Eq. 6), while Sec. 2 does provide some necessary understanding of the neural circuit computation, including

• The geometric understanding of motion planning circuit computation (Fig. 3C).
• The comparison of computational complexity (Table 1). Specifically, we need to refer to the workflow in Sec. 2 (Eqs. 13 and 15) to determine the computational complexity of finding operators by using FFT (lines 140-143; Table 1, middle column). This is necessary to show the feedforward circuit can have lower complexity (lines 215-223; Table 1, right column)

Certainly, we cannot agree more that Sec. 2 would be more relevant if we demonstrated it on a more abstract group operator search (we discuss this in lines 315-323). For example, in searching for operators in the 3D rotation group SO(3), we can use spherical harmonics as the basis function to derive the representation of operators -- the Wigner D rotation matrices. We will include this in the Discussion in our revised manuscripts.

2. “In Eq 19, should it be 𝑑𝑥′ instead of 𝑑𝑥?”

Yes, thank you for pointing it out, it should be $dx’$ instead of $dx$.

3. ” Why did the authors assume equal norm ... maintaining this sort of "balance"?”

The invariant norm/sum of neuronal responses has been widely observed in neural data, which was evident in Drosophila's heading direction systems (E-PG neurons) (e.g., Fig.1G&H in Ref. [31] and Fig.1 in Ref. [32]) and other areas, like in orientation neurons in the primary visual cortex, moving direction neurons in the Medial Temporal area, etc. This invariance assumption is also widely accepted in theoretical modeling, e.g., Fig. 3.8 in the textbook Dayan & Abbott 2001; Zhang, J. Neurosci., 1996; Jazayeri, Nat. Neurosci., 2006; Wu et al., Neural Computation 2008; Zhang, NeurIPS 2022, etc. Specifically, a recent work (Ref. 27) demonstrated that this invariance is necessary for a translational/rotational equivariant representation.

Yes, the invariant sum of neural responses results from inhibitory neurons, which is modeled as divisive normalization (DivNorn) in our model. DivNorm is suggested from PV inhibitory neurons by various experiments. (e.g., Niell, Annual Rev. Neurosci., 2015).

In addition, despite the invariance to certain stimulus features, experiments show that the responses can be modulated by factors such as stimulus intensity, which affects the gain of neuronal responses (Salinas & Sejnowski, 2001)

4. ” Authors assume uniform coding, ... be applicable to other systems?”

First, the uniform coding results from the symmetry of translation/rotation groups (The paragraph above Eq. 7 in Ref. 27), and is approximately true in the neuroanatomy of the heading system of Drosophila in Ref. [31-32]. Removing the uniform coding will break the symmetry and thus the equivariant representation in the neural circuit. It is possible that under other group structures the neuronal distributions on the attractor manifold are not uniform. Indeed, scaling group transformation would require log coding (Esteves, et al, ICLR, 2018).

Besides the systematic non-uniformity, the system can deviate from uniformity due to randomness. It has been suggested that the Continuous Attractor Network (CAN) is robust to noises (Deneve, Nat. Neurosci., 1999; Wu, Neural Computation, 2002), thus an approximate equivariant representation can still be held. Certainly, the heterogeneity will break the perfect equivariant representation. One possibility is that the neural heterogeneity and noise can help the brain implement sampling-based Bayesian inference (e.g., Orban, Neuron 2016; Echeveste, Nat. Neurosci., 2020; Zhang, Nat. Comms., 2023).

At last, the uniform coding (neurons are distributed uniformly in the attractor manifold) is not incompatible with non-uniform coding of neurons (such as cardinal effects in visual systems where more neurons prefer horizontal and vertical bars). One possibility is that we still have translation-invariant recurrent weights (ensuring uniform attractor manifold), but the feedforward weights connecting external inputs and network neurons are non-uniform, which will make non-uniform neuronal response reside in a uniform attractor manifold.

5. ” Section 4, Page 8 introduces a new notation 𝑟𝑠+...I am missing something in the explanation here.

Sorry for the ambiguity, we will revise to clarify the notations. $r_{s\pm}$ here denotes the firing rate of P-EN neurons representing heading rotation speed (Figure 4A), which is driven by DN $r_{v\pm}$ (the output neurons in Figure 3A). As indicated in the diagram, P-EN neurons receive copies of the heading direction (E-PG neurons) signal (Eq. S14 in Supplementary) and project leftward- and rightward-shifted feedback to E-PG neurons (Eq. S12A, last term in Supplementary), which will shift the heading direction representation according to the velocity signal generated by PFL3 and DN neurons (suggested by Refs. [27, 35] ), and assemble an intact sensory-action loop.

6. ” How is 𝑢(ℎ) obtained ... the goal representation?”

We used a separate neuronal population to directly represent the goal signal u(h) (last term in Eq. S13a in Supplementary). In Drosophila, it has been discovered to be represented by FC2 neurons (Ref. [11], Fig1i).

7. ” Minor typo: ....”

Thanks for pointing them out! We will correct them in the revised version.

We hope our reply will address your concerns and potentially lead to a reassessment of our manuscript.