The motion planning neural circuit in goal-directed navigation as Lie group operator search
Authors: Junfeng Zuo
Ying Nian Wu
Si Wu
Wenhao Zhang
Venue: NeurIPS 2024 poster (2024)
Keywords: Lie group equivariance
motion planning
goal-directed navigation
ring attractor network
Primary area: neuroscience_and_cognitive_science
TL;DR: A neural circuit model of motion planning and its formulation as a Lie group operator search problem.

Abstract: The information processing in the brain and embodied agents form a sensory-action loop to interact with the world. An important step in the loop is motion planning which selects motor actions based on the current world state and task need. In goal-directed navigation, the brain chooses and generates motor actions to bring the current state into the goal state. It is unclear about the neural circuit mechanism of motor action selection, nor its underlying theory. The present study formulates the motion planning as a Lie group operator search problem, and uses the 1D rotation group as an example to provide insight into general operator search in neural circuits. We found the abstract group operator search can be implemented by a two-layer feedforward circuit utilizing circuit motifs of connection phase shift, nonlinear activation function, and pooling, similar to Drosophila's goal-directed navigation neural circuits. And the computational complexity of the feedforward circuit can be even lower than common signal processing algorithms in certain conditions. We also provide geometric interpretations of circuit computation in the group representation space. The feedforward motion planning circuit is further combined with sensory and motor circuit modules into a full circuit of the sensory-action loop implementing goal-directed navigation. Our work for the first time links the abstract operator search with biological neural circuits.

The motion planning neural circuit in goal-directed navigation as Lie group operator search
Junfeng Zuo 1.3 zuoj unf zuojunf eng@pku. edu.
Ying Nian Wu? ywuêstat. ucla. edu
Si Wu! siwu@pku edu. cn
Wen-Hao Zhang wenhao zhang@utsoothhvsstenn edu
"Peking-Tsinghhee Center for Life Sciences, Academy for Advanced Interdisciplinary Studies,
School of Psychology and Cognitive Sciences, IDG/McGovern Institute for Brain Research,
Center of Quantitative Biology, Peking University.
Department of Statistics, University of California, Los Angeles.
Lyda Hill Department of Bioinformatics, UT Southwestern Medical Center.
*O' O''Dnnelll Brain Institute, UT Southwestern Medical Center.
Abstract
The information processing in the brain and embodied agents form a sensory-action loop to interact with the world. An important step in the loop motion planning which selects motor actions based on the current world state and task need. In goal-directed navigation, the brain chooses and generates motor actions to bring the current state into the goal state. It is unclear about the neural circuit mechanism of motor action selection, nor its underlying theory. The present study formulates the motion planning rotation group as an example to provide insight into general operator search neural circuits. We found the abstract group operator search can be implemented by a two-layer feedforward circuit utilizing circuit motif's of connection phase shift, nonlinear activation function, and pooling, similar to Drosophila's goal-directed navigation neural circuits. And the computational complexity of the feedforward circuit be even lower than common signal processing algorithms in certain conditions. We also provide geometric interpretations circuit computation the group representation space. The feedforward motion planning circuit is further combined with sensory and motor circuit modules into a full circuit of the sensory- action loop implementing goal-directed navigation. Our work for the first time links the abstract operator search with biological neural circuits.
Lie group operator search problem, and uses the 1D
Introduction
The information processing in the brain forms sensory-action loop interact with the external world (Fig. [TA) [H3]. The sensory-action loop consists of three modules: the sensory neural circuit module forms neural representation of the world state, the motor circuit module produces motor actions to change the world states, and in between there an essential sensorimotor transformation module that plans motor actions based the sensory input and tasks goals 4 5] For example, in gire-directed navigation tasks, the sensorimotor transformation module plans a sequence of motor actions to bring the current sensory state towards goal state. Extensive neuroscience studies have investigated the
Corresponding author.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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A
Brain
B
Equivariant map
C
Goal-directed navigation & D 1D Rotation group
RIF
Motor actions,
Comrloggum Repoonee
Motor to sensary fesibbokk
R
State
state,
Sonsorimoto trandforration
Ri0)
Sensory
response ((s))
Sensory
Goal nourons u(A)
R10)
Figure I: (A) The sensory-action loop. Sensorimotor transformation plans motion actions based on the world state and task goals. (B) The equivariant map of the sensory neurons' responses. (C) A concrete example of ID rotation group acting on periodic state s that can be regarded as the heading direction of a fly. The sensory neurons form a neuronal population code u(æ]ss) uniformly covering the space of s (Eq. D.. (D) Finding a desired operator can be realized by group convolution, i.e.,
finding the peak location of cross-correlation function over the group manifold.
however, it remains far from clear as well as the underlying computational theory. Motion planning is also an essential computation for embodied agents in engineering research, including, e.g., robotic control, reinforcement learning, and machine learning [3] etc. Studying the motion planning neural circuits and their computational theory will help us understand the brain and can provide brain-inspired circuit models for embodied agents.
neural circuit mechanism of motion planning 9
To provide theoretical insight into motion planning, the present study defines the sensory-action loop using the (Lie) group theory. Denote by as a continuous world state, and u(s) = w(r[s) the evoked responses of a population of sensory neurons with x as the neuron index. Suppose the motor system generates the same kind of actions from a Lie Group G (e.g.. translation or rotation) to transform the world state s. The effect of motor action g E G the world state and updated sensory representation us)) can be denoted by (Fig. [A; o denotes the group action in b
u(s') uR, ss) Rg u(s). g E G.
(1)
Ry is the group operator (motor action) changing the world state (Fig. DA green), whose effect on updated sensory responses u(s') can be summarized as a neural operator Ry directly acting on the original response u(s) (Fig- [ blue) that can be regarded as the motor-to- -s-sssoryy neural feedback in the brain [5]. To represent all world states under all group transformations, u(s) must satisfy Eq. ①I for all E G, and is called equivariance with the group G (group homomorphism) (Fig. (B). In the Lie group framework (Eq. [ motion planning in goal-directed navigation can be formulated as finding an operator Ry to bring the sensory response of the current state u(s) into a goal state u(h).
Find Ry: subject to Rg u(s) uhh), g gE6 G.
An intuitive way to find Ag is exhaustive search in the group space: apply every operator to u(s) and select the one transforming u(s) closest to the goal response u(h) (Table SI Supplementary Info. (SI),, corresponding to find the peak location in the group convolution (Fig. D. bottom) [16] 17..
g" arg max L(g); where L(g) = (w(h) u (())((g) = (u(h), Rg о u(s)),
g
then the optimal group operator is Rg.. The * denotes the group 9 convolution, and (u(h), (((h),., Rg o u(s)) f h(=:)))(= o s) is the inner product. In particular, when is 1D variable and R. is a ID translation operator, the group convolution L(g) (Eq. 3 simplifies into the cross-correlation
function that is extensively used in signal processing 18 19
The present study investigates how brain's neural circuits search group operators Rg (action) in goal- directed navigation (Eq. 2) We use the 1D rotation group as a working example to provide insight into the neural circuit mechanism of general operator search. Although finding 1D rotation operators is mathematically simple and can be realized by available signal processing algorithms, its neural circuit implementation has never been explored. We theoretically derive a two-layer nonlinear feedforward circuit for ID rotation operator search, which is composed of circuit motifs of connection phase shift, nonlinear activation function, and pooling, and the derived circuit is similar to Drosophila's goal-directed navigation neural circuits 23). We link every neural circuit computation with

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operations in operator search, and provide geometric interpretations of circuit computation in the group representation space. Moreover, the computational complexity of the derived feedforward circuit for operator search can be even lower than the standard algorithm based on the fast Fourier transform in signal processing in certain conditions. We further assemble the derived feedforward circuit with sensory and motor circuits to form full neural circuit of the whole sensory-action loop.
Significance. The present study is one of the first studies formulating motion planning as the group operator search problem, and deriving a biologically plausible neural circuit implementation with rigorous mathematical analysis. The group operator search formulation of motion planning can provide normative approach to generalize existing motion planning algorithms into different transformations. Moreover, in terms of group equivariant machine learning, the theory and the sensorimotor transformation circuit model developed in the present study are complementary to many existing equivariant networks that correspond to the sensory system (e.g.. 17 24 27
2 1D rotation neural group operator search
We use the ID rotation group U(1) as an example to provide insight into the general principle of group operator search in neural circuits. The U(1) manifold is unit circle on the complex plane (Fig. 24), and can be parameterized by the angle 0 (corresponding to g in general Lie group, Eq.
U(1) fexp(i0), A E [-T, x)}. i= v=V-I.
group element R(0) E U(1) rotates ID stimulus direction E [-T,, , T), or its complex represen- tation es by 0, i.e., s into s + (mod 2T), and is denoted as R(O)) o _ ei0ges = cd(s+0), Based on Eq. rotation-equivariant sensory response u(s) should satisfy (suppress "mod 2x for brevity),
n (R((0) s] a(s 0) (0) us))
(5)
R(0) is the neural operator rotating the sensory representation u(s). Since the rotation operator changes in additive way, i.e, u uz]ss) + u(zls s + 0). it can be checked the equivariant sensory response satisfies u(r(s) = u(z - s) u(s) (28), where the neuron index æ (also called preferred direction) additively interacts with s, implying the neuronal response only depends
the difference x s (Fig. D]. The rotation rotation-equivarian equivariant neural responses u(x s) have been widely used in neural coding studies 31) and are usually called homogeneous neural codes. To make our theory general, we leave the concrete profile of u(s) open and will see how the group structure constrains it.
In the case of 1D rotation, the motion planning in goal-directed navigation task is finding a rotation operator R()) to rotate the sensory response u(s) into the goal direction, u(h). Due to the simplicity of U(1) (Eq. 5). the desired rotation operator is 12(0" h s), in that
R(h s) a u(x - 5) = u|x (s h - s)] = u(x - h),
(6)
whose parameter is the angular difference 6" = h Although computing the angular difference is simple, it is non-trivial to search neural operators A((0) in neural circuits. This is because R((0)) is an abstract math object rather than the numerical value A", even if it is indexed by 8* This distinction reflected by the fact that although the operator's parameter 6* = h is arithmetic subtraction, the neural operator R((0) by means of arithmetic subtraction between neural responses u(h) and u(s).
2.1
The structure and representation of 1D neural rotation operators
Searching abstract neural operators i((0) first requires neural circuits to represent them. Intuitively, the representation means a one-to-one mapping between abstract operators with numerical values (e.g.. neuronal activity), and then searching abstract operators can be converted into common numerical optimizations and further mapped to neural dynamics. Hence, we study the structure of the neural operator R(0) to reveal its neural representation. Consider an infinitesimal rotation R(60) (59 -> 0),
whose effect on the sensory response u(x s) is (using first-order Taylor expansion).
(6)60) u( 3) = u(z 60 s) = u(x s) + 80(-000(())))) uz s) (1 + 60 -ju(zz - s). (7)
j= = 0//r is the 1D rotation generator characterizing the tangent space of infinitesimal rotations, and can be used as the basis of Lie algebra. By using the infinitesimal rotation, the
3

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A
1D rotation group
B
The representation space
E
Two processes of rotating sensory representations
F
Sequential motions
manifold Im
I
Re
C
Neural responses
Desired rolation
Time
operator
Neural space
Rep. space
19=h-1
Goal
Sensory rep.
o(+)
((A)
Re
-3
R
R:00
=(6..)
Unicel
Goal
u(A)
((A)
x(4,)
Figure 2: (A) The ID rotation group manifold. (B) The eigenvalue spectrum of the 1D rotation group operators. (C-D) The representation of neural responses (C) and desired rotation operator (D) in the group representation space spanned by operators' cigenfunctions. (E) Two mathematically equivalent processes of rotating sensory responses into the goal direction. (F) Sequential motion planning.
differential and exponential form of a rotation operator is (see SI. Sec. 1.1
d
i((0) = Jo P((0) db
+ R()) = exp(()),
(8)
The neural rotation operator R(0) is commutative, meaning the composition of two rotations is the same regardless of their order, i.e.. R(0, R((2)) R(()))(())) Commutative group operators share a common set of eigenfunctions that can be used to represent group operators. It can be checked (f) = e/wx/ / 2TT, w E 2} is the normalized eigenfunction set of J,
(9) with each Iw having eigenvalue (J) -i.. The eigenvalue (J) can be regarded the representation of J based on fw. Hereafter, we call the space spanned by the eigenfunctions {fw as the representation space. It is worth noting that { are Fourier bases, which are widely used in frequency analysis in signal processing to extract each frequency component [88 In contrast, the current study uses Fourier bases represent the generator J and operator R((0).
j fw -getwa ke /v2 = (-iw) e buz /V2// = PJ) fw:
Since rotation operators can be composed by the generator j via the exponential map (Eq. 8 the neural operator's representation based on the eigenfunction fw is derived as (details at SI. Sec. 1.2
R((0) fw = e uu0 fw (o) fw:
(10)
whose eigenvalue e --v0 A
(0) is regarded as the representation of rotation operator fe(0).
Comparing Eqs. 9 and I0,
we see the exponential map from the generator to the operator, i.c..
k((0) = exp( also exists in the representation space, i.e., (o) = cxp(- - -iwf) = cxp(0 Pu (J)).
The closed-form formula of p(0) is the inner product between the rotated and the original eigenfunc- tions (multiplying both sides of Eq. (TO) by 1 (f: conjugate), and integrating over x))
(0) (0) fw. fw) S [i((0) fw fi ii
(11)
There is an one-to-one mapping between each operator R((0) and its representation (0). The advantage of using eigenfunctions to represent abstract operators R((0) is searching abstract operators (Eq. 3 can be converted into usual numerical optimization in the representation space.
2.2 Rotation operator search in the representation space
Despite the ansatz of the desired operator (Eq. 6), we still need an algorithm explicitly outputting the operator based on neuronal responses. To derive the numerical computation in the representation space, we decompose neural responses u(s) and u(h) by using operators' eigenfunctions fwr
u(s) (uss, fo = En (w]s) & [
(12)
corresponding to the inverse Fourier transform. Meanwhile the representation (w]s) U(s) (w is suppressed unless confusion), called Fourier coefficient, is calculated via Fourier transform,
U(s) = (u(s), fw) = -_" u(s)fide & Flu(s)],
(13)

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Table 1: Computational complexity. N is the neuron number, i.c., the dimension of u(s).
Group convolution
Representation space
Feedforward circuit
(Eq.) 3 Table
via FFT (Fq. 15 Table S2)
(sequential motions, Fig. 2 Table S3
(N2
(NVlog
O log( // - s()
Since u(s) R(s) u()), we have (s)) Pw (0) = e -- (O), with (O) the representation
of u(ss 0). Similarly, the representation of the goal neurons' response uh)) is (h) Pa (h)
Referring both (s) and (h) by (0),, their representations are (s) and (h) respectively, and are geometrically visualized as two vectors of unit length with angle s and / respectively (Fig. 2
With sensory and goal neuronal responses' representations (Eq. 12). the objective function of abstract rotation operators, L()) = (R(O) o u(s), u(h)) (Eq. 3). simplifies into numerical function as the inner product of representations of neuronal responses and operators (see details in SI. Sec. [.2].
T(()= = >n. (w) (0)) ( [[)
hn) = >.. (w((w)s) (0) (W]]))
(14)
where the Cauchy-Schwartz inequality is used and ||a]] - v Vaat, The Eq. (44 is maximized only if
U(wls)p = (w]h), implying the representation of the required rotation operator is (Fig. 2D).
Pa (0)) = (b) = pa (h) p_ (s) = e -ii(h-)))
0- arg max L(0) = h -. 8. (15)
8
In the representation space, the neural operator is a numerical ratio with a closed-form solution, which provides algorithm to find the operator rather than checking the ansatz (Eq. 60. Moreover, the computational complexity of finding operators in the representation space is much lower than the group convolution (Table SI. Sec. 2 the complexity via the representation space is O(N log N) with -V the neuron number, mainly coming from Fourier transform when using fast Fourier transform (FFT) (Eq. 13) 18. In contrast, the complexity of the group convolution (Eq. 3 is ( Once (a)( ) is found (Fq. 155, it can be multiplied with (s) followed by inverse Fourier transform to rotate the sensory response into the goal direction uhh), i.e., F FF (s)p
(b)) u(h)
(Fig. 2 dashed lines). Nevertheless, this procedure (Fig. 2E, dashed lines) is physically different from actual motor actions, which corresponds to firstly mapping pw (0*) back to the physical operator R((0)) and acting on u(s) directly (Fig. 2 solid line). Hence we explore how the neural circuit finds the operator A(8*) and use it to physically rotate sensory response u(s) (Fig. 2 solid line).
3. Towards a neural circuit of motion planning
3.1 Sequential motion planning strategy
In reality, the motor system (muscles) has power constraints and cannot generate actions with too large amplitude, implying it might not rotate the stimulus direction abruptly within an infinitesimal time period. Rather, the brain decomposes a strategic, complex motion action into a continuous sequence of small actions 4 forming time-continuous process (Fig. 7
ie((0) u(so)= ... (((60))))((66))) ... r((60o) u((s) = [[... ((--0())))) u(s)),
(16)
where su = s is the initial stimulus direction, and 8 = so + -E 80, is the direction after applying rotation sequence with angles 50, to 80, The sequential motion planning imposes a sensory-action loop: after the motor system generates small rotation R(60.) at time t, the sensory response updates from u(s;) to u($+41), followed by another rotation R (() which repeats over time until the stimulus direction 8 is rotated to the goal h. Differentiating Eq. (16) over I, utilizing the differential form of operators (Eq. 8). the dynamics of the sensory responses u(s:) in the sensory-action loop is,
d R((e)) o u(s))= = dl((0)) d0; di; dt We see the rotation dynamics is determined by the rotation speed vt = d0./ There are multiple strategies for generating U, sequence and we consider a strategy performing gradient ascent along the objective function (Eq, 3) i.e., the Ut at time is proportional to the gradient of rotation angle Or.
d dt (u(s)) = dt
o u((s) = vg (R(,) o u(so) = v u(st), (17)
0. A dL.(0.) d0,
d( (()..) u(5o), v(h)) E... [[((([]])))]))))) [inlw/iii/// st)],
(18)
d0,
with À determining the step size. The rotation speed sine function of the direction difference
h 3, and representing rotation group parameter (comparing Eqs. 18 and 15 Fig. 3D)
5

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A Two-layer feedforward circuit for motion planning
B Drosophila's goal-directed
C
The geometry of feedforward
D
navigation circuit Heediin direction E-PG. w(A)
circuit computation
(0,26)
Gradient /d./ whekmaa
Goal response u(h)
P-EN left
P-EN right
PRLL teft
PFLS right
rt/ fw
Im
Rate
COOO00U00 Sensory response
DN
DN right
Cuy T Theory
Tum
FC2, MER
/dd0
Goal dirdiion
Re
Figure 3: (A) The derived feedforward motion planning circuit, composed of circuit motifs of con- nection phase shift (ray neurons receive different connection phases from u(s) and (h))), nonlinear activation function (left-bottom inset), and pooling (2). The difference between two output neurons To+ conveys the rotation speed v,. (B) Drosophila's goal-directed navigation circuit (adapted and modified from (2I). Neurons are arranged by their preferred direction z. For illustration, only four PFL3 neurons (re, neurons in A) are shown. The PFL3 right (green) and left (red) neurons receive heading input u(s) with shifted phases and goal input w(h). Two DN neurons (Fe. neurons in A) pool PFL3 left and right neurons respectively and rotate heading direction. (C-D) The geometry of feedforward circuit computation in the representation space. (C): Two populations of ro_ neurons rotate the (0.) by =AO. The firing rate difference of two output neurons To+ is regarded as the length difference between horizontal green and pink arrows, which is sine function with 0r = h St. the distance to the goal direction h (D).
3.2
A feedforward circuit for motion planning
The closed-form solution of U2 in sequential motion planning (Eq. 18) and the optimal operator (Eq. 15p imply can be computed by a feeedforward circuit in a single propagation of neural inputs, which would be faster and simpler than recurrent circuit. We explore how generic feedforward circuit
computes 1/ (Eq. 18 via receiving sensory response u(s) and goal response u(h) (29).
Ty (cr) = [[ w. (x, x²) d')uu'a' - s)de' + fw (x, x n')u(x' - 1i)dal,
(19)
where F() is a nonlinear increasing activation function. wa (x, and wn (z, x²) are feedforward weights from sensory neuron u(æ s) and goal neuron u(x h). Two issues are to be resolved for it computation in feedforward circuits. One is how the feedforward circuit as a nonlinear function of summed neural inputs (Eq. [9] computes ((0)) as inner product of neural inputs u(s) and uh)
(Eq. 18 Another is computing the derivative dLL/d,, in the feedforward circuit.
We propose the feedforward circuit approximates the derivative dL. d0, as difference form,
Ut R L(O, + A0) L(0, A0)
A [(u(54)) u(h)) - u,(())
(20)
2A0
240
where 84 = (50 + ++) + A0 = 8, + AO. To implement the inner product (Eq. 18 in the feedforward
circuit (Eq. 19 we convert the inner product of two neural inputs into,
(u(s+), u(h)) = (I]u(s+) + u(h)ll - [|u($ - || h)) /)2
(21)
[|u($+)]] )12 = u(x - )+) ddz where the square function is similar to feedforward circuit's nonlinear output (Eq. 19p. By using the form in Eq. (2T). the two inner products difference Eq. (20) is,
(u(s+), u(h)) - (u(s-), u(h)) = (1l u(h) - [| + u(h))12)
(22)
where we used [w(s=)Il2 [2)))] i.e., the norm of neural responses is irrelevant with represented directions. Eq. 22 suggests computing L(B, + A0) - L(0: - Ad) for v (Eq, 28 can be achieved by two-layer feedforward circuit with each layer containing two neuronal populations (Fig. 3A).
First layer: rea (x) = [u(x - $ u(z h))?:
(23)
Sccond layer Tuy = re (z) = [1((s=) + (2))
In the 1st layer, each neuronal population ro.. (xr) computes the square of the sum of sensory and goal inputs, followed by two neurons Try at the 2nd layer pooling responses rea (x) at the 1st layer.

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Receiving rotated sensory input u(s+) ) by +AO can be realized by feedforward weights with shifted connection phase (Fig. 3A ro receives connections from u(s) and u(h) with different phases),
w0t.s x 8(x x' A0);
(24)
where (x, x') is the weight from sensory neuron u(x' - s) to ro (a)). Eventually, the difference of two neurons at the 2nd layer is proportional to the rotation speed 82 (combine Eqs. 20 and 22
(re, Tv [(((++) (u(s=), x(h)]] x er-
(25)
Then each ro= can drive a corresponding effector (c.g., muscle) that generates actual motor actions to rotate the heading direction clockwise counter-clockwise.
General nonlinear activation fuucction. The square function for the first layer neurons rea (2) (Eq. 23p doesn't 't necessarily mean their activation function must be a square function, otherwise our theory and circuit is limited. Instead, the derived feedforward circuit works well with a general nonlinear activation function F()) monotonically increasing with u (Eq. 19). To see the mechanism of general nonlinear activation functions, we expand it at to the second order.
F(u)= ~ F(0) FO())u + F")))²) /2 _ Fo F1 u + F2 u2.
(26)
Then the neuronal responses re (xx,)) can be approximated as,
To (x) R Fo F1 [u(s 8+) + u(x - h)] + F2 [u(x 82)² + u(x - -))),
(27)
+ 2F2 u(x 84 )=(( h),
Finally, the difference of neurons at the 2nd layer is still proportional to the rotation speed Ut.
Tv+ -r__ J re, (z)dz - re_ (x)dz= = 2F22 [(((4), w(h) - (u(s_), x ez. (28)
Again, used the fact that the summed neuronal activities do not depend on the represented direction, i.e., J uu(== - s) s)dr = u(z)dz, and I u(x - d) = J u(z) Notably, the 1st layer neuron ræ must have nonlinear activation function to enable the feedforward circuit to output the rotation speed Ut, otherwise (setting 1/2 = 0 in Eq. 28 the neurons at the 2nd layer, Toy will fully cancel. Overall, the architecture of the derived feedtorward circuit utilizes the connection phase shift, nonlinear activation function, and pooling of neuronal activities to compute the rotation speed et.
Comparison with Drosophila's 's circuit. The derived feedforward circuit is similar to the recently identified Drosophila's 's goal- goal-directed navigation circuit (Fig. 3A-B) 20 22), which also has two- layer feedforward architecture receiving the sensory input u(s) (E-PG neurons) and the goal input w(h) (FC neurons) to compute the rotation speed 14 At the 1tt layer in Drosophila's 's circuit, PFL33 left (right) neuronal population (Fig. BB, red (green) neurons) combines the shifted sensory input u(s) with angle T A respectively with the goal input u(h), and output via a nonlinear activation function, which are similar ret neurons in our feedforward circuit (Fig. A Then the DN left and right neurons at the 2nd layer (equivalent to ro= neurons) pool all activities of PFL3 right and left neurons respectively, and their response difference determines the rotation speed v. 20 22
Neural circuit computational complexity. Considering each (0)(a) neuron only receives one feedforward connection from sensory neurons u(s), i.e., Wop1s is delta function (Eq. 24). which is the case in Drosophila's circuit. Then in each time step during sequential rotations, the Teedforward circuit computes ((N) addition and (N) multiplication (suppose a square activation function). Given a fixed goal direction h, the sequential rotation will take Oloo I/ - =l) time steps to rotate from the original direction s into [19] Hence the total complexity of feedforward circuit during the whole sequential rotations is (NN log lh sl). When the stimulus direction is close to the goal direction h enough, i.c., I/- - s| < N, the computational complexity of the feedforward circuit is even lower than the widely used fast Fourier transform (Eq. [3) with complexity O(N log N) (Table 1.
3.3 The geometry of feedforward circuit computation
Although the feedforward circuit doesn't explicitly use the operator's representation (Eq. [5] Fig. 2 the representation space (Eq. 9) provides clear geometrical interpretation of the feedforward circuit computation. Substituting the representation of sensory response u(st) at time and goal
response u(h) (Fq. 12] into rotation speed neurons Tor (Eq. 27p.
Fi = 2Fg E., [0)]1 (8. F A0) + (6. F A09t] + const, (0. = h - st). (29)
7

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B
s(h) IFC) Goal
r. (DN)
Modion planning
not. speed
Firig rate (a.u.) Direction (rad)
Speed
IPFL3, right)
0
DOOC
Gradient di,
-0.4
Motor
000000000
000000000
Rot. generator J
(P-EN, right
Goal Heading
m
F. Tr_ (normed)
-0.4
Direcilon(rad)
u(+) (E-PG)
280
DN lult
1
Headlin
Gool
Sensory response
DN right
200
Ring attractor
@ gain control
paoling
140
40
120
400 800
Time (ms)
Time (me)
Figure 4: (A) The full circuit of the sensory-action loop. The diagram is simplified from Drosophila's goal-directed navigation circuit to illustrate connections, without influencing the circuit function. Only neurons on the right side are labeled and names in the parenthesis denoting Drosophila's neurons. (B) Top: population response of sensory neurons u(s) in the full circuit. Bottom: The decoded stimulus direction from u(s) moves towards the goal direction. (C) Right and left DN neural activities drive the rotation in (B). (D) Difference between right and left DN linearly matches the moving velocity of sensory representation. (E) Sensory response tracks moving goal direction.
const is a constant irrelevant with angles (from the first two terms at Eq. (27). RHS). Geometrically, frg corresponds rotate the optimal operator's 's representation Par (0)) by T TAO, and sum the rotated operator with its complex conjugate (mirrored by the real axis, Fig. 3), and hence it resides on the real axis. The difference between Tu, and ru_, exhibited by the length difference of pink and green arrows in Fig. 3C, is sine function depending on the optimal operator's 's angle 0, (Fig. 3D).
4 A full circuit model of the sensory sensory-action loop
We further assemble the derived feedforward motion planning circuit (Fig. (3A with concrete sensory and motor circuit modules to construct a full circuit the sensory-action loop (Fig. AA implementing sequential rotations (Eq. 17). For simplicity, the full circuit model only includes the internal motor- to-sensory neural feedback (Fig. MT blue line) rather than the external loop (Fig. [A green lines), with mildly implicit assumption that the actual sensory feedback from the physical world is the same as the internal motor-to-senson temporal dynamics, even if our theoretical derivations consider memory- neurons (Eq. 19.. All connection weights have Gaussian profiles spreading over the neuronal space x (SIEqq ST22). with different peak weights, connection widths and phases. Due to the page limit, we briefly introduce key features of the full circuit here, and its detailed dynamics can be found at SI.Sec. 3
feedback. To be realistic, all neurons in the full circuit have
Sensory and motor circuit modules. The sensory and motor circuit modules are based on a recent theoretical study that analytically linked Drosophila's 's sensory and motor circuit model in its internal compass circuit with the ID translation/rotation group 22 The sensory circuit module uss) is modeled by a ring attractor network that has been experimentally verified in Drosophila's s brain (Fig. AA, blue ring) [32 34]. The ring attractor network uses its cotation-invarian recurrent connections to generate itlation-cquivare
sensory representation u(s) (28). The motor circuit module has two neuronal populations T (I)) (Fig. 4A. P-EN), whose feedback connections to sensory neurons u(s) are shifted by F towards opposite directions (comparing connections from red and green PB neurons to the ring attractor, Fig. [A). It was found that these shifted connections give rise to the rotation generator J and the firing rate difference Exrr, (xr rs (x)) determines the rotation speed U2 of sensory representation u(s). Hence we call rsy as rotation generator neurons hereafter.
Assemble the full sensory-action circuit. The rotation group interpretation of sensory and motor circuits 22 provides a clear interface to connect three circuit modules. Functionally, we use feedforward circuit output neuron rv-. to modulate the gain of rotation generator neuron T... i.c., multiplying each generator neuron rx₁ (x) or r._ (x) by feedforward circuit output neuron response (scalar) re+ ro_. respectively. The gain modulation of rotation generator neurons Ta by rotation speed v, was indeed observed in experiments [35]. In addition, the feedforward circuit receives the instantaneous sensory responses u(x 8) (Eq. generated by the ring attractor network.
Theoretical analysis of the full circuit. We perform theoretical analysis of the full eircuit dynamics to verify whether it implements sequential rotations toward the goal direction (Eq. 17). We perform
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perturbative analysis of the whole circuit dynamics around its attractor states, analytically derive the eigenvector corresponding to the stimulus direction s in the neural dynamics and find the eigenvector of has the largest eigenvalue suggesting the circuit dynamics is dominated by the movements along the stimulus direction. Then throwing away the circuit dynamics along the subspace perpendicular to the eigenvector of s, we find the sensory responses in the ring attractor embedded into the full circuit are approximately reduced to a form similar to the sequential rotation dynamics (Eq. 17
d
dt u(x 81) x Weg.a - -r_)) [[ws,s Ar) o u(x - -t)]
(30)
where the difference of motion planning circuit's output neurons Tv+ To_ determines the rotation speed it (compared to Eq. [7). In Eq. (30), Wa.sa and Ar denote respectively the peak weight and the weight shift (absolute value) of the connection from rotation generator neuron rsg. to the sensory neuron u()) in ring attractor (Fig. 44)) and similarly for wa Further analysis reveals,
(Te, - ) x wwg. >.. [[] sin (wA0) sin[(w(h - s)]l,
(31)
where wa.s and 0 the peak weights from sensory neuron u(s) to neurons rog. and the one from roy to the output neuron Tou respectively. A is the weight phase shift for the weight from u(s)
ro+ (Eq. (4 ). R(w]0)
(0) is the Fourier transformation of r(x) at O. The derivation details can
be found in SI. Sec.
We perform numerical simulations of the full sensory-action loop circuit. We fix all circuit parameters and only change the goal direction /t that determines goal neurons' responses u(h). Fig. 4 shows the represented stimulus direction in the ring attractor's 's population responses u(s) indeed moves towards the goal direction h, which is driven by the activity difference between To+ neurons (Fig. [ The full circuit model can also track a moving direction (Fig. 4D), although some delay exists due to the temporal dynamics of neurons (see Discussion). Numerical simulation also confirms the rotation speed of sensory responses is proportional to the response difference between Trg (DN neurons).
Conclusion and Discussion
Motion planning is important in sensorimotor transformation in the brain and embodied agents. The present study formulates motion planning as a group operator search problem and investigates the neural circuit mechanism of operator search goal-directed navigation. Using the ID rotation group as an example, we analytically derive searching ID rotation operators can be realized by two-layer feedforward circuit with three circuit motif's of connection phase shift, nonlinear activation function, and pooling, which is similar to the recently identified goal- goal-directed navigation circuit in Drosophila's a brain 20 22). We further assemble the feedforward sensorimotor transformation circuit with sensory and motor circuit modules into full circuit of the cunsory-action loop which successfully produces sequential rotation dynamics in tracking goal direction (Fig. 4 Our study provides overarching connections between Lie group operator search with a biologically plausible neural circuit model comparable to Drosophila's 's circuit. gains our understanding of neural circuit computations from a structured computation perspective, and also provides a biologically plausible neural network solution for artificial intelligence research.
5.1 Comparison to other work
Although the derived feedforward circuit is similar circuit models in recent neuroscience studies of Drosophilas 20 22), there are several notable differences. First, recent circuit models required both neural responses u(s) and u(h) to have cosine-profile (pure frequency component at 1, Eq. [3) 20 21). Although the cosine profile is experimentally supported, our theory releases the requirement of neural response profile, e.g., our sensory-actio circuit has Gaussian profile neural responses (Fig. (A). The generalization may reduce the limitation when deploying the neural circuit model in real applications. Second, goal-directed navigation circuit models in Drosophila's 's research 22 haven't 't composed full circuit of the sensory-action loop as in the present study. In addition, from the group equivariant machine learning perspective, the group operator search theory and the motion planning feedforward circuit in the current study correspond to the sensorimotor transformation stage, which is complementary to many equivariant neural networks corresponding to the sensory system (c.g.. 17 24 27) when building embodied agents.

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5.2 Extensions and limitations of the model
Extension to complicated scenarios. For the sake of concision and biological solidity, we only demonstrated the ID rotation case in the main text, but our modeling framework has the potential to extend to more complicated scenarios. The 2D translation group a sufficient example to explain its generality. An important step in motion planning neural circuit is approximating the derivative of the objective function over the transformation amount (Eq. (8) by the spatial difference in the neural circuit (Fq. 200). i.e., the sensory representation is rotated to the positive and negative direction (0, + 2 and 0, A in Eq. 20). This spatial difference strategy can also be used in the 2D case and there are two equivalent circuit solutions. One is considering an allocentric representation with an x-y coordinates, where the sensory representation will be shifted along +z and Hy. forming neuron populations analog to PFL3 left and right neurons in our model, which was also considered in spatial representation circuits [36]. Another strategy operates in an egocentric representation with a polar coordinate, and the sensory representation will be shifted along the clockwise and counter-clockwise directions, requiring neuron populations This strategy is supported by a recent experiment (377). A simulation result is shown in Fig. ST
Non-uniform distribution of neurons. Our model considered neurons are uniformly distributed in the attractor manifolds, simplification widely used in continuous attractor networks [38 41) Although the assumption holds in Drosophila's 's brain, the representation is usually non-uniform in other cases. For non-uniform distributions of neurons in the current circuit model (the in Eq. S12 is irregular), the same neural circuit dynamics (Eqs. S131ST3 in the SI.) can still approximately facilitate motion planning and rotate heading representations. To realize exact computation in the non-uniform case, the recurrent weights (below S11 in SI.) need to be fine-tuned numerically, by using a technique similar to 42 Overall, we think the non-uniform distribution does not alter the neural circuit implementation substantially while it requires new theoretical insights to understand why the system still functions effectively. Besides, the uniform distribution of neurons is only required by the translation symmetry (28). while it can be un-uufformmmmmerfeett in other group structures, e.g., the scaling group would require log coding 43).
False nulling problem. Our feedforward motion planning circuit outputs zero speed when the
heading and goal directions are anti-aligned, which is under debate in experiments (e.g.. [20]
observed maximum speed around the anti-aligned direction while 44 observed the opposite). A
reasonable model should not have the 'false nulling' problem (settling at the opposite direction (20), and one potential circuit solution is introducing PFL2 neurons observed in Drosophila which fire actively at the anti-aligned direction. That is, PFL2 neurons will speed up turning velocity near the opposite direction and provide gain modulation to P-EN to E-PG feedback. Moreover, including PFL2 will only change the À in theoretically defined objective function (Eq. (8].
Future work. From the neurobiology perspective, the direct gain modulation from Tee. (DN neurons) to r.a (P-EN neurons) in the full circuit model (Fig. A Eq (114 needs to be verified by future experiments. likely to be the case Drosophila's 's brain because Y*= neurons are gain modulated by rotation speed [32 34). i From the machine learning point of view, proof of concept, only study the theory and corresponding circuit model for ID rotation operator search, where the group representation theory (Sec. 2.1
2.2) appears unnecessary because ansatz of the optimal operator can be obtained intuitively (Eq. 6). Nevertheless, the group representation theory and the research protocol in the present study are necessary for searching complicated group operators especially non-commutative groups, e.g., SO(3) and SE))), where obtaining the solution of the required operator is non-trivial and no longer intuitive (e.g., [3I.. The group representation theory is a normative method that guarantees to find such an operator and derive corresponding circuit models. Extending the motion planning circuit search more complicated group operators forms our future research.
Acknowledgments
W.H.Z. is supported by the UT Southwestern Endowed Scholars program. Y.N.W. W. is supported by NSF DMS-2015577, NSF DMS-2415226. S.W. is supported by the National Natural Science Foundation of China (No. T2421004), the Science and Technology Innovation 2030-Brain Science and Brain-inspired Intelligence Project (No. 20212D0200004). The authors thank Yue Liu and Gengshuo Tian for feedback on an carly draft of this manuscript.
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Reviews:
[{"review_id": "29oGSpP0Gx", "reviewer": "Reviewer_wau7", "summary": "Intelligence systems, both biological and artificial, need to leverage sensory information and sensory-motor feedback in order to plan their (motor) actions in order to interact with their environment. A specific behavioral scenario is goal-directed navigation, wherein an agent needs to plan its actions based on its current state in order to move towards a specified goal state. The authors formulate this goal-directed naviration behavior as a Lie group operator search problem. Specifically, they argue that given an group-equivariant representation   of the stimulus space, the agent needs to search for the appropriate operator from a Lie group (corresponding to the agent's action space) that transforms its current state observation to the goal state observation. They demonstrate the utility of their framework by considering a 1D rotation group of actions in a goal-directed navigation problem as in the Drosophila head-direction system. The authors demonstrate that their framework of operator search can be implemented by a 2-layer feedforward circuit, that mimics the biological circuit motif in Drosophila, and is computationally simpler than commonly used signal processing algorithms, such as fast Fourier transform-based strategies. Moreover, the authors also propose a complete model of the Drosophila head-direction system by combining the sensory and motor circuit modules, that together implements the goal-directed navigation behavior. This study has the potential of connecting group-theoretical perspectives of representation learning with the development of planning systems in artificial agents, as well as explaining the circuit motifs in biological systems that implement such goal-directed navigation behavior.", "questions": "1. Given the current framing of the introduction and the rest of the paper, I am wondering if Section 2 (albeit useful to have for completeness) is slightly distracting from the core contributions of the work. Specifically, Section 2 helps the reader grasp the theoretical framework of group operator search but isn't really necessary for appreciating the rest of the paper. Perhaps, this section would be more relevant if the authors could demonstrate the utility of their framework in more abstract group operator settings. I do acknowledge that the authors touch on this aspect in their limitations section as well.\n\n2. In Eq 19, should it be $dx'$ instead of $dx$?\n\n3. Why did the authors assume equal norm and equal sum of responses for all stimuli? Preserving the 1st and 2nd moments of responses seems to require a layernorm-like operation. Is there experimental evidence for this statistic in the Drosophila system? Would the authors suspect a role of inhibitory cells in maintaining this sort of \"balance\"?\n\n4. The authors assume uniform coding, i.e. all stimuli are represented with equal fidelity/accuracy. How true is this in the Drosophila head-direction system? Can this assumption be removed for the proposed framework to be applicable to other systems?\n\n5. Section 4, Page 8 introduces a new notation $r_s+$, which isn't grounded/related with quantities introduced before. Can the authors clarify why this was required and what does it correspond to in terms of $u(s)$? As such, why is there a need for this extra population of neurons in the proposed model, when you already have access to $u(s)$ from the ring attractor? Perhaps, I am missing something in the explanation here.\n\n5. How is $u(h)$ obtained from the ring attractor? Is there a separate memory system that stores the goal representation? \n\n6. Minor typo: Line 171 should start as \"computes\". There are also a few referencing issues, wherein the correct equations are not referenced in text. Apart from these minor typos, the paper is very well written.", "limitations": "There are no immediate negative societal impact of this work. But it would be nice to add an impact statement (per NeurIPS guidelines) in the Conclusions.", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The study is well structured and connects group-theory principles to the desiderate of planning systems. The paper also provides a thorough introduction of the theoretical framework and related ansatz that are required to grasp the contributions of the proposed framework.\n\n2. The paper is quite well written and easy to read. Although there are a few typos, and minor referencing issues, I was able to read the paper and follow the key arguments presented by the authors.\n\n3. This work presents a theoretical neuroscience framework for goal-directed navigation that is strongly grounded in the biological circuitry of the Drosophila head-direction system. Therefore, it presents a strong link between theoretical and experimental neuroscience.\n\n4. The proposed framework is interesting and relevant to the larger AI community, especially in grounding concepts in equivariant representation learning and reinforcement learning-based planning systems.", "weaknesses": "1. The key weakness is in the disconnect between the abstract/introduction of the paper and the rest of it. Although the authors argue that their framework is general and present theoretical arguments that can be extended to abstract group operators, the key results and demonstrations hinge specifically on the 1D head-direction system of Drosophila. Therefore, it is not immediately clear how effective the Lie gorup operator search theory can be applicable to other planning systems, including navigation in 2D environments. \n\n2. The current demonstration in the 1D system assumes an equivariant representation of the stimulus space, such that the changes in representation of head direction follows the rotation of the animal. It is unclear how this assumption can be applied to other abstract planning systems, and how the proposed framework can be applied in case the equivariant assumption no longer holds.\n\n3. Related to the previous point, the current study also assumes uniform coding of the stimulus space, i.e. each head direction is represented with equal fidelity. This assumption certainly doesn't hold for the visual system or even 2D space, wherein representations of certain stimuli or locations are more sensitive to changes than others. It is unclear how the group operator search can be implemented in this case. These issues, I feel, limit the applicability of the proposed framework to very specific systems.\n\n4. The numerical simulations/demonstration presented in this work are limited to 1D stimulus case. It would be interesting to see how the framework extends to higher dimensional stimulus cases, even if that requires reduced faithfulness to the underlying biological circuitry. Adding these experiments will definitely improve the relevance of the proposed framework for the NeurIPS community, and the larger AI community.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "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.\n### Reply to review’s questions:\n1.\t>“Given the current framing ... the authors touch on this aspect in their limitations section as well.”\n\nWe 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 \n- The geometric understanding of motion planning circuit computation (Fig. 3C). \n- 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)\n\nCertainly, 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.\n\n2.\t>“In Eq 19, should it be 𝑑𝑥′ instead of 𝑑𝑥?”\n\nYes, thank you for pointing it out, it should be $dx’$ instead of $dx$.\n\n3.\t>” Why did the authors assume equal norm ... maintaining this sort of \"balance\"?”\n\nThe 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.\n\nYes, 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).\n\nIn 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)\n\n4.\t>” Authors assume uniform coding, ... be applicable to other systems?”\n\nFirst, __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).\n\nBesides 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).\n\nAt 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. \n\n5.\t>” Section 4, Page 8 introduces a new notation 𝑟𝑠+...I am missing something in the explanation here.\n\nSorry 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.\n\n6.\t>” How is 𝑢(ℎ) obtained ... the goal representation?”\n\nWe 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).\n\n7.\t>” Minor typo: ....”\n\nThanks for pointing them out! We will correct them in the revised version.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "WUwADUh6Bw", "reviewer": "Reviewer_9EEd", "summary": "The paper presents a formulation of the problem of motor planning as a Lie group operator search. The authors define a 2-layer neural network implementing the transformation from sensory input to rotation direction, which presents similarities to the neural circuits of the Drosophila responsible for goal-directed navigation.", "questions": "* This paper seems to propose a general theory of motion planning, but 1D rotation is the only example provided. How can this theory be generalized to generic trajectory planning?\n\n* The authors mention that this is \"one of the first studies formulating motion planning as the group operator search  problem\", but cite none. Which other studies do the authors refer to? How does this work fit in the literature?\n\n * What is the meaning of the following sentence: \"This distinction is reflected by the fact that although the operator’s parameter θ∗ = h − s is arithmetic subtraction, the neural operator Rˆ(θ) by no means of arithmetic subtraction between neural responses u(h) and u(s).\" (lines 91-93)?", "limitations": "The limitations of this work are adequately addressed. However, the lines 318-323 are unclear to me. What do the authors mean when they say that \"group representation theory (...) guarantees to find such an operator and deriving corresponding circuit model\"?", "rating": 5, "confidence": 1, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The paper's theoretical approach to describing motor planning is interesting. A strength of this work is the attempt to find a plausible implementation of the 1D rotation operator search as a neural network, which could be realistically implemented in the Drosophila. The mathematical formulation of the problem seems sound, and the authors exhibit good knowledge of the tools employed in the study.", "weaknesses": "The potential of this work is largely limited by the poor quality of the exposition. The paper is difficult to read, because the design choices are not well motivated and the results are presented in a confused manner.\n\nFor instance, the introduction of the theory writes: \"Suppose the motor system generates the same kind of actions from a Lie Group G (e.g., translation or rotation) to transform the world state s.\". Why does this assumption make sense? Why should the motor system generate actions through translation and rotation of the sensory input?\n\nResults are often presented with vague sentences, such as \"And the computational complexity of the feedforward circuit can be even lower than the common signal processing algorithms in certain conditions.\" Which signal processing algorithms? Why are we comparing a solution to a motion planning problem to a signal processing algorithm? Which are these \"certain conditions\"? Or, \"The group operator search formulation of motion planning can provide a normative approach to generalize existing motion planning algorithms into different transformations.\" Which algorithms? Which transformations? How can this generalization be done?\n\nThe description of the numerical simulations is limited to 7 lines (279-285), and visualized in half of figure 4. In my opinion, it is impossible to evaluate the correctness of the experiments and the sensibility of the results, given the minimal amount of information provided.\n\nI recognize that the study might describe meaningful results, and that the mathematical derivations of a neural circuit  to transform sensory input into rotations might be relevant for this research field. However, I do not think the paper provides sufficient elements to understand the model, the theoretical derivations and the experiments. Hence, I doubt the paper can have an impact in its current form.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your review! \nWe want to clarify about the writings first:\n> ‘Suppose the motor system ...through translation and rotation of the sensory input?’\n\nIn 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. \nMeanwhile, 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.\n\nA 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. \n\nThe present study only considers the simple 1D rotation group, and eventually lands on concrete neural circuits in flies.\n\n>” And the computational complexity ... these \"certain conditions\"?”\n\nThe 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.\n\nOf course, we will revise our text accordingly to make it less vague. \n\nThe 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.\n\n>”The group operator search formulation ...this generalization be done?”\n\nWe will revise our manuscript and discuss/explain with more details based on our current discussion in Lines 312-324. \nSpecifically, 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.\nIn addition, our Global Rebuttal lists one example of 2D translation operators. We can incorporate this discussion in the revised manuscript.\n\nHere are our answers to the questions:\n\n1.\t>” This paper seems to propose ... generic trajectory planning?”\n\nDue 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.\n\n2.\t>” The authors mention that ... this work fit in the literature?”\n\nTo 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.\n\nIn 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.\nRegarding 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.\n\n3.\t>” What is the meaning ...”\n\nWe 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 $\\theta^*= h-s$. \nThis 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. \n\nIn principle, finding the neural operator involves below steps\n1. Decode the embedded directions $s$ and $h$ from neuronal responses.\n2. Compute the angular difference, i.e., $\\theta^* = h-s$.\n3. Substituting $\\theta^*$ into the expression of the rotation operator (Eq. 8).\n\nThe 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.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JutTWOX1J9", "reviewer": "Reviewer_a5Tr", "summary": "This paper describes how goal-directed navigation in a group-structured space can be framed as group operator search, describes some neural network implementations of this search, and relates one of these networks to the central complex of the famous fruit fly.", "questions": "There's a few questions in the weakness section above. Additionally, how much do these results hold for discrete groups?", "limitations": "Limitations were discussed, though a few more discussion points could be added based on the weaknesses above.", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "The exposition is clear and, for the most part, well-thought through. The figures especially were nice.\n\nThe relationship between goal-directed navigation in continuous group-structured spaces (hopefully that's the right wording), group operator search, and neural implementations of it as found in the fly is, to the best of my knowledge, novel, and definitely interesting.\n\nThe emphasis on removing assumptions, specifically on the structure of the neural response, the shape of the neuronal nonlinearity, through clever analysis was nice.\n\nThe analysis was high quality and thorough, with good knowledge of the biology.\n\nDeriving the linear response result (equation 30) was cool, though I admit, I stepped through the maths without checking the calculations.\n\nI think the results are of definite interest to the neuroscience community.", "weaknesses": "I think there were three main ares of weaknesses, of variable importance.\n\nFirst, it is very puzzling that the model suggests the speed of the orienting response is minimal when the heading direction and the goal are anti-aligned (fig 3D, its not puzzling why the model does this, its puzzling because a reasonable model shouldn't do this). Westeinde et al., who the authors cite, show that the animals don't behave like this, which they convincingly attribute to the behaviour of the PFL2 neurons. These neurons are not included in the authors framework. It would make the paper stronger if they were included, would this be possible? At the very least this fault mode of the algorithm (not re-aligning when you're 180 degrees off, or even realigning very slowly, seems like a definite fault), should be discussed, and the resolution that PFL2 neurons provide should be mentioned, even if it cannot be fit into the framework.\n\nSecond, I don't agree with the claim that $u(x|s) = u(x-s)$ is equivalent to the representation being equivariant. As a simple example, imagine a 2 neuron population coding for a 1D angle, ${\\bf u}(\\theta) \\in \\mathbb{R}^2$, with tuning properties $u_1(\\theta) = cos(\\theta)$, $u_2(\\theta) = a\\sin(\\theta)$, i.e. the neural activity traces out an ellipse as $\\theta$ varies. There is a consistent rotation matrix that can perform the role of $\\hat{R}(\\theta)$, so satisfying the equivariance condition (equation 5), but each neuron does not have a tuning curve that is just a translated version of some canonical curve. [To show this generalises to many neurons just add many neurons with translated but also amplitude shifted tuning, the same point holds] So it seems that assuming $u(x|s) = u(x-s)$ is a strict specialisation of the equivariance assumption, which on its own is very reasonable. Do you agree with this, or have I misunderstood some of the claims?\n\nThird, the contribution is, as the authors say, limited by the existence of many models of the same type. The main novelty with regards to fly neuroscience seems to be in the mathematical framing and analysis of an already well established model. The general framing of the group operator search, before specialising to the only left or right case, in order to match the fly, also seems novel (though I haven't read everything hehe), and is interesting. That said, it is a shame that the more elegant set of results, on pure operator search, are not the ones that match the biology. Are there settings in which you might expect the representation space algorithm to be implemented? Either way, I like the paper, even if large parts of it are simply an elegant mathematical reframing of existing ideas.\n\nSmall point: on figure 4A it shows E-PG responses going to P-EN, then another arrow leading to PFL3, yet I think in the model E-PG goes direct to PFL3? If that is correct could such an arrow be added?\n\nThen finally two related classes of suggestions, on related literature and typos.\n\nThere seemed to be a few bodies of literature either it would be interesting to know if the authors consider relevant. First, there are many existing models of goal-to-motor circuits for angles, see references 7-14 of Westeinde et al. It seems worth pointing out? Second, there are two recent papers on how artificial neural networks empirically perform a related computation, modular arithmetic, which includes comparison of pairs of cosines to produce the correct output (Nanda et al. 2023, Zhong et al. 2023). I think there might be a  similarity in the algorithmic details, but I'm not sure. Most interestingly the two cited papers find that ANNs use their nonlinearities to perform an approximation to multiplication type calculations, I wonder if the way nonlinearities are used in this work is somewhat similar? Third, there's been a seam of papers that use group and representation theory to understand grid cells (Gao et al. 2018, Gao et al. 2021, Dorrell et al. 2023, Xu et al. 2024), how closely related do you think these approaches are (and they also suggest that equivariant ideas have made it into neuroscience beyond the sensory stage, as sometimes seems to be suggested)? Does this suggest you can directly port your algorithms into 2D navigational problems, and does this match any known cell types? Finally, you could use these normative approaches (or ones like Sengupta et al. 2018), and copious experimental evidence to justify what I see as the $u(x|s) = u(x-s)$ ansatz (see weakness 2).\n\nFinally, a few typos:\nLine 171 'computs'\nLine 202 'outputs' -> to output\nLine 156/7 SI verb conjugation incorrect, similar in 158\nLine 297 gains doesn't makes sense, 'it adds to' perhaps?\nLine 313 Missing 'needs to be...'", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your positive review on our work! Following are detailed replies to your comments.\n\n>”First, it is very puzzling ... it cannot be fit into the framework.”\n\nThank you for mentioning PFL2 neurons, and we are happy to include more discussions in the revised manuscript. Let’s discuss some issues here first.\n- 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.\n- 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.\n- 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.\n\n>” Second, I don't agree with ... some of the claims?”\n\nThank you for your deep thinking. \nWe have carefully checked your example, and found that it is also consistent with our conclusion $u(x|s) = u(x-s)$. \nBased on your example, we define \n$$\nu(x|s) = [u_1(s), u_2(s)]^\\top \n= (\\cos s, a \\sin s)^\\top \n= [\\cos (0-s), a \\cos (\\pi/2 - s)]^\\top  \n$$\nThe preferred directions of the two neurons correspond to $x_1=0$ and $x_2=\\pi/2$.\nThen we define a $2 \\times 2$ rotation matrix $R(\\theta)$ with rotation angle $\\theta$ as,\n$$\nR(\\theta) =\n\\begin{pmatrix} \n\\cos \\theta & -a^{-1}\\sin\\theta \\\\\\\\\na\\sin \\theta &  \\cos \\theta  \n\\end{pmatrix},\n\\ \\ \\ \\textrm{where}\\ \\ \\  a \\neq 0. \n$$\nApplying the rotation matrix $R(\\theta)$ to $u(x|s)$ yields,\n$$\nR(\\theta) u(x|s)  \n= \\begin{pmatrix} \n\\cos (\\theta + s) \\\\\\\\  a\\sin (\\theta + s)\n\\end{pmatrix} \n= u(x|s + \\theta).\n$$\n\nComparing $u(x|s) = (\\cos s, a \\sin s)^\\top $ with \n$u(x|s+\\theta) = [\\cos (\\theta + s), a\\sin (\\theta + s)]^\\top$, it can be checked that \n$u(x|s) = u(x-s)$.\n\nMoreover, it can be checked that the above rotation matrix $R(\\theta)$ satisfies the definition of rotation group (Eq. 4 is an equivalent definition), including \n$$\n\\det (R(\\theta)) = 1, \\\\\\\\\nR(\\theta_1)R(\\theta_2) = R(\\theta_1 + \\theta_2), \\\\\\\\\nR(\\theta)^{-1} = R(-\\theta).\n$$\n\nWe 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 $s$. \nMoreover, 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).\n\n\n> \"Third, the contribution is, ... an elegant mathematical reframing of existing ideas. \"\n\nWe 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. \n\nYes, 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.\n\n>” on figure 4A ... such an arrow be added?”\n\nYes, 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. \n\n\n>” There seemed to be a few bodies ... does this match any known cell types?”\n\nThank you for leading us to these valuable works of literature! \n- Certainly, we will discuss and compare these models in the revised manuscript. \n- 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. \n- 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.\n\n>” how much do these results hold for discrete groups?”\n\nUnfortunately, 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.\n\n>” a few typos: ...”\n\nThank you for pointing out the typos, we really admire your accuracy and meticulousness. We will correct them in our revised version.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
## Global Rebuttal
### Generalizability to complex scenarios
All three reviewers asked how the proposed framework can generalize to complex scenarios. Yes, the proposed framework can generalize to complex scenarios in theory, but the challenge comes from the limited amount of experimental evidence in complex scenarios.

We will use, for example, the 2D translation group as an example to explain the generalizability of our framework. An important step in motion planning neural circuit is approximating the __derivative__ of the objective function over the transformation amount (Eq. 18), which is realized by the spatial difference in the neural circuit (Eq. 20), i.e., the sensory representation is rotated to the positive and negative direction ($\theta_t + \Delta \theta$ and $\theta_t - \Delta \theta$ in Eq. 20). This __spatial difference__ strategy can be also used in the 2D case. In principle, there could be 2 different but equivalent strategies. 
1. In an allocentric representation with an x-y coordinate, where the sensory representation will be shifted along four directions, i.e., $\pm x$ and $\pm y$, and then we expect to have four populations of neurons (the 2D counterparts of PFL3 left $r_{\theta-}$and PFL3 right $r_{\theta+}$). Then the pooling operation will be similar to the 1D case while we will have four output neurons (2D counterparts of DN left $r_{v-}$ and DN right $r_{v+}$) which will generate actions along four directions in the 2D plane. This kind of shift was also considered in Burak, Plos Comp. Bio. 2009, although that paper didn’t study the motion planning circuit. Overall, for groups with intrinsic dimension $M$, we need $2M$ populations of neurons where in each dimension there are two neuronal populations shifting the sensory representation along positive and negative of that direction. 
2. In an egocentric representation with $r$-$\theta$ polar coordinate, assume direction is represented in a $2\pi / 2n$ resolution, then each sensory representation will have 2n copies shifted along 2n different directions. Copies along opposite directions can be used as counterparts of PFL3 left $r_{\theta-}$and PFL3 right $r_{\theta+}$ to calculate the derivative along this direction. 

No matter which strategy shown above is used to compute the spatial derivative, the pooling operation will be similar. The difference is just how neurons are arranged in the stimulus feature manifold to find the spatial derivative.  
Both strategies are plausible in the neural system. Vollan et al. (2024) from Mosers’ lab suggests that rodents’ entorhinal cortex probably exists a system similar to the 2nd strategy above. Their results show that a kind of grid cell conjunctively coding location and heading direction projects to pure grid cells, and this projection is shifted by its preferred heading direction.

__We are happy to include a compressed version of the above discussion in the revised manuscript that has an extra page to strengthen our manuscript if reviewers encourage us do to that.__ The reason for not including them in the submitted manuscript is just due to the page limit. 

### The significance and contribution of the present study
The main contribution of the present study is building an __overarching connection__ between the abstract Lie group theoretical framework to concrete fly’s motion planning neural circuits in goal-directed navigation, providing a deep, novel __understanding__ of neural computation, which is acknowledged by the reviewer a5Tr. Even if the 1D rotation appears simple (mentioned by the reviewer wau7), we would like to point out that no previous study has ever linked a concrete 1D goal-directed circuit to the Lie group operator search problem, which resulted from two challenges
1. There was a lack of analytical results of complex nonlinear recurrent circuit dynamics, and thus it is very hard to analytically identify the operator representation in circuit dynamics.
2. There was not sufficient amount of experimental observation to build the link between data and theory.

We believe the present study successfully solved the two challenges by providing analytical results and linking them to recently discovered fly’s circuits.

### Equivariant representation
Equivariant representation is a general framework of neural representation and has been well studied by many previous literatures. Regarding the reviewer’s concern about its application to other planning systems, we would say many methods have been developed by previous studies to build representations equivariant to various transformation groups, including very general ones, such as discrete rotation and reflection groups in Cohen, et al, ICML, 2016, E(2) group in Weiler, et al, NeurIPS, 2019, Gao, et al, NeurIPS, 2021 etc. With all these methods, we do not worry about the equivariant condition in other planning systems.

Meta-review: The paper proposes a novel approach to motion planning in the brain by framing it as a Lie group operator search problem. The majority of reviewers found the paper to be well-written, clear, and providing the necessary theoretical background to understand the proposed framework. The formulation of navigation as a Lie group operator search is particularly interesting.

Based on my reading of the reviewers' feedback and the active engagement between the reviewers and authors in the discussion (which I highly encourage readers to review), most of the comments and criticisms have been addressed or do not prevent the paper from meeting the NeurIPS acceptance criteria.
Decision: Accept (poster)