Title: The motion planning neural circuit in goal-directed navigation as Lie group operator search
Venue: NeurIPS 2024
Accepted: yes
Keywords: Lie group equivariance

motion planning

goal-directed navigation

ring attractor network
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

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 sensoryaction loop implementing goal-directed navigation. Our work for the first time links the abstract operator search with biological neural circuits.

1 Introduction

The information processing in the brain forms a sensory-action loop to interact with the external world (Fig. 1A) [1-3]. The sensory-action loop consists of three modules: the sensory neural circuit module forms a neural representation of the world state, the motor circuit module produces motor actions to change the world states, and in between there is an essential sensorimotor transformation module that plans motor actions based on the sensory input and tasks goals [4, 5]. For example, in goal-directed navigation tasks, the sensorimotor transformation module plans a sequence of motor actions to bring the current sensory state towards a goal state. Extensive neuroscience studies have investigated the

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neural circuit mechanism of motion planning [4-9], 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 [10-13], 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.

To provide theoretical insight into motion planning, the present study defines the sensory-action loop using the (Lie) group theory. Denote by s as a continuous world state, and u ( s ) ≡ u ( x | 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 a motor action g ∈ G to the world state s and updated sensory representation u ( s ) can be denoted by (Fig. 1A; ◦ denotes the group action in below)[14],

R g is the group operator (motor action) changing the world state s (Fig. 1A, green), whose effect on updated sensory responses u ( s ′ ) can be summarized as a neural operator ˆ R g directly acting on the original response u ( s ) (Fig. 1A, blue) that can be regarded as the motor-to-sensory neural feedback in the brain [15]. To represent all world states s under all group transformations, u ( s ) must satisfy Eq. (1) for all g ∈ G , and is called equivariance with the group G (group homomorphism) (Fig. 1B). In the Lie group framework (Eq. 1), motion planning in goal-directed navigation can be formulated as finding an operator ˆ R g to bring the sensory response of the current state u ( s ) into a goal state u ( h ) ,

An intuitive way to find ˆ R g 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 S1, Supplementary Info. (SI)), corresponding to find the peak location in the group convolution (Fig. 1D, bottom) [16, 17],

then the optimal group operator is ˆ R g ∗ . The ⋆ denotes the group g convolution, and 〈 u ( h ) , ˆ R g ◦ u ( s ) 〉 = ∫ u ( x | h ) u ( x | R g ◦ s ) dx is the inner product. In particular, when s is a 1D variable and ˆ R g is a 1D 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 ˆ R g (action) in goaldirected 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 1D 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 [20-23]. We link every neural circuit computation with

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 a 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 a 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 1D 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 a unit circle on the complex plane (Fig. 2A), and can be parameterized by the angle θ (corresponding to g in a general Lie group, Eq. 1),

A group element R ( θ ) ∈ U (1) rotates a 1D stimulus direction s ∈ [ -π, π ) , or its complex representation e is , by θ , i.e., s into s + θ (mod 2 π ), and is denoted as R ( θ ) ◦ s ≜ e iθ e is = e i ( s + θ ) . Based on Eq. (1), a rotation-equivariant sensory response u ( s ) should satisfy (suppress 'mod 2 π ' for brevity),

ˆ R ( θ ) is the neural operator rotating the sensory representation u ( s ) . Since the rotation operator changes s in an additive way, i.e, u ( x | s ) ↦→ u ( x | s + θ ) , it can be checked the equivariant sensory response satisfies u ( x | s ) = u ( x -s ) ≡ u ( s ) [28], where the neuron index x (also called preferred direction) additively interacts with s , implying the neuronal response only depends on the difference x -s (Fig. 1D). The rotation-equivariant neural responses u ( x -s ) have been widely used in neural coding studies [29-31], 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 ˆ R ( θ ) 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 ˆ R ( θ ∗ = h -s ) , in that

whose parameter is the angular difference θ ∗ = h -s . Although computing the angular difference is simple, it is non-trivial to search neural operators ˆ R ( θ ) in neural circuits. This is because ˆ R ( θ ∗ ) is an abstract math object rather than the numerical value θ ∗ , even if it is indexed by θ ∗ . This distinction is reflected by the fact that although the operator's parameter θ ∗ = h -s is arithmetic subtraction, the neural operator ˆ R ( θ ) by no 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 ˆ R ( θ ) 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 ˆ R ( θ ) to reveal its neural representation. Consider an infinitesimal rotation ˆ R ( δθ ) ( δθ → 0 ), whose effect on the sensory response u ( x -s ) is (using first-order Taylor expansion),

ˆ J ≡ -∂ x = -∂/∂x 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

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The neural rotation operator ˆ R ( θ ) is commutative , meaning the composition of two rotations is the same regardless of their order, i.e., ˆ R ( θ 1 ) ˆ R ( θ 2 ) = ˆ R ( θ 2 ) ˆ R ( θ 1 ) . Commutative group operators share a common set of eigenfunctions that can be used to represent group operators. It can be checked { f ω ( x ) = e iωx / √ 2 π, ω ∈ Z } is the normalized eigenfunction set of ˆ J ,

with each f ω having eigenvalue ρ ω ( ˆ J ) ≜ -iω . The eigenvalue ρ ω ( ˆ J ) can be regarded as the representation of ˆ J based on f ω . Hereafter, we call the space spanned by the eigenfunctions { f ω } as the representation space . It is worth noting that { f ω } are Fourier bases, which are widely used in frequency analysis in signal processing to extract each frequency component [18, 19]. In contrast, the current study uses Fourier bases to represent the generator ˆ J and operator ˆ R ( θ ) .

Since rotation operators can be composed by the generator ˆ J via the exponential map (Eq. 8), the neural operator's representation based on the eigenfunction f ω is derived as (details at SI. Sec. 1.2),

whose eigenvalue e -iωθ ≜ ρ ω ( θ ) is regarded as the representation of rotation operator ˆ R ( θ ) . Comparing Eqs. (9 and 10), we see the exponential map from the generator to the operator, i.e., ˆ R ( θ ) = exp( θ ˆ J ) , also exists in the representation space, i.e., ρ ω ( θ ) = exp( -iωθ ) = exp[ θ · ρ ω ( ˆ J )] . The closed-form formula of ρ ( θ ) is the inner product between the rotated and the original eigenfunctions (multiplying both sides of Eq. (10) by f † ω ( † : conjugate), and integrating over x ),

There is an one-to-one mapping between each operator ˆ R ( θ ) and its representation ρ ω ( θ ) . The advantage of using eigenfunctions to represent abstract operators ˆ R ( θ ) 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 f ω ,

corresponding to the inverse Fourier transform. Meanwhile the representation U ( ω | s ) ≡ U ( s ) ( ω is suppressed unless confusion), called Fourier coefficient , is calculated via Fourier transform,

Since u ( s ) = ˆ R ( s ) ◦ u (0) , we have U ( s ) = ρ ω ( s ) U (0) = e -iωs U (0) , with U (0) the representation of u ( s = 0) . Similarly, the representation of the goal neurons' response u ( h ) is U ( h ) = ρ ω ( h ) U (0) . Referring both U ( s ) and U ( h ) by U (0) , their representations are ρ ω ( s ) and ρ ω ( h ) respectively, and are geometrically visualized as two vectors of unit length with angle s and h respectively (Fig. 2C).

With sensory and goal neuronal responses' representations (Eq. 12), the objective function of abstract rotation operators, L ( θ ) = ⟨ ˆ R ( θ ) ◦ u ( s ) , u ( h ) ⟩ (Eq. 3), simplifies into a numerical function as the inner product of representations of neuronal responses and operators (see details in SI. Sec. 1.2),

where the Cauchy-Schwartz inequality is used and ∥ a ∥ = aa † . The Eq. (14) is maximized only if U ( ω | s ) ρ ω ( θ ) = U ( ω | h ) , implying the representation of the required rotation operator is (Fig. 2D),

In the representation space, the neural operator is a numerical ratio with a closed-form solution, which provides an algorithm to find the operator rather than checking the ansatz (Eq. 6). Moreover, the computational complexity of finding operators in the representation space is much lower than the group convolution (Table 1; SI. Sec. 2): the complexity via the representation space is O ( N log N ) with N 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 O ( N 2 ) . Once ρ ω ( θ ∗ ) is found (Eq. 15), it can be multiplied with U ( s ) followed by inverse Fourier transform to rotate the sensory response into the goal direction u ( h ) , i.e., F -1 [ U ( s ) ρ ω ( θ ∗ )] = F -1 [ U ( h )] = u ( h ) (Fig. 2E, dashed lines). Nevertheless, this procedure (Fig. 2E, dashed lines) is physically different from actual motor actions, which corresponds to firstly mapping ρ ω ( θ ∗ ) back to the physical operator ˆ R ( θ ∗ ) and acting on u ( s ) directly (Fig. 2E, solid line). Hence we explore how the neural circuit finds the operator ˆ R ( θ ∗ ) 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 [1, 4], forming a time-continuous process (Fig. 2F),

where s 0 = s is the initial stimulus direction, and s t = s 0 + ∑ t τ =0 δθ τ is the direction after applying rotation sequence with angles δθ 0 to δθ t . The sequential motion planning imposes a sensory-action loop : after the motor system generates a small rotation ˆ R ( δθ t ) at time t , the sensory response updates from u ( s t ) to u ( s t +1 ) , followed by another rotation ˆ R ( δθ t +1 ) , which repeats over time until the stimulus direction s is rotated to the goal h . Differentiating Eq. (16) over t , utilizing the differential form of operators (Eq. 8), the dynamics of the sensory responses u ( s t ) in the sensory-action loop is,

We see the rotation dynamics is determined by the rotation speed v t = dθ t /dt . There are multiple strategies for generating v t sequence and we consider a strategy performing gradient ascent along the objective function (Eq. 3), i.e., the v t at time t is proportional to the gradient of rotation angle θ t ,

with λ determining the step size. The rotation speed is a sine function of the direction difference h -s t and representing rotation group parameter (comparing Eqs. 18 and 15, Fig. 3D).

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3.2 A feedforward circuit for motion planning

The closed-form solution of v t in sequential motion planning (Eq. 18) and the optimal operator (Eq. 15) imply v t can be computed by a feedforward circuit in a single propagation of neural inputs, which would be faster and simpler than a recurrent circuit. We explore how a generic feedforward circuit computes v t (Eq. 18) via receiving sensory response u ( s ) and goal response u ( h ) [29],

where F ( · ) is a nonlinear increasing activation function. w s ( x, x ′ ) and w h ( x, x ′ ) are feedforward weights from sensory neuron u ( x -s ) and goal neuron u ( x -h ) . Two issues are to be resolved for v t computation in feedforward circuits. One is how the feedforward circuit as a nonlinear function of summed neural inputs (Eq. 19) computes L ( θ t ) as an inner product of neural inputs u ( s ) and u ( h ) (Eq. 18). Another is computing the derivative dL/dθ t in the feedforward circuit.

We propose the feedforward circuit approximates the derivative dL ( θ t ) /dθ t as a difference form,

where s ± = ( s 0 + θ t ) ± ∆ θ = s t ± ∆ θ . 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 ± ) ∥ 2 = ∫ u ( x -s ± ) 2 dx where the square function is similar to feedforward circuit's nonlinear output (Eq. 19). By using the form in Eq. (21), the two inner products' difference in Eq. (20) is,

where we used ∥ u ( s ± ) ∥ 2 = ∥ u ( h ) ∥ 2 , i.e., the norm of neural responses is irrelevant with represented directions. Eq. (22) suggests computing L ( θ t +∆ θ ) -L ( θ t -∆ θ ) for v t (Eq. 28) can be achieved by a two-layer feedforward circuit with each layer containing two neuronal populations (Fig. 3A).

In the 1st layer, each neuronal population r θ ± ( x ) computes the square of the sum of sensory and goal inputs, followed by two neurons r v ± at the 2nd layer pooling responses r θ ± ( x ) at the 1st layer.

Receiving rotated sensory input u ( s ± ) by ± ∆ θ can be realized by feedforward weights with shifted connection phase (Fig. 3A, r θ ± receives connections from u ( s ) and u ( h ) with different phases),

where w θ ± ,s ( x, x ′ ) is the weight from sensory neuron u ( x ′ -s ) to r θ ± ( x ) . Eventually, the difference of two neurons at the 2nd layer is proportional to the rotation speed v t (combine Eqs. 20 and 22)

Then each r v ± can drive a corresponding effector (e.g., muscle) that generates actual motor actions to rotate the heading direction clockwise or counter-clockwise.

General nonlinear activation functions . The square function for the first layer neurons r θ ± ( x ) (Eq. 23) doesn'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 ( u ) monotonically increasing with u (Eq. 19). To see the mechanism of general nonlinear activation functions, we expand it at 0 to the second order,

Then the neuronal responses r θ ± ( x v ) can be approximated as,

Finally, the difference of neurons at the 2nd layer is still proportional to the rotation speed v t ,

Again, we used the fact that the summed neuronal activities do not depend on the represented direction, i.e., ∫ u ( x -s ) dx = ∫ u ( x ) dx , and ∫ u ( x -s ) 2 dx = ∫ u ( x ) 2 dx . Notably, the 1st layer neuron r θ ± must have a nonlinear activation function to enable the feedforward circuit to output the rotation speed v t , otherwise (setting F 2 = 0 in Eq. 28), the neurons at the 2nd layer, r v ± will fully cancel. Overall, the architecture of the derived feedforward circuit utilizes the connection phase shift, nonlinear activation function, and pooling of neuronal activities to compute the rotation speed v t .

Comparison with Drosophila's circuit . The derived feedforward circuit is similar to the recently identified Drosophila's goal-directed navigation circuit (Fig. 3A-B) [20-22], which also has a twolayer feedforward architecture receiving the sensory input u ( s ) (E-PG neurons) and the goal input u ( h ) (FC neurons) to compute the rotation speed v t . At the 1st layer in Drosophila's circuit, PFL3 left (right) neuronal population (Fig. 3B, red (green) neurons) combines the shifted sensory input u ( s ) with angle ∓ ∆ θ respectively with the goal input u ( h ) , and output via a nonlinear activation function, which are similar to r θ ± neurons in our feedforward circuit (Fig. 3A). Then the DN left and right neurons at the 2nd layer (equivalent to r v ± neurons) pool all activities of PFL3 right and left neurons respectively, and their response difference determines the rotation speed v t [20-22].

Neural circuit computational complexity . Considering each r θ ± ( x ) neuron only receives one feedforward connection from sensory neurons u ( s ) , i.e., w θ ± ,s is a delta function (Eq. 24), which is the case in Drosophila's circuit. Then in each time step during sequential rotations, the feedforward circuit computes O ( N ) addition and O ( N ) multiplication (suppose a square activation function). Given a fixed goal direction h , the sequential rotation will take O (log | h -s | ) time steps to rotate from the original direction s into h [19]. Hence the total complexity of feedforward circuit during the whole sequential rotations is O ( N log | h -s | ) . When the stimulus direction s is close to the goal direction h enough, i.e., | h -s | < N , the computational complexity of the feedforward circuit is even lower than the widely used fast Fourier transform (Eq. 13) 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. 15, Fig. 2E), the representation space (Eq. 9) provides clear geometrical interpretation of the feedforward circuit computation. Substituting the representation of sensory response u ( s t ) at time t and goal response u ( h ) (Eq. 12) into rotation speed neurons r v ± (Eq. 27),

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const is a constant irrelevant with angles (from the first two terms at Eq. (27), RHS). Geometrically, r v ± corresponds to rotate the optimal operator's representation ρ ω ( ¯ θ t ) by ∓ ∆ θ , and sum the rotated operator with its complex conjugate (mirrored by the real axis, Fig. 3B), and hence it resides on the real axis. The difference between r v + and r v -, exhibited by the length difference of pink and green arrows in Fig. 3C, is a sine function depending on the optimal operator's angle ¯ θ t (Fig. 3D).

4 A full circuit model of the 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 of the sensory-action loop (Fig. 4A) implementing sequential rotations (Eq. 17). For simplicity, the full circuit model only includes the internal motorto-sensory neural feedback (Fig. 1A, blue line) rather than the external loop (Fig. 1A, green lines), with a mildly implicit assumption that the actual sensory feedback from the physical world is the same as the internal motor-to-sensory feedback. To be realistic, all neurons in the full circuit have temporal dynamics, even if our theoretical derivations consider memory-less neurons (Eq. 19). All connection weights have Gaussian profiles spreading over the neuronal space x (SI.Eq. S12), 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.

Sensory and motor circuit modules . The sensory and motor circuit modules are based on a recent theoretical study that analytically linked Drosophila's sensory and motor circuit model in its internal compass circuit with the 1D translation/rotation group [28]. The sensory circuit module u ( s ) is modeled by a ring attractor network that has been experimentally verified in Drosophila's brain (Fig. 4A, blue ring) [32-34]. The ring attractor network uses its rotation-invariant recurrent connections to generate rotation-equivariant sensory representation u ( s ) [28]. The motor circuit module has two neuronal populations r s ± ( x ) (Fig. 4A, P-EN), whose feedback connections to sensory neurons u ( s ) are shifted by ∓ ∆ x towards opposite directions (comparing connections from red and green PB neurons to the ring attractor, Fig. 4A). It was found that these shifted connections give rise to the rotation generator ˆ J and the firing rate difference ∑ x [ r s + ( x ) -r s -( x )] determines the rotation speed v t of sensory representation u ( s ) . Hence we call r s ± as rotation generator neurons hereafter.

Assemble the full sensory-action circuit . The rotation group interpretation of sensory and motor circuits [28] provides a clear interface to connect three circuit modules. Functionally, we use feedforward circuit output neuron r v ± to modulate the gain of rotation generator neuron r s ± , i.e., multiplying each generator neuron r s + ( x ) or r s -( x ) by feedforward circuit output neuron response (scalar) r v + or r v -respectively. The gain modulation of rotation generator neurons r s ± by rotation speed v t was indeed observed in experiments [35]. In addition, the feedforward circuit receives the instantaneous sensory responses u ( x -s ) (Eq. 19) generated by the ring attractor network.

Theoretical analysis of the full circuit . We perform theoretical analysis of the full circuit dynamics to verify whether it implements sequential rotations toward the goal direction (Eq. 17). We perform

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 s 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),

where the difference of motion planning circuit's output neurons r v + -r v -determines the rotation speed v t (compared to Eq. 17). In Eq. (30), w s,s ± and ∆ x denote respectively the peak weight and the weight shift (absolute value) of the connection from rotation generator neuron r s ± to the sensory neuron u ( s ) in ring attractor (Fig. 4A), and similarly for w s ± ,s . Further analysis reveals,

where w θ,s and w v,θ are the peak weights from sensory neuron u ( s ) to neurons r θ ± , and the one from r θ ± to the output neuron r v ± respectively. ∆ θ is the weight phase shift for the weight from u ( s ) to r θ ± (Eq. (24)). R ( ω | 0) is the Fourier transformation of r ( x ) at s = 0 . The derivation details can be found in SI. Sec. 4.3.

We perform numerical simulations of the full sensory-action loop circuit. We fix all circuit parameters and only change the goal direction h that determines goal neurons' responses u ( h ) . Fig. 4B shows the represented stimulus direction s in the ring attractor's population responses u ( s ) indeed moves towards the goal direction h , which is driven by the activity difference between r v ± neurons (Fig. 4C. 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 r v ± (DN neurons).

5 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 1D rotation group as an example, we analytically derive searching 1D rotation operators can be realized by a two-layer feedforward circuit with three circuit motifs of connection phase shift, nonlinear activation function, and pooling, which is similar to the recently identified goal-directed navigation circuit in Drosophila's brain [20-22]. We further assemble the feedforward sensorimotor transformation circuit with sensory and motor circuit modules into a full circuit of the sensory-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 circuit. It 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 to 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 a cosine -profile (pure frequency component at ω = 1 , Eq. 13) [20, 21]. Although the cosine profile is experimentally supported, our theory releases the requirement of neural response profile, e.g., our sensory-action circuit has Gaussian profile neural responses (Fig. 4A). The generalization may reduce the limitation when deploying the neural circuit model in real applications. Second, goal-directed navigation circuit models in Drosophila's research [20-22] haven't composed a 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 (e.g., [17, 24-27]) when building embodied agents.

5.2 Extensions and limitations of the model

Extension to complicated scenarios . For the sake of concision and biological solidity, we only demonstrated the 1D rotation case in the main text, but our modeling framework has the potential to extend to more complicated scenarios. The 2D translation group is 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. 18) by the spatial difference in the neural circuit (Eq. 20), i.e., the sensory representation is rotated to the positive and negative direction ( θ t +∆ θ and θ t -∆ θ 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 ± x and ± y , forming 4 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 2 neuron populations This strategy is supported by a recent experiment ([37]). A simulation result is shown in Fig. S1.

Non-uniform distribution of neurons . Our model considered neurons are uniformly distributed in the attractor manifolds, a simplification widely used in continuous attractor networks [38-41]. Although the assumption holds in Drosophila's brain, the representation is usually non-uniform in other cases. For non-uniform distributions of neurons in the current circuit model (the x in Eq. S12 is irregular), the same neural circuit dynamics (Eqs. S13-S15 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 Eq.S12 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 non-uniform/imperfect in other group structures, e.g., the scaling group would require a 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 λ in our theoretically defined objective function (Eq. 18).

Future work . From the neurobiology perspective, the direct gain modulation from r v ± (DN neurons) to r s ± (P-EN neurons) in the full circuit model (Fig. 4A, Eq.S14) needs to be verified by future experiments. It is likely to be the case in Drosophila's brain because r s ± neurons are gain modulated by rotation speed [32-34]. From the machine learning point of view, as a proof of concept, we only study the theory and corresponding circuit model for 1D rotation operator search, where the group representation theory (Sec. 2.1 - 2.2) appears unnecessary because an 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 (2) , where obtaining the solution of the required operator is non-trivial and no longer intuitive (e.g., [13]). 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 to 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. 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. 2021ZD0200204). The authors thank Yue Liu and Gengshuo Tian for feedback on an early draft of this manuscript.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS:\nThe 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.\n\nWEAKNESSES:\nI 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...'\n\nQUESTIONS:\nThere's a few questions in the weakness section above. Additionally, how much do these results hold for discrete groups?\n\nLIMITATIONS:\nLimitations were discussed, though a few more discussion points could be added based on the weaknesses above.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-06-22"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS:\nThe 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.\n\nWEAKNESSES:\nThe 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.\n\nQUESTIONS:\n* 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)?\n\nLIMITATIONS:\nThe 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\"?", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 1, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-11"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIntelligence 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.\n\nSTRENGTHS:\n1. 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.\n\nWEAKNESSES:\n1. 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.\n\nQUESTIONS:\n1. 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.\n\nLIMITATIONS:\nThere 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.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-20"}