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The motion planning neural circuit in goal-directed navigation as Lie group operator search
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Formulates goal-directed motion planning as Lie group operator search and derives a two-layer feedforward circuit (1D rotation group) resembling Drosophila's navigation circuit, combined into a full sensory-action loop model.
Problem
Unclear neural circuit mechanism and underlying theory for motion planning (motor action selection) in goal-directed navigation, i.e., how neural circuits search for group operators that bring the current sensory state to a goal state.
Approach
Casts motion planning as a Lie group operator search solved in the Fourier representation space, and shows the resulting rotation-speed computation can be implemented by a two-layer feedforward circuit with connection phase shift, nonlinear activation, and pooling, matching Drosophila's PFL3/DN circuit.
Key results
- Feedforward circuit computes rotation speed as difference of two output populations, proportional to dL/dtheta, with analytic derivation.
- Computational complexity O(N log|h-s|) for the feedforward circuit versus O(N log N) for FFT and O(N^2) for group convolution, lower when |h-s| < N.
- Full sensory-action loop simulation: the represented heading direction converges toward the goal direction and tracks a moving goal.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the two-layer feedforward circuit instead of group convolution | U(1) operator search · U(1) operator search | computational complexity: O(N 2) -> O(N log(|h − s|)) | Helped |
| Use the feedforward circuit when |h − s| < N instead of FFT | U(1) operator search · U(1) operator search with |h − s| < N | computational complexity: O(N log N) -> O(N log(|h − s|)) | Helped |
| Use a nonlinear activation function in the first layer (ablation) | Two-layer feedforward circuit · U(1) operator search | rotation speed computation | Helped |
| Replace the nonlinear activation with a linear activation (F2 = 0) (ablation) | Two-layer feedforward circuit · U(1) operator search | rotation speed computation | Hurt |
Surprising findings
- The feedforward circuit's computational cost can be lower than FFT-based search under certain conditions.
Limitations
- Only the commutative 1D rotation group is demonstrated; non-commutative groups such as SO(3) and SE(2) are left for future work.
- Circuit predicts zero speed at anti-aligned heading and goal, i.e., a false nulling problem debated in experiments.
- Analysis assumes uniform neuron distribution; non-uniform case only approximately works.
Experimental setup
Metrics: computational complexity tracking of goal direction in simulation
Baselines: group convolution exhaustive search FFT-based cross-correlation
Builds on: group convolution ring attractor network Drosophila goal-directed navigation circuit (Lyu et al. / Kim et al.)
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