[Verse 1]
Start with one dimension, build a bump that holds its ground
Ring attractor for direction, neural spikes in circular bound
Head direction cells are firing, maintaining where you face
Conjunctive cells bring velocity, offset connections set the pace
When the bump begins to travel, check the speed is calibrated right
Neural populations dancing, keeping track throughout the night
[Chorus]
Grid cells map the spatial world, hexagons emerge from code
Bump attractors, oscillations, different paths down neural road
Torus wrapped with boundaries, periodic domains unfold
Two models fight for dominance, which story will be told
[Verse 2]
Extend dimensions, now we're cooking on a torus wrapped tight
Periodic boundary conditions, edges loop back left to right
Equilibrium patterns forming, read the activity you've made
Hexagonal structure crystallizes, grid cell magic on parade
Integration builds position, neural sheets begin to glow
Spatial representation blooming, watch the hexagons bestow
[Chorus]
Grid cells map the spatial world, hexagons emerge from code
Bump attractors, oscillations, different paths down neural road
Torus wrapped with boundaries, periodic domains unfold
Two models fight for dominance, which story will be told
[Verse 3]
Burgess interference theory, three oscillators in the mix
Sixty degrees apart they're beating, velocity input tricks
Wave patterns crash together, constructive interference builds
Hexagonal firing emerges, spatial landscape that it fills
Teleportation breaks the model, sudden jumps destroy the phase
CAN maintains its grid intact, oscillations lose their ways
[Bridge]
Gardner's paper holds the secret, persistent homology reveals
Toroidal manifolds extracted, what the neural data feels
CAN models show clean structure, continuous attractor states
Interference shows disruption, broken when position jumps its gates
[Verse 4]
Place cells anchor spatial memory, landmarks guide the neural fire
Border cells define the edges, environmental wire
Grid scale varies cross the dorsal, ventral gradient running deep
Theta rhythm synchronizes, spatial coding modules keep
From microcircuit computation to the maze navigation task
Neural geometry unfolds the secrets that we ask
[Chorus]
Grid cells map the spatial world, hexagons emerge from code
Bump attractors, oscillations, different paths down neural road
Torus wrapped with boundaries, periodic domains unfold
Two models fight for dominance, which story will be told
[Outro]
From one dimension to the next, neural computation grows
Medial entorhinal mysteries, the spatial brain bestows