[Verse 1]
Whittington's paper maps the neural maze
TEM architecture in microscopic ways
Band cells encode the boundaries we trace
Object vector cells mark spatial place
Grid cells tessellate in triangular phase
While border cells define the edge's base
Memory traces through entorhinal space
Each module learns its geometric grace
[Chorus]
Successor representation holds the key
Eigenvectors dancing wild and free
Grid patterns emerge naturally
Structure over space, can't you see
MEC encodes topology
Navigation's deep geometry
From rat brains to AI's destiny
Spatial cognition's symphony
[Verse 2]
Build a gridworld, small and neat
Compute transitions, row by row complete
Gamma discount shapes the temporal beat
Matrix eigenvalues make patterns sweet
Hexagonal firing fields repeat
Like honeycomb beneath your feet
Mathematical proof so concrete
Brain and algorithm finally meet
[Chorus]
Successor representation holds the key
Eigenvectors dancing wild and free
Grid patterns emerge naturally
Structure over space, can't you see
MEC encodes topology
Navigation's deep geometry
From rat brains to AI's destiny
Spatial cognition's symphony
[Bridge]
If structure trumps Euclidean ground
What happens when space wraps around?
Hyperbolic surfaces, graphs unbound
Small-world networks, connections found
Test the rats on twisted terrain
Watch their firing patterns change domain
Non-spatial graphs could break the chain
Prove MEC's computational refrain
[Verse 3]
Design experiments with warped geometry
Place electrodes, record activity
If grid cells fire with new symmetry
On hyperbolic reality
Then structure beats pure geography
MEC learns abstract topology
Neural maps of connectivity
Beyond three dimensions we can see
[Verse 4]
Reinforcement learning meets the brain
TD errors drive the plastic change
Temporal difference breaks the chain
Of where we've been and future range
Graph neural networks learn the same
Manifold structure, hidden strange
Deep learning architectures claim
That space and time can rearrange
[Chorus]
Successor representation holds the key
Eigenvectors dancing wild and free
Grid patterns emerge naturally
Structure over space, can't you see
MEC encodes topology
Navigation's deep geometry
From rat brains to AI's destiny
Spatial cognition's symphony
[Outro]
From Whittington's model to the lab
Eigenvectors tell us what we have
Grid cells dancing, patterns grab
The essence of cognitive maps