[Verse 1] One neuron is a pixel, but the story's way more grand, Gardner's crew in twenty-two zoomed out to understand, Every grid cell's hexagons collapse into one shape, Population living on a donut — call it torus tape. Persistent homology counts holes that just won't close, One loop for east-to-west, one loop for how it flows, Betti numbers whisper what the raw noise can't confess: Circle times circle equals torus, nothing less. [Chorus] Тор, крутись, покажи нам путь (spin the torus, show the way) Population geometry — не забудь (never forget it, don't stray) From a single spike to the shape of the mind, Клетка к карте — так рисуем мир (cell to map, we draw the world in kind) [Verse 2] Hafting two-thousand-eight caught something wild in the maze, Grid cells drifting earlier each theta wave phase, Phase precession — spike creeps forward through the cycle's spin, Painting future footsteps before the paw steps in. String those phases up and what you get's a chain, Theta sequences sprinting way ahead of the brain, Eight cycles a second, compressed little films, Replaying the path before you climb the hills. [Chorus] Тор, крутись, покажи нам путь Population geometry — не забудь From a single spike to the shape of the mind, Клетка к карте — так рисуем мир [Verse 3] Now flatten the chaos, UMAP pulls the thread, Thousands of dimensions collapsed to what's ahead: A donut in disguise where each point holds a place, Decode position purely from the population's face. No need for landmarks, no need for a cue, The manifold itself already told you what to do, Latent space geometry, structure hiding in plain sight, Persistent homology proves the torus isn't slight. [Bridge] Возьми датасет Гарднера, сожми его в кольцо (take the Gardner dataset, compress it to a ring) Run the homology, let the barcode show its face, Загрузи, отобрази — увидишь этот тор (load it up, plot it, see this torus) UMAP or Rips complex, either lands you in that place. [Verse 4] So the lesson climbs from single spike to swarming crowd, Geometry was hidden, now it's speaking loud, Precession writes the sentence, torus holds the grammar tight, Decode the population, put the map into the light. Not memorizing neurons, you're reading manifold instead, That's the shift from cell biology to the shape inside your head. [Chorus] Тор, крутись, покажи нам путь Population geometry — не забудь From a single spike to the shape of the mind, Клетка к карте — так рисуем мир [Outro] Betti one, betti two, тор, тор, тор, Population geometry — that's the entorhinal score.
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