[Verse 1] Start with free energy, that's our foundation Gaussian assumptions smooth the equation Minimize surprises, prediction's the game When cortical columns learn without shame Take the derivative, set it to zero That's how we find our computational hero Update rules emerge from this sacred ground Where prediction errors make their sound [Chorus] Free energy down, predictions up high Gaussian bells ringing through the neural sky Three layers dancing, gradients align Predictive coding, your brain by design Error signals flowing backward through time Teaching cortex how to read every sign [Verse 2] Layer one receives the sensory stream Layer two predicts what things should seem Layer three sends context from above This hierarchy's what neurons love Code it up with matrices clean Forward pass shows what you've seen Backward flow adjusts the weights Till prediction compensates [Chorus] Free energy down, predictions up high Gaussian bells ringing through the neural sky Three layers dancing, gradients align Predictive coding, your brain by design Error signals flowing backward through time Teaching cortex how to read every sign [Bridge] But wait, there's a catch in this tale so neat When errors grow large, our math incomplete Linearization breaks at high noise levels Backprop and prediction start to dishevel Test small perturbations, then ramp up the heat Watch equivalence crack under pressure's beat [Verse 3] Compare your gradients, line by line Backprop versus predictive design Small errors show they march in sync Large errors make the connection sink Plot the difference, watch it grow That's where linear assumptions go The cortex knows this boundary well Adaptation's got stories to tell [Verse 4] Bayesian priors set the stage While learning rates turn each page Precision parameters weight the trust In signals clear or noisy dust The brain's own implementation scheme Makes prediction more than just a dream Neural circuits approximate with care The mathematics floating in the air [Chorus] Free energy down, predictions up high Gaussian bells ringing through the neural sky Three layers dancing, gradients align Predictive coding, your brain by design Error signals flowing backward through time Teaching cortex how to read every sign [Chorus] Free energy down, predictions up high Gaussian bells ringing through the neural sky Three layers dancing, gradients align Predictive coding, your brain by design Error signals flowing backward through time Teaching cortex how to read every sign [Outro] Derive, implement, investigate the seams Mathematics meeting biological dreams From functional forms to working code The cortical column's secret mode
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