[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