[Verse 1]
In the cortex where the columns rise like towers
Computing signals through their layered powers
Each neural unit holds a secret key
Mathematics sets the patterns free
When Jacobians turn negative and true
Contraction pulls the system through
The symmetric part must dominate
While chaos learns to dissipate
[Chorus]
Contract and sync, wash it out
That's what cortical math's about
Negative definite keeps it tight
Kuramoto brings the light
Echo states forget the past
Contraction makes the learning last
Hierarchies preserve the rule
Distributed computational tool
[Verse 2]
Phase oscillators dance in Kuramoto's hall
Coupling strength determines if they sync or fall
When frequencies spread too wide apart
Critical thresholds break the collective heart
But push the coupling past the edge
Natural rhythms make their pledge
Synchronization takes the stage
Written on the neural page
[Chorus]
Contract and sync, wash it out
That's what cortical math's about
Negative definite keeps it tight
Kuramoto brings the light
Echo states forget the past
Contraction makes the learning last
Hierarchies preserve the rule
Distributed computational tool
[Bridge]
Recurrent networks hold their memories deep
But echo property won't let them keep
Initial conditions wash away
When weight constraints have their say
Parallel combinations hold the line
Hierarchical structures intertwine
Preservation flows from top to ground
Mathematical truth is always found
[Verse 3]
From single columns to the cortical sea
Distributed units working endlessly
Each computation adds its voice
Mathematics guides the neural choice
Forgetting serves a greater plan
Echo washing clears the span
While contraction keeps control
Of the distributed system's soul
[Verse 4]
Reservoir computing holds the key
To understanding complexity
Input layers feed the beast
While readout layers find the feast
Random weights need not be trained
Only outputs are constrained
Liquid state machines arise
From chaos organized and wise
[Chorus]
Contract and sync, wash it out
That's what cortical math's about
Negative definite keeps it tight
Kuramoto brings the light
Echo states forget the past
Contraction makes the learning last
Hierarchies preserve the rule
Distributed computational tool
[Outro]
In every column, every layer deep
These mathematical promises we keep
Contraction, synchrony, and washing clean
The most beautiful system ever seen
[Final Chorus]
Contract and sync, wash it out
That's what cortical math's about
Negative definite keeps it tight
Kuramoto brings the light
Echo states forget the past
Contraction makes the learning last
Hierarchies preserve the rule
Distributed computational tool