[Verse 1]
In the cortical maze where neurons dance
Strogatz shows us the bistable stance
Two fixed points waiting in phase space tonight
One stable basin pulls neurons left, one pulls right
When signals cascade through synaptic gates
The system decides which attractor it takes
Saddle nodes appear then vanish away
Bifurcations mapping the neural pathway
[Chorus]
Read the phase portraits, trace every flow
Lyapunov functions tell us where systems go
Stable or unstable, the eigenvalues know
In cortical columns where computations grow
Linear approximation breaks down the scene
Jacobian matrices reveal what they mean
Khalil's stability, chapter four's gold
Mathematical secrets the brain's networks hold
[Verse 2]
Limit cycles spinning in neural time
Hopf bifurcations cross the parameter line
When columns synchronize their firing rate
Periodic orbits seal the computational fate
The nullclines intersect where flows collide
Vector fields pointing where solutions hide
Poincaré maps reduce the complex dance
To discrete iterations of neural circumstance
[Chorus]
Read the phase portraits, trace every flow
Lyapunov functions tell us where systems go
Stable or unstable, the eigenvalues know
In cortical columns where computations grow
Linear approximation breaks down the scene
Jacobian matrices reveal what they mean
Khalil's stability, chapter four's gold
Mathematical secrets the brain's networks hold
[Bridge]
Asymptotic stability means trajectories converge
While exponential growth makes solutions diverge
The basin of attraction defines the neural state
Where initial conditions determine columnar fate
Center manifolds organize the borderline case
Where critical eigenvalues slow down the pace
[Verse 3]
Global analysis beyond the linear realm
Lyapunov's direct method takes the helm
Energy functions decrease along each path
Proving stability through dynamical math
When perturbations fade and systems return
To equilibrium points where neural patterns learn
The mathematics maps how cortex computes
Through nonlinear dynamics and their complex routes
[Chorus]
Read the phase portraits, trace every flow
Lyapunov functions tell us where systems go
Stable or unstable, the eigenvalues know
In cortical columns where computations grow
Linear approximation breaks down the scene
Jacobian matrices reveal what they mean
Khalil's stability, chapter four's gold
Mathematical secrets the brain's networks hold
[Outro]
From chaos to order, the equations unfold
How cortical columns process and hold
Information flowing through dynamical space
Where Strogatz and Khalil illuminate grace