[Verse 1]
In the neural circuits where computations dance
Discrete steps forward, x sub t advance
Next state depends on current and theta's call
X t plus one equals f of x t, parameters enthrall
Continuous flows where derivatives reign
X dot equals f, smooth change without strain
Time's river flowing, gradients guide the way
Cortical columns process night and day
[Chorus]
Fixed points standing still, where motion finds its rest
X star equals f of x star, equilibrium blessed
Jacobian eigenvalues tell stability's tale
Negative real parts mean convergence will not fail
Lyapunov functions falling like autumn leaves
Energy landscapes where the system believes
[Verse 2]
Linearization strips away the complex curves
First-order approximation, what the matrix observes
Eigenspectrum painted across the complex plane
Left half stability, right half growing pain
Contraction mappings pull distances together
Banach fixed-point theorem, convergence weather
Lipschitz constant less than one, the golden key
Unique solutions guaranteed, mathematically free
[Chorus]
Fixed points standing still, where motion finds its rest
X star equals f of x star, equilibrium blessed
Jacobian eigenvalues tell stability's tale
Negative real parts mean convergence will not fail
Lyapunov functions falling like autumn leaves
Energy landscapes where the system believes
[Bridge]
When theta shifts the parameter space around
Implicit function theorem, new fixed points are found
Derivatives of equilibria trace the changing scene
How stable states migrate through dimensions unseen
Cortical computation in this framework dwells
Neural populations ringing mathematical bells
[Verse 3]
From Hopfield networks to attractor neural nets
Dynamical systems theory, the foundation it sets
Basins of attraction carve the phase space wide
Where neural patterns settle, where memories hide
Critical bifurcations mark the boundaries clear
Where qualitative changes in behavior appear
Mathematics guiding how the cortex computes
Distributed processing through dynamical routes
[Verse 4]
Saddle points emerge where manifolds collide
Unstable directions, stable subspaces divide
Phase portraits mapping the flow's eternal dance
Nullclines intersecting in mathematical trance
Competitive dynamics shape the neural code
Winner-take-all networks on the cortical road
[Chorus]
Fixed points standing still, where motion finds its rest
X star equals f of x star, equilibrium blessed
Jacobian eigenvalues tell stability's tale
Negative real parts mean convergence will not fail
Lyapunov functions falling like autumn leaves
Energy landscapes where the system believes
[Outro]
In cortical columns, mathematics unfolds
Dynamical stories that the brain's structure holds