[Verse 1] Cortical columns dancing in the brain Coupled oscillators finding their refrain Kuramoto's model shows the way they bind Phase differences vanish, synchrony aligned Each neuron's frequency begins to drift Toward the common rhythm, nature's gift The coupling strength determines how they meet Mathematical harmony, neural beat [Chorus] Sync up, lock in, oscillations align Lyapunov functions keep the system fine Contract and converge, stability's the key Timescales separate, let the patterns be Echo states remember what came before Cortical columns, computational core [Verse 2] Lyapunov functions paint the energy scene Decreasing always, keeping systems clean For coupled networks, finding that sweet spot Where chaos settles down to what it's not The derivative negative, stability's crown Energy landscapes always trending down Basins of attraction pull the states inside Mathematical compass, neural guide [Chorus] Sync up, lock in, oscillations align Lyapunov functions keep the system fine Contract and converge, stability's the key Timescales separate, let the patterns be Echo states remember what came before Cortical columns, computational core [Verse 3] Contraction theory, Lohmiller's elegant view Slotine's framework makes the chaos through Metrics that shrink as time moves along Distance between trajectories can't stay strong Incremental stability, local and tight Global convergence emerges from sight The Jacobian's eigenvalues must behave Negative real parts, stability's wave [Verse 4] Plasticity rules shape the neural dance Hebbian learning gives connections their chance Spike-timing dependent, the windows align Strengthening synapses in patterns divine Homeostatic forces keep the balance true Critical dynamics in networks we view Self-organized criticality finds its way Between order and chaos, neural display [Bridge] Fast and slow, the timescales drift apart Averaging theorems tear complexities apart Slow variables guide while fast ones blur Mathematical magic, patterns that occur Echo state property holds the memory trace Recurrent connections find their proper place Fading echo principle keeps the past alive Neural reservoirs help the system thrive [Chorus] Sync up, lock in, oscillations align Lyapunov functions keep the system fine Contract and converge, stability's the key Timescales separate, let the patterns be Echo states remember what came before Cortical columns, computational core [Outro] From micro-circuits to the global brain Mathematical beauty breaks the neural chain Distributed units compute and store The mathematics opens up the door
← Cortical Harmony's Grand Design | Towers That Contract and Sync →