[Verse 1] Feedforward dies when signals fade away But add recurrence, watch the network play Activity becomes a dynamo state Memory stored in motion, not just weight Hopfield patterns locked in symmetric dance Hebbian rules give every pattern chance Energy function drops to valleys deep Where stored configurations always sleep [Chorus] Attractors pull the chaos into line Point attractors recall what's mine Ring attractors spin around the clock Line attractors integrate and lock Neural states that never let you go Persistent rhythms in the flow [Verse 2] Energy equals negative one half sum Weights times activities, here they come Dynamics descend the energy hill Noisy patterns find their stored refill Pattern completion from a broken cue Error correction pulls the memory through Capacity limit point one four times N Overload spawns spurious valleys then [Chorus] Attractors pull the chaos into line Point attractors recall what's mine Ring attractors spin around the clock Line attractors integrate and lock Neural states that never let you go Persistent rhythms in the flow [Bridge] Continuous families of stable ground Line attractors where integration's found Eye position held by oculomotor grace Evidence accumulates toward decision's place Analog register meets integrator's call Input becomes velocity through it all [Verse 3] Ring attractors close the line around Direction angles neurally profound Head direction system compass true Activity bump shows where you're heading to Central complex in the insect brain Circular memory through sun and rain Working memory holds the active trace Persistent firing keeps thoughts in place [Verse 4] Basin boundaries divide the neural space Competition drives the winner's race Attractor network learns through correlation Storing patterns through association Temperature controls the noise and drift Higher chaos makes the basins shift Critical slowing near the phase transition Memory retrieval needs precision [Chorus] Attractors pull the chaos into line Point attractors recall what's mine Ring attractors spin around the clock Line attractors integrate and lock Neural states that never let you go Persistent rhythms in the flow [Outro] From Hopfield nets to compass rings Attractor dynamics, memory sings Recurrent loops that never die Neural mathematics reaching high
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