[Verse 1] Synapses fire then fade away quick But memory lingers in the network's trick Recurrent loops bring signals back around Creating states where echoes can be found The input dies but dynamics stay alive In circular paths where memories survive Not stored in weights like traditional schemes But held in motion, flowing endless streams [Chorus] Dynamics hold what inputs can't retain States outlive their source, break the chain Point attracts to one, line flows free Ring cycles round in memory Attractors compute what networks need Dynamics plant the memory seed [Verse 2] Point attractors pull all paths to one spot Fixed solutions when the system's hot Line attractors slide along a plane Multiple states but movement's constrained Ring attractors circle without end Oscillations that never bend Each geometry computes its own design Mathematical landscapes so divine [Chorus] Dynamics hold what inputs can't retain States outlive their source, break the chain Point attracts to one, line flows free Ring cycles round in memory Attractors compute what networks need Dynamics plant the memory seed [Verse 3] Strange attractors twist through higher space Chaotic patterns leave their trace Fractal boundaries, infinite loops Where computation forms and groups Torus shapes hold periodic song Limit cycles keep time strong In phase space where the vectors flow Memory patterns learn and grow [Bridge] Trained attractors learn their grooves precise Carved by gradient descent's advice Reservoir computing takes different routes Random dynamics with readout suits Fixed connections, chaotic dance Let the reservoir enhance No training needed in the core Just tap the dynamics, nothing more [Verse 4] Contrast the methods, see the divide Trained systems sculpt attractors inside Reservoirs leave chaos intact Harvest richness, that's the pact Both exploit the temporal flow States that linger, states that grow Memory lives not in the weights But in the dance that never breaks [Chorus] Dynamics hold what inputs can't retain States outlive their source, break the chain Point attracts to one, line flows free Ring cycles round in memory Attractors compute what networks need Dynamics plant the memory seed [Outro] When the signal fades to black Neural echoes circle back In the loops we find the key To computational memory
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