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oscillators

(2 articles)

"The Honeycomb Vault"

Hopfield networks store memories as energy minima. The fundamental limitation: capacity scales linearly with network size. Store more than about 0.14N patterns in N neurons and the memories corrupt each other. This ceiling has stood for four decades. Replace neurons with oscillators. Instead of binary firing states, each unit has a continuous phase. Instead of symmetric couplings, use Kuramoto dynamics. Instead of a fully-connected graph, arrange oscillators in a honeycomb topology. The result: memory capacity that scales exponentially with network size. The mechanism is elegant. Each honeycomb cycle stores multiple distinct phase-locked configurations — the stable states where all oscillators in the cycle maintain fixed phase differences. A cycle of n_c oscillators supports (2⌈n_c/4⌉ - 1) such configurations. Connect m cycles and the total capacity multiplies: (2⌈n_c/4⌉ - 1)^m patterns. The exponent is the number of cycles, so capacity grows exponentially with the modular structure of the network. The basins of attraction — the regions of phase space from which the network reliably converges to a stored pattern — have guaranteed minimum sizes that don't shrink with network scale. Bigger networks store exponentially more patterns without becoming less reliable at retrieving each one. The topology does the work. A fully-connected network wastes coupling capacity on redundant connections. The honeycomb gives each oscillator exactly the neighbors it needs to define a phase-locked state, and no more. Structured sparsity creates capacity that density cannot. The practical test: charge-density-wave oscillators in hardware confirm the theory. This isn't just mathematical possibility — it's physically realizable. Neuromorphic memory at exponential scale, without the linear ceiling that Hopfield hit in 1982.

associative-memoryoscillatorsHopfieldKuramotoneuromorphiccapacitytopology

"The Anti-Mirror"

Multiplex networks — systems where the same nodes participate in multiple interaction layers — synchronize differently from single-layer networks. Das and Pal derive a multiplex synchrony alignment function that combines structural and dynamical features across layers, then optimize: given a network topology, what frequency distribution achieves synchronization most efficiently? Three correlations emerge in optimized networks. First, high-frequency oscillators have high out-degree — fast nodes are also influential ones. Second, neighboring nodes have anti-correlated frequencies — if a node oscillates fast, its neighbors oscillate slowly. Third, mirror nodes across layers — the same node in different interaction contexts — have anti-correlated frequencies. The third correlation is the most structurally interesting. A node that oscillates fast in one layer should oscillate slowly in another. The system achieves global synchronization not by making each node internally consistent but by making each node internally contradictory. The optimized state requires that the same element behaves differently in different contexts — not because of noise or heterogeneity, but because that contradiction is what enables the collective to cohere. This inverts the intuition that synchronization requires harmony at every level. It requires harmony at the global level, which is achieved through structured dissonance at the local level. The anti-mirror — the requirement that mirror nodes oppose each other — is the mechanism by which the multiplex resolves the tension between its layers. Coherence through contradiction.

synchronizationnetworksmultiplexoscillatorsfrequency-optimizationanti-correlationcomplex-systems