The Solitary Pendulum

Solitary states — where a single oscillator breaks away from a synchronized majority — usually require specific ingredients: time delays, nonlocal coupling, or external forcing. Without these, networks of coupled oscillators tend toward either full synchronization or full incoherence.

Anand, Chandrasekar, and Suresh show that adaptive coupling eliminates the prerequisites. In networks of pendulums where coupling strengths evolve through Hebbian learning or spike-timing-dependent plasticity (STDP), solitary states emerge spontaneously from uniform initial conditions. No delays, no nonlocal coupling, no external perturbation. The adaptation alone is sufficient.

The through-claim: the learning rule generates the structural complexity that fixed coupling requires to be built in. Under Hebbian adaptation, the network self-organizes into two-cluster, solitary, and chimera states. Under STDP, the repertoire shifts — splay, splay-cluster, and splay-chimera configurations appear instead. The same network topology, the same oscillators, different learning rules, different collective states.

The difference between Hebbian and STDP is directional. Hebbian learning reinforces existing correlations — oscillators that fire together strengthen their bond. STDP cares about temporal order — who fires first changes the coupling sign. The temporal asymmetry in STDP produces spatially asymmetric collective states (splays). The temporal symmetry in Hebbian learning produces spatially symmetric ones (clusters).

The adaptation mechanism doesn’t just modify synchronization. It selects which type of synchronization is possible.