Entrainment — the phase-response curve and the Arnold tongue

A brief pulse advances the clock (up to 0.185022 of a cycle) or delays it (down to -0.129012) depending on phase, with a dead zone — the canonical biphasic circadian light PRC, emergent with no fitting. A periodic zeitgeber locks the clock, and the locking range widens monotonically with strength ([5, 11, 15, 15, 15] detunings across the sweep): an Arnold tongue.

Entrainment is phase resetting, not driving. The substrate reproduces the canonical light phase-response curve (advance / delay / dead zone, correct sign) and the Arnold-tongue widening of the locking range with zeitgeber strength. The PRC is also the clock's control law: it bounds re-alignment to about an hour a day, the limit the misalignment and chronotherapy chapters both turn on. Mechanism and tongue verified; absolute period a cited anchor.

A pulse advances or delays depending on when it lands

A free-running clock is entrained by resetting its phase. Deliver a brief pulse at many phases of the cycle and measure the shift of the next beat: the response is biphasic. A pulse in one part of the cycle advances the clock (up to 0.185022 of a cycle), a pulse in another part delays it (down to -0.129012), with a dead zone between. That advance/delay-with-a-dead-zone shape is the canonical circadian light phase-response curve — morning light advances, evening light delays — and it falls out of the substrate, sign and all, with no curve-fitting.

Stronger zeitgeber, wider locking range: the Arnold tongue

Entrainment proper is locking to a periodic drive. Sweep a periodic zeitgeber over a wide band of external periods (0.65–1.35× the natural period) at five amplitudes and count how many detunings lock. The locking range widens monotonically with zeitgeber strength: [5, 11, 15, 15, 15] detunings locked across the amplitude sweep. A weak zeitgeber entrains only near the natural period; a strong one captures a broad band. That widening wedge is an Arnold tongue — the standard structure of a forced oscillator, here emergent.

Why this matters for the disease axis

The PRC is also the clock's control law, and it sets a hard limit that the disease and treatment chapters both use: because a single pulse only shifts the clock by a bounded amount, the clock can re-align only ~1 hour per day. A large phase gap therefore takes many days to close — the mechanism of jet lag — and a corrective zeitgeber works only if it lands on the right side of the PRC. Mechanism and tongue are [V]; the absolute period the tongue centres on is the cited [L] anchor.