Strong clocks, a quiet chorus

Miraia S. Chiou

Reading · Neuroscience

Strong clocks, a quiet chorus

Discover how strong cellular clocks can cancel the collective rhythm.

2026-09-08 · 12 min read · 1 interactive demos

Imagine eight violinists playing the same long, smooth note pattern. Each rises and falls with perfect commitment. Yet the sound from the doorway barely rises or falls at all. Half the players swell while the others soften.

Now imagine the violinists are cells. The disappearing population rhythm could mean the clocks inside them have weakened. Or every clock could remain strong while their schedules drift apart. An average recording alone cannot settle the question.

This is a particularly beautiful place to bring the Epistemic Physics habit of keeping amount and direction together. A collection of strong contributions can have a small resultant because its contributions point different ways. For clocks, the direction around the circle tells us when in the cycle a cell currently is.

First, earn the word clock

Before gathering a chorus, we need one player who can keep going.

A leaky response follows an input while gradually losing its previous state. Write its state as $x (t)$ , the input as $u (t)$ , and its relaxation time as $τ > 0$ :

τ \frac{d x}{d t} = u (t) - x (t) .

If the input rises and falls periodically, the response does too. It is smaller and later. If the input then becomes a constant $u_{0}$ , the response becomes

x (t) = u_{0} + [x (0) - u_{0}] e^{- t / τ} .

The oscillation disappears. We have built a follower, with a memory of the recent input. To build an autonomous oscillator, a system that keeps cycling under constant conditions, we need another ingredient.

One possible route is delayed negative feedback. A correction based on an old state can arrive after the state has already passed its target. Picture adjusting a shower whose temperature changes several seconds after you turn the tap. You turn too far, then compensate too far the other way.

A minimal mathematical sketch is

\frac{d x}{d t} = - a x (t) - b x (t - d), a, b > 0.

Here $a$ is the immediate relaxation rate, $b$ the delayed correction rate, and $d$ the delay. Both rates have units of inverse time. Try a disturbance $x (t) = e^{λ t}$ , where the possibly complex number $λ$ describes growth and oscillation. Substitution gives

λ + a + b e^{- λ d} = 0.

At a possible oscillatory stability boundary, set $λ = i ω$ , with $i^{2} = - 1$ and angular frequency $ω > 0$ . Separating real and imaginary parts gives

a + b \cos (ω d) = 0, ω = b \sin (ω d) .

When $b > a$ , these permit $ω = \sqrt{b^{2} - a^{2}}$ and a first boundary at $d = \arccos (- a / b) / ω$ . The delay can therefore change whether small disturbances decay. A linear instability alone does not supply a bounded clock. A suitable nonlinear regulation law must also limit growth and support an attracting cycle. Such a repeating orbit is called a limit cycle: nearby states settle onto it, although they can arrive at different points around it.

System	What keeps the motion going?	What happens under constant input?
Leaky follower	The changing external input	It settles to a constant
Delayed linear feedback	An old state drives the correction	Disturbances can decay or grow, depending on parameters
Nonlinear autonomous oscillator	Internal dynamics support a stable cycle	Cycling can persist
Phase-only clock	Rotation is supplied as a model assumption	Its phase keeps advancing

The laboratory below starts at the last row. It lets us study how clocks coordinate after assuming that each cell already has a clock.

Put a clock on a circle

A clock’s phase $ϕ$ is its location within one cycle. One full turn is $2 π$ radians, just as one turn of an ordinary clock face returns to the same direction. A period $T$ is the time needed for that turn, so a free clock advances at angular rate $ω = 2 π / T$ .

Choose a simple observable, $s (t) = \cos ϕ (t)$ . Its amplitude is one. When the phase advances evenly, this observable rises and falls evenly. The mapping from a cell’s actual biochemistry to an observable can be much richer; this cosine is our chosen teaching signal.

For several clocks, use the phase model

\frac{d ϕ_{j}}{d t} = ω_{j} + \frac{κ}{n} \sum_{k = 1}^{n} \sin (ϕ_{k} - ϕ_{j}) .

There are $n$ clocks. Clock $j$ has its own natural rate $ω_{j}$ . The nonnegative coupling strength $κ$ sets how strongly they adjust toward one another. The sine makes a clock slightly behind its neighbors speed up and one slightly ahead slow down. This is an all-to-all Kuramoto model, a deliberately compact description of phase coordination.

Try this prediction: spread the phases evenly and set coupling to zero. Will the mean signal become small because each clock has shrunk, or because their full-sized signals cancel? Then increase coupling and advance several days.

Phase pairing

Quiet the chorus

Pair opposite clock hands until the gold average shrinks below 0.05. Every clock keeps amplitude 1.

Arrange a quiet chorus

Synchrony R = 0.969 · amplitude of each clock = 1

0 of 4 opposite pairs. Try pairing 5 with 1, 6 with 2, 7 with 3, and 8 with 4.

Choose a clock above, then drag its numbered tip or focus it and use arrow keys. Dial labels show hours within a cycle. Gold is the average hand; blue is the cue phase.

Explore coupling, light cues, and individual rhythms

Clock-to-clock coupling 0.12 / hour

A new light cue

Shift the cue by six hours, then deliver a pulse to clocks 1-3. Other clocks respond later through coupling. A pulse advances some phases and delays others.

Elapsed model hours: 0
Synchrony R: 0.969
Individual amount A: 8
Geometric M²: 3.96

Clock 1 outputMean outputCue phase

The clocks sing together: their average nearly matches an individual rhythm. Individual amplitude stays at 1 throughout.

This chosen Kuramoto phase model supplies autonomous rotation from the start; it does not simulate the cellular chemistry that creates a clock. Coupling acts continuously, light only when you press the pulse button. Editing phases restarts the plotted history. No claim about actual hours of jet-lag recovery is encoded here.

The clocks have slightly different periods, so perfectly even spacing does not persist by itself. Coupling can gather their phases. At every stage each individual cosine still has amplitude one. The missing population rhythm is a question of alignment.

The cone hidden in the chorus

Give clock $j$ a positive weight $w_{j}$ . This might represent its contribution to the measured signal. Define

A = \sum_{j} w_{j}, P = \sum_{j} w_{j} e^{i ϕ_{j}}, R = \frac{| P |}{A} .

The complex number $e^{i ϕ} = \cos ϕ + i \sin ϕ$ is simply a two-dimensional unit arrow. $A$ adds their weights. $P$ adds the arrows. The synchrony $R$ compares the resultant length with the available total, so $0 \leq R \leq 1$ .

Place this beside the familiar amount-direction construction:

Quantity	Clock chorus	Evidence-geometric reading
Positive total	$A = \sum_{j} w_{j}$	Amount retained before cancellation
Resultant	$P = \sum_{j} w_{j} e^{i ϕ_{j}}$	Sum of weighted directional contributions
Normalized length	$R = \| P \| / A$	Degree of directional alignment
Cone surplus	$M^{2} = A^{2} - \| P \|^{2}$	Amount squared left after subtracting the resultant squared

The triangle inequality, $| P | \leq A$ , puts $(A, Re P, Im P)$ inside a positive cone. Consequently,

M^{2} = A^{2} (1 - R^{2}) = 2 \sum_{j < k} w_{j} w_{k} [1 - \cos (ϕ_{j} - ϕ_{k})] \geq 0.

This identity is exact for the chosen positive weights. Aligned phases give $M = 0$ . Equal opposite phases give a zero resultant even though their individual weights remain. For eight unit-weight clocks evenly spaced around the circle, $A = 8$ , $P = 0$ , and $M^{2} = 64$ .

The geometry is the same circle used to encode a heading. Its meaning has changed: north versus east describes a spatial direction; morning versus evening describes a phase. Likewise, this phase clock measures position within a repeating cycle. Epistemic exposure time, built from accumulated trials or tracelet amounts, answers a different question. A geometric correspondence does not identify the two time variables.

A light pulse is a question asked at a particular phase

In the demo, a brief cue at phase $θ$ changes a sensitive clock by

Δ ϕ = ϵ \sin (θ - ϕ),

where $ϵ$ is the pulse strength. This chosen phase response curve specifies how a pulse’s effect depends on arrival phase. The same cue advances one clock, delays another, and leaves an already aligned clock unchanged. A phase exactly opposite the cue is another zero of this simplified sine rule.

Only three clocks receive the pulse directly. The rest learn about it through coupling. Try a six-hour cue shift, pulse once, and advance time. Then repeat with weaker coupling. Distinguish three questions: did the sensitive clocks move; did the population stay together; did its average phase follow the new cue? Strong coupling need not optimize all three for every cue and network.

There is a biological reason to care about this distinction. In mice, Hatori and colleagues studied Lhx1, a gene regulating other genes in the suprachiasmatic nucleus, or SCN, a brain region central to circadian coordination. Their experiments linked SCN-specific loss of Lhx1 to disrupted communication and synchrony among oscillator neurons, alongside altered responses to changes in the light schedule. The study separates properties of individual oscillators from those of their network. Hatori et al., 2014.

That invites a useful exploration: a network may resist a disturbance because its members hold one another near the shared rhythm, yet the same coordination can affect how it reorganizes after a new schedule. The particular outcome depends on who receives the cue, the coupling pattern, and each oscillator’s phase response.

Follow the rhythm into new territory

Start with a recording experiment: compare the amplitude of individual cells with the amplitude of their average. Then examine phase differences. Those measurements distinguish weakened clocks from a desynchronized chorus far better than population amplitude alone.

Next, replace the demo’s equal connections with a network. Do a few light-sensitive cells bring the rest along? Can two well-synchronized groups disagree with each other? Their combined $R$ may be small, while each group’s $R$ is large. A global summary then points us toward a hidden grouping.

A more ambitious model would let the positive weights adapt too. A cell could keep its phase but change its influence on the collective signal. Phase and weight would become separate coordinates of adaptation, rather than asking the average rhythm to tell both stories at once.

And underneath all these adventures waits the cellular clock we assumed at the beginning. Joining a nonlinear biochemical oscillator to a phase-coordination model would let us ask when a perturbation merely changes the time on the clock, and when it changes the clock itself.

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