When a mother holds her baby and their heart rates rise and fall together, what carries it? This model builds three answers from the same pacemaker and breathing cells, then asks which bedside measurements could tell them apart.
Perceived
Mom perceives the infant's state and responds. The bond runs through two nervous systems, with a delay.
Physical
Contact acts on the infant's cells directly. Warmth changes pacemaker kinetics; mom's heartbeat pushes on the infant's pacemaker. Nobody has to notice anything.
Alike
Nothing passes between them. Two bodies built alike share a room, its alarms and its rhythms.
Each world is kept pure so its fingerprint is clean. Real life may mix them; the tests are built so a mixture could be taken apart.
Live
The bedside, simulated
0:00
InfantRR — · node sync —MomRR — · node sync —
Outer ring: sinoatrial pacemaker cells. Inner ring: respiratory rhythm cells. A cell lights when it fires. A heartbeat is the moment the ring fires together. The bar under each ring is arousal, the hidden state the cells respond to.
Heart rate, last 90 secondsInfantMomAnother parent in the pod
Breathing, last 20 secondsInspiration rises, expiration falls
Where mom's heart is at each infant beatlocking — · other parent —
A synchrogram. Each dot is one infant beat, placed by how far mom is through her own cardiac cycle. Scattered dots mean the hearts are timing independently. Horizontal bands mean the infant's beats fall at preferred moments of mom's cycle. The locking index is 0 for random, 1 for locked; the other parent gives the chance level. It looks for 1:1, 2:1 and 3:1 and reports the strongest. A 3:2 rhythm also puts three bands on mom’s cycle, so it shows up as 3:1; ratios such as 4:3 or 5:3 would need a different index.
Infant · mom—
Infant · other parent—
Difference—
Rate synchrony is the peak correlation of 5-second heart rate within ±10 seconds of lag, over the last 90 seconds. The other parent shares the room but not the bond, so the difference is the part the room can't explain, as long as both adults take in the room alike. They are built alike here, but a mom who is asleep reacts less to the alarms and one out of the room misses them, and then the difference averages below zero even in the alike world, where there is no bond, though any single window can still read above zero. One window is noisy. The tests below average many sessions.
Small factors
Perceived world. How strongly mom is moved by the infant's distress and acts on it. Zero means she notices nothing.
Perceived world. Time between the infant's change and mom's response. Timing decides whether a response settles or overshoots.
How fast mom settles her own arousal. Slow to anger, in James's terms.
Physical world. Degrees above the warmer's set point. Skin-to-skin, the infant's temperature drifts toward it over about seven minutes, and the pacemaker runs about 8% faster per degree.
Physical world. How hard each maternal heartbeat pushes on the infant's pacemaker cells, in rad/s at the peak of the pulse. The infant's cells run at about 16 rad/s on their own. Test 5 shows what each value means.
Coupling between sinoatrial cells. Below about 0.3× the cells stop firing together and the rhythm turns irregular.
Resting parasympathetic brake on the pacemaker. Higher tone, slower heart, more room to respond.
The live bench uses these at once. The tests use them when you press Run.
Six tests
What could tell the worlds apart
Each test runs many simulated sessions in all three worlds. A test is useful only if the worlds give different answers. Results below use the default settings until you run them with yours.
Test 1
Shared room or shared bond?
Thirty five-minute sessions per world, skin-to-skin, with alarms and unit noise reaching everyone in the pod. Each dot is one session's peak rate synchrony. The infant's heart rate is compared with three people: mom, another parent in the same pod at the same time, and a mom from a different session (the usual surrogate).
Default settings
At the bedside
Record a second adult in the same pod at the same time: another parent, or a nurse sitting nearby. Pseudo-pairs built from other days remove the room along with the bond, so beating them proves only that the room is shared. Match that adult to mom as closely as you can. Someone calmer, noisier or out of earshot of the alarms correlates differently with the infant even when no bond exists, so subtracting correlations can manufacture a difference or hide one.
Test 2
Who moves first?
The same sessions, now asking about direction. For every lag from −15 to +15 seconds, the infant–other-parent correlation is subtracted from the infant–mom correlation. What remains is the part of the timing the room can't explain, provided both adults take in the room alike (in these sessions they do). Positive lag means the infant changed first.
Default settings
At the bedside
The direction here is an assumption built into the perceived world: the infant's distress reaches mom faster than mom's calm reaches the infant. Real recordings could show the opposite. The test is whether any consistent lag survives after the room is subtracted.
Test 3
The ventilator as a control
On controlled ventilation, the machine sets the infant's breathing. If the bond runs through arousal, it should disappear from breathing and persist in heart rate. Bars show the bond-specific rate synchrony (mom minus other parent) in each channel, with standard error.
Default settings
At the bedside
On the ventilator, breathing synchrony is zero by construction. The informative result is heart-rate synchrony that survives. Ventilation also travels with sedation and severity, so the cleanest comparison is the same infant held before and after extubation, at the same time of day.
Test 4
Is more coupling better?
Perceived world. Responsiveness is swept from 0 to 3. A heel stick lands at 80 seconds. Recovery is the first time the infant's heart rate is back within 10 bpm of baseline and stays there 15 seconds; a later relapse doesn't count. The lower panel shows rate synchrony at each setting. The stored curves use 200 runs per point and a run with your settings uses 30; the band is the middle half of runs. In the physical world this sweep does nothing: responsiveness isn't in its equations.
Default settings
At the bedside
Measure recovery, not harmony. After a heel stick or suctioning, time how long heart rate and saturation take to return to baseline when the infant is held by mom versus lying in the bed. Code mom's response latency from video. Jaffe, Beebe and colleagues found that a mid-range degree of mother–infant vocal-rhythm coordination at 4 months predicted secure attachment at 12 months; higher coordination predicted disorganized or anxious-resistant attachment, and lower coordination avoidant attachment. The model gets a similar shape from two mechanisms of its own: a delayed response that overshoots, and a capacity to soothe that gives out as mom’s own arousal climbs. That shows the shape can arise this way, not that it does. A different measure, how fast the last of the arousal fades from the pair as a whole, is best at a much lower responsiveness (below 0.1 at the default settings). Above that, what fades last is mom’s own arousal: soothing the infant can’t speed it up, and her arousal leaking back to the infant slows it a little. That measure describes the pair’s return after something that stirs mom directly, such as an alarm, not the infant’s heart rate after a heel stick.
Test 5
Beats, not rates
If mom's heartbeat pushes on the infant's pacemaker through contact, the signature isn't in rates. It's in timing: infant beats fall at preferred moments of mom's cycle, near a 2:1 ratio. The phase-locking index measures that from two sets of beat times. Dots: one session each, from the Test 1 sessions, against mom and against the other parent. Then the index against the strength of the push, in the physical world.
Default settings
At the bedside
This needs beat times from both ECGs, not monitor heart rates. Plot the synchrogram and compare against twin surrogates. Van Leeuwen and colleagues did exactly this with maternal and fetal magnetocardiography in six pregnant women breathing at paced rates. They found brief epochs of 3:2, 5:3 and 4:3 locking, and at fast breathing rates more of them than in surrogate data, but no evidence of a direct effect of maternal breathing movements on fetal heart rate. An earlier, larger study of theirs found about as many epochs in surrogates as in real pairs, and a later one concludes the coupling is present but generally weak; how it travels is not established. That is what weak forcing near a ratio produces: intermittent locking, not a locked rhythm. After birth, the medium changes from amniotic fluid and tissue to air and skin.
Test 6
The first fifteen minutes
Three minutes at the bedside, then skin-to-skin for twelve. Top: infant heart rate relative to the pre-contact baseline, averaged over fourteen runs, with mom awake (solid) and asleep (dashed). Middle: how much that heart rate wanders from run to run. Bottom: infant temperature. No alarms.
Default settings
At the bedside
The cheapest discriminating experiment is a sleeping mother. Kangaroo-care sessions where mom falls asleep happen anyway. If the infant's response to contact survives her sleep, it isn't running through her responsiveness. Log a skin temperature probe alongside heart rate, because the heat channel is small and slow and a few beats per minute will hide in noise without it. The kangaroo-care literature also reports the mother's chest temperature adjusting to the infant's; this model doesn't include that, and it would be worth measuring.
How it's built
Three levels, each built from the one below
CellsLevel 1
The sinoatrial node is 24 pacemaker cells (16 in the batch tests, to keep them fast). Each is a phase oscillator with its own natural rate (about ±7% scatter), joined to the others by gap junctions. A heartbeat is the moment the population completes a cycle together. Breathing comes from 12 respiratory rhythm cells (8 in the tests) built the same way. Vagal tone slows the pacemakers and sympathetic drive speeds them. Each inspiration briefly withdraws vagal tone, which produces respiratory sinus arrhythmia without being programmed in. Temperature scales every pacemaker's rate at Q10 = 2.2.
PersonLevel 2
One hidden variable, arousal, drives vagal withdrawal, sympathetic drive and breathing rate. It relaxes toward baseline (about 30 s for the infant, 20 s for an adult) and carries its own noise. A heel stick raises the infant's arousal. An alarm raises everyone's. Asleep, mom's arousal settles lower and she perceives nothing.
Pair, perceivedLevel 3
Mom perceives the infant's distress after a delay. It raises her own arousal (contagion) and drives a soothing action that lowers the infant's arousal over about four seconds. Her capacity to soothe falls as her own arousal climbs. Her arousal also leaks back to the infant (mirroring). All worlds share a slow unit noise (light, sound, activity) and the alarms. Singing or prayer adds a small calming input and raises the infant's vagal tone by 12%, a placeholder for the channel a music therapy study would measure.
Pair, physicalLevel 3
Two channels, both needing skin-to-skin contact and neither needing anyone to notice. Heat: the infant's temperature drifts toward mom's chest with a time constant of about seven minutes, and the pacemaker cells run faster as they warm. Pulse: each maternal heartbeat is a brief push on every infant pacemaker cell, shaped like a pressure pulse. In this world the perceived paths are cut; in the perceived world these are.
cell k dθₖ/dt = ωₖ·(1 + 0.25·S)(1 − 0.4·V)·Q10^(T/10) + K_gap·r·sin(ψ − θₖ) − ε·P(ψ_mom)·sin θₖ
V = V₀·clamp(1 − 0.8·A, 0, 1.5)·(1 − RSA·insp), S = 0.1 + 0.5·A⁺, A⁺ = max(A, 0), P(ψ) = e^{2(cos ψ − 1)} − mean
infant dAᵢ/dt = (−Aᵢ + 0.8·U)/30 − C + g·0.04·Aₘ(t−τ) − 0.02·Aᵢ³ + noise
mom dAₘ/dt = (−(Aₘ − A₀) + 0.8·U)/(20/steadiness) + g·0.10·Aᵢ(t−τ) − 0.02·Aₘ³ + noise
soothing dC/dt = (g·0.30·Aᵢ(t−τ)·cap(Aₘ) − C)/4, cap = 1/(1 + (Aₘ⁺/0.8)²)
voice with mom present, adds −0.006 to dAᵢ/dt and −0.004 to dAₘ/dt, and multiplies the infant’s V₀ by 1.12
heat dT/dt = contact·(T_chest − T)/400 + (1 − contact)·(0 − T)/400
perceived g = responsiveness × presence × awake, ε = 0, heat off
physical g = 0, ε = pulse × contact, heat on
alike g = 0, ε = 0, heat off
locking 2:1, to first order, when |ω_infant − 2·ω_mom| < ε·a₂/2, a₂ = 0.19 (second harmonic of P)
the exact range is solved numerically for one forced phase (Test 5)
A note on scale
A cell is mostly water, and it is tempting to reach for Navier–Stokes. But at a cell's size the Reynolds number is about 10⁻⁵. The inertial term that makes those equations hard vanishes, and what remains is Stokes flow: linear, time-reversible, no turbulence. At the cell, water matters as the solvent for the ion gradients that set the pacemaker's rate, and the right equations are Hodgkin–Huxley, not Navier–Stokes. Navier–Stokes is the physics of the infant's aorta, where the Reynolds number runs into the thousands and the flow is pulsatile. It describes each body from the inside. After birth no fluid connects the two bodies; what they share is heat, pressure, sound, light and air.
Limits
What this model can't tell you
Which world is real. All three are built to be equally plausible. The model shows which measurements could separate them and roughly how many sessions it might take. Only recordings can decide.
Any number here as a clinical value. Every parameter was chosen to give realistic resting heart and breathing rates, not fitted to patients. The temperature coefficient and the chest warmth are assumptions to check.
How strong the pulse channel is. The slider's default is set so the live synchrogram shows something; Test 5 says what strength full locking would need and why that is far more than a pressure wave through two chest walls could deliver.
The direction of influence. Test 2's lag follows from assumptions about contagion and soothing.
Much of the physiology. Arousal is one number. Real infants carry pain, sedation, cortisol and illness. An infant with trisomy 18 and a cardiac lesion would change nearly every parameter.
Cellular detail. The cells are phase oscillators. They capture synchronization, not ion channels, action potential shape or conduction.
Life before birth. The fetus floats in amniotic fluid. In the placenta, maternal blood bathes the fetal villi across a thin barrier that passes heat, gases, water and nutrients. The two bloods are normally kept apart, though a few fetal cells cross into the mother. Van Leeuwen's prenatal locking was recorded in that setting, before this model begins.
Ratios other than n:1. The locking index checks the first three harmonics of mom's cycle, so it looks for 1:1, 2:1 and 3:1. In effect it counts the bands the infant's beats form on mom's cycle: a 3:2 rhythm reads exactly like 3:1, and 4:3, 5:3 or 4:1 read near zero. Naming the actual ratio in real recordings needs n:m analysis, as in the prenatal work.
Whose good the response serves. The equations describe how two bodies move each other. Nothing in them distinguishes care from skilled manipulation.
If one of these worlds is the real one, the math can describe the breath. It still won't be the life the breath carries.