5D Library › The Default Mode Network, the Entropic Brain and Neural Oscillations · 中文
Chapter 6 Entrainment: Forty Hertz and Binaural Beats
Can rhythm be imposed from outside?
Chapter 2 showed that coupled oscillators can fall into step. A rhythmic stimulus, a light that flickers or a sound that pulses, is a coupling from outside: it can, in principle, pull neural populations towards its own frequency. Brain responses that follow a rhythmic stimulus are well established; the questions are how far the effect spreads beyond the sensory areas that first receive the stimulus, and whether it changes anything that matters.
Forty hertz
Evidence
In mice engineered to develop Alzheimer’s-like pathology, Li-Huei Tsai’s group at MIT found that driving gamma activity at 40 Hz, first optogenetically and then non-invasively with light flickering at 40 Hz, reduced amyloid-β in the visual cortex and changed the shape and activity of microglia (Iaccarino et al. 2016). Later studies by the same group combined 40 Hz light and sound.
Evidence
An independent group reported that 40 Hz light flicker did not entrain native gamma oscillations beyond the visual cortex in two mouse models, and that one hour or seven days of it did not significantly change amyloid load or microglial shape (Soula et al. 2023).
Evidence
In people, a feasibility study found 40 Hz light and sound stimulation safe and able to entrain activity in cortical and some deeper regions; a small randomised pilot in 15 people with mild Alzheimer’s dementia, stimulated daily for three months, reported less brain atrophy and some differences in connectivity and memory test performance in the active group, described by the authors as exploratory results that need a larger trial (Chan et al. 2022).
Taken together, the 40 Hz work is a serious line of research with conflicting results in animals and early, small results in people. Larger human trials have been started. It is not yet a treatment, and devices or audio sold as “40 Hz brain enhancement” for healthy people rest on none of this evidence.
Binaural beats
When a tone of 200 Hz is played to the left ear and 210 Hz to the right ear through headphones, many people hear a slow throb at 10 Hz. Unlike the ordinary beats heard when two tones are mixed in the air, this throb is not present in either ear’s sound. It is produced by the brain, in the brainstem nuclei that compare the timing of sound at the two ears, chiefly the superior olivary complex, with responses also recorded further up the auditory pathway (Oster 1973; Ingendoh et al. 2023).
Calculation
Tones of 200 Hz to the left ear and 210 Hz to the right. Neither ear’s signal contains any energy at 10 Hz; the envelope of the two tones mixed in one ear does. With headphones, what cycles at 10 Hz is the difference in phase between the ears.
import math
fs, n = 4000, 4000 # one second at 4 kHz
fl, fr = 200.0, 210.0 # tone to the left ear, tone to the right ear
t = [i / fs for i in range(n)]
left = [math.sin(2 * math.pi * fl * u) for u in t]
right = [math.sin(2 * math.pi * fr * u) for u in t]
def power_at(x, f):
"""Squared magnitude of the discrete Fourier component at f Hz."""
re = sum(v * math.cos(2 * math.pi * f * i / fs) for i, v in enumerate(x))
im = sum(v * math.sin(2 * math.pi * f * i / fs) for i, v in enumerate(x))
return (re * re + im * im) / len(x) ** 2
mixed = [a + b for a, b in zip(left, right)] # both tones in one ear
envelope = [v * v for v in mixed] # a crude rectifier
print(f"10 Hz component, left ear alone: {power_at(left, 10):.4f}")
print(f"10 Hz component, right ear alone: {power_at(right, 10):.4f}")
print(f"10 Hz component, envelope of the mix: {power_at(envelope, 10):.4f}")
# With headphones the two tones never mix in the air: the 10 Hz 'beat' exists
# only as the interaural phase difference, which the brainstem compares.
for ms in (0, 25, 50, 75, 100):
ipd = (2 * math.pi * (fr - fl) * ms / 1000) % (2 * math.pi)
print(f" t = {ms:3d} ms: right ear leads by {math.degrees(ipd):5.0f}°")
# 10 Hz component, left ear alone: 0.0000
# 10 Hz component, right ear alone: 0.0000
# 10 Hz component, envelope of the mix: 0.2500
# t = 0 ms: right ear leads by 0°
# t = 25 ms: right ear leads by 90°
# t = 50 ms: right ear leads by 180°
# t = 75 ms: right ear leads by 270°
# t = 100 ms: right ear leads by 0°
This matters for claims of “brainwave entrainment”. A monaural or loudspeaker beat is a real amplitude modulation of the sound, which the auditory system follows. A binaural beat is a subtle interaural phase difference that has to be computed by the brainstem; the brain response to it is weaker. Whether it then pulls cortical rhythms towards the beat frequency is the question that has been tested.
Evidence
A meta-analysis of 22 studies found an overall medium effect of binaural beats on outcomes in memory, attention, anxiety and pain (Hedges’ ), with larger effects when exposure came before the task and lasted longer (Garcia-Argibay et al. 2019). A systematic review of 14 studies that measured brain oscillations found results consistent with entrainment in five, contradicting it in eight and mixed in one, and described the evidence as inconclusive (Ingendoh et al. 2023).
Calculation
What an effect size of 0.45 means in practice, and the tally of the review of EEG studies.
from statistics import NormalDist
Phi = NormalDist().cdf
def superiority(g):
"""Probability that a random person from the stimulated group scores
better than a random person from the control group (equal variances)."""
return Phi(g / 2 ** 0.5)
def overlap(g):
"""Share of the two distributions that overlaps (Cohen's OVL)."""
return 2 * Phi(-abs(g) / 2)
for label, g in (("small", 0.2), ("binaural-beat meta-analysis", 0.45), ("large", 0.8)):
print(f"g = {g:.2f} ({label}): better than control {100 * superiority(g):.0f} % "
f"of the time; distributions overlap {100 * overlap(g):.0f} %")
# EEG entrainment studies in one systematic review (see the text)
studies = {"consistent with entrainment": 5, "contradictory": 8, "mixed": 1}
total = sum(studies.values())
for k, v in studies.items():
print(f" {k:28s} {v:2d} of {total}")
# g = 0.20 (small): better than control 56 % of the time; distributions overlap 92 %
# g = 0.45 (binaural-beat meta-analysis): better than control 62 % of the time; distributions overlap 82 %
# g = 0.80 (large): better than control 71 % of the time; distributions overlap 69 %
# consistent with entrainment 5 of 14
# contradictory 8 of 14
# mixed 1 of 14An effect of this size means that a person who listened to binaural beats would do better than a person who did not about 62 % of the time, not every time; the two groups’ scores overlap by about four-fifths. And the evidence that the effect works by entraining brain rhythms, rather than by relaxation, expectation or simply listening quietly to sound, is weak. That is why the experiment in Chapter 8 compares binaural beats with another form of quiet listening, rather than with doing nothing.
Hypothesis · not established
Not established. The behavioural effects reported for binaural beats are explained by quiet, focused listening and by expectation rather than by entrainment at the beat frequency. The hypothesis predicts that binaural beats perform no better than an equally engaging non-beating sound or a breath-focused listening practice of the same length, on attention and mood measures, when participants are blind to the hypothesis. It would be refuted if binaural beats outperformed such active comparisons by a worthwhile margin in a pre-registered trial.
References
Fan, Y. (2026). The Default Mode Network, the Entropic Brain and Neural Oscillations: 5-HT2A Receptor Kinetics, Pineal Calcite Microcrystals and the Computational Neuroscience of Meditative Attention [默认模式网络、熵脑与神经振荡] (Version 1.0.0). 5D Library. https://doi.org/10.5281/zenodo.23697320 · Licensed under Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0).