And for the unclean they shall take of the ashes of the burning of the purification from sin, and running water shall be put thereto in a vessel.Numbers 19:17
In my essay “Paradox of the Red Heifer,” I proposed a mechanism by which the ashes of the red heifer remove the impurity of death, where I analogized the procedure with resuscitation using a defibrillator. In this essay, I propose an additional mechanism based on the phenomenon of entrainment, which I will explain below.
In physics and biology, entrainment refers to the synchronization of two or more rhythmic cycles. This concept can apply to various systems, from physical phenomena to biological rhythms. Here are a few examples:
Physics. In mechanical systems, entrainment refers to the process whereby two interacting oscillating systems assume the same period. An example of entrainment is Huygens’ synchronization of clocks, where two pendulum clocks on a common wall will oscillate in synchrony. Indeed, this was the earliest documented observation of entrainment made in the 17th century by Dutch scientist Christiaan Huygens. Huygens, credited as the inventor of the pendulum clock, noticed that two of his clocks hanging on the same wall would synchronize their motion if they were placed close to each other. This phenomenon, now known as “Huygens’ entrainment” or “Huygens’ synchronization,” is one of the earliest examples of entrainment.
Biology and Physiology. Entrainment is also seen in the synchronization of organisms to the natural rhythm of the environment. For instance, many organisms have a circadian rhythm—a roughly 24-hour cycle of biochemical, physiological, or behavioral processes. This rhythm can be entrained by natural light-dark cycles over a 24-hour period. An example would be the human sleep-wake cycle, which is generally synchronized with the day-night cycle.
Biological rhythms and their entrainment to environmental factors have been explored for more than a century. Early observations of biological rhythms can be traced back to the 18th century, with Jean-Jacques d’Ortous de Mairan noting the circadian rhythm in plants in 1729. The understanding of biological entrainment developed further in the 20th century with the work of scientists like Colin Pittendrigh, who studied the entrainment of fruit flies to light-dark cycles.
Neuroscience and Psychology. Entrainment is in neuroscience used to describe the process of synchronizing a neural oscillation to an external rhythm. This can be seen in the synchronization of human brainwave activity to certain frequencies of auditory or visual stimuli, a phenomenon used in various types of neuromodulation therapy.
The concept of neural entrainment has emerged more recently with the development of technologies like electroencephalography (EEG) that allow researchers to observe brainwave patterns. Research in the 20th century revealed that brainwave patterns could synchronize with rhythmic sensory input, a phenomenon known as “steady-state evoked potentials” (SSEP).
In our research, we use brain entrainment to “transplant” brain states from one person to another to help achieve the desired brain state, such as sleep, for example (see endnote 6).
Music and Dance. In a social context, entrainment can describe the synchronization that occurs between individuals as they adapt their movements to the rhythm of the music or the movements of others.
In general, entrainment describes the process of a rhythmic or oscillating system influencing another, leading to the synchronization of their rhythms.
The physics of entrainment
The physics of entrainment can be understood in terms of coupled oscillators. An oscillator is a system that moves back and forth between two positions. When two oscillators are coupled, meaning they can influence each other’s motion, under certain conditions, they can become entrained. In this state, the oscillators synchronize and move in a coordinated way.
Let us use a classic example to illustrate this: two pendulum clocks mounted on the same wall. Here is how entrainment works in this system:
1. Coupling: The clocks are coupled because they’re mounted on the same wall. When one clock’s pendulum swings, it slightly shakes the wall, influencing the motion of the other clock’s pendulum.
2. Synchronization: Over time, these tiny influences can cause the clocks to synchronize. The precise mechanism depends on the initial state of the clocks. They will synchronize in-phase if they’re close to in-phase (swinging in the same direction simultaneously). They will synchronize in anti-phase if they are initially close to anti-phase (one swinging left while the other swings right).
3. Stability: Once synchronized, the system is stable. Any slight disturbance will be corrected over time. For example, if one clock is slightly disturbed, it will be brought back into sync with the other by the small forces transmitted through the wall.
This phenomenon isn’t limited to mechanical systems like pendulum clocks. It can also occur in biological systems (like fireflies flashing in unison), chemical systems (like the Belousov-Zhabotinsky reaction), and many others.
The mathematics of entrainment can be quite complex and usually involves differential equations. A critical factor in these equations is the coupling strength: how strongly one oscillator influences the other. If the coupling is too weak, the system won’t synchronize. If it’s too strong, the system can behave in more complicated ways, potentially including chaos.
The concept of the Arnold tongue comes from the field of nonlinear dynamics. It is particularly relevant when studying the behavior of coupled oscillators, such as in synchronization and entrainment.
The Arnold tongue describes a region in parameter space where a driven (or slave) oscillator is phase-locked (i.e., entrained or synchronized) with a driving (or master) oscillator.
The concept is named after Russian-Jewish mathematician Vladimir Arnold. The “tongue” is a region of synchronization characterized by its boundaries in a graph that plots the frequency of the driven oscillator on one axis and the strength of the coupling (or the amplitude of the forcing term) on the other axis.
Each tongue represents a different synchronization ratio between the driving and the driven oscillators. The shape of these regions resembles a tongue, which is how the phenomenon gets its name.
The Arnold tongue structure reflects the fact that strong coupling or nearness in natural frequencies can lead to synchronization. Synchronization is less likely to occur when the coupling is weak and the frequencies are dissimilar.
These structures are essential in understanding the transition from non-synchronization to synchronization in many complex systems, including biological rhythms, engineering systems, and even climatic patterns. The shapes and sizes of the Arnold tongues can provide insights into the nature and stability of the synchronization in these systems.
The Red Heifer
The Torah instructs us to mix the ashes of the red heifer (parah adumah) with the living waters (mayim ḥaym—spring waters). What is the symbolism of this mixture?
In my essay “Paradox of the Red Heifer,” I explained that, based on Kabbalah and the Hasidic philosophy of Chabad, every person’s soul oscillates in to and fro movements called raẓo vashov (or ratzo v’shuv, that is “running and returning”). This rhythm synchronizes the body with the frequency of the divine light. The (spiritual) impurity of death (tumat met) is a spiritual disease in which, due to contact with a dead body, the body’s rhythm slows down and gets out of sync with the divine rhythm resulting in the loss of spiritual sensitivity. This diagnosis suggests the cure—raising the frequency of the body’s rhythm to get it in sync with the rhythm of the soul.
As we have just learned, entrainment can come in handy. After all, the soul is an oscillator, the body is an oscillator, and they are coupled. All that is needed is to raise the frequency of the body to the frequency of the soul.
It seems to me that the ritual of purifying a person from the impurity of death, involving sprinkling the person with ashes of the red heifer mixed with the living waters, does just that. The ingredients used in this procedure are full of symbolism. The ashes of the red heifer symbolize the initial low frequency associated with the dead body. The living waters symbolize the target frequency associated with the living person and with the person’s soul. As Rashi notes, the cedar tree, which used to be burned together with the red heifer, symbolizes growth. There you have it: the initial slow rhythm of the body growing to reach the target frequency of the healthy soul—all highly symbolic of entrainment.
To illustrate this mystical ritual with a tangible example, in my Neuroenhancement Lab, we are developing a technology to bring comatose patients out of coma using brain entrainment. It is based on exactly the same idea as above–we stimulate the patient’s brain with stimuli (such as light, sound, tACS, or TMS) having the initial frequency coinciding with the slow frequency of the patient’s brainwaves (delta band) to form phase locking (Arnold tongue), after which we slowly increase the frequency of the stimuli to the desired frequency (alpha or beta band). It seems this is precisely what happens spiritually during the ritual with parah adumah, awakening a person afflicted with the impurity of death from the spiritual coma.
Alas, today, we do not have the ashes of the red heifer. Consequently, we are all infected by the impurity of death. Tumat met (the impurity of death) may be the primary reason for the loss of spiritual sensitivity, let alone prophecy, characteristic of the time of exile. What can we do today to minimize this impurity and raise our spiritual sensitivity?
One remedy available to us is immersion in a mikvah—a pool of water connected to mayim ḥaym—living waters. While married women are required to immerse in a mikvah every month to remove the impurity of niddah, it is not mandatory for a man. Prophets and kabbalists have always used immersion in a mikvah to purify themselves before engaging in divine mysteries. It has become customary among ḥasidim to immerse daily, while many non-ḥasidic pious Jews immerse at least once a week before Shabbat. Mikvah does not have the ashes of the red heifer (para adumah), but it does have mayim ḥaym—living waters—one of two ingredients required for purification. (Recall that for Arnold tongue to form, the initial frequency of the master (driving) oscillator must be close to the frequency of the slave (driven) oscillator. However, due to the absence of the ashes of the red heifer, which are symbolic of the slow frequency associated with the impurity of death, the Arnold tongue cannot form, and the entrainment is weak. Therefore, while highly beneficial, mikvah does not remove the impurity of death). As one of the Hassidik masters put it, mikvah is not a miẓvah, and depression is not a sin. Nevertheless, no sin can do more harm to the soul than depression, and no miẓvah can do more good to the soul than mikvah!
Another remedy available today is attaching oneself to a ẓadik (“righteous” man) who is completely attuned to the divine rhythm. This is why ḥasidim emphasize the yiskashrut (“attachment”) to their Rebbe. The Rebbe “entrains” his ḥasidim to the divine rhythm.
In the meantime, we are eagerly awaiting the restoration of the Temple in Jerusalem (Bet Hamikdash), after which it will be possible to prepare the ashes of a red heifer to remove the impurity of death from the Jewish people. When that happens., godliness and spirituality will no longer be abstract ideas but a palpable reality.
 See, for example, Pikovsky, A., Rosenblum, M., & Kurths, J., Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge (University Press, 2001).
 Bennett, M., Schatz, M. F., Rockwood, H., & Wiesenfeld, K., Huygens’s clocks. Proceedings: Mathematical, Physical and Engineering Sciences, 2002, 458(2019), 563-579.
 De Mairan, J. J., Observation Botanique. Histoire de l’Académie Royale des Sciences, 1729.
 Pittendrigh, C. S., On temperature independence in the clock system controlling emergence time in Drosophila. Proceedings of the National Academy of Sciences, 1954, 40(10), 1018-1029; see also Buck, J., & Buck, E., Mechanism of rhythmic synchronous flashing of fireflies: Fireflies of Southeast Asia may use anticipatory time-measuring in synchronizing their flashing. Science, 1968, 159(3821), 1319-1327.
 Thut, G., Schyns, P.G., & Gross, J., Entrainment of perceptually relevant brain oscillations by non-invasive rhythmic stimulation of the human brain. Frontiers in psychology, 2011, 2, 170.; Poltorak Alexander, “Replicating Cortical Signatures May Open the Possibility for “Transplanting” Brain States via Brain Entrainment,” Frontiers in Human Neuroscience, 2021, 15, available online at https://www.frontiersin.org/articles/10.3389/fnhum.2021.710003.
 Adrian, E. D., & Matthews, B. H., The Berger rhythm: potential changes from the occipital lobes in man. Brain, 1934, 57(4), 355-385; Galambos, R., Makeig, S., & Talmachoff, P. J., A 40-Hz auditory potential recorded from the human scalp. Proceedings of the National Academy of Sciences, 1981, 78(4), 2643-2647.
 See, for example, Clayton, M., Sager, R., & Will, U., In time with the music: The concept of entrainment and its significance for ethnomusicology. ESEM Counterpoint, 2004, 1(1), 1-45.
 See Strogatz, S., From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D: Nonlinear Phenomena, 2000, 143(1-4), 1-20;
 See Pikovsky, A., Rosenblum, M., & Kurths, J., Synchronization: A Universal Concept in Nonlinear Sciences, (Cambridge University Press, 2001); Strogatz, S. H., Sync: The Emerging Science of Spontaneous Order. (Hyperion, 2003).
 Blekhman, I. I., Synchronization in Science and Technology, (ASME Press, 2008); Wiesenfeld, K., Colet, P., & Strogatz, S. H., Frequency locking in Josephson arrays: Connection with the Kuramoto model. Physical Review E, 1998, 57(2), 1563; Acebrón, J. A., Bonilla, L. L., Vicente, C. J. P., Ritort, F., & Spigler, R., The Kuramoto model: A simple paradigm for synchronization phenomena. Reviews of Modern Physics, 2005, 77(1), 137.
 The Belousov-Zhabotinsky (BZ) reaction is a well-known example of a non-equilibrium chemical oscillator. This reaction is named after Boris Pavlovich Belousov and Anatol Markovich Zhabotinsky, who are the two main scientists associated with its discovery and understanding. The BZ reaction is perhaps the best-known example of a class of reactions called oscillating reactions. In the BZ reaction, a mixture of several chemicals oscillates between different color states (usually clear to amber to a deep blue) over , a period of time when left undisturbed. The significance of the BZ reaction extends beyond chemistry; it’s been used as a model system for studying pattern formation and wave propagation, phenomena that are also seen in various biological, physical, and geological systems. The oscillations and spiral waves produced in the reaction are examples of “dissipative structures,” which occur in far-from-equilibrium systems and were a key concept in the development of the field of complexity science. The Belousov-Zhabotinsky reaction serves as a physical example of a system that displays temporal oscillations and spatial patterns, similar to the rhythms and waves seen in certain biological systems (like the heart), population dynamics, and even climate models. See Zhabotinsky, A. M., A history of chemical oscillations and waves. Chaos, 19911(4), 379-386; see also Epstein, I. R., & Pojman, J. A., An Introduction to Nonlinear Chemical Dynamics: Oscillations, Waves, Patterns, and Chaos, (Oxford University Press, 1998).
 Arnold, V. I., Geometrical Methods In the Theory of Ordinary Differential Equations, (Springer, 1983).
 See Kuramoto, Y., Chemical oscillations, waves, and turbulence. (Springer-Verlag, 1984).
 Rabbi Schneur Zalman of Liadi, Tanya, Likutei Amarim, Chapter 36. See also Rabbi Sholom Dovber Schneersohn (Rebbe Rashab), Yom Tov Shel Rosh HaShanah : 5666 (a.k.a. Hemshch Samach Vav).
 Ezekiel (1:14).