17. The Interdisciplinary Journey of Oscillators: From Transatlantic Navigation to Neurophysiology

Abstract

Oscillators have proven profoundly versatile, weaving through numerous disciplines to unlock innovations. In this chapter, we explore how oscillators have had a significant impact on the advancement of science and technology throughout history. Both mechanical and electromagnetic oscillators, along with the mathematical models that describe them, have played a crucial role in various fields such as navigation, communications, and the study of neurophysiology. Key contributions in the science of oscillators that have left a lasting influence on other areas are highlighted, and the influence of figures like John Harrison, James C. Maxwell, Heinrich Hertz, Balthasar van der Pol, Alan Hodgkin, and Andrew Huxley is emphasized. We discuss how they continue to be essential in current research and the development of advanced interdisciplinary approaches.

Introduction¶

Physics encompasses a number of notable concepts that have shaped the course of scientific and technological progress. The ideal gas and the black-body radiator, for example, have had a significant impact on various branches of physics and inspired the development of novel technologies. However, the harmonic oscillator is arguably the most profound and versatile concept of all.

The study of oscillators has a long and rich history, weaving together ideas and discoveries from diverse disciplines including physics, mathematics, engineering, biology, and medicine. Research on oscillators has unlocked fundamental insights about the natural world while also enabling transformative technologies.

This chapter will trace the winding journey of oscillator research over centuries, highlighting interdisciplinary collaborations that produced groundbreaking innovations. We will explore how the quest for accurate timekeeping at sea drove innovations in clock engineering, and how this converged with discoveries in electromagnetism to enable radio technology. Electrical oscillator models would come full circle, providing key insights into biological oscillators underlying heart rhythms and nerve impulses. By tracing this vivid narrative, we will reveal how the incessant human drive to understand and manipulate oscillatory systems has profoundly reshaped civilization. Those interested in learning more are referred to the following sources Pol (1926), Pol & Mark (1928), Bremmer et al. (1960), Sobel (2007), Brown (2020), Parry (1992), Hodgson (1977), Cipolla (2003), Baigrie (2006), Maxwell (2011), Maxwell (2011), Clarke (2011), Shamos (1987), Izhikevich & FitzHugh (2006).

The Longitude Problem and the Development of Clocks¶

In the 2nd century BCE, the Silk Road was established as a network of trade routes linking China with the Mediterranean region, fostering the exchange of goods, ideas, and technologies between Asia and Europe. However, after the fall of Constantinople in 1453, the Silk Road was disrupted, making trade routes to Asia and the Middle East more cumbersome and expensive for Europeans. To circumvent this challenge, explorers set out to discover direct sea routes to Asia, leading to voyages such as Christopher Columbus’ discovery of the Americas in 1492, Vasco da Gama’s expedition that reached India by sailing around Africa in 1498, and the voyage of circumnavigation initially commanded by Ferdinand Magellan and completed in 1522 by Elcano Parry (1992). These journeys were soon followed by others from various European powers, marking the beginning of the Age of Exploration, which spanned from the 15th to the 18th century and produced significant changes in global trade, culture, and politics.

The contributions of Islamic scholars during the Islamic Golden Age, which spanned from the 8th to the 14th century, in the fields of mathematics, astronomy, medicine, and geography are partly responsible for the accomplishments of the Age of Exploration Hodgson (1977). They contributed significantly to the translation and preservation of works from the Greek and Roman eras, which had a major impact on philosophers of the European Renaissance. Since navigation relied heavily on the observation of celestial bodies prior to the 18th century, the contributions in astronomy and mathematics were particularly notable. However, astronomical methods only allowed sailors to determine their latitude, or their north-south position, while determining longitude was a more challenging task. This challenge came to be known as the longitude problem.

Accurate determination of longitude was vital for ensuring safe navigation, particularly during lengthy voyages. Vessels capable of precisely calculating their position faced reduced risks of grounding, collision, or becoming lost at sea. This not only protected the lives of sailors but also safeguarded valuable cargo, minimizing losses for ship owners and merchants. Additionally, efficient navigation reduced travel time, enabling ships to undertake more voyages and enhance profitability. To promote advancements in this crucial area, the British government introduced the Longitude Prize, also referred to as the Longitude Act, in 1714. This competition aimed to encourage the creation of a dependable and feasible method for determining longitude at sea Sobel (2007).

John Harrison, a British clockmaker, successfully tackled the problem of determining longitude. Harrison recognized the importance of accurate timekeeping in solving this problem. Rather than relying on celestial observations, Harrison proposed that a meticulously crafted timepiece could allow sailors to keep track of the time at a reference location and compare it to the local time. By comparing the local and reference times, it was possible to convert the difference in hours, minutes, and seconds into a reliable indicator of longitude.

John Harrison was born in 1693 in Yorkshire, England. From a young age, he displayed exceptional skill as a carpenter and clockmaker. In the 1720s, Harrison moved to London and began designing innovative clocks, including long-case clocks with novel mechanisms. Interested in the Longitude Prize, he set out to design a marine chronometer that could maintain accuracy under the harsh conditions encountered on transoceanic voyages.

Following on the steps of scientists and clock-makers like Christiaan Huygens, Harrison crafted a series of marine chronometers. The first, known as H1, was finished in 1735. It pioneered innovations such as the use of “grasshopper escapement” and temperature compensation through a bimetallic strip. However, issues with operation at sea meant further refinements were needed.

Over the next few decades, Harrison went on to develop the H2, H3 and finally the H4 models, incrementally improving the timekeeping accuracy and durability. The H4 chronometer, completed in 1759, was a compact, high-performance device that fully met the demands of marine navigation. In rigorous sea trials, it kept time to within a few seconds per day, allowing longitude to be calculated to within half a degree.

The Board of Longitude was reluctant to fully award Harrison the prize money, leading to a prolonged dispute. However, his chronometers were widely recognized as a monumental achievement. Their unprecedented accuracy revolutionized navigation and enabled the great voyages of exploration that expanded global knowledge.

At its core, Harrison’s ingenious solution to the longitude problem required developing a sustained mechanical oscillator that could remain perpetual and constant for months under the harsh conditions of ocean voyages. The marine chronometer’s clockwork mechanism is precisely an oscillator in which energy dissipated through friction is continually replenished. This allows the periodic motion to be maintained indefinitely, enabling accurate timekeeping over long periods.

Harnessing Oscillators for Wireless Communication¶

The chronometer’s sustained mechanical oscillations enabled advances in global navigation and mapping. However, the nineteenth century ushered in a new paradigm as scientists sought to harness the power of electromagnetic waves. Just as the perpetual motion of Harrison’s clocks arose from carefully sustaining a harmonic system, steady electrical oscillations would prove vital for powering modern wireless communication.

The pioneering experiments on electricity and magnetism conducted in the eighteenth and early nineteenth centuries laid the foundation for understanding oscillatory electrical phenomena Baigrie (2006). Notable contributions came from Hans Christian Ørsted’s discovery of the connection between electricity and magnetism in 1820, demonstrating that an electric current produces a magnetic field, and the efforts of Michael Faraday to quantitatively relate electric and magnetic forces. However, it was James Clerk Maxwell who, through his seminal treatise published in 1873, unified the previously fragmented theories into a comprehensive framework Maxwell (2011), Maxwell (2011).

Maxwell synthesized prior knowledge within a system of equations describing the interrelation between electric and magnetic fields. A key insight was that these fields could propagate through space as waves, spreading energy outward from a source. One remarkable conclusion was that light itself was simply a high-frequency electromagnetic wave. However, generating such waves to study their properties proved challenging. It was not until 1887 that Heinrich Hertz conclusively demonstrated the existence of electromagnetic waves Shamos (1987).

In his laboratory, Hertz sought to generate and detect the elusive electromagnetic waves predicted by Maxwell’s equations. To achieve this, he devised an electrical oscillator circuit connected to a dipole antenna. When operated at high voltages, oscillations in the circuit induced corresponding oscillations in the antenna, launching electromagnetic radiation. To detect the waves, Hertz used a separate loop antenna connected to another resonant circuit. When placed in the path of the radiation, oscillations were induced in this receiver loop, providing experimental evidence of propagating electromagnetic waves.

The apparatus engineered by Hertz definitively confirmed Maxwell’s theories and opened up a new paradigm in physics. Additionally, Hertz measured properties of the waves including polarization, reflection, refraction and interference, verifying that they obeyed the same rules as visible light. This cemented light’s electromagnetic basis.

Hertz’s experiments proved that coordinated electrical oscillations could be harnessed to generate and receive wireless signals. However, the capabilities of Hertz’s apparatus were limited. While suitable for laboratory studies, more powerful and efficient oscillators would be needed to enable practical applications like radio communication.

Early radio transmitters used crude ore detectors and inefficient spark gap generators to produce electromagnetic radiation. Generating steady, high-frequency electrical oscillations was key to enabling continuous wave radio broadcasts. This relied on certain analogies with mechanical oscillators like those used in chronometers. A resonant LC circuit can exhibit oscillations, but losses due to resistance will dampen the oscillations over time. To sustain continuous oscillations, energy must be continually fed into the system, analogous to rewinding the chronometer’s drive spring. In an electronic oscillator, amplifying elements like vacuum tubes are used to replenish the energy dissipated in the resonant tank circuit. This negative resistance precisely counteracts losses, allowing persistent high-frequency oscillations. By maintaining energy balance in an electrical harmonic system, electronic oscillators enabled modern wireless communication.

The parallels between mechanical and electrical oscillators highlight the continuity in physics concepts across disciplines. Whether weighing a clock’s escapement or balancing amplifiers and dissipation in a radio transmitter, the goal of sustaining an oscillator despite losses united these efforts. Powered by electronic oscillators, radio technology would fundamentally reshape society in the 20th century Clarke (2011).

Modelling Neural Excitation with Oscillators¶

The transmission of signals in nerves and muscles relies on the propagation of electrical impulses known as action potentials. Understanding the nonlinear dynamics underlying excitation in neurons and cardiac cells would require integrating concepts from physics, mathematics and biology. Once again, oscillator models would provide vital insights, this time by mimicking the spikes and rhythms produced by biological cells.

The Dutch physicist Balthasar van der Pol joined Philips Research Laboratories after receiving his doctorate in 1913. During his time there, he made significant contributions to the field of electronics, especially in radio technology. Van der Pol was a key figure in the development of the Philips radio receiver, which was a huge success at the time. According to one of his biographers, “Radio might have remained a field of haphazard empiricism along with wild commercial ventures, but for the influence of men like van der Pol who stressed the need for a more scientific approach” Bremmer et al. (1960).

In 1926, van der Pol derived a nonlinear differential equation to describe the behavior of vacuum tube circuits used in early radios. While analyzing his model equations, van der Pol made an intriguing discovery. When the value of a parameter known as the damping coefficient was small, the system exhibited the familiar traits of a harmonic oscillator. However, as the damping coefficient increased to larger values, the model solutions diverged from the conventional behavior of harmonic oscillators. Instead of following smooth periodic sinusoidal oscillations, the model solutions displayed alternating periods of rapid and slow changes. Van der Pol coined the term “relaxation oscillations” to describe this behavior Pol (1926).

The spiking behavior of Van der Pol oscilator was analogous to the action potentials transmitted along neurons and the cells of the heart pacemaker. Building on this, van der Por created a circuit with three oscillators to reproduce the electrical behavior of the heart. Using this method, he successfully replicated specific cardiac arrhythmias, demonstrating the model’s ability to capture the complex dynamics of cardiac electrical activity Pol & Mark (1928).

In parallel to the oscillator modeling work, new technologies enabled direct measurements of neural electrical signals. The development of operational amplifiers (op-amps) in the early 1940s was a key advance. Op-amps can perform high-gain voltage amplification and mathematical operations like addition, subtraction, integration and differentiation. This made signal processing and analysis much more feasible. In 1949, Kenneth Cole and Howard Curtis invented the voltage clamp technique using an op-amp based feedback circuit to control the voltage across a cell membrane Brown (2020). This allowed the underlying ionic currents to be precisely measured for the first time.

Fundamental discoveries in thermodynamics and physical chemistry by pioneers like Max Planck and Walther Nernst also contributed to understanding the electrical behavior of cell membranes. Nernst’s work led to his eponymous equation in 1889, relating an ion’s equilibrium potential to its concentration gradient. The Nernst-Planck equations described the motion of charged particles, extending diffusion to include electrostatic effects.

Also important was Cole’s adoption of the squid giant axon as a model for studying membrane electrical properties in 1936, following J. Z. Young’s suggestion. The size and spacious lumen of the squid axon made it more amenable to experimental techniques such as intracellular recording, which were previously impractical with smaller axons.

While collaborating with Cole in the 1930s, Hodgkin realized the potential of using the squid giant axon to record intracellular action potentials. Returning to Cambridge in 1938, he recruited the undergraduate Andrew Huxley. Huxley’s outstanding physics and mathematics background paired with Hodgkin’s neurophysiology expertise. Their collaborative project resulted in a series of papers providing quantitative insights into the biophysical basis of the action potential Brown (2020).

The culminating paper in the Hodgkin and Huxley series established the field of quantitative membrane biophysics, integrating mathematical modeling with empirical measurements. Hodgkin and Huxley developed a model of interconnected differential equations and performed painstaking calculations to match the model output to their voltage clamp data. Lacking access to early computers, Huxley relied on a desktop mechanical calculator, achieving remarkable precision.

The Hodgkin-Huxley model marked a breakthrough in neuroscience, offering the first quantitative, mechanistic picture of neural excitability. Their integrative approach combining math and biology yielded insights into ion channel function while spurring new fields like computational neuroscience. This achievement emerged from interdisciplinary collaboration, as experts in physics, mathematics and physiology united around the common goal of demystifying the action potential. The Hodgkin-Huxley collaboration highlights the power of bridging disciplinary divides to propel discovery.

The complexity of the Hodgkin-Huxley model posed challenges for broader usage in an era before digital computers. In the 1960s, Richard FitzHugh at the NIH sought to investigate the model’s mathematical properties using nonlinear dynamics techniques. To solve the equations across parameters, FitzHugh collaborated with John Moore to build an analog computer from op-amps, multipliers and plotters Izhikevich & FitzHugh (2006). This allowed them to graphically visualize solutions, though operating the intricate analog simulator required sophisticated engineering and math skills. The model complexity and lack of digital computing motivated efforts to derive simplified models of neural excitation.

Guided by Cole’s insights, FitzHugh modified van der Pol’s relaxation oscillator equations to distinguish key features of the Hodgkin-Huxley model. The aim was to separate the dynamics of sodium and potassium ion flow across the membrane from the regenerative excitation process. Originally called the Bonhoeffer-van der Pol equations, these were later renamed the FitzHugh-Nagumo equations because, around the same time, Japanese engineer Jin-Ichi Nagumo invented an electronic circuit using tunnel diodes that reproduced the key cubic nonlinearity. The simplified FitzHugh-Nagumo model provided an accessible approximation to the complex Hodgkin-Huxley system. Reconfiguring the analog computer to solve these reduced equations enabled more extensive mathematical analyses of neural excitation dynamics.

The quest to understand the electrical basis of neural signaling exemplifies the potency of cross-disciplinary pollination. Integrating oscillator models from physics and engineering with emerging techniques to probe nerve impulses yielded insights into the mechanisms of neural excitation. Constructing fruitful analogies between biological and man-made oscillators led to accessible mathematical representations that captured the essence of spiking dynamics. By breaking down barriers between physics, mathematics and physiology, pioneers transformed understanding of the fundamental processes underlying thought itself. The intertwined narrative of biological and technological oscillators underscores how synthesizing diverse perspectives propels discovery.

Discussion¶

The history of oscillators unveils the intricate relationship between science and technology. Driven by practical needs, advancements in timekeeping brought about significant changes in navigation. Not only this pursuit had profound political and economical effects but also led to fundamental scientific discoveries.

Likewise, electronic oscillators played a crucial role in powering wireless communication systems that transformed society. Yet, when examined through mathematical models, they also shed light on the electrical rhythms of the nervous system. Time and again, the manipulation of oscillatory systems for engineering purposes unintentionally deepened our scientific understanding, crossing disciplinary boundaries.

The enduring significance of oscillators lies in their versatility as model systems. The concept of the harmonic oscillator provides an approximate description for a wide range of systems, including physical, biological, and engineering domains. This commonality across different areas facilitates analogies that enhance the process of discovery.

By tracing the history of oscillations through the centuries, we gain insight into the very evolution of science itself. While curiosity guides exploration in specialized fields, it’s the process of synthesis that breathes life into the deepest ideas. The journey of oscillators highlights how collaborative human effort, aimed at understanding natural rhythms, regardless of their origin, propels us toward a deeper understanding of nature and contributes to the shaping of our civilization.

Finally, oscillators still play a crucial part in today’s science and technology. High-frequency electronic oscillators act like the heartbeat of modern computers, making sure that different parts work together and share information at the right time. In computer science, they facilitate complex calculations and data processing by synchronizing clock cycles and enabling high-speed data transfer within computer systems. In the internet, they play a central role in precise data transmission, ensuring the uninterrupted flow of information during digital interactions. These unsung heroes quietly contribute to the seamless operation of computer technology and the interconnected world we rely on daily.