What travels at 12 metres per second, is 6-12 metres long and can go on indefinitely before people get bored of watching it? The answer: a Mexican wave. Mexican waves are generated when a few dozen individuals initiate an excited state in others. This excitation spreads locally between adjacent people to produce a wave of standing people that we observe travelling around the stadium. Waves like this are not unique - they occur every millisecond in our bodies and are responsible for the beautiful examples of coordination we observe in nature. But what are waves, what mechanisms produce them, and are there different forms of waves?

Waves are observed when an individual’s behaviour causes a change in behaviour of another individual. We observe a wave when individuals next to each other behave more similarly to each other than other members of the group. For example, A Mexican wave forms when individuals sitting next to each other adopt a different posture to others in the stadium. In oceanic waves, the wave is formed when water molecules transfer energy between adjacent particles. In both these waves, the individuals/particles involved in the wave do not move very far from where they are positioned. Nevertheless, a wave is formed.

These types of waves are called travelling waves and they are common in nature. Our nerves use waves of electrical signals to tell us what part of our body we should move next. These pulses spread when one section of the neuron membrane depolarises a successive section, passing the electrical signal along the nerve. Similar effects are observed in corals, where stationary polyps withdraw into their skeleton, creating a wave of retraction that travels across the colony. Groups of Aphids and Giant honeybees produce spectacular travelling waves across their colonies when individuals adjacent to each other progressively move their abdomens up and down to ward off attacks from predatory wasps. Starling flocks display highly coordinated movements when under attack from predatory falcons. Following an attack, individuals turn away from the threat and waves of turning bodies are observed travelling across the flock.  Fireflies produce waves of light that travel along riverbanks when males synchronise the timing of their flashes to attract females. We have also previously reported on waves of sound that are produced when male cicadas synchronise their calls.

Whilst many waves are produced without individuals moving very far, other waves involve the mass displacement of individuals.  In a recent study, we investigated these types of waves in schooling fish. We found that information on a detected threat was propagated directionally between the first individuals to detect the threat and the rest of the group. The individuals to detect the threat increased their speed and turned around so to travel in the opposite direction to the threat. As they did this, they met oncoming group members. These individuals, observing the change and direction of the startled neighbours, changed their direction too, and started travelling in the same direction as their startled counterparts. This resulted in a dense band of individuals travelling in the opposite direction to the threat. In this system, like the starlings, there is mass displacement of individuals that follow the wave of turning individuals. The figure below shows this:From top to bottom, each progressive frame shows how individuals turn away from the threat. The time between frames is about 0.7 seconds. You can see that the group initially moves towards the threat (the dark stick like object). Then the threat moves out to attack them. Individuals positioned further away from the threat gradually turn away from it, followed by a dense band of fleeing fish.

Waves are ubiquitous in nature and create beautiful displays of coordination in different levels of biological organisation, from cells to animal groups. By examining these waves, we gain important insights into the mechanisms behind self-organisation in biological systems. Moreover, the similarity in waveforms between different systems highlights the robustness of self-organising principles in physiological and behavioural interactions.