Back in the 1990’s, chaos was all the rage. I’m not talking about riots or fashion or music, I’m talking about mathematics. Several popular science books were published on the subject, including ‘Chaos: Making a New Science’ by James Gleick and ‘Does God Play Dice?: The New Mathematics of Chaos’ by Ian Stewart. Chaos theory even made an appearance in the original Jurrasic Park feature film.
One of the main characters in the movie was Dr Ian Malcom, played by Jeff Goldblum, who was a mathematician with expertise in chaos theory. In one particular scene, Dr Malcom explains chaos using an illustration of water droplets moving down the hand of palaeontologist Ellie Sattler. What he was attempting to do, apart from sweet talk Sattler, was to illustrate a fundamental defining property of chaotic systems called ‘sensitivity to initial conditions’. This is the starting point in our brief journey into the mind-blowing world of mathematical chaos. But first, let me explain what chaos has got to do with the brain.
Do you have a chaotic brain?
Neuroscience provides a good deal of evidence that the activity of your brain is chaotic. This may not have come as a surprise to some of us who often feel that what goes on in our brains is chaotic! But the neuroscience evidence is not referring to the common meaning of the term ‘chaos’, which is normally used to describe something which is highly disorganised. On the contrary, in this context we are using the term ‘chaos’ in its specialist mathematical sense to describe something which is highly organised, deeply complex and difficult to predict.
Neuroscientists have measured the activity of neurons in animals in response to various stimuli (e.g. smells, tastes etc.) and observed that this activity has properties that align with those of chaotic systems. It is important to note, though, that there is no conclusive evidence that the activity in the brain is chaotic; the complexity of the brain makes it difficult to draw such conclusions. Also, it is difficult to be sure that the activity you are observing in the brain is driven by chaos rather than randomness. In many respects these two look similar.
What’s the difference between chaos and noise?
Chaos and randomness are quite different from each other. Chaos is deterministic (i.e. predicable with perfect knowledge of system and state), randomness is not. Also, unlike randomness, a chaotic system is sensitive to initial conditions. But because chaos is, by definition, a non-repeating pattern of activity, it can appear to be random to the observer.
Nevertheless, some neuroscientists are convinced that chaotic dynamics underlie the signalling patterns of the neurons on the mammalian brain. Much work has been done in recent years to discover the advantages that chaotic dynamics might bring to the information processing capabilities of the brain. I have contributed to some of this work through studies that I conducted in the field of computational neuroscience. Computational neuroscience involves the creation of biologically plausible computer models of the brain, and in particular of neurons, and studying their behaviour under different conditions. I will introduce some of this work in future posts. But for now, let us return to understanding some of the properties of chaos.
The Birth of Chaos
Chaotic dynamics are a feature of many physical and abstract systems. Classically, the weather is widely regarded as being governed by chaotic dynamics. But the weather is too complex to use as an illustration here. Let’s stick with Malcom’s much simpler illustration of water by imaging that we have a dripping tap. Starting from the off position, if you turn the tap on slightly, enough to allow a small amount of water through, drips will start to form on the tap outlet, and will eventually drop into the sink. To begin with, these might be evenly timed drips: “drip . . . drip . . . drip”. This forms a nice, regular drip pattern which mathematician’s call a period one oscillation.
If you turn the tap on just a little bit more, what you then find is the pattern changes, often to a “drip-drip . . . drip-drip” pattern: two drips in quick succession, followed by a pause, then another two drips. In mathematical terms, the period of this sequence has doubled. If it were possible to have fine-grained control over the amount of water flowing out of the tap, you would find that as you turn the tap on a little bit more, the period doubles again to four, and then 16. After just a few more turns of the tap something remarkable happens, the period of the dripping goes to infinity. In other words, the pattern of time gaps between drips never repeats. The dripping water has moved into chaotic dynamics – a never repeating pattern of activity. What you have observed in this tap experiment is what is called the period doubling route to chaos.
Sensitivity to Initial Conditions
Many other systems exhibit these chaotic dynamics, including irregular heart rhythms, ocean turbulence and the stock market. What unifies them all are the characteristics of chaos, the most significant of which for our purposes is sensitivity to initial conditions. This characteristic is often used to measure the presence and extent of chaos in the dynamics of a system. The property of sensitivity to initial conditions is, as its name suggests, where the evolving state of the system can be radically changed by even the smallest change in its initial state.
To illustrate, let us return to the weather. Back in the 1960’s, meteorologist Edward Lorenz developed a set of mathematical equations for modelling the phenomena of atmospheric convection – the cyclic movement of volumes of air in the atmosphere. In particular, he developed a set of equations for studying the circulation of fluid in a shallow layer that was heated from the bottom and cooled from above. The fluid was assumed to circulate in both horizontal and vertical directions and be contained within a rectangular boundary. Lorenz used three variables to model the state of the fluid: x was related to the rate of convection, y to the variation in horizontal temperature, and z to the variation in vertical temperature.
One of the ways dynamical systems like this are studied is to plot the values of the variables against each other as they evolve in time using what is called a phase plot (see Animation 1). If you create such a plot for a chaotic system like the Lorenz equations, you find that the values of the state variables will converge onto a surface within the phase plot which is called an attractor. In both animations the blue lines indicate the shape of the attractor.
Animation 2 shows a phase plot of the Lorenz attractor demonstrating the sensitivity to initial conditions property of chaos, plotted in the x and z dimensions. The time evolution of the system from two nearby starting points (red and blue) are shown, illustrating that a small change in initial conditions (from red to blue) ultimately results in radically different evolutions of the system.
In future posts I will explain how this property of sensitivity to initial conditions leads to two somewhat surprising features of chaotic systems that are extremely useful to the brain.
If you would like to dig deeper into the topic of chaos theory, then I can highly recommend ‘Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering‘ by Steven Strogatz, and his lecture series which is available on YouTube.