# 2013 Yuletide Lecture: A Physicist’s View of Complexity by Prof Henry D.I. Abarbanel

Below are note from the meeting:

What is complexity and what is simple? The key issue is the number of degrees of freedom.

At the end of the 19thC phenomena looked complex, because we had no understanding of those issues. We studied the simple systems first – hydrogen spectroscopy before iron. Helium was found in the sun before it was found on Earth based on the emerging understanding from hydrogen spectroscopy.

Complex outcomes have their origins in the dynamics of non-linear systems. Complex outcomes should have underlying physical, biological or other principles.

Today we should understand our complex problems by looking for simple representations, not unlike the role played by hydrogen in the 19thC.

Newton’s method for finding the cube roots of unity reveals the beautiful fractal basis of attraction to the three roots. A simple question yields a complex structure.

Edwin Colpitts invented an oscillator in 1918 – took 11 years to get the US Patent (1624537). These oscillators are ubiquitous and cheap – used in electronic key fobs etc. Very stable outputs.

The Colpitts oscillator has three differential equations. Only one point of non-linearity (an exponential term). Voltage output looks very complex. A tiny perturbation makes a huge difference (voltage as a function of time).

The weather and your brain are not Colpitts Oscillators.

Hydrogen is to iron as the Colpitts Oscillator is to our brain or the weather.

There are not enough measurements of complex systems to determine the state of the complex system. That statement is not true of the Colpitt’s Oscillator, but the Colpitts Oscillator can help with our understanding.

By experimenting on the Colpitts Oscillator we should be able to determine all the parameters of its state. It is not easy, but it is possible. Solving it will help us with the more complex problems of the weather and our brains.

We make 12 million weather measurements every 12 hours – we need at least ten times more than that to make a model.

Tuning such a model is like tuning a radio. If you aren’t tuned you don’t stand a chance to gain understanding.

The trick is to be able to tune so we can extract useful information.

Tuning the radio is tuning to a stable transmission, that is not true in a non-linear system. Changing the tuning settings can change the system.

Finding the parameter values for the Colpitts Oscillator is essentially impossible – an impossible noise of local minima. Appears impossible to stabilise to find a minimum on which one can converge, but actually it can be done.

Stabilise your searches (an exercise in minimisiation) is difficult, but vital if you want to understand a complex system.

By measuring the emitter to base voltage in the transistor based Colpitts Oscillator can one infer the parameters of the circuit? Short term model predictions work, but soon lose registry with the data (after approximately 10ms).  Any slight mismatch between best system estimate and reality will induce divergence of reality and the model.

You need to know the start state to have a chance for even short term predictions. In  order to have a chance of making predictions you need an ongoing series of measurements.

Errors are key (in model parameter guesses, the data and the equations themselves). In fact one needs to deal with probability estimates. That allows us to get a handle of how far in the future one can predict.

Now to the weather, it’s all about liquids. We have known about liquids for 250 years. What we do with weather is divide the map into grids typically (T63) of 180km on a side. On that coarseness of a grid one needs 8 million dimensions of the model.

The weather/climate system is complex!

If we reduce the grid size to 50km then we have 104 million dimensions, but we could resolve the weather in Birmingham as distinct from London. Real state of the art weather models use 80 million to 5 billion variables depending on resolution. Between 11 and 13 million measurements are made every 12 hours. It is probably not enough. We are presently only measuring about 15% of the variables. We do not know the necessary number of measurements.

Computers are key to all this. ENIAC in 1946 computed at 35 bits per second. Over the yars we have used our better computing to give better spatial resolution in weather prediction (smaller grids). We haven’t focussed on pulling in more measurements. Technology from Liquid Robotics in California allows robust ocean data collection at modest cost.

Onto the brain …

Nervous systems are based on non-linear oscillators known as neurons. The basic oscillator equations have been known since the 1950s (Hodgkin and Huxley).

The extracellular medium circuit is not so different from the Colpitts oscillator. Millions of neurons and even more connections. It is worse than the weather. The zebra finch brain is easier to study than the human brain. It’s not hydrogen – it’s lithium (a joke).

Zebra finch has only about one million neurons and perhaps 100 million connections (c.f. 10e15  in the human brain).

Zebra finch smallest known animal with cultural behaviour (birdsong).

Study a neuron in vitro and drive it with a small current and study the circuit output. Can study each for a few seconds before the neuron dies.

One must get the initial state of the neuron, but then one can start to make predictions.

One cannot live with just understanding universal macroscopics one  needs to understand the details.

Q&A

1. If only one neuron why not use human neurons?
2. If we do the finch we might, just might, be able to build a model that can sing.

1. As concerns the weather – we get more measurements with time. Is the history useful? Is it simply not long enough? How much historical data do we need?
2. I don’t know. We don’t need decades worth of data to predict tomorrow’s weather. We need good quality recent data. History can help us test models but we are not in that position.

1. You chose to study the finch’s brain – is the difference to humans at the neuronal level?
2. No. Neurons are similar between species. What we learn across species is all about neuron connectivity that is where the difference is.

1. Does it matter that we can’t predict if we know what we can’t predict – that knowledge is very powerful.
2. Why does the BBC pay the Met Office for weather predictions? The goal is to predict. That’s the job to be done – fitting is easy – predicting is very hard.

Engineering is the face of physics in society – but we must get to the underlying basics