# When Interaction Matters

At Imperial, quantitative modelling using methods from a wide range of mathematics, statistical mechanics and network theory is applied in collaborative efforts within projects in neuroscience, evolutionary biology, medicine, engineering and sociology producing new insights of fundamental importance and of immediate relevance to applications.The coherent combination of applied science will, when performed in collaboration with research in fundamental aspects of complex systems theory, provide the ideal forum to move complexity science beyond the confines of idealised modelling.

The Complexity and Networks programme is situated in the Institute for Mathematical Sciences and brings together researchers from across the college and from other universities in UK and abroad. The simplest way to characterise Complexity Science at Imperial is as quantitative modelling of emergent phenomena at a macroscopic level caused by the many interactions taking place at a microscopic level.

For this we use various types of statistical mechanics. In its current form statistical mechanics does not hold all the answers for all complex systems, however, we think that broadly speaking it is the best methodology available to obtain a quantitative description of complex systems. By systematically applying it to fields outside its traditional range of application in physics, statistical mechanics can be developed further, and at the same time contribute to the understanding of those fields. The importance of this feedback loop cannot be overestimated. It can also provide a starting point for the possible development of new mathematical techniques and approaches. This is at the moment happening within the theory of large networks, as e.g. encountered in neuroscience, engineering and social science.

# Dynamics of Complex Systems

Dynamics is essential. Everything changes with time and when dealing with complex systems the changes can be dramatic. Often a stationary state doesn’t exist for relevant time scales. Typically the overall state and properties may change abruptly through abrupt events separated by long periods of quiescence. We have found that the same statistics and slow decrease in activity is present in seemingly very different situations.

The Tangled Nature model of evolution was inspired by observations suggesting intermittent bursts of extinction and creation events in the fossil record. The model reproduces these observations and describes the statistics of the events in terms of record statistics. It is found that the same statistics and observed slowing down applies to a range of seemingly unrelated systems. We found for example in model studies of superconductors that the magnetic field relaxes in the same way. This behaviour is also seen in the type of magnetic materials call spin glasses and in polymer glasses. Most surprisingly our collaborators at Bristol University have found in their experiments on ant colonies that the times at which ants leave the nest exhibit the same statistical behaviour. This leads to the anticipation that slow, directed dynamics, during which the system’s properties change significantly, is fundamental to complex systems.

The microscopic mechanisms underlying this type of dynamics is related to a slow, decelerating but spasmodic release of an intrinsic strain or tension. The strain arises from local frustration. As the strain is released through “quakes,” some system variable undergoes record statistics with accompanying log-Poisson statistics for the quake event times. It is interesting to note that to check this scenario, one just need a time series of an appropriate observable to check if the statistics of the abrupt changes has the properties of a Poisson process as function of logarithmic time. A better understanding of this very general behaviour might help us to improve our understanding of catastrophic events such as systems failure or financial crashes.

# Emergence in Neuroscience

Analysis of fMRI data is used to extract the nature of the time-averaged correlations across the entire brain. Further analysis is applied to investigate causality relations between different parts of the brain. One very remarkable finding is that the time averaged correlation structure is scale free and hence entirely different from what one would anticipate from standard functional imaging.

In a related research effort, EEG data sampling is used to assess the working of the brain during music making and the degree of creativity involved. Since a measure of creativity doesn’t exist, the performer and musical judges evaluate the rendition; their classification is then compared with the EEG analysis. The hope is in particular to elucidate what happens when creative leaps and ingenuity occur during play. There are indications that during particular creative play, the right hemisphere becomes strongly correlated internally while the remaining part of the brain briefly de-correlates.

### List of Interest

- Regularities in Firm Dynamics: The Basic of the Complex Economy
- Self-Organised Criticality
- Synchronisation by time delay
- Paul Expert's research
- Tim Evans' Research
- The Acupuncture on Economic Network

System analysis from the perspective of complexity is an emerging field of science. In order to understand the whole picture, one must look from different angles and combine different approaches. The vision of the project is to find a better way to evaluate the state of an economic network, and hence to determine possible signs of instability, and identifying the acupuncture points which control the evolutionary direction of such network back to normal state to avoid catastrophic events. Analysis to the interconnection of a network has a more promising insight than examining merely a falling component. It is believed that there exist a leading mechanism in real economic evolving network.