Filed under: Uncategorized — frm @ 10:20 pm

Besides the many research papers I am getting through for my research, I have several books on the go, each providing inspiration and useful background material…

  • Frontiers of Complexity, Peter Coveney and Roger Highfield
    A fascinating, wide-ranging and accessible overview of where things stand in the science of complexity
  • Introduction to the Theory of Neural Computation, John A. Hertz, Anders Krogh and Richard Palmer
    Very useful, comprehensive textbook on neural networks.
  • From Cells to Consciousness and From Neurons to Behaviour, The Open University
    Books 1 and 4 of the OU’s set of six textbooks for their course ‘Biological Psychology: Exploring the Brain’. Generally rather well-written overviews of neurology; good information to keep in mind when working on models inspired by the human brain.
  • Dynamical Systems and Fractals, Karl-Heinz Becker and Michael Dörfler
    A delightful book written in 1986, translated by Ian Stewart in 1989. Fun for the insight it provides into the first years of experimental mathematics becoming an important force, and the immense enthusiasm engendered by the discovery of things like the Mandelbrot set. Nice as a compendium of useful information on a variety of complex systems, as well.
  • Advanced Ecological Theory, edited by Jacqueline McGlade
    Work in ecology, including research by McGlade with Peter Allen, provided part of the inspiration for Lesser’s mathematical model of the mind as an interest system. This may not be directly relevant, but I thought it would be as well to acquaint myself with some of the ways similar models are used in quite different fields.
  • Dealing with Complexity, edited by Mirek Kárný, Kevin Warwick and Vera Kurková
    A collection of interesting-looking research papers, many of them relevant to my research to various degrees.

I am open to other suggestions…

‘Net of Indra’ Report

Filed under: Uncategorized — frm @ 9:33 pm

A report on the work I conducted with Mike Lesser of Autism and Computing

Statement of problem

Autism has often been seen as a puzzling disorder, with those affected exhibiting a range of symptoms with no obvious connecting thread: Sensory abnormalities, obsessive tendencies, difficulties with social interactions and often problems using language. In recent decades a number of competing theories have been advanced which attempt to provide a coherent explanation of this complex of psychological differences, some more successful than others.

The idea of monotropism, advanced by Dinah Murray and Mike Lesser, suggests that the central feature of autism, from which all or almost all of its common features arise, is a different way of distributing attention. Those on the autistic spectrum exhibit a tendency, which we call monotropism, to focus their attention very tightly and intensely on only one or two things at a time – they are conscious of only a very small set of interests at any given time. By contrast, most people are polytropic, with several interests aroused at any moment, but seldom with the intense concentration of the autistic.

To model these different strategies of attention use, Murray and Lesser propose what they call the interest system. A computer model of this system uses reaction-diffusion type equations to model the dynamics of attention use over time. The model demonstrates complex behaviours, the spectrum of attention-use strategies being represented by variations in the rate of diffusion.


Explanations of Autism

Three main alternative ideas have been proposed for the ‘core cognitive deficit’ in autism .

  1. ‘Theory of Mind’ deficit. See eg. Baron-Cohen et al (1985). This posits that those on the autistic spectrum have a weakened ‘theory of mind module’, with the rest of their cognitive abilities left intact. This fails to account for many features of the condition outside of the social sphere, and has largely been abandoned.
  2. Impaired executive function. According to Hill (2003) ‘Executive function is an umbrella term for functions such as planning, working memory, impulse control, inhibition, and shifting set, as well as for the initiation and monitoring of action.’ It is a theorised system thought to control other processes, whose impairment, it is argued, could lead to autism.
  3. Weak Central Coherence theory. See Frith (1989). Posits a detail-focused processing style, with difficulty understanding context; shares various features with the monotropism hypothesis, but takes difficulty in understanding context to be the fundamental difference in autistic individuals, as opposed to its being a consequence of different use of attention.

The approach proposed by Murray et al was published in Autism, Vol. 9, No. 2, 139-156 (2005).

The Model

The model can be seen as a floating-point cellular automaton, or a spatially discretised differential equation. It is essentially a reaction-diffusion system, and is derived in part from an earlier model of predator/prey interaction inspired by the Lotka-Volterra approach.

The model is intended as an analogue of the human mind. Attention is treated as a scarce resource, which is competed for by various tasks or potential actions. Action depletes attention. An interest, or concern, is a local clustering of attention; an interest with enough attention will usually lead to action. Interests are auto-catalytic, and also feed into each other. In some circumstances they are known to arise or divide spontaneously.

The original model conceived by Mike Lesser can be described by the following equations:

(click to see full-sized)


N is total available attention
xi,j is interest
yi,j is activity
b is the rate at which interest is excited
s is the rate at which interest leads to activity
mx is the rate at which arousal of interest decays
my is the rate at which arousal of activity decays
w is the rate of positive feedback
f is the rate of associational excitation of interests
ρ is the decay factor in resource overlap with distance
d(i,j:i’j’) is the distance between xi,j and xi’,j’

At Mike’s request I extended this to include synapse-like dendritic connections, diffusing attention in one direction from each cell to several others, their location chosen at random on initialisation. I also simplified the term representing resource depletion, removing its dependence on distance, and added a negative cubic term to represent the exhaustion of resources on a local level. This gives us the following equations:

(click to see full-sized)


c sub x sub i,jrepresents the dendrites attached to xi,j
is the strength of the dendrite connections
q is the cubic term

Reaction-Diffusion Systems

Filed under: Uncategorized — frm @ 4:44 pm

A wide variety of reaction-diffusion systems are found in nature. A candle flame is one classic example, where evaporating wax forms an excitable medium, dynamic structures being produced through its reaction with and diffusion into the surrounding air.

Alan Turing was the first to formulate reaction-diffusion equations as such, in his 1952 landmark paper on morphogenesis – the process by which living things derive their shape, structure and function. He proposed that many varieties of spots, stripes and other markings found in nature, among other things, could be explained by reaction-diffusion equations.

In the same year Andrew Huxley and Alan Hodgkin published a system of equations describing the propagation of action potentials, the central nervous system’s chief method of propagating signals electrically. Based on empirical observation of in vitro giant squid axons, their reaction-diffusion model proved seminal, winning them the Nobel prize for Medicine, forming the basis of most neurobiological models for four decades and inspiring similar models of electrical activity in the heart.

Perhaps the single phenomenon best known as a reaction-diffusion system is the Belousov-Zhabotinsky (BZ) reaction. First discovered by Boris Pavlovitch Belousov in the middle years of the twentieth century, the surprising nature of the reaction – oscillating periodically between one colour and another – was initially met with incredulity by the scientific establishment. At least one editor rejected Belousov’s manuscript outright on the grounds that it was clearly impossible, even before the identification of its best-known and most remarkable features – spontaneous self-organisation into a beautiful range of stripes, rings and spiral waves. When Zhabotinsky continued exploring the reaction, years later, he finally perservered through the entrenched scientific scepticism and published a series of papers on the subject; in the decades since it has attracted huge amounts of scientific interest.

Reaction-diffusion systems provide one example of dissipative structures in far-from-equilibrium systems, for which the simple laws of traditional thermodynamics prove quite inadequate. Systems of this sort frequently give rise to complexity which must be described using the language of nonlinear dynamics; seldom amenable to precise analytical solutions, it is only with the rise of computer models that their in-depth study has become possible.

There are some fun reaction-diffusion animations here, and a rather delightful interactive simulator here.

The information in this post is largely drawn from:

  • Encyclopedia of Nonlinear Science, edited by Alwyn Scott
  • Frontiers of Complexity, by Peter Coveney and Roger Highfield

‘Biologically-Inspired Reaction-Diffusion Networks’: An Introduction

Filed under: Uncategorized — frm @ 12:59 pm

Welcome to my new academic blog, where I plan to record my thoughts, readings and progress on my PhD at the University of Paisley, the title of which is ‘Biologically-Inspired Reaction-Diffusion Networks’.

Depending on how technical an explanation seems called for, I’ve been telling people my PhD is on ‘computer brains’ or ‘neural modelling’, and/or telling them the official title, to see if it makes any sense to them at all. Loosely speaking, I am trying to gain some insight into how thoughts arise using models based on reaction-diffusion equations. I am based in the School of Computer Science, but my background is mainly in physics, and I expect to be drawing on many different branches of science in my work.

The project is closely related to something I was working on a couple of years ago with my mother, Dinah Murray, and her colleague Mike Lesser, at Autism and Computing – a model designed to illustrate and shed light on their conception of the mind as a system of interests competing for attention. This idea explains autism as a difference in attention-using strategy, labelled ‘monotropism’; where most people have a range of interests aroused at any particular moment, those on the autistic spectrum have a tendency to focus their attention very tightly on only one or two concerns at any particular time. Switching the focus of attention is often uncomfortable, and interactions with autistic people can be made much easier and more fruitful by engaging with their existing attention tunnel rather than trying to wrench it onto another subject.

The mathematical model of the ‘interest system’ treats it as a reaction-diffusion system, with interests feeding on one another and action exhausting interest. Different calibrations of the model show features of monotropic attention use or its converse, polytropism.

For reference, this was the description of the PhD project in the advertisement which caught my attention:

The reaction-diffusion systems, being capable of pattern formation, describe numerous natural phenomena. In particular, they are known to model selforganizing behaviour of living brains. Neural selforganization implementated as bistable reaction-diffusion medium bears several disadvantages. First, neurons are not integral part of reaction-diffusion medium but distinctive from it components. Second, this medium requires three-molecular or variable rate two-molecular reactions, which is, in fact, rare phenomenon in nature.

In this project, we are going to develop realistic, purely reaction-diffusion models of neural activity. Mechanisms of formation of neural activity patterns and their effect on brain’s information processing will be investigated.