In the end, the PhD didn’t work out; to give the briefest possible explanation, my supervisor and I wanted to pull it in too-different directions. I may well return to the research I was doing in my spare time, but my thought at the moment is that I will start using this blog for general research notes and ponderings, probably on a range of topics including physics, maths and computer graphics.
A lot has happened since my last post, but it’s never really felt constructive to post about here. I’m currently looking at diffusion as a mechanism for temporal memory formation.
This is a round-up of work by Principe, Euliano et al on dynamic topographic maps and temporal self-organisation.
- Principe, Euliano and Garani (2002), Principles and networks for self-organization in space-time
Applying R-D-type dynamics to SOMs (SOMTAD – Self-Organising Maps with Temporal Activity Diffusion) and GasNets to achieve spatiotemporal memories. Talks about the possible role of NO in memory formation, includes a review of relevant work in this area.
The basic approach of TAD is for winning neurons to send out waves of activity making neurons close to recent winners more likely to fire next. This enables the network to form noise-resistant memories of temporal sequences. Positive results are obtained here for robot navigation and (with the GASTAD) speech classification.
Leaves open further theoretical work on R-D type systems: ‘Although there are clear links to the R-D paradigm (the generation of traveling waves), we were unable to derive a cost function and learning rule from first principles’. Also suggests exploring the use of diffusion to train other sorts of networks, such as MLPs and RBFs.
- Neil R. Euliano (1998), Temporal Self-Organization for Neural Networks (PhD thesis)
Principe’s supervisee and collaborator explores three related approaches for temporal self-organisation in ANNs. SOTPAR and SOTPAR2 seem to correspond to SOMTAD and GASTAD in the above paper. Additional results are reported on phoneme sequences with SOTPAR and time series prediction with SOTPAR2. The third approach explored is ‘Dynamic Subgrouping of RTRL [real-time recurrent learning] in Recurrent Neural Networks’, an approach to supervised dynamic learning.
- Neil R. Euliano, Jose C. Principe (1996), Spatio-temporal self organizing feature maps
Some of the work leading up to Euliano’s thesis, above.
- Neil R. Euliano, Jose C. Principe (2000), Dynamic Subgrouping in RTRL Provides a Faster O(N2) Algorithm
Further work on RTRL.
- Jose C. Principe, Neil R. Euliano and W.Curt Lefebvre (1999) Neural and Adaptive Systems: fundamentals through simulations
A 672-page book which I have not yet had a chance to see, though the National Library of Scotland has a copy.
- Cho et al (inc. Principe) (2007), Self-organizing maps with dynamic learning for signal reconstruction
Signal reconstruction for brain-machine interfaces, compressing and reconstructing neural spike data.
Eagle-eyed readers will have noticed that contrary to what I may have suggested I was going to do in my previous ‘Return of the Blog‘ entry, I haven’t actually updated this blog in months. Due to personal circumstances, I had to take a break from academic work entirely, and I’ve hardly stopped to think about my studies since November, when I temporarily suspended my registration at the University of the West of Scotland.
However I have now returned to my PhD, and I am in currently in the process of reviewing my progress so far and planning out what to do next. Watch this space…
Several features of reaction-diffusion systems make them particularly promising candidates for computing by chemical reaction: They are capable of exhibiting varied and complex, ordered behaviour; different forms of r-d systems can sustain both stationary structures and information transmission over long distances with little loss, by means of excitation waves. Another useful property of certain r-d systems is their ability to cause some reagent to migrate towards areas where that chemical is already concentrated, making it possible for those systems to select for the highest among several peaks of concentration, which is crucial for the modelling of a Kohonen network, among other tasks.
One reason for the interest in chemical computation is simply the possibilities opened up by moving away from the digital computational paradigm in which almost all computer science to date has been conducted. Nobody knows where this might lead, but it is not unreasonable to suspect it might take us somewhere interesting. Another reason is the importance of chemical systems in the information processing performed by real brains and nerve cells.
The propagation of electrical signals within neurons is known to work by reaction-diffusion, and there is some reason to believe that substantial information-processing takes place on this level. In addition to this, the interaction between neurons in the brain and modulatory chemicals which spread by diffusion may be modelled by reaction-diffusion equations. It has become clear only relatively recently that these diffusing neuromodulators play a major role in brain function, and the details of this role are still being worked out. We have so far only seen the beginnings of work exploring the coupling between digital analogues of these chemicals and traditional neural networks.
I haven’t been keeping this blog up to date, partly because I wasn’t sure if anyone was reading it, but I’ve now decided to maintain it for my own benefit, regardless. All being well, I may even make daily updates on my progress.
What I’ve invested the most time and effort into these last few months has been using a reaction-diffusion system to implement self-organising feature maps (Kohonen maps). This is working fairly well, and it’s a novel enough approach that I hope to be able to get a publishable paper out of it. I will post here about this soon, to summarise where I’m at so far and discuss progress as I go along.
Another project I’ve worked on is implementing ‘clustering’ with diffusion, on which topic I had a poster presentation accepted for a conference on Dynamics of Learning Behavior and Neuromodulation at the European Conference on Artificial Life 2007, although for complicated reasons, in the end I was not able to attend the conference myself.
As for this blog, I’ve just implemented a plugin to allow me to use LaTeX, in order to easily include equations. It’s called mimeTeX, and several different versions are available; I ended up with mimeTeX 1.1.2, which seems to work nicely except that you need to edit a line of the code to make it work – there’s a note about it on the page, which is fine except that when I copied and pasted the line in question it didn’t work, thanks to ’smart quotes’ – I needed to replace them with apostrophes by hand.
Self-organising feature maps are a way of making computers classify input data according to arbitrary features of that data, with little input from human beings.
Also known as Kohonen networks, the maps are created by a form of neural network. This consists of a series of input neurons, connected to an array of output neurons. Each output neuron stores a vector, corresponding to a point in space. The vectors are initially assigned at random, but eventually they arrange themselves into a representation of the input space; neurons corresponding to related or similar inputs are clustered together in this resulting map.
This is useful firstly because the map that is produced can often bring out some of the order in the inputs which may not be obvious by a simple examination of the inputs themselves (which may in some cases have many dimensions); it can also be used to classify fresh inputs in terms of inputs already received. The algorithm has been applied, for example, in speech and handwriting recognition systems, and has also proven its worth in retrieving other images which resemble any given input image.
The self-organisation of the feature map works by repeatedly finding which of the neurons currently matches a given input most closely; the vector associated with this neuron is then updated to bring it closer to the input which it matched, and crucially the vectors of neighbouring neurons are similarly updated, but to a lesser degree. It is this updating of neighbouring neurons which leads to similar features being clustered together in the final map.
Usually, in the early stages of running the algorithm, many neurons are updated by a large factor to bring them closer to the input vector; later on, the size of the ‘neighbourhood’ of neurons slowly shrinks, and so does the size of each update. This allows the map to initally arrange itself into roughly the same sort of shape as the input space, followed by closer matching of any fine details.
- Neurocomputation by Reaction Diffusion, by Ping Liang in Physical Review Letters (1995)
This is the closest I’ve found to the approach taken by Lesser et al in modelling the mind/brain, mathematically speaking. Explores seriously the computational possibilities of diffusion-based (non-syaptic) mechanisms, which is still quite uncommon in neural modelling. Very interesting work, although there seems to have been surprisingly little direct follow-up. However, see also:
- Neural information processing using network-in-a-field by Ping Liang in Neural Networks, 1996., IEEE International Conference on
Which explores coupling between synaptic and diffusive information transmission, as does
- Flexible Couplings: Diffusing Neuromodulators and Adaptive Robotics by Andy Philippides, Phil Husbands, Tom Smith and Michael Oâ€™Shea in Artificial Life 11: 139â€“160 (2005)
This goes into the role of nitric oxide as a neurotransmitter, and the possibilities of models inspired by its action in an evolutionary robotics setting.
- Modeling Complex Systems by Reaction-Diffusion Cellular Nonlinear Networks with Polynomial Weight-Functions by Frank Gollas and Robert Tetzlaff
This is related to Ping Liang’s work, as above, and probably worth exploring. The same authors have also worked on modelling and predicting epileptic seizures using R-D systems, an intriguing possibility, but I so far haven’t been able to download that paper.
- Dynamical approaches to cognitive science, by Randall Beer
Discusses three examples of dynamical systems in cognitive science, with some dicussion of the merits of this approach over other strategies of cognitive modelling. A very useful review.
- The dynamical systems approach to cognition by Wolfgang Tschacher and Jean-Pierre Dauwalder (eds) (2003)
The introduction is available for download at the above page, and substantial portions of the book are available on Google Books – however, I look forward to seeing the book itself, which I will have to get on inter-library loan.
- On What Makes Certain Dynamical Systems Cognitive: A Minimally Cognitive Organization Program by Xabier Barandiaran in Adaptive Behavior, Vol. 14, No. 2, 171-185 (2006)
An interesting piece about how one might make a convincing case for the cognitive properties of a dynamical system.
- The dynamical hypothesis in cognitive science by Tim van Gelder in Behavioral and Brain Sciences (1998), 21: 615-628 Cambridge University Press
Van Gelder has evidently been a major player in pushing the idea that a dynamical systems approach to understanding cognition could be fruitful. He is referenced widely, though some aspects of his work have come in for substantial criticism.
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 DÃ¶rfler
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 KÃ¡rnÃ½, Kevin Warwick and Vera KurkovÃ¡
A collection of interesting-looking research papers, many of them relevant to my research to various degrees.
I am open to other suggestions…
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 .
- ‘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.
- 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.
- 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 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:
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
Ï is the decay factor in resource overlap with distance
d(i,j:iâ€™jâ€™) is the distance between xi,j and xiâ€™,jâ€™
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:
represents the dendrites attached to xi,j
D is the strength of the dendrite connections
q is the cubic term
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.
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