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|could do in the disk controller example, where relationships between bits in a particular disk drive command may be entirely arbitrary.) For it may be the case, for instance, that something like SIC is a property required by neural systems capable of making the kinds of fine discriminations displayed by the olfactory cortex. While the higher level order may be non-chaotic, the lower level order may be essentially chaotic, ordered on the logical depth measures but not on KCS.|
Alternatively, finding a chaotic signal, or even a signal which was random on both the KCS and logical depth measures, could be akin to finding apparently patternless signals on a parity bit line. We might be measuring an apparently random byproduct of a functionally relevant, logically deep process. Both these possibilities for interpreting chaotic or even random (on the logical depth measures, as opposed to just KCS) signals in the brain suggest that without much more information there is very little we can conclude about the r™le the signal plays in the brainÕs functioning.
This brings us to the more difficult question about complexity measures applied to brain functioning: how can we move from an analysis of complexity of a representation of brain functioning (such as binary representations of spiking frequencies) to an analysis of complexity in the brainÕs Ônative representationÕ? In particular, how can we begin to draw conclusions about the presence of chaos in the brain and how this could be related to nondeterministic noise? This is akin to moving from an analysis of disordered individual disk controller commands to the functionally relevant and ordered use to which these commands are put. More dramatically, it is akin to moving from the level of describing the less ordered precise positions in 3-space of particles of toner on a white sheet of paper to the more ordered level of the actual text being represented by those particles of toner.
Answering this question is by no means easy, and it is one on which we shall here make only rudimentary headway. Some people might like to say that there canÕt really be functionally relevant chaos in the brain because otherwise peopleÕs behaviour would be chaotic. Others, like Smith, appeal to the KCS definition of complexity to put chaos on the same footing as nondeterministic noise; on this view, it is hard to see how chaos in the brain could play any important r™le at all. But both these approaches are na”ve and unable to accommodate the kinds of observations we have been making. They do not begin to answer the question of how to analyse chaos in the brain in itÕs Ônative representationÕ. So letÕs take note of some very general points that can be made.
First, it is possible that all chaos in the brain, if analysed in terms of the uses to which it is put by the brain itself, would be something like the patternless codes which might be sent to a disk drive. In other words, we could theoretically change the chaotic pattern to any other pattern, and as long as we preserved the functional r™le of the new pattern with respect to the old, nothing would change. Even if it werenÕt physically possible to achieve such a change with a real biological neural system, perhaps because neurons are structurally disposed to produce chaotic spiking behaviour, it could still be true that any artificial rendition of an intelligent system could get by with a nonchaotic pattern in place of the chaotic behaviour of the real brain. Moreover, in this case, a chaotic pattern could just as easily be wholly nondeterministic noise (as long as bits of such noise were reproducible by recording or other means).
Alternatively, all chaos in the brain might turn out to be nothing but a byproduct of interactions between separate nonchaotic subsystems. In this case, it might be impossible to achieve the same kind of information processing with neurons without creating the same chaotic byproduct. Along similar lines, it might be impossible to build a wood fire for cooking food without producing smoke, but the special properties of the smoke might be entirely irrelevant to the cooking process. If the chaos were a result of peculiarities of biological neurons but not of simulated intelligences (in the sense that smoke is a property of burning wood but not of an electric hob), it might be possible to implement the relevant information processing artificially without creating the byproduct chaos (just as it is possible to cook food with a smokeless electric hob).
Finally, it could be that the special properties of chaotic systems are functionally essential at the level where they are present. I suggested earlier that SIC might be important for any neural network capable of recognising fine differences between odours. Such sensitivity could also be important for sensing changes in the position of a motor control system, and topological transitivity might be important for rapidly ÔsearchingÕ all neighbourhoods of an area of phase space. Dense coverings of phase space with periodic points could be important for a system able to make creative nonlogical ÔleapsÕ. Yet all these capabilities might at the same time be parts of a process which was essentially nonchaotic at a higher level. None of these capabilitiesÑrecognising fine differences in odour, sensing positions for motor control, searching large spaces, or being creativeÑimply in any way that higher level processes which appeal to them will themselves be chaotic. This takes care of the concern stated above that real biological systems canÕt be relevantly chaotic because then their behaviour would be chaotic: a chaotic process may simply provide input to a higher level process which was not noticeably chaotic.
In the end, of course, conclusions about the r™le of chaos in biological intelligent systems and its relationship to nondeterministic noise will have to wait for more empirical data. It is primarily a job not for philosophers but for experimentalists. Philosophers have an important job in analysing the possibilities and developing testable theories which can guide experimental explorations, but they have no place in prejudging the whole question by appealing to some abstract measure of complexity. We have come a long way from the original view that chaos was essentially little different from noise and that all that really mattered in terms of analysing real physical systems were the kinds of limited precision simulations we can run on digital computers. Chaos as it relates to questions of mind is an area ripe for experimental exploration and theoretical guidance, and it cannot be dismissed so easily.
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1 Research in this area is too extensive to list within the paragraph; to name a few of the highlights: Chapeaublondeau 1993; Fan and Holden 1993; Pollack 1992; Wilson and Bower 1992; King 1991; Freeman 1991, 1989; Ambros-Ingerson, et al 1990; Kolen and Pollack 1990; Yao and Freeman 1990; Li and Hopfield 1990; Sompolinsky and Crisanti 1988; Choi and Huberman 1983.
2 Some of the material in this section is based very loosely on an earlier paper (Mulhauser 1993a) in which the representational schema I will describe was set within a context of fuzzy mathematics. In the present approach, I have adopted a cleaner mathematical framework in which fuzzy logic is made superfluous by observations about the topological relationships between the representational spaces.
3 The material about chaos is adapted from a small section of Mulhauser 1993b; this paper receives more attention later on.
4 A fixed point of a given period