Gregson, Goertzel, and Tsuda each raise related criticisms to our claim that electrocortical activity exhibits microchaos while macroscopically behaving as a linear, near-equilibrium system. The most powerful counterargument advanced is the work of Kaneko (1990). This showed that far-from-equilibrium global chaos emerges from coupled chaotic maps. We respond that the effect of noise in the Kaneko model brings its findings into line with our own.
2. Gregson (1994) has taken us to task, partly for failure to address a set of problems we had deliberately avoided, namely, the knotty difficulties of characterising EEG signals as chaotic or otherwise, using estimates for correlation dimension, Kolmogorov entropy, etc. It is by no means clear that these measures, as commonly applied, accurately reflect underlying attractor properties (Rossler & Hudson, 1989; Nunez & Srinivasan, 1994; Dunki, 1991).
3. We neither contend nor deny that EEG time series recorded at macroscopic scale may exhibit chaos, and we agree that it is common for physical phenomena to exhibit different dynamic features at different scales. Molecular motion, sound waves, and the weather are perhaps the most widely discussed examples of this kind.
4. Rather than attempting a review in this problematic field, we concentrated on models which, whatever their drawbacks in other respects, make definite and contrasting claims that EEG exhibits particular dynamic properties at particular scales, namely, far-from-equilibrium cellular and small cellular group interactions, contrasting with near-equilibrium dynamics and linear wave superposition at the macroscopic level. It remains for us an open question whether the macroscopic EEG time series preserves chaotic dynamical features carried through from the underlying neural events driving the wave-motion. Conversely, this is not the only way in which such macroscopic chaos might emerge, for the interaction with ongoing sensory input and the interaction of cortex with brainstem must also be considered, as we have endeavoured to indicate.
5. Gregson does not see the distinction of dynamics by scale as useful as we do. Interaction between scales seems to us to underlie the mysterious qualities of localisation and nonlocalisation exhibited in brain function; these have posed a paradox since the work of Lashley. We shall return to this important issue when replying to Tsuda.
6. Goertzel (1994) considers us shaky on a number of grounds. He has some reservations about our briefly outlined estimates of cell-to-cell coupling asymmetry. We hope he will be reassured when he has had a chance to read the more detailed reports of these calculations (Liley & Wright, 1994) and the simulation of EEG dynamics based upon them (Wright & Liley, 1994) which are currently in press. Without needing to appeal to any selective neurochemical effects, we have, we believe, achieved a rather precise fit to a body of quantitative anatomical data, on the basis of purely stochastic intracortical connectivity, as well as a good consequent fit to known EEG properties. Not, of course, that we deny the presence of selective neurochemical tropisms.
7. We are unable to say whether the resulting model of electrocortical dynamics is compatible with Edelman's (1988) evolutionary group selection views, but we agree that there is a need to make such comparisons. Part of our manifest difficulty in doing so lies in the fact that the details of dynamics in real neurons seem to need sorting out before any learning rule can be properly chosen. And before we could claim to understand the dynamics of cortical neurons we need to understand how their dynamics differ with scale and how they interact across scale -- the point of our target article. We have, we admit, not yet succeeded in properly dealing with the problems of interaction across scale.
8. That Hopfield networks are not completely appropriate models of real neural networks is not in contention, nor did we intend to force a parallel to global dynamics to the degree Goertzel seems to object to. We wished to highlight the point that symmetry of connection and defined system energy are mutually dependent ideas. This permits us to treat (1) macroscopic energetic considerations as partly defining the trajectory of the global system and (2) the interaction across scales as analogous to ongoing input from the microscopic to the global system, and vice-versa. While such an approach is not definitive, it does offer the prospect of dividing the modelling problem into two parts, to be later related. Which brings us to the commentary of Tsuda.
9. Tsuda (1994) does not accept our treatment of macroscopic EEG dynamics as linear and near-equilibrium and cites an important counterexample in the work of Kaneko. This work was not previously known to us. Tsuda points out that a set of chaotic maps with diffusion couplings can preserve macroscopic chaos and can defy the Law of Large numbers and even the Central Limit Theorem; thus it must exhibit far-from-equilibrium properties. Tsuda argues persuasively that such coupled maps are so general, and the far-from-equilibrium behaviour so ubiquitous, that there appears no escape from contradiction in our own theorising. That this is not so can be seen by comparing Kaneko's paper with one of our own source papers (Wright, 1990).
10. Kaneko (1990) reports that with the introduction of uncorrelated noise to N coupled maps, the Law of Large numbers is restored, in the sense that the mean square deviation of the field again decreases with increasing N. Mutual information of the individual maps diminishes. The concept of correlation distance is thus restored by the noise. The model for macroscopic dynamics outlined in our target article depends on the presence of noise, assumed (Wright, 1990) to be introduced to the cortex via the reticular formation. The effect of even small amounts of noise upon microscopic neural activity of high Lyapunov exponent may be such as to render the mutual information between macroscopic pools of neurons very low. Equivalently, instantaneous natural frequencies, damping factors, and coupling coefficients describing the dynamics of small pools of coupled neurons are stochastically independent; it is this stochastic independence upon which our hypothesis was built. Thus noise leaves the Law of Large Numbers applicable to both formulations, and in turn to the conclusion that macroscopic wave motion can obey superposition and is near-equilibrium. We outlined the tests for this conclusion in the target article. They support our hypothesis, so we hold to our view.
11. Tsuda's criticism of us for the use of AR in the tests of hypothesis appears misguided, since the AR technique was used empirically to make measurements which tested our hypothesis. There was no circular use of a linear technique to falsely imply that the underlying signals were therefore linear. Tsuda's final comments on learning in chaotic nets are of great intrinsic interest but they do not appear to contradict our position.
12. Clearly the Kaneko findings and our own are not in contradiction, and may be complementary. Indeed, the "Kaneko effect" would appear to be maximised for coupled maps of small separation in the presence of noise. This leads to the intriguing possibility that local coherence among pools of cells might be generated from subsets of cells currently engaged in high amplitude microscopic chaos, the local pool then providing a driving signal to the linear and near-equilibrium macroscopic dynamics. This mechanism appears to offer a further means of bridging dynamic interactions across scale.
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