Nunez's (1994a) views on EEG modelling raise procedural issues about the construction of models of EEG criticising Wright et al. (1993). We reply that simulation based upon anatomical estimates of cell connectivity is not a metaphoric approach. Our approach is in line with Nunez's canons of science, and our results in line with much, if not all, of his own theory. We question the superiority of analytic tools to attack the problem of interaction across scale in the brain. In our view, two discrete scales (cellular and macroscopic) rather than a continuum of scales, are relevant.
1. In his commentary on our target article (Wright et al., 1993), Nunez (1994a) is in substantial agreement with us, but he is critical about what he feels may be our advancing a metaphor, such that experimental tests become tests of metaphor, rather than tests of rigorous theory. As criteria for such rigorous theory formation he makes reference to techniques applied in physics to bridge explanation across multiple scales (e.g., Ingber & Nunez, 1990; Ingber, 1991). He deplores work which incorporates free (arbitrary) parameters, and emphasises the importance of rigorous analytic technique, based upon anatomical and physiological quantitative data.
2. We agree with these criticisms inasmuch as they bear upon the construction of EEG models. However, the implication that we have erred in these ways is not correct. We do not claim to have solved the problem of interaction across scales in the brain, on which we focused. We have not used rigorous analytic methods at all points. But we have used techniques of equivalent effectiveness, and perhaps of greater potential in this class of problem.
3. We took as given, only anatomically established measurements of cortical neuronal connectivity, physiological estimates of synaptic rise-times, etc. The foundation work on anatomical connectivity is currently in press (Liley & Wright, 1994; Wright & Liley, 1994). Dendritic properties were drawn from experimental work by Freeman (e.g., Freeman, 1975). We thus avoided the use of free parameters in the ways criticised by Nunez.
4. Our crucial assumption, which Nunez focuses on as a metaphor, was that the behaviour of any small ensemble of connected cells could be described as a noisy and/or chaotic process, in at least the sense that their states are decreasingly predictable with time, from any initial conditions. This notion was drawn from Freeman (1991). It is not a proven conjecture, just a plausible one. Given that neurons exhibit sharply nonlinear response, and that our calculations showed them to be highly asymmetrically coupled, and considering that they are activated by diffuse input from reticular and monoaminergic pathways, this is a very plausible assumption indeed, in our view.
5. We then argued that when such locally chaotic neural networks are coupled, they can be described as a coupled system of stochastic second-order oscillators with independently time-varying parameters. Consequently, the macroscopic EEG should exhibit linear and near-equilibrium properties (Wright, 1990). Aspects of these conclusions are implicit in the way our macroscopic simulation was then formulated. The work of Kaneko (1990), the commentary of Tsuda (1994), and our reply to Tsuda (Wright et al. 1994) are directly relevant to this issue. This approach may be less than fully formal, but it is not metaphorical. It involves simile, as does all theory formation.
6. This said, we are admirers of the work of Ingber and Nunez cited above, in which they introduce the techniques of group renormalisation to the dynamics of the brain. But we are bound to point out that this technique has yet to yield any strong predictions testable on EEG recordings. In addition, we believe that the development of such theories to the point where the effects of noise from subcortical systems and asymmetry of connectivity are fully accounted for, will either require making assumptions equivalent to those we have ourselves made, or deriving those assumptions from more fundamental properties.
7. Nunez (1994b) himself makes assumptions in formulating theory. These include linearisations about operating points, restriction of critical parameters to limited range when these are known from experiment only with lesser precision, and elimination of terms describing branches of dispersion relations with higher spatial damping. These necessary manoeuvres are well justified, but we take the opportunity to note their similarity to the fitting of free parameters of which Nunez is so critical in other (unspecified) cases. Systems of analytic equations which make such necessary simplifications, can lead to erroneous conclusions, even when every approximation is good step-by-step.
8. In passing, we apologise for any misdefinition of symbols in Nunez's fundamental equations, as cited in our target article, to which Nunez rightly draws attention. We assure him that this error did not reflect a basic lack of comprehension.
9. Nunez's theory, as it stands, depends very strongly on the specific size of the human brain to explain the basic EEG rhythms as resonant modes of global activity, imposed by periodic boundary conditions. This raises a problem in the explanation of the close spectral similarity of the EEG of species such as cats and rats to that of humans. We are not able to discriminate EEG from all of these species by spectral means. Nunez deals with this by invoking local models to explain the generation of EEG observed in small brains. We agree, and note that our simulation at millimetric scale does generate a spectral output which resembles the EEG of all these species.
10. In recent experiments we have attempted to scale up our simulation to that of the human brain by including realistic lags due to axonal conduction in the fashion of Nunez. Preliminary results confirm many predictions of the Nunez model, in particular, the large-scale propagation velocity of about 7 m/sec he predicts, which has been experimentally confirmed (Burkitt, 1994). This velocity must be clearly distinguished from the local velocity of about 0.5 m/sec predicted in our millimetric simulation, and also found by experiments (Lopes da Silva & Storm van Leeuwen, 1978; Wright & Sergejew, 1991). Results of the global simulation will be reported elsewhere. In the present discussion of method, we note that the match of the simulation to the Nunez theory suggests an equivalence of method.
11. In his commentary, Nunez also emphasises the importance of scale, tabulating the scales, and guessing the interconnectivities of the minicolumn, macrocolumn, module, Brodmann area, etc. Although there is anatomical ground to discretise the brain in this way, there is little ground to build up models on the basis of any of these as favoured units (Braitenberg & Schuz, 1991). The neurons themselves are fundamental. For this reason, we prefer to consider dynamics at only two scales -- the microscopic, and the macroscopic. We consider that the connectivity of any neuron with its surrounding cells is so colossal, and involves smoothing of activity over such a large ensemble of other cells, that it may be regarded as a "fluid" interaction at macroscopic scale, to use Nunez's term. This does not prevent the isolation of any particular group of cells to describe interactions between them, but it does rule out considering then as a closed system. It imposes the need to consider the cellular dynamics as taking place within a "fluid" surround, and contributing to the surround. This is the problem we attempted to approach in our target article, and which we feel lends itself to piecewise solution at two scales.
12. Finally, we again emphasise that our choice of a two-scale view of brain dynamics is partly conditioned by the applicability of energy concepts at one level (the macroscopic) in contrast to the cellular level, where different attractor dynamics appear applicable (Amit, 1989). We accept that many higher order scales must be considered within the general macroscopic domain, but we feel that these higher scales are of lesser functional importance, since they reflect the scale of the observer, or at least of the observers electrodes, rather than any fundamental difference of dynamic type.
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