Estimates of phase velocity using scalp recorded data are shown to be consistent with a global theory of EEG in which waves are partly transmitted along corticocortical fibers. General features of neocortical dynamics and their implications for theoretical descriptions are considered. Such features include multiple scales of interaction, multiple connection lengths, local and global time constants, dominance of collective interactions at most scales and periodic boundary conditions. Cognitive events may be directly related to the continuous forming and reforming of local and regional circuits which are functionally disconnected from tissue involved in global operation.
1. Wright et al. (1993) use frequency-wavenumber spectra to obtain phase velocity estimates. A more direct approach (which does not, of course, yield spectral details) is simply to measure phase versus distance in any particular direction. Such methods have been applied by the Melbourne group to steady-state visual evoked potentials (unstructured flicker). About 40% of subjects exhibit a smooth variation in phase along the midline in scalp recorded EEG and MEG (Burkitt 1994; Silberstein 1994). Such phase velocity estimates are generally in the 10-20 ms range along the scalp or about 5 to 10 ms along the folded cortical surface. These estimates are close to the known propagation velocities in corticocortical fibers, generally in the 6-9 ms range (Katznelson, 1981; Nunez, 1994).
2. Our group has estimated phase velocity of alpha rhythm recorded in Melbourne using a slightly different method. Linear regression is obtained for bipolar phase versus distance along the midline in short (100 ms) segments of scalp EEG data. Questions addressed by this method are concerned with:
(i) What fraction of segments exhibits correlation coefficients at the 1% significance level? That is, in how much of the data can we define apparent phase velocities in either posterior-anterior or anterior-posterior directions?
(ii) What is the range of these phase velocity estimates? About 10 to 15% of such data segments pass the significance test as compared with much fewer than 1% in simulation studies involving random phases of source activity (but accounting for volume conduction effects). Thus, the data exhibit evidence for an underlying characteristic velocity, again in the 5-10 ms range along the cortical surface.
3. These data support the general theoretical idea that EEG is composed of traveling waves which partly combine (interfere) to also form standing waves (accounting for the low percentage of pure traveling waves), and that these waves are partly propagated along corticocortical fibers (Nunez, 1972; 1981). These are the ideas underlying Eqn (6) of the target article. However, the existence of waves at such large scales does not preclude the simultaneous existence of waves at several smaller scales in which propagation can be adequately described in terms of exclusively intracortical interactions. The latter phenomena are apparently of short wave length and cannot be observed from the scalp (Nunez, 1989; 1994) as suggested by data presented in the target article.
4. Whereas the modern concept of low dimensional chaos may eventually prove to be important in brain studies, I am not aware of any convincing current evidence that this is the case. The term "chaos" in modern nonlinear systems literature is associated with exponential divergence of trajectories in phase space. The revolution in our thinking about chaos over the past two decades has mostly to do with a new appreciation that apparently simple systems (i.e., those with few degrees of freedom) can exhibit chaotic behavior. However, it has long been appreciated that complex systems (i.e., those with large numbers of degrees of freedom) often exhibit chaotic behavior (although the label "chaos" is new). The brain is certainly not a simple system, that is, we expect many brain state variables to exhibit high dimensional chaos so this is hardly a novel idea.
5. It does appear that phenomena characteristic of many complex nonlinear systems (e.g., self-organization, interactions at multiple temporal and spatial scales, stable spatial structure in the presence of temporal chaos, etc.) occur in neocortex and may be closely aligned with cognitive processing. This is suggested in the experimental work of Freeman (1991; see also references in the target article), and the theoretical work of Ingber (1982,1984,1991). The target article presents some of these general ideas metaphorically, although many of the concepts that are critical for connections between variables recorded at different scales are omitted. (In fairness, a comprehensive review of this area would be quite difficult to construct).
6. The point made in the target article that linear or quasilinear phenomena at one scale can coexist with highly nonlinear phenomena at another scale is an important one for brain theories. Again, this point is made metaphorically rather than theoretically, for example, by deriving distribution functions at one scale and integrating to form more macroscopic variables (refer to Ingber, 1982; Ingber & Nunez, 1990). An example from the physical sciences is provided by simulations of star dynamics in a galaxy (Miller, 1992). Radial standing wave phenomena (large scale mass distributions) are predicted to coexist with chaotic "microstructure" (the trajectories of individual stars).
7. The authors focus on the idea that connections between assumed functional units at different scales (e.g., neurons, minicolumns, macrocolumns) can be symmetric or asymmetric, depending on the spatial scale of such units. I would agree that the nature of connections at different scales is a critical theoretical issue (Ingber, 1982; 1994; Ingber & Nunez, 1990). However, features other than symmetry of such interactions may be more important, for example, the density of connections.
8. Consider, for example, physical phenomena for which collective interactions are important (e.g., hot plasma, stars in a galaxy, etc.). These differ fundamentally from systems with only nearest-neighbor interactions. An example of the latter is a neutral gas which involves mostly two body interactions. By contrast, each electron in a hot plasma may interact simultaneously with 10**5 or more other electrons in a manner somewhat analogous to the interactions of a neocortical neuron with 10**5 local neurons.
9. Since neocortical dynamic variables may behave quite differently at different scales, it is perhaps useful to construct a table outlining some characteristics of these scales. The following is based mainly on the works of Mountcastle, Braitenberg, Szentagothi and Abeles (summary in Nunez, 1994). The "interconnectivities" (number of units directly connected) in the right most column are mostly guesses.
TABLE I: Spatial Scales of Human Neocortex
Name Scale (mm) No. of Neurons Interconnectivity (guess)
Soma 10**-3 1 10**5
Minicolumn 3x10**-2 10**2 10**3
Module 3x10**-1 10**3-10**4 10**2
Macrocolumn 1 10**5-10**6 10**4
Broadman 50 10**8 50
Lobal 170 10**9 10
10. It should also be noted that there are important differences in the nature of these interactions between units, for example, excitatory or inhibitory, long range (up to 20 cm) or short range (mm) and origins in different cortical layers. One can evidently view neocortical dynamic function as the simultaneous interaction of 10**10 neurons, 10**8 minicolumns, 10**6 modules (i.e., corticocortical columns), etc. Furthermore, interactions across scales are likely to be important, as emphasized by Ingber (1994). The brain can be compared with the human global system, which involves simultaneous interactions of people, cities, nations, etc. Furthermore, both top-down (e.g., nations-people) and bottom-up (e.g., states-nations) interactions are important.
11. Scalp recorded data occurs at the lobal (conventional EEG) or Broadman (high resolution EEG) scales. Thus, any theory of EEG constructed at smaller scales must be coarse grained before comparisons are made with scalp data. We have illustrated some of these ideas with a simple, nonlinear mechanical system (Nunez & Srinivasan, 1993) in which either chaotic or quasiperiodic behavior may be observed depending on recording strategy (e.g., the spatial filter implicit in the experimental methods).
12. One way to avoid some of the extreme complexity (including a large number of unknown physiological parameters) is to construct a "fluid- like" theory of mass action which applies only to very large scales. This is my approach using Eqn (6) of the target article, which involves three dependent variables: he(r,t), hi(r,t) and g(r,t). The first two are synaptic action densities, whereas the third is action potential density (note the error in the description of the latter). Thus, a third equation is required; this is provided at the local (columnar) scale (Nunez, 1989). The local equation g=g(he,hi) is generally expected to be highly nonlinear; however, if the local scale is sufficiently large (e.g., macocolumnar or larger), I have postulated an approximate linearization of the function g which is applicable to some fixed physiological states.
13. The resulting system of equations is linear, contains no free parameters and results in a number of correct qualitative and semiquantitative predictions which are observed in the EEG, including standing waves with frequency in the 10 Hz range (within a factor of about two or three), increasing frequency with maturation of the alpha rhythm due to myelination of corticocortical axons, negative correlation between brain size and alpha frequency, and mode scanning during sleep and anesthesia states (Katznelson, 1981; Nunez, 1994). Furthermore, an approximate quasilinear solution predicts global limit cycle behavior by individual spatial modes (Nunez, 1994).
14. The views expressed here are in general qualitative agreement with those expressed in the target article; however, I would add several ideas which may or may not be in agreement:
15. Based on experience with complex physical phenomena which appear to have important similarities to neocortex, the following appear to be essential features of neocortical dynamics: multiple scales of interaction with different rules (e.g., equations) operating at different scales, both top-down and bottom-up interactions, multiple connection lengths, both local and global time constants (e.g., PSP rise/decay times and delays due to finite propagation velocity of action potentials in corticocortical fibers), dominance of collective interactions at most scales, and periodic boundary conditions.
16. In addition, lateral inhibition provides a mechanism by which local and regional circuits can continuously form and reform with different regions functionally disconnected from other tissue (a form of self-organization). In the global theory, the switching between more local and more global operation is mediated by local and global control parameters, which are assumed to change due to the influences of various neuromodulators. One may speculate that such events are directly connected to cognitive processing (Gevins, 1994; Silberstein, 1994).
17. Given these apparently critical contributions to neocortical dynamics, the usual neural network methods, which contain almost none of these features to significant degree, are likely to have very limited application to successful theories of neocortical function.
18. It would appear that successful theories must either approximate interactions between neural masses at some scale consistent with the scale of the experiment (e.g., electrode size and location) that they attempt to describe (e.g., the global theory outlined in paragraphs 12 and 13), or they must use modern statistical methods to express variables at experimentally interesting scales in terms of integrals over more microscopic variables and their distribution functions at smaller scales (Ingber, 1982; 1994).
19. Metaphorical descriptions, which may be based on analogs to known physical systems, can be of great value in designing both theory and experiment. However, they should be distinguished from genuine neocortical theory, which is based on real anatomy and physiology (as understood at the time) and contains no free (arbitrary) parameters. Scientific advancement is greatest when agreement between theory and experiment is achieved. Unfortunately, the agreement most often claimed is actually between metaphor and experiment; this is at best a modest achievement, owing to the ease with which free parameters and loosely defined concepts can be fudged to effect superficial accord.
20. In my view, the main value of Wright et al.'s target article lies in communicating several general concepts, which are apparently critical to brain dynamic function, to disparate fields. I am less enthusiastic about some of the details.
Burkitt, G. (1994) Steady-state Visually Evoked Potentials and Traveling Waves. Ph.D. Dissertation, Swinburne University of Technology, Melbourne, Australia.
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