Paul L. Nunez and Ramesh Srinivasan

Print publication date: 2006

Print ISBN-13: 9780195050387

Published to Oxford Scholarship Online: May 2009

DOI: 10.1093/acprof:oso/9780195050387.001.0001

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Neocortical Dynamics, EEG, and Cognition

Chapter:
(p.486) 11 Neocortical Dynamics, EEG, and Cognition
Source:
Electric Fields of the Brain
Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780195050387.003.0011

Abstract and Keywords

EEG dynamic behavior is closely correlated with cognitive processing. If a (partly) valid physiologically-based dynamic theory of EEG is developed, an indirect connection between physiology and psychology will be achieved. This chapter presents a general conceptual framework in which cell assemblies are pictured as embedded in the global synaptic action fields generating EEG. A simple standing wave model of neocortical global fields is outlined that is consistent with both large-scale (EEG) data and established physiology and anatomy. The global model fits naturally with complementary network models, consistent with the suggested co-existence of networks and global fields. “Binding by resonance” is discussed, in which several networks and/or global fields strongly interact in select frequency bands even when they are only weakly connected. This general view of the “binding problem” of brain science fits well with observations of cognitive events revealing EEG signatures in select frequency bands.

1 Neocortical Dynamic Properties for the Millennium

For more than a century, neuroscientists have pursued the Holy Grail of connecting psychology with physiology, but such achievements have been quite difficult to accomplish. Many robust EEG links to psychology have been established over the past 80 years. General states of consciousness, brain pathology, and specific cognitive processes have been shown to be moderately to strongly correlated with EEG dynamic measures (Kellaway 1979; Gevins and Cutillo 1995; Silberstein 1995a; Uhl 1998; Klimesch 1999; Niedermeyer and Lopes da Silva 1999; Aminoff 1999). Thus, an indirect approach to establishing connections between psychology and physiology is to link EEG data to the underlying physiology as indicated in fig. 11-1; this chapter provides several views at this issue from a theoretical perspective. Several preliminary mathematical theories are proposed so that stronger links may be added to the serial chain physiology–EEG–psychology.

New relations between cognitive events and EEG measures are discovered on a regular basis. We have not described many such connections in this book. Rather, a few cognitive experiments are cited mainly as a means to demonstrate the data analyses that seem effective in revealing dynamic properties of EEG. Naturally, we would also like to focus on critical cognitive experiments that are most likely to endure, but who can say what cognitive science will look like in 50 or 100 years? By contrast to the cognitive theories that motivate many of today’s experiments, EEG dynamic properties are determined by natural selection; they will be just as valid in a century or so as they are today. We guess that future cognitive experiments, whatever form they may take, will benefit from such (p.487)

Figure 11-1 Indirect connections between cognitive science and physiology may be achieved through the established EEG–cognitive links (solid arrow) and new links between EEG and physiology (dashed arrow) facilitated by mathematical theory. In the lower box, synaptic and action potential fields are shown above networks suggesting that such fields are more directly related to EEG data than neural networks (or probably more accurately, cell assemblies).

dynamic knowledge. An abbreviated summary of some of today’s established dynamic EEG properties is as follows.
1. (i) Eyes closed, resting alpha rhythms with frequencies in the 8 to 13 Hz range are easily recorded from the scalp in perhaps 95% of the adult population. Alpha rhythms have been recorded from nearly the entire upper surface of exposed neocortex (Jasper and Penfield 1949). High-density scalp EEG also reveals widespread alpha rhythms over the entire scalp as discussed in chapter 10 (Nunez et al. 2001). Cortical and scalp recordings have revealed the existence of multiple alpha rhythms in different parts of cortex (Pfurtscheller and Neuper 1992; Andrew and Pfurtscheller 1997; Florian et al. 1998; Sarnthein et al. 1998; Pfurtscheller and Lopes da Silva 1999). Some are blocked by eyes opening; some are not. Some are blocked by mental activity; some are not. Scalp recordings of alpha rhythm are space averages of apparent multiple processes. As a result of this spatial filtering, scalp potentials are biased towards cortical global activity; that is, activity in the low end of the spatial frequency spectrum (Nunez 1974b; Wingeier et al. 2001). Comparison of temporal frequency spectra obtained from high-resolution and conventional scalp recordings reveals that alpha rhythms show both global and local dynamic behavior (Nunez et al. 2001). That is, high-resolution EEG methods spatially filter out both the global source dynamics and volume conducted potentials leaving only mid-scale dynamics (which is still much larger scale than cortical recordings). These scalp EEG studies are fully consistent with Grey Walter’s early ECoG observations.

2. (p.488)
3. (ii) When recorded from the scalp, the low and high frequencies of the alpha band have somewhat different spatial distributions. The high alpha band has more relative power at higher spatial frequencies (Nunez 1974b, 1995; Shaw 1991; Wingeier et al. 2001; Wingeier 2004). Furthermore, power and coherence changes with transitions between resting and cognitive states (mental tasks) can occur in opposite directions in the upper and lower alpha bands (Klimesch et al. 1999; Nunez et al. 2001; Wingeier 2004). Again, these data are consistent with Walter’s ECoG observations of multiple alpha rhythms.

4. (iii) Scalp recordings during deep sleep and some coma and anesthesia states produce large-amplitude, widespread delta (0 to 4 Hz) activity over the scalp. With halothane anesthesia, it is possible to “tune” the brain (Stockard 1996; Nunez et al. 1976, 1977, 1995). Halothane-dominant oscillation frequency can vary from about 4 to 16 Hz depending on inspired concentration; high concentrations result in lower frequencies and larger amplitudes. Multimode oscillation frequencies go up and down together with inspired concentration as shown in the example of fig. 11-2.

Figure 11-2 Anesthesia time-frequency plot. Multimode oscillation frequencies go up and down together with inspired concentration of halothane (sine wave modulated shown at right). EEG is large amplitude (≈50–100 μV) over the entire scalp. All eight subjects in the study showed similar EEG behavior, but with differences in number and intensity of apparent modes. Reproduced with permission from Stockard (1976) and Nunez et al. (1977).

5. (p.489)
6. (iv) Differences in observed ECoG dynamic behavior between cortical areas disappear during anesthesia based on visual inspection of raw time series (Bickford 1950; Penfield and Jasper 1954). That is, transitions from normal waking to anesthesia states appear to correspond to transitions from more local to more global dynamic states. A large variety of EEG behavior may be observed depending on depth and type of anesthesia as well as type of coma. These include sinusoidal oscillations and complex waveforms (combinations of oscillations) in the delta, theta, alpha, and beta bands. However, as a general rule of head, lower temporal frequency oscillations tend to occur with larger amplitudes in a wide range of brain states (Barlow 1993).

7. (v) Mental activity of various sorts tends to enhance EEG power in certain frequency bands and suppress power in other bands. Many of the details are subject and task dependent, but increased frontal power in the theta band (4–7 Hz) during mental activity is a common finding (Gevins et al. 1997; Klimesch 1999). In some subjects, power increases may also occur in upper alpha (10–13 Hz) and perhaps beta bands (>13 Hz), while power in the lower alpha band (8–9 Hz) decreases (Petsche et al. 1997; Petsche and Etlinger 1998; Klimesch et al. 1999; Nunez et al. 2001). Based on evidence from intracranial studies in animals (Singer 1993; Bressler 1995) and scalp experiments in humans (Lachaux et al. 1999), there are reasons to believe that mental activity is also associated with changes in the 40 Hz range (and possibly higher); however, the high probability of muscle artifact contamination has limited interpretations of gamma recordings from the human scalp.

8. (vi) Changes in the long-range covariance of transient event-related potentials are associated with correct performance on various mental tasks (Gevins and Cutillo 1995). These covariance patterns are presumably due to selective source synchrony at the relatively low temporal frequencies of the theta and alpha bands. These data suggest that cognition is associated with rapidly shifting patterns of statistical interdependency between (often) remote cortical locations.

9. (vii) Amplitude, phase, and coherence changes of 13 Hz steady-state visually evoked potentials (SSVEPs) are correlated with performance on mental tasks (Silberstein 1995a; Silberstein et al. 2001, 2003, 2004). Long-range coherence between some regions increases during mental activity (suggesting formation of regional networks) while coherence between other regions decreases. The latter effect can be interpreted as indicating dissolution of irrelevant networks, reduction in global field effects (discussed in section 3) or some combination.

10. (p.490)
11. (viii) In studies of binocular rivalry, where two incongruent images are flickered (one to each eye) at 7–12 Hz, steady-state evoked magnetic fields and SSVEPs show that conscious perception of only one of the two images (a unitary consciousness) is associated with increased inter-hemispheric coherence (Srinivasan et al. 1999; Edelman and Tononi 2000).

12. (ix) Mental activity of various sorts tends to enhance EEG coherence in certain frequency bands and electrode pairs and suppress coherence in other bands and electrode pairs (Nunez et al. 1997, 1999). Many of the details are subject dependent, but increased coherence in frontal electrodes in the theta band (4–8 Hz) during mental activity is a common finding (Nunez et al. 2001). Coherence changes may be either coincident or occur independently of power changes (Petsche et al. 1997, 1998), apparently depending on the spatial scale of coherent source activity as discussed in chapter 9 and Nunez (2001). One study has reported coupling of theta and gamma band activity during human short-term memory processing, as measured by bicoherence (Schack et al. 2002). These data have direct relevance to the resonance phenomena outlined in sections 7 and 8.

2 A Tentative Framework for Brain Dynamics and Several Conjectures

The framework outlined in chapter 1 (see fig. 1-8) appears to be well supported by these general properties of EEG data together with the various theoretical considerations discussed throughout this book, including both dynamic and volume conduction theory. That is, neocortical source dynamics may be viewed in terms of rapidly changing cell assemblies (or networks) embedded within a global environment. We choose to express this global environment in terms of synaptic and action potential fields. Scalp potentials evidently provide signatures of some (generally unknown) combination of the synaptic action fields and network activity, but are strongly biased towards dynamics with low spatial frequencies, suggesting substantial (or perhaps dominant) contributions from globally coherent synaptic fields. We further conjecture that both bottom-up (networks to global) and top--down (global to networks) interactions provide important contributions to the neocortical dynamics of behavior and cognition. Such two-way interactions appear to substantially facilitate so-called brain binding, the ability to coordinate separate functions into unified behavior and consciousness. Our proposed framework is consistent with the following descriptions by other scientists.

1. (i) In the following quote by Mountcastle (1979) we may substitute local networks or small-scale cell assemblies for “systems”: (p.491)

The brain is a complex of widely and reciprocally interconnected systems and the dynamic interplay of neural activity within and between these systems is the very essence of brain function.

2. (ii) Based on extensive studies of evoked potential covariance associated with mental activity, Gevins and Cutillo (1995) state:

…many (cortical) areas probably are involved in a constellation of rapidly changing functional networks that provide the delicate balance between stimulus-locked behavior and purely imaginary ideation.

3. (iii) Based on synchronization and other studies in human and nonhuman primates, Bressler (1995) conjectures:

…elementary functions are localized in discrete cortical areas, whereas complex functions are processed in parallel in widespread cortical networks. Control processes, operating at cortical and sub-cortical levels by a variety of mechanisms, dynamically organize and regulate large-scale cortical networks.

4. (iv) The following view is expressed by Edelman and Tononi (2000) in the context of a quantitative complexity measure associated with consciousness and brain binding:

…high values of complexity correspond to an optimal synthesis of functional specialization and functional integration within a system. This is clearly the case for systems like the brain—different areas and different neurons do different things (they are differentiated) at the same time they interact to give rise to a unified conscious scene and to unify behaviors (they are integrated).

In this latter view complexity (and by implication, cognition) tends to maximize between the extremes of isolated networks and global coherence. We find this to be a compelling working hypothesis. It then follows that both local network dynamics and interactions between networks are important for brain function. However, one of the central messages of this book is that different experimental designs and methods of data analysis can bias EEG (or MEG or fMRI or PET) physiological interpretations in either local or global directions.

Silberstein (1995b) has taken this general local versus global dynamic picture a step further in the physiological and clinical directions by suggesting several ways in which brainstem neurotransmitter systems might act to change the coupling strength between global fields and local or regional networks. He outlines how different neurotransmitters might alter the coupling strengths by selective actions at different cortical depths, thereby changing resonance properties. He further conjectures that several diseases (Parkinson disease, some schizophrenias) may be manifestations of hypercoupled or hypocoupled dynamic states brought on by the faulty actions of the neurotransmitters. With this background in mind, we examine possible properties of networks and global fields in the following sections.

(p.492) 3 Multiscale Dynamic Theory Illustrated with a Metaphorical Field

Throughout this book, we have emphasized the importance of spatial scale in both experimental and theoretical studies of brain dynamics. In order to facilitate better understanding of relationships between dynamic variables defined at different scales and their association with putative brain networks, a metaphorical theory is outlined here. Our general conceptual framework is expressed by fig. 1-8, in which we imagine cell assemblies (or probably less accurately neural networks) immersed in global fields of synaptic action. These synaptic fields are distinguished from the electric and magnetic fields that they generate. The existence of these fields is noncontroversial. For example, all we mean by the excitatory synaptic action field Ψe(r, t) is simply the number density of active excitatory synapses in some tissue voxel located at r, defined over the entire cortical surface. Such definition implies a coarse graining over small time and space scales so that Ψe(r, t) varies relatively smoothly in time and space. The only possible controversy is whether the introduction of such field concepts is helpful to neuroscience.

Modern theories of large-scale neocortical dynamics are likely to be field theories (or mean-field theories) for both theoretical and experimental reasons. In order to facilitate better communication between biological and physical scientists and between experimentalists and theoreticians, we here consider a fanciful field theory of human alcohol consumption over the earth’s surface. Suppose our first choice of dependent variable is Ψa(r, t), the volume of alcohol consumed at surface location r and time t. This choice causes immediate problems for theory development. The variable Ψa(r, t) can only be nonzero at the discrete locations of human drinkers, and humans are continually on the move. Also, the drinking process also tends to be discontinuous in time. Thus, our variable Ψa(r, t) must fluctuate widely in both space and time in a manner similar to very small-scale (perhaps fractal) measurements of synaptic current source activity. Such discontinuous dynamic behavior creates substantial problems, not only for theory development, but also for experimental attempts to measure alcohol consumption and check the theory. For these reasons, we may define a set of coarse-grained variables (fields) in terms of time and space averages, that is

(11.1)
$Display mathematics$
Here the (x, y) are surface coordinates and Γ(x, y) is some weighting function. The spatial and temporal scales of the theory or experiment are (p.493) given by the coarse graining parameters (X, Y, T). The weighting function (or distribution function or kernel) Γ(x, y) is constant (equal to one) if the field Ψ is a simple space-average of alcohol consumption. Alternatively, Ψ might represent another large-scale variable that depends on small-scale consumption, say alcohol-related traffic accidents, in which case the integral weighted by Γ(x, y) would effect the transformation from small-scale variable Ψa(r, t) to the new field. Of course, finding the appropriate weighting function Γ(x, y) typically requires a separate theory. Some poorly informed sociologist working at the small scales of individual family behaviors might display scale chauvinism by labeling the macroscopic field Ψ as an epiphenomenon, displaying an attitude found in several scientific fields. A more enlightened viewpoint would consider the large-scale variable Ψ(r, t) to result from a bottom-up interaction across spatial scales. Even this would not be fully accurate if the small-scale drinking rate Ψa(r, t) were substantially influenced by (say) global laws or police action resulting from excessive alcohol-related accidents, a top-down interaction across spatial scales. This system would then experience circular causality as discussed in section 10, or symbolically Ψa(r, t) ↔ Ψ(r, t).

A theory developed for particular spatial and temporal scales, say the (1 mile, 1 day) scales might provide some prediction of the rate of alcohol consumption in neighborhoods on a day-by-day basis. Such theory might predict “fast” sinusoidal oscillations in time with a constant period of one week and “spikes” at holidays, but would be unable to predict hourly oscillations. These particular scales might be chosen in the theory with the aim of matching experimental data, perhaps obtained by contacting neighborhood merchants every morning. Alternatively, we might imagine that alcohol data are available only at quite different scales, say at city or state levels. Naturally, we want our theory to match these experimental scales. Just as in the case of synaptic fields, any global alcohol function Ψ must satisfy two fundamental mathematical conditions: Ψ must be finite at all earth surface locations, and it must be a single-valued function of surface coordinates. The latter condition requires that Ψ satisfy periodic boundary conditions. By contrast, strictly local theories do not require global boundary conditions. We might, for example, just prohibit drinking in the countryside, that is set Ψ = 0 at the boundaries of our cities of interest.

Several physiological examples of this kind of coarse graining are provided in this book. One such example (but in three spatial dimensions) is the definition of the mesosource function in terms of the synaptic microcurrent sources (4.26). In this case, the weighting function Γ is simply the location vector w of microsources within the tissue mass and no time averaging is needed. Another example, is the expression for (macroscopic) scalp potential in terms of the mesosource function given by (2.2), where the weighting function Γ is the Green’s function containing all information about the head volume conductor. The standard scalp-recorded evoked (p.494) or event-related potential represents an experimental coarse graining of cortical potential with space averaging forced by the head volume conductor. Finally, we note that investors in financial markets are intimately familiar with the coarse graining of experimental data. The usual running average of market indices over perhaps T = 90 or 200 days provides the time averaging, while the market index itself (say the S&P500) is essentially the space average, calculated with the market capitalizations of individual stocks forming the weighting function Γ.

4 A Simple Model for Global Fields

In order to distinguish the various theories of large-scale neocortical dynamics, we adopt the label local theory to indicate mathematical models of cortical or thalamocortical interactions (feedback loops) for which corticocortical propagation delays are assumed to be zero. The underlying timescales in these theories are typically postsynaptic potential (PSP) rise and decay times. Thalamocortical networks are also “local” from the viewpoint of a surface electrode, which cannot distinguish purely cortical from thalamocortical networks. Finally, these theories are “local” in the sense of being independent of global boundary conditions dictated by the size and shape of the cortical–white matter system.

By contrast, we adopt the label global theory to indicate mathematical models in which delays in the corticocortical fibers forming most of the white matter in humans provide the important underlying timescale for the large-scale EEG dynamics recorded by scalp electrodes. Periodic boundary conditions are generally essential to global theories because the cortical–white matter system of one hemisphere is topologically close to a spherical shell as indicated in figs. 11-3 and 3-12. The most recent theories of neocortical dynamics include selected aspects of both local and global theories, but typically with more emphasis on one or the other.

Figure 11-4a indicates the two halves of the cerebral cortex with several modular columns of diameter ≈ 300 μm, each containing about 1,000 to 10,000 neurons (Szentagothai 2004). Several axons are shown crossing through the corpus callosum (the total number is of the order of 108 in the human brain), one corticocortical axon (total number about 1010) and two corticothalamic axons (total number about 108). figure 11-4b shows just a few network details in a single column through the entire cortex (with transverse diameter stretched to show more detail). In this section, we imagine a neocortex entirely devoid of networks. That is, all preferential cell interactions are ignored and only global synaptic fields due to simple input–output relations in cortical tissue masses are considered. We do not suggest that this is a realistic picture for all (or even most) brain states. Rather, we conjecture some approximate correspondence with sleep, resting, or anesthesia states. We further suggest that global fields may play (p.495)

Figure 11-3 Periodic boundary conditions are generally essential to global theories because the cortical–white matter system of one hemisphere is topologically nearly identical to a spherical shell. Here a computer algorithm progressively inflates each brain hemisphere and transforms them into spheres. The figure was generated by Armin Fuchs and kindly transmitted to Nunez for this publication. The computer algorithm was developed by Dale et al. (1993, 1999) and Fischl et al. (1999).

(p.496)

Figure 11-4 (a) Several corticocortical columns (modules) with diameters ≈200 to 300 μm through the entire cortex are shown. Column widths are greatly exaggerated to show interiors. The modules are defined as the width of aborization of corticocortical afferents, that is, the input scale for long-range cortical connections. Several callosal and one corticocortical axons are shown. The human brain has about 1010 corticocortical and perhaps 108 thalamocortical fibers (Braitenberg and Schuz 1991). (b) Some details of networks inside one column are shown. Each corticocortical column has about 1,000 to 10,000 neurons and most neurons send an axon into the white matter to connect to other cortical regions or to thalamus. The hourglass shape was chosen to suggest the dynamic distortion of a cylindrical module by the tendency for lateral (intracortical and corticocortical) excitation to spread to neighboring columns from layers I and VI (white arrows) with inhibitory interactions act mainly on middle layers III, IV, and V (black arrows). Reproduced with permission from Szentagothia (2004).

(p.497) important roles in facilitating communication between separate networks. Mainly, this basic global theory should be considered a crude limiting approximation that suggests some general global properties to look for in experimental EEG data. For purposes of the simplest version of the global brain theory we assume the following:
1. (i) Excitatory action potentials are transmitted along intracortical and the corticocortical axons that form most of the white matter layer directly below neocortex. Since axon propagation velocities are finite, action potentials at one cortical location produce synaptic activity at distant locations after time delays that are proportional to separation distance on an inflated (idealized) smooth surface.

2. (ii) The intracortical and corticocortical axons are parceled into M excitatory systems. For each fiber system (m = 1, M), the density of connections between any two cortical regions falls off exponentially with separation distance. The characteristic lengths of the individual exponential decays are given by $λ m − 1$. An example three-fiber system is depicted in fig. 11-5.

Figure 11-5 In the global theory, the assumed fall-off in excitatory fiber density between any two cortical locations separated by distance |xx 1| is given by a sum of M exponential decays corresponding to M fiber systems with characteristic lengths $λ m − 1$. This example pictures M = 3 fiber systems: a short-range intracortical system (recurrent collaterals, λ1 > 10 cm−1) represented by the short arrow, an intermediate length system (U fibers and somewhat longer corticocortical fibers, λ2 < 1 cm−1) represented by the dashed line, and a long corticocortical fiber system (λ3 ≈ 0.1 to 0.2 cm−1) represented by the solid line. Only efferent fibers exiting the gray column are indicated in this picture except for dynamic input from the thalamus (vertical gray arrow). All cortical tissue masses are assumed to have both afferent and efferent fibers. In the simplest version of the global theory, the influences of the shorter excitatory fibers (larger λm values) and all inhibitory fibers are lumped into the cortical excitability parameter β, which acts to control neocortex by means of chemical and electrical input from subcortical regions (dashed white arrow). This input is assumed to act on much longer timescales (seconds to minutes) than the millisecond-scale dynamic input from the thalamus. See appendix L and Nunez (1995) for details.

(p.498)

Figure 11-6 A mesoscopic cortical tissue mass (say millimeter scale) is represented with excitatory synaptic input Ψe(x, t) from other cortical tissue indicated by the gray arrow (intracortical and corticocortical). As a result the tissue mass produces a certain number of action potentials (per volume or cortical surface area): the tissue mass output Θ(x, t). The linear approximation of the global theory postulates that small changes in this input δΨe(x, t) cause proportional changes in output δΘ(x, t). The text uses the simplified notation δΨe(x, t) ≡ Ψ(x, t). The proportionality constant β depends on tissue excitability as determined by chemical and electrical cortical input that occurs on timescales much longer than synaptic field cycle times. In the quasi-linear approximation proposed in this chapter, it is conjectured that large excitatory fields |Ψe(x, t)| cause recruitment of additional inhibitory input to prevent instability (epilepsy) in healthy brains.

3. (iii) An incremental change in number density of action potentials δΘ(r, t) produced in a mesoscopic mass of cortical tissue increases roughly in proportion to small increases in number density of excitatory synaptic inputs δΨe(r, t) as suggested in fig. 11-6. Similarly, action potential density tends to decrease in approximate proportion to small increases in number density inhibitory synaptic inputs δΨi(r, t). This is the basic linear approximation that ignores nonlinear feedback as well as all local network effects expected in most brain states.

4. (iv) The magnitudes of the action potential changes in (iii) are determined by a single control parameter β, the background excitability of neocortex. This excitability parameter β, which partly determines brain state, may be changed by unspecified chemical and electrical input from midbrain or neocortex. In the simplest version of the global theory, influences of the inhibitory input δΨi(r, t) and shorter corticocortical fibers (excitatory) are lumped into β. Changes in β occur on much longer timescales (seconds to minutes) than EEG cycle times.

5. (v) As the excitatory synaptic action density |Ψe(r, t)| in some neocortical region r becomes progressively larger, extra inhibitory input to the local tissue mass is recruited in healthy brains as a result of negative feedback from contiguous cortex, thalamus, or both. This is a quasi-linear approximation that may cause limit cycle-like oscillations of global modes, but still ignores the local network effects expected in most brain states.

The model cortex outlined above has been considered in a series of theoretical papers (Nunez 1972, 1974, 1989, 1995, 2000a, b; Katznelson (p.499) 1981). A short summary of the one-dimensional version appears in appendix L. The general predictions of this theory are as follows.

The modulation of excitatory neocortical synaptic action may be expressed as a weighted sum of contributions of the form

(11.2)
$Display mathematics$
Here the functions ξn(t) are called the order parameters in the field of synergetics, the science of cooperation (Haken 1983). The general form of the expression (11.2) applies to a large variety of complex physical systems. In some quasi-linear approximations, the order parameters ξn(t) in the sum may be approximated by
(11.3)
$Display mathematics$
The spatial functions (eigenfunctions) ψn(r) are severely restricted by cortical boundary conditions; that is, by the size and shape of the cortical surface. For example, the theory predicts standing waves in a spherical shell (Katznelson 1981; Nunez 1995), in which case the ψn(r) are the spherical harmonics Y nm(θ, φ). Here we outline the simplest version of this work in which oscillation frequencies of multiple modes occur in a closed one dimensional loop representing the anterior–posterior circumference of one cortical hemisphere. As outlined in appendix L, each spatial mode oscillates with its characteristic mode frequency
(11.4)
$Display mathematics$
The neocortical parameters in (11.4) and their probable ranges are as follows:
1. (i) Characteristic velocity (peak in the velocity distribution function) for propagation in corticocortical fibers (review in Nunez 1995):

(11.5)
$Display mathematics$

2. (ii) Effective front-to-back circumference of one cortical hemisphere after inflation to smooth the surface as shown in fig. 11-3. For waves on a spherical surface with area equal to 1500 cm2, the effective radius is about 11 cm, corresponding to a circumference of about 70 cm. However, each brain hemisphere is shaped more like an eccentric prolate spheroid with a long (smooth) circumference in roughly the range

(11.6)
$Display mathematics$

3. (p.500)
4. (iii) Nondimensional cortical excitability control parameter β. The excitability control parameter is roughly proportional to the incremental increase in action potential density δΘ(r, t) produced within a cortical mass element (the output) due to an incremental increase in excitatory synaptic input, that is

(11.7)
$Display mathematics$
A plot of action potential output Θ versus excitatory input Ψe is expected to have a sigmoid shape if there are no local circuit effects. In this case, β is proportional to the local slope of the sigmoid. In the linear limiting case, β>1 causes all modes to become unstable. However, we assume here that natural selection has provided for enhanced negative feedback from thalamus or contiguous cortex to prevent such instability in healthy brains. We do not now have accurate estimates for the range of β, which is expected to vary with brain state, but a plausible guess is something like 0<β<10 (Nunez 1995). As discussed below, ignorance of β does not prevent us from obtaining several rough theoretical predictions.

5. (iv) The longest corticocortical fiber system density is assumed to fall off according to $exp ⁡ [ − λ M / r − r 1 / ]$ between any two cortical locations r and r 1 (or x and x 1 in the one-dimensional version), as indicated in fig. 11-5. The effects of short excitatory fibers (larger λm values in both intracortical and corticocortical systems) are included (approximately) in the excitability parameter β (for details see pp. 488–490 of Nunez 1995). This assumed exponential fall off in fiber connection density is chosen for mathematical convenience rather than physiological reality in order to facilitate analytic solutions. Nevertheless, by fitting a slightly more complicated function to the measured (and scaled) fall-off of corticocortical fiber density in mouse brain, very similar theoretical results are obtained (for details see pp. 506–511 of Nunez 1995). We simplify the notation using λM → λ. An estimated range for the average (long) fiber length λ−1 for an inflated cortex is 5 to 10 cm (Nunez 1995). Thus, the following nondimensional parameter range is suggested:

(11.8)
$Display mathematics$
(p.501) By applying these estimates to the predicted angular frequencies (11.4) yields
(11.9)
$Display mathematics$
In this case of one-dimensional waves around the circumference, the n = 1 mode appears to be nonoscillatory (imaginary frequency). Both the n = 1 and n = 2 modes appear to be roughly consistent with the average spatial distribution of scalp alpha rhythms. To consider a (partly arbitrary) numerical example, the parameter choices
(11.10)
$Display mathematics$
yield the following predicted mode frequencies:
(11.11)
$Display mathematics$
While the exact frequencies cannot be taken seriously, one general prediction is multiple global oscillatory modes with progressively higher frequencies. These overtones are not harmonics, which are normally associated with nonlinear effects. We have ignored many details; for example, the mode frequencies of standing waves in a closed surface (a prolate spheroidal shell, for example) tend to occur in clusters near the frequencies of the major modes associated with the largest circumference (Nunez 1995). Thus, we might expect to find several modes clustered near 10 Hz due only to the shape of the two-dimensional surface. The frequencies predicted by (11.4) are generally in the EEG range, suggesting that the general global picture of standing waves warrants deeper study. However, parameter uncertainty prevents quantitative verification of the theory based only on frequency estimates.

Each hemisphere of neocortex is topologically very close to a spherical shell as indicated in fig. 11-3. The choice of the eigenfunctions ψn(r) is severely restricted in closed systems for two fundamental physical reasons. First, because they represent genuine (measurable) variables, the functions ψn(r) must be finite everywhere on the closed surface. Second, they must be single-valued functions of surface coordinates. For example, suppose we choose some function to represent population density over the earth. The population of (say) New York predicted by our function cannot depend on whether the coordinate path is measured east to west or west to east. The most common eigenfunctions used on spherical surfaces are the spherical harmonics Y nm(θ, φ). These functions satisfy both physical constraints and arise naturally as a result of the spherical geometry and (p.502)

Figure 11-7 Traveling waves in a spherical shell. Wave packets originate at eight cortical locations perhaps as a result of subcortical input. Dark and light patches indicate regions of positive and negative variations of the excitatory synaptic action field δΨe(r, t) ≡ Ψ(r, t), respectively. Empty spaces show regions with excitatory synaptic action close to the background field Ψe(r, t). Constructive and destructive interference is observed in all but the first plot. These plots represent generic weakly damped waves; they do not depend on any specific model. Reproduced with permission from Nunez (1995).

the Laplacian operator, which occurs in the equations of physics that are most frequently applied to physical systems as well as in volume conduction in the head.

Nondispersive and weakly damped traveling waves in a thin spherical shell are illustrated in fig. 11-7. These example wave packets are generated at eight discrete locations and spread out in the shell in a manner similar to water waves due to raindrops in a pond or electromagnetic waves due to multiple lighting strikes in the atmosphere. Overlapping wave packets exhibit interference phenomena; that is, regions with negative field values tend to cancel regions of positive field values. After the external input stops, the fields settle into standing wave patterns for some additional time (depending on the magnitude of damping) as in the examples shown in fig. 11-8. A similar interference phenomenon is postulated in neocortical tissue at mesoscopic scales. That is, if a portion of a wave packet having excess excitatory synaptic action Ψ(r, t) ≡ δΨe(r, t), that is, a positive perturbation about the background level, encounters a wave portion with locally reduced synaptic action, we expect partial cancellation of synaptic action in tissue masses containing large numbers of neurons. This interference effect may or may not be approximately linear depending partly on the magnitudes of the perturbations δΨe(r, t).

(p.503)

Figure 11-8 Standing waves in a spherical shell similar to the Schumann resonances discussed in chapter 3. The surface waves were driven by 100 point sources, random in both location and time (for example, lightning strikes in the atmosphere). The resulting spatial patterns (analogous to fig. 11-7) are quite complicated and not shown here. Rather we Fourier transform these data and show magnitude (upper row) and cosine (phase) plots (lower row) for the broad frequency bands associated with the sum of several of the spherical harmonic functions. (Left column) (n = 2 through 5, labeled l in the plot). (Right column) (n = 2 through 7) showing more contributions from higher spatial frequencies. Reproduced with permission from Srinivasan (1995) and Nunez (1995).

Predicted global modes for standing brain waves in a spherical shell fig. 3-12) were obtained by Katznelson (1981) as a solution to the global wave equation (Nunez 1972, 1974, 1981). The model assumptions listed above are all unchanged except that the closed loop becomes a spherical shell of radius R (reviewed and updated in Nunez 1995). The nondimensional mode frequencies (ωn R/v) and damping (γn R/v) are plotted versus the spherical control parameter βS in fig. 11-9 (mode n = 1), fig. 11-10 (mode n = 2), and fig. 11-11 (mode n = 3). With the choice of parameters (L, R, λ, v) = (80 cm, 12.7 cm, 0.1 cm−1, 750 cm/s), the fundamental mode f 1 is about 9 Hz when the nondimensional frequency ω1 R/v equals one. With this choice of λR = 1.27, the two-dimensional control parameter used here (βS) is roughly equivalent to the one-dimensional version β (for details see pages 521–528 of Nunez 1995). Mode damping is independent of mode number in the simplest version of the one-dimensional linear theory; all modes become unstable for β>1. By contrast, the spherical model exhibits a mode scanning property in following sense. As βS increases from values less than one, lower modes become more weakly damped and oscillate at progressively lower frequencies. (p.504)

Figure 11-9 The fundamental mode (n = 1) in the spherical global model. (Upper) The nondimensional global mode damping (γ1 R/v) is plotted versus the spherical cortical excitability parameter βS, chosen here to match approximately the equivalent one-dimensional parameter β for the case λR = 1.27. (Lower) The nondimensional global mode frequency (ω1 R/v) is plotted versus the spherical cortical excitability parameter βS. See Katznelson (1981) and Nunez (1995) for more detail.

At still larger βS these lower modes become unstable (in the strictly linear theory); their frequencies fall to zero and become nonoscillatory.

An example of this mode scanning feature (adopting the above parameters for ease of the description) is as follows. For very small βS, the fundamental mode (n = 1) oscillates at about 9 Hz, but is strongly damped as shown in fig. 11-9. We might guess that this mode can be observed in EEG only if the global field is coupled to local oscillatory networks with similar resonance frequencies (perhaps generating the multiple alpha rhythms discussed in section 1). As βS increases towards one, the fundamental mode frequency falls sharply while its damping decreases. The fundamental mode becomes nonoscillatory for βS>1 as shown in fig. 11-9. The second mode (first overtone, n = 2) is strongly damped for small βS, but damping falls to zero for βS ≈ 1.6 as indicated in fig. 11-10. At this point the second mode oscillates with a frequency of about 13 Hz. Larger values of βS then cause the frequency of the second mode to fall while instability increases (in the strictly linear theory). The second mode becomes nonoscillatory at βS ≈ 3.2. When βS reaches about 1.7, the third mode becomes unstable and oscillates at about 23 Hz, as indicted by the solid curves in fig. 11-11. As βS increases further, the (p.505)

Figure 11-10 The first overtone (n = 2) in the spherical global model. (Upper) The nondimensional global mode damping (γ2 R/v) is plotted versus the spherical cortical excitability parameter βS. (Lower) The nondimensional global mode frequency (ω2 R/v) is plotted versus the spherical cortical excitability parameter βS.

frequency of the third mode progressively decreases and becomes nonoscillatory at βS ≈ 6.1. This general behavior is repeated for higher modes with additional increases in βS except that multiple branches of the dispersion relation appear as indicated in the fig. 11-11; the (dashed) branches shown are strongly damped. In summary, at about the same time that lower modes disappear from the field oscillations, higher modes become progressively more weakly damped and oscillate with lower frequencies. As in the one-dimensional version of the global theory, we conjecture enhanced negative feedback in healthy brains to prevent instability.

5 More Realistic Approximations to the Neocortical Dynamic Global Theory

Because of the extreme oversimplification inherent in the “toy brain” outlined in section 4, its predictive value is expected to be quite limited. Nevertheless, it may provide a genuine “entry point” to more comprehensive brain theory. One obvious place to start is to re-examine the assumed linear relation between action potential output Θ(x, t) and excitatory synaptic input given by (11.6). By backtracking one step in the (p.506)

Figure 11-11 The second overtone (n = 3) in the spherical global model. (Upper) The nondimensional global mode damping (γ3 R/v) is plotted versus the spherical cortical excitability parameter βS. (Lower) The nondimensional global mode frequency (ω3 R/v) is plotted versus the spherical cortical excitability parameter βS. The solid and dashed lines indicate two branches of the dispersion relation.

one-dimensional global theory (see appendix L) we obtain equation (A.10) of the appendix of Nunez (1995) or equation (46) of Jirsa and Haken (1997), both of which are of the form
(11.12)
$Display mathematics$
Here Ψ(x, t) ≡ δΨe(x, t) and δΘ(x, t) are the modulations of the excitatory synaptic action and action potential fields about their background levels, respectively, and Z(x, t) includes additional cortical input, such as sensory input. In the simplest case of no local circuit effects, we may expect a sigmoid relation between excitatory input and action potential output (Freeman 1975). A straightforward mathematical approach to this general idea is to expand the sigmoid function in a Taylor series (Nunez 1995). By suitable linear transformation of Ψ(x, t) and keeping the first nonzero nonlinear term, Jirsa and Haken (1997) obtain the simple relation
(11.13)
$Display mathematics$
(p.507) With the approximation (11.13), equation (46) of Jirsa and Haken (1997) is of the form
(11.14)
$Display mathematics$
In the linear limit (α → 0), the spatial-temporal Fourier transform of (11.14) recovers equation (A.10) of Nunez (1995) and the dispersion relation (11.4). The most important effect of the nonlinear terms is to prevent the instability that occurs in the one-dimensional linear theory when β>1. Equation (11.14) may be solve numerically and/or one may seek approximate solutions of the form
(11.15)
$Display mathematics$
Equation (11.15) is the one-dimensional version of (11.2) with the complex spatial function chosen to satisfy periodic boundary conditions; that is, forcing Ψ(x, t) and its spatial derivatives to be continuous functions of x everywhere in the cortical loop (−L/2, L/2). Note that only integer waves are allowed in the closed loop, in contrast to also allowing half integer waves in (say) a violin string fixed at each end.

While (11.14) is apparently an improvement over the linear theory, treatment of Z(x, t) simply as a known forcing function neglects local circuit effects. Furthermore, our conjecture that natural selection may have provided multiple mechanisms for negative feedback to prevent instabilities in healthy brains suggests that the assumption of the sigmoidal input/output relation is probably inadequate. Thus, a more accurate dynamic theory might couple the equation for the excitatory modulation field Ψ(x, t) ≡ δΨe(x, t) given by (11.12) with separate equations for the inhibitory modulation field δΨi(x, t) and action potential field δΘ(x, t), for example

(11.16)
$Display mathematics$
(11.17)
$Display mathematics$
where S(x, t) is due only to sensory input. Various models of large-scale neocortical dynamics have been published that mostly fit this general approach as outlined in section 9.

6 Experimental Connections to Global Theory

Any neocortical dynamic theory that attempts to include cortical or thalamocortical network effects must contain several (more likely many) physiological parameters that vary with brain state and are probably (p.508) poorly known (if at all), as suggested by fig. 11-4. This provides motivation to start simply in our attempts to connect theory to EEG data. Thus, we consider several experimental implications of (11.4) that may enjoy some very approximate connections to brains in their more globally dominated states: apparently anesthesia, deep sleep, and the more globally dominant components of resting alpha rhythms. The reader should note that we only claim connections not comprehensive explanations to complex physiological processes!

2. (ii) The larger the cortex, the slower the global mode frequencies. Suppose we were to compare a large number of brains in which the longest corticocortical fiber tracts scale with cortical size, that is, with λL ≈ constant. Suppose further that axon diameters and (p.509) myelination do not tend to increase with brain size (no change in axon velocity v). In this case, each global mode frequency ωn would be inversely proportional to characteristic cortical size L. We are aware of only one human study in which this predicted size–frequency relationship was tested (Nunez et al. 1978; Nunez 1995). Participants were chosen through newspaper advertisements seeking volunteers with either very large or very small heads. Brain size is strongly correlated with skull volume (correlation coefficient = 0.83) (Blinkov and Glezer 1968). Head sizes and peak alpha frequencies were measured (blind) in 123 subjects with identifiable spectral peaks in the alpha band. Linear regression revealed a small (r = −0.206) but significant (p = 0.02) negative correlation between peak alpha frequency and head size. In addition, a “maximum frequency” was defined for each subject by the peak power histogram. That is, for each four-second epoch, the single frequency within the alpha band with the largest power was identified. A similar correlation between maximum frequency and head size was obtained (correlation coefficient = −0.233; p = 0.01). Strong correlations cannot be expected in such studies because brain sizes vary over only a small range, and many other parameters are expected to influence oscillation frequencies.

3. (iii) Both standing and traveling waves with long spatial wavelengths should occur across neocortex and be measurable on the scalp. As background, note that several studies have reported propagating activity when recording from animal cortex with small electrodes (Petsche et al. 1984, 1988; Lopes da Silva and Storm van Leeuwen 1978). These cortical spatial wavelengths are in the mm range and phase velocities are generally in the 1 mm/s range. Such waves apparently propagate by means of intracortical processes. These short-wavelength waves cannot be recorded from the scalp; volume conduction removes essentially all power at the mid and high ends of the spatial spectrum. That is, wave components with wavelengths shorter than a few centimeters cannot be recorded on the scalp. The data cited in chapter 10 show that (long-wavelength) traveling scalp waves can indeed be recorded in several experimental conditions.

4. (iv) Phase velocities measured at the scalp should be in the general range of the characteristic corticocortical propagation speed v. This is confirmed; supporting data are outlined in chapter 10.

5. (v) Higher temporal frequencies should be associated with higher spatial frequencies above the fundamental mode. Group velocities should be in the general range of v. Phase and group velocity are defined by

(11.18)
$Display mathematics$
(p.510) Application of (11.4) to the definitions (11.15) yields
(11.19)
$Display mathematics$
The data presented in chapter 10 are consistent with the (approximate) existence of dispersion relations for oscillations above the low end of the alpha band. Both phase and group velocities apparently occur in some (globally dominant) brain states and the velocities are in the general range of v. Our experimental estimates are not sufficiently refined to check (11.19) to see if group and phase velocities change in opposite directions. In any case, this test may be asking too much of such a crude theory.

6. (vi) The ECoG should contain more high-frequency content above the low end of the alpha band than the corresponding EEG. We know that volume conduction causes low-pass spatial filtering of potentials passing from cortex to scalp. We also know that the amplitudes and phases of scalp potentials generated by implanted dipole sources are unaltered by source frequency; tissue is purely resistive at large scales (Cooper et al. 1965). If our proposed global wave picture is correct, higher temporal frequencies should tend to occur with higher spatial frequencies in ECoG (above the low end of the alpha band). In this case, spatial filtering by the volume conductor implies temporal filtering of frequencies above the lower alpha band between cortex and scalp, as reported in a number of studies (Penfield and Jasper 1954; Delucchi et al. 1975; Pfurtscheller and Cooper 1975; Nunez 1981).

7. (vii) Higher frequency oscillations should tend to have lower scalp amplitudes. Scalp EEG amplitudes depend on two factors: the magnitude of the cortical mesosource function P(r, t) and the phase synchrony of the oscillations. Source function magnitude is roughly proportional to the modulation depth of the excitatory synaptic field Ψ(r, t). Source synchrony over large distances (long effective correlation lengths) is closely associated with large power in the low end of the spatial frequency spectrum. Thus, if the magnitude of the cortical mesosource function is fixed and higher temporal frequencies are associated with higher spatial frequencies (only above the lower end of the alpha band), we expect high-end alpha, beta, and gamma oscillations to have progressively lower scalp amplitudes. This is roughly consistent with observations.

However, we cannot apply these arguments to delta and theta bands. First, we have no evidence of wave dispersion for frequencies lower than the lower end of the alpha band. More importantly, the large frequency difference between alpha and (p.511) delta rhythms cannot be accounted for by changes in the spatial spectrum (by means of some putative dispersion relation). Nevertheless, increases in the cortical excitability parameter β is predicted to lower frequency in (11.4) and increase the amplitude of the synaptic action field modulation in (11.14). While the details of these changes depend on model specifics, several versions predict larger Ψ and lower mode frequencies with increasing β (Nunez 2000a).

8. (viii) Globally dominated EEG generally consists of multiple modes (frequency components) that increase and decrease together as the cortical excitability parameter β changes. The general effect of several modes going up and down together is demonstrated for one subject under varying concentrations of halothane anesthesia in fig. 11-2 and in fig. 11-12. These oscillations have large amplitudes over the entire scalp, generally in the range of

Figure 11-12 Anesthesia time-frequency plot. One subject is anesthetized to varying depths with halothane. Inspired concentration is shown as a function of time at right side. Increased halothane concentrations cause frequency reductions and amplitude increases in several modes. The oscillations generally have large amplitudes over the entire scalp, but different modes are emphasized by electrode pairs with different locations and orientations as shown. In the case of standing waves, placing both electrodes near a (cortical) nodal line of one mode is expected to reduce substantially the scalp amplitude of that particular mode, but not the amplitudes of other modes. Data recorded in connection with a separate cardiovascular study by Nunez et al. (1976). Reproduced with permission from Katznelson (1981).

(p.512) approximately 50–100 μV at moderate inspired concentrations of halothane. However, different modes are emphasized by different electrode pairs as expected of standing waves and indicated in fig. 11-12 (Nunez 1995). Furthermore, the inverse relationship between amplitude and frequency expected of limit cycle modes was observed in all eight subjects (Nunez et al. 1976).

Equation (11.4) may be differentiated with respect to β to predict fractional changes in mode frequency as a function of excitability changes (due perhaps to halothane concentration changes), that is

(11.20)
$Display mathematics$
To provide a numerical example, we choose the parameters (11.10) leading to f 2 = 10.5 Hz and suppose a brain state change causing a Δβ = 0.2. In this case (11.14) predicts a frequency reduction of the nth mode Δf n ≈ 3.5 Hz, that is, from 10.5 Hz to 7.0 Hz. Of course, this is just a numerical example; we have very little idea of the magnitude of β and no idea of the size of Δβ associated with any state change. However, a more accurate quasi-linear approximation might predict the relationship between the amplitudes of (conjectured) limit cycle-like modes C n and their frequencies (Nunez 2000b), that is
(11.21)
$Display mathematics$
For example, a combination of coordinate transformations and numerical solutions of (11.14) suggests the following approximation:
(11.22)
$Display mathematics$
for β somewhat larger than one. In summary, brain state changes provide experimental amplitude and frequency changes (ΔC n, Δf n). In principle, equations like (11.21) and (11.4) provide partly independent equations to estimate β, Δβ, or both. A measure of the consistency of these estimates provides a rough quantitative test of theory. For example, by combining (11.20) with (11.22), we obtain an estimate for β based only on fraction changes in field amplitude and frequency:
(11.23)
$Display mathematics$
(p.513) The first factor on the right-hand side of (11.23) is of the order of one if f n is in the alpha band. In principal, the fractional changes of amplitude and frequency may be estimated in transitions involving hyperventilation or drugs that alter alpha frequency and amplitude (alcohol, for example) or anesthesia. A lot of assumptions are required to obtain (11.23) so this estimate should be viewed mainly as providing ideas for more sophisticated studies.

9. (ix) Transitions between states due to increasing the parameter β can involve both decreases and increases in dominant frequencies. Consider the following EEG behavior of a patient after introduction of cyclopropane anesthesia (Sadove et al. 1967); these general changes are common to the actions of many anesthetics (Stockard and Bickford 1975). (30 seconds after anesthesia induction: 20 μV resting alpha rhythm disappears and is replaced by low voltage “fast activity” ≈ 5 to 10 μV), (90 seconds: low voltage fast activity superimposed on 4 to 8 Hz dominant rhythm ≈ 30 μV), (100 seconds: low voltage fast activity superimposed on 3 to 4 Hz dominant rhythm ≈ 35 μV), (120 seconds: deep anesthesia, 3 Hz irregular rhythm ≈ 40 μV), (3 minutes: respiration depressed, 1 Hz irregular waveforms more−“nonlinear looking” ≈ 40 μV), (Very deep anesthesia: progressively longer intervals of near-isoelectric EEG with occasional bursts of fast activity in the 5 to 10 μV range).

The actions of anesthetics on neural tissue are quite complicated so that corresponding EEG changes are believed to reflect the actions of many different neurotransmitters on both cortical and subcortical tissue. Therefore, any attempt to “explain” EEG in anesthesia by the single parameter β (or βS) is likely to be viewed as farfetched. Nevertheless, the simple global theory provides us with some general ideas of how the observed EEG behavior during progressively deeper anesthesia might occur. Consider the plots of mode frequency versus cortical excitability βS in figs. 11-9 through 11-11. The initial abrupt transition from alpha to low voltage fast activity is not well explained by the purely global theory; perhaps the inclusion local alpha networks that interact with the global field might accomplish this. However, imagine a progressive increase in βS associated with deepening anesthesia. For small βS, the higher modes are strongly damped. As βS increases lower mode frequencies become progressively slower until they become unstable in the linear theory. Again, we conjecture that, in healthy brains, additional negative feedback is recruited to prevent instability as in the example of (11.14). Brains that fail to accomplish this may suffer excessive net positive feedback, perhaps resulting in epilepsy. At the same time the lower modes become nonoscillatory, higher modes become more weakly damped and begin to appear as high-frequency oscillations in the total signal. Thus, for each β (or βS) we expect (p.514) a characteristic mixture of slow and fast frequencies, roughly consistent with EEG observations in anesthesia.

7 Weakly Connected Resonant Oscillators and Binding by Resonance

Here we outline a general mathematical theory by Hoppensteadt and Izhikevich (1998) and Izhikevich (1999) to suggest how “soft-wired” brain networks might continually interact and disconnect on roughly 10 on 100 ms timescales. A very general class of weakly connected oscillators was considered. The main attraction of this approach is that very minimal restrictions need be placed on the oscillators—the results are largely independent of specific network model. An arbitrary number N of semiautonomous oscillators is assumed to be pair-wise weakly connected to themselves and to a central oscillator X 0. By this we mean that the system is described in terms of vector dynamic variables X n of the form

(11.24)
$Display mathematics$
The vector functions F n and G nj are largely arbitrary. For example, simple mechanical or electrical oscillators, whether linear or nonlinear, might each be described by two scalar variables x n and y n (say position and velocity or current and voltage) so that X n = (x n, y n). When disconnected from the larger system (the limit ε 0), each oscillator (n) is assumed to undergo quasi-periodic motion consisting of a set of discrete characteristic frequencies f n1, f n2, f n3…. For example, the coupled van der Pol oscillators in chapter 9 fit these criteria. Hoppensteadt and Izhikevich (1998) showed that the individual oscillators cannot substantially interact (that is, interact on a time scale of the order of 1/ɛ) unless certain resonant relations exist between the characteristic frequencies of the autonomous oscillators.

To consider one example, suppose a central oscillator with dependent variables X 0, perhaps representing tissue in thalamus, is assumed to have a single characteristic frequency f 01. The central oscillator is weakly connected to a pair of other oscillators (X 1, X 2) also weakly connected to each other. This later pair of oscillators may represent two cortical networks of arbitrary size, as long as the conditions of semiautonomy and quasi-periodic oscillations are satisfied. Assume that each of the two cortical networks has a single characteristic frequency (f 11 and f 21). In this example, the cortical networks substantially interact (that is, on a timescale 1/ɛ) only when the three frequencies (f 01, f 11, f 21) satisfy the resonant relation

(11.25)
$Display mathematics$
(p.515) where (m 1, m 2) are any combination of nonzero integers and m 0 is any integer including zero. Two simple cases are
(11.26)
$Display mathematics$
The first case is the well-known resonant interaction between two oscillators with the same resonant frequency. The second case is illustrated by the following example. Suppose that two cortical networks composed of groupings of columns (at any scale) are formed as a result of strong internal interconnections. Let the two networks with characteristic gamma frequencies f 21 = 37 Hz and f 11 = 42 Hz be weakly connected to each other and to the thalamus (f 01). The two cortical networks may substantially interact, perhaps even forming a temporary single network in some sense, when the thalamic characteristic frequencies satisfy (11.26); several examples are f 01 = 2.5, 5, 15.67, 23.5 Hz.

Suppose several networks (X 1,X 2,…X N) each have several characteristic frequencies, that is (f 11, f 12,…f 1a), (f 21, f 22,…f 2b),…(f n1, f n2,…f nz), where the subscripts a, b,…z indicate the number of characteristic frequencies associated with each network. The central network X 0 is then able to control interactions between the peripheral networks by changing its own characteristic frequencies (f 01, f 02,…f 0a). Furthermore, the central network X 0 may allow cortical network X n to interact with several other networks X j, X k,…that do not interact among themselves, that is, by multiplexing. Following the analysis of Izhikevich (1999), we conjecture that cortical columns or larger networks may use rhythmic activity to communicate selectively. Such oscillating systems may not interact even when they are directly connected. Or, the systems may interact even with no direct connections, provided their characteristic frequencies satisfy the appropriate resonance criteria. Functional coupling between small elements or networks is then pictured as dynamic rather than hard-wired. Coupling strengths may easily change on short timescales.

8 Synaptic Action Fields and Global (Top--Down) Control of Local Networks

This section begins with yet another metaphor, but readers are reminded that metaphor is not theory. Our metaphor is used only to facilitate communication between the global theory of sections 1-4 through 1-6 and the oscillator theory of section 1-7 and to suggest new or modified physiologically based theory. A metaphor that roughly describes the putative local and global brain processes suggested by fig. 1-8 involves sound in an opera hall, analogous to the system composed of neocortex and corticocortical axons (Nunez 2000b). The synaptic and action potential fields [Ψe(r, t), Ψi(r, t), Θ(r, t),] are analogous to physical fields in (p.516) the opera hall like air pressure, density, temperature, molecular velocity, and so forth. Note that we comfortably use the label “field” for these physical variables even though they all originate with the dynamics of individual molecules (active synapses).

We replace the opera singers (pacemakers) by external sound sources (subcortical input). Global sound resonance occurs at multiple frequencies (fundamental and overtones) depending on sound speed (corticocortical axon propagation speed) and the opera hall size and shape. To avoid physiologically unrealistic reflective boundary conditions at the walls of a normal opera hall, let the opera hall take the shape of a torus or spherical shell with sound-absorbing walls. External sound sources (say from speakers on the wall) cause traveling waves in the air that interfere because of the periodic boundary conditions due to the hall’s size and shape. Thus, certain spatial wavelengths (and corresponding temporal frequencies) dominate the sound (pressure) field. These frequencies and their corresponding spatial patterns (the eigenfunctions) are called the normal modes of the opera hall, the resonant frequencies of air pressure modulations around background pressure (analogous to short time synaptic field modulations).

Our imagined opera hall contains many water glasses of different sizes and shapes (local and regional networks) that vibrate when driven by global sound waves at glass resonant frequencies. Valves (neurotransmitters) control the amount of water in each glass, thereby controlling local resonant frequencies. Sensors on an outside wall of the hall (scalp electrodes) record only the long-wavelength part of the internal sound because of the opera wall’s physical properties (CSF, skull, scalp) and physical separation of sensors from air molecules (active synapses). A purely global opera hall theory (the purely global theory of section 4) might attempt to predict resonant frequencies of air pressure modulations around background pressure (synaptic field modulations) by ignoring all influences of the internal glass structures as a first approximation. The normal modes of the opera hall can be controlled by heating the air (changing the molecular velocities), but here our metaphor breaks down since the neocortical global mode frequencies in section 4 are controlled by the background excitability parameter β.

In the next approximation, an oversimplified local/global theory of the opera hall might attempt to consider some of the effects of air–glass interactions. For example, we might generally expect selective top-down interactions between the sound (pressure field or synaptic field) and the water glasses. Each water glass should respond to sound waves at one of its particular resonant frequencies (interaction F-A in fig. 1-8). We also expect bottom-up interactions. That is, the resonating glasses will generally modify the air waves (interaction A-F in fig. 1-8). In this manner, widely separated glasses with at least one resonant frequency in common can become parts of the same network if driven by a sound wave field containing substantial power in a “matching” (in the sense of section 7) frequency band. A particular glass having several resonant frequencies could easily (p.517) participate simultaneously in multiple networks. Adding water to each glass (long-timescale neuromodulation by chemical input to cortex) changes its resonant frequencies so that it participates in a different collection of networks, perhaps contributing to a global phase change (brain state change).

To make the opera hall a more realistic metaphor for neocortex, hierarchical interactions that appear critical to neocortical dynamic behavior may be included (Freeman 1975; Ingber 1982, 1995). Replace the water glasses by complex networks of test tubes and beakers connected by glass rods that fill a substantial part of the space in the opera hall. Imagine rods inside tubes inside small beakers inside larger vessels with overlapping structures (cortical columns at various scales), the chemistry laboratory from hell! One can imagine progressively more complications of opera hall glass networks. However, if an important source of experimental data for this system is externally measured sound, the idea of macroscopic waves should be maintained, despite the enormous complexity of dynamics within the glass networks. One obvious reason is that the externally measured sound will be substantially biased towards low spatial frequencies (scalp potentials), and with nearly any wave phenomenon, low spatial frequencies imply low temporal frequencies. Thus, our external measurements (sound, EEG) will be biased towards more towards global fields than local networks.

These arguments suggest that we should retain synaptic field concepts as long as EEG is an important data source, even as discrete networks believed to underlie behavior and cognition become better understood. Separation of synaptic field and network concepts, even when they interact strongly and the separation is somewhat artificial, helps to simplify a very messy picture.

Given the general picture provided by the opera hall metaphor, we now consider how these general ideas might apply to the oscillator theory outlined in section 7. In that description, we imagined a central oscillator X 0, perhaps a small midbrain structure like thalamus or hippocampus. However, the mathematical analysis summarized in section 7 is more general; it does not restrict the size or location of this oscillator. Suppose now we identify X 0 as the synaptic action field in (11.1) which has multiple global resonant frequencies given (in this very crude approximation) by (11.4), that is

(11.27)
$Display mathematics$
Suppose also that several brain networks (X 1, X 2,…X N) are embedded within this global field, and each network has its own set of characteristic frequencies determined by local control parameters. In this imagined system, the global field X 0 is able to control interactions between the local networks by changing its own characteristic frequencies; that is, by (p.518) changing β. Furthermore, the global field X 0 may allow cortical network X n to interact simultaneously with several other networks Xj, Xk,…that do not interact among themselves. Given this general framework, it is not so implausible to conjecture that diffuse chemical input to the cortex may change global dynamic behavior, especially characteristic global frequencies, by changing β. Similarly, we imagine networks with local control parameters that modify local resonant frequencies (van Rotterdam et al. 1982; Nunez 1989, 1995). This global neocortical dynamics can perhaps act (top down) to influence functional coupling between specific networks embedded within the global neocortical/corticocortical system. In principle, these networks could be cortical, corticothalamic, or any other combination of brain structures, large or small.

9 Relationships to Other Theoretical Models and Criticisms of the Global Theory

Perhaps a dozen serious large-scale neocortical dynamic theories have been published over the past four decades that attempt to explain various aspects of EEG dynamic behavior. Many more excellent small-scales theories of interacting neurons have been developed, but our restriction to the adjective “large scale” is critical if our goal is to explain EEG dynamics recorded with large electrodes on the scalp and cortex. Some published theories are competitive with the global theory of section 4 while others are more complementary; many have both features. To the best of our knowledge, no general survey of these works has been published. It is not hard to see why. Any comprehensive survey would require quite a large effort and could easily fill an entire book—a book that would probably be read by only a few of today’s scientists, some fraction of EEG scientists having appropriate mathematical and theoretical backgrounds. Future generations with strong training in both physical science and neuroscience will be required to fill the gap.

We do not attempt any survey of neocortical dynamic theories here; however, several general citations seem appropriate. One is Ingber’s (1982, 1995) ambitious statistical mechanics of neocortical interactions, which derives large-scale dynamic variables from the smaller scales in the spirit of our alcohol metaphor of section 3. This work identifies multiple stable firing patterns that are candidates for storing short-term memory. A widely cited paper is Wilson and Cowan’s (1973) quasi-linear treatment of coarse-grained synaptic and action potential fields predicting local limit cycle behavior in corticothalamic cell assemblies. Their coarse graining operation is also similar to our metaphor in section 3. Similar theories of note were published by Freeman (1975, 1992), van Rotterdam et al. (1982), and Zhadin (1984).

The theory of van Rotterdam et al. (1982) and Lopes da Silva (1991, 1995) is based on thalamocortical feedback with PSP delays. Nunez (1989, (p.519) 1995, 2000a,b) showed that the original linear global theory (Nunez 1972, 1974) combines naturally with the linear local theory of van Rotterdam et al. (1982) so that observed oscillation frequencies may occur naturally as a result of both local synaptic and corticocortical axon delays. The idea is based on differential equations for local tissue networks of the form

(11.28)
$Display mathematics$
Here D is some local differential operator acting on the modulation of action potential density δΘ(x, t) with the synaptic action modulation function f[Ψ(x, t)] appearing as a forcing function. Equation (11.28) may then replace (11.13) so that (11.12) and (11.28) provide a coupled set of differential equations, thereby allowing for both global and local contributions to oscillation frequencies. Linear approximations of (11.28) lead to multiple local-global branches of dispersion relations (see pages 494–498 and 694–698 of Nunez 1995). Local networks and global fields interact in both directions (bottom up and top down) in contrast to the usual pacemaker idea (exclusively bottom up).

In more complicated versions of this approach, inhomogeneous local properties may be modeled by varying the parameters in (11.28). For example, local resonant frequencies may vary because of neurotransmitter-based differences in local or regional feedback gains (Silberstein 1995b). Because of the many idealizations of genuine tissue, we expect most details to be wrong even if the general approach is correct. Nevertheless, this model provides a compelling argument that macroscopic fields of synaptic action and local or regional neural networks coexist naturally. Different dominant frequencies are expected at different cortical locations, but local dynamic behaviors (including local oscillation frequencies) are due to combined local and global mechanisms. We further propose that local networks are bound to each other by the global field.

Comprehensive theoretical work by Jirsa and Haken (1997) and Haken (1999) advanced this idea of complementary local and global processes by showing that the Wilson and Cowan local model and the Nunez global model are fully compatible. These scientists conclude that the brain may act as a parallel computer at small scales by means of local or regional neural networks, while simultaneously producing global field patterns at macroscopic scales.

Haken’s (1999) work also showed that the global field equations have a general character, in the sense that the synaptic field dispersion relation for long-wavelength dynamics is relatively insensitive to corticocortical fiber distribution, thereby adding additional support to similar studies (Nunez 1995). The latter work also considered effects of distributed axon propagation velocities, which tends to eliminate higher modes by increasing their damping. Some effects of inhomogeneity of dynamic parameters on a linear cortex, say β → β(x) can be anticipated from studies (p.520) of waves in physical media (Morse and Ingard 1968; Nunez 1995). For example, each mode frequency can be expected to oscillate with a range of spatial frequencies, thereby “smearing” simple dispersion relations. In an inhomogeneous closed loop

(11.29)
$Display mathematics$
where the wavenumbers (spatial frequencies) k n are still required to satisfy the periodic boundary conditions. However, even with smeared version of the dispersion relation, we expect some correspondence between higher spatial and temporal frequencies, for example in the frequency–wavenumber spectra shown in chapter 10.

Jirsa and Haken’s local-global theory was used to describe evoked magnetic field behavior (Jirsa and Haken 1997; Kelso et al. 1999; Fuchs et al. 1999; Jirsa et al. 1999, 2002). Subject performance on a motor task identified a brain state change or phase transition in the parlance of complex physical systems. The spatial-temporal MEG dynamics were described in terms of a competition between two spatial modes, with time-dependent amplitudes as order parameters as in (11.2). The first order parameter ξ1(t) (the first time-dependent spatial mode coefficient) dominated the pretransition state and oscillated with the stimulus frequency. The second order parameter ξ2(t), having twice the stimulus frequency, dominated the posttransition state. A differential equation was derived with auditory and sensory cortices considered as local circuits embedded within the global field of (11.12). The theoretical model was able to reproduce essential features of MEG dynamics, including the phase transition. Thus, a triple correspondence was achieved relating behavior, MEG data and physiologically based theory. In later work from this group, spreading of wave fronts in folded cortex and scalp have been generated with a dynamic brain model coupled to a volume conductor model (Jirsa et al. 2002).

Robinson et al. (1997, 1998, 2002) have argued that the isolated cortex is relatively stable, leading to strongly damped waves and minimal influence of boundary conditions. When corticothalamic feedback is included, weakly damped waves become possible at low frequencies and near the alpha frequency; these are sensitive to boundary conditions and can display modal structure (Robinson et al. 2003). These authors have obtained a number of analytic results for their model in its linear regime. In the context of the local-global framework advocated in this chapter, these scientists are essentially saying that weakly damped waves require local network contributions. In another version of this work (Robinson et al. 2004), EEG data are used to “work backwards” to determine ranges of the model’s physiological parameters that might plausibly account for experimental observations. More studies of this kind can be expected in the future by attempting to include progressively more of observed spatial temporal dynamics on the scalp and cortex, thereby narrowing the plausible range of theory parameters.

(p.521) The work of Liley et al. (2002, 2003) and Bojak et al. (2003, 2004) is also a field theory that minimizes the effects of boundary conditions. Here EEG rhythms emerge as a consequence of reverberant activity within and between cortical excitatory and inhibitory neuronal populations, rather than through thalamocortical or corticocortical delays. This model predicts a variety of cortical oscillations by focusing on different magnitudes and time courses of postsynaptic excitation and inhibition; it is mostly a “local” theory in our chosen parlance. A range of EEG effects induced by sedatives and anesthetic agents is predicted by the model based on the experimentally determined influence of the agents on postsynaptic potentials. For example, insights into some of the effects of anesthesia outlined in section 5 and a basis for benzodiazepine-induced accelerations of resting EEG are suggested.

The works described above rely on a large number of physiological parameters. As in the case of all models, questions arise about the sensitivity of model predictions to parameter uncertainty. Nevertheless, we regard these studies as plausible approaches to (mainly) local theory and consistent with the idea of local networks immersed in a global environment, although the authors may argue (correctly) that such separation of the dynamics into two distinct parts is somewhat artificial (Liley 2000). The counter argument is that the purely global theory outlined above is able to make perhaps a dozen testable qualitative (and some semiquantitative) predictions that are a relatively robust to parameter uncertainty. Furthermore, many of the dynamic effects generated by local models are not directly observable at the scalp because of severe spatial filtering. Thus, we argue that separation of the dynamic behavior into two parts provides convenient entry into more complex dynamic models and generally facilitates our primitive attempts at thinking about thinking. Similar separations have proved quite convenient in the physical sciences. The separation of (global) longitudinal waves due to particle motion from (local) particle collisions in hot plasma theory comes to mind in this context.

A plausible criticism of the global theory outlined in section 4 is that genuine operating ranges of the physiological parameters cause propagating cortical waves to be spatially overdamped (Robinson et al. 1997; Liley et al. 1999; Wright 2000; Wright et al. 2001). If this criticism is correct, EEG may owe its origins more to local networks: cortical, thalamocortical, or both. In this view long-range corticocortical interactions are not required to explain EEG phenomena. It then follows that global boundary conditions may be neglected as in the local theories developed by these same scientists. Our answers to this criticism are as follows.

1. (i) We readily acknowledge the importance of local network effects on EEG and that cortical waves may be strongly damped in many brain states. However, our experimental focus for testing the global theory has been directed to brain states of minimal cognition where cortical waves may be weakly damped. (p.522) We contrasted EEG recorded in these resting states with recordings during mental calculations (see chapters 9 and 10). We also note that theories attempting contacts with intracranial recordings naturally focus more on local network effects that may dominate any possible global behavior in the recorded data.

2. (ii) Spatial damping is proportional to temporal damping through the group velocity of the wave packet. The two parameters involved with global damping are the fall-off rate λ−1 of the long corticocortical fiber system and the cortical excitability parameter β. Longer fibers and higher excitability reduce damping and produce instability in the purely linear theory or perhaps limit cycle-like modes in quasi-linear approximations; λ is fixed by the anatomy and is in the appropriate range for undamped cortical wave propagation provided that β is sufficiently large. A very crude estimate of β suggests it can be in the range to produce undamped or unstable waves (Nunez 1995, see pages 492–494 with B ≡ 2β). However, we are skeptical that accurate knowledge of the effective β range for cortical tissue will be available anytime soon, noting the vast complexity of local cortical networks.

3. (iii) The question of cortical wave damping is addressed experimentally by measurement of EEG propagation at the scalp. As shown in chapter 10, such propagation is observed in several brain states, and phase and group velocity estimates match corticocortical axon speeds. Of course, critics may point out that weak damping of cortical waves could require local network influences.

4. (iv) EEG recorded on the scalp is spatially filtered by the volume conductor. Thus, scalp EEG is strongly biased towards the low end of the spatial frequency spectrum. In the case of resting alpha rhythm, most scalp power occurs in the first few spherical harmonics as discussed in chapters 7 through 10. This has several implications. One question concerns the neglect of global boundary conditions when describing a phenomenon whose dominant wavelengths are comparable to brain dimensions. Local theories appear quite appropriate for local alpha rhythms, but only the parts of such fields with large-scale phase synchrony are observed at the scalp. The corticocortical fibers provide a compelling means to facilitate such synchrony as shown clearly in studies of EEG maturation (Thatcher et al. 1987; Srinivasan 1999; Thatcher 2004) and split brain subjects (Nunez 1981). It may be a bit implausible to recruit corticocortical fibers into a theory to obtain the required synchrony, and at the same time neglect the influence of their propagation delays (were this to be proposed by competing theories).

5. (p.523)
6. (v) The relative importance of corticocortical versus thalamocortical dynamics appears to be much higher in humans than lower mammals that provide much of the intracranial data (see fig. 1-2). That is, the number of thalamocortical axons entering (or leaving) a typical patch of the underside of human cortex is only a few percent of the number of corticocortical fibers (Braitenberg and Schuz 1991). Theoretical strategies that focus on the (minority) thalamocortical interactions but neglect the (majority) corticocortical interactions require compelling justification. This issue is discussed in more detail in the context of possibly stronger thalamocortical interactions (at least in primary sensory cortex) in Nunez (1995).

While we disagree with this criticism of the basic global theory for the reasons given above, such issues raised by critics are essential to genuine scientific progress and will likely be debated and (hopefully) tested experimentally in the future.

10 Summary

We have outlined the rationale for our conceptual framework in which cell assemblies are embedded within synaptic and action potential fields as summarized in fig. 1-8. The inclusion of the synaptic fields in our conceptual framework is motivated first by the close relationship of synaptic fields to scalp potentials, a connection more easily justified than putative (direct) relationships between neural networks and scalp potentials. Furthermore, this conceptual framework does not appear restrictive. It does not prejudge issues like the relative importance of local versus global delays to EEG dynamics or whether various behavioral and cognitive states are better associated with functional localization or integration. Such questions may be conveniently addressed experimentally in the context of the chosen framework.

Some of the most robust dynamic properties of scalp recorded EEG are summarized here. A global theory of large-scale neocortical dynamics is shown to have some limited predictive value for EEG despite its neglect of all network effects. This “toy brain” is presented first as a plausible entry point to more realistic theory in which cell assemblies (or networks) play a central role in cognition and behavior. Second, we conjecture that the synaptic action fields of the global theory may act (top-down) on local networks in a manner analogous to human cultural influences on social networks (Nunez 2000a). Such interactions across spatial scales are generally expected in a wide range of complex systems (for some simple examples, see Nunez 1995). The ubiquitous phenomenon of top-down/bottom-up interactions across scales in complex systems has been labeled circular causality by Haken (1983, 1987) and studied widely under the (p.524) rubric synergetics. Systems in which circular causality forms an essential part of dynamic behavior include weather, magnetic materials, DNA dynamics, combat, societies, and financial markets (Ingber 1995). The preeminent complexity of human brains suggests that circular causality should be treated as a central issue in both EEG and cognitive theory.

Our conjecture that cell assembly interactions may be substantially facilitated by resonance effects is based on three known phenomena: (i) cognitive and behavioral events are associated with power changes in certain preferred EEG frequency bands; (ii) coherence and other measures of phase synchronization change during mental tasks, also in preferred frequency bands; and (iii) resonance interactions are critical in a wide range of physical and biological systems. Circuit resonance of analog filters, resonant interactions between the strings and wooden bodies of violins, and quantum wave function resonance associated with chemical bonds provide prominent examples.

Several local EEG theories are cited. We argue that much of this work complements the global theory because local networks must operate while immersed in a global field environment. There is no shortage of physiological mechanisms able to predict many of EEG’s dynamic properties; rather, the difficulty is picking the right ones. The large number of parameters that must be included in most serious brain theories presents a major obstacle to the cornerstone of genuine theory: the opportunity for experimental falsification. We emphasize the global theory here because of its ability to make several correct experimental predictions of general EEG properties, while acknowledging its vast oversimplification of genuine brain dynamics. Perhaps the greatest shortcoming of the global theory is that it is difficult to falsify. We could, for example, conjecture that any dynamic behavior not predicted by the global theory is due to embedded networks not included in the theory. Nevertheless, we adopt the strategy of searching for brain states and other experimental conditions that appear to minimize network effects; thereby providing opportunities to check the global theory in these limited circumstances. Our emphasis on (low spatial frequency) scalp-recorded data and resting alpha and anesthesia states supports this goal. With this strategy in mind, “falsification” may be associated with failing to find brain states in which the simplified global theory has substantial predictive value.

Finally, we note that local characteristic timescales like rise and decay times of postsynaptic potentials and global timescales like the corticocortical transmission times across the entire brain appear to be in the same general range. Perhaps this is no coincidence. The dynamic properties of individual neurons and small neural assemblies appear to be quite variable and critically dependent on the details of the experimental environment. Thus, we offer one additional conjecture—top-down, multiscale, neocortical dynamic plasticity—in which individual neurons and networks adjust their time constants for (perhaps resonant) compatibility with other (p.525) networks and the global environment. By “top-down plasticity” we imply that fixed global boundary conditions might constrain global mode frequencies, thereby forcing networks at multiple spatial scales to conform to the global field in healthy brains. Perhaps nonconforming networks become schizophrenic and these brains tend to be eliminated by natural selection. Who knows? Maybe consciousness is a resonance phenomenon and only properly tuned brains can orchestrate beautiful music of sentience.

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