Below is the unedited penultimate draft of:

Nunez, Paul L. (2000) Toward a Quantitative Description of Large Scale Neocortical Dynamic Function and EEG Behavioral and Brain Sciences 23 (3): XXX-XXX.

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Toward a Quantitative Description of Large Scale Neocortical Dynamic Function and EEG

A general conceptual framework for large-scale neocortical dynamics based on data from many laboratories is applied to a variety of experimental designs, spatial scales and brain states. and mathematical theories, but includes experimental predictions from both inside and outside EEG. Partly distinct, but interacting local processes (eg, neural networks) arise from functional segregation. Global processes arise from functional integration and can facilitate (top down) synchronous activity in remote cell groups that function simultaneously at several different spatial scales. Simultaneous local processes may help drive (bottom up) macroscopic global dynamics observed with EEG (or MEG).

A local/global dynamic theory is outlined that is consistent with EEG data and the proposed conceptual framework.. The theory is neutral on properties of neural networks, embedded within macroscopic fields. Nevertheless, the purely global part of the theory makes several qualitative and semi-quantitative predictions of EEG measures of traveling and standing wave phenomena. A more general "meta-theory" also suggests what large-scale quantitative theories of neocortical dynamics may be like when more accurate treatment of local and non-linear effects is achieved.

The theory describes the dynamics of fields of excitatory and inhibitory synaptic action fields. EEG and MEG provide large-scale estimates of modulation of these synaptic fields about background levels. Brain states are determined by neuromodulatory control parameters. Purely local states are dominated by local feedback gains and rise and decay times of post-synaptic potentials. Dominant local frequencies vary with brain region. Other states are purely global, with moderate to high coherence over large distances. Multiple global mode frequencies are due to a combination of delays in cortico-cortical axons and neocortical boundary conditions. Global frequencies are identical at all cortical regions. But most states are local/global, involving dynamic interactions between local networks and the global system. Observed EEG frequencies may involve "matching" of local resonant frequencies with one or more of the multiple, closely spaced global frequencies.

Keywords: EEG, neocortical dynamics, standing waves, functional integration, spatial scale, binding problem, synchronization, coherence, cell assemblies, limit cycles, pacemakers

1. Introduction and prologue

The conceptual framework proposed in this target article is based on the idea of neocortical dynamic behavior at multiple spatial scales, ranging from single molecules to neurons to overlapping local and regional cell groups of different size to global fields of synaptic action density. Interaction across these hierarchical levels (or spatial scales) may be essential to the dynamics (and by implication to behavior and cognition), similar to hierarchical interactions in human social systems. For example, a social network requires preferential interactions between individuals, but its dynamic behavior is influenced (top down) by the global social environment. Thus, remote social networks with no direct connections can exhibit correlated activity. These ideas do not entirely contradict classical neurophysiological views of focal control (bottom up) mechanisms. Rather, they suggest dynamics which dynamics that are more fully integrated across spatial scales, potentially using the full range of (bottom up and top down interactions, analogous to multi-scale social interactions among persons, families, neighborhoods, cities, and nations.) For example, the US Federal Reserve Chairman Alan Greenspan is analogous to a small-scale local network. His words and actions strongly influence global economic behavior (bottom up). However, Greenspan is himself influenced (top down) by many larger scales, including groups of private persons, the US Congress, and the global economy.

Considering the enormous complexity of brains and our experience with successful models of relatively complex physical systems, distinct mathematical theories are required for each level (spatial scale). Such theories are directly connected only to data recorded at the same level, although good theories strive for some overlap with adjacent scales to show qualitative and hopefully semi-quantitative cross-scale connections. across scales. For example, electrophysiological data span about five orders of magnitude of spatial scale, ranging from microelectrode tip (0.0001 cm) to EEG scalp electrode (1 cm), which records neural source activity space-averaged over regions much larger than the electrode due to passive current spread in the volume conductor. A major accomplishment of 20^th century neurophysiology was the connection of membrane recordings to theory. New large-scale theories must exploit these data, but may use quite different mathematical methods in order to obtain connections to data obtained at the same large scales. Large-scale theories typically contain "control parameters," which depend on neurotransmitter action at smaller scales, but may not be derived from smaller scales without the introduction of new organizing principles.

Many physics sub-fields are concerned with multi-scale theory. For example, theoretical connections have been derived between the distribution of molecular velocities in a gas and its pressure, temperature, viscosity, or electrical resistivity (in the case of ionized gas or plasma). With certain assumptions, these macroscopic variables may be calculated from microscopic variables; however, such calculations require verification with macroscopic experiments. at the appropriate scale. Another physical example involves theoretical connections between microscopic and macroscopic electromagnetic fields. The precise laws of electromagnetics at microscopic scales (Maxwell’s microscopic equations) were first developed using the approximate macroscopic data available at the time, although the validity of such extrapolation to microscopic scales was far from obvious (Jackson 1975).

An important part of this process was the development of distinct connections between

macroscopic electromagnetic fields and space averages over molecular fields. A critical aspect of modern electromagnetic theory is the identification of macroscopic control parameters describing magnetic, dielectric, and resistive properties of large "assemblies" of these molecules. Thus, we are able to define the electrical resistance of a macroscopic mass of material simply, even though macroscopic resistance actually depends on very complicated actions of small-scale electromagnetic fields.

Here we focus mainly on a small part of this the brain’s dynamic complexity, the very large -scale fields of synaptic action density, of which scalp EEG is believed to provide a crude measure. Such "field" may consist of an excess of excitatory or inhibitory synaptic action (eg, active inhibitory synapses per mm³ at time t), providing perturbations about some background level of synaptic excitation or inhibition in large tissue masses containing tens of millions of neurons. These synaptic fields of membrane current sources are distinguished from the electric and magnetic fields (EEG and MEG) that they produce. The synaptic fields may be both strongly influenced by and act back on smaller scale activity, eg, the synaptic fields can interact with neural networks embedded within the same tissue. No direct influence of weak electric or magnetic fields (EEG or MEG) on neural activity is suggested here; however, the action of macroscopic (eg, space averaged) fields of synaptic activity on local or regional neural networks is proposed. This idea and other contributions to a "physiology of neural mass action" or "macroscopic physics of neocortex" may ultimately help to provide some insight into the dynamics of smaller scales, eg.,, the cooperative workings of local and regional cell groups. However, distinct theories are required at such smaller scales.

Since entering the neuroscience community with an with an engineering physics Ph.D.   in 1971, I have both struggled against and been educated by the multiple, often conflicting sub-cultures of neuroscience. Conflicts have often involved the proper role of mathematics in explaining data obtained at distinct spatial scales. Such conflicts may reflect profound scientific questions or merely "scale chauvinism" and they occur in many fields (eg, economics and physics) as well as in neuroscience. This experience provides part of the motivation for this target article: to consider relations between mathematical theory and experiment generally and to apply these ideas to multi-scale neocortical dynamics, EEG, and cognitive events. The intended audience is a broad (but not necessarily mathematically sophisticated) one; a primary goal is facilitating communication between disparate subfields. The reader is asked not to prejudge this effort, based perhaps on negative views of mathematics or reductionism in neuroscience. A largely neutral view of mind-brain issues is adopted here (e.g.,, reductive materialism or "emergent dualism," as phrased by Scott (1995), are fully consistent with the proposed conceptual framework). Emphasis is placed on practical strategies to encourage fertile marriages of theory with experimental data. Some of these ideas are explored in more depth in my books (Nunez 1981a, 1995a).

Interdisciplinary studies of brains are likely to proliferate, and genuine brain theory requires a firm footing in experimental data. But so called "theories" can cover a wide range of approaches, including analyses of membranes, neurons, artificial neural nets, and even ambitious attempts to connect quantum effects to consciousness. EEG data may be used in support of such work. If so, this semi-quantitative description of neocortical dynamics, spatial scale, and EEG may provide a useful gauge of putative experimental connections.

Consider the following experiment. Place two electrodes on a subject in one room and feed this EEG signal to a computer display in an isolated location. Monitor the subject's state of consciousness over several days and provide this information to a person someone following the unprocessed oscillations of scalp voltage. (with axes showing amplitude and time scales). Even a naive observer, unfamiliar with EEG, will soon recognize that the voltage record during deep sleep has larger amplitudes and contains more low frequency content. Slightly more sophisticated monitoring (but still with only a few channels) and training allow the observer to accurately identify distinct sleep stages, depth of anesthesia, and seizures. Still more advanced methods reveal robust connections of EEG to more detailed cognitive events.

We are now so accustomed to these EEG/brain state correlations that we may forget just how remarkable they are. The scalp EEG (or MEG) provides a very large -scale measure of neocortical dynamic function. A single electrode provides estimates of synaptic action averaged over tissue masses containing something between 10 million and 1 billion neurons (Nunez 1995a). Most human studies are limited to extracranial recordings, with space averaging a fortuitous data reduction process, due to passive current spread in the head volume conductor. Much more detailed local information may be obtain from intracranial recordings. However, the number of intracranial electrodes implanted in living brains is very small compared to anything approaching full spatial coverage, even for recordings at intermediate spatial scales. Thus, in practice, intracranial data provide different information, not more information, than is obtained from the scalp .scalp, in apparent contrast to the views of some physiologists.

Intracranial recordings provide smaller scale measures of neocortical dynamics, with scale dependent on electrode size. A mixture of coherent and incoherent sources generates the small and intermediate scale intracranial data. The smaller the scale of intracranial data, the more likely such data will appear independent of scalp data, which are due mostly to coherent sources with special geometries that encourage superposition of fields generated by many local sources (Nunez 1981a). Intracranial EEG is often uncorrelated or only weakly correlated with cognition and behavior, which are more easily observed at large scales. We are lucky. The technical and ethical limitations of human intracranial recording force us to emphasize scalp recordings, and these methods provide estimates of synaptic fields at the macroscopic large scales closely related to cognition and behavior.

Although cognitive scientists have good reason to be partly content with the low spatial resolution obtained from scalp EEG data, explorations of new MEG and EEG methods to provide somewhat higher spatial resolution continue. A reasonable goal is to record averages over "only" 10 million neurons at the one-cm scale in order to extract more details of the spatial patterns correlated with cognition. This resolution is close to theoretical limits caused by the physical separation of sensor and brain current sources.

There exists a voluminous literature concerning relations between EEG and cognition. Here I focus on high resolution coherence studies that are more closely connected to the central theoretical issue of this paper--locally versus globally dominated dynamics. Coherence is a specific quantitative measure of functional relations between paired locations. Coherence is a squared correlation coefficient; it measures phase consistency recorded at paired locations, at for each frequency component in the EEG. For example, if two regions exhibit an EEG coherence of 0.36 at some frequency, a large-scale dynamic correlation coefficient of 0.6 is implied between these regions at this frequency. Such correlated neocortical activity can result from direct connections between the two regions, common input from thalamus and other neocortical regions, or both.

Paired locations in dynamical systems may exhibit high coherence in some frequency bands and, at the same time, low coherence in other bands in the same set of data. Furthermore, coherence depends on measurement scale. Calling on our analog sociology, we expect that correlations between human activity in New Orleans and Paris will depend on whether correlations between individuals, averages over entire city populations, or something in between is measured. Thus, the qualitative idea of "synchrony," often used in the EEG literature, requires a more substantive definition to be fully useful as a descriptor of dynamic behavior. However,

Although coherence may appear to be an ideal measure of brain function, interpretations of experimental EEG coherence are often confounded by technical limitations. Rraw scalp coherence between electrode sites closer than about 8 to 10 cm is typically large or moderate due only to passive current spread volume conduction and reference electrode effects, even when the underlying cortical sources are uncorrelated (Nunez 1995a; Nunez et al 1994, 1997). Thus, measured EEG coherence changes as a function of brain state may be small. This can magnify interpretation problems due, for example, to artifact and low statistical significance of coherence/brain state correlations. But, erroneous high coherence can be largely eliminated using high resolution EEG to estimate dura potential from dense scalp arrays before calculating coherence. As a result, fractional changes in coherence with brain state changes are larger, more robust, and more dependent on specific electrode pair than with the usual EEG methods of low spatial resolution (Nunez et al 1999).

Coherence measured at small scales with intracranial electrodes (eg, 2 mm diameter) is often zero at all frequencies for separation distances greater than a few cm (Bullock et al 1995). Given the problems with scalp coherence measures, it is not surprising that the validity of moderate to high scalp coherence has been questioned. However, earlier studies by the author estimated coherence from selected periods of high amplitude alpha rhythm using pairs of bipolar electrodes (Nunez 1974b, 1995a). The use of close (1-3cm) bipolar electrodes eliminated reference electrode and mostly eliminated volume conduction contributions to coherence at distances greater than about 4 or 5 cm. Anterior/posterior coherence over one hemisphere between locations separated by about 20 cm was often greater than 0.7 or 0.8 at the peak alpha frequency in these high amplitude alpha data. However, coherence outside narrow bands (typical widths of 1 to 2 Hz) centered at the alpha peak typically fell off sharply. Coherence estimates for the full 8-13 Hz alpha band were much lower (Nunez and Pilgreen 1991). This suggests that studies of broad band coherence, historically the most common approach in EEG (eg, de Munck et al 1992), can result in erroneous low estimates of underlying source correlations.

In a recent experiment, the author alternated one minute periods of resting (slowly counting breaths to facilitate relaxation) with summation of series like (1 + 2 + 3 +...), up to sums of several hundred. In 60 channel data there are 60*59/2 = 1770 coherences to be followed. Coherences at 9 Hz (near the peak frequency), calculated from neck-referenced EEG, are plotted versus electrode separation distance in Fig 1. The upper plot shows coherences obtained during three alternating minutes of the eyes closed, resting state. The lower plot shows coherences for three alternating minutes of eyes closed, mental calculation (the "cognitive state"). High spatial resolution coherences obtained from the same data sets are shown in Fig 2. The high resolution coherences are mostly lower than neck reference coherences and exhibit larger percentage changes between states since all reference electrode and most volume conduction contributions have been eliminated. The solid lines are estimates of scalp (Fig 1) and dura (Fig 2) potential coherence due to uncorrelated, widely distributed radial dipole sources at the approximate depth of cortical gyri (Srinivasan 1995, Srinivasan et al 1996, 1998; Nunez 1995a).

We made reasoned attempts to control for confounding effects like arousal, task familiarity, etc. (Nunez 1995a; Nunez et al 1999). However, for purposes of this paper, the main point is that robust relationships between distinct brain states and large EEG coherence changes were demonstrated, independent of precise interpretations of these states. Such coherence data provide important quantitative measures of local versus more global dynamic behavior at large scales. As such they are closely related to the conceptual framework and specific theory presented here.These  data indicated the following:

1.2.1. Brain states with high EEG coherence are common. In the study described here, coherence outside the 9-10 Hz band centered at the alpha peak was substantially lower than peak coherence. This suggested that volume conduction and reference electrode contributions to erroneous high coherence were, in fact, largely eliminated by high resolution methods since volume conduction is independent of frequency in this frequency range (refer to Nunez 1995a; Nunez et al 1997, 1999). It should be noted that each high-resolution electrode estimates source activity averaged over dura surfaces in the 10 cm² range, whereas conventional methods involve surfaces of perhaps 50 to 100 cm² (Nunez 1995a). Thus, conventional and high-resolution methods estimate coherence at somewhat different spatial scales. These data show that coherent brain states are not unusual, with the strongest evidence for moderate to high resting coherence obtained at the 10 cm² scale since volume conduction is small and reference electrode effects are absent.

Comparisons of unprocessed (reference) EEG data with various simulations suggest that large-scale coherence is also much larger in the resting state (Nunez et al 1999), but this low-resolution evidence is somewhat weaker than the high-resolution evidence. In Fig.1a, resting reference coherence at large distances (eg, 15 to 25 cm, where volume conduction effects are small) ranges from about 0.2 to 1.0, depending on paired locations. In Fig. 2a, resting spline-Laplacian coherence at the same large distances varies from about 0.0 to 0.8 in the same data. But, such high-resolution methods may underestimate dura coherence because they remove the long wavelength part of the spatial spectrum of dynamic activity, which cannot be easily distinguished from volume conduction (Nunez et al 1997). Thus, coherence values intermediate between the estimates shown in Figs (1) and (2) may provide a more accurate resting coherence picture for the broad spatial scale of roughly 10 to 100 cm². Resting coherence values greater than 0.5 (dynamic correlation coefficients greater than 0.7) are therefor quite common at large distances near the alpha peak frequency.

1.2.2. Alpha coherence decreased and theta coherence increased during mental calculations. There are were large, robust decreases in coherence for most many paired sites between states of resting and mental calculation in the 9-10 Hz alpha band. This is was observed for both raw and high resolution coherences, but percentage changes are were largest with high-resolution coherences since there were no reference effects and volume conduction was minimal with high-resolution methods. Coherence in the 4-5 Hz theta band between electrode pairs at many widespread locations were consistently higher during mental calculations than resting coherence (p < 0.005), although the magnitudes of these changes were much smaller than alpha coherence changes. Anterior/posterior theta coherence increases were especially robust. Similar theta (4-7 Hz) coherence increases between prefrontal and posterior cortical association areas were recently reported during working memory retention using a digitally linked-ears reference (Sarnthein et al 1998).

Such coincident coherence changes in opposite directions for different frequency bands may challenge simplistic interpretations of "EEG synchrony." But, the local/global mathematical theory outlined in Section 5 suggests that such dynamic behavior can occur naturally in the context of known physiological mechanisms. For example, one may conjecture specific neural networks operating at theta frequencies during mental calculations. If such phenomena occur at sufficiently large spatial scales, they may produce larger EEG coherence between regions that include parts of the same network. The fact that such theta coherence changes are small relative to alpha coherence changes may be due mainly to poor spatial resolution. For example, if two parts of a neural network each underlie 1 cm² of dura surface, but each electrode records EEG from sources under 10 cm², only small EEG coherence changes can be expected, even when coherence changes in the network are large. Different neuromodulatory influences (and corresponding control parameters) may be more efficient for different frequency bands, perhaps because of their actions in different cortical layers (Silberstein 1995b). If so, the general large-scale reduction in alpha coherence may coincide with more specific (and perhaps smaller scale) increases in theta coherence.

1.2.3. Alpha coherence changes are were largest for frontal to frontal electrode pairs. The effect shown here is much more detailed than traditional "alpha blocking." reported in many early studies involving mental calculations. Alpha amplitude is was lower during mental calculations as expected, with the largest state changes with amplitude occurring over posterior regions. However, the alpha rhythm persisted in both states, and the largest coherence state changes involved frontal-to-frontal sites, consistent with generally accepted ideas about the important role of frontal cortex for higher mental processing. Of course, these large scale coherence reductions could can easily coexist with very specific coherence increases changes in either direction at scales too small to observe from the scalp, e.g., in synchronous cell groups (Bressler 1995; Silberstein 1995b). In our analog social system, coherence of croissant consumption for the whole cities of Paris and New Orleans may decrease when the state of the world changes. But these data do not necessarily tell us about croissant coherence between specific pairs of smaller scale human assemblies, eg, neighborhoods, persons, etc in these same cities.

1.2.4. Complicated patterns of coherence and coherence state changes occur in narrow frequency bands. Coherence patterns were often very specific to narrow bandwidths of 1 to 2 Hz and electrode pair, especially with high-resolution coherence. Such specificity of coherence patterns was obtained over the 1-20 Hz range. For example, 3, 5, and 7 Hz coherence behaved differently, as did 13, 15, and 17 Hz. Of course, one has less confidence in beta coherence measures because of possible muscle artifact contributions, although this was not evident in the raw data. But, these data generally support the idea that studies of broad band coherence can miss important properties of the dynamics.

These data appear inconsistent with the very simple pictures often proposed, e.g, where EEG sources consist of a few isolated "alpha generators" or "dipoles." Apparently, there are robust and very detailed changes in the strength of "binding" at macroscopic scales between brain states. By analogy, if Parisians and New Orleanians follow similar eating habits, croissant consumption coherence will be high in a band of frequencies near 1/24 hr. But, coherence may be high or low in other frequency ranges, depending, for example, on differences of supply or holiday habits. One may guess that neocortical dynamics is not simpler than croissant dynamics. Thus, the complexity of coherence patterns is not surprising.

Another recent coherence project used a 148-channel MEG system to record steady-state visually evoked magnetic fields in order to study binocular rivalry (Tononi and Edelman 1998; Srinivasan 1999). The subject viewed incongruent images with each eye, but was conscious of only one image (perceptual dominance periods) or a mixed image (non-dominance periods). The image that was perceptually dominant alternated every few seconds. For purposes of this paper, two general results stand out. First, for sensors separated by more than 20 cm, coherences at the driving frequency (7.41 Hz) were mostly moderate to high (roughly 0.4 to 0.9). Second, interhemispheric, and to a lesser extent, intrahemispheric coherences were generally larger during periods of perceptual dominance. Again, very complicated robust patterns of coherence were correlated with brain state.

1.2.5. Stable structure with nodal lines occurs in both states. High resolution plots of magnitude and phase over the scalp reveal regions separated by several cm, with voltage oscillations 180 deg out of phase. Since there are multiple regions, "zero phase lag" conditions easily occur over large distances (eg, 20 cm or more). These regions appear to be delineated by nodal lines (lines of zero amplitude), in a manner expected of standing waves and predicted by the global theory presented here and in earlier papers (Nunez 1972, 1974a,b, 1981a,b,c, 1989, 1995a; Nunez et al 1977; Katznelson 1981, 1982). The magnitude and phase patterns do not typically exhibit large changes over time, except for magnitude reductions in the 9-10 Hz band during the statesperiods of mental calculation.

By contrast, magnitude and phase patterns may be sensitive functions of frequency. This was demonstrated in several subjects with 64-channel steady-state evoked potential studies, in which magnitude and phase patterns (projected to the dura surface with high-resolution methods) at the stimulus frequency exhibited large changes with only one Hz change in stimulus frequency (Silberstein 1995a; Nunez 1995a). I suggest that such changes are due to the dynamic response of neocortex to input at different temporal frequencies, rather to changes in the spatial distributions of extra-cortical input. Again, these data support the concept of standing neocortical waves.

Currently, work is under way at the Brain Sciences Institute in Melbourne to study spatial-temporal properties of steady-state visual evoked potentials, recorded with a 131-channel system. The goal of this work is to obtain robust data on dynamic responses of neocortex, and to connect these data to theoretical models. For example, Silberstein (1995b) has suggested that resonant behavior near 10 Hz may be more pronounced in "hypercoupled" brain states (corresponding to more global behavior). By contrast, 40 Hz resonant behavior may occur more easily in "hypocoupled" states (corresponding to more local behavior). It should be possible to manipulate the effective strengths of different cortical coupling mechanisms with various drugs and test this postulate. In the context of the dynamic theory presented here, such manipulation of brain chemistry alters local and global neocortical control parameters.

1.2.6 Skeptical of EEG data? Neuroscientists may be (and should be) skeptical of EEG reports, given that, as a whole, the scientific reputation of EEG lags that of many scientific fields. One problem that has plagued many cognitive EEG studies has been failure to convincingly separate brain rhythms from artifact. Whereas the typical peaked alpha spectrum is quite distinct from any known artifact, low amplitude scalp EEG frequency components above about 15 Hz are typically indistinguishable from muscle activity (refer to review by Gevins and Cutillo 1986). However, EEG data are far too important to neuroscience to throw out the proverbial "baby." In fact, many sophisticated experiments in cognitive science have shown that EEG and evoked potentials are strongly correlated with specific cognitive tasks (Gevins and Cutillo 1986, 1995; Silberstein 1995a).

In the later work, artifact problems were substantially reduced using a steady-state evoked potential visual "probe" (a 13 Hz flicker), and obtaining brain responses in a narrow frequency band covering the 13 Hz driving frequency. Since artifact power is typically distributed over a wide frequency band, total artifact power in narrow bands (eg, less than 0.1 Hz) can be expected to be much lower than signal power in the same band if the brain produces a moderate to high amplitude response at the driving frequency (Regan 1989). This has been recently demonstrated in studies involving purposeful addition of known artifact (Silberstein et al 1992) and in studies of narrow coherence peaks in the 40 Hz steady-steady evoked potential (Silberstein et al 1998). When such critical technical issues were properly addressed, EEG EEG is proved to be a robust measure of large-scale neocortical dynamic function,. and EEG is moderately to strongly correlated with brain state. This basic property should encourage vigorous consideration of experimental EEG data and related theoretical tools by neuroscientists from multiple sub-fields.

The following outline of a semi-quantitative conceptual framework for neocortical dynamic behavior  is sufficiently general to embrace brain theories applied to different experimental designs, spatial scales, and brain states. The framework draws on work by many scientists and is directly applicable to large-scale neocortical dynamics evidenced by EEG recorded from human scalp. It also has much broader implications for brain information processing. The proposed framework includes the following ideas, generally progressing from the most easily defended to the most speculative.

There can be several motivations for developing mathematical theories of macroscopic, neocortical dynamic behavior eg., the spatial-temporal patterns exhibited by brain state dependent field variables. For example, so called "brain theories" may originate from a general interest in complex systems and non-linear mathematics. But such work is often divorced from genuine neuroscience. For example, mathematical "theory" or neural network models often contain "free" (non-physiological) parameters and lack experimental predictions. Although this kind of work can provide useful metaphorical connections between real and model systems, it should be distinguished from genuine theory.

Genuine, physiologically- based neocortical dynamic theory is essential to rapid development in cognitive science and neurology, even though such theory may provide only very crude descriptions of real brains. Without some quantitative framework, it is difficult for neuroscientists from different backgrounds to communicate subtle ideas or even to form well-posed questions about brain information processing. For example, the theory proposed here helps to pin down concepts like "binding," "functional integration," "synchronous activity," "zero phase lag," "hierarchical level," etc., thereby suggesting refinements for qualitative ideas, and hopefully leading to specific quantitative explanations for data recorded at the appropriate scale. As such, the theory should prove useful, even if later proved wrong.

Quantitative dynamic theory is essential for making more informed experimental choices; that is, picking the small subset of experiments that, with limited resources, can ever be carried out. In addition, experimental design involves many small micro-decisions biased by the experimenter's often semi-private view of brain operation. Reference to theory provides a means by which such biases may be known and evaluated in the context of new experimental designs.

Many robust correlations between large-scale EEG dynamic, cognitive, and behavioral measures have been established over the past 70 years. There are strong motivations to establish physiologically- based quantitative theories of EEG by treating brain tissue in terms of dynamical systems, consisting of neural mass elements interacting by means of interconnecting fibers. Such developments are likely to mimic methods used in several physical science fields and may largely ignore cognitive data during early stages. However, once robust plausible connections between EEG dynamics and the underlying physiology are established, these data will should naturally combine with other known correlations between EEG and cognitive/behavioral data. In this indirect manner, quantitative physiological bases for cognitive/behavioral data will can be obtained.pursued.

Several theoretical models based on genuine physiology/anatomy suggest that neural networks at multiple microscopic and mesoscopic scales (eg.,, local and regional networks) exist naturally in a background of macroscopic field activity (eg.,, synaptic action density), and that important hierarchical interactions take place across these distinct scales (Wilson and Cowan 1973; Ingber 1982, 1983, 1995a,b; Nunez 1974a, 1981a,b,c, 1989, 1995a,b; Ingber and Nunez 1990; Harth 1993; Freeman 1995; Silberstein 1995b; Wright and Liley 1996; Jirsa and Haken 1996, 1997; Robinson et al 1997, 1998, 1999; Uhl and Friedrich 1998; Uhl et al 1998; Haken 1999, Kelso et al 1999). The macroscopic fields, for which EEG provides a crude measure, can act to synchronize or otherwise bind distinct neural networks (Nunez 1995a,b, 1997; Jirsa and Haken 1997), thereby providing possible connections between to cell assembly theory (Hebb 1949; Singer 1993; Phillips and Singer 1997). and "field " concepts from Gestalt psychology (Scott 1995). But, in contrast to Gestalt fields, fields are defined here in terms of well-established neurophysiological mechanisms

A specific quantitative theory of neocortical dynamics is outlined here (Nunez 1972, 1974a,b, 1989, 1995a). Although many details may be overthrown by future work, the theory embraces several general concepts that are more likely to endure: Brain state is described in the context of theory by physiologically-based control parameters which change as a result of neuromodulatory action. These control parameters also partly determine dominant EEG frequencies. Neural network activity is believed to operate in a background environment of standing waves of "synaptic action" (modulation of the number of active synapses per unit volume). Excitatory and inhibitory synaptic action and action potential density are treated as macroscopic(the field variables), influenced by the global boundary conditions, as determined by the size and geometry of neocortex and cortico-cortical connections.

The word "field" may be confusing to non-physical scientists, although its use here is fully consistent with standard use in the physical science. I am not describing putative interactions between the small electric and magnetic fields (measured with EEG and MEG) and neural firing patterns. The "fields" described here are simply the numbers of active excitatory or inhibitory synapses in large tissue masses. For example, suppose multiple neural networks with natural frequencies f_j are embedded in the global neocortical system. Such networks may be formed by Hebbian mechanisms and may physically overlap other networks having different natural frequencies. Each network may be viewed as embedded in inhibitory and excitatory macroscopic fields of synaptic action due to the combined activity of all other networks plus tissue not part of any network. Most networks may be inactive in any particular brain state as the result of inhibitory dominance. Activation of each network may require a threshold level of modulation of the excitatory synaptic action field (relative to the inhibitory field) at one of the natural frequencies of the neural network. Furthermore, activation of a network may "drive" global field modulations at matching natural frequencies (in the manner of "internal pacemakers"). Thus, these fields of synaptic action may facilitate synchronous activity in remote cell assemblies having no direct connections. In this view, EEG dynamic behavior involves a state-dependent mixture of "local" (more dominant functional segregation) and "global" (more dominant functional integration) processes (Nunez 1981a,b,c, 1989, 1995a; Ingber and Nunez 1990; Ingber 1995a; Friston et al 1995, Tononi and Edelman 1998) .). Faulty control parameters may lead to "hypocoupled" or "hypercoupled" brain states eg., neurological and neuropsychiatric diseases (Silberstein 1995b; Bressler 1995).

The theory provides explanations (mostly qualitative, some quantitative) for perhaps a dozen or so 30disparate experiments., described in Nunez (1995a). Taken in isolation, any one of these observations could have multiple origins. However, taken as a whole, the experimental data provide strong support for the general, semi-quantitative framework proposed here, and moderate support for the specific quantitative local/global theory. In summary, the theory provides quantitative support for was developed in the same spirit as earlier ideas on "neural mass action" (Lashley 1931; Freeman 1975) and the idea that "dynamic interplay of neural activity within and between its subsystems is the very essence of brain function" (Mountcastle 1979). The cortico-cortical fiber systems, which provide massive positive feedback between close and distant regions (Braitenberg 1972, 1977) are an essential part of the global aspects of the theory.

.The current version of this dynamic theory is incomplete even in the limited context of macroscopic fields. For example, the theory is silent on most cognitive questions because it was developed specifically to describe EEG, not behavioral or cognitive correlates. Another problem is that "cognitive theories" often contain ill-defined concepts that are hard to pin down experimentally. But, the limited dynamic theory outlined here makes predictions about EEG data in several experiments specifically designed to test global predictions of the theory. If proven only partly valid, it should influence future cognitive theories.

Many physical and biological processes can be represented by some field variable f(x,t), where x represents a vector location (in one, two, or three coordinates) in some medium and t is time. The word "field" can be used to describe nearly any mathematical function of space and time; however it is most useful to science when related to measurable variables. For example, macroscopic EEG electric fields (which are easily measured) are believed generated by modulations of large-scale synaptic action fields (which are not generally measured). The spatial-temporal patterns of fields developed in a medium constitute its dynamic behavior. A dynamic theory may involve many such fields and associated coupled equations, only some are directly accessible to experimental measure. Here I focus on "genuine physiological theory" typically a which I define as a set of equations for fields containing physiological parameters, in which at least one of the fields can be measured. Example fields from analog physical systems include seismic, ocean, sound, electromagnetic, and probability waves (either classical or quantum), in which properties of the medium are described by the appropriate equations. In some cases, the dependent variables (fields) of these equations are directly measurable (e.g.,, pressure, ocean surface displacement). In other systems, measurable quantities must be derived from the dependent variables using separate equations. For example, the dynamics of plasma in a fusion reactor (either human made or star) is often described in terms of probability density functions P(x, v, t), where v is electron or ion velocity. However, measurable quantities like pressure and temperature involve integrals of such distribution functions over the velocity variable.

In brain science, neocortical theory may predict certain spatial-temporal distributions of current source activity (e.g.,, synaptic action fields), but such sources brain sources must be related to measurable electric fields through properties of the head volume conductor. Although the appropriate relationship (Poisson's equation) is well- known and exact solutions are available, these solutions depend on tissue boundaries and electrical resistivities that are known only approximately. Thus, even a perfect theory of neocortical dynamics (eg, describing synaptic fields) can produce ambiguous experimental connections to EEG, especially when such experiments are not specifically designed to test theory. To minimize this problem, use should be made of our general understanding that the head volume conductor acts as a low pass spatial filter of brain source activity. That is, only the long wavelength part of the spatial spectrum of source activity is recorded on the scalp. Thus, scalp EEG is generated by a selective collection of "synchronous" neural sources, and can be quite different from intracranial data. Scalp electric fields (or potentials) reflect only a small part of the full spatial spectrum of synaptic fields.

In system science (eg, in electrical engineering), the system (or medium) input g(x, t) is related to its output f(x, t) by rules that generally vary with the state of the system. For this formulation to be useful in experimental science, measurable variables must be identified. That is, the medium operates on input in some way to create output that is partly predictable. Such predictions may involve specific mathematical output functions only of time f(t), as in the case of input/output in the simple electrical or mechanical systems studied by engineering students. students. Or, predictions may be evolve the statistical properties of an ensemble of spatial-temporal functions fj(x, t) in more complex systems. We may represent such a system by a "grey box," with the bold face symbol D depicting a mathematical operator, representing the dynamic properties of the system that determines input/output relations. Such medium properties are often modelled by differential, integral, or integro-differential equations, in which case the relation between input g(x,t) and output f(x,t) in the model system is governed by an equation of the form (Morse and Feshbach 1953)

^D^[f(x, t)] = g(x, t) (1)

Here ^D^ is the appropriate mathematical operator, and the hat symbol distinguishes the model system from the genuine system, which is governed by the (typically unknown) operator D. To the common charge that "physicists oversimplify biology," a guilty plea must be entered. However, engineers and physicists also oversimplify complex physical systems, and such simplifications have played a critical role in the advance of science and technology over the past several centuries. One reason for this success is that Eq (1) may strongly influence experimental designs. Another reason is that it forces discussion to a controlled quantitative context. For example, scientists must advocate better-defined positions on controversial issues and are barred from many opportunities to "weasel out" of deny past positions, later proved inconsistent with experiment. An important aspect of this process, which typically separates genuine theoreticians from mathematicians, is choice of the approximations needed to obtain ^D^ , appropriate for the practical problem at hand. Thus, a large artistic element is required for theoretical science, just as for experimental science.

The system operators ^D^ include "control parameters" which describe the state of the system (Haken 1983; Nunez 1995a). For example, an electrical resistance may be a control parameter in a circuit designed to act as a tuner. Some tuners are used to pick out narrow frequency bands of antenna current due to a broad spectrum of electromagnetic fields transmitted from many locations. One "controls" the TV picture [eg, output information f(x,t)] by changing the resistance of a variable resistor in the tuner. In an analogous manner, a simple view of neocortex is that of a system with input action potentials and EEG output, with neocortical state controlled by different kinds of input from the midbrain. Of course, neocortex has many other "outputs," including behavior. But, here we focus on the EEG, which is a cognitive and behavior correlate and has many spatial-temporal properties that are just beginning to be explored in detail.

The control parameters determining cortical state involve neuromodulatory (chemical and electrical) input from the midbrain. Such control may itself be influenced by neocortical dynamic behavior (just as TV viewer may be influenced to change channels); however, one may reasonably seek approximate theory in which such chemical and electrical feedback mechanisms take place much more slowly than EEG oscillations. In this case it appears reasonable to neglect such complications for fixed-state theory.

I emphasize that the formalism of Eq (1) is based only the assumption that input and output are related in some systematic way. There is no assumption here that the system is linear or otherwise "simple." In the neocortical example, we need only note that by holding brain state approximately fixed, observed EEG is partly predictable rather than random. That is, we are often able to make relatively accurate predictions about the general statistical behavior of EEG spatial-temporal patterns associated with distinct behavioral or physiological states. This principle is well established within the EEG community, although outsiders may rightly question the quality and repeatability of published EEG studies, especially those which claim to quantify fine distinctions between similar brain sub-states or with data likely to contain substantial artifact. For example, we can easily distinguish deep sleep from REM, but would be skeptical of claims that current EEG methods could reliably determine dream content.. Or, we are more likely to believe results based on frequency bands with high signal to noise ratio (eg, alpha) than studies involving unknown signal to noise (eg, scalp beta).

We can identify several limitations on attempts to check a proposed theoretical operator ^D^; that is, to verify a neocortical dynamic theory with experimental data:

3.3.1 The measured output f^(x, t) is only an approximation to the actual output f(x,t). In the case of scalp EEG, potentials are typically sampled at between 20 and 131 scalp locations over upper brain regions. The electrodes are located about 1 cm from the nearest neural sources. Scalp potentials are spatially low-pass filtered versions of dura potentials as a result of this electrode separation and the poorly conducting skull, which act to spread currents and smear potential patterns. Magnetoencephalography (MEG) avoids tissue distortion but not distortion caused by sensor separation (Wikswo and Roth 1988; Nunez 1995a; Srinivasan et al 1999).

The information content of intracranial recordings is also severely limited by the small numbers and fixed sizes of electrodes, which allow sampling of potentials (space-averaged over the volume of the electrode tip) at only a few locations. Thus, each experimental measure applies to a particular spatial scale, and experimental connections are appropriate only to theory ^D^, derived at the same scale. Many publications of "brain mathematics" have avoided this central issue of matching experimental with /theoretical scales; thus they should be categorized as metaphorical descriptions rather than genuine brain theory.

This failure is perhaps not surprising since connecting the dependent variables of equations to experimental measures is often far more difficult than solving the equations. I believe this point is often poorly appreciated by both neuroscientists and mathematicians often fail to fully appreciate this point. The former  tend to place all studies involving mathematics in a single category. The latter often fail to distinguish between mathematical methods (or computer simulations) and genuine theory. Neuroscientists have also been guilty of "scale chauvinism." For example, dynamic behavior in isolated neurons or small networks has been extrapolated to large networks without justification. In extreme cases, EEG has been dismissed as an "epiphenomenon."

3.3.2. The input g(x,t) may be unknown or known only approximately. For example, the input function to the ocean surface due to wind forces is, at best, known only approximately (Kinsman 1965). Can we check theories of ocean waves ^D^ when measured output depends on both D^ and the input g(x,t), which is only partly known? In the case of evoked (or event-related) potentials, we have substantial, but still imperfect, knowledge of the cortical input function g(x,t). Spontaneous EEG involves input that is largely unknown. Since the measured output f(x,t) depends on both the input g(x,t) and system properties D^, how can one verify a neocortical dynamic theory? In extreme cases of bad theory, the assumed input g^(x,t) might be adjusted to make output f(x,t) appear to confirm ^D^, the classic case where conclusions of a "theory" are simply the summation of its assumptions.

3.3.3. At our current stage of knowledge, any brain or neocortical theory ^D^ can, at best, be only a crude approximation to the real system D. We might well consider such approximate theory a resounding success if it were able to describe a few brain states with only moderate (but quantitative) accuracy. An important feature that appears to distinguish brain states is the relative importance of local versus global contributions to the large scale dynamic behavior measured with EEG. This is determined by competition between functional segregation and functional integration in neocortical tissue. Dynamic and behavioral brain states are believed to be controlled by electrical and neurotransmitter input from the midbrain which occurs on much longer time scales than EEG oscillation periods.

3.3.4  Are our theoriesy good enough to be wrong? Given these limitations, can we develop falsifiable theories of neocortical dynamics? Or in the famous parlance of physicist Wolfgang Pauli, can we create theories "good enough to be wrong?" For many complex physical systems, which also have similar or even more severe limitations on verification, the answer is clearly "yes." In comparing such systems to neocortex, there is both good and bad news. One the negative side, brain tissue is much more complex than "complex" physical systems. However, in contrast to many physical systems, experimental EEG data obtained in multiple brain states at macroscopic through several smaller scales are abundant. This provides reason for optimism. Given the practical success of theories developed in engineering and physics (even when such theories later proved to be very limited or even completely wrong), I suggest in the following section that neuroscience tentatively mimic some of the proven methods of physical science and evaluate the results, without deciding in advance how far to carry this paradigm.

Brain science has experienced a long debate concerning localized versus distributed function. But, neuroscientists now generally agree that while specific cell groups perform elementary functions, complex functions require integrated interaction of many areas throughout both hemispheres (Luria 1966; Bressler 1995; John et al 1997). The issue of local versus global dynamics, as measured with scalp and intracranial electrodes of different sizes, parallels the brain function issue. There have been many suggestions of brains operating between the extremes of locally versus globally dominated dynamics (Nunez 1981a,b, 1989, 1995a,b; Ingber and Nunez 1990; Tononi et al 1994; Silberstein 1995b; Ingber 1995; Friston et al 1995; Andrew and Pfurtscheller 1996; Jirsa and Haken 1997; Tononi and Edelman 1998; Haken 1999). If this is so, any comprehensive dynamical brain theory must include local networks and global fields simultaneously. But, no such comprehensive theory, containing realistic, multiple neocortical networks, is likely to be developed soon. Even if developed, such theory would necessarily contain many unknown control parameters, so that anything approaching full verification appears impossible in the near future.

It is far easier to test brain states that are close to extreme ends of this local-global gamut of brain dynamics with limited versions of theory that appear to match such states approximately. In this paper I focus on comparisons between global theory and EEG data obtained in states of relatively high coherence, which are more likely to have strong global contributions. In addition, I show how local neural networks can fit naturally with global theory, while acknowledging that local network properties are mostly unknown. However, if local effects can be minimized by selective experiment, perhaps verification of global aspects of the theory may be obtained. In a similar manner, local theories can be checked with experiments in brain states or for other conditions in which global effects may be neglected. Such experiments are more likely to involve intracranial electrodes, which are sensitive to locally dominated dynamics.

Complicated spatial-temporal patterns and their relations to experimental data are often greatly be simplified using Fourier transform methods. That is, the dynamics of many systems is such that its behavior is most naturally expressed in the spatial - temporal frequency domain. In particular, a system's input g(x,t) and output f(x,t) may be represented by their multi-dimensional Fourier transforms G(k,[omega]) and F(k,[omega]), respectively. Here k is vector wave number (or spatial frequency) and [omega] is temporal frequency. In the case of potentials measured on the dura or cortical surface, k is a vector with two components (kx, ky), representing spatial frequencies in two surface coordinates. For purposes of these general remarks, we consider neocortex and its potential fields to be isotropic so that we need not worry about directional-dependent properties. In this case, we interpret the vector wave number k as simply its magnitude (kx² + ky²)^1/2.

In order to visualize the general spatial-temporal dynamics described here, imagine the dynamics of a rough ocean surface. The surface is composed of a wide range of waves, each with wavelength equal to 2p divided by wave number (k), ranging from ripples to intermediate length wind-driven waves to tides. Note that this description of EEG (say at the dura surface) in terms of such mathematical “waves” of many wavelengths is fully general. That is, it does not depend on the nature of the dynamics, eg, whether genuine "wave phenomena", as this term is used in the physical sciences, actually occur or whether the dynamical system is linear. It is simply a mathematical transformation that has proven useful in a variety of physical and biological systems.

Dynamics on a closed surface may be represented more naturally by special functions of the spatial coordinates like the spherical harmonic functions (essentially functions of latitude and longitude on a sphere) rather than the spatial sine and cosine functions associated with the wave numbers kx and ky. However, such special mathematical functions are qualitatively similar to sines and cosine functions. Thus, this general, semi-quantitative discussion here is not sensitive to characterization of spatial properties in terms of "spatial frequencies" or "wave numbers." Brain waves in a spherical shell are discussed in Katznelson (1981), Nunez (1995a), and the Appendix to this article.

Several motivations for working with the Fourier transforms of input g(x,t) and output f(x,t) are evident. One is that, in the  case of linear systems with slowly varying control parameters, transformed input G(x,t) and output F(x,t) are related by the simple multiplication

F(k,[omega]) = ^H (k,[omega]) G(k,[omega]) (2)

Here the transfer function ^H(k,[omega]) depends on the mathematical operator^ D^ of Eq(1). Thus, a complicated differential or integral equation (1) relating input to output may be replaced by the simple algebraic equation (2), providing an enormous conceptual simplification. This relation tells us quite a bit about the likely output f(x,t) or its Fourier transform F(k,[omega]), even when we possess minimal knowledge of the input g(x,t) or G(k,[omega]). For example, Eq (2) shows that F(k,[omega]) tends to be large over ranges of spatial or temporal frequency where ^H(k,[omega]) is large, except for unusual inputs G(k,[omega]) that happen to be very small in these same frequency ranges. Also, when the input approximates spatial-temporal white noise, i.e., when G(k,[omega]) fluctuates about a constant level over abroad bands of spatial and temporal frequencies, the predicted output F(k,[omega]) is approximately proportional to the theoretical transfer function, thereby allowing quantitative verification of the theory (H or ^D^). Freeman (1975) has pioneered applications of these methods of linear systems theory to time-dependent data recorded in neural tissue at relatively small scales. If input and output dependent variables are time, but not space dependent, as in the case of most common electric circuits, the transfer function depends only on temporal frequency, ie, H^(k, [omega]) may be replaced by H^([omega]).

A second for motivation for using Fourier transformed variables, which clearly applies to studies involving scalp EEG, is that a relatively simple relationship (Poisson's equation) is known to relate the measured output [scalp potential, F^(k,[omega])] to actual dynamical system output [eg, dura potential, cortical current density, or synaptic action, F(k,[omega])]. In an idealized brain medium (Nunez 1981a, 1995a, Katznelson 1982) this relationship may be expressed as

F^(k,[omega]) = V^(k,[omega]) F(k,[omega]) (3)

Here ^V^(k,[omega]) is the transfer function of the head volume conductor, which depends on Poisson's equation, tissue resistivities, and tissue boundaries. While obtaining progressively more accurate head models used to determine ^V^(k,[omega]) provides an important ongoing engineering challenge, the qualitative behavior of V(k,[omega]) is fairly well understood. In the low frequency range of EEG scalp potentials, V(k,[omega]) can be accurately approximated by V(k); that is, tissue volume conduction is approximately independent of temporal frequency (Cooper et al 1965; Plonsey 1969; Nunez 1981a; Malmuvino and Plonsey 1995). Furthermore, passive tissue properties (especially skull) and large electrode-source separations combine to spatially low-pass filter the cortical output F(k,[omega]). Thus, V(k) decreases sharply even at moderately high spatial frequencies (Nunez 1995a; Srinivasan 1995; Srinivasan et al 1996, 1998). In some brain states, higher temporal frequencies above the alpha peak are observed to have more power at higher spatial frequencies. In such cases, three-dimensional plots of F^(k,[omega]), and perhaps by implication H(k,[omega]), contain "mountain ranges" depicting such relations. (Imagine k and [omega] as axes plotted on a surface, with F plotted out of the plane and being large along a particular pathway across mountain peaks, as shown in Fig. 3).

One important engineering challenge in EEG is to obtain more accurate volume conductor models ^V^(k) and computer algorithms so that dura potential [F(k,[omega]) or f(x,t)] can be reliably estimated from discrete scalp measurements [F^(k,[omega]) or f^(x,t)]. Such approaches have been labeled "spatial deconvolution," "software lens," "deblurring," or and "cortical or dura imagining" (Cadusch et al 1992; Le and Gevins 1993; Gevins et al 1994; Nunez et al 1997, 1999; Gevins and Cutillo 1995). The surface spline-Laplacian of the scalp potential field provides an alternate, relatively robust and model-independent method that provides estimates of dura potential that are typically similar to model-dependent dura imaging (Perrin et al 1987, 1989; Nunez 1995a; Nunez et al 1997, 1999; Babiloni et al 1996; Srinivasan et al 1996, 19987). The spline-Laplacian should not be confused with the far less accurate nearest-neighbor (eg, Hjorth) Laplacian.

A third compelling reason for following Fourier transformed variables is that the transfer functions H(k,[omega]) of many dynamic phenomena tend to be "contained" in relatively small regions of the (k,[omega]) space, even though a very wide variety of complicated output behavior f(x,t) may occur. For example, dynamic outputs of musical instruments (string, membrane, or air column displacement) and other wave phenomena typically obey dispersion relations [omega] = [omega](k), even though a wide range of musical sounds or other oscillatory phenomena may be produced by the same system. Thus, Fourier transformed variables typically behave more simply that un-transformed variables, although there are important exceptions like spatial-temporal chaos, eg, in turbulent fluids. For example, when discussing electromagnetic radiation in a vacuum, engineers typically take such dispersion relations for granted by referring to wavelength (2p/k) and temporal frequency ([omega]) interchangeably, since the simple dispersion relation is [omega] = ck, where c is light velocity. Thus, one may speak unambiguously of "microwaves" in relation to longer waves (at lower frequencies) called "radio waves." A one-to-one correspondence between frequency and wavelength is taken for granted with electromagnetic waves in a vacuum. However, when electromagnetic waves travel through material media, field dynamics can be far more complicated. In extreme, but not unusual cases, dispersion relations may not exist. This is more likely to occur in non-linear media, but can also occur in linear media, e.g, linear plasma. Another example is illustrated in Fig. 3, which shows an idealized transfer function ^O^(k,[omega]) for ocean waves over a wide range of wavelengths (2p/k), from ripples to long wind-driven waves to tides. The water wave dispersion relation relating temporal frequency ([omega]) to wave number (k)

[omega] = k(g/k + bk)^1/2 tanh(kh) (4)

depends on three control parameters, g(the acceleration of gravity), b(related to surface tension), and h(ocean depth). The term on the far right is the hyperbolic tangent function. A wide variety of ocean wave dynamics is possible, depending on the input function G(k,[omega]) and control parameters. However, Newton’s laws and the properties of water, when subjected to surface or gravitational forces, effect a substantial restriction on the dynamics, as given by Eq (4). Thus, even if the input wind forces are unknown, we may expect ocean wave dynamic behavior F(k,[omega]) to behave somewhat similarly to that shown in Fig. 3, at least at large distances from sources of surface disturbance. That is, any sailor knows that water surface dynamics can be very complicated when un-transformed data are followed in the spatial-temporal domain. However, the water medium obeys Newton’s laws, which place relatively severe limitations on dynamic behavior likely to be observed. Such restrictions are not so apparent to sailors observing the un-transformed sea surface, but reveal themselves to oceanographers in the Fourier transformed variables. If the input wind forces approximates spatial-temporal white noise, i.e, G(k,[omega]) relatively constant but with random fluctuations, the high mountain range in Fig. 3 is expected to rise above the lower mountains associated with input forces that may be random.

To imagine dynamic behavior with no dispersion relation, consider a plot similar to Fig. 3, but with multiple peaks and valleys similar to a portion of the Rocky Mountains. In this example, a particular choice of wave number k can be associated with a wide range of temporal frequencies. For example, a storm in the ocean may drive ocean wave dynamics. The surface within the storm system (called the "near field") will generally satisfy no dispersion relation; ie, both the surface and its Fourier transform will look like rocky mountains. By contrast, at some distance from the storm (called the "far field" or "wave field"), the non-wave parts of the surface disturbance are removed as a result of water dynamics. We are unlikely to use the term "wave" to describe spatial-temporal dynamics lacking well-defined dispersion relations. Rather, words like "turbulence" or "spatial-temporal chaos" may be more appropriate. That is, "waves" are very special spatial-temporal fields for which the dynamics is severely restricted when viewed in Fourier transformed space, although little hint of such restriction may be evident from un-transformed space-time observations.

Another common characteristic of waves is that [omega] is a monotonically increasing function of k for most frequencies, wave numbers, and wave media (Exceptions are labeled "anomalous dispersion"). If this holds true in brain dynamics, high temporal frequencies will be attenuated in the observed scalp potentials F^(k,[omega]), as an indirect result of spatial filtering by the volume conductor rather than any dependence of V(k) on temporal frequency, an experimental finding that as has been reported for alpha and beta rhythms (Pfurtscheller and Cooper 1975). An example simulation plotted in Fig. 6 illustrates this idea.

Fig. 6. Solutions of the system of global mode equations (22), derived in the Appendix. The system consists of five coupled, second order ordinary differential equations, each with about 40 mode coupling terms. The vertical axis represents normalized synaptic action density at cm scales. The horizontal axis is normalized time [time (sec)/t _g], where t _g = 1/l v is the characteristic global delay time, apparently in the 6 to 16 ms range based on long cortico-cortical fibers. If t _g = 10 ms, the period shown is one second and dominant frequencies are in the 11 Hz range. The upper plot is the sum of modes F _n(t), n = 1, 5, crudely representing dura potential. The lower plot is the same solution, but with modes n =3,4, and 5 removed to crudely simulate spatial filtering by the head volume conductor between dura and scalp. Other numerical solutions (not shown) indicate that increases in the linear feedback parameter B cause amplitudes to increase with minimal frequency changes for small to moderate B. Large B values produce solutions with larger amplitudes and lower frequencies. The parameters for this numerical solution are B = 2.2, A = 0.0667, k_n² = 1.5, 1.95, 2.70, 3.75, 5.10 (n = 1,5, consistent with the rectangular geometry discussed in the Appendix).

4.4 Extension to non-linear systems

A possible objection to the approach outlined above is that, "the brain is nonlinearnon-linear." That is, if the dynamic operator ^D^ of Eq (1) is nonlinearnon-linear, there is no simple definition of a transfer function, and Eq (2) is generally not valid. However, in engineering practice, Eq (2) is often applied to complex non-linear systems, not because of "linear psychosis," but rather because it provides a useful bridge to deeper theoretical and experimental study. For example, in physical media, a nonlinear relation between input g(x,t) and output f(x,t) can often be approximated by one or more linear relations, each applicable to a narrow range of the input function. Or, the relation (or operator) may be approximated by a linear relation in certain states of the system, perhaps when a control parameter multiplying a nonlinear term in ^D^ is small. Linearization also often may makes a theory more robust over more applications, even though it may be less accurate in specific applications.

Given our current limited knowledge, any dynamic theory of neocortical function (linear or non-linear) can, at best, be expected to provide a very crude approximation ^D^ to the actual system D. Thus, it may make sense to develop preliminary linear theory, partly to guide new experimental work, with the goal of later development of more accurate, non-linear theory. My own bias has Led me to seek some crude success with experimental connections to linear theory before substantial effort is directed to nonlinearnon-linear theory. One obvious reason is that a non-linear theory must depend on the specific kind of non-linearity employed. It is not enough to say that a system is "non-linear;" a theoretical model must choose the type of non-linearity from a number of potential candidates. This choice (or, more likely, often a relatively arbitrary assumption) must involve additional control parameters that are probably unknown for brain tissue. However, several extensions of this global theory using various non-linear approaches are described in Sec. 5.3.8 and the Appendix in the context of EEG experiments.

One ambitious, fully non-linear theory of neocortical dynamics aims to derive rules of interaction in tissue at intermediate (mesoscopic) scales from rules at smaller scales (Ingber 1982, 1983). This mesoscopic approach of non-linear, non-equilibrium statistical mechanics bears a complementary relation to the macroscopic local/global theory proposed here, similar to the relationship between the kinetic theory of gasses and fluid mechanics. By contrast to many artificial neural networks, Ingber's statistical theory is based on genuine neocortical physiology. One interesting prediction is that the number of neural firing patterns that can simultaneously persist for several seconds is in the range of 5 to 10. Such patterns may store short-term memories that are known to be limited to 7+-2 items, eg., for auditory patterns (Ingber 1985). Another prediction involves the 4+-2 rule for visual pattern storage. Stability, duration, and statistical interactions of firing patterns appear consistent with short-term memory data. These apparent connections to memory may be fortuitous; however they suggest fertile ground for future work. For example, preliminary attempts to reconcile the microscopic/mesoscopic statistical theory with the macroscopic global theory  with the macroscopic global theory are noted (Ingber and Nunez 1990; Ingber 1995a).

In this section I outline a preliminary theory of the large-scale neocortical dynamics appropriate for verification, modification, or falsification with scalp EEG. The theory emphasizes a combination of local and global physiological mechanisms. I use the term "local theory" to include dynamic descriptions based on intracortical (and possibly thalamocortical) feedback "circuits," with signal delays perhaps due mostly to postsynaptic potential rise and decay times. Negative and positive feedback mechanisms at millimeter scales are critical to such theories. I also include dynamics at the single neuron level in the category of "local theory," for example intrinsic membrane oscillations (Jahnsen and Llinas 1984a,b). This unequal division of dynamic behavior into a single "local" category, encompassing multiple spatial scales, and a single-scale "global" category is constructed to facilitate contact with large scale EEG data. The dynamics of blood flow provides an analogy. We may categorize blood as a mixture of fluid and cells. The cells encompass dynamic behavior at many small scales not subject to macroscopic measurements. However, a biomedical engineer's practical description of the fluid-blood system is likely to contain only large-scale cell characteristics (representing aggregate small-scale properties) subject to macroscopic measurements.

Local theories are most compatible with functional segregation, and global boundary conditions are typically neglected to simplify analyses. These "infinite brain" approximations that neglect boundary conditions essentially assume that the spatial spectrum of neural activity at long wavelengths (comparable to a circumference of neocortex) contains negligible power. Thus, boundary conditions have no influence on predicted dynamic behavior, eg, on the interference of propagating neocortical synaptic fields. Local field potential theories having apparent connection to EEG include work by Wilson and Cowan (1972, 1973), Lopes da Silva et al (1974), Freeman (1975), van Rotterdam et al (1981), Zhadin (1984), and Liley et al (1999). Any such theory (linear or non-linear) can be represented by the "Local Network" box in Fig 4. Linear versions can be described by the local transfer function L([omega];Q).

By "global theory" I mean that signal delays are mainly due to finite propagation of action potentials along cortico-cortical fibers that provide positive feedback between multiple cortical regions. Furthermore, periodic boundary conditions due to the closed neocortical surface typically exert important influence on the dynamics because of the required interference of synaptic fields. This approach emphasizes functional integration in neural tissue. Global theory of EEG has been published by Nunez (1972, 1974a,b, 1981a,b,c, 1995a), Katznelson (1981, 1982), Srinivasan (1995), Jirsa and Haken (1997), and Haken (1999); it is represented by the "Non-local Neocortex" box in Fig 4.

The term "regional circuits" has been used to describe dynamic processes intermediate between local and global (Silberstein 1995b). An important example might be a neural network in which both synaptic and propagation delays contribute to its dynamic behavior. Such "regional circuits" (eg, large neural networks) fit naturally within the general conceptual framework on this article, but are not included in the specific theory outlined here. Inclusion of such regional networks within global fields may be an important part of 21^st century neuroscience (Ingber 1995a, Jirsa and Haken 1997, Tononi and Edelman 1998, Haken 1999).and Robinson et al (1997).

5.1 Competition between functional segregation and integration

Both local and global mechanisms are well established in neuroscience. However, their relative importance to neocortical dynamic behavior and EEG is controversial. Attempts to combine local and global effects and to evaluate their relative importance have been published by Nunez (1981a,b,c, 1989, 1995a,b, 1997), Nunez and Srinivasan (1993), Srinivasan and Nunez (1993), Tononi (1994), Silberstein (1995b), Friston et al (1995), Wright and Liley (1996), Jirsa and Haken (1996, 1997), Robinson et al (1997), Tononi and Edelman (1998), Haken (1999), and Liley et al (1999). Local/global theory includes interactions between the two boxes in Fig 4.

Here, I favor neither side of this controversy, although my own work has been mostly on the global side where more contact with large scale (scalp) EEG data has been obtained. Rather, I suggest that given our current meager understanding of physiological control parameter ranges, we should focus on the likelihood that both local and global mechanisms generally contribute to dynamic behavior. One or the other may be more dominant in different brain states, or perhaps over different wave number ranges of the dynamics. By analogy, surface tension, gravity forces, or bottom effects may dominate ocean waves, depending on control parameters and wavelength, as shown by Eq (4). With this approach, local versus global arguments of brain function and EEG reduce to issues of relative magnitudes of local and global control parameters occurring in each brain state. By analogy to our social system, the dynamics of croissant consumption may be due to both local (eg, family) and global (eg, television) influences.

A local/global neocortical transfer function, derived from known physiological processes, but with seemingly appropriate linear approximations to input/output in neural tissue at centimeter scales, follows directly from Fig 4 (Nunez 1989, 1995a). That is, action potential output from the local network (G) is the product of the estimated local/global transfer function (H^) and subcortical input (U), where

The dynamic transfer function H^ takes on this relatively simple form, in which it depends on only two control parameters (B,Q), provided that the ratio of characteristic global (cortico-cortical, t _g) to local (eg, PSP rise, t _l) times is fixed. Also, the wave number (k) and frequency ([omega]) are non-dimensional variables here, normalized in terms of characteristic intracortical and cortico-cortical interaction lengths and propagation speed, also assumed to be fixed parameters in this introductory discussion. Here ^L^(k,[omega]; Q) is a local transfer function dependent on a local control parameter Q. A semicolon separates the independent variables k and [omega] from control parameters. In several local theories, Q is proportional to the product of intracortical inhibitory and excitatory feedback gains in local circuits. The global transfer function ^T^(k,[omega]; B) depends on a global control parameter B, which is proportional to positive feedback gains established by cortico-cortical fibers. Both B and Q are independent of delay times in this model. It is assumed they change with brain state as a result of neuromodulatory influences on the strengths of local (positive and negative) and global (positive) feedback. Equation (5) is based only on linear processes. However, a set of non-linear differential equations, which include cubic non-linear negative feedback mechanisms, is proposed by Eqs (23) and (24) in the Appendix. These equations describe much more complicated input/output relations, but many of the general ideas are adequately illustrated by the linear description.

A derivation of Eq (5), definitions of non-dimensional variables, estimates of the physiological parameters, a semi-quantitative solution, and studies of the effects of various assumptions are presented in Nunez (1995a). This work is too lengthy to repeat fully here. However, the basic idea is illustrated by Fig 4. Synaptic action from the midbrain U(k,[omega]) provides synaptic input to a local mass of neocortical cells (box at left). "Local" positive and negative feedback by intracortical fibers (and possible thalamocortical loops, depending on specific local theory) process this input to produce output action potentials G(k,[omega]). We expect this process to be non-linear; however, if linearity is assumed in this elementary version, the local system acts according to the local transfer function L(k,[omega]; Q). The local output action potentials G(k,[omega]) then provide excitatory input to the entire non-local neocortex through cortico-cortical fibers. The global neocortical system integrates this input according to the global transfer function T(k,[omega]; B) to produce excitatory synaptic output F(k,[omega]), indicated by the box at right. Complications like the distribution of cortico-cortical fibers must be considered to derive T(k,[omega]; B). However, this second stage is believed to be fundamentally linear since it essentially assumes that the number of active synapses (output) is proportional to number of action potentials (input) at earlier times. The global synaptic output F(k,[omega]) provides additional input to each local mass. While there are many physiological details to be studied in the future, the main point for this paper is that Eq (5) was derived from plausible physiology and contains separate, but complementary local and global contributions to dynamic behavior.

In the case of dynamics on a closed (e.g, spherical) surface, we might choose to express ^G^(k, [omega]) as ^G^(n, m, [omega]), where the (n, m)  indices label spatial frequencies in two surface coordinates (e.g, latitude and longitude in spherical coordinates). However, the simpler wave number idea is sufficient for this general outline. This theory differs from metaphorical approaches to "model" brain dynamics in that it is based physiology and anatomy that is known, at least in outline. It is also based on a number of approximations that could turn out to be wrong or perhaps valid only for a very limited number of brain states. Readers interested in the mathematical details should consult the Appendices of this article and the Nunez (1995a) book. However, for purposes of this article, the central issue is not so much whether the theory is right or wrong, but whether it provides a useful, semi-quantitative environment to discuss the general conceptual framework outline here. If the theory can accomplish this goal, is should provide guidance to future theoretical and experimental work.

5.3 Special (limiting) forms of the transfer function

Equation (5) illustrates several predicted aspects of EEG dynamics that may hold, even when more accurate theories are later developed, because they these features appear to have general validity, i.e, they appear to be largely independent of detailed physiological assumptions of this specific theory:

5.3.1 Local/global resonances occur in several ways. As formulated here, the magnitude of the local transfer function L^ is less than one at all spatial and temporal frequencies since it converts synaptic input into action potential output. By contrast, the global transfer function T^ is larger than one since it converts action potential input to synaptic output. The product L^T^ is less than or the order of one. The full (local/global) dynamic transfer function ^H^ is tends to be large if the local function L is large. H^ will also be large if (L^T^) is close to one. This latter condition defines, with physiologically -based mathematics, a "matching" of local and global resonances, which is a possible mechanism to facilitate interaction between remote cell assemblies, as discussed in Sec 5.3.6. Regions of (k,[omega]) space where ^H^ is large ("mountain ranges") correspond to multiple branches of dispersion relations for the putative "brain waves." Such dynamics may exhibit large changes due to changes in the parameters (B,Q), which that control the state of this model cortex.

5.3.2. Local properties at long (e.g, scalp) wavelengths are may be independent of wavelength. Equation (5) may be used with several different local theories in which the relation between synaptic input and action potential output has been approximated as linear. Most intracortical fibers are shorter than a few mm. Thus, characteristic interaction lengths of such theories are much shorter than wavelengths of scalp recorded EEG. For this reason, we might expect local transfer functions to be approximately independent of wave number in In the regions of (k,[omega]) space where wavelengths (2p/k) are large. This approximation has been derived mathematically in at least one such local theory (van Rotterdam et al 1982). are much larger than typical intracortical connection lengths (up to a few mm), the local transfer function is approximately independent of wavenumber. That in this case, ^L^(k,[omega]; Q) may be replaced by ^L^([omega]; Q). This , where the approximation applies to all scalp recordings and all intracortical recordings using electrodes with diameters larger than several mm.

5.3.3. Purely local phenomena may occur. If long range positive feedback is negligible (e.g, there are no cortico-cortical fibers, or the global control parameter B is small due to specific neuromodulatory influences), the global transfer function in Eq (5) is small. In this limiting case, the full transfer function is essentially the local transfer function, i.e

H^( k,[omega]; Q) = L^(k,[omega]; Q) (6)

This is may be a valid approximation for some brain states. In this basic version of the theory, the cortical/white matter system is assumed to be homogeneous, but in more detailed versions, the parameters Q and B may be functions of cortical location. Thus, we generally expect many local cell groups with different properties, together with regional networks at various scales that can perhaps be studied with intracranial recordings. Such mathematical and physiological complications will provide many future challenges, but are unnecessary to illustrate the central ideas of this paper.

I will describe one published local transfer function (van Rotterdam et al 1982), although other local theories (eg, Freeman 1975; Liley et al 1999) could also be used to illustrate the general conceptual framework. The dominant frequency band depends on rise and decay times of PSPs, and the local feedback gain Q. The magnitude of this gain is not known for genuine tissue. However, if the oscillations are to be weakly damped, Q must lie in a confined range. In this case, the predicted local non-dimensional frequency [omega] is approximately one ([omega] = 2p ft _l = 1), or f = 1/2p t _l (Hz), where t _l is a characteristic local delay time given by the square root of the product of EPSP and IPSP rise times. If EPSP and IPSP rise times (with matching decay obtained from cable theory) are about 10 and 20 ms., respectively, the dominant frequency range is near f = 11 Hz. However, published EPSP and IPSP rise times are typically shorter (Rall 1967), perhaps 3 and 5 ms, respectively. With these shorter delays, the [omega] = 1 case represents higher frequencies, in the range f @ 41 Hz. But, one may guess that local oscillations over relatively broad frequency ranges are possible in different tissue masses due to variations in local membrane properties, feedback gains, PSP rise times, and other fixed anatomical and variable neuromodulatory influences that occur in different brain states (Lopes da Silva 1991, 1995; Wright and Liley 1994; Liley et al 1999). A version of local input/output relations is given by the partial differential equation (14) in the Appendix.

5.3.4 Purely global phenomena may occur. In regions of (k,[omega]) space where periods (2p/[omega]) are much longer than local delay times (rise and decay of PSPs), the local transfer function varies slowly with frequency and no local resonance occurs. In this frequency range, input to the local network passes through without further processing and the full transfer function in Eq (5) is approximately

H^(k,[omega]; B) = Constant / [1 – Constant T^(k,[omega]; B)] (7)

Neocortical resonance is by exclusively global mechanisms in this frequency range. By setting the denominator of Eq (7) to zero, we obtain the global dispersion relation, [omega]=[omega](k). Again we may conjecture that this is a valid approximation for some brain states. The most likely corresponding EEG states may occur under anesthesia, the three Hz spike and wave of epilepsy, and to a lesser extent some sleep stages and, and perhaps to a lesser extent the awake alpha rhythm. e.g, these states of minimal cognitive processing often exhibit widespread, spatially coherent EEG that is most likely associated with globally dominated dynamics, as partly illustrated in the upper parts of Figs 1 and 2. For relatively small values of the control parameter B, mode frequencies are not sensitive to moderate changes in B. Rather, dominant frequencies depend mainly on cortico-cortical propagation velocity and wave number. Allowed wave numbers depend on global boundary conditions in the cortical shell as discussed below. The equivalent linear partial differential equation (11) and a related set of non-linear ordinary differential equations (22) based on cubic non-linear feedback are given in the Appendix.

5.3.5. Global oscillations are partly determined by neocortical boundary conditions. Global boundary conditions (periodic) allow oscillatory dynamics to persist only for specific wave numbers (or the n,m indices on closed surfaces). For example, excitatory and inhibitory synaptic action fields must be single valued functions of surface coordinates if they are genuinely measurable fields.. Due to the interference of propagating waves of synaptic activity on the cortical surface, only certain discrete wavelengths of standing waves can persist, as in analog physical systems like musical instruments, atoms, chemical compounds, resonant cavities, etc. For example, destructive interference is expected when regions of enhanced excitatory synaptic activity interact with regions of enhanced inhibitory activity. Over time, such destructive interference tends to remove synaptic action fields that fail to match global boundary conditions.

Such fundamental connections between interference, boundary conditions, standing waves, and resonances occur throughout physical science and engineering. For example, in simple one-dimensional systems like string or wind musical instruments, only wavelengths equal to twice the length of the system divided by an integer (n=1,2,3,…) can persist. In more complex geometry, inhomogeneous, or anisotropic systems, and/or systems closed on themselves, the allowed "wavelengths" (or quantum numbers n,m) are typically limited by more complex rules; however the basic principle of wavelength restriction by boundary conditions still applies. Examples include quantum wave functions in atoms, electromagnetic radiation generated by random lightning strikes in the resonant cavity formed by the earth's surface and the bottom of the ionosphere ("Schumann resonances", Jackson 1975), and as suggested here, oscillations of synaptic action (and by implication EEG) in the closed neocortical/white matter "shell." The specific geometry of neocortex is not critical to these general arguments. It is only necessary that synaptic action fields propagate in multiple directions as a result of action potentials in intracortical and cortico-cortical fibers. Such fields must either damp out quickly or interfere on the cortical surface. Such interference between excitatory and inhibitory fields may be linear or non-linear. In either case, we expect boundary conditions to force the occurrence of standing waves.

When the wave number restriction is combined with the dynamic restrictions implied by the transfer function (or dispersion relation), preferred temporal frequency ranges are predicted. In other words, such systems act as band pass spatial-temporal filters. These special (or resonant) frequency ranges emerge from a combination of tissue (or other media) properties and global boundary conditions. When periodic boundary conditions for the neocortical surface are combined with the global transfer function, Eq (7), a series of global modes (ie, global resonant frequencies) is predicted. Detailed discussion of parameter ranges and the effects of various physiological parameters is presented in Nunez (1995a). However, the basic idea is simple. The purely global theory predicts lowest frequencies (eg, fundamental and lower overtones) in the general range f = 1/2p t _g (Hz), where t _g is a characteristic global delay time. t _g is roughly 10 to 30 ms, as a result of action potential propagation (6 to 9 m/sec) along the longest cortico-cortical fibers (10 to 20 cm), which have strongest influence on dynamics predicted by the global theory. Thus, the predicted frequency range for the lowest (fundamental) mode is roughly f = 5 to 16 Hz.

Of course, much more detailed analyses are required to include the effects of distributed cortico-cortical fiber lengths, propagation speed distribution, feedback gain, etc. Most of these details do not appear to alter the general semi-quantitative results outlined here (Nunez 1995a). However, when the global feedback gain parameter B is increased above certain mode-dependent critical values, mode frequencies (fundamental and overtones) can decrease sharply, in a manner suggestive of transitions from the awake to sleeping states or varying depths of halothane anesthesia (Nunez 1995a). When the parameter B is small to moderate, the lowest (fundamental) global mode is roughly in the 10 Hz range, within a factor of perhaps two or three. By contrast to the local oscillations discussed in Sec 5.3.3, a large number of global mode overtones can occur, but these are equal at all cortical locations. Such overtones can be very closely spaced; they are not generally harmonics of the fundamental.

5.3.6. Most brain phenomena involve combined local and global effects. I have

conjectured extreme cases when local, Eq (6), or global, Eq (7), processes dominate neocortical dynamics. However, most brain states probably involve both large-scale functional integration (facilitating global contributions to the dynamics) and functional segregation (facilitating local effects). Figure 5 illustrates two examples of local/global dynamic behavior in the model system, described by the transfer function, Eq (5). The non-dimensional frequency [omega] is normalized with respect to the global delay time t _g. In Fig. 5a, the local delay parameter t _l is chosen so that both local and global processes contribute to preferred frequencies roughly in the 0 to 10 Hz range. By contrast, Fig. 5b, shows an example where only global mechanisms contribute substantially to the lower frequency range. But, in addition, local and global mechanisms combine to provide activity at a higher frequency range, perhaps f = [omega]/2p t _g = 33 to 46 Hz. While the physiological parameters which determine the specific shapes of the transfer function parameters, which determine the specific shapes of the transfer function and mode frequencies, are not known with sufficient accuracy for close quantitative comparison with EEG data, all parameters in these examples are in plausible physiological ranges. That is, in contrast to metaphorical approaches to modeling, the theory contains no "free" parameters. These parameters have a physiologic basis. Thus, future adjustments based on new, independent experiments can be anticipated.

(a)

(b)

The matching of local and global resonances may be illustrated with a physical example involving purely global modes of simple (eg, non-dispersive sound or electromagnetic) waves in a prolate spheroid shell (roughly the shape of a rugby football, Nunez 1995a). These differ substantially from the putative "brain waves," but resonances are obtained from a dispersion relation equivalent to setting the denominator of Eq (7) to zero. Assuming propagation speed and surface area that roughly match neocortical/cortico-cortical parameters (near 6 to 9 m/sec and 1500 to 2000 cm², respectively), global mode frequencies (Hz) are something like 10, 12, 17, 18, 21,…, 39, 40, 42, 45,…These specific numbers are not critical since they are not rigorously derived from physiology. The main point is that many closely spaced global modes may occur, which are in the general range of EEG frequencies. These preferred (or "resonant") frequencies refer to modulations of synaptic action and action potential fields about background levels; there is no guarantee they will be observed in EEG. However, according to Eq (5), the local/global transfer function H^ will be large when both local L^ and global T^ transfer functions are large so that the product L^T^ is close to one. For example, if local networks at two locations x₁ and x₂ have resonances in the 10 and 40 Hz range, respectively, both will tend to be enhanced by global modulations of synaptic action in their respective frequency ranges. On the other hand, the many global modes not matching local modes may be too weak for observation as EEG.

5.3.7. EEG pacemakers. The old idea of thalamic pacemaker origins of EEG can be extracted as a very special case of the general picture presented here. For example, if only one of the local networks depicted in Fig 4 produces resonant frequencies, its dynamic behavior might be forced on the entire global system by its output G. For purposes of these general arguments, it doesn’t matter if the local network is located in the thalamus, a region of neocortex, or both. The essential idea of a pacemaker is that its dominant frequencies are determined by internal mechanisms, not by feedback from external tissue masses. Such pacemakers might occur at membrane levels (Steriade and Llinas 1988) or at multiple network levels. However, "pacemakers" (at any spatial scale) must have very restricted input in order to preserve autonomy. The accuracy of such membrane autonomy (Freeman 1992, Lopes da Silva 1995) and thalamic network autonomy (Lopes da Silva 1991, 1995) has been critically questioned.

Another problem is that the pacemaker’s target system can be expected to respond most strongly when the target’s input frequency matches one its own resonant frequencies (Nunez 1995a). But, in this case, the distinction between target system and pacemaker is likely to be lost, eg, we can just as easily think of the target driving the pacemaker as the pacemaker driving the target. In one insists (as some physiologists do) of crediting all EEG to "pacemakers," the local theory presented here could be described in terms of many such "pacemakers" associated with many local networks (Fig 4). However, in a brain with dense thalamocortical, intracortical, and cortico-cortical connections, the dynamics of each "pacemaker" is likely to be altered by many other "pacemakers," so this label appears to provide a generally unsatisfactory description of dynamic brain behavior. If a genuine autonomous pacemaker of a particular brain rhythm were actually to be verified for a small tissue mass, the arguments put forward here strongly discourage unsubstantiated extrapolation of this mechanism to other EEG phenomena.

5.3.8. Frequency estimates, instability, and quasi-linear theory. Numerical frequency estimates applied to the non-dimensional frequency plotted in Fig 5 are based partly on the assumption that the local and global control parameters B and Q are close to values for which the linear theory predicts instability. This choice of parameters is consistent with weakly damped oscillations, which may be physiologically unrealistic. However, larger control parameters do not necessarily lead to instability (eg, epilepsy), only breakdown of the linear theory. In order to account approximately for nonlinearity, action potential density in a neural mass was assumed to be a sigmoid function of synaptic input (Nunez 1995a), as suggested by Wilson and Cowan (1973) and Freeman (1975). The sigmoid function describes a system where increased excitatory synaptic input at low levels causes a steep rise in the number of output action potentials, but only small increases in output at high excitatory input. This could occur at very high excitatory synaptic input due to saturation of the neural mass, perhaps in epilepsy. But, it could also be part of normal brain operation, occurring at much lower input levels due to recruitment of additional local cortical or thalamic inhibitory feedback not included in the linear theory. This assumption leads to prediction by the purely global theory that EEG consists of a linear combination limit cycle or perhaps chaotic (in time, but not space) global modes as shown in the Appendices of this article and the Nunez (1995a) book.

Limit cycle behavior is well established in many fields, eg, mathematics, physical science and ecology. Limit cycles (or "self excited oscillations") occur when damping functions change sign at large amplitudes, typically due to selective external energy input in physical systems (eg, electric circuits). The global limit cycle modes discussed in the Appendix can coexist with local temporal chaos or local limit cycles, eg, those predicted by Wilson and Cowan (1973). The main conclusion of this very crude, semi-quantitative theory is that quasi-linear approaches to global theory are fully consistent with the idea of complementary local and global neocortical dynamics, as shown in the Appendix.

The following equation for limit cycle frequencies f_nm (Hz) corresponding to spatial modes (n,m) provides a semi-quantitative idea of the general predictions of the global theory:

f_nm = (v/2p R) [S_nm – b_nml ²R²]^1/2 (8)

Here v (cm/sec) and l ^-1(cm) are characteristic action potential speed and fall-off distance in the density of cortico-cortical fibers, respectively. R(cm) is a linear scale factor for the model neocortical system, eg, circumference/2p for a closed loop, radius of a sphere, or semi-major axis of a prolate spheroid. Surface shape determines the indices (or "quantum numbers") S_nm. The case of extreme anisotropic cortico-cortical fibers (eg, only in anterior-posterior directions in each hemisphere) may be crudely represented by a one-dimensional closed loop of cortex. For the closed loop, S_nm = n², where n=1,2,3, ...independent of m. For the sphere with isotropic, homogeneous cortoco-cortical fibers, S_nm = n(n+1), where the index n numbers the fundamental (n=1) and overtone frequencies. The prolate spheroid and other more complicated surfaces or anisotropic or inhomogeneous fiber systems are expected to have limit cycle overtones that depend on both (n,m) indices. The mode dependent parameters b_nm are related to the global control parameter B in Fig 3; they are generally expected to increase as the strength of cortico-cortical feedback is increased. If these parameters are mode independent, b_nm = (B-1), as shown in the Appendix. Limit cycles are predicted in this crude quasi-linear approximation for modes with b_nm greater than one, but sufficiently small so the term in the square root in Eq (8) is positive.

It is emphasized that the exact form of Eq(8) cannot be taken too seriously. It is derived in the Appendix only for the one-dimensional loop with mode coupling assumed to be negligible. However, in non-linear systems, spatial modes (n,m) generally interact with each other to complicate dynamic behavior, although there is evidence in some physical systems that boundary conditions can force more regular behavior (eg, Bishop et al 1983; Nunez and Srinivasan 1993; Srinivasan and Nunez 1993). But, in this crude estimate, each limit cycle mode oscillates independently, and synaptic fields are linear combinations these spatial modes. Generally, this approximation is not valid and mode coupling is expected. However, for values of b_nm greater than but close to one, Eq(8) can perhaps provide some general ideas of how global oscillations should behave. The reason for optimism about qualitative and perhaps semi-quantitative predictions is that Eq(8) originates from genuine physiology, not metaphor. If this approximate picture appears plausible, it should motivate more accurate studies of both physiology and mathematical approximations by talented scientists in the context of the global theory. In Sec 6, Eq(8) is used to make some predictions of global EEG phenomena that appear to be relatively robust, ie, somewhat independent of simplifying assumptions used to derive them.

In order to provide more quantitative demonstration of these ideas, non-linear versions of the global and local/global theories, which include approximations to mode coupling, are derived in the Appendix. Figure 6 shows two numerical solutions to the purely global equations. The upper plot consists of the sum of five spatial modes in a rectangular medium of length three times the width. This geometry was chosen mainly for mathematical simplicity, but does have some rough correspondence to the geometry of a single hemisphere as discussed in the Appendix and in Nunez (1995a). The upper plot crudely simulates dura potential. The lower plot originates with the same solution, but includes only the sum of the two lowest spatial modes or "order parameters" [F ₁(t) + F ₂(t), defined in the Appendix]. That is, the volume conductor is assumed to spatially filter higher modes between brain and scalp as indicated by Eq (3). Thus, the lower plot crudely simulates the corresponding scalp potential. But, because the underlying dynamics associates higher temporal frequencies with higher spatial frequencies (in a manner similar to linear waves satisfying a dispersion relation), spatial filtering causes the temporal filtering evident in the lower plot. Genuine properties of the volume conductor were not used here to construct the spatial filter; however the qualitative behavior shown here is probably valid as discussed on pages 57-63, 79-81, and 383-387 of Nunez (1995a). The time axis is normalized with respect to the parameter t _g, which appears to be roughly in the 10ms range. In this case, the period shown is about one second and the dominant frequency is near 11 Hz. These oscillations are due exclusively to global delays.

Example solutions of non-linear local/global equations derived in the Appendix are shown in Fig 7. Local frequencies [q = q(Q), perhaps determined by local PSP rise and decay times and local feedback gain Q] progressively increase down the page. With t _g equal to 10ms, the period shown is 2 seconds and dominant frequencies vary from about 10 to 18 Hz as q increases. These four example oscillations are due to a combination of fixed global and variable local delays.

However, in some experiments, physiological parameters maybe estimated with one set of experiments, and these parameter estimates used in the equations to predict outcomes

6. Experimental connections to theory

The truism "any theory must ultimately stand or fall based on experiments" provides a necessary, but insufficient paradigm for neurodynamic theory verification. First, no rational neuroscientist is likely to is likely to suggest that any current brain theory approaches the quantitative accuricy and general applicability of modern physical theories. Rather one may hope to develop identify crude theory that approximately describes EEG observed in one or (hopefully) several brain states. Second, it is often very difficult to interpret experimental results EEG in the context of any particular theory, especially if such since very few EEG experiments were designed to test the theory. Very few EEG studies were so designed. Thus, questions of experimental verification of our the proposed local/global theory often do not often have simple yes or no answers.

For example, local EEG theories typically contain unknown control parameters. Several neuroscientists continue the struggle to connect such theory more closely to genuine physiology. This work should be encouraged, but several barriers must be overcome, including estimating influences of multiple neurotransmitters on control parameters, eg, local feedback gains. Furthermore, when purely local theories are tested (eg, with intracranial electrodes), the influence of global dynamics acting (top down) on local systems is nearly always unknown. Similarly, when global theory is tested, we cannot be sure of the importance of local effects (eg, EEG "pacemakers"). However, we appear to know much more about the physiological parameters of the purely global theory. Thus, if we are able to identify brain states in which local effects are minimal, qualitative and semi-quantitative predictions of the purely global theory can be achieved.

A critic may rightly complain that this approach leaves the global theoretician with an easy way to avoid falsification of global theory. Any data found to be inconsistent with global theory can always be blamed on local effects not included in the theory. This criticism is valid; I know of no fully satisfactory way to overcome this problem in the near future. This is also a common problem with experimental verification in complex physical systems. The world’s weather system comes to mind in this context. Dynamic theories of such complex fluid systems are typically successful only in limited contexts, eg, in tightly controlled experiments designed to eliminate confounding influences on simplified theories.

I suggest that success of the global theory depends on its ability to predict outcomes of an expanding class of disparate experiments involving coherent EEG phenomena. But, in absence of such theory, it is unlikely that such experiments will ever be carried out. By "successful" I mean "useful" and partly valid for some brain states. But even such relatively modest success should provide substantial influence on neuroscience. Thus, the the following summary of several dozen apparently correct semi-quantitative experimental predictions of the local/global theory are offered, together with conjectures on the possible contributions of local and regional networks. More detailed discussions of these connections may be obtained from Nunez (1995a).

Human cortico-cortical propagation speeds (v) are distributed with distribution apparently peaked in the 6 to 9 m/sec range (Katznelson 1981; Nunez 1995a). The scale factor (R) for human neocortex can be estimated from its surface area. Inter-hemispheric fiber connections (mostly callosal fibers) are much fewer than intra-hemispheric connections, suggesting that a single hemisphere may be the more appropriate wave medium. On the other hand, a hemisphere is elongated, somewhat like a prolate spheroid for which R is the semi-major axis. These two effects tend to move R estimates in opposite directions. But, a conservative surface area range of 1000 to 3000 cm² yields R = 9 to 15 cm. Thus, the term (v/2p R) in Eq (8) has a magnitude in the range 6 to 16 Hz.

For cortico-cortical fibers that span large portions of the model cortex, l R is of the order of one. Neuromodulation (state-dependent) and cortico-cortical fiber density (fixed) determine the parameters b_nm. For example, if these parameters were mode-independent, all b_nm would equal (B-1) in the one-dimensional closed loop geometry, as shown in the Appendix. This feedback gain parameter B can be crudely estimated from physiology (Nunez 1995a), but has a large error range with the state of current knowledge. Each b_nm parameter in Eq (8) must be greater than one for the corresponding limit cycle to occur, but not so large that the quasi-linear approximations are invalid. An example of waves on a prolate spheroid surface of small eccentricity illustrates the main ideas. The surface shape determines the indices S_nm = n(n+1) + a m², where a depends on the eccentricity of the prolate spheroid, and the indices run n = 1,2,3,…; m = -n,+n. Suppose l R = 1, (v/2p R) = 11 Hz, a = 0.3, and b_nm = 1.3 for all (n, m). The two lowest limit cycle frequencies (n = 1, m = 0,1) are approximately 9 and 11 Hz. If neuromodulatory input causes the b_nm to increase to 1.8, frequencies of these modes decrease to 5 and 8 Hz, respectively. For b_nm greater than 2.0 but less 2.3, than the mode (n = 1, m = 0) becomes non-oscillatory, and the mode (n = 1, m = 1) occurs in the delta frequency range. Still larger b_nm values cause higher overtones to decrease in frequency.

The b_nm par