On the Problem of Anomalous Dispersion in Chaoto-Chaotic Phase Transitions of Neural Masses, and Its Significance for the Manag

Walter J. Freeman Journal Article e-Reprint

On the Problem of Anomalous Dispersion in Chaoto-Chaotic Phase Transitions of Neural Masses, and Its Significance for the Management of Perceptual Information in Brains

W.J. Freeman

University of California, Berkeley, CA 94720, USA

Reprinted with permission from, Synergetics or Cognition, Springer Series in Synergetics, Vol. 45 , Springer-Verlag Berlin, Heidelberg 1990, Editor. H. Haken and M. Stadler

There is recent recognition that gamma oscillations In the visual cortical EEG may reveal a phase locking mechanism for the assembly of visual pattern information in the cortex. This is plausible in homology to a similar mechanism known to operate in the paleocortex of the olfactory system. Among the major problems to be solved in exploiting this breakthrough involve determinations of the basic form of the oscillations as the emergent properties of cooperative neural masses in visual cortex; the kinds of neuronal feedback that give rise to the gamma oscillations; the mechanisms for registration of geniculate input into the oscillations; the neural mechanisms required for readout; and the essential role played by periodic or chaotic oscillations in neocortical information management. Answers to these problems in the olfactory system are put forward for possible relevance to the still unsolved problems in visual cortical physiology.

1. INTRODUCTION

For over three decades visual physiologists have been immersed in the unit paradigm with its attendant grandmother cell hypothesis, exploiting in full measure the technological breakthroughs in recording axonal discharges of single cortical neurons that were opened by the work of Jung [1], Hubel & Wiesel [2], and others. Despite on-going studies in visual cortical evoked potentials and EEGs by psychologists and clinicians, that were founded on classical studies by Bishop and Clare [3] and reviewed by Mitzdorf [4], scant attention was paid to dendritic fields of electrical current. Such has been the disregard and mistrust directed at so-called "local field potentials" by the hundreds of workers dedicated to "unit spikes" that, until lately, not one among them has opened the pass band of his amplifiers to observe low frequency components, despite repeated urgings of them to do so [5-9].

Now a new technology has emerged. that is based on computerized data processing and the body of theory provided through synergetics, nonlinear dynamics and the concepts of chaos. By this paradigm new observations have been made, that link oscillations in dendritic potential to visual conditioned stimuli in the waking monkey by Freeman and van Dijk [101, and to unit activity induced by shining bars of light onto the retina in the anesthetized cat [11-131. This discovery brings visual cortical physiology to a threshold. that was crossed in olfactory electrophysiology some thirty years ago.

From the perspective afforded by this long experience with neuronal oscillatory phenomena, five key questions are raised by these now data, which must be answered as quickly as possible in order to bring the new paradigm up to speed. These questions concern the fundamental nature of the oscillations. whether periodic or chaotic; the nature of the neuronal feedback that gives rise to them; the means by which incoming information is encoded in them, and by which it is stored and retrieved; the mechanisms by which it is read out of the visual cortex; and the functional role that oscillations play in physiological information processing.

The purpose of my presentation is to offer some hypotheses on possible answers to these questions. that are based on my experience in observing olfactory and visual neuronal fields of potential.

2. WHAT IS THE BASIC NATURE OF THE CORTICAL OSCILLATIONS?

The first question to be answered is whether the oscillations are the property of single neurons acting as "pacemakers", which are entrained or "coupled" to yield the observed widespread phase looking. or they arise as local mean field potentials by virtue of feedback interactions in pools of neurons that in their normal functional dynamic range are intrinsically nonoscillatory. By this is meant that in the absence of interaction with other neurons, each neuron on Impulse driving generates a postsynaptic potential that decays monotonically to baseline. Experimentally this is found under deep anesthesia.

A univocal answer comes from interval histograms of "spontaneous" units from cortical neurons. These typically reveal the exponential curve of the Poisson process, except for the refractory period; that is, the pulse train is aperiodic, and, units the only means by which each neuron communicates its immediate states to others, the neuron cannot be viewed as an oscillator that can entrain others to itself. Moreover, the firing rate of single cells is much lower than the gamma frequencies observed in oscillatory EEGs (20-100 Hz).

It follows that the oscillations are an emergent property of interactive non-oscillatory neurons. Tens of thousands must cooperate in each local neighborhood to provide the density of pulses needed. The cortical neuropil being a continuum without segregation of nests of neurons, it is reasonable to expect that the cooperative domains extend in spatial continua out to distances not yet known. Our data from the cat and rabbit show that the entire bulb acts as an element, 100-200 sq. mm. in area, and our data from the monkey suggest that the visual cortical cooperative domain was larger than the electrode array window that we used (20 mm x 40 mm).

Importantly, in the bulb of the rabbit we have proven that the density of the odor-specific information is spatially uniform, in much the way that the information density is uniform in a hologram or on a TV screen. The participation of neurons is expressed in the amplitude of their wave (Figure 1) and unit firing modulation at the common instantaneous frequency. The fraction of the variance in the activity of each neuron that is covariant with the whole Is 1 part In 1,000 - 10,000 in the olfactory system. The recent data on hand from the visual cortex indicate that that fraction may be far higher for many neurons, which will greatly facilitate the study of neocortical oscillations, owing to their accessibility with ease by spiketriggered time averaging of EEGs. This procedure Is not usually so robust in the olfactory system.

Oscillations are most easily dealt with when they are periodic. The classic near-sinusoidal BEG burst In olfaction has for years given us aid and comfort, being so amenable to Fourier and linear analysis, and It appears that Mother Nature has smiled on us once again in providing bursts of near periodic waves in the visual cortex. It is altogether easy to be seduced into discarding irregular waveforms as non representative, when, indeed, we must look to the general case and learn to deal with aperiodic, chaotic waveforms (Figure 2). We have concluded that all oscillatory activity in the olfactory system is chaotic; and the 1/f type spectra we have obtained from our visual cortical EEGs support this postulate there as well (Figure 3). Hence, the rapidity of decay of spike triggered averaged EEG waveforms speaks not so much to a weakness of coupling as to the variability of the carrier frequency.

Figure 1. Contour plots show values of the average factor loadings Of the first principal component of the visual cortical EEGs, that were recorded simultaneously from 16 electrodes over the left cortical surface. The electrodes were chronically implanted at intervals averaging 0.6 cm forming a "window" about 1.4x2.4cm. Channel 8 is closest to the foveal projection. The left edge faces the midline; the upper edge faces anteriorly. Dashed contours are negative; solid contours are positive. Before 400 msec the monkey perceives a conditioned stimulus; In the next 200 msec it looks to Its reward, and after 600 msec It looks for the next stimulus. The EEG spatial patterns change accordingly, indicating that the cortical Information Is related to perception and not to sensation. From [10].

Measurement of the duration of oscillatory bursts is easy in the olfactory system owing to respiration, but in visual cortex it poses a vexing problem. In our studies of the monkey we found that a useful index was the ratio of the variance of the spatial ensemble average waveform over a time segment of 50 msec (corresponding to a frequency of 20 Hz or a half-cycle at 10 Hz) to the average variance of amplitude over the array. This measure is based on the promise that within a burst there is common to all electrodes a carrier waveform that has a high peak-to-peak amplitude, but that between bursts there is low carrier amplitude and high spatial variance. Although the index works empirically with a high-pass filter set at 20 Hz (Figure 4). it varies unpredictably for other filter settings. Unlike the case for olfaction. we do not know the temporal or spatial spectral ranges that carry behavioral information in oscillations, nor do we know the statistics of the durations of event-related EEG segments and their behavioral correlates, e.g. microsaccades, Dietrich Lehmann's [14] global dipole, the alpha rhythm, etc.

Figure 2. Sixteen traces were simultaneously recorded for a period of 1,500 msec, during the performance of a conditioned response by a rhesus monkey. The common aperiodic waveforms were extracted by principal components analysis. The unaveraged traces best revealed the chaotic oscillations in the gamma range after high pass filtering to remove all activity below 20 Hz. From [10].

These data support the concept that the oscillations manifest what we have called a "wave packet" [71, which is a macroscopic neural activity pattern covering several sq. cm. (Figure 1). and lasting on the order of 0.1 to 0.2 sec. The failure of any given unit to show evidence for oscillation in its output cannot be used as evidence that it is not part of a cooperative. any more than the absence of silver grains in a pixel of a photograph can be said to show that the pixel does not belong in the picture. Both black and white are needed, that Is, some neurons that fire and others that do not.

3. WHAT IS THE NEURAL FEEDBACK MECHANISM OF THE OSCILLATIONS?

Given the premise that oscillation is an emergent property, it follows that interaction exists among excitatory and inhibitory neurons giving rise to negative feedback. Elementary feedback theory requires that two populations of neural discharge coexist in the mass, that have common complex frequencies and that differ in phase on the average by one quarter cycle of the common Instantaneous frequency. This is because each peak of activity of the excitatory limb must precede each peak of activity in the inhibitory limb by a time unit corresponding to the dominant time constant of the neurons (typically the passive membrane time constant). The key to this problem lies in understanding the phase relationship between unit activity and dendritic field potentials. We have proven that, for laminar cortex in which the neurons that generate the field are aligned perpendicular to the surface, the EEG oscillatory wave and the spike-triggered averaged EEG are in phase with the pulse probability wave (Ch. 4 In [7]), provided that surface-negative implies excitation (EPSPs) of the generating cells. This condition appears to hold for all of the visual cortical neurons reported In the recent literature on oscillations.

Figure 3. At left is shown the spatial ensemble average of the 16 time series prior to high pass filtering. In the center Is seen the natural logarithm of the power spectrum. At right the spectrum is shown after subtraction of the 1/f trend line computed over the range of 1 -75 Hz. One or more peaks of power were always seen in the gamma range between 25 and 50 Hz but with variation in frequencies between successive time segments. From [10].

No evidence has thus far surfaced concerning the requisite inhibitory neurons and their properties in visual cortex. The quest may be far from simple. In the prepyriform cortex, for example. the pyramidal cells are numerous and large, and they generate the In-phase EEG [71, whereas the inhibitory interneurons are small, difficult to find and hold, and proven only by their quarter-cycle phase lag (Figure 5). In the olfactory bulb the inhibitory neurons lack axons and generate no extracellularly detectable spikes, but they do generate the EEG [71, so the excitatory neuron discharge leads the EEG by a quartercycle (Figure 6). In the hippocampus the EEG and the evoked potential both lead the excitatory discharge by a quarter-cycle [151, under the convention of "positive upward" (Figure 7). It may be that other relations will be found in visual cortex.

Clearly visual cortical physiologists have their homework to do in determining which cells are excitatory, which inhibitory, which generate pulses, which generate open (dipole) fields and which have closed (monopole) fields, which polarity means excitation and which means inhibition. This assignment of state variables to component neurons is just the beginning. Thereafter come measurement of the open loop time and space constants. determination of the connectivities of the feedback loops, and estimation of the feedback strengths from measurement of the closed loop rate constants. Then comes the evaluation of transmitter chemistries and the evaluation of changes in synaptic gains with associative learning. There is sufficient work here to occupy a substantial number of theses and careers [7, 16].

4. HOW IS SENSORY INFORMATION ENCODED IN OSCILLATIONS?

This is a trick question, because the most honest answer Is that it Is not. The sensory information is replaced by perceptual information that is freshly created in the cortex from the residues of past and present input.

Most cortical physiologists will continue to believe until they die that cortical percepts are assembled from visual primitives that are expressed in the discharge of point, edge and bar "feature detectors" in visual cortex, perhaps in much the way that a cartoonist thinks he constructs an image of a face from lines, curves and shaded surfaces. While undeniably these raw sensory data are injected by afferent axons into the cortex and are essential for the formation of each percept, we believe that their role is not to serve as building blocks, but rather as instructions that suffice to place a cortical mechanism into an appropriate basin of attraction, so that it might rapidly evolve a structured spatial pattern of cortical activity that supplants the raw sensory data with a percept.

Figure 4. (LEFT) A high pass filter at the designated frequency, f, was applied to the 16 time series. Then a moving mean square value, V(t), was computed along the spatial ensemble average with a window of half the cycle duration of the filter frequency, 1/2f. At each time step, t, (4.15 msec) the value V(t) was divided by the sum of squares of the 16 values v(t) of potential to give a ratio R of the time variance to the spatial variance. The ratio R was used as an empirical index to locate "bursts" of EEG, which were defined as episodes of sustained high temporal variance and low spatial variance. The index proved to be sensitive to the value of f in unpredictable ways.

Figure 5. (RIGHT) In the prepyriform cortex the units of the excitatory feedforward neurons (type A) fire in synchrony with the local field potential oscillation, whereas the inhibitory feedback neurons (type B) show oscillation at the same frequency and decay rate but with a quarter cycle phase lag. The solid curves are averaged evoked potentials from impulse electrical stimulation of the cortex, and the dots show the poststimulus time histogram of the units. From [7].

In our view the "feature detector" story is an anthropocentric teleological overlay of interpretation that fails to address the proper place of the pre-processing operations in the visual system. The complex log topographic projections from the retina through the geniculocalcarine radiations are requisite largely for the extraction of LOCAL gradients, that is. the temporal and spatial derivatives of retinal Images, leading in the main to contrast enhancement in its many forms. Contrariwise, the extraction of "figure" from "ground", which is the essence of perception, and which depends on past experience and present expectation, is a GLOBAL integrative process over very large numbers of responding cortical neurons that carry equivalent and largely redundant sensory information.

This mechanism in vision remains almost entirely unknown. From our experience in olfaction we suggest that in visual cortex it involves a state transition, that constitutes a jump in cortical dynamic state constituting a type of bifurcation. The model requires that cortex be intrinsically unstable and liable to sudden transitions under the appropriate stimuli. The conditions that facilitate controlled instability include a high level of cortical activity and of excitability, which is achieved under the neurochemical states of behavioral arousal and motivation; synaptic facilitation such as is postulated to occur in cortical synapses during learning under reinforcement (one type of which is the Hebb synapse in its many forms); and an input-dependent nonlinear gain (Figure 8), which was first discovered in the olfactory bulb [17-19], and which has since been demonstrated in other parts of the olfactory system [20], and in visual cortex [12]. In a suitably aroused animal that expects a certain stimulus, the arrival of the stimulus sought can induce neural activity that serves as a bifurcation parameter (akin to a temperature increase) and also as specific Information that serves to guide the mechanism into the expression of a global pattern. Thus a pattern is created rather than retrieved. and a classification can occur without matching, completion, correlation, or other cumbersome computational stratagems of the artificial world.

Given that the olfactory bulb (and the visual cortex) can respond discriminatively to a vast array of identifiable stimuli, and flexibly so. we at first supposed that the neurodynamic system might maintain a separate near-limit cycle or chaotic attractor for each discriminable stimulus. This would require that each state transition involve either a multiple-type bifurcation from the basin of one chaotic attractor to another, or the warping and re-shaping of the dynamic landscape so as to re-create an attractor through each delivery of input. Our current view is that a global chaotic attractor exists at all times with multiple wings, one for each learned stimulus, and that the role of a stimulus is to restrict the state within a wing of the attractor. which leads to its expression by a spatial pattern of amplitude of the common chaotic waveform of the oscillation [21 , 22]. We have had some success in elaborating this as a model for pattern recognition with a chaotic generator In software simulations [23].

Figure 6. In the olfactory bulb the excitatory feedforward neurons generate units, shown here in a poststimulus time histogram (above), but make little or no contributions to the EEG. The inhibitory feedback neurons have no axons or action potentials, but they do generate powerful field potentials shown here as an averaged evoked potential (below). The experimental values (plotting symbols) were were from simultaneous unit and EEG recordings. They were fitted with an equation for a damped cosine having the same frequency for both data sets. In accordance with theory, the peaks of unit activity lead the peaks of EEG activity by a quarter cycle on the average. From [25].

Figure 7. In the hippocampus the peaks of unit activity precede the peaks of positive field potential by a quarter cycle. The time course of the poststimulus time histogram is proportional to the time first derivative of the averaged evoked potential. The shaded areas indicate the time segments in which the level of evoked unit activity exceeds the mean background level. For all of the units the phase relationship of the oscillation in relative firing frequency to that of the field potential is invariant over changes in response frequencies, that are induced by changes in the stimulus intensity. From [15].

5. WHAT IS A POSSIBLE MECHANISM FOR READOUT OF OSCILLATIONS?

The major problem for such a mechanism to solve is that of Interfacing between Input and output connections. On the one hand. visual information Is registered into the primary cortex In a dozen or more parcellated areas relating to color. contours, motion, binocularity, etc., some of them topographically organized with respect to the retina, others not. On the other hand information on visual patterns must be sent on to numerous parts of the brain, Including the lateral geniculate, Pulvinar, superior colliculus, the inferotemporal cortex, frontal eye fields, and the frontal pole. Some projections such as the callosal appear to be topographically organized, but this aspect is not so well documented. How has evolution solved the problem of connecting and switching within the cortex, so that the correct information is routed to where it Is needed within the tenth of a second allowed by a visual percept?

Figure 8. The static sigmoid nonlinearity that dominates the dynamics of the olfactory system has been experimentally evaluated from the dependence of the pulse probability conditional on the EEG amplitude [7,17-19]. The nonlinear gain is given by the first derivative with respect to amplitude. The equation has been derived from a statistical mechanical generalization of the Hodgkin-Huxley equations. Three examples illustrate the effects of increasing arousal and motivation on the gain. The maximal gain is to the excitatory side of rest, so that gain is increased by sensory input. This destabilizes the cortex and can cause bifurcation to occur, especially when the synaptic interconnections of the neurons receiving the Input have been strengthened by learning. From [17].

An answer suggested by olfactory studies Is that during the high density neuronal interactions underlying burst formation, the information contributed by every "feature detector" is disseminated over the entire interactive domain. Thereafter It is transmitted along with all other perceptual information to every target of the cortical projections. The selection of specific aspects of the global information content is presumably done In each of the several targets by their local synaptic modifications with associative learning [7, 19, 22, 23].

This answer is based on the experimental finding that the information density in the bulb Is spatially homogeneous, and on analogy with the hologram, each part of which contains an entire scene under Fourier transformation albeit at resolution proportional to its fraction of the total area. An experimental test in vision will be to repeat our experiments in classifying multichannel EEG segments with respect to visual conditioned stimuli, so as to determine whether any specific channels are more or less important for the classification than the average. We predict that none will be. It will likewise be essential to show that, when either the visual cues or their reinforcement contingencies are changed. the spatial patterns of the EEG change in their entirety, not merely in one pattern, or in parts of one pattern.

One further aspect of readout must be noted, which Is the nature of Its "vector", using the word in its biological sense meaning "carrier". Input is brought to the cortex in the form of action potentials on labeled lines; hierarchically the input vector is at the neuronal or microscopic level. Through the process of bifurcation the intracortical vector assumes the macroscopic form of local mean field activity, which is best accessed through extracellular recording of dendritic potentials. Readout Is by a process of extraction whereby the local mean field intensity Is sampled and transmitted by axons using pulse frequency modulation. The output vector thereby is of the same microscopic character as the input vector, and the problems of management of that vector as Input for the several targets of the bulb and cortex can be treated in the same way as those of the olfactory bulb and primary visual cortex in managing their input vectors.

6. WHAT ARE THE OSCILLATIONS FOR?

Whereas the information density is spatially uniform, the appearance of the cooperative activity, that is, the time function of the local mean field intensity, most certainly is not. On the average the variance in the common carrier wave form does not exceed 50% of the total variance of bulbar EEG activity nor 65% of the visual. At the neuronal level, as already noted, the fraction of the variance carrying the output "signal" Is exceedingly small. owing to the high degree of independence of cortical units. The question is, how might the targets extract that small "signal" from the "noisy" carrier?

An answer proposed from olfactory studies is that the extraction is done by spatial ensemble averaging in real time [21]. Each receiving neuron in the target accepts synaptic input from axons originating in many parts of the primary visual cortex, by virtue of divergence in the projection pathways [7]. Under real time summation the only activity that will be enhanced is that having a common instantaneous frequency, which is, in fact, the cooperative activity and therefore the signal. Activity at all other frequencies not shared across the cooperative domain tends to average to zero. However, this mechanism for "laundering" the "signal" can only work if the carrier is oscillatory, irrespective of whether it is periodic or chaotic.

This model for readout imposes another severe constraint on the cortical mechanism, which is that the time dispersion of the carrier wave, when expressed as a phase dispersion at the dominant frequency of oscillation, cannot much exceed a quarter cycle without degradation of the "signal" under global spatial integration. The question is, how is a common oscillatory pattern established over a large area of cortex without significant phase dispersion?

An answer, again provided from olfaction, comes through our experimental measurement of the phase patterns in space from an array of 64 electrodes placed on the bulb. In brief, the isocontours of phase for each burst are concentric circles (Figure 9) forming a cone, showing that the phase pattern resembles that of waves in water from a dropped pebble [24]. The location of the apex of the cone is a random variable unrelated to stimuli, and so also is its sign (randomly alternating maximal lead or lag), demonstrating that a "pacemaker" explanation for the phase is unsatisfactory. The phase gradient is a function of frequency, such that, when expressed as a velocity, its mean fits the estimated conduction velocity of small axonal branches that run through the depth of the bulb parallel to the surface.

These data suggest that the cortical mantle can be likened to a relativistic medium, in which there is a limit on the rate of transmission of information within it. The problem here is that the velocity is too high, because the small axon branches on the average extend no more than a tenth of the distance across the bulb. When synaptic delays are accounted for (Figure 10), then the velocity of wave transmission is far less than the values obtained by measurements of EEG bursts (251. and it is not compatible with the psychophysical constraints on the speed of perceptual processing. Indeed, our experimental data on wave spread allow only a few tenths of a millimeter for each half cycle of the oscillation (Figure 11), and we estimate [6, 7] that dissemination across the bulb would require many cycles of oscillation, far exceeding psychophysical estimates of the time durations sufficing for perception in olfaction.

We re-affirm that transmission across the cortex is by axonal propagation and synaptic communication, but that the velocity of state transitions may greatly exceed local wave transmission. Our hypothesis for the bulb is that the phase gradient of a burst is formed at the moment of bifurcation under sensory Influx, that the location of the apex of the cone demarcates a site of nucleation, and that the pattern is frozen into the burst for Its duration. From this perspective it appears that the group velocity of the state transition far exceeds the wave velocity of synaptic propagation. The physiological mechanism for this anomalous dispersion is not known. We postulate that perhaps a small number of long-distance collaterals from tufted cells in the bulb suffice to trigger bifurcations, when the chaotic generator has been carried by Input close to the threshold for a transition.

The problem Is of course compounded for the visual cortex, In which the surface area of a cooperative domain is at least an order of magnitude larger. But here lies a mechanism readily at hand, which is provided by the deep layers of neocortex. The outer three layers of neocortex are similar to the three layers of paleocortex. but the Inner three layers added on by neuronal migration from the basal ganglia In the embryos of mammals, provides for long distance transmission at high velocities. Briefly the functional significance of the six-layered cortex may be precisely to provide for high velocity spread of state transitions, so as to establish an oscillatory common carrier wave with minimal phase dispersion, allowing real time spatial integration and the greater enhancement of the cortical signal:noise ratios. It is unlikely, however, that simple conical phase gradients will be found in neocortical EEGs, owing to the spotty character of corticocortical projections, resembling a Levy distribution more than the Gaussian.

Figure 9. An example is shown of the locations of phase maxima (solid dots) and minima (open dots) projected Into the flattened surface of the olfactory bulb of a rabbit. The 64 phase values from each burst, after low pass filtering of the real and imaginary parts from the amplitude and phase values of the cosine fitted to the 64 simultaneously recorded EEG traces, were fitted with a cone in spherical coordinates, recovering 65% of the variance on the average. The sketch is a projection of the outline of the bulb as it would appear on looking through the bulb onto the surface of the electrode array (the 4x4 mm square). A representative set of isophase contours Is shown at 0.25 radians. The locations of the apices of the cone are shown in polar coordinates as angle and azimuth from one pole of the sphere (2.5 mm in radius) at the center of the array to the antipode. The standard error of measurement (SEM) of each point is twice the radius of its plotting symbol. From [241.

The spatial patterns of phase in olfactory EEGs are crucial bits of evidence for the existence of macroscopic neural active states, and for those neurodynamic mechanisms by which they emerge. The study of oscillations in the visual cortex cannot be considered mature until spatial phase patterns of its EEGs have been identified and precisely measured. This task is difficult to accomplish, owing to the low amplitudes of the neocortical EEG (about 1/5 to 1/10 those of paleocortex owing to the poor alignment of the generating neurons in neocortex). to their less periodic appearance (in the awake monkey), and to the likelihood that multiple overlapping macroscopic states may coexist over the relatively vast domains of contigous neocortex. The challenge for visual physiology is clearly laid down by the high standards of measurement of phase and amplitude patterns (Figure 12), that have been painstakingly achieved for the paleocortical systems [7, 16 , 26].

Figure 10. The location is mapped of units in the bulb that were Influenced by impulse stimulation of the input tract at a focal point on the sheet. Initial excitation ranged from minimal + to maximal ++++, and initial inhibition ranged from barely detected - to very strong or no change, 0, at the indicated sites. The ellipses indicate the average extent of activation during El and of I1 respectively. The synaptic wave does not spread in a continuous manner like a wave in water, but with successive polarity reversals, each having a larger radius than the one before by a decreasing amount of change. From [25].

Figure 11. The phase interference pattern shows the predicted form of synaptic wave spread of activity in the excitatory neuron population of the olfactory bulb. starting with Impulse excitation of two synaptic nests called "glomeruli" separated by 900 microns. The pattern is superimposed on a histological section through the layer of glomeruli to show their mosaic appearance, each averaging 0.135 mm in diameter. The center circle of excitation during the first peak of oscillation El is surrounded by inhibition I1, and this in turn by the second peak of excitation E2, typically 20 msec later (at 50 Hz). By peak E5 the synaptic wave has spread 1.2 mm in 100 msec (.01 m/sec) and has covered about 1% of the bulbar surface area. This contrasts with the group velocity of 1.8 m/sec, which covers 100% of the area within 6 msec or one quarter cycle. From [7].

Figure 12. It is essential in working with oscillations to have accurate measurements of frequency, phase, and amplitude, and to have accurate estimates of standard errors of measurement (SEM) and their dependence on signal:noise ratios (SNR). The system developed for deriving phase and amplitude patterns from paleocortical EEGs was calibrated by fitting known cosines In added colored noise. The ordinates at the left show the standard deviations (SDs) of phase and amplitude of simulated data from their known 64 values. "Edited" refers to deleting values for which the correlation between the trace and the ensemble average was less than 0.2. "Filtered" refers to the use of low pass spatial filtering [241. A further test was to measure 100 to 300 BEG segments by fitting curves to them and expressing the SNR as the ratio of the variance of the fitted curve to the variance of the residuals. The right ordinate shows the cumulative % of segments reaching or exceeding the SNR on the abscissa. The curves A - D reflect improvements in the algorithms for fitting the curves to the EEG data, that is, for extracting the information and reducing the SEM to 0.1 SD of amplitude and 0.1 radian (6 degrees) of phase. From [26].

ACKNOWLEDGEMENT

The research was supported by a grant MH06686 from the National Institute of Mental Health, United States Public Health Service

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