Analysis of Spatial Patterns of Phase in Neocortical Gamma EEGs in Rabbit

Analysis of spatial pat- terns of phase in neocortical gamma EEGs in rabbit. J 84: 1266–1278, 2000. Arrays of 64 electrodes (8 3 8, 7 3 7 mm) were implanted epidurally on the surface of the visual, auditory or somatosensory cortex of rabbits trained to discriminate conditioned stimuli in the corresponding modality. The 64 electroencephalo- graphic (EEG) traces at all times displayed a high degree of spatial coherence in wave form, averaging . 90% of the variance in the largest principal components analysis component. The EEGs were decomposed with the fast Fourier transform (FFT) to give the spatial distributions of amplitude and phase modulation (AM and PM) in segments 128 ms in duration. Spatial (2-dimensional) and temporal (1-dimensional) ﬁlters were designed to optimize classiﬁcation of the spatial AM patterns in the gamma range (20–80 Hz) with respect to discriminative conditioned stimuli. No evidence was found for stim- ulus-dependent classiﬁcation of the spatial PM patterns. Instead some spatial PM distributions conformed to the pattern of a cone. The location and sign (maximal lead or lag) of the conic apex varied randomly with each recurrence. The slope of the phase gradient varied in a range corresponding to that of the conduction velocities reported of axons to extend parallel to the cortical surfaces. The durations and times of recurrence of the phase cones corresponded to those of the optimally classiﬁed spatial AM patterns. The interpretation is ad- vanced that the phase cones are manifestations of state transitions in the mesoscopic dynamics of sensory cortices by which the intermit- tent AM patterns are formed. The phase cones show that the gamma EEG spatial coherence is not due to volume conduction from a single deep-lying dipole generator nor to activity at the site of the reference lead on monopolar recording. The random variation of the apical sign shows that gamma AM patterns are self-organized and are not imposed by thalamic pacemakers. The half-power radius of the phase gradient provides a useful measure of the soft boundary condition for the formation and read-out of cooperative cortical domains responsi- ble for binding sensory information into the context of prior experience in the process of perception. troids and the Euclidean distance was measured in 64 space from the two centroids to each of the patterns remaining in the second two subgroups (A 2 , B 2 ). If a patterns was closer to its centroid than to the opposite centroid (i.e., a pattern from subgroup A 2 was closer to centroid A 1 than it was to centroid B 1 ), then it classiﬁed correctly. This classiﬁcation was repeated by calculating centroids from the second two subgroups and classifying the patterns from the ﬁrst subgroups. A binomial probability pattern was used to determine whether or not a certain level of classiﬁcation was signiﬁcant. This method yielded a time series of probability values showing the times when the CS 2 and CS 1 spatial AM patterns could or could not be separated.

Freeman, Walter J. and John M. Barrie. Analysis of spatial patterns of phase in neocortical gamma EEGs in rabbit. J Neurophysiol 84: 1266 -1278, 2000. Arrays of 64 electrodes (8 ϫ 8, 7 ϫ 7 mm) were implanted epidurally on the surface of the visual, auditory or somatosensory cortex of rabbits trained to discriminate conditioned stimuli in the corresponding modality. The 64 electroencephalographic (EEG) traces at all times displayed a high degree of spatial coherence in wave form, averaging Ͼ90% of the variance in the largest principal components analysis component. The EEGs were decomposed with the fast Fourier transform (FFT) to give the spatial distributions of amplitude and phase modulation (AM and PM) in segments 128 ms in duration. Spatial (2-dimensional) and temporal (1-dimensional) filters were designed to optimize classification of the spatial AM patterns in the gamma range (20 -80 Hz) with respect to discriminative conditioned stimuli. No evidence was found for stimulus-dependent classification of the spatial PM patterns. Instead some spatial PM distributions conformed to the pattern of a cone. The location and sign (maximal lead or lag) of the conic apex varied randomly with each recurrence. The slope of the phase gradient varied in a range corresponding to that of the conduction velocities reported of axons to extend parallel to the cortical surfaces. The durations and times of recurrence of the phase cones corresponded to those of the optimally classified spatial AM patterns. The interpretation is advanced that the phase cones are manifestations of state transitions in the mesoscopic dynamics of sensory cortices by which the intermittent AM patterns are formed. The phase cones show that the gamma EEG spatial coherence is not due to volume conduction from a single deep-lying dipole generator nor to activity at the site of the reference lead on monopolar recording. The random variation of the apical sign shows that gamma AM patterns are self-organized and are not imposed by thalamic pacemakers. The half-power radius of the phase gradient provides a useful measure of the soft boundary condition for the formation and read-out of cooperative cortical domains responsible for binding sensory information into the context of prior experience in the process of perception.

I N T R O D U C T I O N
Five premises are adopted here for analysis of the spatiotemporal integration of neural activity within and between areas of neocortex. First, neurons communicate with each other over short distances by ionic diffusion of synaptic neurotransmitters/modulators and over long distances only by propagated action potentials that incur delays. This premise disallows appeals to instantaneous communication by ephaptic linkages through fields of electrical current (Bullock 1997;Bullock et al. 1995;Prechtl et al. 2000;Terzuolo and Bullock 1956), electromagnetic resonance (Adey 1969(Adey , 1993, or quantum coherence (Hameroff 1998). Second, communication between parts of neurons is predominantly by ionic loop current, which is governed by the cable equation and which, like diffusion, imposes attenuation of amplitude with distance (Hodgkin and Rushton 1946). Local sources of energy for the action potential surmount this attenuation but at the cost of distance-dependent delay. Therefore neurons that are distributed in a network or mass and that fire simultaneously can achieve simultaneity in the arrival times of their action potentials at selected points in the network only under severe constraints. Third, spatiotemporal integration of the activity of afferent axons by the cablelike dendrites of receiving neurons requires some degree of synchrony in the multiple inputs. This means that some component of the activity of each of the transmitting axon terminals must be at or near zero time lag with respect to the activity of all. The span for "zero lag" is defined here as the modal duration of an action potential, ϳ1 ms, and "near" as the duration of a passive membrane time constant, about Ϯ5 ms.
Fourth, synchronization is essential for integration of the activity of "feature detector" neurons in the process of perception, whereby the action potentials of relay neurons that are activated by sensory neurons having receptor fields for features of a stimulus are combined to form a representation of the entire stimulus. How much synchrony is required and how is it achieved in the face of axonal delays? This constitutes what has been called the "binding problem" (Hardcastle 1994;Milner 1974;Schillen and König 1994;Singer 1993;von der Malsburg 1983). When the problem is formulated in this way, the description of the neural mechanism requires measurement of that fraction of the variance of the activity of individual neurons or local ensembles that is covariant over the whole network or mass. The solution also requires an explanation of how that fraction of the total variance that is covariant over the ensemble can constitute or carry a neural signal and a description of a neural mechanism by which that covariant fraction can be extracted by the target neurons receiving spatial activity distributions that have been transmitted by axons with obligatory propagation delays.
The fifth premise is that the activity of cortical neurons is synchronized by their synaptic interaction in closed loops. Ana-tomical evidence shows that each neuron forms synapses on thousands of others within its axonal arbor and receives axon terminals from thousands of other neurons within its dendritic arbor, but cell counts indicate that each neuron connects only with on the order of 1% of the neurons that lie within its reach. Anatomists such as Sholl (1956) and Braitenberg and Schüz (1991) have concluded that the likelihood of a feedback connection between any designated pair of neurons may be as low as one in a million. If the connections of most neurons are as sparse as these estimates indicate and if the impact of single action potentials is as minute with respect to the thresholds for neural firing of recipient neurons as estimated by biophysicists (Amit 1989), one of the feedback loops by which each neuron can interact with other neurons is provided by its local neighborhood. This premise complements the significant roles of interactions among delimited subsets of neurons forming discrete networks in information processing (Arieli et al. 1995;Jagadeesh et al. 1992;Nicolelis 1997;Nicolelis et al. 1998;Tsodyks et al. 1999;Zhu and Connors 1999), which can give rise to "zero lag" correlations between the gamma oscillations of neurons separated by millimeter distances (Roelfsema et al. 1997) despite the axonal transmission times that are expected to cause phase shifts between oscillations. Usher, Schuster and Niebur (1993) and Schillen and König (1994) have modeled this phenomenon by assuming that the feedback delay within each target matches the transmission delay between targets in an excitatory feedback network. Traub et al. (1996) have overcome some of the rigidity of those models by invoking doublet firing of single neurons (Freeman 1975), which we and they show is enhanced in states of high-amplitude gamma.
The fifth premise provides an alternative mechanism for spatiotemporal integration in cortex through a distinction between the microscopic functions of individual neurons and the mesoscopic functions of neural masses in local neighborhoods (Freeman 2000), reserving the term macroscopic to refer to the much larger ensembles that are visualized with whole brain imaging techniques such as functional magnetic resonance imaging, single photon emission computer tomography (SPECT), and optical recording (Arieli et al. 1995). The dendritic currents of cortical neurons give rise to extracellular fields of potential, which, owing to the laminar architecture of cortex, sum in the volume conductor and constitute the main source of the EEG. These are the same currents that determine the firing probabilities of the neurons (Freeman 1975;Mitzdorf 1985Mitzdorf , 1987Nunez 1981). This causal relation is manifested in the statistical dependence of pulse probability of cortical neurons conditional on the amplitude of the EEG in the same local neighborhood (Eeckman and Freeman 1990;Freeman 1975). The relation is weak, because ϳ10,000 pulses are required from each individual neuron in the local mass to reveal the statistical regularity (Freeman 1992). Estimates of the fraction of the variance of individual spike trains that is covariant within an ensemble of simultaneously recorded cortical spike trains are ϳ0.1% (Abeles 1991;Aertsen et al. 1989).
The mesoscopic EEG activity that results from distributed synaptic interactions in each local neighborhood provides a measure of the local mean field intensity of that neighborhood. What makes the EEG important is that while it is the local average of electric potentials established by the vector sum of currents across the cortical impedance, that sum is the reflection of the mesoscopic state formed by the cooperative actions that result from synaptic interactions, mainly by excitatory synapses on excitatory neurons that are broadly but sparsely distributed. A counterexample is a compound nerve action potential formed by the sum of action potentials triggered by an electrical stimulus of a peripheral nerve. It spreads with time and distance, forming the "A" and "C" peaks, because the axons do not interact to bind together the action potentials. The currents sum in the volume conductor, but they do not reveal a local mean field because the axons do not interact. Moreover in cortex, the EEG currents are too weak to influence directly the firing rates of the individual neurons (Freeman 1962(Freeman , 1975(Freeman , 1992Freeman and Baird 1989). The cooperative interactions that are revealed by the EEG typically are aperiodic oscillations in the spectral range from 1 to 100 Hz with intermittent peaks in well known ranges labeled theta, alpha, beta, gamma, etc. The fast Fourier transform (FFT) (Press et al. 1988) of relatively long epochs of recording (Ͼ1 s) yields spectra that show a near linear fall in log power with increasing log frequency at a slope near 2, a "1/f 2 " distribution that is characteristic of Brownian motion and telegraph noise. Short segments (Ͻ0.1 s) typically reveal at least one spectral peak at a frequency that differs seemingly at random in successive segments. This variation reflects the FM and aperiodicity that characterize the neocortical EEG.
Simultaneous recordings of EEG from 8 ϫ 8 electrode arrays (7 ϫ 7 mm) placed subdurally on the visual, auditory, or somatosensory cortices of trained rabbits have revealed a high degree of spatial coherence in these spatial windows as previously reported (Barrie et al. 1996;Freeman and Viana Di Prisco 1986). After temporal band-pass filtering to extract the activity in the gamma range (20 -80 Hz in the waking rabbit), on average, Ͼ90% of the variance was found to be contained in the first component of principal components analysis (PCA) as compared with ϳ50% when EEGs were recorded simultaneously from multiple areas and filtered in the same way. The shared gamma wave form in trained rabbits revealed spatial distributions of AM in short time segments. These segments were classifiable with respect to conditioned stimuli (CS) that the rabbits had been trained to discriminate (reinforced CSϩ from unreinforced CSϪ) in the sensory modality corresponding to the cortex under observation (Barrie et al. 1996). Therefore they were identified as "AM patterns." The spatial coherence and the relatively narrow spectral peak of the carrier wave in the gamma range in these same brief AM patterns have made it possible using the FFT to measure the phase of the EEG at 64 sites in short time segments with respect to the phase of the spatial ensemble average of the 64 traces at the shared peak frequency. The square of the cosine of the phase for each trace gives a basis for estimating its level of shared variance with respect to the whole. The present report describes the spatial patterns of phase modulation (PM) that have been found in the EEGs from the primary sensory cortices, an interpretation of the mechanism by which the PM patterns form and a suggestion on how the spatial patterns are transformed by output pathways and received by the targets of transmission. Note should be taken that every multichannel EEG segment of any length gives an amplitude distribution and a phase distribution; the term "pattern" is reserved here for AM patterns that can be classified with respect to CS and for PM patterns that correspond to the direction and conduction velocities of axons in a specific tract in the brain (Bressler 1987) or that can be fitted with a cone over time durations corresponding to the time durations of AM patterns as determined by classification efficacy (Fig. 10 in Barrie et al. 1996).
These AM and PM patterns provide crucial evidence that the spatially coherent wave form of the epipial EEG is not due to a deep-lying generator having a broad point spread function (PSF, a term used in optics to denote the distribution in a plane of observation of light from a point source), nor is it due to oscillation in potential at the reference lead in monopolar recording but instead to an array of equivalent point dipoles in a plane at some depth below the plane of recording (Freeman 1975). Direct measurement of the PSF is an arduous task that has only been done for the olfactory bulb, but equivalent information can be obtained from the spatial spectrum of the EEG, and this has been done for the bulb, prepyriform, visual, auditory, and somatic cortices in rabbits and the superior temporal gyrus in humans .

Chronic implantation of an epidural electrode array
Eighteen female New Zealand White (NZW) rabbits (2.5-4.5 kg, ϳ2 yr old) were implanted (aseptically, full surgical anesthesia by 4% isoflurane/O 2 mixture) with a prefabricated electrode array onto the left hemisphere. Arrays were 0.25-mm stainless steel wires in an 8 ϫ 8 square matrix (0.79-mm interelectrode distance to limit spatial aliasing, 7 ϫ 7-mm window). Reference and ground leads were placed in the skull adjacent to the site of each placement over the visual, auditory, somatic, or olfactory cortex. The arrays were placed in accordance with descriptions of the locations of the primary sensory neocortices in rabbits by means of mapping with evoked potentials (Galli et al. 1971;Gould 1986;Hollander and Halbig 1980;McMullen and Glaser 1982). Subjects were maintained 2-4 yr, then killed with 120 mg/kg pentobarbital, perfused with 10% formalin, and autopsied. All procedures were conducted according to protocols approved by the University of California at Berkeley Animal Care and Use Committee with veterinary supervision by the Office of Laboratory Animal Care.

Experimental paradigm
For recording, each subject was placed into a restraining carrier in an electrically shielded, sound-resistant dark chamber. The electrodes were connected to World Precision Instruments ISO 4/8 differential amplifiers. A pneumograph was attached to the chest, skin clips for the unconditioned stimulus (US) were placed onto the left cheek, and white noise (72 dB) was introduced into the chamber. After 3 wk of familiarization, each subject was trained using classical aversive conditioning to discriminate between two stimuli corresponding to its implant site: one of two 10-ms full-field flashes varying only in intensity (3.6 vs. 2.8 ft.-cd); or binaural (stereo earphones at 72-84 dB above the 72 dB white noise) 100-ms sinusoidal tones (500 vs. 5,000 Hz); or a 3-s air puff onto the cheek versus the hindquarters ( Barrie et al. 1996). One of the conditioned stimuli (designated CSϩ) was reinforced with a weak electric shock to the cheek adjusted initially to elicit an orienting response (Freeman and Viana Di Prisco 1986), and the other conditioned stimulus (designated CSϪ) on randomly interspersed trials was not reinforced. For each experiment, 40 6-s artifactfree EEG records were recorded, filtered with single-pole, first-order RC filters (6 dB/octave falloff) at 100 Hz (3 dB point) and 0.1 Hz, digitized at 500 Hz, phase corrected for the time lag introduced by multiplexing (10 s), and digitally stored to magnetic disk in binary data files (ϳ16 MB/file). Each record was divided into a 3-s prestimulus period and a 3-s poststimulus period. The sequence of CSϩ/CSϪ trials (20 each) was randomized across the set of 40 trials, and the time interval between each stimulus was randomized between 30 and 120 s. Three replicates were made for each experiment (Barrie et al. 1996).

Data analysis
All data analysis was done off-line on a Macintosh PowerPC using original software specifically developed for this purpose. Every data set was reviewed and edited for bad recording channels and EEG artifacts (bad connections, animal movement, 60 Hz noise). EEG records were segmented by either of two methods. The first method involved using a sliding window to parse each record into m overlapping, 128-ms EEG segments (where w indexed each segment) separated by 2-ms intervals (Barrie et al. 1996). Each individual 128-ms EEG segment was then converted into a root-mean-square (RMS) spatial AM distribution, after band-pass filtering (at the optimal band-pass filter setting), resulting in a series of 1 ϫ 64 column vectors (V ៝ ) where R is the RMS amplitude of the EEG for each channel (e). (A list of variables used in this paper is provided in the APPENDIX.) The second method was based on locating stable phase cones in the spatial distributions of phase of neocortical EEGs. Conic phase patterns had been found in EEGs of the olfactory bulb Freeman and Baird 1987). Here each multichannel EEG record was parsed into m overlapping, 128-ms segments separated by 2-ms intervals. Each segment was decomposed by FFT (Press et al. 1988), into 50, 64-dimensional spatiotemporal distributions of phase (P ៝ ) at fixed frequency intervals ( f ) ranging from 2 to 100 Hz. Phase differences in (P ៝ ) at high spatial frequencies were inferred to be due to noise (Fig. 6 in Freeman and Viana Di Prisco 1986), so they were attenuated with a spatial filter in each 8 ϫ 8 phase distribution by transforming the frame into the frequency domain with the twodimensional (2-D) FFT (embedding in a matrix of 32 ϫ 32 zeroes and omitting Hamming and Hanning), passing the real and imaginary components through digital 2-D highcut and lowcut filters (Barrie et al. 1996;Freeman and Baird 1987) and using the inverse 2-D FFT to recover the smoothed values. The two optimal cutoff frequencies in cycles/mm were determined by repetition to give a tuning curve ( Fig.  1) to identify the value giving the most conic-like phase segments. A low cutoff spatial filter was used to minimize the risk of what appeared to be an artifact from excessive highcut filtering and to take advantage of the usefulness already demonstrated of spatial band-pass filtering (Freeman and Baird 1987). Each spatially filtered phase distribution was then fitted by nonlinear regression with a 2-D conic surface (C ៝ ) in planar coordinates P ៝ w,f ϭ ͕P e,f ͑w͒ : w ϭ 1, 2, . . . , m; f ϭ 2, 4, . . . , 100; e ϭ 1, 2, . . . , 64͖ where P is the spatial phase distribution, w is the window index (for m windows), f is the frequency index, e is the electrode array channel index, P ៝ is the 1 ϫ 64 phase vector, C ៝ is the 1 ϫ 64 cone vector FIG. 1. This graph illustrates calculation of a spatial filter tuning curve. The real and imaginary components of the fast Fourier transform (FFT) decomposition were iteratively spatially low-pass filtered using different highcut filter settings, with the criterion of the number of poststimulus (3,000 -3,300 ms) stable phase segments (40ϩ ms in length) in a 40-record experiment. The count for no filter was 12. Values were optimized in the range of 0.2-0.5 cycles/mm. These settings were used for each experiment analyzed. The peak with the highcut spatial filter set at 0.029 cycles/mm was interpreted as an artifact.
representing the conic pattern, and is the conic function regressed onto the phase distribution. Every cone had a slope, b (in mm/radian) and an apex (location in x and y mm from the center of the recording array), which was not restricted within the area of the recording area. The residual between the original phase distribution and the conic regression was minimized by nonlinear regression (3) and expressed as the percentage of total power for each frequency and segment. Since every 128-ms EEG segment contained a phase distribution at 50 different frequencies, the regressions with large residuals (meaning the original phase distribution did not resemble a cone) were removed by flagging all apices of cones with low residuals. If there were multiple peaks in the phase pattern (at a single frequency), then the residual of the conic regression increases (i.e., bad fit and no cone is identified). There could be multiple, simultaneous phase cones existing at different frequencies. For example, there could be a conic phase pattern at 10 Hz and another conic phase pattern at 40 Hz. However, all attempts to fit the sum of two conic surfaces to phase distributions at a single frequency resulted in singular matrices, that is, nonlinear regression failed.
Conic apices were separated into two categories: apices with Ͻ20% (4) and apices with Ͻ40% residual (A ៝ Ј) (5) The numbers of apices within each segment from conic regressions with R Ͻ 20% and R Ͻ 40% were recorded as nЉ w and nЈ w , respectively. Both matrices of apices were subsequently sorted from lowest to highest residual (i.e., A ៝ Љ 1,1 indexed the apex of the phase cone with the lowest residual R Ͻ20%) and ending (always a cone with R Ͼ40%) of particular segments of temporally contiguous phase cones with apices that are constant in location and sign (i.e., "phase segments") ( Barrie et al. 1996).
Once all of the individual phase patterns resembling cones were located and marked (by Eqs. 4 and 5), the search for phase segments began. This strategy had two parts: the location of the initial site of nucleation (this was always a cone with R Ͻ 20%) and the location of subsequent temporal points (always consecutive phase cones with R Ͻ 40%) which were indicative of the temporal duration of a cortical event. After locating the initial point, called the site of nucleation, two distances were calculated and The first distance indicated how far (in mm) an apex (located 2q ms in time from the initial phase cone) was from the initial site. The second distance indicated how far any two consecutive phase cones were from each other. Three criteria were used to identify whether a sequence of phase cones formed a continuous segment that began at A ៝ Љ wϩ0,i : ⌬A 1 Ͻ 1.2 mm, which constrained an entire sequence of cones within a segment to a particular area of the cortex); ⌬A 2 Ͻ 0.5 mm, which ensured that successive cones lay close to the previous location; and uniformity in the signs of the slopes of a group of phase cones, all of which had to be either lead or lag. If all of the above criteria were satisfied for q Ն 25 contiguous time steps (t ϭ 50 ϩ ms), then a stable phase segment was demarcated. In summary, a stable phase segment was defined as a pattern of phase that had q Ͻ 20% at time t ϭ 0, no shift in location Ͼ1.2 mm from the initial point, an apex that did not move Ͼ0.5 mm during any 2-ms interval, and a phase value at the apex that was either positive or negative for the entire 2q-ms segment. Stationarity over the measurement window is empirically fulfilled because we are able to track similar patterns across 2-ms epochs. Figure 5B illustrates the search for stationary phase segments. The first point (i.e., phase cone apex) within the grouping labeled "segment I" represents the site of nucleation. Each subsequent point is located at 2-ms intervals and satisfies all of the preceding criteria until the spatial jump into the positive quadrant; that was the end of the first stable phase segment. A 1 ϫ q vector (Z ៝ ) demarcated each stable phase segment (8) where T was the time in milliseconds beginning at window w and extending to window w ϩ q (the time when no further phase cones could be found to satisfy the preceding search criteria), for each phase segment (s) within every experimental record (r). There were N r individual segments per record. Each of these segments was then converted into a series of RMS spatial AM patterns (as in Eq. 1).
To determine whether the demarcated phase segments recurred in specific time intervals with respect to the CS, a histogram (H ៝ ) of the phase segments was composed for the incidence in the 6-s trials over the 40 trials in each session By calculating the FFT of H ៝ , a spectrum of frequencies was computed, reflecting the mean recurrence rates of stable phase cones. Two other important values obtained by this analysis included the EEG phase velocity (S) in M/s S M s Ϫ1 ϭ ͑b mm rad Ϫ1 ϫ 2f rad s Ϫ1 ͒/1000 (10) where b is the slope of the conic surface regressed onto the phase distribution decomposed at frequency f, and the modal diameter (D) in cm is defined as twice the distance from the apex to the half-power level of the EEG The absolute phase difference of the oscillation at the common frequency increases monotonically from 0°at the apex with increasing distance of the recording site. The fall in amplitude of the common component is given by the cosine of the phase. At some distance the commonality must fall to zero, and in fact the correlation between simultaneous recordings from different primary sensory cortices is much less than that within each cortex, indicating that each domain of spatial coherence must have a soft boundary. We estimate the location of the boundary from the radius where the absolute phase difference has increased to 45°, where the cosine is 0.707, and the power of the shared wave form has decreased to the square of the cosine, 0.5.
After an experiment had been segmented (by either fixed-length windows separated by fixed-length time intervals (Barrie et al. 1996) or by locating segments demarcated by stable phase cones) into a matrix of spatial AM pattern vectors, a method of cross-classification was employed to determine whether or not the CSϪ patterns were significantly different from the CSϩ patterns. Briefly, for contiguous temporal epochs, all patterns within that epoch were grouped as either type A (CSϪ) or B (CSϩ). Each type was further subdivided into equal groups A 1 , A 2 , B 1 , B 2; the first and second halves of group A/B became subgroups A1/B1 and A2/B2, respectively. The patterns within the first two subgroups (A 1 , B 1 ) were averaged to form cen-troids and the Euclidean distance was measured in 64 space from the two centroids to each of the patterns remaining in the second two subgroups (A 2 , B 2 ). If a patterns was closer to its centroid than to the opposite centroid (i.e., a pattern from subgroup A 2 was closer to centroid A 1 than it was to centroid B 1 ), then it classified correctly. This classification was repeated by calculating centroids from the second two subgroups and classifying the patterns from the first subgroups. A binomial probability pattern was used to determine whether or not a certain level of classification was significant. This method yielded a time series of probability values showing the times when the CSϪ and CSϩ spatial AM patterns could or could not be separated.

Paleocortical versus. neocortical EEGs
Since the goal of this research was to assess a method, originally used to characterize the EEG from the olfactory system (Freeman and Baird 1987), for parsing the neocortical EEG, a comparison of the characteristics of the olfactory and neocortical EEG was compiled as a reference. The olfactory EEG was strongly correlated to the respiratory rhythm of each rabbit ( Fig. 2A, top). It was punctuated by a sequence of high-frequency (20 -80 Hz), high-amplitude bursts typically referred to as gamma oscillations (Fig. 2B, left). Such olfactory bursts were time-locked to the inhalation phase of the respiratory cycle and were separated by 1/f-type (20 -100 Hz), broad spectrum interbursts. Previous research utilized the presence of these gamma bursts as an EEG marker for stimulus-induced, endogenously generated events (Freeman 1978(Freeman , 1991. A cursory comparison of the olfactory and visual cortical EEG revealed significant differences. The visual EEG was not correlated to the respiratory rhythm ( Fig. 2A, bottom), did not have a distribution of gamma oscillations, and only contained an evoked potential marking the arrival time (3,000 ms) of the afferent volleys of action potentials into the cortex from the periphery. An examination of the average frequency spectra of visual cortical neuroactivity further demonstrated that (Fig. 2B, right) log power was uniformly distributed across the temporal spectrum as a 1/f m function of log frequency. This relation also held for the spatial spectra (Barrie et al. 1996).

Measurement of AM patterns in neocortices
One property that the olfactory and neocortical EEGs shared was the presence of a common carrier wave at any one instant. This meant that the frequency spectrum of the EEG using the FFT was spatially coherent across the cortex (over distances covered by the 7 ϫ 7 mm arrays) at any one time. Coherence was revealed by a complex wave form shared on the multiple EEG traces having a common instantaneous frequency, that changed continually and unpredictably. Short time segments (64 -256 ms) had prominent peak frequencies, which changed in successive segments, reflecting a continuous modulation in frequency and/or phase. Power spectral analysis of longer FIG. 2. A: these traces represent single-channel electroencephalographic (EEG) recordings (-) and respiratory rhythms (---) from a record of a classical conditioning experiment from the prepyriform cortex (PPC, top) and the visual cortex (bottom). The olfactory trace illustrates a series of high-frequency EEG bursts time-locked to the respiration. The visual trace does not show any periodic oscillatory activity, but it does have an evoked potential marking the arrival of the stimulus, t ϭ 3,000 ms ( Barrie et al. 1996). B: temporal frequency spectra. These are averaged frequency spectra (-) and confidence intervals (---) from the PPC (left) and visual (right) corticies. Each spectrum was averaged across 40 records from a Hamming smoothed, 500-ms poststimulus window. Regression curves (-) of the form 1/f were fit to the 10-to 50-Hz and 20to 80-Hz regions of the PPC and visual spectras, respectively. These figures illustrate that there was an excess of neuroactivity in the 50-to 80-Hz band of the PPC spectra although the log power in the visual EEG was uniformly distributed as 1/f across the entire log frequency range (Barrie et al. 1996). segments usually revealed a broad 1/f m power spectral density, in which log power decreased linearly with increasing log frequency at slope m ϳ 2.
Spatial coherence observed in plots of multiple time series (Fig. 3A) was supported by the incorporation of Ͼ90% of the variance in the first component of PCA ( Barrie et al. 1996). The frequency spectra of averages of EEG segments had a 1/f distribution, but the spectra for individual EEG segments typically centered around one frequency relative to the overall spectra (Barrie et al. 1996). That frequency component revealed spatial AM (AM) over the 64 channels in a spatial distribution. Owing to the high covariance of the 64 traces the same AM distribution was found in the root mean square amplitudes. Each AM distribution could be represented by a contour plot (Fig. 3B) and also by a point in n space, where n was the number of electrodes and EEG traces. Similar AM distributions formed clusters of points in n space, and multiple AM distributions gave a set of clusters. Assignment of each segment to a class was based on the Euclidean distance of its point to the center of gravity of the nearest cluster.
The AM patterns were modified by classically conditioning the animals to respond to stimuli in the cortex of the corresponding modality; for example, animals with implants over the primary visual cortex (V1) (Hollander and Halbig 1980; Hughes and Vaney 1982;Thompson et al. 1950) received visual stimuli. As the animals were trained to discriminate between one reinforced conditioned stimulus (CSϩ) and another unreinforced (CSϪ), new AM patterns appeared and prior patterns were subtly altered. After completion of training, the AM patterns reappeared on delivery of the corresponding stimuli to which the animals had learned to respond. The locations and durations in time of the AM patterns in neocortical EEGs were not visible as an AM over time. The most effective method for detecting them was to pass a moving window, 64 -256 ms in duration, in 2-ms steps along the set of 20 CSϩ trials and 20 CSϪ trials, and to determine when the AM patterns differed significantly between the CSϪ and the CSϩ trials. The 64-to 256-ms duration sliding window was only used for data exploration. The 128-ms window length was most effective and was used throughout this manuscript. The shorter window lengths did not have enough spectral resolution for successful phase segment tracking. The longer window lengths yielded unsatisfactory results, possibly because the longer windows included EEGs that were unrelated to the activity being tracked (i.e., noise was included in the FFT decomposition of the EEG). The binomial probability of a difference by chance offered a convenient measure for when the two sets of distributions differed. As shown in Fig. 4, the AM patterns distinguishing CSϩ and CSϪ trials occurred shortly after the arrival of the CS and at irregular intervals a few hundred milliseconds thereafter.
This statistical assay, which served effectively to classify AM patterns from visual, auditory, somatic, or prepyriform cortical EEGs, did not serve to locate the AM patterns in time on single trials. The decline in significance of AM pattern separation with elapsed time from the stimulus presentation indicated that the locations in time of successive events varied across trials. The cortical distribution of phase modulation (PM) at the same frequency was given by the FFT. The PM distributions were measured in the same stepped windows to determine whether they could be localized on single trials.

Measurement of PM patterns in neocortices
The discovery of PM patterns was based on the finding of spatially coherent oscillation in multichannel EEGs with spectra in simultaneously recorded short segments having local maxima at the same frequency somewhere in the gamma range. We restricted the selection of the local spectral peak to the gamma range of 20 -80 Hz because this was the spectral range in which the AM patterns could be classified with respect to CS ( Fig. 9 in Barrie et al. 1996). The spatial coherence was demonstrated by use of PCA to extract the dominant component for pattern classification on every data set (Fig. 8 in Barrie et al. 1996) to show that the results of classification were the same using RMS, FFT, or PCA amplitudes. All of the cortical areas showed that the dominant component captured between 90 and 98% of the total variance despite the lack of zero lag synchrony (Fig. 3A), with no significant differences between areas. The 64 spectra of the EEG traces correspondingly had peaks at the same frequency even as that frequency shifted unpredictably from one segment to the next. Therefore when that frequency was identified or when any other frequency was selected, the 64 phase values defined at that frequency provided a spatial phase distribution constituting PM about the phase of the spatial ensemble average. Comparisons could then be made between PM distributions from differing cortices in search of varieties of PM patterns. Stable patterns of phase had already been identified in the prepyriform cortical EEGs, which were clearly determined by the velocities and directions of propagated action potentials in the lateral olfactory tract (Boudreau and Freeman 1963;Bressler 1987Bressler , 1995. Comparable phase patterns were sought in EEG segments from the primary sensory neocortices, first by visual inspection of contour plots of phase in search of regularities and next by attempts to find distinctive locations of peaks of phase lead on CSϩ versus CSϪ trials on the assumption of topographic organization of activity driven by thalamic pacemakers. Peaks were occasionally found, but the locations varied unpredictably and without relation to CS, and when they were present in the array, the sign varied seemingly at random. Then a plane was fitted to the 8 ϫ 8 phase values in search of broad phase gradients such as those found in array recordings of scalp EEGs (Walter 1953) and magnetoencephalograms (MEGs) (Llinás and Ribary 1993), on the premise that large wave fronts might be approximated by a plane with no curvature over the local 7 ϫ 7 mm covered by the epidural arrays. The phase gradients in some segments did appear to conform to a plane, but the orientation and steepness varied randomly. The patterns were eventually resolved by fitting a cone to the 64 values of phase, giving consistently lower residuals than a plane fitted to the same phase data and allowing a close fit to the phase extrema that happened to be located within the arrays. The FFT of the 64 traces (128-ms windows stepped at 2-ms intervals) yielded the spatial distributions of amplitude (Fig. 3B) and phase (Fig. 3C) at frequencies in the gamma range. Conic functions were fit (in planar coordinates) to each phase distribution (Fig. 3D) after band-pass filtering (0.2-0.5 c/mm) of the phase values, by which the standard error of measurement of phase was minimized (Fig. 6 in Freeman and Viana Di Prisco 1986). A phase cone apex was flagged (Fig.  5A) when the variance incorporated by the fitted conic surface exceeded 80% of the total variance. That conic pattern was tracked in successive window steps (defined by Eqs. 4 and 5) as illustrated in Fig. 5B. PM patterns lasting 25 steps (50 ms) were labeled as stable phase cones. The distributions that could be fitted with a plane could be explained as having conic apices located well outside the arrays. No significant differences were seen in PM patterns from the three neocortical sensory areas. The relative constancy of the sign of the phase extremum, and of its location, and of the frequency at which the phase was defined, together constituted evidence for stationarity of the dynamic states over the time durations shown in Fig. 7A. Stationarity is defined here to mean cortical dynamics holding in an attractor basin between two successive state transitions, irrespective of the oscillations within that basin, including the trajectories of entry and exit.
The locations and signs of the apices of successive stable phase cones varied unpredictably without relation to the CS. The duration of identified stable phase cones averaged 80 ms with a long tailed distribution. Neocortical phase velocities in M/s (Eq. 10) had skewed distributions from 0.5-4.0 M/s (Fig.   FIG. 4. Validation of the significance of spatial AM patterns in EEGs depends on showing that they change with learning (Gray et al. 1986), and EEG segments can be classified with respect to conditioned stimuli (CS) that are reinforced (CSϩ) or not (CSϪ) on randomly alternated trials. The graph shows estimates of the probability that the degree of correct classification could have occurred by chance when comparisons are made using a 128-ms window stepped at 2 ms along the 6-s recording period in 20 trials of each stimulus (Freeman and Baird 1987). 5C). The modal value of 0.6 -2.1 M/s was consistent with measurements reported on the conduction velocity of large groups of cortical axons parallel to the pia. No attempt succeeded to define edges of the coherent domains, so an appropriate criterion of size was adopted: the half power diameter across the apex (Ϯ cos 45°), using Eq. 11. Diameter values were skewed from 5 to 20 mm or more with a mode near 0.5 cm (Fig. 5D).
The dependence of phase lag in radians on the distance between recording electrodes in millimeters was first found without spatial filtering by calculating the absolute phase differences between channels at all frequencies in many windows and many trials. In prior measurements of PM patterns in the olfactory bulb, the conic basis function led to an invariant measure, which was the steepness of the phase gradient expressed in M/s (Freeman and Baird 1987). The gradients were measured in radians/mm, and they varied with frequency, but when they were converted to M/s by use of the frequency of the oscillation (Eq. 10), they converged to the conduction velocity estimated for the axons of the bulbar axon collaterals running parallel to the surface of the bulb. A similarly invariant relationship was found in EEGs from the prepyriform cortex (Boudreau and Freeman 1963), so it was sought in the EEG data from the neocortices. Unlike the invariant relation for the olfactory bulb, the phase velocity of the neocortical EEG increased on average with increasing frequency. The breakdown by frequency of the distributions of phase velocities on conversion to M/s using Eq. 10 is shown in Fig. 6 (top) and Table 1 for three frequency bands. The modal phase velocities were consistent with the conduction velocities reported for short intracortical axons over distances Յ3 mm, as distinct from corticocortical, callosal, and efferent axons that are much faster (Langdon and Sur 1990;Lohmann and Roerig 1994;Swadlow 1994;Swadlow and Weyand 1981). In the neocortical EEGs, a new invariant was found in the modal half-power diameter (Eq. 11) of the identifiable stable phase cones across neocortical locations and frequency bands (Table 1; Fig. 6, bottom).
Stable phase cones identified by these criteria were found at irregular intervals throughout each trial (Fig. 7A). Pre-and poststimulus time histograms of the stable phase cones showed higher than average probabilities of cones after the time (3,000 ms) of stimulus arrival (Fig. 7B). The FFT of histograms calculated from all neocortices yielded peaks in the 4-to 80-Hz range (Fig. 7, C-E), indicating a degree of rhythmicity in the theta range for the arrival and duration times of stable phase cones. The AM patterns were derived from segments demarcated by stable phase cones in visual, auditory and somatic EEGs as in Fig. 3B. These AM patterns also revealed a significant level of correct classification of CSϩ and CSϪ trials, and in comparison to the fixed steps, the AM patterns accompanying stable phase cones gave higher levels of significance later in trials (Fig. 8, A-C), indicating that detection of phase cones improved the identification of the locations of behaviorally significant EEG events in the CS-CR interval.

D I S C U S S I O N
EEGs were measured by decomposing them with appropriate basis functions, which in the present case were cosines giving the phase and amplitude of frequencies in the gamma range. Whereas accurate measurement of amplitude was relatively easy, measurement of the phase was subject to large errors of measurement, owing to the brief duration of segments, the small number of cycles in each segment, the strong tendency to FM and AM about center frequencies in both time and space, and the mix of multiple frequencies in spectra of most segments, especially from neocortices with their tendency to "1/f 2 " spectra. The standard error of measurement was estimated by measuring cosines at unit amplitude and zero phase embedded in varying levels of random numbers to simulate white noise (Freeman and Viana Di Prisco 1986). The results gave standard deviations averaging Ϯ60°for raw EEGs, which FIG. 5. A: locations of apices of phase cones are projected onto the visual neocortical surface with the 64-channel recording array (▫). Each point represents an apex from 1 phase cone at a best frequency. B: clusters formed when the next apex, 2 ms later, was Ͻ1.2 mm from the site of initial detection, Ͻ 0.5 mm from the preceding apex, and had the same sign (lead or lag). The minimum duration was 48 ms; few exceeded 100 ms. A pattern of transition is illustrated as "segment I" and "segment II." C: distribution of phase velocities from pooled segments. D: distribution of half-power diameters.
were reduced to Ϯ6°after application of appropriately designed spatial and temporal filters.
The requirement for validating the measurements of spatiotemporal AM patterns was met by relating them to behavior. This criterion led step by step (Barrie et al. 1996;Freeman 1975Freeman , 1992Freeman and Schneider 1982) to decomposition of EEG time series with cosine basis functions, measurement of AM and PM patterns having the same instantaneous frequency in the arrays, and classification of the AM distributions with respect to sets of CSϩ and CSϪ trials after discriminative conditioning. The PM patterns consistently failed to classify with respect to locations and signs of apices, but the AM patterns extracted from the segments demarcated as stable phase cones did classify well above chance levels at intermittent times between the CS and CR onsets.
The main problem encountered in correlation of neocortical EEGs with behavior (as compared with olfactory EEGs) was temporal segmentation. The identification of spatial AM patterns in olfactory EEGs was facilitated by the prominent respiratory wave in the theta range and the associated temporal AM of the gamma activity giving "bursts" of oscillation. In contrast, visual inspection of the neocortical EEGs gave no indication of where the stationary segments might start or end, either by any near-periodic wave in the theta and alpha ranges or by any temporal AM in the gamma range. Therefore a fixed-duration segment was stepped along the multiple EEG traces on every trial, and classifications of AM or PM patterns from CSϩ and CSϪ trials were made between segments on different trials all from the same time step with respect to stimulus onset. The results showed that the selection of AM patterns that was guided by determination of the temporal locations of the stable phase cones improved the reliability of segment classification in the CS-CR intervals. The consistency and reliability of the results from olfactory, visual, auditory, and somatosensory EEGs led to the conclusion that the radial phase gradients are not an artifact of the measurement algorithms and that they manifest an important biophysical property of the neural masses that generate the gamma oscillations of the EEG.
Several inferences follow from this conclusion. First, the common wave form on the 64 electrodes cannot be ascribed to activity at the site of the reference electrode on monopolar recording or to volume conduction from a deep dipole generator far under the array, not only because of the AM (which might conceivably be explained by local variations on cortical specific resistance) but also because of the phase modulation  Summary statistics for the average and standard deviation modal phase velocity (M/s) and modal phase cone diameter (cm) for three different frequency ranges . The percentage of change of the modal values between the first and last frequency ranges is also expressed. Each modal value, for a particular cortical area, was calculated by locating stable phase segments from 18 different experiments using animals with visual, somatic or auditory implants. This table illustrates that the phase velocity increases relative to the frequency of the phase cone, while the conic diameter remains an invariant across frequencies. Values are means Ϯ SD; n ϭ 6 for visual, somatic, and auditory.
(the reactive component of the cortical impedance vector is much too low to give the observed range of differences) (Freeman 1975). Nor is it due to decorrelation with distance over a distribution of noise generators smeared by volume conduction (Elul 1972) because that would not give radial phase gradients. Nor does it manifest entrainment of coupled oscillators because the phase gradients persist through the segments, probably owing to the sparseness (Braitenberg and Schüz 1991) of the local connection densities, which fails to support convergence of activity into synchrony.
Second, the findings can explain the zero time lag correlation reported between the pulse trains of neurons separated by distances less than the array size (Brosch et al. 1995;Engel et al. 1992;Gray 1994;König and Schillen 1991;Roelfsema et al. 1997;Singer 1993), when that relation has been derived between pulse trains of pairs of neurons by time ensemble averaging over multiple trials. When the technique is used because there are too few pulses on any one trial, the unpredictable variation in location and sign of the conic apices randomizes the phase relations on repeated single trials, and the resulting average can only approach zero.
Third, the random variation of the sign of the apices cannot be explained by an intracortical or thalamic pacemaker because those drivers could only give apices with phase lead whether acting by excitation or inhibition. It is compatible with a symmetry breaking state transition such as a saddle node bifurcation. The property of mesoscopic states that makes them interesting is the capacity they give to an ensemble of neurons for rapid changes in the global spatiotemporal distributions of FIG. 7. A: solid bars show phase cones stable for Ͼ48 ms in 40 trials. When the EEG within a certain time window is decomposed into phase patterns, there is 1 pattern per frequency bin. Because it was observed that there are often 2 conic-like patterns of phase existing within the same temporal window at different frequencies, the trial number is half the number on the ordinate to display overlapping segments at different frequencies. We explain the sustained level of incidence in the occurrence of phase cones by inferring that perceptual processes occur before as well as after the presentation of a CS and that any brain activity pattern that is related to or underlies perception must be found in both prestimulus and poststimulus segments. B: histograms of the preceding segments. C-E: spectra of the preceding histograms from the visual (above), somatic, and auditory sessions.
FIG. 8. The probability of correct classification of AM patterns on discriminative CSϩ and CSϪ trials by chance is shown as time series for the visual, auditory and somatic EEGs over the 6-s duration of the trials, as in Fig. 4. In this figure the EEG segments in which the AM patterns were found were determined from the presence in the window of a stable phase cone in the visual (A), somatic (B), and auditory (C) sessions. organization and function of ensembles. Some well-known examples are the transitions between waking and sleeping states, between vocalizing and swallowing, and between walking and running by which the neurons distributed in the brain and spinal cord shift their firing from one coordinated pattern to the next globally in a few milliseconds.
We postulate that in primary sensory cortices a state transition is enabled by a volley of action potentials on afferent axons, which is initiated by a sensory event and gated by thalamic relay nuclei. Gated input transiently increases the level of activity in the cortex. Owing to the asymmetry of the sigmoid curve governing the output of the cortical neurons receiving the volley (Freeman 1992), the forward gain (sensitivity of the trigger zones of the excited neurons) is increased. Owing to the predominance of intracortical excitatory synapses to other excitatory neurons, the triggered output goes mainly to excite those other neurons and sensitize them as well. In the ensemble, some of the activity returns to the initially excited neurons, still further increasing their activity and sensitivity. This constitutes regenerative feedback at the mesoscopic level that is equivalent to the nonlinear local response of axons approaching their thresholds (Hodgkin and Rushton 1964). Just as microscopic axons are bistable (either above or below threshold) (Izus et al. 1998), we infer that mesoscopic ensembles of cortical neurons are bistable or polystable. Above some threshold an ensemble can become unstable and jump into a high-intensity oscillatory state. Owing to the amplitude-dependent gain of the population (Freeman 1992), the pattern generated in the new stable state that has been triggered by the input is dominated by the intracortical synaptic connections, which are subject to modification by learning during previous experience (Emery and Freeman 1969), particularly at the excitatory synapses on the dendritic spines of excitatory neurons as described by Rall (1995) and others. Hence the AM patterns could reflect the past and present contexts imposed by behavioral constraints, not the specific forms imposed by the sensory input volley. This could explain the fact that the AM patterns are not invariant with respect to the stimuli but are modified by new learning, such as that which occurs with changes in reinforcement contingencies (Freeman 1991;Freeman and Grajski 1987).
Fourth, evidence from physical distributed systems shows that state transitions do not start simultaneously throughout the systems but begin at a site of nucleation and spread radially, as in the formation of a snowflake around a dust particle. The velocity of spread indicated by the radial phase gradient is compatible with the conduction velocities reported for axons running parallel to the pia (Langdon and Sur 1990;Lohmann and Roerig 1994;Swadlow 1994;Swadlow and Weyand 1981). Those axons extending more than a millimeter (Hellwig 1993;Read et al. 1997) could play an important role, because there is insufficient time to allow for serial multisynaptic transmission over the predominantly short axons (Sholl 1956) to the distances covered by observed phase cones. We infer that the phase cones manifest a group property, the velocity of a state transition, not a transfer of information ). Physicists call the spread of change at different velocities in media anomalous dispersion. An example is hitting a metal rod on one end with a hammer; the sound wave gets to the other end of the rod before the impulse does. Anomalous dispersion in the cerebral cortex may account for the rapidity with which state transitions can spread over large distances faster than can be achieved by serial synaptic transmission, relying instead on small percentages of long axons in corticocortical projections and on the extreme sensitivity of the cortical ensemble as it approaches the border of its existing basin of attraction, analogous to the "local response" that sensitizes axons brought near threshold (Hodgkin and Rushton 1946). We conclude that the radial phase patterns in the EEG provide strong evidence that AM patterns form by self-organizing cortical state transitions that direct the cortices through a landscape of attractors, each yielding a reproducible AM pattern (Freeman 1992;Freeman et al. 1995).
Fifth, the delays in axonal propagation manifested in the phase cones may serve to delimit the spatial boundaries of neocortical AM patterns during both construction and read-out. The neocortical neuropil forms a continuous sheet, as shown by the phenomenon of spreading depression of Leaõ (Bures et al. 1974), which stops only at the borders of the neocortex with the archicortex and the callosum. The submillimeter microscopic architecture of cortex is spatially coarse-grained by its input projections into cortical columns and barrels, which are smaller by an order of magnitude than the mesoscopic AM patterns. The EEGs of cortical areas that are separated by macroscopic distances lack high spatial coherence, giving evidence that the mesoscopic AM patterns must have soft boundaries. The obligatory axonal propagation delays may provide the neocortical dynamic boundary conditions, which are required to give the different areas a degree of autonomy, while not freezing them into anatomically fixed arrangements. Within each area the spatially coherent EEGs manifest cooperative interactions among millions of neurons but cooperativity must weaken with phase dispersion. The radial phase gradients can serve to attenuate mesoscopic synaptic interactions with distance. With no hard edges, the half-power radius (Ϯ cos 45°) can serve to define the functional boundaries and sizes for local cooperative domains, at the interface between microscopic neural activity and cortical mesoscopic states. The cosine values then provide the desired measure of the degree of relatedness of the gamma activity throughout a temporal segment of the EEG in a mesoscopic cortical area.
Sixth, the classification of AM patterns with respect to CS reveals that the classificatory information is homogeneously distributed in space; no recording channel in an array is any more or less important than any other (Barrie et al. 1996). This property shows that the information relating to the topographic mapping of the sensory input has been spatially disseminated by the dynamic operation of constructing AM patterns. The independence of the mesoscopic pattern from the details of the input-dependent cortical architecture may be critical for the integration of multisensory percepts in which the local spacetime gradients peculiar to the retinal, cochlear, and cutaneous mappings are no longer relevant. This operation also pertains to the read-out of AM patterns. The olfactory bulb transmits through the divergent-convergent lateral olfactory tract that imposes a spatial integral transformation. Each bulbar neuron disseminates its pulses broadly, and each target neuron integrates over input from a wide area of the bulb. This operation sums only the transmitted activity having minimal frequency and phase dispersion, and it attenuates the activity that is not spatially coherent. In effect, it defines the self-organized bulbar mesoscopic activity as signal, and the sense-dependent micro-scopic activity as noise. Whether the outputs of neocortical primary sensory cortices likewise undergo spatial integral transformation is unknown, but the fact that the neurons in the secondary sensory areas have very large receptor fields gives evidence that a comparable mesoscopic operation may be performed on the outputs of neocortices. In effect, every local population sends the same signal to all targets of a transmitting cortex, in the manner that each fragment of a broken hologram has the entire picture, and each receiving target selects what is appropriate to it. An alternative hypothesis, holding that discrete microscopic networks of selected neurons embedded in the neuropil have been synthesized by large local synaptic modifications during learning, suffers from combinatorial explosions in matching input and output topographic maps, and it does not readily offer a neural code that is common to all perceptual ports beyond the early stages of sensory information processing but prior to multisensory percept formation.
The prepyriform cortex provides an important exception because its phase gradients are not conic but conform instead to the propagation delays imposed by the directions and velocities of the axonal branches in the lateral olfactory tract or occasionally to axonal projections in the opposite direction (Freeman 1973(Freeman , 1999. This finding indicates that this cortex lacks the autonomous dynamics required for the self-organization of AM patterns and is directly driven by the bulb (Boudreau and Freeman 1963;Bressler 1987). This difference offers a distinction between two types of cortex, one that constructs AM patterns, the other that integrates them selectively. These two types may work pairwise in reciprocity. They might be sought among neocortical areas by measurement of their PM patterns, so that their respective roles might be better understood. A P P E N D I X Variable list V ៝ 1 ϫ 64 RMS amplitude column vector (V ៝ ) R RMS amplitude of the EEG for each channel e EEG recording channel index P ៝ spatial phase distribution w window index m number of overlapping, 128 ms EEG windows separated by 2 ms intervals f frequency index C ៝ 1 ϫ 64 cone vector representing the conic pattern regressed onto the phase pattern conic function regressed onto the phase distribution A ៝ Љ apices of conic regressions with Ͻ20% residual A ៝ Ј apices of conic regressions with Ͻ40% residual nЉ numbers of apices within each segment from conic regressions with R Ͻ 20% nЈ numbers of apices within each segment from conic regressions with R Ͻ 40% i index j index A 1 distance indicating how far (in mm) an apex (located 2q ms in time from the initial phase cone) was from the initial site distance indicating how far any two consecutive phase cones were from each other q number of consecutive time points in a stable phase segment 1 ϫ q vector demarcating each stable phase segment time in milliseconds beginning at window w and extending to window w ϩ q s phase segment index record index number of segments per record (r) histogram of phase segments S EEG phase velocity in M sec Ϫ1 b slope of the conic surface regressed onto the phase distribution D modal diameter in cm-defined as twice the distance from the apex to the half-power point (the square of cosine 45°ϭ 0.5) of the EEG