INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

In turtles, visual stimulation evokes a complex wave of depolarization within the cerebral cortex (Prechtl 1994; Prechtl et al. 1997, 2000; Robbins and Senseman 1998; Senseman 1996, 1999; Senseman and Robbins 1999a). Within the dorsal cortex (D), a primary visual area that receives a direct afferent projection from the lateral geniculate complex of the thalamus, visually evoked waves propagate at a relatively uniform velocity of 0.01-0.04 m/s from the rostral pole to the caudal pole. However, as the wave travels dorsomedially, the advancing wavefront slows significantly as it leaves the primary visual area and passes through a secondary visual area, the dorsomedial cortex (DM).

A useful first step in ascertaining the possible informational content of cortical waves evoked by sensory stimulation is to separate complex cortical waves into their intra- and intercortical wave motions. In favorable situations, systems exhibiting complex spatiotemporal behaviors can be characterized by decomposing their responses into simpler components using orthogonal coordinate rotations. Principal component analysis (PCA) and Fourier analysis are familiar examples of such transformations (Glaser and Ruchkin 1976). These decomposition techniques are often applied to waveform data such as action potentials (Abeles and Goldstein 1977), evoked field potentials (Chapman and McCrary 1995; Kisley and Gerstein 1999; Musial et al. 1998), and synaptic potentials (Astrelin et al. 1998). In each case, the goal is to represent a collection of waveforms as a linear sum of basis waveforms. In Fourier analysis, the basis waveforms are predetermined as sine and cosine functions. In PCA, the basis waveforms are specific to each data set and are computed from temporal correlations in amplitude change across the signal collection.

The term PCA is often used interchangeably with singular-valued decomposition (SVD) and Karhunen-Loéve (KL) decomposition. In common practice, however, PCA usually refers to the technique when the mean is removed from the data set prior to the calculation of the basis functions, whereas mean removal may or may not be performed in either SVD or KL decomposition. Because we retain the mean in our calculations, we have avoided the appellation "PCA." Between the two remaining names, we have selected the term KL decomposition by virtue of its historical use in obtaining separable approximations for systems whose behavior can be described by partial differential equations. A more detailed discussion of KL decomposition and mean removal is contained in the APPENDIX.

Instead of analyzing time relationships of the waveform data, we have used KL decomposition to study spatial relationships of the response (Robbins and Senseman 1998; Senseman and Robbins 1999a). When applied to a movie of VSD images, the analysis decomposes the evoked response into a series of KL basis images (Everson et al. 1997, 1998) that capture spatially coherent changes in signal amplitude. Because changes in the amplitude of the turtle cortical VSD signals correspond closely to changes in the dendritic membrane potential of the pyramidal cells (Senseman 1996), one might conclude that KL basis images show regions where coherent changes in membrane potential occurred during wave propagation. However, as we shall demonstrate in this paper, care must be taken in associating neuronal activity patterns with the shapes of these images.

The present study examines how the static basis images generated by KL decomposition interact dynamically to capture propagating cortical waves. Some of these results have appeared previously in abstract form (Senseman and Robbins 1999b, 2000). In addition to data obtained in our laboratory from an in vitro eye-brain preparation stimulated with brief light flashes, we have also applied the technique to a previously published data set (Prechtl et al. 1997) acquired from an in vivo turtle preparation stimulated with a looming white ball.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Experimental methods

Details concerning data acquisition for the in vivo turtle preparation can be found in Prechtl et al. (1997). Experiments that utilized the in vitro eye-brain preparation were performed on adult (shell length, 10-21 cm) pond turtles (Pseudemys scripta elegans) using protocols approved by the University's Institutional Animal Care and Use Committee (IACUC). Details concerning the surgical procedures, the imaging of VSD signals, and the analysis of these signals by KL decomposition have been described previously (Robbins and Senseman 1998; Senseman 1996, 1999; Senseman and Robbins 1999a). Briefly, VSD signals were imaged from an in vitro preparation consisting of the complete CNS with one eye attached (Fig. 1A). The eye's anterior chamber was removed to produce an eyecup to facilitate visual stimulation. To provide a mechanically stable and optically flat surface for VSD imaging, a U-shape incision was made in the contralateral cerebral cortex and the medial wall of the cortical sheet. The freed tissue was reflected laterally and pinned securely to the chamber bottom with pins fashioned from tungsten wire (0.025 mm diam). It should be noted that this procedure does not compromise the normal visual input to the cortex because the geniculocortical afferents are carried by the lateral forebrain bundle (Lfb) that transverses the lateral cortical wall (Fig. 1A) (Kreigstein 1987). It is also noteworthy that unfolding the cortical sheet does not disrupt the extensive intracortical circuitry within the area being imaged in contrast to the extensive damage that occurs in conventional brain slice preparations.

View larger version (19K): [in this window] [in a new window]   Fig. 1. Experimental setup. A: the image shows a photograph of the experimental preparation in which the left cerebral cortex has been unfolded for voltage-sensitive dye (VSD) imaging. Top left inset: the dark circular object at the top left of the preparation is the attached eye. The anterior chamber has been removed, forming an eyecup, to expose the retina. Large- or small-caliber fiberoptic light guides were used to provide diffuse or spot illumination of the retinal surface, respectively. A U-shape incision in the cortical sheet allowed the medial wall to be reflected laterally and pinned flat against the bottom of the recording chamber. Main figure: a coronal section through the brain shows how fibers from the dorsal lateral geniculate complex reach the visual cortex through the lateral forebrain bundle (Lfb). The dotted line shows the position of the cortical tissue prior to its lateral reflection. To improve visualization, the anterior dorsal ventricular ridge (ADVR) was removed. The cortex is illuminated from below by a 700-nm light. The ependymal surface of the visual cortex is focused by a ×4 objective on a 464-element photodiode array. B: a single element in the photodiode array measures the average of electrical activity in a 150 × 150 × 750 µM volume containing ~600-700 spiny pyramidal neurons (P) and a smaller number of aspiny inhibitory neurons (I) in cortical layers I and III. Notice that most of the active membrane of the pyramidal cells (>90%) is composed of the apical dendrites in cortical layer I and the basal dendrites in layer III. C: correspondence between the VSD signal and the compound postsynaptic potential evoked by visual stimulation in a single experimental trial. The VSD amplitude has been scaled so that the waveforms have comparable peak values.

The preparation was stained for 1-2 h at 10-12°C by submersion in turtle Ringer solution [which contained (in mM) 96.5 NaCl, 2.6 KCl, 4.0 CaCl₂, 2.0 MgCl₂, 31.5 NaHCO₃, and 10 D-glucose, gassed to pH 7.6 with 95% O₂-5% CO₂] containing either the merocyanine dye NK-2495 or a close analogue, NK-2761, obtained from Nippon Kankoh-Shikiso Kenykyusho (Okayama, Japan). VSD signals were recorded at 720 ± 30 nm for preparations stained with NK-2495 and at 700 ± 25 nm for preparations stained with NK-2761. The time interval between successive frames is 2.83 ms unless otherwise noted. A ×4 water-immersion objective (0.13 numerical aperture) projected a real image of the cortical sheet on to the surface of a 464-element silicon photodiode array (model MD464, Centronic, Newbury Park, CA) mounted on the video port of a large binocular microscope (model UEM, Carl Zeiss, Oberkochen, Germany) (Fig. 1A). At this magnification, each photodiode element monitored a 150 ×150 × 700 µM tissue volume containing ~600-700 spiny pyramidal cells (Fig. 1B). Based on the results of concurrent intracellular microelectrode recordings and VSD imaging, VSD signals recorded at this wavelength appear to largely reflect intracellular membrane potential changes occurring within the apical and basal dendrites of spiny pyramidal cells (Fig. 1C) (Senseman 1996).

Visual stimulation

A Grass Instruments photostimulator (model PS22D) coupled to a flexible light guide was used to deliver brief light flashes (~20 µs duration) to the eyecup (Fig. 1A, inset). For diffuse illumination, the tip of a relatively large caliber (2.5 mm diam) metal-clad light guide was positioned 1-2 cm from the eyecup to evenly illuminate the entire retinal surface. Spot illumination was achieved by bringing a smaller bore (800 µM diam), metal clad, fiber optic light guide to within 0.5-1 mm of the retinal surface. At this distance, the spot subtended ~15° of visual angle on the retinal surface, which is relatively small given an acceptance angle of 165192° for the turtle eye (Northmore and Granda 1991).

Normalization and data exclusion

The photodiode array has 464 detectors arranged in a hexagonal pattern within a 24 × 24 bounding box. Each data set consisted of 576 snapshots of the photodiode array output measured at intervals of 2.83 ms. KL decomposition was performed on the 464-element vectors that were time snapshots of the response. The results were then embedded into a 24 × 24 array and padded with zeroes to correctly depict the positions of the responses for display. Pixel intensity was proportional to a detector's normalized, instantaneous photocurrent. VSD signals obtained in single trials had sufficiently large signal-to-noise ratios that averaging or filtering was not required. However, to compensate for spatial inhomogeneities in the bright field illumination, the amplitude of each VSD signal was normalized to its resting light level measured prior to the experimental trial (Senseman and Rea 1994). This normalization scheme worked well except in cortical areas in which the overlying remnants of the dorsal ventricular ridge scattered a significant portion of the transmitted light. VSD signals from these areas had very low amplitudes and, when normalized, proved to be excessively noisy. To prevent these signals from corrupting the analysis, we excluded the output of any element if its resting light level was <5% of the median of the 20 largest resting light levels in that trial. We also excluded the contributions of approximately four detectors that had become defective over time and generated only noise. Elements excluded from analysis had their VSD signal values set to zero.

Contour maps of wave front latency

To quantify the cortical wave front velocity, algorithms were developed to determine response latency directly from the VSD signal waveforms, without user intervention. Figure 2 shows a typical VSD signal with three possible latency measurements relative to the time of stimulus onset: onset latencythe time at which the response first becomes visible, half-height latencythe time at which the signal reaches half of its absolute signal maximum, and peak latencythe time at which the signal reaches its absolute maximum value. In practice it proved difficult to automatically select onset and peak latencies with a high degree of accuracy. Onset latency requires a subjective determination of when the signal first emerges from baseline noise. This determination could be based on objective criteria such as changes in the statistical character of the signal or be calculated by stepping back from the peak latency to the time at which the signal last exceeds a threshold noise level directly prior to ascending to its peak. However, in either case, the selection is still highly dependent on the exact criterion selected for response initiation as well as the signal-to-noise ratio. Peak latency also proved difficult to determine unequivocally because the initial response often exhibited several peaks due to oscillations at the maximum. For the trace shown in Fig. 2, the latency calculated from the absolute signal maximum (peak) is almost twice the value obtained using the first relative maximum (first maximum). Half-height latency can be determined algorithmically with a relatively high degree of temporal accuracy. Because the initial upstroke of the VSD signals occurs rapidly, the time to reach half-height is largely independent of which relative maximum the algorithm selected as the peak. The algorithm is applied to the VSD signal from each detector as follows. The maximum value of the signal at the detector is determined. Once the peak value is found, the algorithm steps forward from the time of stimulus onset to determine the time the detector signal first reached half of this maximum value. The times to half-height are stored in an array and displayed as a contour map using Mathematica. The displays in this paper use a progressive gray-scale coloring of the contours. The pixels at which the signal first reaches half-height are colored light gray, while the pixels with the highest signal latency are colored black.

View larger version (16K): [in this window] [in a new window]   Fig. 2. Latency measurement. Half-height latency is the time required for the VSD signal to 1st reach half of its maximum (peak) height following stimulus onset (). Other possible latency measurements, such as time to the signal maximum or time of first visible onset are also shown.

KL decomposition of VSD images

KL decomposition (Everson et al. 1997, 1998; Robbins 1998; Robbins and Senseman 1998; Senseman and Robbins 1999a; Sirovich 1987; Sirovich et al. 1996) was used to find spatially coherent changes in visually evoked cortical activity. Each photodiode image of the VSD stained cortex is treated as an N-dimensional vector, where N is the number of pixels in the image. Changes in cortical activity that are spatially correlated can be found by rotating the coordinate system for this N-dimensional space so that the first axis (most important basis image) resolves the greatest amount of total variance among the response images (i.e., passes through the center of the vectorized image cloud), the second basis image resolves the greatest amount of the residual variance, and so on until all the variance has been accounted for.

Senseman and Robbins (1999a) observed that the dominant basis images have a modal structure and that the most important basis image is always unimodal and ovoid in shape. The other important basis images can be roughly characterized by the alignment of the maxima and minima relative to the principal axes of the most important basis image. We denote the most important basis image by M_1,1 to indicate that it has a single extrema along its major and minor axes. Basis image M_2,1 has two extrema (a maximum and minimum) separated by a line of antisymmetry that runs approximately along the major axis of M_1,1. Similarly basis image M_1,2 has two extrema (a maximum and minimum) separated by a line of antisymmetry that runs approximately along the minor axis of M_1,1. The notation is motivated by the structural similarity that the basis images have with the normal modes of a drum. The APPENDIX explains in more detail the correspondence among M_1,1, M_2,1, and M_1,2 and the results of KL decomposition.

Each axis or basis image resolves spatially correlated changes that occurred in the cortex. In the two extreme cases, only a single basis image is needed if all the cortical images are exactly alike, and N basis images (1 for each movie frame) if there is zero correlation among the movie frames. The response is generally considered "low-dimensional" if a few basis images account for most of the total variance. While there is no generally accepted criterion for what constitutes low dimensionality, visually evoked cortical waves in the turtle are clearly low-dimensional even by the most conservative standards. The most important basis image captures 70-80% of the response energy, with an additional 10-20% captured by the two next most important basis images (Senseman and Robbins 1999a). As a result, the response f(t,x) is well-approximated by

(1)

where the time histories {a_i,j(t)} and the spatial "KL" functions {M_i,j(x)} are fully determined by the data.

Computation

KL decomposition of VSD cortical images was performed on a SUN Microsystems workstation (model Ultra60) using custom software. Various figures and computer animations were generated on an Intel Pentium III-based personal computer (PC) using custom software, Mathematica, MatLab, Adobe Photoshop and Adobe Premiere. A MatLab script for decomposing VSD data files into their basis images and displaying them is provided without charge as part of the supplemental materials that accompany this publication.¹ (See the APPENDIX for additional details.)

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

The results presented below were based on the analysis of >100 experimental trials obtained in six separate eye-brain preparations.

Intra- and intercortical wave motions

The initial spread of depolarization forms a front of excitation that spreads anisotropically across the cortical surface. The half-height latency provides an estimate of how long it takes for the excitation front to reach a particular cortical location and is closely related to the wave front velocity. Senseman (1999) manually measured wave front latency of VSD signals along linear transects and found that the turtle cortex responded to brief light flashes with waves of depolarization that propagated rostrocaudally through the dorsal cortex and dorsomedially into the adjoining dorsomedial cortex. We have re-examined the wave front dynamics by determining wave front latency over the entire cortical surface monitored by the photodiode array. Figure 3 shows wave front latency displayed as contour maps for left and right cerebral cortical preparations. The area encompassed by each contour map has been shaded on a line drawing of the preparation shown at the upper right. To the left of the line drawing is a photomicrograph of the reflected cortical sheet with a superimposed image showing the recording locations for the 464 elements (pixels) in the photodiode array.

View larger version (63K): [in this window] [in a new window]   Fig. 3. Intra- and intercortical waves propagate at different velocities as illustrated by the response of left and right cortices from different experimental preparations. Top left inset: an outline of the photodiode array is shown superimposed on a photomicrograph of the preparation. Top right inset: semi-diagrammatic drawing of the experimental preparation (cor34 for the left cortex and cor32 for the right cortex) showing the recording location of the 464-element array on the unfolded left cortex. Main figure: contour map of wave front latency. Dashed lines in the main figure and insets indicate the approximate boundary between the dorsal cortex (D) and the dorsomedial cortex (DM). Solid line in the main figure indicates residue tissue from the anterior dorsal ventricular ridge (ADVR).

The wave front latency maps exhibit four spatial characteristics that were observed in all trials in which cortical activity was evoked by visual stimulation. First, the cortical wave activity was initiated in the rostral pole of the dorsal cortex within the area showing the lightest shading. This location is approximately where the geniculocortical afferents first establish their synaptic contacts (Desan 1984; Mulligan and Ulinski 1990). Second, the cortical wave propagates into the caudal pole of the dorsal cortex at a relatively constant velocity. The constancy of this wave front velocity is reflected in the even spacing between contour lines along the rostrocaudal axis. Third, the wave front slows significantly along the dorsomedial axis as it moves into the dorsomedial cortex. The dashed-line indicates the approximate boundary between these two cytoarchitecturally distinct cortical areas. Like the second visual area in mammals (V2), dorsomedial cortex does not receive a direct geniculocortical input, but rather receives visual information via its reciprocal connections with dorsal cortex (Desan 1984). Finally, the wave front velocity decreases less in rostral areas than in more caudal areas. Collectively, these results confirm and extend the earlier observations of Senseman (1999), providing a more complete view of the spatial heterogeneity of the wave front velocity as it spreads within and between the turtle's primary and secondary visual cortical areas.

Correspondence between KL basis images and wave front latency

In an effort to determine a possible relationship between wave front propagation and the basis images produced by KL decomposition, we superimposed wave front latency maps on to the three largest KL basis images (Fig. 4). The spatial organization of KL basis image M_2,1 appears to be clearly related to wave front latency. The minimum of M_2,1 (blue-purple area) is largely congruent with dorsomedial cortex at the point where the wave front velocity experiences its most pronounced deceleration. The line separating M_2,1's maximum from its minimum largely follows the dorsal/dorsomedial cortical boundary, placing the maximum (yellow-orange-red area) within dorsal cortex and the minimum within dorsomedial cortex.

View larger version (49K): [in this window] [in a new window]   Fig. 4. KL basis images and wave front contour maps share a similar spatial organization. The wave front latency contour maps shown in Fig. 3 (trial cor3409 for the left cortex of experimental preparation cor34 and trial cor3207 for the right cortex of experimental preparation cor32) have been reproduced at the top of the figure. The 3 largest KL basis images calculated for the corresponding trial (M1,1, M1,2, and M2,1) are shown in the left column below. Right: the superposition of the 3 KL basis images on to the wave front latency contour map.

The spatial organization of mode M_1,2 is also related to wave front latency. The maximum of M_1,2 was largely congruent with the rostral area within the dorsal cortex exhibiting the earliest wave front latencies. The line separating the maximum of M_1,2 from its minimum runs roughly perpendicular to the rostrocaudal axis, and the minimum of M_1,2 falls within the more caudal areas of dorsal cortex and dorsomedial cortex. Although the major axis of the basis image M_1,1 is aligned with the dorsal/dorsomedial cortical boundary, its contours of constant response did not appear to be aligned with the wave front latency maps.

The basis images generated by KL decomposition reveal cortical areas where correlated changes in neuronal activity occurred. These changes may be in the same direction ("positive") or in the opposite or anti-correlated direction ("negative"). The color scheme used for the basis images in Fig. 4 uses "warmer" colors (yellow, orange, and red) to indicate positive correlations and "cooler" colors (blues and purples) to indicate opposite correlations. Areas left uncolored are areas in which the correlation was calculated to be near zero.

One potential problem with this coloring scheme is that encourages the view that different cortical areas are "positive" or "negative" in an absolute sense. This view is incorrect. The assignment of which correlation peak should be considered "positive" (a maximum) or "negative" (a minimum) is completely arbitrary. To ensure consistency, we have assigned polarities to the basis images so that M_1,1 has a positive polarity, M_1,2 has its maximum in the rostral pole of dorsal cortex and M_2,1 has its minimum in dorsomedial cortex.

Once a polarity has been assigned, however, the contribution of that basis image to the response is fixed. In other words, the polarity of a basis image determines the sign or polarity of its projection. Reversing the polarity of the basis image simply inverts the sign of its projection. Consequently, a given basis image always "adds" or "subtracts" from the overall response in exactly the same manner regardless of which polarity has been assigned to its correlation peaks. The polarity scheme we have selected orients the projections for the three basis images so they initial move in a positive direction during the onset of wave propagation.

Reconstruction of the response

Inherent in the concept of decomposition is the ability to reconstruct the original response by linearly combining the basis images. If the contributions of all of the basis images returned by the analysis are included, an exact copy of the original response can be reconstructed. In a low-dimensional system, only a few basis images are needed to produce a relatively good representation of the response. In this section we evaluate how well the low-dimensional approximation of Eq. 1 reproduces the wave motion. The simplest measure of quality is the amount of data set energy captured by the basis images. Table 1 gives the fraction of total data set energy accounted for by each basis image for three representative trials (cor3204, cor3207, and cor3208). In each case, three basis images account for >97% of the energy. Movie 1² of the supplemental materials compares the quality of the original data set with a data set reconstructed using only KL basis images M_1,1, M_1,2, and M_2,1.

Another way to evaluate the quality of these reconstructions is to determine how well they reproduce the wave front latency maps. Figure 5 compares the wave front latency maps for representative trials recorded in the left (cor34) and right (cor32) cortices. The top pair of images shows the latency maps calculated from the original data sets. The middle pair shows maps calculated from data sets reconstructed from the first 10 KL basis images, and the bottom pair from data sets reconstructed using only KL basis images, M_1,1, M_1,2, and M_2,1. Not unexpectedly, the wave front latency map for the data sets reconstructed from ten basis images looks more similar to the original data set map than the map from the data set reconstructed with only three. Nevertheless, the three-image data set generated latency maps that exhibit the most prominent characteristics of the intra- and intercortical wave propagationnamely the constant velocity of the intra-cortical rostrocaudal wave and the pronounced slowing of the inter-cortical wave as it passes from dorsal cortex into dorsomedial cortex.

View larger version (52K): [in this window] [in a new window]   Fig. 5. A comparison of latency maps calculated from original and reconstructed data sets for the left (cor3409) and right (cor3207) cortices. Top: the front latency map calculated from the original data set. Middle: the latency map calculated from a reconstruction using the 10 most significant KL basis images. Bottom: the latency map calculated from a movie reconstruction using only the 3 most significant KL basis images.

It was not possible to reconstruct data sets that exhibited both intra- and intercortical traveling waves using less than three basis images (Fig. 6). In Fig. 6A, the column labeled latency shows the three wave front latency maps for data sets reconstructed from only the largest basis image, M_1,1 (top), from basis images M_1,1 and M_1,2 (middle), and from basis images M_1,1 and M_2,1 (bottom). The boxed images in the center of Fig. 6A show superpositions of the basis images with their corresponding wave front latency maps. In the column labeled advance, only the positive part (maximum) of the mode has been superimposed on the latency map while in the column labeled delay, only the negative part (minimum) has been superimposed.

View larger version (36K): [in this window] [in a new window]   Fig. 6. Traveling waves can be generated by the dynamic interaction of 2 (or more) basis images. A: correspondence among the 3 largest KL basis images and wave front latency contour maps from data sets reconstructed using only 1 or 2 KL basis images. Far left: the 3 largest KL basis images, M1,1, M1,2, and M2,1 for trial cor3207; far right: the wave front latency contour maps for 3 different reconstructed data sets. The top contour map is for a data set reconstructed using only KL mode M1,1; middle contour map: a data set reconstructed using KL basis images M1,1 and M1,2; and bottom contour map: a data set reconstructed using KL basis images M1,1 and M2,1. Middle left and middle right: superposition of the KL basis images with the wave front latency contour maps shown at the right. Middle left (advance): only the positive portion of the KL mode is superimposed; middle right (delay): only the negative portion of the KL mode is superimposed. (Note: mode M1,1 has only a positive part.) Ba: projections of the data set on the 3 largest basis images are plotted as a function of time. The amplitudes have been scaled to produce similar sized curves. The bar at the right of each projection indicates the magnitude of the scaling. When the projection is positive (t = t1), M1,2's maximum (the red, orange, yellow area) is located rostrally in visual cortex and its minimum (the blue, purple area) is located caudally in visual cortex. When the projection is equal to 0 (t = t2), M1,2 makes no contribution to the overall response. When the projection is negative (t = t3), the effects of the maximum and minimum reverse so that the maximum now subtracts from the response in the rostral pole and the minimum adds to the response in the caudal pole. Bb: schematic of how a traveling wave in the cortex can be generated from the interaction of 2 standing waves. The x axis corresponds to space (not time) and represents a linear transect along the rostrocaudal axis of the cortex. Left: the amplitude of mode M1,1's () and mode M1,2's ( · · · ) projection across the cortical sheet. For simplicity, the projection amplitude as a function of distance has been drawn as a sine wave and mode M1,1's projection has been held constant. Right: the linear addition of the 2 images. Notice the change in position of the maximum to a more caudal location at later time steps.

Wave front latency maps calculated from data sets that were generated from one or two basis images are clearly different from those generated from three or more basis images. The map calculated from the data set reconstructed using only basis image M_1,1 is simply an outline devoid of internal structure because all areas encompassed by the image reach half of their maximum at the same moment 190 ms after stimulus onset. On the other hand, the map calculated from a data set reconstructed using basis image M_1,1 and one of the smaller basis images, either M_1,2 or M_2,1 shows different cortical areas reaching their half-height latencies either before (light gray) or after (black) the 190-ms latency of the single basis image data set.

As can be seen, the lightest areas in the wave front latency maps coincide with the maxima of basis images M_1,2 and M_2,1, whereas the darkest areas coincide with their minima, suggesting a physical interpretation. KL decomposition decomposes the response into a large standing wave within the area encompassed by basis image M_1,1. Within this area, depolarization and repolarization occurs at the same rate and at the same time (but not necessary by the same magnitude) so that all areas reach their half-maximum at the same instant (i.e., 190 ms after stimulus onset).

The two smaller basis images, M_1,2 and M_2,1 can be viewed as serving to transform this standing wave into two separate traveling waves. Very early in the response, the maxima of M_1,2 and M_2,1 serve to accelerate the rate of rise of the response while their minima work in the opposing direction. Relative to the standing wave generated by basis image M_1,1, the maximum of basis image M_1,2 advances the response in the rostral pole of the dorsal cortex by 30 ms, whereas the minimum delays the response in the caudal pole by 30 ms. Similarly, the maximum of basis image M_2,1 advances the response within the dorsal cortex by 10 ms while the minimum delays the response within the dorsomedial cortex by 174 ms. As the response progresses, the contributions of the two smaller basis images change in a predictable fashion to generate the intra- and intercortical traveling waves.

The contribution of each basis image to the overall response can be empirically determined by projecting the basis image onto the original data set. Figure 6Ba shows projections of basis images M_1,1, M_1,2, and M_2,1 over the entire experimental trial. The squared magnitude of a basis image's projection is a quantitative measure of how much response energy that image captured on a frame-by-frame basis. For bimodal basis images such as M_1,2 and M_2,1, the polarity of the projection defines how the maximum and minimum contribute to the response. Using basis image M_1,2 as an example, when its projection is positive (t = t₁), the maximum (yellow-orange-red area) increases the response in the rostral pole of the dorsal cortex, whereas the minimum (blue-purple area) decreases the response in the caudal pole by the same relative amount. However, when the projection becomes negative (t = t₃), the maximum now serves to decrease the response in the rostral pole while the minimum serves to increase the response in the caudal pole. When the projection is zero (t = t₂), basis image M_1,2 makes no contribution to the response.

A graphical demonstration of how basis images M_1,1 and M_1,2 interact to establish the cortical wave along the rostrocaudal axis is provided in Fig. 6Bb. For simplicity, the waveforms have been depicted as one-dimensional sine functions. The standing wave corresponding to basis image M_1,1 is shown at a constant amplitude, whereas the basis image M_1,2 is shown when its projection is positive (t₁), zero (t₂), and negative (t₃). The peak of the summed waveform can be seen to travel from its initial rostral position caudally as the sign and magnitude of M_1,2's projection change. Movie 5 shows an animation of this interaction.

Another way to infer the dynamic contribution of different basis images is to compare movies of original data sets with reconstructed movies using various combinations of the most important basis images. Movie 1 shows an animation of the original data sets cor3409 and cor3207 along side movies reconstructed using the three most important of their respective basis images, M_1,1, M_1,2, and M_2,1 (Eq. 1). Movie 2 shows an animation reconstructed from the same data sets only using a_1,1(t) M_1,1(x). Movies 3 and 4 show reconstructed movies using a_1,1(t) M_1,1(x) + a_1,2(t) M_1,2(x) and a_1,1(t) M_1,1(x) + a_2,1(t) M_2,1(x), respectively. Selected frames from these movies, spanning the time interval over which the various cortical regions reached their half-maximum value, are presented in Fig. 7. The frames in the top row, recorded at 156 ms poststimulation, capture the moment when the first cortical region had reached its half-maximum value while the frames in the bottom row, recorded at 275 ms poststimulation, capture the moment when the last cortical region reached its half-maximum.

View larger version (108K): [in this window] [in a new window]   Fig. 7. A progression of frames showing the wave captured by different levels of data set reconstruction for trial cor3207. Each row of 4 side-by-side images consists of the original frame, the 3-basis image construction a1,1(t) M1,1(x) + a1,2(t) M1,2(x) + a2,1(t) M2,1(x), the 2 image reconstruction a1,1(t) M1,1(x) + a1,2(t) M1,2(x), and the 2 image reconstruction a1,1(t) M1,1(x) + a2,1(t) M2,1(x). The number at the top left of each row shows the time in milliseconds because the onset of a diffuse light flash to the contralateral eye. The frames in the top row show the moment in time when the first cortical region reached its half-maximum value, the frames in the bottom row show when the last cortical region reached this value.

Projections provide information on wave direction and velocity

Propagating cortical waves, evoked by electrical stimulation in neocortical brain slices, have been observed to reverse direction under certain conditions (Chagnac-Amitai and Connors 1989). In two back-to-back trials in the current study, we observed a visually evoked cortical wave reverse direction after reaching the caudal boundary of the dorsal cortex. In both instances, the reflected wave propagated caudorostrally at 0.23 m/s or approximately six times the normal velocity of visually evoked cortical waves.

Figure 8A shows selected frames from one trial in which a reflected wave was observed. For this movie, the level of membrane depolarization captured by the VSD imaging is displayed as a surface contour. To emphasize the anisotropy of wave propagation, a color scheme is used for each pixel based on its half-height latency, with dark red indicating cortical areas of earliest wave front arrival. Initially, all pixels are displayed in gray. When the response amplitude at a pixel exceeds its half-height, the pixel is displayed in its latency color. When its amplitude falls below its half-height, the pixel's color is returned to gray. Movie 6 shows the complete movie of this response.

View larger version (46K): [in this window] [in a new window]   Fig. 8. Projections provide information on the speed and direction of wave propagation. Aa: primary wave propagation in the rostrocaudal direction. Five selected frames from a movie showing the primary spread of depolarization evoked by visual stimulation. VSD signal amplitudes are represented as a surface contour plot using the following color scheme. The surface area corresponding to a particular photodiode element was colored gray whenever the VSD signal amplitude was less than half of its peak value. When the VSD signal reached its half-maximum, the color of the surface was changed from gray to 1 of 12 color values. Areas that first reached their half-maximum were colored dark red, areas that reached their half-maximum last were colored dark purple and areas reaching their half-maximum between these extremes were assigned an intermediate color. The assigned color remained fixed as long as the VSD signal amplitude remained at, or above, the half-maximum value. It was changed back to gray whenever the signal amplitude dropped below the half-maximum threshold. Ab: secondary reflected propagation in the caudorostral direction. The 5 selected frames are a continuation of the movie shown in a. B: M1,2's projection inverts when the cortical wave reverses direction. Three trials from the same experimental preparation (cor32) are shown. Left: data from a trial (cor3204) in which the initial depolarization was followed by a slow secondary depolarization. Middle (b) and right (c): data from 2 trials (cor3207 and cor3208, respectively) in which the initial depolarization was followed by a fast secondary depolarization that reflected off of the caudal pole of the visual cortex and propagated in a caudorostral direction. The data shown in Bb came from the same experiment as shown in A of this figure. The VSD signals have been recorded from a rostral (red) and a caudal (green) location as indicated on the photomicrograph of the preparation shown at the left of the row. The bottom 3 rows show the projections of 3 data sets on the global basis images shown at left. The basis images are displayed with over bars to indicate that they were computed by concatenating all of the trials (26) obtained from the preparation prior to KL decomposition. The arrows point to the inversion of mode M1,2's projection during the reflected wave.

Figure 8B compares VSD signal waveforms and the basis image projections for a trial without a reflection (a) with two trials that have reflections (b) and (c). Because KL decomposition computes a unique coordinate system for each data set analyzed, it would inappropriate to directly compare the projections obtained in separate trials even though the trials come from the same preparation. This limitation can be overcome, however, by simply concatenating the VSD movies obtained in separate trials end-to-end and performing KL decomposition on the combined data set to obtain a common coordinate system or global basis. (Despite a superficial similarity, finding a global basis by concatenation is distinctly different from, and should not be confused with, averaging multiple trials.) The global basis used in Fig. 8B was calculated by performing KL decomposition on a single data set obtained by combining all 26 trials performed in this preparation. Overall, the 26 trials included 9 trials with diffuse light flash stimuli of varying intensity and 17 trials with spot stimuli administered at different retinal positions. The trials displayed in Fig. 8B have diffuse light flash stimuli of the same intensity. Table 2 shows the fraction of energy captured in these trials by the three dominant global basis images. The global basis images themselves are shown in the first column of Fig. 8B and are denoted with overbars to distinguish them from the basis images computed from a single data set. Notice that the global basis images have the same modal structure as the self-basis images.

The two VSD waveforms shown in the top graph of each column were taken from the cortical locations indicated on the photomicrograph of the preparation shown at the left. The VSD signal recorded at the more rostral location is shown in red; the signal recorded from the caudal location is shown in green. With respect to the initial depolarization, the rostral location depolarizes before the caudal location in all three trials. This is as expected from the initial wave front propagating in a rostrocaudal direction. In the trial shown in a, the initial depolarization is followed by a smaller secondary depolarization with a slow rise-time and an even slower decay. Like the initial depolarization, the secondary depolarization also appears first at the rostral location indicating that it was caused by a secondary wave traveling in the same rostrocaudal direction.

For the trials shown in b and c, the initial depolarization is followed by a large secondary depolarization, exhibiting both a fast rise time and equally fast decay. In both trials, the peak of the fast secondary wave occurred first at the caudal location and then at the more rostral location indicating caudorostral propagation. By visual inspection of the basis image projections it is possible to deduce both the direction and the velocity of wave propagation. Inversion of the projection for basis image M_1,2 (straight arrows, Fig. 8B) indicates the reflected wave traveled along the same trajectory, but in the opposite direction, as the primary wave. The rapid propagation velocity of the reflected wave is signaled by the rapid time course of the inverted peak in the projection.

Comparison with data obtained in vivo

Even though the surgical procedure used to "unfold" the cortex did not overtly damage the geniculocortical pathway or the intra- and intercortical pathways within the cortical area imaged by the photodiode array, it is still possible that unfolding the cortical sheet and pinning it to the chamber bottom distorted the shape of the basis images. To resolve this issue, we performed KL decomposition on VSD movies recorded from the intact cerebral cortex, in vivo by Prechtl et al. (1997), who studied the same turtle species.

One difficulty in applying KL decomposition to these in vivo data sets was the presence of large-amplitude noise in some detectors. For example, the optical signals recorded by detectors 339 (green) and 340 (red) in Fig. 9 appear to be generated by a nonneural source because they are largely mirror images of each other and they exhibit large amplitude fluctuations prior to stimulus onset. By comparison, the optical signal recorded by detector 305 (blue) appears be relatively free of noise contamination and more closely resembles the responses we recorded in vitro.

View larger version (39K): [in this window] [in a new window]   Fig. 9. KL decomposition for removing correlated noise. A: optical signals at 3 locations (indicated in the diagram at left) generated in an in vivo preparation in response to a looming stimulus. The red and green pixels near dorsal/dorsal-medial boundary exhibited a high-amplitude mirrored response during both the pre- and poststimulus period. B: KL decomposition of the prestimulus response yielded a single basis image that captured >89% of the prestimulus response. After the contribution of this prestimulus basis image was removed from the entire data set, the problem detectors appeared to reflect the response signal.

In an attempt to isolate the noise contamination, KL decomposition was performed only on the VSD images recorded during the prestimulus interval. The largest prestimulus basis image, basis image 1, captured ~90% of the response energy prior to stimulus onset (Fig. 9B). By simply removing this "noise basis image" from the data set prior to performing KL decomposition on the entire movie (pre- and poststimulus intervals), it was possible to significantly improve the signal-to-noise ratio of the in vivo response without resorting to signal averaging of multiple trials (Fig. 9B).

After "denoising," we found that KL decomposition of the VSD movies recorded by Prechtl et al. (1997) generated basis images for the intact cerebral cortex that were similar in three respects to those generated for the unfolded cerebral cortex (Fig. 10). First, the largest basis image (denoted by M_1,1) was unimodal with the highest degree of coherence (red area) located within the dorsal cortex in both preparations. Second, the maxima and minima of basis image M_1,2 were largely confined to the dorsal cortex in both preparations. Third, the boundary separating the dorsal cortex from the dorsomedial cortex also separated the maxima and minima of basis image M_2,1 in both preparations. (Note that the locations of the dorsal and dorsomedial cortices appear reversed in the in vitro preparation because the cortex has been unfolded and is being viewed from its subpendymal surface.) These similarities are even more striking in view of the fact that the unfolded cortex was stimulated with a diffuse light flash, while the intact cortex was stimulated with a looming white ball.

View larger version (47K): [in this window] [in a new window]   Fig. 10. Comparison of the basis images computed for response from intact (in vivo) and folded (in vitro) preparations. The intact response was generated by a looming stimulus, whereas the folded response was generated by a flash stimulus. The dashed line separates the primary visual area, the dorsal cortex (D) and the secondary visual area, the dorsomedial cortex (DM). Note that in labeling the cortical areas in their in vivo preparation, Prechtl et al. (1997) followed the nomenclature suggested by Desan (1984) in which the visual cortex was subdivided in areas D1 and D1 (see Fig. 1C in Prechtl et al. 1997). Here we have followed the revised nomenclature proposed by Ulinski (1999) in which area D2 is now D and area D1 is now DM.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

KL decomposition has been widely used to separate the spatial behavior of complex physical systems from their temporal behavior (Aubrey et al. 1991; Sirovich and Everson 1992). In neuroscience, its use has been largely restricted to the static analysis of intrinsic signals recorded from the mammalian primary visual cortex (Everson et al. 1997, 1998; Gabbay et al. 2000; Sirovich et al. 1996). More recently, we have shown that KL decomposition can also be used to analyze dynamic activity in the turtle cortex (Robbins and Senseman 1998; Senseman and Robbins 1999a). This work demonstrated that the cortical behavior was low-dimensional with >95% of the response energy being captured by the first three basis images. Moreover, the spatial organization of these basis images superficially resembled the normal modes of a rectangular drum.

Although it was clear from our earlier work that the first three basis images did a relatively good job of capturing the cortical spread of depolarization, it was not obvious how these static images captured the dynamics of the wave motion. In other words, was there a reasonable physical interpretation for the basis images or were they merely abstractions generated by the statistical analysis? The main result of the current research is that the KL analysis decomposes the compound wave into intra- and intercortical wave components. The intracortical wave motion arises from an interaction between basis images M_1,1 and M_1,2, whereas the intercortical wave motion arises from an interaction between basis images M_1,1 and M_2,1 (Fig. 6). This result is plausible because KL basis images represent areas in which change occurs in a coherent manner, and the intracortical wave motion is clearly different from the intercortical wave motion (Fig. 3). Moreover that each wave motion required the interaction of two basis images follows directly from the fact that KL decomposition separates the cortex's spatial behavior from it temporal behavior. In other words, a basis image has no time dependency. The relative contribution of a basis image to the response (i.e., its magnitude) can change over time (Fig. 8B), but its shape and spatial location must remain fixed. Consequently, a single basis image is incapable of representing a traveling wave whose spatial location is obviously moving. On the other hand, a single basis image can represent a standing wave whose amplitude can change over time but whose spatial position remains fixed. Thus KL decomposition captures a traveling wave through the dynamic interaction between two (or more) standing waves each represented by a different basis image (Fig. 6Bb). It should be noted that the need to represent a traveling wave by a minimum of two basis functions is not a limitation of KL decomposition per se but extends to all separable decompositions including Fourier analysis.

Physiological significance of the projection trajectory

Once the physical significance of the basis images is recognized, it becomes possible to directly visualize wave dynamics by simply plotting their projection trajectories. Because the projection of the largest basis image, M_1,1, closely resembles the waveform of the largest VSD signals measured at individual pixels (Fig. 8B), it might seem plausible that this correspondence should extend as well to the smaller basis images, M_1,2 and M_2,1. However, the projections of M_1,2 and M_2,1 have trajectories that make little sense if viewed as being VSD signal waveforms, i.e., as time-dependent changes in intracellular membrane potential. Why, for example, should the projection of M_1,2 invert when the secondary depolarization is reflected but not when the secondary response propagates in the normal rostrocaudal direction (arrows, Fig. 8B)? Clearly the membrane potential of the cortical cells doesn't depolarize when the wavefront propagates in one direction and hyperpolarize in the other. Moreover, why is it that only the projection of M_1,2 inverts and not M_2,1 during the propagation of a reflected wave?

The answer to these questions lies in the fact that basis image projections, including those of M_1,2 and M_2,1, are not simple reflections of time-dependent changes in membrane potential as captured by the VSD signals, but rather a frame-by-frame weighting of a particular basis image in a reconstruction the overall response. For example, with normal, rostrocaudal wave propagation, the projection of basis image M_1,2 initially starts out positive (t = t₁), goes to zero (t = t₂), and eventually becomes negative (t = t₃, Fig. 6Ba). The intracortical wave begins within the rostral pole of the dorsal cortex because this is where the maximum of M_1,2 adds to the response while the minimum, located in the caudal pole, subtracts from the response. When the intracortical wave has reached the end of the dorsal cortex, the projection of M_1,2 reaches its most negative value. Because of the reversal in sign, the minimum now adds to the response in the caudal pole while the maximum now subtracts from the response in the rostral pole (Fig. 6Bb). If a secondary depolarization then propagates through the dorsal cortex in the same rostrocaudal direction, the projection should once again start out positive and end up negative as the wave reaches the caudal pole (Fig. 6Ba, left).

Now consider what happens during propagation of a reflected wave. For the wave to begin in the caudal pole, the projection of M_1,2 must start out negative so that the minimum adds to the response in the caudal pole while its maximum subtracts from the response in the rostral pole. As the reflected wave propagates caudorostrally, the projection should become less negative and eventually turn positive so that the maximum now adds to the response in the rostral pole and subtracts from the response in the caudal pole. (This would be equivalent to simply reversing the t = t₁ and t = t₃ in Fig. 6Bb.) In other words, the projection should appear inverted when the direction of wave travel reverses.

Why does the projection for M_1,2 invert during the caudorostral propagation of reflected waves but not the projection of M_2,1? As the reflected wave propagates caudorostrally, excitation from the dorsal cortex would still be expected to spread into the dorsomedial cortex. The crucial point is that the direction of intercortical spread is still in the same direction, i.e., from the dorsal cortex into the dorsomedial cortex. The "sign" of the trajectory peak defines the direction of wave travel, while the slope defines its speed of propagation.

Waveform decomposition and the analysis of visual processing

Because the dorsal and dorsomedial cortical areas in turtles exhibit significant differences in their anatomy and physiology (cf. Ulinski 1990), it is likely they play different roles in visual processing. The ability of KL decomposition to separate visually evoked cortical waves into intracortical wave motions largely confined to the dorsal cortex, from intercortical wave motions that propagate between these two cortical areas may help to clarify these differences.

Although dorsal cortex establishes both afferent and efferent connections with a number of subcortical areas, dorsomedial cortex does not. Its connections are restricted to other cortical areas. Besides a commissural projection to its contralateral homologue, dorsomedial cortex only makes a reciprocal connection laterally with dorsal cortex and medially with medial cortex (Desan 1984), giving it the structural and functional attributes of an association area (Ulinski 1990). Consequently, intercortical wave motions extracted by KL decomposition and captured by the projection trajectory of M_2,1 should reflect, at least in part, large-scale information flow between the dorsal and medial cortical areas. In contrast, the intracortical wave motions extracted by KL decomposition and captured by the projection trajectory of M_1,2 largely reflect rostrocaudal wave motions within the dorsal cortex. Such wave motions not only reflect the arrival of afferent visual information from the lateral geniculate but also signal the generation of efferent flow from the dorsal cortex to its subcortical targets including the lateral geniculate, the septum, the anterior dorsal ventricular ridge, and the nucleus rotundus (Ulinski 1990).

The generation of efferent flow from the dorsal cortex to the nucleus rotundus is of particular interest within the context of visual motion perception. Ulinski (1999) has proposed that cells within the dorsal cortex act as global motion detectors, integrating the output of motion-sensitive retinal ganglion cells that serve as local motion detectors in the turtle. The majority of these ganglion cells have axons that bifurcate within the optic tract. One axonal branch terminates within the lateral geniculate giving rise to the geniculocortical pathway to the dorsal cortex. The other branch terminates within the optic tectum, which in turn, projects to the nucleus rotundus of the thalamus. Like the pulvinar in mammals, the turtle's nucleus rotundus is well positioned to compare information about eye movements arriving from the tectum with movement of the visual image arriving from the cortex (cf. Grieve et al. 2000). The ability to make such comparisons would seem to be an essential prerequisite for basic visual orientating tasks such as stabilizing the visual world and avoiding objects during locomotion as well as orienting to and tracking objects moving in visual space (Ulinski 1999). Insight into how corticofugal interactions perform such comparisons might be obtained through the deconvolution of cortical and tectal wave activity evoked by motion stimuli and recorded concurrently with VSD imaging. As a point of departure, such an analysis might compare the trajectory of cortical basis image M_1,2 to the trajectories of different basis images extracted from the tectal VSD signals.

Application of KL decomposition to other experimental preparations

We believe KL decomposition to be a generally useful approach for the analysis VSD signals recorded in a variety of experimental preparations besides the turtle. When VSD imaging is performed in vivo, contamination of the VSD signals by vibrational noise arising from pulsatile blood flow and/or pulmonary activity is often problematic even when "novel" voltage-sensitive dyes are used (Shoham et al. 1999). Under such circumstances, KL decomposition may be able to improve the signal-to-noise ratio of the VSD signals (Fig. 9) without the need to average multiple trials, frequency filter the data or to digitally subtract "blank trials." Similarly, using KL decomposition to reduce noise might also prove useful for VSD recording from cardiac tissue where motion artifacts are a serious problem (Morley and Vaidya 2001).

In preparations that exhibit more complex wave motions such as the turtle olfactory bulb (Lam et al. 2000) or the hippocampal trisynaptic pathway (Nakagami et al. 1997), KL analysis might be used to look at behavior in low-dimensional spaces. We have successfully applied KL decomposition to the analysis of pharmacologically induced wave propagation in mouse horizontal brain slices. As in turtle cortex, the direction and velocity of waves propagating between temporal cortex and entorhinal cortex can be determined directly from the projection trajectories of the basis images (Senseman and Robbins 2001).

KL basis images reflect anatomical organization not "neurobiological distinct" processes

The idea that KL basis images might capture different types of integrative activity occurring in different cortical microcircuits has been criticized by Mitra and Pesaran (1999). They argue that because KL decomposition and related statistical techniques rely on orthogonal coordinate rotations, they are ill suited for the analysis of cortical optical signals because "there is no reason why the neurobiologically distinct basis images in the data should be orthogonal to each other."

We believe that this argument is not directly relevant to the analysis of wave motion. In turtle cortex, the KL basis images are not capturing "neurobiologically distinct" processes because their location and shape do not change appreciably when the cortex is activated by qualitatively different visual stimuli (Senseman and Robbins 1999a). Rather, KL basis images clearly reflect the anatomical shape and location of the primary and secondary visual areas and their reciprocal connectivity (Fig. 4). The centers of the maximum and minimum for basis image M_1,2 were always located within the primary visual area, and the line separating the maximum and minimum of basis image M_2,1 was always congruent with the boundary between the primary and the secondary visual areas. These correspondences were observed in both the data obtained in the in vitro preparation, where the cortical sheet was unfolded, as well as in the in vivo preparation, where the cortical sheet was left intact. Perhaps more significantly, the same relation between the basis images and the location of the dorsal and dorsomedial cortex held whether the cortex was stimulated by a diffuse 20-µs light flash or a looming stimulus (Fig. 10).

Finally, KL decomposition offers a solution to the problem of how to compare dynamic responses captured by VSD movies recorded across different experimental trials. Because the dynamic behavior of the response is encapsulated within the projection trajectories of its basis images, differences between trials can be easily seen, and quantitative analyzed, by simply co-plotting the basis image projections (Fig. 8B). For this to be a meaningful comparison, however, it is essential that all the projections be plotted on the same coordinate system, i.e., plotted on a common basis. This can be readily accomplished by simply concatenating the movies from the various trials, end to end, and performing KL decomposition on the combined data to produce a set of global basis images. Comparable projection trajectories can then be obtained by projecting the global basis images on to the individual VSD movies recorded in the different trials (Fig. 8B).

Comparison with other techniques

Fourier analysis expresses a function in terms of predefined set of orthogonal basis functions (sines and cosines), and the power spectrum produced by Fourier analysis is a representation of the energy captured by each of these fixed basis functions. Because the basis (coordinate system) is fixed, the Fourier representations of different data sets can be meaningfully compared and particular spectral features (such as peaks at related frequencies) can be directly interpreted.

If the macroscopic behavior of cortical wave motion closely approximately sine waves, Fourier analysis would be ideally suited. In most cases, however, the macroscopic behavior of neuronal populations is not sinusoidal nor is its spectral energy confined to a few narrow energy bands. More commonly, the energy is widely distributed along a continuum showing only small, and relative broad peaks centered at several frequencies (e.g., Fig. 1F in Prechtl et al. 1997). Consequently, in the Fourier domain, the representations of the response must be high dimensionala relatively large number of basis functions (corresponding to different spatial frequencies) are required to capture most of the energy in the response.

An additional problem with Fourier analysis (but not KL decomposition) is that the basis functions (sines and cosines) are not localized in either space or time. This problem can be addressed through the use of wavelets (Mallat 1998) or multitaper filters (Percival and Walden 1993; Prechtl et al. 1997). Both techniques use basis functions that are localized in space, time, and/or frequency. However, the localization comes at the price of requiring an even larger set of basis functions to represent the response.

In contrast, KL decomposition computes a set of basis functions that are the best separable approximation (sum of products of functions of space and time) for a data set (Schmidt 1907). The KL representation, consequently, minimizes the number of basis functions needed to capture a specified amount of data set energy and opens the possibility of a low-dimensional representation. It is precisely the low dimensionality that has allowed the basis image M_1,2 to be associated with intracortical wave propagation and basis image M_2,1 to be associated with intercortical wave propagation.

Summary

The long-term goal of this research is to identify the possible information capacity of cortical waves (e.g., Ermentrout and Kleinfeld 2001). As a first step, we have demonstrated how KL decomposition, a simple and widely used statistical procedure in the physical sciences, can be adopted to decompose complex cortical waves into simpler traveling wave motions. The KL basis images returned by this analysis are easily recognizable as reflecting the anatomical shape and location of the turtle's primary and secondary visual areas. The projection trajectory of basis image M_1,2 provides a direct way of assessing the velocity and direction of intracortical waves, while the projection trajectory of basis image M_2,1 provides a direct way of assessing the velocity and direction of intercortical waves. The procedure works equally well for recordings performed in vitro or in vivo. A MatLab script for performing KL decomposition is provided