8.2.1 Alignment of maps
Typically, the frame of reference of recordings slightly varies during an experiment.
In order to compare the maps from the different recordings, their cortical coordi-nates must be aligned. To achieve this, we morphed all maps onto a reference map
chosen to be the map recorded immediately before ICMS (Pre map). We used a linear morphing
z(x) = z x0+bΩαx0
(8.1) with the 2-dimensional rotation matrixΩα, a scaling factorb ≈1, and the transla-tionx0= (x0,y0). To find the 4 morphing parameters for a given map we used two methods. The first method aims to maximize the cross-correlation between maps, the second to align the position of radial vessels in images from the cortex.
The correlation method is applicable only if the map remains sufficiently similar during an experiment. For instance, if the changes induced by ICMS are too large, an alignment based on the structure of the map may become questionable. Practi-cally, correlation coefficients down to 0.3 were tractable. For efficiency of parameter search, we used three steps of successive refinement. First, a relative wide param-eter region ofα = 0±4.8◦, b = 1±0.05, x0 = 0±0.3mm, y0 = 0±0.3mm was scanned using maps subsampled by a factor of 4. For 1.6·104randomly chosen pa-rameter combinations equally distributed within these intervals, we calculated the cross-correlation within the overlap region of both maps under the condition that the overlap was sufficiently large (we required 50% of the area of each map). The parameter combination maximizing the cross-correlation was further optimized by a gradient ascent on cross-correlation. In each subsequent step the resulting param-eters were used as an initial guess. Parameter ranges were decreased by a factor of 2, the number of tested combinations decreased by a factor of 4 while the size of the maps increased by a factor of 2.
The method using radial vessels works independently of the structure of the map. It is based on the vasculature image showing the cortex with superficial blood vessels in the region of the map (Fig. 8.2). Whereas these vessels can move consid-erably between subsequent recordings, the location where they enter the cortical layers, called radial vessels, remain fixed. By visual inspection, we identified a number of radial vessels within each vasculature image. The function chosen to be minimized by morphing was the average nearest neighbor distance between radial vessel locations in the reference and the morphed map. For the sake of effi-ciency, we used rescaling as for the correlation based method. Though in principle the method of choice, it suffers from technical difficulties concerning the recording devices. With the present equipment in the labs were the experiments were con-ducted, displacements of the map relative to the vasculature image are frequently.
In fact, by means of local cross-correlation displacements of more that 0.6mm were observed rendering the application defective given that the map has a typical scale of≈1.2mm.
Thus, we used the radial vessel method wherever it was possible and the corre-lation method in all other cases. The radial vessel method was used for k109 and k341, the correlation method for k060, k116, k242, k343, k119 and all acute experi-ments. A mix of both methods was used for k114 and k242.
Figure 8.2: A method for ensur-ing for each map identical cor-tical coordinates with the refer-ence map (Pre). Maps were mor-phed such that the radial ves-sels (marked by yellow symbols in each image) were aligned with those in the Pre-map (left image).
8.2.2 Preprocessing
After morphing all maps from a given experiment into the same coordinate sys-tem, a common region of interest (ROI) was defined excluding regions containing major vessels, boundary or reflectance artifacts, and regions with a low signal-to-noise ratio. High- and low-pass filtering was applied in the Fourier domain using a Fermi-filter as described in Chapter 4. We usedλhp = 1.7mm,λl p =0.6mm, and β = 0.1 for the chronic experiments (Magdeburg) in accordance with the typical column spacing of Λ ≈ 1.2mm observed. Larger column spacings of Λ ≈ 1.5 were found in the animals from the acute experiments for which we used λhp = 2.5mm,λl p =0.7mm.
8.2.3 Power spectrum
The power spectrum|z˜(k)|2 of an orientation mapzwas calculated by transform-ing the unfiltered map into the Fourier domain, multiplytransform-ing with the high- and low-pass Fermi-filter kernels, and finally taking the squared absolute value. The resolution ink−space was artifically increased by a factor of 4 by zero wrapping in the spatial domain.
8.2.4 Local similarity
Maps of local similarity where calculated as described in Chapter 7. To reduce fluctuations over time, maps were averaged over predefined time intervals before calculating the local similarities. We used the intervals [post, 3h], [3h, 9h], [9h, 15h], [>15h], each containing typically 2-3 maps in each experiment.
8.2.5 Pinwheel analysis
Pinwheels were analyzed based on the Methods developed in Chapters 4 and 5. To estimate pinwheel locations and densities also at times in between the recordings,
we interpolated between subsequently recorded orientation maps. To avoid fur-ther assumptions we used linear interpolation, sampled with a sufficiently large number of equally distributed maps. Usually, m = 20 maps were sufficient for our purpose leading to a total of m = 64 maps in the case of an experiment with recordings at 4 different times.
For each of these maps we calculated the pinwheel density using the fully au-tomated method described in Chapter 4 using 50 low-pass cutoff wavelengthsλl p equally spaced in [0.2, 1.18]mm with βl p = 0.05. Local column spacings Λ(x) were calculated using wavelets with 25 scales li equally spaced in the interval d = [0.8, 2.0]mm with 12 wavelet orientations θi = {0π/12, . . . , 11π/12}. The calculation was performed on a grid with a spatial resolution of 0.1mm. Since we are primarily interested in a changing of the number of pinwheels and not of col-umn spacing, we used for all maps the local colcol-umn spacingΛ(x)of the Pre map assuming that the changing of column spacing under ICMS is negligible.
For tracing individual pinwheels, we used the method described in Chapter 5.
Nearest pinwheels in subsequent maps were assigned if their distance did not ex-ceed ∆x = 0.2mm. The pinwheel locations in a given (real or interpolated) map were calculated with the low-pass filtering resulting from the pinwheel density analysis. With this filtering, the low-pass cutoff wavelengthλ(x) generally varies for different locations. As discussed in Chapter 4, the cutoff wavelengthλ(x) at a given location is constraint by the range[λ0,λ0+∆λ]of the plateau of the filter de-pendent pinwheel densityρ(x,λ). For the ICMS data we used a cutoff wavelength λ = λ0+∆λ/4. Consistency was checked by comparing the pinwheel density in this optimally filtered maps with the pinwheel densityρfrom the above analysis.
8.2.6 Softness
To calculate the softness κ+ we transformed the unfiltered map into the Fourier domain and multiplied it with the Fermi filter kernels. By this procedure, low and high frequency amplitudes are eliminated. Furthermore, at low and high frequen-cies this also truncates the contribution from the Fourier transform of the boundary of the map (approximately varying as sinx/xover many frequencies) with which the power spectrum of the map is convolved. Neglecting the contribution of many spurious modes with small amplitudes, we truncated the sum in Eq. ( 8.3) by in-cluding only the n = 100 largest amplitudes. Finally, we normalized the softness by its average value to enable a comparison between different maps.