# Analysis and Model of Cortical Slow Waves Acquired with Optical Techniques

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Procedures and Data Processing

^{2}(5 mm × 5 mm) area and offers a perspective from above on the brain cortex of two transgenic Thy1-GCaMP6f mice (hereafter, referred to as Keta1 and Keta2). The total collection time for each mouse was about 5 minutes (eight sets of 1000 frames, each set corresponding to an observation period of 40 ms × 1000 = 40 s). Modern high-density electrode arrays for ECoG recordings can interrogate up to 32 sites per hemisphere in mice and can reach a sampling frequency of 25 kHz when the signal from each channel is digitized [27,28,29]. Compared to this, the resolution of wide field optical microscopy is much worse from a temporal point of view (25 Hz for the optical technique, up to 25 kHz for the electrode grid), but spatially higher (for the optical technique, each pixel is 50 $\mathsf{\mu}$m × 50 $\mathsf{\mu}$m, to compare with the 500 $\mathsf{\mu}$m electrode spacing for the grid that corresponds to a worsening of the spatial inspection by a factor 100; the spacing of the electrodes can be smaller than 500 $\mathsf{\mu}$m for ultra-thin flexible arrays [30], approaching the spatial resolution of imaging recordings).

#### 2.2. Data Analysis Pipeline

**Image initialization**. Each .tif image is imported and translated in a matrix of floats. Matrices are then assembled so that a collection of images belonging to the same recording set is stored in a 3D-array: 2 dimensions are spatial (representing the cortex surface) while the third is time (that is quantized in 40 ms steps). Second, data are manipulated in order to keep informative content only, as already illustrated in Figure 1. Matrices are cut in the spatial dimensions, isolating the portion of interest of the dataset (e.g., eliminating parts of the right hemisphere). Afterwards, in order to remove the non-informative black background, we make use of a masking process. The edges of the mask are obtained with the find_contours function (skimage.measure Python package) which adopts the marching squares algorithm; each pixel outside the mask is forced to take the Not A Number (NaN) value. Since images in the same recording set share the same perspective on the cortex, the cropping process is applied en bloc to each dataset. The cropping process (cutting and masking) is the step of the analysis procedure that cannot be fully automatized at this stage of software development, because it depends on the quality of the image set under study.- Background subtraction and spatial smoothing. Once the informative parts of the data are isolated, the pipeline proceeds with signal cleaning. The constant background is estimated for each pixel as the mean of the signal computed on the whole image set; it is then subtracted from images, pixel by pixel. Fluctuations are further reduced with a spatial smoothing: pixels are assembled in $2\times 2$ blocks (macro-pixels); at each time, the value of a macro-pixel is the mean signal of the 4 pixels belonging to it (from now on, macro-pixel is meant each time pixel is mentioned). This step may be unnecessary given the cleanness and regularity of the datasets we have used to test and develop the analysis pipeline, but our goal when devising the analysis procedure was to arrange a set of methods that could have been applied to any dataset, including noisy cases. The current implementation with macro-pixels corresponds to a factor $1/4$ of reduction in the number of inspected channels, with an equivalent grid step of 100 $\mathsf{\mu}$s for the pixel array, smaller than or comparable with the size currently accessible with surface multi-electrode arrays (MEAs), keeping this analysis still competitive in term of spatial resolution.
- Spectrum analysis and time smoothing. In order to identify the dominant frequency band in each dataset, a real Fourier transform (RFT) is computed for each pixel. A unique spectrum for each dataset is then obtained as the mean of single pixels’ RFT. Given the mean spectrum, the frequency band of maximum intensity is identified and the signal is further cleaned by applying a 6° order band-pass Butterworth filter (provided by the scipy Python package, Scientific Python, https://www.scipy.org).

- Search of minima. For each pixel, the signal’s minima are identified, comparing the intensity value at each instant (frame) to its previous and next one. This operation corresponds to evaluate the difference quotient of the data, and it used as the starting point for the subsequent step of interpolation.
- Parabolic interpolation of minima. A parabolic fit is evaluated around each minimum. For the parabolic fit, five data points are used. Analytically, if a fit with a second order polynomial is tried (three parameters), 5 points are enough for the estimate of the fit parameters. We evaluated also the options of using a higher order polynomial (a quartic, five parameters), for which five points are theoretically enough, but practically seven or nine would be requested for a more accurate estimate of the function profile. However, under the assumption that each minimum corresponds to the passage of a wave, and that distinct waves are separated in time at least by 250 ms (neglecting the bordeline case of two waves in antiphase), using five points (i.e., inspecting 160 ms around the minimum, with the sampling time of 40 ms) should allow to isolate each wave contribution, while with seven or nine points (240 and 320 ms respectively) an overlap of more than one wave cannot be excluded. The three parabola’s parameters are saved into a proper data structure. The time of the minimum (interpolated between original frames) is obtained from the vertex of the parabola. With this information, assumed as the timestamp of the transition, it is possible to reconstruct the activity of the passing waves, following the movement both in time and space of the minima, as we will show in Results (Section 3).

^{®}, The MathWorks, Inc., Natick, MA, USA, https://www.mathworks.com/products/matlab.html) pipeline, already employed to analyze and elaborate data from electrodes [9,31,32,33] and recently revised and refined [29]. On this, it is notable that the analysis pipeline that we devised nicely connects, after only minor adaptation efforts, with a well established analysis workflow, that was developed for addressing the study of completely different experimental data. This fact is already a result, since it represents a first successful step towards a plan of delivering a set of tools and methods, to extract quantitative information from electrophysiological signals, and to compare experiments, overcoming differences in the data taking and any systematic effects that may eventually depend on the data acquisition techniques.

- Excitability of the neuronal populations. For each minimum, the quadratic coefficient of the corresponding parabolic interpolation is taken. It is proportional to the concavity of the parabola, therefore it contains information on the excitability that the respective neuronal population (i.e., pixel) exhibits as a consequence to the wave that is passing. In detail, the quadratic coefficient is proportional to the second derivative of the function, and it is a measure of the speed of variation of the first derivative; the first derivative indicates a variation of the function; the function is a measure of the luminosity, and it is related to the state of the tissue (low/high emitting because of the variation in the calcium concentration). The first derivative can identify the minima of the function, and thus the timestamps at which a most significant variation (a change in the state, a Down-to-Up transition) is occurring; the fastest is such variation (monitored by the values of the second derivative), the fastest is the state transition, and thus, in our description, the most reactive (i.e., excitable) is the tissue underneath. The unit of the excitability is $\left[{t}^{-2}\xb7\frac{\mathcal{F}}{{\mathcal{F}}_{\mathit{max}}}\right]$. In order to study the distribution of excitability, values of excitability are collected within a histogram reporting statistics of the entire sample of minima. In addition to this, an Excitability Map, representing the average excitability for each pixel, is produced (we point out that the information on the excitability can be obtained without splitting the collection of minima into global waves, i.e., before the WaveHunt step of the pipeline, that is necessary only for origin points and velocity. This is clearly shown in the flowchart at the end of this section (Figure 2), in which the logic of the whole pipeline is represented in a schematic form).
- Origin Points of the waves. Once that the collection of waves is obtained, each wave in the collection is examined separately. For each wave, there is a sequence of pixel activation; the first N pixel in the sequence are identified as the origin points (we set $N=30$ in this work; this value, given the number of non-NAN pixels in the images and the request of globality set at $75\%$, corresponds to having the subset of origin points with less than $3\%$ of the pixels involved in each wave). Some pixels more often appear in this ranking, and are the ones that identify the dominant spatial origin of the waves.
- Wave velocity. In order to obtain the average speed of each wave, the wave velocity is calculated on different points. If the passage time function of the wave $T(x,y)$ is known, the speed of a wave on a point $(x,y)$ can be defined as the inverse of the module of the function’s gradient,$$v(x,y)=\frac{1}{\left|\mathsf{\nabla}T\right(x,y\left)\right|}.$$(the passage time function indicates, for each position $(x,y)$, the time at which that position has been reached by the wave during its propagation). Computing the gradient and taking its module, we obtain:$$v(x,y)=\frac{1}{\sqrt{{\left(\frac{\partial T(x,y)}{\partial x}\right)}^{2}+{\left(\frac{\partial T(x,y)}{\partial y}\right)}^{2}}}.$$It should be noted that the function $T(x,y)$ represents the wave passage time (that we identify with the time of the minima) in the spatial continuum, bur we have this information only for the discrete points corresponding to the pixels. For this reason, the partial derivatives that appear in (2) have been calculated as finite differences, with the distance between two pixels denoted as d:$$\left\{\begin{array}{c}{\displaystyle \frac{\partial T(x,y)}{\partial x}}\simeq {\displaystyle \frac{T(x+d,y)-T(x-d,y)}{2d}}\hfill \\ {\displaystyle \frac{\partial T(x,y)}{\partial y}}\simeq {\displaystyle \frac{T(x,y+d)-T(x,y-d)}{2d}}.\hfill \end{array}\right.$$In other words, for any pixel identified by the pair of coordinates $(x,y)$, the velocity of the wave on that pixel is computed using the information collected on the four adjacent pixels. Therefore, given the above calculation rule, for each wave only pixels with four valid neighbours are taken into account: these are pixels for which it is possible to evaluate both the $\frac{\partial T(x,y)}{\partial x}$ and the $\frac{\partial T(x,y)}{\partial y}$. By averaging the values obtained over pixels that meet this condition, we obtain the average speed of a wave; this procedure is repeated for each wave, in order to create a histogram of average speeds.

#### 2.3. Toy Model

## 3. Results

#### 3.1. Analysis of the Experimental Data

#### 3.2. Analysis of the Simulation

#### 3.3. Comparison between Experimental Samples

#### 3.4. Comparison between Experimental and Simulated Data

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Representation of the mouse cerebral cortex with overlapping names of the different areas: MOT = motor; SOM = somatosensory; Au = auditory; VIS = visual; PtA = associative parietal; RSD = retrosplenial dysgranular. The zoom shows which part of the cortex is visible in the raw data at our disposal, after selecting the left hemisphere and before the removal of the black background.

**Figure 3.**Lognormal functions for different set of parameters $\mu $ and $\sigma $. The pair $\left\{\mu =2.2,\sigma =0.91\right\}$ (in orange) is selected for the emulation of the response of GCaMP6f.

**Figure 5.**Sequence showing the different steps of the signal cleaning process. (

**a**) Example of a $100\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}100$ pixels raw image. (

**b**) Image cropping to exclude the non-informative area: only the left hemisphere has been selected (image size: $100\times 80$ pixels) and a mask has been created (specifically, 5564 out of 8000 original pixels were within the contour, to be considered as signal sources). (

**c**) Background, computed as the average signal, was subtracted from masked images. (

**d**) Spatial smoothing: blocks of $2\times 2$ pixels constitute a macro-pixel; the number of informative channels was reduced to a factor $1/4$ of the previous amount (in this specific case, 1391 macro-pixels were obtained).

**Figure 6.**Frequency spectrum, averaged across the pixels, for a set of 1000 frames, corresponding to 40 s of recordings (mouse referred to as Keta1).

**Figure 7.**Comparison, for a given pixel, between raw signal (

**a**) and filtered signal (

**b**), for the same 8 s of acquisition.

**Figure 8.**Neuronal activity evaluated as $\frac{\mathcal{F}}{{\mathcal{F}}_{\mathit{max}}}$ in different macro-pixels, for 15 frames (corresponding to $15\times 40$ ms = 600 ms of recordings). The miniplot graphic layout reproduces the pixels’ spatial location on the hemisphere. In this visualization, the pixel in the centre is assumed as a reference; the signal is in blue, the minimum is fitted with a parabola (in orange), the green vertical line indicates the position of the minimum (from the vertex of the parabola). In the other plots, the red vertical line shows the time shift of the minimum with respect to the central pixel.

**Figure 9.**Series of images—spaced by 40 ms time step—showing the propagation front of the minima. The global and correlated wave activity is evident. The pixels activated in the 80 ms preceding the time shown below each image are illuminated. The color represents the time elapsed from the up-ward transition: it goes from the dark blue for the pixels that have just turned on, to the green for those that have been activated almost 80 ms before the image capture time, up to the yellow for all the pixels whose neuronal population has not transited in the last 80 ms.

**Figure 10.**Raster plot of a time interval for a given data set. The presence of a global and repeated activity of the system is clear, although each signal is characterized by different dynamics, reflecting differences in the propagation patterns.

**Figure 11.**(

**a**) Firing rate generated by a single Neuron during a simulation. (

**b**) Signal obtained after the convolution with a Lognormal kernel of parameters $\mu =2.5$ and $\sigma =0.5$, and including the discard of the transient time, the subtraction of the mean signal and the normalization to the maximum value.

**Figure 12.**Comparison between the average frequency spectra of Keta1 (

**a**) (same plot as in Figure 6) and Keta2 (

**b**). Both samples present a peak at similar frequency values in the delta band.

**Figure 13.**Maps of the point of origin of the experimental waves for Keta1 (

**a**) and Keta2 (

**b**). The color code represents the number of times a single pixel has been involved in the birth of a global wave, normalized to the total number of waves in the collection. The “birth set” is defined as the first $N=30$ pixels on which each wave passes; given the request of globality for the waves included in the collection, and taking into account the different numbers of informative pixels in the two datasets, the birth set constitutes at most the $3\%$ of the wave.

**Figure 16.**Maps of the cortex showing the average excitability per pixel for Keta1 (

**a**) and Keta2 (

**b**).

**Figure 17.**Average frequency spectrum, comparison between the simulation (

**a**) and the experimental data that have been used for the generation of the activation states in the simulation (

**b**) (same plot as in Figure 12b).

**Figure 18.**Comparison between the signal generated by the simulation, in red, and the related experimental signal, in blue, for a single pixel.

**Figure 19.**(

**a**) Histogram of excitability for artificial data. (

**b**) Histogram of wave propagation speed for artificial data.

Parameter | Description | Value [Unit] |
---|---|---|

${t}_{\mathrm{end}}$ | Total time to simulate | Defined in input [s] |

${t}_{\mathrm{step}}$ | Simulation time step | 40 [ms] |

${\tau}_{\mathrm{up}}$ | Neuron permanence time in active state | 200 [ms] |

$\frac{{\mu}_{\mathrm{up}}}{{\mu}_{\mathrm{down}}}$ | Ratio of firing rates (active and idle states) | $5[\xb7]$ |

${N}_{\mathrm{pixel}}$ | Neuron number for each Pixel | $N\simeq \mathcal{N}\left(\mu =10,\sigma =2\right)\phantom{\rule{4pt}{0ex}}[\xb7]$ |

$\mu $ | $\mu $ parameter of the LogNorm Kernel | $2.2\phantom{\rule{4pt}{0ex}}[\xb7]$ |

$\sigma $ | $\sigma $ parameter of the LogNorm Kernel | $0.91\phantom{\rule{4pt}{0ex}}[\xb7]$ |

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**MDPI and ACS Style**

Celotto, M.; De Luca, C.; Muratore, P.; Resta, F.; Allegra Mascaro, A.L.; Pavone, F.S.; De Bonis, G.; Paolucci, P.S. Analysis and Model of Cortical Slow Waves Acquired with Optical Techniques. *Methods Protoc.* **2020**, *3*, 14.
https://doi.org/10.3390/mps3010014

**AMA Style**

Celotto M, De Luca C, Muratore P, Resta F, Allegra Mascaro AL, Pavone FS, De Bonis G, Paolucci PS. Analysis and Model of Cortical Slow Waves Acquired with Optical Techniques. *Methods and Protocols*. 2020; 3(1):14.
https://doi.org/10.3390/mps3010014

**Chicago/Turabian Style**

Celotto, Marco, Chiara De Luca, Paolo Muratore, Francesco Resta, Anna Letizia Allegra Mascaro, Francesco Saverio Pavone, Giulia De Bonis, and Pier Stanislao Paolucci. 2020. "Analysis and Model of Cortical Slow Waves Acquired with Optical Techniques" *Methods and Protocols* 3, no. 1: 14.
https://doi.org/10.3390/mps3010014