Calculation of Detection Regions and Estimation of Neuronal Activity

The regions in which a simulated neuron would be detected is based on a simulation of a single action potential. After the simulation, EAPs were calculated (as described in section 2.2.1.1) on a 3D spatial grid around the neuron: the grid spacing was 5µm for L2-4 and smooth interneurons, and 10µm for pyramidal cells in L5-6 (chosen to reflect the size of the detectable region (as illustrated

in Figure 4.16). I chose an average minimum threshold of 80µV for detection and clustering, based on my own recording and clustering results (described below), but also analyzed results for higher thresholds.

From the grid of calculated EAPs I used Matlab to generate an isosurface for which the peak EAP amplitude was greater than the minimum threshold. The interior of this region was considered to be the detectable region. The volume of the detection region was determined by dividing volume enclosed by the isosurface into pyramids using custom Matlab scripts (the isosurface routine returned a triangulated surface), and calculating the volume for each pyramid. After the volume of the detection region was computed I describe it by computing the radius of an equivalent sphere having the same volume; although the regions were usually not exactly spherical. Volumes that were typically 1/10,000 or 1/100,000 of a mm3 _{are difficult to interpret and an equivalent radius (in}

µm) provides not only a convenient description but a useful analytic approximation. I refer to the equivalent radius as the detection range, or detection radius.

Because the EAP amplitude depends linearly on the extracellular resistivity (conductivity) it is important to control for this variable. As illustrated in Figure 4.1, the resistivity of cortex is highly variable but has a mean and median both very close to 250 Ωcm (also see [Hoeltzell and Dykes, 1979]). For this reason I used 250 Ωcm as the standard value for my calculations, but also looked at the impact of resistivities ranging from 150 Ωcm to 400 Ωcm.

After calculating the detection range for each cell, I calculated average detection ranges for cells in each layer. The averages were weighted to correct for differences between my morphology sample and what I know of the true distribution of cell sizes, as described in section 4.2.1.2. This average detection range in a given layer is interpreted as the average maximum distance from the recording electrode tip at which neurons in that layer could be detected based on the approximation of spherical detection regions. When the volume of the detection radius sphere is multiplied by the packing density of neurons within each layer it gives the average number of neurons within range for that layer.

[Bealieau and Colonier, 1983] provides cell packing densities separately for the monocular and binocular regions of layers 2, 3A, 3B, 4A, 4B, 5, 6A and 6B. For my calculations I always use the average value of the monocular and binocular regions, and for layers 2/3 I use the average of layers 2, 3A and 3B. For layer 6 I use the packing density and thickness of layer 6A only: my morphology sample consists only of pyramidal neurons of the type found in layer 6A, and histological measurements suggest that my recording penetrations never reached the deeper layer 6B. [Bealieau and Colonier, 1983] provides the packing density of neurons without distinguishing between spiny and smooth cells. Therefore for my packing densities I further multiply by the typical percentage of neurons which are spiny, 80%. In order to determine an average packing density for smooth interneurons, I take the average neuron packing density over all layers and multiply by 20%.

Figure 4.1: Histogram of cortical resistivity measurements. Measurements from monkey somatosensory cortex. Courtesy of Nikos Logothetis.

Having calculated the average number of neurons within range for each layer, the fraction of neurons active in layeriis:

αi=

Nrecorded

i

N_irange (4.1)

with N_irecorded the average number of units recorded from each recording site in layer i, and

N_irange the model calculation for the number of units within range for layeri.

Note that the simulation detection ranges calculated by this method are radii from thecenter of the soma, not from the soma surface. This is because the detection region volume does not correct for the volume occupied by the soma itself (a relatively small component at the 80 µV threshold.) Consequently this method is appropriate for calculating the neurons within range based on an average packing density, at a low detection threshold. However, care must be taken when applying the same method to a higher threshold: if not corrected the LSA will calculate the potentials that would occur within the confines of the soma itself as if the neural currents were in fact located on a line near its center. This could erroneously provide a non-zero detection region located “inside” the boundaries of the soma at high threshold, when in fact valid potentials calculated outside of the region of the soma are below this threshold. An exact correction for this problem would be

a complex exercise in computational geometry, and would require determining the intersection or enclosure of the detection isosurface with the soma surface. I used a simplification: if the volume of the calculated detection region exceeds the volume of the soma it was interpreted as a valid detection radius from the center of the soma. If the volume of the detection region was less than the volume of the soma it was considered to be zero, meaning no detectable region for that threshold. This was the case for some or all of the neurons in each class at higher thresholds.

4.2.1.2 Correction for Diameter Bias in Morphological Data

Because my goal is to estimate the average detection range for cells in each layer, it is important to ask whether my sample of 68 neuron morphologies based on careful reconstruction is, in fact, representative. [Gabbott et al., 1987] provides the equivalent area diameters for 1350 pyramidal cells in layer 5 of cat V1 sampled randomly with a light microscope. I used this as a basis for comparison with my own sample and the results are summarized in table 4.2. I found that my sample of morphologies is in fact highly biased: more than 50% of neurons in my sample were in the top 10% of size based on the larger sample. This bias most likely results from the process of intracellular labeling with a sharp electrode, which will naturally tend to pick out bigger cells.

To remedy this bias when calculating the average detection region radius for L5 neurons I used a weighted average based on groupings of the cells by size. I grouped the neurons as “small/medium”, having diameters smaller than 20µm, and as “large” having diameters greater than 20 µm. I then calculated an average detection range within each group. For the overall average for L5 neurons I used a weighted average as suggested by the data in [Gabbott et al., 1987]: 90% of the weight went to the small/medium group and 10% of the weight went to the large group.

Size Diameter Literature Data Data

Category Range % Samples %

Small 7-14 12% 2 15%

Medium 14-20 78% 4 31%

Large 20+ 10% 7 54%

Table 4.2: Layer 5 area equivalent diameters in the literature and morphological data

Diameter Range: Range of equivalent area diameters, inµm. Literature %: Percentage of neurons in this category in [Gabbott et al., 1987]. Data Samples: Number of cells in this

category for my available morphological data. Data %: Percentage of cells in this category for the available morphological data.

Unfortunately, I could not find large, unbiased samples of the average neuron size for other layers of cat V1. But based on the results for the layer 5 neurons, I have every reason to expect that my morphology sample for other layers will be similarly biased. Consequently I made an approximate correction by following the results for layer 5: neurons in each layer were divided into

a “small/medium” group containing roughly the smaller 50% of the neurons in my sample and a “large” group containing the larger 50%, as described in table 4.4. When calculating average detection ranges for each layer 90% of the weight was given to the average for the “small/medium” group and 10% of the weight was given to the large group. The impact of the correction is discussed below in section 4.3.7 (in particular see Figure 4.21).

All Small/Medium Large

Std. Std. Std.

Layer N Mean Dev. N Mean Dev. N Mean Dev.

L23 13 17 3.1 6 14.1 1.6 7 19.4 1.3

L4 20 16.9 4.3 8 12.7 1.4 12 19.1 3.7

L5 13 21.7 5.8 6 16.9 2.8 7 25.8 3.4

L6 12 15.8 1.9 6 14.8 1 6 17.5 1.7

Smooth 10 18.5 4.9 5 14.6 3.2 5 22.5 2.1

Table 4.4: Area equivalent diameters in the morphological data

Area Equivalent diameter is the sphere diameter having the same surface area as the actual soma.

All: All neurons in each layer and also for smooth interneurons from all layers. Small/Medium: The neurons comprising the “small/medium” category receive 90% of the weight in determining the average detection range. Large: The neurons comprising the “large” category receive 10% of the weight in determining the average detection range.