![](\"attachment:hippo_cells.png\")
\n",
"\n",
"![](\"ellips_eigenv.png\")
\n",
"\n",
"**Fig 2. “Positional-Morpho-Anatomical” modelling**. CA1 PC cells are indicated by orange circle."
]
},
{
"cell_type": "markdown",
"id": "404cb8c6",
"metadata": {},
"source": [
"## Ellipsoids \n",
"\n",
"Assuming that any quadratic function $f (x_1, . . . , x_n)$ can be written in the form $ X ^\\mathsf{T} Q X $, where $ Q$ is a symmetric matrix ( $ Q = Q^\\mathsf{T} $ ), given a system of eigenvectors that diagonalize the symmetric matrix, any ellipsoid can be described as a volume oriented in the direction set by the eigenvectors and elongated along the semi-axis as set by the eigenvalues.\n",
"\n",
"\n",
"Considering an orthonormal system of eigenvectors $ \\mathbf{v_1, v_2, v_3}$ associated, respectively, with the eigenvalues $ \\lambda_1, \\lambda_2, \\lambda_3$ of a $3 \\times 3$ symmetric positive matrix $M$. If \n",
"$$\n",
"V = [ \\mathbf{v_1, v_2, v_3} ] ,\n",
"$$ \n",
"Then\n",
"$$\n",
"V^\\mathsf{T} M V = \\Bigg[ \\begin{matrix}\n",
" \\lambda_1 & 0 & 0 \\\\\n",
"0 & \\lambda_2 & 0 \\\\\n",
"0 & 0 & \\lambda_3 \\\\\n",
"\\end{matrix} \\Bigg] = D \\,[\\lambda_1, \\lambda_2, \\lambda_3 ]\n",
"$$\n",
"\n",
"| \n", " | PlacementPyr_x | \n", "PlacementPyr_y | \n", "PlacementPyr_z | \n", "MinDistCA3 | \n", "MinDistCA3_x | \n", "MinDistCA3_y | \n", "MinDistCA3_z | \n", "MinDistSub | \n", "MinDistSub_x | \n", "MinDistSub_y | \n", "... | \n", "r_ApicalDend | \n", "nn | \n", "MinDistOriens | \n", "MinDistOriens_x | \n", "MinDistOriens_y | \n", "MinDistOriens_z | \n", "MinDistLacun | \n", "MinDistLacun_x | \n", "MinDistLacun_y | \n", "MinDistLacun_z | \n", "
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | \n", "8125.370117 | \n", "2469.269775 | \n", "9085.161133 | \n", "600.477478 | \n", "7744.009766 | \n", "2885.100098 | \n", "8879.679688 | \n", "959.296570 | \n", "8988.040039 | \n", "2366.489990 | \n", "... | \n", "75.562119 | \n", "1.0 | \n", "44.070309 | \n", "8102.819824 | \n", "2440.929932 | \n", "9110.271484 | \n", "268.089783 | \n", "8122.830078 | \n", "2677.699951 | \n", "8916.571289 | \n", "
| 1 | \n", "8176.560059 | \n", "3510.365967 | \n", "9620.210938 | \n", "376.325958 | \n", "7882.220215 | \n", "3629.330078 | \n", "9418.139648 | \n", "1015.934021 | \n", "9184.879883 | \n", "3528.199951 | \n", "... | \n", "72.826775 | \n", "2.0 | \n", "108.608330 | \n", "8128.389648 | \n", "3473.450928 | \n", "9710.281250 | \n", "308.827515 | \n", "8224.769531 | \n", "3671.210938 | \n", "9361.021484 | \n", "
| 2 | \n", "7935.819824 | \n", "5826.859863 | \n", "8160.361328 | \n", "37.691891 | \n", "7916.029785 | \n", "5796.759766 | \n", "8149.270020 | \n", "1309.069458 | \n", "9080.230469 | \n", "5367.250000 | \n", "... | \n", "90.401756 | \n", "3.0 | \n", "101.279541 | \n", "7905.229980 | \n", "5898.060059 | \n", "8225.571289 | \n", "51.631149 | \n", "7980.679688 | \n", "5810.830078 | \n", "8140.451172 | \n", "
| 3 | \n", "8230.169922 | \n", "2692.069824 | \n", "9283.851562 | \n", "608.663940 | \n", "7816.259766 | \n", "3013.169922 | \n", "8973.940430 | \n", "937.358215 | \n", "9137.700195 | \n", "2683.639893 | \n", "... | \n", "58.591160 | \n", "4.0 | \n", "72.143684 | \n", "8245.179688 | \n", "2652.079834 | \n", "9341.991211 | \n", "211.220276 | \n", "8314.849609 | \n", "2787.929932 | \n", "9115.761719 | \n", "
| 4 | \n", "8718.200195 | \n", "4007.935791 | \n", "9805.201172 | \n", "800.595398 | \n", "8240.089844 | \n", "3964.889893 | \n", "9164.490234 | \n", "489.271667 | \n", "9188.759766 | \n", "4043.209961 | \n", "... | \n", "65.722137 | \n", "5.0 | \n", "78.015656 | \n", "8762.959961 | \n", "3949.673828 | \n", "9831.441406 | \n", "308.917267 | \n", "8701.339844 | \n", "4021.460938 | \n", "9497.041016 | \n", "
| ... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "
| 216430 | \n", "8022.179688 | \n", "2303.029785 | \n", "8942.380859 | \n", "609.587769 | \n", "7711.689941 | \n", "2799.010010 | \n", "8771.509766 | \n", "978.513672 | \n", "8861.049805 | \n", "2161.629883 | \n", "... | \n", "54.149960 | \n", "216431.0 | \n", "49.564491 | \n", "8041.060059 | \n", "2308.499756 | \n", "8987.880859 | \n", "272.917908 | \n", "8151.359863 | \n", "2368.969971 | \n", "8711.191406 | \n", "
| 216431 | \n", "8197.729492 | \n", "5001.349609 | \n", "9580.071289 | \n", "540.179443 | \n", "7890.080078 | \n", "4773.049805 | \n", "9199.250000 | \n", "937.382385 | \n", "9113.049805 | \n", "5022.910156 | \n", "... | \n", "73.821106 | \n", "216432.0 | \n", "63.861050 | \n", "8155.500000 | \n", "4957.020020 | \n", "9561.911133 | \n", "401.460388 | \n", "8352.639648 | \n", "4764.562988 | \n", "9295.281250 | \n", "
| 216432 | \n", "8316.330078 | \n", "3736.258789 | \n", "9763.931641 | \n", "524.106201 | \n", "7952.270020 | \n", "3733.139893 | \n", "9386.919922 | \n", "884.767029 | \n", "9190.419922 | \n", "3823.790039 | \n", "... | \n", "73.559975 | \n", "216433.0 | \n", "107.298393 | \n", "8282.729492 | \n", "3687.611816 | \n", "9853.471680 | \n", "258.790009 | \n", "8321.149414 | \n", "3801.478271 | \n", "9513.541016 | \n", "
| 216433 | \n", "8199.200195 | \n", "2605.719971 | \n", "9225.611328 | \n", "613.186035 | \n", "7816.259766 | \n", "3013.169922 | \n", "8973.940430 | \n", "949.058411 | \n", "9106.450195 | \n", "2624.780029 | \n", "... | \n", "63.066402 | \n", "216434.0 | \n", "80.917030 | \n", "8231.530273 | \n", "2564.829834 | \n", "9287.500977 | \n", "234.244354 | \n", "8334.459961 | \n", "2703.499756 | \n", "9061.250977 | \n", "
| 216434 | \n", "6990.308105 | \n", "1737.569824 | \n", "7572.471191 | \n", "470.250702 | \n", "6677.490234 | \n", "2086.320068 | \n", "7613.140137 | \n", "1175.725464 | \n", "7978.209961 | \n", "1884.729980 | \n", "... | \n", "52.314598 | \n", "216435.0 | \n", "52.514091 | \n", "7010.776855 | \n", "1691.729858 | \n", "7557.061523 | \n", "241.867889 | \n", "7150.609863 | \n", "1907.689819 | \n", "7510.321289 | \n", "
216435 rows × 27 columns
\n", "| \n", " | PlacementPyr_x | \n", "PlacementPyr_y | \n", "PlacementPyr_z | \n", "MinDistCA3 | \n", "MinDistCA3_x | \n", "MinDistCA3_y | \n", "MinDistCA3_z | \n", "MinDistSub | \n", "MinDistSub_x | \n", "MinDistSub_y | \n", "... | \n", "r_ApicalDend | \n", "nn | \n", "MinDistOriens | \n", "MinDistOriens_x | \n", "MinDistOriens_y | \n", "MinDistOriens_z | \n", "MinDistLacun | \n", "MinDistLacun_x | \n", "MinDistLacun_y | \n", "MinDistLacun_z | \n", "
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | \n", "7392.106934 | \n", "1747.999878 | \n", "6732.710449 | \n", "890.852295 | \n", "7059.520020 | \n", "2480.439941 | \n", "7115.509766 | \n", "481.840271 | \n", "7805.290039 | \n", "1948.609985 | \n", "... | \n", "75.934700 | \n", "1.0 | \n", "0.0 | \n", "0.0 | \n", "0.0 | \n", "0.0 | \n", "234.771942 | \n", "7299.580078 | \n", "1944.149902 | \n", "6822.609375 | \n", "
| 1 | \n", "7621.520020 | \n", "1876.859863 | \n", "8444.201172 | \n", "628.380432 | \n", "7357.609863 | \n", "2436.050049 | \n", "8332.309570 | \n", "1056.485596 | \n", "8534.780273 | \n", "1838.369995 | \n", "... | \n", "71.809219 | \n", "2.0 | \n", "0.0 | \n", "0.0 | \n", "0.0 | \n", "0.0 | \n", "327.164154 | \n", "7824.839844 | \n", "2055.999756 | \n", "8260.880859 | \n", "
| 2 | \n", "8242.309570 | \n", "4284.031738 | \n", "9922.581055 | \n", "604.455139 | \n", "7884.509766 | \n", "4205.689941 | \n", "9441.740234 | \n", "976.706604 | \n", "9181.580078 | \n", "4373.419922 | \n", "... | \n", "79.149597 | \n", "3.0 | \n", "0.0 | \n", "0.0 | \n", "0.0 | \n", "0.0 | \n", "449.711182 | \n", "8482.290039 | \n", "4196.378906 | \n", "9552.491211 | \n", "
| 3 | \n", "7902.299805 | \n", "1866.799805 | \n", "8285.681641 | \n", "760.362732 | \n", "7498.049805 | \n", "2510.729980 | \n", "8295.030273 | \n", "733.929565 | \n", "8534.780273 | \n", "1838.369995 | \n", "... | \n", "65.585480 | \n", "4.0 | \n", "0.0 | \n", "0.0 | \n", "0.0 | \n", "0.0 | \n", "158.995499 | \n", "7922.120117 | \n", "2017.859863 | \n", "8240.210938 | \n", "
| 4 | \n", "8585.219727 | \n", "2023.969849 | \n", "8492.821289 | \n", "944.984985 | \n", "8081.180176 | \n", "2794.449951 | \n", "8279.980469 | \n", "253.315735 | \n", "8810.830078 | \n", "2034.680054 | \n", "... | \n", "86.468346 | \n", "5.0 | \n", "0.0 | \n", "0.0 | \n", "0.0 | \n", "0.0 | \n", "252.800308 | \n", "8631.629883 | \n", "2248.109863 | \n", "8385.511719 | \n", "
| ... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "
| 45403 | \n", "8710.139648 | \n", "3970.956787 | \n", "9976.451172 | \n", "927.806396 | \n", "8133.100098 | \n", "3930.250000 | \n", "9251.059570 | \n", "563.599426 | \n", "9205.879883 | \n", "3958.399902 | \n", "... | \n", "67.252625 | \n", "45404.0 | \n", "0.0 | \n", "0.0 | \n", "0.0 | \n", "0.0 | \n", "479.555908 | \n", "8534.519531 | \n", "4050.527832 | \n", "9537.361328 | \n", "
| 45404 | \n", "8492.479492 | \n", "1886.529907 | \n", "8391.851562 | \n", "993.687744 | \n", "8043.640137 | \n", "2757.699951 | \n", "8227.469727 | \n", "284.303741 | \n", "8700.129883 | \n", "1938.589966 | \n", "... | \n", "69.881683 | \n", "45405.0 | \n", "0.0 | \n", "0.0 | \n", "0.0 | \n", "0.0 | \n", "329.694336 | \n", "8392.959961 | \n", "2137.069824 | \n", "8202.051758 | \n", "
| 45405 | \n", "7346.531738 | \n", "1707.459839 | \n", "6708.384277 | \n", "919.362610 | \n", "6983.200195 | \n", "2498.370117 | \n", "7004.490234 | \n", "532.282715 | \n", "7805.290039 | \n", "1948.609985 | \n", "... | \n", "73.362648 | \n", "45406.0 | \n", "0.0 | \n", "0.0 | \n", "0.0 | \n", "0.0 | \n", "266.971924 | \n", "7299.580078 | \n", "1944.149902 | \n", "6822.609375 | \n", "
| 45406 | \n", "6590.809570 | \n", "1857.409912 | \n", "6773.599121 | \n", "324.349274 | \n", "6431.700195 | \n", "2124.780029 | \n", "6865.250000 | \n", "1213.702148 | \n", "7727.000000 | \n", "2113.169922 | \n", "... | \n", "68.168320 | \n", "45407.0 | \n", "0.0 | \n", "0.0 | \n", "0.0 | \n", "0.0 | \n", "244.541473 | \n", "6804.740723 | \n", "1888.679810 | \n", "6659.335449 | \n", "
| 45407 | \n", "8286.349609 | \n", "1671.939819 | \n", "8027.061523 | \n", "1049.507446 | \n", "7832.270020 | \n", "2617.790039 | \n", "8052.459961 | \n", "305.399933 | \n", "8509.259766 | \n", "1774.050049 | \n", "... | \n", "67.491074 | \n", "45408.0 | \n", "0.0 | \n", "0.0 | \n", "0.0 | \n", "0.0 | \n", "327.380371 | \n", "8327.570312 | \n", "1993.229858 | \n", "8074.511230 | \n", "
45408 rows × 27 columns
\n", "