S-38074 Löttorp, Sweden
KL Chem AB, 23734 Bjärred and Camurus Lipid Research
S-22370 Lund, Sweden
Lund University Children’s Hospital
S-22185 Lund, Sweden
John C Fiala
Department of Biology, Boston University
Boston, MA 02215 USA
Based on extensive electron microscopy (EM) analysis, the general features of the packing arrangement of the neurons of the brain have been ascertained (cf. Synapse Web). Only a few neurons have been involved, and we show here that there are some short range packing principles. It is demonstrated that certain rod packing arrangements provide good models of certain observed neuron organizations. The rod packing alternatives are analyzed based on a new mathematical method. A case of four- coordinated packing which agrees with the experimental data is demonstrated. This new mathematics – “the exponential scale” – is proposed to provide a powerful tool in order to describe packing geometry and synaptic connections in modeling of neuron organization and the ultrastructure in the brain.
For mathematical modeling of neurons, fractal mathematics provides a powerful method. In order to describe the lateral coordination and the three-dimensional organization of neurons, however, other approaches are needed. We have recently demonstrated how biological structures can be modeled based on the so-called exponential scale mathematics . We also illustrated potential applications to neuron membranes and synaptic couplings.
Here we have continued this approach in order to model cortical arrangements of neurons. It is known that the six layers of neurons exhibit preferred orientations perpendicular and parallel to the surface. Thus there seems to be geometric laws defining the packing arrangement implicitly expressed by the signaling events that direct the growth zones of the axons.
Synapse Web has shown beautiful ultrastructural arrangements of neurons based on three-dimensional electron microscopy reconstruction that we will use here in order to model the organization bases on rod packings.
We start with a study of rod mathematics and continue with the application to arrangements of axons, dendrites and synapse connectivity in space.
Axons or dendrites grow in the shape of rods, which seem necessary for the building of typical neuron cells.
Axons and the dendrites from at least two different neurons form different kinds of rod systems that by growing interpenetrate into each other without intersections. As one rod in one system sees rods from the other system as next neighbors the situation is ideal for making synapses that connect the two systems – the background for the formation of neural circuits, and for communication between neurons. The interpenetrating structure – a circuit – becomes mechanically very stable via the synapse connections. Maximum contact possibilities are obtained via interpenetration. We see from 3D EM reconstructions [1,9- 11] that rods of axons, dendrites, and axons + dendrites cluster to a geometry we call rod packing.
We continue now with the geometrical principles of rod packings, which we have shortly reviewed in our last book . There are also detailed descriptions in refs [4-7]. To our knowledge the first description of the so-called garnet rod packing, the fundamentally most important, was made by Belov .
We start with a short description of rod packing with special emphasis on interpenetration. We shall use mathematics from the exponential scale, and then the so-called Gauss Distribution (GD) mathematics well developed in references [1,8]. You may as well use the cyclic analogs – still on the exponential scale - getting an infinite number of rods. The GD mathematics is easier to use. You may use as many rods as you like, and just add or substract rods after your own taste. We give the equations for each structure below and you shall realize that this is not difficult.
The three important permutations in space are shortly x, x+y, x+y+z and they give the arch types of rod packing, which are used below.
First we show the two simplest of rod packings in figures 1 a and b.
Fig 1: a. Tetragonal rod structure. b. Cubic rod structure.
Figure 1a gives a tetragonal structure of parallel rods. Of this one we take three structures, but with only two rods in each structure and make them interpenetrate after the three cartesian axes. The result is the beautiful structure in figure 1b, also called fence packing, and its equation is in 1.
(1, in Mathematica)
The hexagonal one is famous for being most common, and also the best space filler of all rod packing arrangements when the rods touch each other. We show this structure of parallel rods in two different orientations in figure 2 a - b.
Fig 2: a. Hexagonal rod structure. b. Seven rods as in a but oriented after a cube diagonal.
When four such structures oriented as in figure 2b interpenetrate each other after the directions of the four diagonals in the cube we obtain the structure in figure 3a - b with the equation in 2. This is called body- centered cubic, or garnet packing of rods.
(2, in Mathematica)
Fig 3: a. Rod structure as in 2b but with four identical hexagonal systems interpenetrating each other. b. Projection of a to show one complete hexagonal bundle.
It is also possible to combine the mathematics behind figures 2 - 3 as shown in figure 4 and equation 3 .
(3, in Mathematica)
Fig 4: The two rod structures from above interpenetrate.
In figure 5a from  there are two dendrites and two axons (in white) that are connected via axonal boutons and dendritic spines to form a neural circuit. In figure 5b there is a part of the fence packing of rods from figure 1b, with two thicker rods corresponding to the dendrites. The equation for the packing is in 4.
(4, in Mathematica)
Fig 5: a. Real packing of rods . b. Fence packing from Fig 1b.
In the beautiful picture of five axons in figure 6a from , we trace a part of the body centered cubic packing of rods given in figure 3, as shown here in figure 6b. The equation is in 5.
Fig 6: a. Five axons pack . b. Corresponding rods.
(5, in Mathematica)
The most important feature of the lateral neuron organization is the coordination, which provides prerequisites for formation of synaptic connections. The coordination can be analyzed by matching with a particular model of rod packing as illustrated here. These rod packing arrangements can be successively bent, and dilatation in two dimensions, as the geometry in cortex, is also possible.
We leave the cubic symmetry and go to the tetragonal system of rod packing. In figure 7 with equation in 6 there is one central and thick rod called goke, surrounded by eight thinner, on two different levels to show a possible periodicity. The eight could now be axons that are surrounding a central dendrite.
(6, in Mathematica)
Fig 7: Tetragonal rod packing.
We now propose that this tetragonal rod packing is useful in order to describe the remarkable structure found by Shepherd and Harris in rat hippocampal slices [10,11]. They found a 3D structure here given in figure 8a, containing eight axons that ‘travel in many different directions’. We compare with the tetragonal rod packing as given in figure 8b, which is then identical geometry as in figure 7. But we use now cosine space in order to easily scan a region (see the boundaries in figure) where there is good agreement between mathematics and the experimental findings. Equation describing this packing without goke is given in 7.
Fig 8: a. Eight axons after . b. Corresponding tetragonal rod structure.
(7, in Mathematica)
Finally we do a variation of this tetragonal packing after equation 8 to obtain a double-helical arrangement of rod units circumscribing a center rod (goke) as shown in figure 9a. Similar arrangements are readily found in the dense neuropil where large numbers of axons are passing through the dendritic arbors of pyramidal neurons. Here it is shown for a lateral dendrite of a CA1 pyramidal neuron from the hippocampus of the rat (figure 9b). A perfect match can however hardly be expected in view of the dynamics and plasticity of the brain. Disturbances in ordered rod packing arrangements may even be consequent to information storage.
(8, in Mathematica)
Fig 9: a. Double helix of axons surrounding a dendrite goke as produced by equation 8. b. A reconstructed dendrite segment (gray) with a sample of eight axons (colored) that touch it.
We have demonstrated that certain rod packings can provide good models of certain neuron organizations in space. This representation can be used to model the three-dimensional relationship of axons and dendrites, potentially providing a concise mathematical description of synaptic connectivity within a volume.
7. Lidin, S., Jacob, M., and Andersson, S., J. Solid State Chem. 114 36 (1995).
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