Abstract
Neurons innervate space by extending axonal and dendritic arborizations. When axons and dendrites come in close proximity of each other, synapses between neurons can be formed. Neurons vary greatly in their morphologies and synaptic connections with other neurons. The size and shape of the arborizations determine the way neurons innervate space. A neuron may therefore be characterized by the spatial distribution of its axonal and dendritic “mass.” A population mean “mass” density field of a particular neuron type can be obtained by averaging over the individual variations in neuron geometries. Connectivity in terms of candidate synaptic contacts between neurons can be determined directly on the basis of their arborizations but also indirectly on the basis of their density fields. To decide when a candidate synapse can be formed, we previously developed a criterion defining that axonal and dendritic line pieces should cross in 3D and have an orthogonal distance less than a threshold value. In this paper, we developed new methodology for applying this criterion to density fields. We show that estimates of the number of contacts between neuron pairs calculated from their density fields are fully consistent with the number of contacts calculated from the actual arborizations. However, the estimation of the connection probability and the expected number of contacts per connection cannot be calculated directly from density fields, because density fields do not carry anymore the correlative structure in the spatial distribution of synaptic contacts. Alternatively, these two connectivity measures can be estimated from the expected number of contacts by using empirical mapping functions. The neurons used for the validation studies were generated by our neuron simulator NETMORPH. An example is given of the estimation of average connectivity and Euclidean pre- and postsynaptic distance distributions in a network of neurons represented by their population mean density fields.
Highlights
Because synapses can form only when axons and dendrites are in close proximity, the connectivity in neuronal networks strongly depends on the three-dimensional morphology of the constituting neurons
ESTIMATION OF THE CONNECTIVITY BETWEEN AN AXONAL AND A DENDRITIC NEURON USING POPULATION MEAN DENSITY FIELDS For the application of the method the morphologies of a number of 50 neurons were generated with the simulator NETMORPH, using a parameter set optimized on a set of rat layer 2/3 pyramidal cells obtained from the Svoboda data set in the NeuroMorpho.org data base (Figure 2)
The axonal and dendritic density fields are calculated by dividing the “mass” at a given height and radius r from the symmetry axis by the perimeter (2πr) of the circle with radius r, under the assumption that the mass is distributed uniformly over the circle centered at the symmetry axis
Summary
Because synapses can form only when axons and dendrites are in close proximity, the connectivity in neuronal networks strongly depends on the three-dimensional morphology of the constituting neurons. The morphology of neurons is complex, with branches of varying orientations and diameters bifurcating at different lengths In reconstructions this complex morphology is usually approximated in a piece-wise linear fashion, i.e., by a number of line pieces or cylinders (the latter when the diameter is measured). These reconstructions in continuous space preserve the details of the arbor structures of the neurons. Another way of characterizing the spatial structure of neurons is by discretizing space by means of a grid of voxels and defining in each voxel the neuronal “mass” (i.e., the length or the volume of a branch in that voxel). The density field of a single neuron fully reflects the arbor structure of the neuron, with non-zero densities in voxels occupied by arbors and zero densities elsewhere
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