BA developed the theoretical model and wrote the paper.
The author has declared that no competing interests exist.
The structure of local synaptic circuits is the key to understanding cortical function and how neuronal functional modules such as cortical columns are formed. The central problem in deciphering cortical microcircuits is the quantification of synaptic connectivity between neuron pairs. I present a theoretical model that accounts for the axon and dendrite morphologies of pre- and postsynaptic cells and provides the average number of synaptic contacts formed between them as a function of their relative locations in three-dimensional space. An important aspect of the current approach is the representation of a complex structure of an axonal/dendritic arbor as a superposition of basic structures—synaptic clouds. Each cloud has three structural parameters that can be directly estimated from two-dimensional drawings of the underlying arbor. Using empirical data available in literature, I applied this theory to three morphologically different types of cell pairs. I found that, within a wide range of cell separations, the theory is in very good agreement with empirical data on (i) axonal–dendritic contacts of pyramidal cells and (ii) somatic synapses formed by the axons of inhibitory interneurons. Since for many types of neurons plane arborization drawings are available from literature, this theory can provide a practical means for quantitatively deriving local synaptic circuits based on the actual observed densities of specific types of neurons and their morphologies. It can also have significant implications for computational models of cortical networks by making it possible to wire up simulated neural networks in a realistic fashion.
Each neuron communicates signals via synaptic connections simultaneously to several hundreds of neighboring neurons forming a synaptic circuit. Determining the pattern of synaptic connections between local neurons is crucial for understanding a specific cortical function implemented by a synaptic circuit. The connectivity between a pair of neurons is affected by their axonal/dendritic morphologies and relative spatial locations. Although neuroscientists have precise tools to measure neuronal activity caused by the flow of signals between circuit neurons, there are still considerable difficulties in the direct experimental measurement of local synaptic connectivity, which actually determines the underlying activity. This paper presents a theoretical approach to synaptic connectivity accounting for the morphologies of pre- and postsynaptic neurons and providing the average number of synaptic contacts formed between them as a function of their relative locations. An important aspect is the decomposition of the complex structure of an axonal/dendritic arbor into a small number of basic structures. The theory is in very good agreement, within a wide range of cell separations, with empirical data on axonal–dendritic contacts of pyramidal cells and somatic synapses formed by the axons of inhibitory interneurons. The current approach can provide a practical means for quantitatively deriving local synaptic circuits based on the actual observed densities of specific types of neurons and their morphologies.
Unraveling intrinsic cortical circuitry—the pattern of synaptic connections between neurons within a local region—still is one of the most difficult challenges faced by researchers in neuroscience. The structure of intrinsic circuitry is the key to understanding cortical function and how neuronal functional modules such as cortical columns are formed [
To form a synaptic contact it is necessary that the presynaptic axon comes to a close spatial apposition to a certain cellular site (dendrite, soma, axon hillock, etc.) located on the postsynaptic neuron, establishing a physical contact. For the purpose of synaptic connectivity, it is useful to distinguish morphological features of neurons at large and small spatial scales. Large-scale features, such as the characteristic shape and size of a volume occupied by axonal/dendritic ramifications, set the limits of spatial separation between a pair of neurons within which they can potentially establish physical contacts and thus affect how the synaptic connectivity changes as a function of their relative positions.
Small-scale morphological features, such as the length and local curvature of axonal branches, can reveal cellular site specificity of synaptic contacts. The question of whether synaptic connectivity is random or specific has been addressed in several studies [
Morphological properties of the remaining (20%–30%) neocortical cells, which are mostly inhibitory interneurons [
Quantitative studies of synaptic connectivity are usually restricted to the case of nonspecific connections, when geometries of axons and dendrites are assumed to be mutually independent [
In this paper I present a theoretical model of synaptic connectivity that strives to bring the strong aspects of each of these two different approaches into a single framework. For example, it takes into consideration large-scale morphological features of pre- and postsynaptic cells. However, it does not require 3D reconstructions of neurons; the necessary structural parameters of the underlying arbors can be directly estimated from plane, two-dimensional (2D) drawings of axons and dendrites. Importantly, such drawings are already available in literature for many different types of neurons. On the other hand, this theory does not explicitly consider small-scale morphological features of arbors but rather, in the spirit of several previous studies [
Synaptic connectivity between cortical neurons takes place at short-range (<103 μm) and long-range (>103 μm) scales [
Suppose that one is able to record the spatial position of each synapse formed by the axon branches of a given presynaptic cell belonging to a particular morphological type
One can describe this structure in terms of the volume density of dots—synaptic sites—averaged over a large number of contributing cells of the same morphological type
where 〈·〉 indicates ensemble averaging.
The consideration above can be carried out, likewise, for synapses formed on the dendrite branches of postsynaptic cells. This would result in a corresponding synaptic density field
Consider now a pair of cells separated by a displacement vector
In the framework of this approach,
I now make two simplifying assumptions. First, I assume that
This means that, at any given spatial location
where
Using
Integrating
I assume that at any given spatial location, the ensemble distribution of the number of synaptic contacts within an element of volume Δ
The evaluation of the average number of synaptic contacts
Synaptic density fields have cylindrical symmetry. The axis of symmetry traverses the cell soma and is oriented vertically, orthogonal to the cortical layers. Thus, the density is isotropic in the horizontal dimension, parallel to the layers. This assumption is motivated by the laminated structure of the cortex [
The arborization structure of a given morphological type
Likewise, the average total number of synaptic contacts between a pair of cells is the sum of the contributions
where
(cf.
Synaptic density field of each cloud is illustrated by a set of concentric ellipsoids of different weights. An ellipsoid represents the equal-synaptic-density surface, whereas its weight represents the magnitude of the density. The outer ellipsoid, in addition, encloses the spatial extent of cloud ramifications. Yellow dots depict cell somata. The horizontal ℓ|| and vertical ℓ⊥ dimensions of one of the clouds as well as the displacement
(A) A drawing of the dendritic arbor typical for L3 pyramidal neurons.
(B) A drawing of the axonal arbor typical for L2 pyramidal neurons.
The drawings of arbors are based on data representations in [
Equal synaptic-density surfaces of a given elementary cloud form a continuum of concentric, similar ellipsoids that are aligned at the cloud center. This assumption is motivated by the observation that the contours of the spatial spread of axonal and dendritic clouds often have an ellipsoidal shape (
Synaptic density falls off exponentially along the longitudinal and transverse axes of the ellipsoids. Specifically, I assume that the elementary cloud density field is given by
where
Such a choice of the density field function is motivated by the work of Sholl [
Given that synaptic density fields are determined by
where
where
where
The central point of the present theory is to reduce the detailed picture of the complex branching patterns of axonal/dendritic processes and the locations of individual synapses along them to a simple but adequate (for the explanation of local synaptic connectivity) representation that is described by a small number of phenomenological parameters. To that end, the synaptic field of a given arbor is represented as a superposition of the synaptic fields of elementary clouds. For each elementary cloud
Here I exploit Hellwig's work [
Based on the layer of origin of the cell soma, Hellwig distinguished two different types of axons and two different types of dendrites. I designate them as P2A and P3A for the axons and P2D and P3D for the dendrites of neurons in L2 and L3, respectively (
(A–D) Images representing the average structures of axons (A and B) and dendrites (C and D) of pyramidal neurons originating from L2 (A and C) and L3 (B and D). Yellow dots depict cell somata. Ellipsoids capture the spatial extent of the synaptic clouds identified from these images. The dimensions
(E–H) Average number of contacts between pre- and postsynaptic neurons as a function of the distance between them. The type of axonal–dendritic connection is shown on each plot. Empirical curves [
For each type of axon and dendrite there were four 3D reconstructions. By averaging over 32 possible combinations of pre- and postsynaptic cell pairs of the same
First, I visualized average spatial structures of the reconstructed axons and dendrites. To that end, individual arborization drawings based on data representations in [
Second, visually inspecting these images, I identified elementary clouds of axons and dendrites, and enclosed them in distinct ellipses capturing the spatial extent of individual cloud ramifications (
Third, these measurements were linked to the parameters of the theory. Specifically, I assumed that the space constants of a given cloud
Fourth, the parameters
Finally, predictions of the theory were compared against independent experimental data. Specifically, the two remaining empirical curves,
One can see that overall there is very good agreement between the theory and experiment. Note that a single fixed pair of parameters
In this example I use experimental data obtained by Kisvárday and colleagues [
(A) Radial distribution of the average number of postsynaptic somata contacted by the axon. Dots with drop-lines show empirical distribution obtained by pooling data from the two cells [
(B) Image representing the average structure of the clutch cell axon. Yellow dot depicts cell soma. The ellipsoid captures the spatial extent of the synaptic cloud identified from the image. The dimensions
To compare these experimental data with the theory, which provides ensemble averaged quantities, I first pooled data from the two cells and obtained the average observed radial distribution
First, utilizing the assumption of cylindrical symmetry, an image representing the average spatial structure of the clutch cell axon was obtained. Specifically, individual drawings based on data representations in [
Second, based on this image, the morphology of the clutch cell axon was described by a single cloud. The horizontal
Third, as in example 1, it was assumed that the axonal field space constants are proportional to the dimensions of the enclosing ellipse:
where
Finally, the parameters
This example, illustrated in
(A and B) Connectivity map showing the average number of contacts formed between the presynaptic cell positioned at the origin and the postsynaptic cell at location
(C and D) Images representing the average structures of dendrites (C) and axons (D) of pyramidal neurons in L5. Yellow dots depict cell somata. Ellipsoids capture the spatial extent of the synaptic clouds identified from these images. The dimensions
To compare these results with my theoretical model, I first visualized, as in previous examples, the average spatial structures of the underlying arbors using drawings based on data representations in [
The theoretical average number of contacts
In this work I proposed a simple theoretical model of local synaptic connectivity between a pair of cortical neurons that takes into account the morphological structure of axons and dendrites and the relative spatial locations of the pre- and postsynaptic somata. To understand the implications of the underlying simplifying assumptions, the theoretical number of synaptic contacts was compared with the number of contacts estimated empirically in quantitative studies of synaptic connectivity [
The present approach relies on the assumption that the interactions between axons and dendrites are negligibly small and, therefore, their morphological properties can be treated independently. This is adequate, particularly, for the axons of pyramidal cells that form nonspecific axonal–dendritic contacts (examples 1 and 3). In addition, I demonstrated that the same formalism can be extended to the case of highly specific contacts such as somatic synapses formed by the axons of inhibitory interneurons (example 2), and thus the present approach, unlike the previously suggested method [
An important aspect of the theoretical framework is the “linearization” of the complex structure of an axonal/dendritic arbor of a given morphological type
In the present consideration it was assumed that the parameter
It is noteworthy that although the number of 3D reconstructed neurons is growing, the existing empirical methods [
The significance of this work, however, goes beyond the derivation of an analytical expression describing synaptic connectivity between morphologically distinct neuronal pairs. For example, the present approach could be used for deciphering the structure of local synaptic circuitry (i.e., the pattern of connections between neurons) in a cortical region of interest. In particular, one could estimate the individual contributions from diverse types of neurons distributed across cortical layers to the net synaptic input received by a neuron of a given type
Recently, Stepanyants, Hof, and Chklovskii [
The framework of potential synapses could be also used in different contexts, providing insights into different aspects of synaptic connectivity. In the present approach the specificity of synaptic connections is determined by geometrical factors such as the layout of axonal and dendritic branches and the relative spatial positions of pre- and postsynaptic cells. Can the specificity of synaptic connections go beyond the geometry, without major remodeling of dendritic or axonal arbors? This is possible if the number of potential synapses as defined in [
Is this potential for pyramidal neuron local synaptic specificity actually realized in the cortex? Until a short time ago, this was an open question. In a recent paper, Kalisman, Silberberg, and Markram [
Thus, the specificity in synaptic connectivity without major remodeling could occur at least at two levels. While the geometry of axons and dendrites and relative cell positions define the coarse level of specificity, recent work [
In conclusion, the phenomenological approach to local synaptic connectivity described in this paper provides a remarkably simple way for extracting the relevant structural parameters of axons and dendrites from 2D arborization drawings. It was demonstrated that a crude approximation of axonal and dendritic arbors as a superposition of a set of ellipsoids is satisfactory for the purpose of quantitative estimation of synaptic connectivity between specific types of neurons as a function of their relative locations. Since for many types of neurons 2D drawings are available from literature, the present approach could be of principal significance for the practicality of deciphering synaptic microcircuits of a given cortical region based on the actual observed densities of specific types of neurons and their morphologies. It could also have significant implications for computational models of cortical networks by making it possible to wire up simulated neural networks in a realistic fashion.
The aim of this section is to explain how I evaluated the average number of synaptic contacts
where
Note now that the above integral is a convolution of two functions:
To evaluate
Next, note that
Exploiting the symmetry present in the problem, let us change to conventional cylindrical coordinates:
The integral (
where
where
Consider a case
Using now the higher symmetry of this case, one can change to spherical coordinates:
and, after integrating over
where
Making the change of variable
This last integral is evaluated using the same method of contour integration as in the general case considered above. Again, the integrand has two poles of order two in the upper-half of complex
I dedicate this paper to the memory of Alexander Lukashin, my teacher, friend, and an inspiring scientist.
I thank Apostolos P. Georgopoulos for his constant support, stimulating conversations, and encouragement. I am grateful to Bernhard Hellwig for his permission to use arborization drawings of L2 and L3 pyramidal neurons of rat visual cortex, Zoltán Kisvárday for his permission to use arborization drawings of clutch cells in L4 of cat visual cortex, and Henry Markram and Gilad Silberberg for their permission to use the connectivity map and arborization drawings of L5 pyramidal neurons of rat somatosensory cortex.
This work was supported by the United States Department of Veterans Affairs and the American Legion Brain Sciences Chair.
two-dimensional
three-dimensional
layer [number]