Optimization principles of dendritic structure
© Cuntz et al; licensee BioMed Central Ltd. 2007
Received: 26 March 2007
Accepted: 08 June 2007
Published: 08 June 2007
Dendrites are the most conspicuous feature of neurons. However, the principles determining their structure are poorly understood. By employing cable theory and, for the first time, graph theory, we describe dendritic anatomy solely on the basis of optimizing synaptic efficacy with minimal resources.
We show that dendritic branching topology can be well described by minimizing the path length from the neuron's dendritic root to each of its synaptic inputs while constraining the total length of wiring. Tapering of diameter toward the dendrite tip – a feature of many neurons – optimizes charge transfer from all dendritic synapses to the dendritic root while housekeeping the amount of dendrite volume. As an example, we show how dendrites of fly neurons can be closely reconstructed based on these two principles alone.
The anatomy of the dendritic tree is one of the major determinants of synaptic integration [1–6] and the corresponding neural firing behaviour [7, 8]. Dendrites come in various shapes and sizes which are thought to reflect their involvement in different computational tasks. However, so far no theory exists that explains how the particular structure of a given dendrite is connected to their particular function. Because dendrites are the main receptive region of neurons, one common requirement for all dendrites is that they need to connect with often wide-spread input sources such as elements which are topographically arranged in sensory maps . This implies that the distance of different synaptic inputs to the output site at the dendritic root may vary dramatically from one synapse to the other. As a result, the impact of different synapses on the neural response would be expected to be highly inhomogeneous. Some neuron types seem to cope with this problem by increasing the weights of distal synapses [10–12], but see . The intrinsic structure of dendrites, with thinner dendrites (larger input impedance) at more distal sites, however plays a crucial role in compensating for the charge loss from distal synapses [14–16]. In the present study we show how the effort of homogenizing synaptic efficacy can completely characterize the fine details of dendritic morphology, using the dendrites of lobula plate tangential cells of the fly visual system as an example. These interneurons integrate visual motion information over a large array of columnar elements arranged retinotopically as a spatial map . By observation, their planar dendrites which spread across the lobula plate to contact the columnar input elements within their receptive fields are regarded as being anatomically invariant  suggesting a rather strong functional constraint.
Results and Discussion
V root (x) = V ratio (x)·R IN (x)·I syn
We therefore investigated whether such an inverse proportionality between the voltage ratio and the local input resistance exists. As can be seen from Figure 1B–D for tangential cell dendrites, the input resistance does indeed increase in a similar way to the voltage ratio drop off throughout the dendrite. The inverse proportionality between R IN and V ratio is reflected in their relationship to each other (Figure 1D). This observation holds true when strong full-field visual stimulation increases the membrane conductance drastically (see Additional file 1, Figure S2) and when peak or integral values of the charge are considered for time varying synaptic currents. This feature of the passive dendritic structure represents a homogenous backbone on which active properties could sensibly implement non-linear computations. However, in the case of the cells analysed here, responses correspond to graded potential shifts only moderately further modulated by active non-linearities. In the following we will explain this behaviour of the passive dendritic tree by first considering the effect of diameter tapering and then examining the topological features.
Diameter tapering related electrotonic homeostasis
The increasing input resistance in distal dendrites producing an almost homogenous current transfer could be a simple consequence of the decrease in dendrite diameter with distance from the root . In a symmetrical dendritic tree corresponding to a cylinder of constant diameter, the increase of R IN with distance, as well as the attenuation factor can be computed analytically . There, R IN and the attenuation factor are not inversely proportional since their ratio depends on cosh(L), L being the electrotonic length (in units of the space constant, λ). This implies that tangential cells and other neurons which optimize current transfer from synapses to dendritic root achieve this by utilizing different principles.
Synapse-targeted topological properties of dendrites
Aside from adjusting dendritic diameters to optimize synaptic efficacy, dendrites could also follow some optimization principles with respect to their branching structure. To describe the topology of dendrites, graph theory provides an appropriate framework. In this context, the branching structure of a dendrite is regarded as a network connecting all points at which synapses are located. After assigning vertices to particular locations in space according to putative synapse positions, the branching structure is defined as the set of directed edges between these locations leading away from the dendritic root. From a purely topological point of view, maximal proximity of each synapse to the dendritic root is achieved by a direct connection in a fan-like manner. This would minimize the path lengths with respect to individual synapses since each indirect connection would correspond to a detour on the way from the synapse to the dendritic root. However, such a fan-like structure is not usually observed in real dendrites.
Lobula plate tangential cells exhibit a rather invariant anatomy from one animal to the next . They are interneurons whose function it is to integrate over an array of local columnar elements distributed retinotopically over the surface of their receptive fields. Here we propose that optimizing synaptic efficacy at the root leads to the stereotyped nature of their dendritic structures. We show that dendritic diameter tapering towards the terminal tips optimally equalizes current transfer from all synaptic locations to the dendritic root. This could correspond to the finding that dendritic morphology can be described in a diameter dependent manner . The optimal course of tapering is a quadratic decay. It will be interesting to further investigate this electrotonic feature of dendrites and cables in general. In addition to its optimized diameter tapering, the dendritic tree is optimally branched to keep synaptic contacts close to the dendritic root whilst minimizing the total dendritic wiring. Our analysis has therefore re-affirmed the importance of wiring cost to which several morphological and organizational principals in the brain were attributed previously [21, 25]. Together, these represent fundamental principles for shaping dendrite structure. Both monotonic tapering diameters and homogenous integration of spatially distributed inputs are characteristic of many dendrites; these principles may well therefore be applicable for many other systems. In recent years, a number of approaches have been taken to describe dendritic morphologies based on local branching statistics and on only a few branching rules [26, 27]. In contrast to these studies, we show here the possibility of setting neuroanatomical reconstruction into the context of their function: synaptic integration.
Electrotonic investigation on dendrites as graphs
when the current matrix I is chosen to be the identity matrix. The resulting symmetric matrix V corresponds to the potential distributions throughout all nodes in each column when current is injected in the node corresponding to the column index. The local input resistances in the different branches of the dendritic tree can therefore be read in the diagonal of V. In order to obtain the electrotonic measurements in the tangential cell model used in Figure 1, we converted the neuroanatomical description of a compartmental model of an HSS cell  into a sparse adjacency matrix and sparse matrices containing length and diameter of each compartment in the diagonal. The inverse of the matrix G obtained from Equation (2) is shown for this compartmental model (consisting of 2251 compartments) in Figure S1B (Additional file 1). The specific passive properties (membrane resistance of 2000 Ωcm2 and axial resistance of 40 Ωcm constant in all models) were adopted from .
(N = 7, number of segments including the appended axon) was minimized using the built-in MATLAB function fminsearch. Results were supported by corresponding simulations in NEURON . Since segments of up to 500 μm are not isopotential, the adjacency matrix required a stretching extension to divide the seven segments into several compartments. A complete investigation of the current transfer optimization in the example of the unbranched cable showed similar results under a variety of simulation settings (Additional file 1, Figure S2). In all cases the diameter tapered in a quadratic manner starting at different initial diameters depending on the settings of the bounding axonal segment.
With continuous matrix multiplication on the directed adjacency matrix, as in A r , the (i, j)-entry represents the number of distinct r-walks from node i to node j in the graph. Therefore, some elementary statistical properties, e. g. path lengths, can readily be accessed using the graph representation of the dendritic tree. To be able to compare topologies between different dendrites and assign them to an equivalence class we developed an ordering scheme based on conventional graph sorting. After assigning a root index, the remaining indices were first sorted by path length to the root and if those were the same then by level order (summing up the path lengths to the root of all child branches). Indices were then sequentially reassigned just next to their parent index following the sequence of the above order. This resulted in dendrograms in which the 'heavier' sub-tree is always on the left.
Optimizing topological features
The extended minimum spanning tree algorithm to obtain the adjacency matrix in an optimized wiring scheme for a given set of points followed the principles described by Prim . Starting with the root, the set of connected points was compared to the set of non-connected points. One at a time, the closest point from the non-connected set (the distance measure included the total path length to the root with a balancing factor bf) was connected to its partner in the set of connected points. In order to keep the total path length of each new point P x to the root P 0 small, we simply added a term to the distance measure D weighted by a factor bf:D x,i = |P i P x |+ bf|P 0 → P x |
bf was chosen to be 0.2 to reproduce best the topology of the tangential cell dendrite (for the choice of bf see Additional file 1, Figures S4 and S5). This represents a crude definition of the distance constraint and can be refined in further studies. The algorithm was run on homogenously distributed points in a random way confined to the convex hull around the dendrite of the original tangential cell (Figures 4AB, and Additional file 1, Figures S4 and S5). Alternatively, the branching and termination points of the original tangential cell were chosen (Figures 4CD, 5 and Additional file 1, Figure S6).
In order to apply diameter tapering on the constructed topology for Figure 5, the diameters corresponding to the optimized quadratic tapering along all normalized paths from root to terminal points were averaged for each compartment. In this way a monotonic tapering could be attributed to any type of branching structure. Validation of this procedure and comparison to the monotonic tapering in real tangential cells is shown in additional file 1, Figure S6. All computations were performed in MATLAB.
- V root :
voltage response at dendrite root
- I syn :
constant steady synaptic current
- V ratio :
- R IN :
- A :
directed adjacency matrix.
We would like to thank J. van Pelt and A. van Ooyen for fruitful discussions. H.C. was funded by a Minerva scholarship and by a post-doctorate fellowship from the Interdisciplinary Center for Neural Computation, the Hebrew University, Jerusalem Israel.
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