we have shown previously that by extending the minimum spanning tree ( mst ) algorithm it is possible to connect a set of point targets which are distributed in space to satisfy biological optimization costs of dendritic trees ( cuntz et al . , 2007 ) . the required cost function is composed of two individual costs ( figure 1a ) : the total amount of wiring ( cable length cost ) and the cost of signal conduction ( path length cost : the sum of the length of all paths along the tree from any target point to the root ) corresponding well with costs originally proposed by ramn y cajal ( 1909 ) . when a set of target points is adequately selected , any dendritic tree morphology can be reproduced after connecting the target points to a tree using the mst algorithm . this approach represents a method of morphological modeling useful for generating realistic synthetic dendritic trees ( cuntz et al . , 2010 ) . the sketch in figure 1b shows a synthetic starburst amacrine cell which was generated by distributing target points stochastically according to a geometrical arrangement of densities . target points are then connected to a tree structure whilst observing the wiring costs mentioned above . this simple procedure highlights the usefulness of a ddf , here representing the density of target points per area , to describe a dendrite s morphology . the ddf and the one parameter balancing the two costs for wiring mentioned above are thereby sufficient to describe the dendrite type . in fact , we have previously shown that for fly interneurons the spanning field is the most informative element for classifying dendritic trees into their respective cell types ( cuntz et al . , , ddfs can be obtained directly from reconstructions of real neurons . by superimposing their soma locations and rotating them such that they lie symmetrically around the y - axis and then scaling their width , height , and depth , a set of reconstructions of dentate gyrus granule cell ( data from rihn and claiborne , 1990 ) dendrites an appropriate approximation of the ddf is then obtained directly from the density of branch and termination points from the reconstructions ( figure 1c ) . when apical and basal dendrites are considered separately , the ddfs of pyramidal cells can be calculated in a similar way ( l5 pcs , figure 1d ; data from wang et al . , 2002 ) . ( a ) from left to right the same set of unconnected target points ( green ) were connected to a starting point ( large black dot ) according to the minimum spanning tree ( mst ) algorithm with increasing balancing factor bf between the two wiring costs : total cable length and sum of all paths . ( b ) geometric description of the ddf of a starburst amacrine cell ( left ) . by randomly selecting target points according to the ddf and subsequently connecting them using the mst algorithm ( right ) to a starting node ( large black dot ) ( c ) the ddf obtained directly from reconstructions of dentate gyrus granule cells with its characteristic cone - like shape . ( d ) separate ddfs for l5 cortical pyramidal cell basal trees ( red ) and apical trees ( green ) . while the dendrites of starburst amacrine cells , dentate gyrus granule cells , and many other neurons exhibit complex ddfs , some simpler ones such as pyramidal cell basal dendrites observe basic principles which can be characterized by studying the statistical moments of their density distributions ( snider et al . , 2010 ) . stevens and colleagues have shown that the cable density distributions in dendrites roughly follow separate gaussian distributions for each dimension in space which are cut off at 2 standard deviation . in synthetic dendrites generated using the mst , homogenously distributed carrier points lead to a homogeneous cable density . the mst alone can therefore not be responsible for the observed gaussian cable density distribution . intuitively , the shape of a ddf should be determined by the set of axons which are a dendrite s potential , i.e. , anatomical connection partners . when input target points were distributed along one dimension , the mst was simply a straight line connecting these points ( figure 2a ) . when these target points were extended to parallel lines as in figure 2b , and the mst was set to connect to each axon once , the resulting synthetic dendrite was not changed . out of cable and conduction time minimization , a straight and direct connection remained ideal . when more axons were added and noise was introduced to their trajectories ( figure 2c ) , the mst dendrite started to branch out . in a similar but more realistic setting , a dendrite connecting to a set of parallel axons traversing space was flat according to the mst ( figure 2d ) . a flat dendrite which lies perpendicular to a set of parallel axons was shown to be the overall ideal geometrical arrangement given cajal s wiring costs ( wen and chklovskii , 2008 ) . such a planar dendritic structure can be observed in cerebellar purkinje cells for example which reach out to the molecular layer to collect their parallel fiber inputs . these being arranged entirely in parallel the dendrite must grow in a planar way perpendicular to the parallel fibers to connect to them most efficiently . ( a ) target points ( green ) lying on one line are connected to a starting node ( large black dot ) using the mst algorithm . ( b ) same as in ( a ) but targets are parallel lines ( axons , green ) rather than points . the optimal dendrite connecting to these axons is the same as in ( a ) . ( c ) same as in ( b ) but the number of target axons is increased and the axon trajectories are slightly jittered : the mst is slightly jittered and a few branch points appear . ( d ) synthetic dendrite ( black ) connecting a starting point ( large black dot ) in 3d space to a set of parallel axons ( green ) : the resulting mst is flat . ( e ) target axons ( green ) are arranged in parallel planes but are of isotropic orientation within these planes . the axons are connected using mst ( black tree ) to a starting point ( large black dot ) located below the layer of axons . ( f ) ddf of 50 synthetic granule cells grown as in ( e ) ; compare with figure 1c . left : side view of density profile ; right : mesh representation of the same density distribution . ( f ) but the starting point was moved to the center of the layer to reproduce the characteristic shape of the basal tree of a cortical pyramidal cell . ( h ) same as in ( e ) but the mst connects to axons located in two parallel layers . ( i ) ddf of 100 synthetic pyramidal cells grown as in ( h ) ; compare with figure 1d . ( j ) one such synthetic pyramidal cell where diameter values were mapped onto the dendritic segments and spatial jitter was added along the dendrite . most neural systems however are not entirely optimized for the connection between only one set of axons and one dendrite type . axon distributions are therefore more complex . in order to illustrate the potential of this constructive approach , i demonstrate in the following how a few sample input axon sets would impact on the resulting mst dendrites and therefore on their average ddf . the first case scenario that i chose exhibits a set of axons which were located in parallel neighboring planes forming a neuronal layer . within their respective planes , the axons were isotropically oriented and the starting point , the root of the dendrite , was positioned below the layer of axons . only the axons passing in close vicinity with a cutoff at half of the thickness of the layer were considered ( figure 2e ) . in such a configuration the distribution of axons was homogeneous . however , the resulting mst dendrite grew a shape reminiscent of dentate gyrus granule cell dendrites and exhibited an inhomogeneous ddf ( figure 2f ) . when the starting point was moved to the center of the axonal layer , this resulted in an isotropic ddf similar to a pyramidal cell basal dendritic density profile ( figure 2 g ) . it is possible that such a ddf follows similar features as the dendritic cable densities measured previously ( snider et al . , 2010 ; teeter and stevens , 2011 ) . note that the precise relation between density of topological points in a dendrite and dendrite cable density has yet to be studied in detail . in any case , it will be possible to find the type of arrangement of input axons which generally reproduces their measures , thereby unveiling general principles of axon arrangements and connectivity patterns . to finish , i would like to suggest a last sample configuration in which two separate layers of axons similar to the one previously described were both connected to a single starting point located in the middle of the lower of the two layers ( figure 2h ) . under these conditions , the natural shape of a pyramidal cell , including its apical dendrite and its two separate ddfs ( figure 2i ) were a natural consequence . when biological jitter and diameter values were mapped onto the corresponding tree structure ( as in cuntz et al . , 2010 ) , this resulted in a biologically realistic pyramidal cell ( figure 2j ) . i have shown that morphological models can provide more than just anatomical profiles for realistic neural network simulations . they can also be a tool to understand how dendrite morphology comes about and a tool to test our knowledge about the local connectivity in the brain . this short exercise provides once more evidence that the major determinant for a dendrite s morphology is its role in the connectivity . it is well known that morphology particularly in cortical pyramidal cells plays a role for the intrinsic computation that a single neuron performs on its inputs ( vetter et al . , 2001 ; polsky et al . , 2004 ; sjstrm and husser , 2006 ; branco et al . , 2010 ) . the role of morphology on single neuron computation , and vice versa , can also be studied using morphological modeling . in the morphological model , neuronal computation the main morphological parameter which impacts neuronal computation is the balancing factor bf between the two wiring costs ( figure 1a ) . we have previously shown that a large bf promotes a higher electrotonic compartmentalization of the resulting synthetic dendritic tree ( cuntz et al . , 2010 ) . this adds to the natural increase in conduction time speeds which is favored with increasing bf to the detriment of wiring economy . secondly , however , only electron microscopy reconstructions ( e.g. , kubota et al . , 2011 ) will resolve the precise role of diameter distributions on the electrotonic properties of a dendrite . lastly , to implement a given computation , the single neuron can select a subset out of a large variety of possible ion channel combinations which may be adapted to suit the underlying morphological template ( prinz et al . , 2003 ) . interestingly , the toy model of a pyramidal cell presented here does not build on any functional differences between the two layers that it connects . the location of the starting point alone determines which of the layers will yield an apical dendrite and which one a basal dendrite . this dendrite root corresponds to the summation point of the input signals but more importantly it is the coordinate from which the axon , the output of the neuron , exits the cell . the fact that the location of the exiting axon determines the neuron s shape was at the center of cajal s axopetal polarity theory . in the cortex , most long - range axons project toward the white matter away from the pial surface . we would therefore predict a stereotypic polarity of pyramidal cell morphology : basal dendrites of pyramidal cells should be on the inside closer to the white matter , whereas the apical dendrites should be on the pial side . this indeed is the case . at the example of the pyramidal cell dendrite i showed that it is possible to predict both their ddf and their precise branching structure by making assumptions about the input axon distribution . in this way it might for example be possible to link regionalized specializations in pyramidal cell dendrite shape ( elston , 2003 ) with differences in local connectivity . to summarize , i have shown that the ddf is a promising attribute of dendrite shape which enables direct conclusions on the arrangement of a neuron s axonal inputs . the methods presented here have been discussed in depth in previous reports ( cuntz et al . , 2007 , 2008 , 2010 , 2011 ) and the corresponding code is available at www.treestoolbox.org . the original mst algorithm was implemented as an iterative process which connects unconnected target points one at a time to an existing tree structure following the wiring costs mentioned in the text ( figure 1a ; see cuntz et al . , 2007 ) . when a target point is connected to the tree , the remaining target points in the same group become unavailable for the algorithm in the further process . target input axons were then implemented as labeled groups of target points distributed in close intervals along the trajectory of the respective axons . this was tested in figures 2b  d on simple parallel axon trajectories including a slight spatial jitter in figure 2c ( 50 axons ) and 2d ( 100 axons ) . the resulting mst dendrite grown to connect a set of parallel axons was flat in all cases ( figure 2d ) . for the dentate gyrus granule cell model in figures 2e , f , the set of axons was generated by randomly selecting x and y coordinates and a random angle for each axon within a 1.5-mm    1.5-mm plane and drawing straight lines with these coordinates . three hundred axons were generated in this way and a random z - value between 0 and 250  m was associated to each axon . the starting point for the dendrite was located 25  m below the axonal layer and in the center of the xy plane . only axons passing in 125  m of the center of the axonal layer were selected resulting on average in a set of around 60 axons . for the pyramidal cell basal dendrite in figure 2 g , the procedure for figures 2e , f was reproduced after moving the starting point to the center of the axonal layer . for the complete pyramidal cell model in figure 2h  j , two layers similar to the granule cell axonal layer from figures 2e , f were generated with 200 axons each . around 3040 axons were selected by the same 125  m sharp cutoff criterion . the starting point for the dendrite the author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest .
much is known about the computation in individual neurons in the cortical column . 
 also , the selective connectivity between many cortical neuron types has been studied in great detail . however , due to the complexity of this microcircuitry its functional role within the cortical column remains a mystery . 
 some of the wiring behavior between neurons can be interpreted directly from their particular dendritic and axonal shapes . 
 here , i describe the dendritic density field ( ddf ) as one key element that remains to be better understood . 
 i sketch an approach to relate ddfs in general to their underlying potential connectivity schemes . as an example 
 , i show how the characteristic shape of a cortical pyramidal cell appears as a direct consequence of connecting inputs arranged in two separate parallel layers .