* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Simulation of signal flow in 3D reconstructions of an anatomically
Neuroeconomics wikipedia , lookup
Binding problem wikipedia , lookup
Dendritic spine wikipedia , lookup
Long-term depression wikipedia , lookup
Artificial general intelligence wikipedia , lookup
Environmental enrichment wikipedia , lookup
Electrophysiology wikipedia , lookup
Neural engineering wikipedia , lookup
Clinical neurochemistry wikipedia , lookup
Axon guidance wikipedia , lookup
Artificial neural network wikipedia , lookup
Neuromuscular junction wikipedia , lookup
Neural oscillation wikipedia , lookup
Neural modeling fields wikipedia , lookup
Mirror neuron wikipedia , lookup
Caridoid escape reaction wikipedia , lookup
Multielectrode array wikipedia , lookup
Activity-dependent plasticity wikipedia , lookup
Molecular neuroscience wikipedia , lookup
Premovement neuronal activity wikipedia , lookup
Circumventricular organs wikipedia , lookup
Holonomic brain theory wikipedia , lookup
Neural coding wikipedia , lookup
Single-unit recording wikipedia , lookup
Metastability in the brain wikipedia , lookup
Central pattern generator wikipedia , lookup
Stimulus (physiology) wikipedia , lookup
Apical dendrite wikipedia , lookup
Recurrent neural network wikipedia , lookup
Convolutional neural network wikipedia , lookup
Neuroanatomy wikipedia , lookup
Optogenetics wikipedia , lookup
Nonsynaptic plasticity wikipedia , lookup
Neurotransmitter wikipedia , lookup
Pre-Bötzinger complex wikipedia , lookup
Neuropsychopharmacology wikipedia , lookup
Feature detection (nervous system) wikipedia , lookup
Development of the nervous system wikipedia , lookup
Types of artificial neural networks wikipedia , lookup
Synaptogenesis wikipedia , lookup
Channelrhodopsin wikipedia , lookup
Biological neuron model wikipedia , lookup
Synaptic gating wikipedia , lookup
Neural Networks ( ) – Contents lists available at ScienceDirect Neural Networks journal homepage: www.elsevier.com/locate/neunet 2011 Special Issue Simulation of signal flow in 3D reconstructions of an anatomically realistic neural network in rat vibrissal cortex Stefan Lang a,b , Vincent J. Dercksen c , Bert Sakmann a , Marcel Oberlaender a,∗ a Digital Neuroanatomy, Max Planck Florida Institute, 5353 Parkside Drive, MC19-RE, Jupiter, FL 33458-2906, USA b Interdisciplinary Center for Scientific Computing, University of Heidelberg, INF 368, 69120 Heidelberg, Germany c Department of Visualization and Data Analysis, Zuse Institute Berlin, Takustrasse 7, 14195 Berlin, Germany article info Keywords: 3D anatomy Full-compartmental models Numerical simulation Cortical column Barrel cortex abstract The three-dimensional (3D) structure of neural circuits represents an essential constraint for information flow in the brain. Methods to directly monitor streams of excitation, at subcellular and millisecond resolution, are at present lacking. Here, we describe a pipeline of tools that allow investigating information flow by simulating electrical signals that propagate through anatomically realistic models of average neural networks. The pipeline comprises three blocks. First, we review tools that allow fast and automated acquisition of 3D anatomical data, such as neuron soma distributions or reconstructions of dendrites and axons from in vivo labeled cells. Second, we introduce NeuroNet, a tool for assembling the 3D structure and wiring of average neural networks. Finally, we introduce a simulation framework, NeuroDUNE, to investigate structure–function relationships within networks of full-compartmental neuron models at subcellular, cellular and network levels. We illustrate the pipeline by simulations of a reconstructed excitatory network formed between the thalamus and spiny stellate neurons in layer 4 (L4ss) of a cortical barrel column in rat vibrissal cortex. Exciting the ensemble of L4ss neurons with realistic input from an ensemble of thalamic neurons revealed that the location-specific thalamocortical connectivity may result in location-specific spiking of cortical cells. Specifically, a radial decay in spiking probability toward the column borders could be a general feature of signal flow in a barrel column. Our simulations provide insights of how anatomical parameters, such as the subcellular organization of synapses, may constrain spiking responses at the cellular and network levels. © 2011 Elsevier Ltd. All rights reserved. 1. Introduction The different functions of neural circuits emerge from the morphology and connectivity patterns established among diverse neuronal cell types. The single neuron represents the elemental functional unit of these networks. Depending on their dendrite morphology, as well as their synaptic innervations and conductance distributions, neurons perform (non-) linear computations that generate a variety of electrical responses (Hausser & Mel, 2003). To understand how complex neuronal output drives perception and behavior, one has to investigate the synaptic ‘wiring’ between neurons, identify the microcircuits that they form and understand how these microcircuits participate in neural networks. When it becomes possible to monitor propagation of electrical activity through identified networks, the relationships between neuronal structure, brain function and behavior will be revealed. ∗ Corresponding author. Tel.: +1 5619729182; fax: +1 5619729001. E-mail address: [email protected] (M. Oberlaender). 0893-6080/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.neunet.2011.06.013 At present, observations of neuronal ensemble activity in the cortex, by optical or multi-electrode techniques, are limited. An alternative, reverse engineering approach is to reconstruct anatomically realistic neural circuits, to populate these circuits with measured spike activities and finally, to compare responses of these in silico ensembles with responses measured in the in vivo circuit (here, in silico refers to numerical simulations of neural network activity that correspond with experiments done in vitro and in vivo). This allows investigating how anatomical parameters of individual neurons, such as dendrite morphology, soma location or synaptic innervation patterns, may influence the signal flow within neural circuits. Several approaches aim to reconstruct wiring diagrams of complete neural circuits (for reviews, see Arenkiel & Ehlers, 2009; Helmstaedter, Briggman, & Denk, 2008). These approaches may, in general, be divided into two main categories, (i) dense circuit reconstructions that aim to trace all neurons and synaptic contacts at electron microscopy resolution (Bock et al., 2011; Briggman & Denk, 2006; Briggman, Helmstaedter, & Denk, 2011) and (ii) reconstructions of average circuits by determining axon–dendrite overlap of individual neurons reconstructed at light 2 S. Lang et al. / Neural Networks ( ) – Fig. 1. Overview of the network reconstruction and simulation pipeline. The pipeline comprises three blocks. First (blue), we review custom-designed methods that allow acquiring anatomical data necessary to assemble anatomically realistic 3D neural networks (i.e. 3D neuron soma distributions, complete 3D dendrite–axon tracings and determination of neuronal cell types). Second (green), we introduce a novel approach, NeuroNet, that allows assembling the structure and synaptic wiring of average neural networks. In contrast to determining synaptic contacts by geometrical proximity of axons and dendrites, we estimate connectivity by reconstructing the 3D distribution of all synapses (i.e. boutons) from a presynaptic cell type and statistically placing these contacts onto dendrites. Third (red), we introduce a novel framework for Monte Carlo simulations, NeuroDUNE, that allows investigating structure–function relationships by monitoring signal propagation in anatomically realistic networks of fullcompartmental neuron models. microscopic levels (Helmstaedter, de Kock, Feldmeyer, Bruno, & Sakmann, 2007; Markram, 2006; Wickersham, Finke, Conzelmann, & Callaway, 2007; Wickersham & Lyon et al., 2007). Dense circuit reconstructions certainly represent the most favorable solution, but remain as yet limited to volumes of about 200 µm cubes (Briggman & Denk, 2006). To construct wiring diagrams that include long-range projections, for example such as those between different cortical layers, or even between different cortical fields and subcortical neuron ensembles, reconstruction of average networks by using conventional filling of neurons (Horikawa & Armstrong, 1988; Margrie, Brecht, & Sakmann, 2002; Pinault, 1996), or molecular and genetic staining techniques (Arenkiel & Ehlers, 2009; Lein et al., 2007; Tsien et al., 1996), remains presently the primary approach to reconstructing anatomically realistic neural networks. We present a pipeline of novel, custom-designed tools that allows the reverse engineering of 3D anatomically realistic circuits and the simulation of the electrical activity propagating through them. We review previously reported tools that allow for fast and automated acquisition of 3D neuron soma distributions and reconstructions of dendrites and axons from in vivo labeled cells. Then, we introduce NeuroNet, a tool for assembling the 3D structure and synaptic wiring of average neural networks. Finally, we introduce a simulation framework, NeuroDUNE, to investigate structure–function relationships within networks of full-compartmental neuron models at subcellular, cellular and network levels. As an example we illustrate the application of this pipeline to excitatory connections between whisker-specific neurons in thalamus (ventral posterior medial division, VPM) and L4ss neurons located in an anatomically defined cortical ‘barrel’ column in rat vibrissal cortex (S1). 2. Methods 2.1. Tools for acquisition of 3D anatomical data Fig. 1(a) summarizes the first block of the pipeline to assemble realistic neural networks. It comprises previously reported tools for the acquisition of anatomical data and provides the input to the tool for reconstructing the 3D structure and synaptic wiring of average neural networks. One essential requirement to reconstruct the 3D structure of average neural networks is to determine the number and 3D distribution of neurons within the respective brain region. Briefly, we label slices for NeuN to specifically visualize the location of all neuron somata (Mullen, Buck, & Smith, 1992). Using 3D S. Lang et al. / Neural Networks ( confocal microscopy and automated soma detection software, NeuroCount, (Oberlaender et al., 2009) we obtain the number and 3D distribution of all neuron somata within relatively large volumes of tissue. Specifically, we determined the number and 3D distribution of neuron somata within an average barrel column in rat vibrissal cortex (Meyer, Wimmer, & Oberlaender et al., 2010; Oberlaender et al., submitted for publication). A second requirement comprises 3D reconstructions of dendrite and axon morphologies that would ideally cover the variability of neuronal cell types in the circuits of interest. CameraLucida-based manual reconstructions represent the state-of-theart tracing technique for neuron reconstructions, but are tedious and time-consuming (He et al., 2003). In case of axon morphologies, manual reconstructions of a large variety of neurons may so far even be unfeasible (Svoboda, 2011). We developed a semiautomated tracing pipeline, NeuroMorph, that allows obtaining reliable 3D dendrite and axon morphologies in less time when compared with manual reconstructions (Dercksen, Oberlaender, Sakmann, & Hege, in press; Dercksen et al., 2009; Oberlaender, Broser, Sakmann, & Hippler, 2009; Oberlaender, Bruno, Sakmann, & Broser, 2007). Neurons are labeled in vivo with biocytin (Horikawa & Armstrong, 1988) using either, juxtasomal- or whole-cell recordings (Margrie et al., 2002; Pinault, 1996) and are reconstructed from consecutive 50–100 µm thick tangential vibratome sections. Using the NeuroMorph tool, we were able to reconstruct the complete 3D intracortical axon morphologies of individual VPM neurons and dendrites of nine excitatory cell types (e.g. L4ss neurons) in a rat barrel column (Oberlaender et al., submitted for publication). A third requirement for network reconstructions is the definition of a 3D reference frame for registering soma distributions and dendrite–axon morphologies into the same standardized coordinate system. This allows an average network to be generated from anatomical data obtained from different animals. In case of a barrel column, we use the pia surface and the stereotypic organization of the barrels in L4 to register all anatomical data into a standardized coordinate system (Oberlaender et al., submitted for publication). Briefly, the origin of the coordinate system is set at the center of the barrel containing the neuron’s soma. The y-axis is chosen to point to the center of the first neighboring barrel in the rostral direction. The z-axis points dorsally, approximately parallel to the vertical column axis. The tool NeuroConv converts the dendrite–axon morphologies into the ‘‘hoc’’ format that is used by NEURON simulation software (Hines & Carnevale, 1997). The last anatomical prerequisite is determining the number of cell types and their location within the network. We recently reported objective classifications of cell types in a barrel column by dendrite geometry (Oberlaender et al., submitted for publication) using a density-based cluster algorithm (Ankerst, Breunig, Kriegel, & Sander, 1999). This classification yielded nine cell types of excitatory neurons and objective anatomical parameters that distinguished between them. Because neuron reconstructions are registered, the vertical extent of the somata of each cell type allowed estimating cell type borders and overlap ratios in regions where somata of several cell types intermingle (Oberlaender et al., submitted for publication). 2.2. NeuroNet—reconstruction of 3D average anatomically realistic neural networks The 3D anatomical data represents the input to the second block of the pipeline. NeuroNet is a custom-designed tool that is integrated into Amira visualization software (Stalling, Westerhoff, & Hege, 2005) and allows anatomically realistic networks to be assembled (Fig. 1(b)). The input to NeuroNet comprises the following anatomical data: (i) the 3D neuron soma distribution, (ii) ) – 3 Fig. 2. Overview of NeuroNet. (a) Assembly of neural networks is based on 3D density distributions of all neuron somata within the network of interest (left panel), definition of cell types and cell type borders (center panel) and representative samples of complete 3D dendrite–axon morphologies of all respective cell types (right panel). (b) Somata are placed within each voxel of the soma density distribution (left panel). The definition of cell type-specific borders allows assigning a cell type to each soma (center panel). Finally, somata are replaced by dendrite–axon tracings of the respective cell type. (c) All dendrite morphologies within the network are transformed into 3D spine distributions (i). All individual spine distributions are summed to yield the 3D distribution of all cell types that are connected to a presynaptic type (here: neurons of type A and B are postsynaptic partners of type B) (ii). All axon morphologies of a presynaptic cell type are transformed into a 3D bouton density distribution. This yields the number and 3D distribution of available synapses from this cell type (iii). Synapses are not placed on dendrites by geometrical proximity to axons. In turn, for each individual postsynaptic neuron, synaptic connectivity is determined as an innervation probability ((iv), see example calculation). The resultant number of contacts per voxel is then randomly placed on dendritic branches within the respective voxel. Anatomical connectivity is thus not a fixed result of the network assembly process, but a statistical parameter (constraint by structural data) for simulations of network function. the number and location of neuronal cell types, (iii) the fraction of somata of each cell type in overlap regions, (iv) neuron soma distributions to correct for missing cell types, (v) representative 3D neuron morphologies for each cell type, (vi) cell type-specific spine and bouton densities and (vii) the definition of connections (i.e. pre- and postsynaptic partner cell types). 2.2.1. Cell type-specific 3D neuron soma locations The soma distribution is presented as a grid of 50 × 50 × 50 µm voxels, with density values given in somata per cubic millimeter (Fig. 2(a)). The 50 µm resolution resembles the accuracy of the standardized reference frame. Within each voxel, 4 S. Lang et al. / Neural Networks ( the respective density value is used to place neuron somata accordingly. According to the number and location of cell types within the network, each soma is assigned to a respective cell type (Fig. 2(b)). In areas where somata of several cell types intermingle, the supplied overlap ratios are satisfied. For the example presented here, only excitatory cells were included. The distribution of inhibitory interneurons was subtracted from the distribution of all neuron somata (Meyer et al., submitted for publication). 2.2.2. Cell type-specific 3D dendrite and axon networks In the next step, 3D neuron reconstructions are inserted at each computed soma location (Fig. 2(b)). These reconstructions must be of the same cell type as specified by the soma location. The morphologies are translated such that their soma centers coincide with the computed soma centers. Only morphologies with a vertical soma location not further than ±50 µm away from the computed soma location are eligible for placement. Thus, we avoid including cells that are located far away from their original location. Further, to ensure that the orientation within the standardized framework is preserved, reconstructions are rotated around the vertical axis (z-axis) such that the same side of a neuron faces the z-axis before and after the placement. The rotation step is of particular importance for cell types that display polar dendrite morphologies, such as L4ss neurons, which preferentially point toward the barrel column center (BCC, (Egger, Nevian, & Bruno, 2008)). Rotation ensures that such L4ss neurons will always point toward the BCC. Finally, to ensure that after placement no compartment of the cell extends beyond the vertical boundaries of the network (i.e. pia surface) or projects to areas it did not before, morphologies are scaled in the z-direction by the following value: scalez = Znew Zorig , where znew and zorig denote the vertical distance to the pia of the new and original soma location, respectively. Since znew and zorig are at maximum 50 µm apart, the maximal scaling factor is given by: scalez = Zorig,min + 50 Zorig,min =1+ 50 Zorig,min , where zorig,min denotes the shortest possible distance between the soma and the pia surface (e.g. 600 µm for L4ss (Oberlaender et al., submitted for publication) → scalez ,L4ss < 1.09). After all somata are replaced by dendrite and, if available, axon reconstructions, the networks comprise cell type-specific ensembles of dendrite and axon morphologies. Even though the sample size of morphologies per cell type may be limited, this approach ensures that at each location in the network, realistic morphologies with correct orientations are present. If axon morphologies originate from cells located outside the network (i.e., long-range projections, e.g. VPM), the number of neurons for these cell types can be specified separately. 2.2.3. Cell type-specific 3D structural overlap and anatomical connectivity Order of magnitude estimates of the number and subcellular distribution of synaptic contacts for individual neurons may be determined by structural overlap between its dendrites and a presynaptic axon distribution (Meyer, Wimmer, & Hemberger et al., 2010; Oberlaender et al., submitted for publication; Peters, 1979, see also discussion). Here, structural overlap is computed with 50 µm precision. Specifically, dendrite and axon morphologies are converted into 3D dendrite and axon density distributions with 50 µm voxel resolution, respectively (Fig. 2(c)). Because dendrites and axons extend usually further ) – than the soma distribution, the original grid is resized to cover all neuronal processes. Further, each dendrite and axon distribution is multiplied by a cell type-specific spine and bouton density, obtained by manually measuring inter-spine and interbouton distances along dendrites and axons in different parts of multiple neurons (e.g. L4ss: 0.5 spines per micron (Larkman & Mason, 1990), VPM: 0.33 boutons per micron (Oberlaender et al., submitted for publication)). Summing up these cell-specific distributions results in total spine and total bouton distributions for the entire network. The 3D distribution of potential putative synaptic contacts for each cell is then calculated as follows: ci,j (x, y, z ) = f (x, y, z ) ∗ si (x, y, z ) ∗ bj (x, y, z ) S (x, y, z ) , where ci,j is the synapse distribution of neuron i with presynaptic cell type j, S is the spine distribution of neuron j, Si is the total spine distribution of all neurons in the network and bj is the bouton distribution of presynaptic cell type j. Here, f denotes an optional term to correct for missing neuron populations (e.g., inhibitory interneurons). Synapses are then randomly placed on dendritic branches within a respective 50 µm voxel. 2.3. Simulation of signal flow in 3D anatomically realistic neural networks The following approach to investigating network dynamics and function is based on Monte Carlo (MC) simulations (Binder & Heermann, 1979; Metropolis, Rosenbluth, Rosenbluth, Teller, & Teller, 1953). Because of the slow convergence of MC simulations, the number of network realizations per in silico experiment typically needs to be in the order of 500. Thus, simulating the response of even a single postsynaptic (e.g. L4ss) and a single presynaptic (e.g. VPM) ensemble is time-consuming. High-performance computing (HPC) is therefore advantageous in performing MC simulations of neural networks that comprise thousands of full-compartmental single neuron models with millions of synaptic contacts. The third building block of our pipeline (Fig. 1(c)), the NeuroDUNE simulation environment (Lang, 2011), has been custom-designed to accommodate such computational needs, allowing efficient access (Bastian & Lang, 2004; Lang & Wittum, 2005) to state-of-the-art HPC capabilities. 2.3.1. The NeuroDUNE simulation framework NeuroDUNE is hierarchically structured into four modules: (i) NeuronGrid, (ii) Neuron, (iii) NeuralNetwork, and (iv) BrainUnit. The four modules are implemented using object-oriented template classes using the C++ programming language. The NeuronGrid module represents the topological and geometrical information of the reconstructed 3D neuron morphologies. An individual NeuronGrid is initiated for any neuron morphology in the network. Initial grid locations are chosen at branch points, end points and synapse locations, thus providing topologically complete representations of neuron morphology and connectivity. Grids consist solely of unbranched segments and are minimal in terms of unknowns. Each segment can be refined individually by adding uniform bisections. The Neuron module encapsulates the NeuronGrid module and provides physiological information necessary to establish a mathematical neuron model allowing for numerical simulations of signal processing upon the grid topology. At present, NeuroDUNE provides a choice of three different model types: (i) full-compartmental neuron models that are based on the cable equation with Hodgkin–Huxley type channel kinetics (FC Neuron), (ii) integrateand-fire neuron models (IF Neuron), and (iii) spike-generating neuron models (SP Neuron). SP Neuron refers to cells that only provide S. Lang et al. / Neural Networks ( input to the network. Here, such input is based on in vivo measurements of spike probability and timing in VPM (Brecht & Sakmann, 2002b; Bruno & Sakmann, 2006). Multiple Neuronmodules can be grouped into cell type-specific units, NeuralNetwork modules, which share anatomical and/or functional properties. Finally, BrainUnit modules are composed of multiple NeuralNetwork modules. 2.3.2. Numerical methods in NeuroDUNE Passive signal propagation in neurons can be approximated by the cable equation (Rall, 1969; Segev, Rinzel, & Shepherd, 1995), a second-order, linear partial differential equation, describing the development of the membrane potential in a dendritic compartment. Active excitability of the neurons’ membrane can be captured by additional reaction equations of voltage-dependent dynamics of ionic currents in individual channels. Further, synapses can be modeled by time-dependent equations (Destexhe, Mainen, & Sejnowski, 1994) adding additional point sources of current. The resultant nonlinear, coupled, reaction–diffusion systems are usually referred to as Hodgkin–Huxley type equations (Hodgkin & Huxley, 1952; Koch, 1999). Describing the dynamics of single FC neurons, these systems are numerically solved in NeuroDUNE by using a finite-volume scheme (FV) (Knabner & Angermann, 2003) that discretizes the diffusion component of the time-dependent cable equation. In contrast to other compartmental simulation environments (Bower & Beeman, 1998; Hines, 1984; Traub, Miles, & Wong, 1988), which use finite-difference schemes (FD) with fixed spacing of the unknowns, the FV scheme offers two primary advantages. First, impedance matching at branch points is achieved automatically, because the integral FV approach guarantees current conservation in all circumstances. Second, the FV scheme is capable of precisely resolving irregularly spaced synapses and thus guarantees secondorder convergence for anatomically realistic and non-uniformly placed putative synaptic contacts. The FD method, in turn, displays only first-order accuracy for irregularly spaced synapses. Consequently, the FV scheme leads to a significant decrease in simulation time while providing the same level of accuracy as the FD method. The time-dependent equation systems are further implicitly discretized by Crank–Nicholson or Backward–Euler schemes to avoid constraints in time step size due to the Courant–Friedrichs– Lewy (CFL) condition (Courant, Friedrichs, & Lewy, 1928). Thus, only a linear system has to be solved during each time-step, which can be achieved in linear time by using a specific ordering of the unknowns (Hines, 1984). During simulation, NeuroDUNE can output data per time-step. In addition to the raw data and activation statistics of synapses, contiguous values of membrane potential can be written for postprocessing and analysis purposes. We have chosen an established format (Visualization ToolKit VTK) to describe data associated with unstructured grids or networks. For more information on NeuroDUNE, validation of the FV scheme and comparison with NEURON, see www.neurodune.org. 2.3.3. Generation of full-compartmental single neuron models A morphologically realistic and biophysically detailed model is specified within NeuroDUNE using a single constructor that defines the modeling space N of FC Neuron modules by a 5-tuple N = (M , P , Ch , Co , S ), where M is the neurons’ morphology, P is the cell type-specific set of passive parameter functions, Ch are the cell type-specific channel models that define the nonlinear dynamics of voltage-dependent membrane properties, Co are the cell typespecific models of additional ionic concentrations (e.g. Ca2+ or Mg2+ ) and S denotes the cell-specific synapse locations that are provided as 3D probability distributions. ) – 5 Further, the presence of spines is taken into account. Therefore the relative spine surface per unit length is added to the dendritic surface per unit length, assuming a uniform distribution of spines along the dendritic compartment (Bush & Sejnowski, 1993; Holmes, 1989; Stuart & Spruston, 1998). The factor F to determine the modified length and diameter is calculated using the following formula: Ashaft + Aspine F = Ashaft where, Ashaft and Aspine are the surface area of the dendrite and spine, respectively. The constructor generates the Neuron module in six steps: (i) the neuron morphology is established by interpreting the geometrical and topological data, (ii) specific synapse locations are generated for an individual morphology in accordance with the given 3D probability distribution (i.e. determined by NeuroNet), (iii) a minimal computational grid is derived, (iv) the grid is enriched by adding further points to achieve a favorable approximation of the continuous equations, (v) the elliptic component of the equation system is discretized (at this stage the neuron is set up to simulate the temporal dynamics) and finally (vi) the reaction–diffusion system is implicitly solved for each time-step. 2.3.4. Simulation of networks of full-compartmental neuron models The reconstruction of average neural networks and the simulating of their electrical activity using a MC approach, introduce several stages of statistical variability. For instance, anatomical connectivity is given by a 3D probability distribution and determined at random within 50 µm voxels. Further, convergence and divergence ratios between pre- and postsynaptic populations (i.e. functional connectivity) introduce additional variability. Statistical parameters may affect the simulation results to different degrees. Thus, to determine the contribution of each parameter to the variability of MC simulation results, we make a sensitivity analysis for each statistically independent parameter (i.e. only one parameter is changed over a series of 500 simulations, while the other parameters are kept at fixed values). Here, we independently varied four network parameters (i) anatomical connectivity, (ii) functional connectivity, (iii) synchrony of input and (iv) synaptic efficacy. 3. Results 3.1. Thalamocortical networks between VPM and L4ss neurons in rat vibrissal cortex Briefly, sensory information after passive deflection of a single whisker (passive touch) is mediate by the whisker-specific lemniscal pathway ((Lubke & Feldmeyer, 2007), Fig. 3(a)). Specifically, segregated structures in VPM, referred to as barreloids (Land, Buffer, & Yaskosky, 1995), relay touch information (Yu, Derdikman, Haidarliu, & Ahissar, 2006) to potentially all neurons located within the respective barrel column in the vibrissal cortex ((Brecht, Roth, & Sakmann, 2003; Brecht & Sakmann, 2002a; Manns, Sakmann, & Brecht, 2004), Fig. 3(b)–(d)). Innervation by VPM axons is most dense within L4, delineating the horizontal and vertical extents of the barrels ((Wimmer, Bruno, de Kock, Kuner, & Sakmann, 2010), Fig. 3(e)). Further, paired recordings in VPM and L4 in vivo (Bruno & Sakmann, 2006) yielded estimates on functional connectivity and synaptic strength for this pathway. In addition, whisker-evoked spiking after passive touch has been characterized in VPM (Brecht & Sakmann, 2002b) and L4ss (de Kock, Bruno, Spors, & Sakmann, 2007) previously. Here, we use this functional data to constrain the average model of the VPM-to-L4ss pathway (Fig. 3(f)) and to activate it with realistic input, measured in vivo. 6 S. Lang et al. / Neural Networks ( ) – Fig. 3. Overview of the whisker system in rats. (a) The whisker system in rats is somatotopically organized. Whisker-specific signals (such as touch) from the follicle (1) are conveyed by anatomically segregated pathways via the brainstem (2) to the thalamus (i.e. VPM, 3) and then to functional modules, referred to as cortical barrel columns (4), in rat vibrissal cortex. (b) The spatial organization of the barrel columns resembles the layout of the whisker pad on the animal’s snout and is determined by the VPM innervation pattern (image modified from (Wimmer et al., 2010)). (c) Input to a cortical barrel column is provided by excitatory neurons located in anatomically segregated, whisker-specific structures, called barreloids, in VPM. VPM neurons potentially target all (excitatory) neurons in the cortical column. (d) In the present study, we illustrate the pipeline of reconstructing average neural networks and simulating signal flow by activating an ensemble of L4ss neurons by VPM input. (e) 3D axon (blue) and dendrite (red) morphologies of individual neurons were reconstructed and assembled to an anatomically realistic network of 2770 L4ss neurons that were interconnected to 285 VPM cells (Oberlaender et al., submitted for publication). (f) In NeuroDUNE this network is converted into a numerical model, where L4ss neurons are converted into full-compartmental models that receive input from VPM neuron models. Network connectivity (NC) is determined by NeuroNet and measurements of functional connectivity (i.e. convergence/divergence ratios (Bruno & Sakmann, 2006)). 3.1.1. Anatomical connectivity between VPM and L4ss neurons in rat vibrissal cortex The ensemble of L4ss neurons within an average barrel comprises 2752 ± 46 dendrite morphologies. This ensemble is innervated by 285 ± 13 thalamocortical axons from the respective VPM barreloid, which results in a total number of 697,065 VPM synapses and an average number of 246 ± 123 VPM synapses per L4ss neuron. These numbers were obtained by using NeuroNet (Oberlaender et al., submitted for publication) and resemble results from a variety of previous studies that investigated connectivity between thalamus and L4ss neurons in the cortex (Bruno & Sakmann, 2006; da Costa & Martin, 2011; Meyer, Wimmer, & Hemberger et al., 2010). Fig. 4(a)–(c) illustrates the procedure of determining numbers and subcellular distributions of synapses, using NeuroNet, exemplified by a single L4ss neuron. As schematically illustrated in Fig. 2, the dendrite morphology is multiplied by a cell type-specific spine density value and then converted into a 3D spine density distribution (Fig. 4(a)). This distribution is multiplied by the total distribution of VPM boutons and then divided by the total distribution of available spines (i.e. from all other L4ss neurons and neurons of other cell types connected to VPM, Fig. 4(b)). The resultant 3D synapse distribution yields first-order estimates of (i) the total number of potential VPM contacts for this neuron (i.e. sum over all voxels, 399 for this example neuron), and (ii) the subcellular innervation pattern with 50 µm resolution (Fig. 4(c)). The 3D synapse distribution is then used to determine specific locations of VPM synapses by randomly placing the number of available contacts on dendritic branches located within a respective voxel (Fig. 4(d)). The average path length distance between VPM synapses and the soma is 112.9 ± 98.3 µm (for 17 reconstructed L4ss neurons, Fig. 4(e)), which is in good agreement with previous reports of subcellular innervation by thalamocortical synapses on L4ss neurons (da Costa & Martin, 2011). In addition to reproducing average numbers and subcellular organization of thalamocortical synapses, applying NeuroNet to the available data provides evidence that the number of synapses per L4ss neuron depends strongly on the location of the neuron’s soma within the barrel. Neurons with somata located within a 50 µm radius around the BCC have on average 347 ± 107 synapses, while neurons located at the barrel column borders receive only 155 ± 78 contacts. Further, L4ss neurons located close to L3 and L5 receive, on average, less contacts than neurons in the barrel center, 198 ± 64, 191 ± 153 and 260 ± 103 respectively (Oberlaender et al., submitted for publication). 3.1.2. Structure–function relationships in the VPM-to-L4ss network at single neuron level NeuroNet yields an average network of 3D dendrite morphologies of L4ss neurons and the distribution of thalamocortical VPM synapses for each registered cell. NeuroDUNE transforms this anatomical data into a network of full-compartmental neuron models that allows investigating structure–function relationships at single neuron level by simulating the activity evoked by individual or subsets of VPM synapses (Fig. 5). Due to the relatively simple and stereotypic morphology of L4ss dendrites, we chose a passive dendrite model for the L4ss neurons. The Neuron constructor in NeuroDUNE thus comprises only a 3-tuple (M , P , S). The constructor transforms the neuron’s dendrite morphology (M) and VPM synapse distribution (S) into a grid that minimally resolves the respective dendrite topology and synapse locations. Passive membrane parameters (P) were chosen in agreement with previously reported experimental data for L4ss neurons in rat vibrissal cortex (Bannister, Nelson & Jack, 2002; Binshtok, Fleidervish, Sprengel, & Gutnick, 2006; Feldmeyer, Lubke, Silver, & Sakmann, 2002; McCormick, Connors, Lighthall, & Prince, 1985; Staiger et al., 2004) : CM = 0.75 µF/cm2 , RM = 12 k cm2 , RA = 150 cm. The resting potential was chosen as Vrest = −70 mV. The time course of VPM synapses was modeled by a double exponential function (Destexhe et al., 1994; Zador, Koch, & Brown, 1990) with rise and decay times τ1 = 0.36 ms and τ2 = 1.12 ms, respectively. Maximal synaptic conductance gmax was chosen as 0.52 nS, and a reversal potential Es = 0 mV was used. S. Lang et al. / Neural Networks ( ) – 7 Fig. 4. Thalamocortical innervation of L4ss neurons determined by NeuroNet. (a) The dendrite morphology of one individual L4ss neurons is transformed into a 3D spine distribution (left panel). (b) Superimposing complete reconstructions of VPM axons and extrapolating the number of axons to 285 neurons results in the 3D distribution of VPM boutons that originate from a single barreloid (top panel, (Oberlaender et al., submitted for publication)). Summing the spine distributions of all excitatory neurons in a barrel column results in the 3D distribution of all spines within the column (lower panel, (Oberlaender et al., submitted for publication)). (c) The three types of distributions are combined to determine the 3D VPM synapse distribution for every postsynaptic neuron. (d) Locations of synapses on dendrites are determined randomly but meet the constraints of the 3D synapse distribution at 50 µm resolution. Here, we show one possible realization illustrating how 399 VPM synapses may be distributed across the dendrites. In every simulation trial these locations change. (e) Calculating the average path length distance (in 25 µm bins) between the VPM synapses and the soma results in a Gaussian distribution for L4ss neurons (n = 17). The mean path length distance is 112.9 µm, which is in good agreement with a previous study that combined confocal imaging and serial electron microscopy (da Costa & Martin, 2011). Fig. 5(a) shows the distribution of unitary excitatory postsynaptic potentials (uEPSPs) evoked by each of the 399 synaptic contacts, respectively, for the example neuron shown Fig. 4. In this example, the uEPSP amplitudes and peak times vary significantly between 0.29–1.62 mV and ∼3–9 ms, respectively (Fig. 5(b)–(c)). More importantly, the uEPSP amplitudes and peak times display bimodal distributions, which are observed for the majority of L4ss neurons. The two parameters are coupled: longer peak times correspond to lower peak amplitudes, reflecting dendritic attenuation and thus the path length distance between the respective synapse and the soma. Specifically, the longer the path length distances, the lower the somatic uEPSP amplitudes and the longer the times to the peak. The bimodality of both distributions suggests that VPM synapses on L4ss neurons may be pooled into ‘proximal’ and ‘distal’ contacts. Subsequently, synapses that evoke peak amplitudes lower than 1 mV are regarded as distal, the remaining as proximal contacts (for definition see Section 3.3). In addition to model subthreshold responses evoked by individual synapses, we simultaneously activate subsets of VPM synapses to investigate spiking responses (Fig. 5(d)). To model generation of an action potential spike (AP) in L4ss neurons an additional active spike initiation segment was implemented (Mainen, Joerges, Huguenard, & Sejnowski, 1995). This initial segment (IS) comprises a fast inactivating Na+ channel and an increased leakage conductance (Farinas & DeFelipe, 1991; Mainen et al., 1995). The spike shape was tuned to match previously reported experimental data of spike duration and amplitude (Staiger et al., 2004). A single ex- 8 S. Lang et al. / Neural Networks ( ) – Fig. 5. Structure–function relationships at single neuron level. The L4ss neuron shown in Fig. 4 was converted into a full-compartmental model in NeuroDUNE. (a) Activating each of the 399 determined VPM synapses individually, allows measuring the respective uEPSP amplitudes at the soma (plots for 10 exemplary contacts are shown). The maximal synaptic conductance is kept at a fixed value for all contacts. The peak amplitude and the time of the peak are co-dependent in that higher peaks relate to shorter peak times, which reflect the dendrite path length distance between the contact and the soma, i.e. due to dendritic attenuation more distal synapses result in lower peaks at later points in time than proximal contacts. (b)–(c) For this example L4ss neuron, the histograms of the uEPSP amplitudes and peak times display bimodal distributions. The shape of these histograms depends on dendrite morphology and soma location of the respective neuron and may thus be regarded as a structure–function characteristic for each neuron. VPM synapses that evoke uEPSP amplitudes smaller than 1 mV are here referred to as proximal (for definition see Fig. 8). (d) An example distribution of active synaptic contacts is shown, where 195 (green) of the 399 (blue) synapses were activated simultaneously. (e) The functional response to this synaptic input pattern can be investigated in 3D at any point in time. NeuroDUNE allows relating the functional output of any single neuron to the 3D spatial and temporal pattern of synaptic inputs. (f) The parameters of the active initial segments of the L4ss neuron model were tuned to result in spike shapes that resemble previously measured ones for this cell type (Staiger et al., 2004). ample spike, that was initiated by the synchronous activation of 195 (of 399) synapses at t = 1.0 ms, is shown in Fig. 5(e)–(f). 3.2. Spiking responses of L4ss neurons are determined by functional connectivity and synchrony of the VPM input Prior to investigating the subthreshold and spiking responses of L4ss neurons at the level of cell ensembles, we performed a sensitivity analysis of four statistically independent parameters (507 simulation trials per parameter) that may strongly influence the results of the MC simulation approach. Fig. 6(a) shows the spiking response of an ensemble of 2770 L4ss neurons, when only the locations of VPM contacts change between simulation trials (synaptic efficacy was set to 1.0). The standard deviation (SD) of the ensemble’s spiking responses due to changes in anatomical connectivity is around 5% of the mean response. Fig. 6(b) shows the spiking response, when anatomical connectivity remains unchanged, but functional connectivity is altered between simulation trials. Active synapses are established using convergence and divergence ratios of 0.43 reported previously for VPM-to-L4ss connections (Bruno & Sakmann, 2006). Thus, input from 43% of the VPM neurons (i.e. 123 of 285) converge on a single L4ss neuron and 43% of the L4ss neurons (i.e. 1191 of 2770) receive input from the same VPM neuron. The pre- and postsynaptic partners are determined randomly for each simulation trial. The SD of the ensemble’s spiking responses due to changes in functional connectivity is around 13%. Fig. 6(c) shows the spiking responses, when anatomical and functional connectivity remain fixed, but the timing of the VPM input is changed between simulation trials. The spiking response of VPM neurons after passive touch has been described previously (Brecht & Sakmann, 2002b). The timing (i.e. onset latency of VPM spikes) of this realistic thalamocortical input is not perfectly synchronous. Specifically, 65% of the VPM neurons (i.e. 185 of 285) generate a single spike after passive touch. Onset latencies of VPM spikes follow a Gaussian distribution (mean: t = 9.97 ± 1.71 ms; passive touch at t = 0 ms). The resultant SD of the ensemble’s spiking responses due to changes in VPM synchrony is around 8%. Finally, the synaptic efficacy is varied between 0.7 and 1.0 in steps of 0.1, while keeping the VPM synchrony, as well as anatomical and functional connectivity fixed. Fig. 6(d) shows that the SD introduced by randomly selecting synapses that are able to generate an uEPSP is small compared to the SDs introduced by varying the other three parameters. However, lower efficacy values systematically decrease the overall spiking response of the L4ss ensemble, while slightly increasing the variability. To evaluate the contribution of each parameter to the variability in L4ss spiking response, we simultaneously varied anatomical and functional connectivity, as well as VPM synchrony, while keeping synaptic efficacy at 1.0 (Fig. 6(e)). The SD of the ensemble responses due to varying all parameters is around 13%. Interestingly, superimposing the distributions obtained by varying only functional connectivity (Fig. 6(b)) and VPM synchrony (Fig. 6(c)) results approximately in the distribution obtained when all parameters were changed simultaneously (Fig. 6(e)). To quantify these observations, we relate the spiking probability of an individual L4ss neuron to the average path length distance of its active synapses (i.e., synapses that generate an uEPSP after VPM input, Fig. 6(f)). The relationship between these two quantities follows a Gaussian distribution. The mean path length distance S. Lang et al. / Neural Networks ( ) – 9 Fig. 6. Sensitivity analysis of statistical parameters in Monte Carlo simulations. The present approach of assembling and simulating an average network between VPM and L4ss neurons introduces four statistical parameters: (i) anatomical connectivity (i.e. synapse locations), (ii) functional connectivity (i.e. locations of active synapse, constrained by convergence/divergence ratios), (iii) timing of the input (i.e. synchrony) and (iv) synaptic efficacy (i.e. percentage of synapses that can elicit an uEPSP). To investigate the contribution of each parameter to the variability of MC simulations we performed a sensitivity analysis by varying one parameter across 507 simulation trails, while keeping the remaining three at fixed values. (a) Varying anatomical connectivity results in ∼5% (i.e. SD/mean) variability in the network response (i.e. number of spikes). (b) Varying functional connectivity results in ∼13% variability in the network response. (c) Varying the VPM input results in ∼8% variability in the network response. (d) Varying synaptic efficacy from 1 to 0.7 (right to left) results in a decrease of the overall network response, while slightly increasing its variability. (e) Varying all three parameters simultaneously results in ∼15% variability in the network response, which can be account for by the variability in functional connectivity and VPM synchrony (dashed plot). (f) The relationship between the average path length distance (in 50 µm bins) of active VPM synapses and the spiking probability of the respective L4ss neuron follows a Gaussian distribution. This illustrates that spikes may occur if a few proximal or many distal VPM synapses are active, but spikes will most likely occur if a mix of proximal and distal synapses are active (i.e. mean path length distance of active synapses equals mean path length distance of all synapses). This illustrates the strong contribution of functional connectivity to the variability in network response. (g) The response of the L4ss network decreases with increased SD in the timing of the VPM input (i.e. less synchronous). This significant correlation (p < 0.0001) illustrates the strong contribution of VPM synchrony to the variability in the network response. (h) The network response decreases within decreasing efficacy values in a slightly supra-linear. This may reflect a stronger reduction of proximal VPM synapses toward the barrel borders, when compared to distal contacts (see Fig. 8). of active synapses (113.4 ± 57.3 µm) is similar to the average path length distance determined for the anatomical connectivity (i.e. 112.9 ± 98.3). For the current model, this analysis shows that L4ss neurons may spike when relatively small numbers of proximal or a larger number of distal synapses are active. This result explains the large contribution of functional connectivity to the variability in spiking of the L4ss ensemble. Spiking is most likely to occur when a mix of proximal and distal synaptic contacts is activated and the average path length distance of active synapses is equal to the average path length distance of all synapses. Further, we relate the number of spikes per L4ss ensemble to the SD in onset latency of the VPM input (Fig. 6(g)). We find that spike rate and VPM synchrony are significantly correlated (Pearson’s correlation coefficient: −0.38, p < 0.0001). Larger SD in onset latency of the VPM input results in lower numbers of spikes. This explains the large contribution of VPM synchrony to the variability in spiking of the L4ss ensemble. Finally, we relate the number of spikes in the ensemble to synaptic efficacy values between 0.5 and 1.0 (Fig. 6(h)). Surprisingly, this relationship is supra-linear. This finding may reflect location-specific differences in the numbers of proximal and distal VPM contacts, which may introduce nonlinearity to the network response (see discussion). In summary, the sensitivity analysis suggests that a lower efficacy value of VPM synapses may decrease the overall spiking probability of L4ss neurons in a column in a slightly supra-linear manner. Further, spiking output of an ensemble of L4ss neurons is primarily determined by functional connectivity and synchrony of the VPM input. In turn, the contribution of varying synapse locations within 50 µm voxels to spiking output is small. This may suggest that the approach presented here of determining thalamocortical connectivity by structural overlap within 50 µm voxels is sufficiently accurate to describe thalamocortical wiring between VPM and the relatively simple dendrite morphologies of L4ss neurons. 3.3. Spiking responses of L4ss neurons are location-specific Assembling the 3D structure of an average L4ss network and estimating synaptic wiring by structural overlap with VPM axons in NeuroNet predicts that the number of thalamocortical synapses per L4ss neuron decreases with increasing distance between the soma and the BCC. We use a MC simulation approach to investigate potential consequences of location-specific anatomical connectivity for functional responses of L4ss neurons. Fig. 7(a) shows a top view of all L4ss neuron somata located within an average barrel column. In this example, spiking neurons (green) are preferentially located around the BCC. Similar spatial spike distributions are observed in all simulation trials (synaptic efficacy: 0.8, the other three parameters change between simulation trials). Fig. 7(b) quantifies this observation. While subthreshold responses display only a small decrease in EPSP amplitudes toward the horizontal column borders (19% decrease between BCC and borders), the number of spiking neurons shows a strong radial decay (60% 10 S. Lang et al. / Neural Networks ( ) – Fig. 7. 3D structure–function relationships at the network level. NeuroNet predicts that the number of VPM synapses per L4ss neurons decreases toward the column borders and toward L3 and L5. Simulating the response of the L4ss network to realistic VPM input, measured in vivo, allows investigating how location-specific connectivity may translate into location-specific functional responses at the subthreshold and spiking levels. (a) Top view onto the distribution of L4ss somata in a barrel column. Spiking neurons are shown in green and are preferentially located around the barrel column center (BCC). Similar spatial patterns are observed in all simulation trials. (b) Quantification of the radial dependence of the subthreshold (i.e. EPSP amplitude at the soma) and spiking responses for 5 simulation trials. While the subthreshold response displays only a weak radial decay (∼19%) toward the barrel borders, spiking responses decay dramatically (∼60%). (c) The radial decay in spiking follows the radial decay of (active) VPM synapses (light gray: all synapses, dark gray: active synapses). The decay in spiking at the barrel borders even exceeds the decay of active synapses. This may reflect a stronger reduction of proximal VPM synapses toward the barrel borders, when compared to distal contacts (see Fig. 8). (d) Side view of the distribution of L4ss somata shown in (a). (e)–(f) Similar relationships between the number of active synapses and the subthreshold and spiking responses are observed along the vertical column axis. Subthreshold responses remain largely unchanged, while spiking follows the distribution of active synapses. decrease between BCC and borders). When compared to the number of (active) VPM synapses per L4ss neuron (Fig. 7(c)), the decrease in spiking probability follows the radial decay of VPM synapses toward the column borders (49% decrease between BCC and borders). Similar relationships between location-specific anatomical connectivity and subthreshold and spiking responses are also observed along the vertical column axis Fig. 7(d)–(f). The number of VPM synapse decreases toward L3 and L5, resulting in lower spike rates in these regions, while subthreshold responses remain largely unchanged. Surprisingly, spiking probability at the barrel borders is even lower than expected by the radial decrease of VPM synapses. We thus investigated whether the subcellular organization of VPM synapses also changes as a function of soma location. The average path length distance of active VPM synapses does not change toward the barrel borders. Additionally, we investigate the uEPSP histograms as a function of soma location. This may yield a better criterion to compare functional implications of different subcellular synaptic innervation patterns. Fig. 8(a) shows the uEPSP histogram averaged across all L4ss neurons with somata S. Lang et al. / Neural Networks ( ) – 11 Fig. 8. Location-specific subcellular organization of VPM synapses. (a) We determined the uEPSP histograms for all L4ss neurons with somata located within 50 µm from the BCC. Each histogram was normalized to its respective peak (i.e. normalized frequency). The average histogram displays a bimodal distribution, as was illustrated for one example L4ss neuron in Fig. 5(b). (b) Fitting two Gaussian distributions allows approximating the fraction of ‘proximal’ and ‘distal’ VPM contacts. We defined an uEPSP amplitude of 1 mV to discriminate between the two groups (i.e. approximately mean + SD of the distal fits). (c)–(d) We obtained the uEPSP histograms and fits for L4ss neurons with somata located 150–200 µm away from the BCC. While the peak of the distal fit remains unchanged, the proximal fit decreases. (e)–(f) We determined the histograms and fits for somata with 0–50, 50–100, 100–150 and 150–200 µm distance to the BCC and determined the fraction of distal and proximal synapses (i.e. summing up the blue and green bins in (a) and (c), respectively). While the relative fraction of distal VPM synapses decreases by 5% toward the barrel borders, the fraction of proximal synapses decreases by 14%. These differences in the subcellular organization of VPM synapses on L4ss dendrites may introduce nonlinearity to the L4ss network responses. located within 50 µm from the BCC. As illustrated for one example neuron in Fig. 5, most L4ss neurons display bimodal uEPSP distributions, suggesting that VPM synapses may be pooled into proximal and distal contacts. We fitted two Gaussian distributions onto the average uEPSP histogram to approximate the relative fraction of the two groups. We defined an uEPSP amplitude of 1 mV to distinguish between proximal and distal contacts (i.e. approximately mean + 1SD of distal fits). We further determined the uEPSP histograms and Gaussian fits for L4ss neurons with somata located between 50–100 µm, 100–150 µm and 150–200 µm (Fig. 8(c)–(d)), respectively. Surprisingly, we found that the peak of proximal synapses displays a stronger decrease toward the lateral barrel borders than the peak of distal synapses. Further, the relative number of distal synapses (fraction of VPM synapses per neuron with uEPSP amplitudes <1 mV) decreases less (5%, Fig. 8(e)) when compared to the relative number of proximal synaptic contacts (14%, Fig. 8(f)). Consequently, the nonlinear relationship between the radial decrease in spiking probability and VPM synapses may be caused by the difference in numbers of proximal and distal VPM synapses. 4. Discussion 4.1. Average neural networks and anatomical connectivity The approach presented here to reconstructing average neuronal networks relies on two critical assumptions. First we assemble the average 3D network structure of a barrel column from a relatively small set of morphological reconstructions and secondly, synaptic wiring is determined by structural overlap between axons and dendrites. We argue that extrapolation to neuronal ensembles yields valid order of magnitude estimates of the 3D structure of neural networks if four prerequisites are met. (i) The number and 3D distribution of all neuron somata within the network of interest need to be known. (ii) Representative samples of complete 3D dendrite morphologies of all cell types within the network of interest need to be reconstructed. The reconstructions allow estimating the number of neurons per cell type, by determining cell type-specific borders and overlap ratios in areas where somata of several cell types intermingle. (iii) Representative samples of complete 3D axon morphologies need to be reconstructed. (iv) A 12 S. Lang et al. / Neural Networks ( standardized 3D reference frame has to be defined that allows combining anatomical data from different animals. Once these prerequisites are satisfied, NeuroNet is able to combine this data and generate an average 3D neural network. Such network models (Oberlaender et al., submitted for publication) may be regarded as anatomically realistic, because the number of neurons per cell type as well as the 3D structure of the cell type-specific dendritic and axonal networks resembles anatomical data from a variety of studies at subcellular (Petreanu, Mao, Sternson, & Svoboda, 2009), cellular (Bruno & Sakmann, 2006; da Costa & Martin, 2011) and network levels (Meyer, Wimmer, & Hemberger et al., 2010). Given that the 3D structure and distribution of somata, dendrites and axons within the network model resembles the average structure of the real anatomical network, estimating synaptic wiring by structural overlap remains to be discussed. The validity of predicting synaptic connectivity by axon–dendrite overlap (commonly referred to as Peter’s rule (White, 1979)) is arguably a matter of scale. It has been demonstrated that, in general, proximity of axons and dendrites does not predict synaptic connectivity at the (sub-) micrometer scale (e.g. da Costa & Martin, 2011; Mishchenko et al., 2010). Further, identification of higher-order connectivity patterns, or clustering of synaptic inputs on specific neurons or dendrite compartments may only be observed by circuit reconstructions at the electron microscopy level (Bock et al., 2011; Briggman et al., 2011). Consequently, the present approach of estimating synaptic locations is not based on geometrical proximity (i.e. touch) of axons and dendrites, which was suggested previously (i.e. The Blue Brain Project, (Kozloski et al., 2008)). Instead, overlap-based approaches at larger scales (e.g. 50 µm) yielded valid order of magnitude estimates of synaptic innervation (e.g. Binzegger, Douglas, & Martin, 2004; Lubke, Roth, Feldmeyer, & Sakmann, 2003; Meyer, Wimmer, & Hemberger et al., 2010; Meyer, Wimmer, & Oberlaender et al., 2010; Oberlaender et al., submitted for publication). Thus, we derive anatomical connectivity by (i) determining realistic numbers of presynaptic boutons and postsynaptic spines, (ii) estimating innervation probabilities with 50 µm precision, (iii) randomly placing synapses onto dendrites within 50 µm voxels and (iv) changing the synapses’ locations during MC simulations. Specifically, we determine the number of neurons per presynaptic cell type and reconstruct their 3D bouton (i.e. axon) distributions. Since boutons in the cortex are associated with synaptic contacts (De Paola et al., 2006), a 3D bouton distribution can be regarded as an order of magnitude estimate of the number and 3D distribution of synapses that originate from the respective cell type. To determine the postsynaptic targets of these boutons, we reconstruct complete 3D dendrite morphologies, measure their spine densities and determine the number of neurons of all postsynaptic cell types. Synaptic wiring is then estimated by applying two statistical steps. First, we assume that all spines in a voxel have equal probability of being the postsynaptic target of the respective boutons. For example, 100 neurons may contribute 1 spine to a voxel that may contain 50 boutons. Consequently, each of the 100 neurons receives 0.5 synaptic contacts in this voxel. In a second step, the synapses are randomly placed on dendritic branches within the respective voxel. In this example, 1 synapse will be placed at a random dendritic location within the voxel in 50% of the network realizations and no synapse will be placed in the remaining ones. Thus, the number of synaptic contacts and their locations along the dendrites, as well as the active preand postsynaptic partner neurons are not fixed results of the network assembly process (generated by NeuroNet), but represent simulation parameters that are constraint by realistic cell typespecific numbers and distributions of neuron somata, boutons and spines. ) – In conclusion, our approach takes into account realistic, quantitative structural bounds of synaptic wiring at 50 µm voxel resolution. This allows investigating structure–function relationships in large neural networks at the single cell level, by changing the subcellular organization of anatomical and functional connectivity (e.g. clustering of synaptic inputs) during MC simulations. 4.2. Monte Carlo simulations of structure–function relationships To illustrate how MC simulations of average anatomically realistic neural networks may help to gain insights into structure–function relationships at the network level, we generated an average network of an ensemble of L4ss neurons in a barrel column in rat vibrissal cortex and connected this ensemble to a population of thalamic neurons located in a barreloid in VPM. Activating the L4ss network with realistic VPM input, measured after passive touch in vivo, should allow investigating the influence of structural parameters, such as numbers and locations of VPM synapses, to subthreshold and spiking responses. These may then be compared to in vivo measurements of neuronal activity in a barrel column. Our simulation results suggest potential anatomical mechanisms that may constrain the functional responses of L4ss neurons evoked by VPM input. Reconstructing the network structure and synaptic wiring predicted that the number of VPM synapses may strongly depend on the soma location of the L4ss neurons. Specifically, the number of VPM synapses decreased with increasing distance from the BCC toward the barrel borders. Activating the ensemble of L4ss neurons by VPM input, results in similar locationspecific spiking responses. Spiking probability at the barrel borders was ∼60% lower than in the BCC. A similar radial decay in spiking probability has been observed for L2/3 neurons in a barrel column of mouse vibrissal cortex using 2-photon Ca2+ imaging (Kerr et al., 2007). There, spiking at the barrel borders was ∼52% lower than in the BCC. L2/3 neurons represent the major target population of L4ss neurons (Feldmeyer et al., 2002; Sarid, Bruno, Sakmann, Segev, & Feldmeyer, 2007). Thus, our simulations suggest that the radial decay in spiking probability of L2/3 neurons may be a direct consequence of the 3D structure of the VPM-to-L4ss network. Further, the VPM-to-L4ss pathway is regarded as one major starting point of cortical processing (Lubke & Feldmeyer, 2007). The radial dependence in spiking probability may therefore be a general feature that underlies the whisker-evoked flow of excitation in a barrel column. In addition to this structure–function relationship at the cellular level, our simulation results predict a second mechanism at the subcellular scale. The relative numbers of proximal and distal VPM synapses per L4ss neuron may also be location-specific. Toward the barrel borders, the number of proximal contacts may decrease more than the number of distal contacts. Consequently, spiking probability at the barrel borders may even be lower than predicted by the radial decay in VPM synapses per cell. This subcellular difference in VPM innervation may also result in several functional nonlinearities. For example, it may account for the observed supralinear decay of spiking with decreasing values of synaptic efficacy. Further, it may also result in a decrease in correlation between pairs of L4ss neurons toward the barrel borders. This de-correlation was previously described for the population of L2/3 neurons (Kerr et al., 2007). The structure–function relationships described here may in part explain previous observations in spiking probability and correlation. However, interpretation of the simulation results has several caveats. The pipeline described here is meant to illustrate the new methods and to give an example how our approach may help to investigate structure–function relationships at subcellular, cellular and network scales. Our simulation of thalamocortical activation of L4ss neurons after passive touch is clearly a S. Lang et al. / Neural Networks ( simplification of the situation in vivo. Intracortical excitation (Feldmeyer, Egger, Lubke, & Sakmann, 1999; Feldmeyer et al., 2002; Feldmeyer, Roth, & Sakmann, 2005), feed-forward inhibition (Sun, Huguenard, & Prince, 2006) or active conductance models (Izhikevich & Edelman, 2008) are examples of aspects of the L4 network neglected here. The number of evoked spikes (de Kock et al., 2007) and the dynamics of the network response are thus beyond scope of the present simulation. However, the pipeline of tools is modular and the VPM-to-L4ss model will be extended by incorporating, for example, intracortical connections (using 3D reconstructions of cortical axon morphologies (e.g. Oberlaender et al., 2011)), inhibition and active dendrite models. Despite the simplified nature of the VPM-to-L4ss network, our simulations illustrate that the structure of neural circuits may not only constrain their function, but that fundamental features in neuronal information processing may emerge from the 3D cellular and subcellular structure of the network itself. The pipeline of tools opens one new way to investigate these potential structure–function relationships. Author contributions M.O. and B.S. conceived and designed the project. M.O. provided all anatomical data and built the anatomical model. M.O. and V.J.D. designed NeuroNet. V.J.D. implemented NeuroNet. S.L. designed and implemented NeuroDUNE and performed the simulations. M.O. and S.L. designed and performed the analysis and all authors were involved in writing the paper. Acknowledgments This work was supported by the Max Planck Society and BMBF under the grant 01GQ0791 (NeuroDUNE). We thank Christiaan P.J. de Kock and Randy M. Bruno for providing data to reconstruct neuron morphologies, Hanno-Sebastian Meyer for data to determine neuron soma distributions, Moritz Helmstaedter for his work on registration of neuron morphologies and Peter Bastian for his ongoing support. Special thanks to the FSU, Scripps Florida and Miami University for access to their high-performance computing facilities and to the American Journal Experts for editing the manuscript. References Ankerst, M., Breunig, M., Kriegel, H.P., & Sander, J. (1999). OPTICS: Ordering points to identify the clustering structure. In ACM SIGMOD’99 int. conf. on management of data. Philadelphia. Arenkiel, B. R., & Ehlers, M. D. (2009). Molecular genetics and imaging technologies for circuit-based neuroanatomy. Nature, 461(7266), 900–907. Bannister, N. J., Nelson, J. C., & Jack, J. J. (2002). Excitatory inputs to spiny cells in layers 4 and 6 of cat striate cortex. Philosophical Transactions of the Royal Society B: Biological Sciences, 357(1428), 1793–1808. Bastian, P., & Lang, S. (2004). Couplex benchmark computations obtained with the software toolbox UG. Computational Geosciences, 8(2), 125–147. Binder, K., & Heermann, D. W. (1979). Monte Carlo simulation in statistical physics: an introduction. Berlin: Springer. Binshtok, A. M., Fleidervish, I. A., Sprengel, R., & Gutnick, M. J. (2006). NMDA receptors in layer 4 spiny stellate cells of the mouse barrel cortex contain the NR2C subunit. Journal of Neuroscience, 26(2), 708–715. Binzegger, T., Douglas, R. J., & Martin, K. A. (2004). A quantitative map of the circuit of cat primary visual cortex. Journal of Neuroscience, 24(39), 8441–8453. Bock, D. D., Lee, W. C., Kerlin, A. M., Andermann, M. L., Hood, G., Wetzel, A. W., et al. (2011). Network anatomy and in vivo physiology of visual cortical neurons. Nature, 471(7337), 177–182. Bower, J. M., & Beeman, D. (1998). The book of GENESIS: exploring realistic neural models with the general neural simulation system. New York: Springer-Verlag. Brecht, M., Roth, A., & Sakmann, B. (2003). Dynamic receptive fields of reconstructed pyramidal cells in layers 3 and 2 of rat somatosensory barrel cortex. Journal of Physiology, 553(Pt 1), 243–265. Brecht, M., & Sakmann, B. (2002a). Dynamic representation of whisker deflection by synaptic potentials in spiny stellate and pyramidal cells in the barrels and septa of layer 4 rat somatosensory cortex. Journal of Physiology, 543(Pt 1), 49–70. Brecht, M., & Sakmann, B. (2002b). Whisker maps of neuronal subclasses of the rat ventral posterior medial thalamus, identified by whole-cell voltage recording and morphological reconstruction. Journal of Physiology, 538(Pt 2), 495–515. ) – 13 Briggman, K. L., & Denk, W. (2006). Towards neural circuit reconstruction with volume electron microscopy techniques. Current Opinion in Neurobiology, 16(5), 562–570. Briggman, K. L., Helmstaedter, M., & Denk, W. (2011). Wiring specificity in the direction-selectivity circuit of the retina. Nature, 471(7337), 183–188. Bruno, R. M., & Sakmann, B. (2006). Cortex is driven by weak but synchronously active thalamocortical synapses. Science, 312(5780), 1622–1627. Bush, P. C., & Sejnowski, T. J. (1993). Reduced compartmental models of neocortical pyramidal cells. Journal of Neuroscience Methods, 46(2), 159–166. Courant, R., Friedrichs, K., & Lewy, H. (1928). Partial differential equations of mathematical physics. Mathematische Annalen, 100, 32–74. da Costa, N. M., & Martin, K. A. (2011). How thalamus connects to spiny stellate cells in the cat’s visual cortex. Journal of Neuroscience, 31(8), 2925–2937. de Kock, C. P., Bruno, R. M., Spors, H., & Sakmann, B. (2007). Layer and cell type specific suprathreshold stimulus representation in primary somatosensory cortex. Journal of Physiology, 581(1), 139–154. De Paola, V., Holtmaat, A., Knott, G., Song, S., Wilbrecht, L., Caroni, P., et al. (2006). Cell type-specific structural plasticity of axonal branches and boutons in the adult neocortex. Neuron, 49(6), 861–875. Dercksen, V.J., Oberlaender, M., Sakmann, B., & Hege, H.C. Interactive visualization— a key prerequisite for reconstruction of anatomically realistic neural networks. In Proceedings of the 2009 workshop on visualization in medicine and life sciences. VMLS09 (in press). Dercksen, V.J., Weber, B., Guenther, D., Oberlaender, M., Prohaska, S., & Hege, H.C. (2009). Automatic alignment of stacks of filament data. In IEEE int. symp. on biomedical imaging: from nano to macro ISBI (pp. 971–974). Destexhe, A., Mainen, Z. F., & Sejnowski, T. J. (1994). Synthesis of models for excitable membranes, synaptic transmission and neuromodulation using a common kinetic formalism. Journal of Computational Neuroscience, 1(3), 195–230. Egger, V., Nevian, T., & Bruno, R. M. (2008). Subcolumnar dendritic and axonal organization of spiny stellate and star pyramid neurons within a barrel in rat somatosensory cortex. Cerebral Cortex, 18(4), 876–889. Farinas, I., & DeFelipe, J. (1991). Patterns of synaptic input on corticocortical and corticothalamic cells in the cat visual cortex. II. The axon initial segment. Journal of Comparative Neurology, 304(1), 70–77. Feldmeyer, D., Egger, V., Lubke, J., & Sakmann, B. (1999). Reliable synaptic connections between pairs of excitatory layer 4 neurones within a single ‘barrel’ of developing rat somatosensory cortex. Journal of Physiology, 521(Pt 1), 169–190. Feldmeyer, D., Lubke, J., Silver, R. A., & Sakmann, B. (2002). Synaptic connections between layer 4 spiny neurone-layer 2/3 pyramidal cell pairs in juvenile rat barrel cortex: physiology and anatomy of interlaminar signalling within a cortical column. Journal of Physiology, 538(Pt 3), 803–822. Feldmeyer, D., Roth, A., & Sakmann, B. (2005). Monosynaptic connections between pairs of spiny stellate cells in layer 4 and pyramidal cells in layer 5A indicate that lemniscal and paralemniscal afferent pathways converge in the infragranular somatosensory cortex. Journal of Neuroscience, 25(13), 3423–3431. Hausser, M., & Mel, B. (2003). Dendrites: bug or feature? Current Opinion in Neurobiology, 13(3), 372–383. He, W., Hamilton, T. A., Cohen, A. R., Holmes, T. J., Pace, C., Szarowski, D. H., et al. (2003). Automated three-dimensional tracing of neurons in confocal and brightfield images. Microscopy and Microanalysis, 9(4), 296–310. Helmstaedter, M., Briggman, K. L., & Denk, W. (2008). 3D structural imaging of the brain with photons and electrons. Current Opinion in Neurobiology, 18(6), 633–641. Helmstaedter, M., de Kock, C. P., Feldmeyer, D., Bruno, R. M., & Sakmann, B. (2007). Reconstruction of an average cortical column in silico. Brain Research Reviews. Hines, M. (1984). Efficient computation of branched nerve equations. International Journal of Bio-Medical Computing, 15(1), 69–76. Hines, M. L., & Carnevale, N. T. (1997). The NEURON simulation environment. Neural Computation, 9(6), 1179–1209. Hodgkin, A. L., & Huxley, A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. Journal of Physiology, 117(4), 500–544. Holmes, W. R. (1989). The role of dendritic diameters in maximizing the effectiveness of synaptic inputs. Brain Research, 478(1), 127–137. Horikawa, K., & Armstrong, W. E. (1988). A versatile means of intracellular labeling: injection of biocytin and its detection with avidin conjugates. Journal of Neuroscience Methods, 25(1), 1–11. Izhikevich, E. M., & Edelman, G. M. (2008). Large-scale model of mammalian thalamocortical systems. Proceedings of the National Academy of Sciences USA, 105(9), 3593–3598. Kerr, J. N., de Kock, C. P., Greenberg, D. S., Bruno, R. M., Sakmann, B., & Helmchen, F. (2007). Spatial organization of neuronal population responses in layer 2/3 of rat barrel cortex. Journal of Neuroscience, 27(48), 13316–13328. Knabner, P., & Angermann, L. (2003). Numerical methods for elliptic and parabolic partial differential equations. Springer. Koch, C. (1999). Biophysics of computation: information processing in single neurons. New York: Oxford University Press, New York. Kozloski, J., Sfyrakis, K., Hill, S., Schurmann, F., Peck, C., & Markram, H. (2008). Identifying, tabulating, and analyzing contacts between branched neuron morphologies. IBM Journal of Research and Development, 52(1–2), 43–55. Land, P. W., Buffer, S. A., Jr., & Yaskosky, J. D. (1995). Barreloids in adult rat thalamus: three-dimensional architecture and relationship to somatosensory cortical barrels. Journal of Comparative Neurology, 355(4), 573–588. 14 S. Lang et al. / Neural Networks ( Lang, S. (2011). www.neurodune.org. Lang, S., & Wittum, G. (2005). Large-scale density-driven flow simulations using parallel unstructured grid adaptation and local multigrid methods. John Wiley & Sons, Ltd. Larkman, A., & Mason, A. (1990). Correlations between morphology and electrophysiology of pyramidal neurons in slices of rat visual cortex. I. Establishment of cell classes. Journal of Neuroscience, 10(5), 1407–1414. Lein, E. S., Hawrylycz, M. J., Ao, N., Ayres, M., Bensinger, A., Bernard, A., et al. (2007). Genome-wide atlas of gene expression in the adult mouse brain. Nature, 445(7124), 168–176. Lubke, J., & Feldmeyer, D. (2007). Excitatory signal flow and connectivity in a cortical column: focus on barrel cortex. Brain Structure and Function, 212(1), 3–17. Lubke, J., Roth, A., Feldmeyer, D., & Sakmann, B. (2003). Morphometric analysis of the columnar innervation domain of neurons connecting layer 4 and layer 2/3 of juvenile rat barrel cortex. Cerebral Cortex, 13(10), 1051–1063. Mainen, Z. F., Joerges, J., Huguenard, J. R., & Sejnowski, T. J. (1995). A model of spike initiation in neocortical pyramidal neurons. Neuron, 15(6), 1427–1439. Manns, I. D., Sakmann, B., & Brecht, M. (2004). Sub- and suprathreshold receptive field properties of pyramidal neurones in layers 5A and 5B of rat somatosensory barrel cortex. Journal of Physiology, 556(Pt 2), 601–622. Margrie, T. W., Brecht, M., & Sakmann, B. (2002). In vivo, low-resistance, wholecell recordings from neurons in the anaesthetized and awake mammalian brain. Pflugers Archiv, 444(4), 491–498. Markram, H. (2006). The blue brain project. Nature Reviews Neuroscience, 7(2), 153–160. McCormick, D. A., Connors, B. W., Lighthall, J. W., & Prince, D. A. (1985). Comparative electrophysiology of pyramidal and sparsely spiny stellate neurons of the neocortex. Journal of Neurophysiology, 54(4), 782–806. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., & Teller, E. (1953). Equation of state calculations by fast computing machines. Journal of Chemical Physics, 21(6), 1087–1092. Meyer, H.S., Schwarz, D., Wimmer, V.C., Schmitt, A.C., Kerr, J.N.D., & Sakmann, B. et al. Inhibitory interneurons in a cortical column form hot spots of inhibition in layers 2 and 5A (submitted for publication). Meyer, H. S., Wimmer, V. C., Hemberger, M., Bruno, R. M., de Kock, C. P., Frick, A., et al. (2010). Cell type-specific thalamic innervation in a column of rat vibrissal cortex. Cerebral Cortex, 20(10), 2287–2303. Meyer, H. S., Wimmer, V. C., Oberlaender, M., de Kock, C. P., Sakmann, B., Helmstaedter, M. L., et al. (2010). Number and laminar distribution of neurons in a thalamocortical projection column of rat vibrissal cortex. Cerebral Cortex, 20(10), 2277–2286. Mishchenko, Y., Hu, T., Spacek, J., Mendenhall, J., Harris, K. M., & Chklovskii, D. B. (2010). Ultrastructural analysis of hippocampal neuropil from the connectomics perspective. Neuron, 67(6), 1009–1020. Mullen, R. J., Buck, C. R., & Smith, A. M. (1992). NeuN, a neuronal specific nuclear protein in vertebrates. Development, 116(1), 201–211. Oberlaender, M., Boudewijns, Z. S., Kleele, T., Mansvelder, H. D., Sakmann, B., & de Kock, C. P. (2011). Three-dimensional axon morphologies of individual layer 5 neurons indicate cell type-specific intracortical pathways for whisker motion and touch. Proceedings of the National Academy of Sciences USA. Oberlaender, M., Broser, P. J., Sakmann, B., & Hippler, S. (2009). Shack–Hartmann wave front measurements in cortical tissue for deconvolution of large threedimensional mosaic transmitted light brightfield micrographs. Journal of Microscopy, 233(2), 275–289. Oberlaender, M., Bruno, R. M., Sakmann, B., & Broser, P. J. (2007). Transmitted light brightfield mosaic microscopy for three-dimensional tracing of single neuron morphology. Journal of Biomedical Optics, 12(6), 064029. ) – Oberlaender, M., de Kock, C.P.J., Bruno, R.M., Ramirez, A., Meyer, H.S., & Dercksen, V.J. et al. Cell type-specific three-dimensional structure of thalamocortical networks in a barrel column in rat vibrissal cortex (submitted for publication). Oberlaender, M., Dercksen, V. J., Egger, R., Gensel, M., Sakmann, B., & Hege, H. C. (2009). Automated three-dimensional detection and counting of neuron somata. Journal of Neuroscience Methods, 180(1), 147–160. Peters, A. (1979). Thalamic input to the cerebral cortex. Trends in Neurosciences, 2, 1183–1185. Petreanu, L., Mao, T., Sternson, S. M., & Svoboda, K. (2009). The subcellular organization of neocortical excitatory connections. Nature, 457(7233), 1142–1145. Pinault, D. (1996). A novel single-cell staining procedure performed in vivo under electrophysiological control: morpho-functional features of juxtacellularly labeled thalamic cells and other central neurons with biocytin or Neurobiotin. Journal of Neuroscience Methods, 65(2), 113–136. Rall, W. (1969). Time constants and electrotonic length of membrane cylinders and neurons. Biophysical Journal, 9(12), 1483–1508. Sarid, L., Bruno, R., Sakmann, B., Segev, I., & Feldmeyer, D. (2007). Modeling a layer 4-to-layer 2/3 module of a single column in rat neocortex: interweaving in vitro and in vivo experimental observations. Proceedings of the National Academy of Sciences USA, 104(41), 16353–16358. Segev, I., Rinzel, J., & Shepherd, G. M. (1995). The theoretical foundation of dendritic function. MIT Press. Staiger, J. F., Flagmeyer, I., Schubert, D., Zilles, K., Kotter, R., & Luhmann, H. J. (2004). Functional diversity of layer IV spiny neurons in rat somatosensory cortex: quantitative morphology of electrophysiologically characterized and biocytin labeled cells. Cerebral Cortex, 14(6), 690–701. Stalling, D., Westerhoff, M., & Hege, H. C. (2005). Amira: a highly interactive system for visual data analysis. In The visualization handbook. Elsevier. Stuart, G., & Spruston, N. (1998). Determinants of voltage attenuation in neocortical pyramidal neuron dendrites. Journal of Neuroscience, 18(10), 3501–3510. Sun, Q. Q., Huguenard, J. R., & Prince, D. A. (2006). Barrel cortex microcircuits: thalamocortical feedforward inhibition in spiny stellate cells is mediated by a small number of fast-spiking interneurons. Journal of Neuroscience, 26(4), 1219–1230. Svoboda, K. (2011). The past, present, and future of single neuron reconstruction. Neuroinformatics. Traub, R. D., Miles, R., & Wong, R. S. (1988). Large scale simulations of the hippocampus. IEEE Engineering in Medicine and Biology Magazine, 7(4), 31–38. Tsien, J. Z., Chen, D. F., Gerber, D., Tom, C., Mercer, E. H., Anderson, D. J., et al. (1996). Subregion- and cell type-restricted gene knockout in mouse brain. Cell, 87(7), 1317–1326. White, E. L. (1979). Thalamocortical synaptic relations: a review with emphasis on the projections of specific thalamic nuclei to the primary sensory areas of the neocortex. Brain Research, 180(3), 275–311. Wickersham, I. R., Finke, S., Conzelmann, K. K., & Callaway, E. M. (2007). Retrograde neuronal tracing with a deletion-mutant rabies virus. Nature Methods, 4(1), 47–49. Wickersham, I. R., Lyon, D. C., Barnard, R. J., Mori, T., Finke, S., Conzelmann, K. K., et al. (2007). Monosynaptic restriction of transsynaptic tracing from single, genetically targeted neurons. Neuron, 53(5), 639–647. Wimmer, V. C., Bruno, R. M., de Kock, C. P., Kuner, T., & Sakmann, B. (2010). Dimensions of a projection column and architecture of VPM and POm axons in rat vibrissal cortex. Cerebral Cortex, 20(10), 2265–2276. Yu, C., Derdikman, D., Haidarliu, S., & Ahissar, E. (2006). Parallel thalamic pathways for whisking and touch signals in the rat. PLoS Biology, 4(5), e124. Zador, A., Koch, C., & Brown, T. H. (1990). Biophysical model of a Hebbian synapse. Proceedings of the National Academy of Sciences USA, 87(17), 6718–6722.