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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
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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,
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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.
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)
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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.
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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-
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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
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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%
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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
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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
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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.
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