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VERTEX
From Scholarpedia
Christopher Thornton et al. (2020), Scholarpedia, 15(6):53365.
[3]doi:10.4249/scholarpedia.53365 revision #193523 [[4]link to/cite this article]
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Post-publication activity
(BUTTON)
Curator: [7]Marcus Kaiser
Contributors:
1.00 -
[8]Christopher Thornton
1.00 -
[9]Frances Hutchings
1.00 -
[10]Richard John Tomsett
[11]Hans E Plesser
[12]Thomas Kreuz
* [13]Dr. Christopher Thornton, Newcastle University, Newcastle upon Tyne, UK
* [14]Dr. Frances Hutchings, Newcastle University, Newcastle upon Tyne, UK
* [15]Dr. Richard John Tomsett, IBM Research, Hursley, United Kingdom
* [16]Dr. Marcus Kaiser, Newcastle University, Newcastle upon Tyne, UK
The Virtual Electrode Recording Tool for EXtracellular potentials (VERTEX) is a
[17]Matlab tool for simulating extracellular potential recordings in spiking
[18]neural network (SNN) models. It uses a forward modelling approach to
calculate extracellular potentials in a model given the position of the neurons
relative to the virtual electrodes. VERTEX makes use of established theory on
extracellular potential generation, modern simulation methods and recent
developments in simplified neuron modelling to simulate [19]local field
potentials (LFPs) from large neuronal network models encompassing over 100,000
neurons [1]. Applied electric fields can also be incorporated into the ongoing
neuronal [20]dynamics, allowing VERTEX to simulate the effect of electric field
stimulation [2]. VERTEX is therefore best suited for simulations that seek to
model a particular experimental output using realistic tissue geometry and neuron
density. Example simulations may be:
* Modelling the LFP generated by pharmacologically induced neuronal
[21]oscillations in an in vitro [22]brain slice preparation.
* Modelling the field potential generated by electrical stimulation in cortical
grey matter.
* Modelling the effect of non-invasive brain stimulation techniques such as
transcranial direct current stimulation.
Contents
* [23]1 Motivation
* [24]2 Overview
* [25]3 What comprises a VERTEX simulation
+ [26]3.1 Tissue Parameters
+ [27]3.2 Neuron Parameters
+ [28]3.3 Connection parameters
+ [29]3.4 The simulation process
+ [30]3.5 Parallelisation
+ [31]3.6 Neuron and synapse models
+ [32]3.7 Electric field stimulation
* [33]4 Analysing simulation output
* [34]5 Performance
* [35]6 Potential Applications
* [36]7 Comparison with similar software tools
* [37]8 Acknowledgements
* [38]9 References
* [39]10 Internal links
* [40]11 External links
* [41]12 See also
Motivation
Extracellular potentials are a robust and commonly used measure of neural
activity. They can be related to a multitude of underlying processes, in
particular synchronised synaptic activation, but their interpretation is not
always obvious. This interpretation is usually based on what we know of the
anatomy and physiology of the system we are recording from. In simple systems a
conceptual model or simple theoretical model may suffice but in more [42]complex
systems such as the neocortex computational models can be used. Computational
modelling using VERTEX can help by integrating our knowledge of the underlying
anatomy and physiology of the circuit to make a prediction of the LFP that can
then be compared directly with the LFP measured in the experiment, creating a
tighter coupling between theory and experiment.
Overview
VERTEX is specifically set up to simulate layered structures - each neuron type
specified by the user has a layer in which its soma will be placed. Synaptic
connections are specified between neuron groups and each connection has a synapse
type, which may indicate whether the connection should have a form of plasticity.
Currently short term plasticity models and a [43]spike timing dependent
plasticity model are available. VERTEX saves the results of a simulation to files
which can then be loaded into MATLAB and visualised using the functions provided
or analysed using home made functions written by the users themselves.
Figure 1: Overview of the VERTEX simulation software. A Simulation workflow. The
user provides parameters as Matlab structures to setup the neuron populations,
position them in layers, connect them together, and simulate their dynamics and
the resultant LFPs. Functionality to export to NeuroML is currently under
development. B Example program structure. The main simulation program only
requires calls to the initNetwork() function and the runSimulation() function,
with the information required to set up the simulation specified in separate
script files.
What comprises a VERTEX simulation
To run a VERTEX simulation the user must first specify parameters to build the
network over which the simulation will run. These parameters can be split into
the following groups: neuron parameters, connection parameters, and tissue
parameters. Figure 1 (A) illustrates the overall workflow of a VERTEX simulation,
this has three components: specifying the parameters, running the simulation, and
analysing the results. Figure 1 (B) shows an example of the MATLAB code used to
specify the parameters.
Tissue Parameters
The tissue parameters group includes the dimensions of the tissue to be
simulated, as well as the layer boundaries (specifying the number and size of
each layer in the structure), neuron density, and conductivity. The tissue
parameters also include any stimulation applied to the tissue, such as the
electric field and stimulation timings. The following example specifies a
single-layered slice of tissue that is 2500 x 400 x 200 microns in volume, with a
neuron density of 20000 and a conductivity of 0.3 Sm^-1:
TissueParams.X = 2500;
TissueParams.Y = 400;
TissueParams.Z = 200;
TissueParams.neuronDensity = 20000;
TissueParams.numLayers = 1;
TissueParams.layerBoundaryArr = [200, 0];
TissueParams.numStrips = 10;
TissueParams.tissueConductivity = 0.3;
TissueParams.maxZOverlap = [-1 , -1];
Neuron Parameters
The neuron parameters group includes the details required to simulate the
intrinsic dynamics of every cell in the network. Each neuron group has its own
set of neuron parameters. These include the proportion of the entire network that
this group comprises, the layer that the group is in, the details of the dendrite
compartments (how many there are and their dimensions), and the passive
properties of the neuron (the membrane resistance, the axial resistance, the
capacitance, and the leak potential). A typical VERTEX simulation uses a neuron
structure that is a caricature of the cell type it wishes to represent to limit
the number of compartments and therefore the computational [44]complexity of the
simulation. VERTEX comes with a simple tool called the Neuron Model Builder,
which allows one to build or modify neuron models for VERTEX simulations. VERTEX
also can load neuron geometries from neuron markup [45]language (NeuroML) files.
The model used to simulate the spiking mechanism of the group is also included
here, the default neuron model used in VERTEX is the [46]Adaptive exponential
integrate-and-fire model (AdEx), and so this would also require the parameters
that this model uses. The following example creates an AdEx neuron group with 8
compartments as the first neuron group in the simulation (additional neuron
groups could be added to the NeuronParams structure as subsequent entries into
the array):
NeuronParams(1).neuronModel = 'adex';
NeuronParams(1).V_t = -50;
NeuronParams(1).delta_t = 2;
NeuronParams(1).a = 2.6;
NeuronParams(1).tau_w = 65;
NeuronParams(1).b = 220;
NeuronParams(1).v_reset = -60;
NeuronParams(1).v_cutoff = -45;
NeuronParams(1).numCompartments = 8;
NeuronParams(1).compartmentParentArr = [0, 1, 2, 2, 4, 1, 6, 6];
NeuronParams(1).compartmentLengthArr = [13 48 124 145 137 40 143 143];
NeuronParams(1).compartmentDiameterArr = ...
[29.8, 3.75, 1.91, 2.81, 2.69, 2.62, 1.69, 1.69];
NeuronParams(1).compartmentXPositionMat = ...
[ 0, 0;
0, 0;
0, 124;
0, 0;
0, 0;
0, 0;
0, -139;
0, 139];
NeuronParams(1).compartmentYPositionMat = ...
[ 0, 0;
0, 0;
0, 0;
0, 0;
0, 0;
0, 0;
0, 0;
0, 0];
NeuronParams(1).compartmentZPositionMat = ...
[ -13, 0;
0, 48;
48, 48;
48, 193;
193, 330;
-13, -53;
-53, -139;
-53, -139];
NeuronParams(1).axisAligned = 'z';
NeuronParams(1).C = 1.0*2.96;
NeuronParams(1).R_M = 20000/2.96;
NeuronParams(1).R_A = 150;
NeuronParams(1).E_leak = -70;
NeuronParams(1).somaID = 1;
NeuronParams(1).basalID = [6, 7, 8];
NeuronParams(1).apicalID = [2 3 4 5];
Connection parameters
Using the tissue and neuron parameters VERTEX creates the neuron models and
position each cell randomly within its layer. The connection parameters include
all the information required to connect the network and to simulate the synaptic
dynamics. This includes the number of connections between neuron groups, the
neuron compartments onto which the synapses form, and the spatial model used to
generate the connections (the default in VERTEX is a Gaussian model, requiring a
radius and limit). VERTEX uses MATLAB's built-in random number generation such
that the specific network realisation is unequivocally identified by the random
seed used. The final connection parameter is the synaptic model between each pair
of neuron groups. This includes the base synapse model (current based,
conductance based, exponential, alpha), as well as any plasticity (short term
plasticity, spike timing dependent plasticity) that may be applied. Synaptic
parameters, such as the weight or decay constants, can be specified as a single
value for all synapses in a particular connection or as a distribution, where
each synaptic parameter is selected from the distribution. The following code
creates connections from neuron group 1 to neuron group 1 and 2, with a current
based exponential synapse, and specifies the particular compartment groups which
the synapses will target. Common parameters for all connections from neuron group
1 are also specified, these include the axon arbor spatial model and its
parameters (the radius and limit), as well as the conduction speed and release
delay:
ConnectionParams(1).numConnectionsToAllFromOne{1} = 1700;
ConnectionParams(1).synapseType{1} = 'i_exp';
ConnectionParams(1).targetCompartments{1} = {'basalID', ...
'apicalID'};
ConnectionParams(1).weights{1} = 1;
ConnectionParams(1).tau{1} = 2;
ConnectionParams(1).numConnectionsToAllFromOne{2} = 300;
ConnectionParams(1).synapseType{2} = 'i_exp';
ConnectionParams(1).targetCompartments{2} = {'dendritesID'};
ConnectionParams(1).weights{2} = 28;
ConnectionParams(1).tau{2} = 1;
ConnectionParams(1).axonArborSpatialModel = 'gaussian';
ConnectionParams(1).sliceSynapses = true;
ConnectionParams(1).axonArborRadius = 250;
ConnectionParams(1).axonArborLimit = 500;
ConnectionParams(1).axonConductionSpeed = 0.3;
ConnectionParams(1).synapseReleaseDelay = 0.5;
The simulation process
After building the network one can proceed with simulating its dynamics. This
requires the specification of some simulation settings (e.g. the duration of the
simulation and the time step) and recording settings (e.g. which variables to
record and from which neurons to record them, sampling rate, etc). The simulation
loop (the code executed at each time step) involves updating any externally
applied electric field, updating all neuron and synapse variables, recording the
value of all variables to be recorded, and processing all spikes (this will also
involve updates of synaptic variables when plasticity is used). Shown below is an
example of how to set up the recording settings structure to record the LFP using
a grid of electrodes, as well as recording the membrane potential of every tenth
of the first 2000 neurons. The sample rate is set to 5 kHz and the maximum
recording time is set to 500 ms indicating that after 500 ms of recording a new
file will be created. The simulation settings are set up to simulate for 500 ms
with a time step of 0.03125 ms, and the simulation will be run in parallel.
RecordingSettings.saveDir = '~/VERTEX_results/';
RecordingSettings.LFP = true;
[meaX, meaY, meaZ] = meshgrid(0:1000:2000, 200, 600:-300:0);
RecordingSettings.meaXpositions = meaX;
RecordingSettings.meaYpositions = meaY;
RecordingSettings.meaZpositions = meaZ;
RecordingSettings.minDistToElectrodeTip = 20;
RecordingSettings.v_m = 1:10:2000;
RecordingSettings.maxRecTime = 500;
RecordingSettings.sampleRate = 5000;
SimulationSettings.simulationTime = 500;
SimulationSettings.timeStep = 0.03125;
SimulationSettings.parallelSim = true;
Parallelisation
VERTEX makes use of MATLAB's Single Program Multiple Data (SPMD) parallel
programming construct to take advantage of multiple cores. This requires the
MATLAB parallel computing toolbox and involves assigning each neuron to a worker
on which its neural and synaptic dynamics will be computed. Spikes are
transmitted between workers during each simulation step.
Neuron and synapse models
Figure 2: The class hierarchy relevant to conductance based exponential synapses.
Neuron and synapse types are described using inheritance to avoid the duplication
of functionality. The abstract NeuronModel class describes the functionality
provided by all multi-compartment neurons. It contains the properties of the
membrane potential, external potential, and axial current (the currents that flow
between compartments as a result of the difference between their membrane
potentials), as well as the functionality required to integrate these. The
integration of the stimulating field is included as an additional step during the
calculation of the axial currents and is performed at each time step when the
stimulation is turned on. It is part of the core functionality of the abstract
NeuronModel class. Classes with specific mechanisms then inherit from this, e.g.
the NeuronModel_passive class provides a simple wrapper on top to allow a neuron
with no active channels. The NeuronModel_adex adds the adaptive exponential
integrate and fire mechanism to the soma, allowing the cell to generate action
potentials. Here each instance of a class would represent a group of neurons in
the same layer and of the same type. This allows employing MATLAB's vectorised
operations when updating variables so that for example the membrane potential
variable (v_m) holds the membrane potentials of all neurons in this group as a
matrix. This also enables us to utilise the object-oriented design advantages
without the overhead that would come from storing each neuron or synapse as its
own object. The integration of the axial current involves a loop over all
possible neighbouring compartments with an operation vectorised for each
compartment. The class hierarchy relevant to [47]conductance-based exponential
synapses (SynapseModel_g_exp) is shown in Figure 2. Here, multiple inheritance is
used to efficiently describe many combinations of synapse types. Synapse models
have a base synapse type (defining how the synapse operates without plasticity,
e.g. g_exp as a conductance-based exponential synapse), it can then also have
short term plasticity (ab for the Abbott model or mt for the Markram and
Tsodyks), spike timing dependent plasticity (STDP), or both. The plasticity
models are defined as separate classes from which the synapse model can inherit.
Synapse Class Description
SynapseModel_i_exp Current based, exponential synapse
SynapseModel_g_exp Conductance based, exponential synapse
SynapseModel_g_exp_mt Conductance based, exponential synapse with Markram and
Tsodyks [3] short term plasticity
SynapseModel_g_exp_ab Conductance based, exponential synapse with Abbott model
[4] of short term plasticity
SynapseModel_g_exp_stdp Conductance based, exponential synapse with STDP
SynapseModel_g_exp_mt_stdp Conductance based, exponential synapse with STDP and
Markram and Tsodyks [3] short term plasticity
SynapseModel_g_exp_ab_stdp Conductance based, exponential synapse with STDP and
Abbott model [4] of short term plasticity
Electric field stimulation
Incorporating the effect of an electric field generated by a stimulating
electrode is relatively straightforward in VERTEX. The first step is to model the
electric field, then the MATLAB [48]Partial Differential Equation (PDE) toolbox
can be used to model most simple electrode setups, and the output of this model
attached to the tissue parameter structure, forming part of the input to the
VERTEX simulation. The only further information needed is the times at which to
apply the stimulation, which are specified as a list of stimulation onset and
offset times. The following code will set the stimulation field and the times at
which to apply it:
TissueParams.StimulationField = stimfield; % stimfield is either a StationaryResults obj
ect
or a TimeDependentResults object from the P
DE toolbox.
TissueParams.StimulationOn = [600 700]; % pulse interval of 100 ms
TissueParams.StimulationOff = [600.5 700.5]; % pulse width of 0.5 ms
To generate an appropriate field (either as a stationary or time-dependent field)
follow the guide on the MATLAB PDE Toolbox website. Time-dependent fields that
are shorter than the duration over which they are applied will be looped, this
allows oscillating fields to be generated only for a single period, saving
[49]memory.
Analysing simulation output
Figure 3: Example outputs of VERTEX simulations. A Spike raster of oscillatory
activity. Black dots represent excitatory cells firing, pink dots represent
inhibitory cells firing. B Soma positions of a network, along with the
stimulation field. C LFP generated by applying the stimulation field in B, as a
paired-pulse.
After a VERTEX simulation the user can load the simulation results and perform
analysis, either using functions provided with VERTEX or custom code. The output
files of the simulation are MAT files and so if the user wished to use other
software to perform analysis then they would be required to export the data to
their required format. VERTEX provides functions to plot a spike raster of the
activity or to plot the LFP recorded at a particular electrode. It also provides
functions to visualise the weights of synapses in the network, either on a group
to group basis, or for each individual connection. The following code will load
the results from the directory they were saved in, and plot a spike raster of the
activity:
Results = loadResults(RecordingSettings.saveDir);
rasterParams.groupBoundaryLines = [0.7, 0.7, 0.7];
rasterParams.title = 'Spike Raster';
rasterParams.xlabel = 'Time (ms)';
rasterParams.ylabel = 'Neuron ID';
rasterFigure = plotSpikeRaster(Results, rasterParams);
This code will plot the stimulation field stored within the TissueParams struct,
and then plot within this the soma positions of the cells in the network:
pdeplot3D(TissueParams.StimulationModel, 'ColorMap', TissueParams.StimulationField.NodalSo
lution, 'FaceAlpha', 0.8)
hold on;
pars.colors = rasterParams.colors;
plotSomaPositions(Results.params.TissueParams,pars);
Figure 3 shows some example visualisations of VERTEX output, a spike raster (A),
the cell positions and stimulating electric field (B), and the response LFP (C).
Performance
The VERTEX simulator can run simulations at the scale of a cortical column on a
modern desktop computer, and at the scale of an in vitro brain slice preparation
on a high-performance computing node. Parallelisation significantly improves the
performance of large VERTEX simulations. Figure 4 shows the simulation times for
a small and large network with increasing numbers of processing cores available.
Increasing the number of electrodes from which to record an LFP also increases
the simulation time.
Figure 4: Parallel simulation performance with increasing numbers of Matlab
workers (i.e. parallel processes). Top row: Model initialisation times for A the
9881 neuron model and B the 123,517 neuron model. Bottom row: Simulation times
for 1 s of biological time for C the 9,881 neuron model and D the 123,517 neuron
model. Thick black lines denote linear speed scaling; legends indicate the number
of electrodes used in each simulation run. The sub-linear speed-up in the small
model is due to the decreased relative performance influence of code
vectorisation for smaller matrices.
Potential Applications
VERTEX is well suited to simulating brain oscillations recorded in vitro. For
example, [1] build a simulation of persistent [50]gamma oscillations recorded
from macaque neocortex, making direct comparisons between the LFP recorded in
simulation and experiment. This allows them to make predictions about the
importance of specific intra-laminar connections in the generation of gamma
oscillations. VERTEX may also be used to build simulations of the response LFP
evoked by electrical stimulation. For example, [2] build a simulation of the
response to focal electric field stimulation in rat neocortex. Using the
simulation they predict the specific postsynaptic currents that contribute to the
response LFP, based on what we currently know of the rat neocortical
microcircuit. This then allows them to inform the interpretation of the changes
seen in the LFP response as a result of short term and spike time-dependent
plasticity.
Comparison with similar software tools
Several simulation packages offer similar capabilities to VERTEX.
The [51]Brian simulator [5], a Python based spiking neural network simulator, has
good support for synaptic plasticity, both short term plasticity models and spike
timing dependent plasticity models. Brian is easy to install and use, and also
very extensible in terms of neuron and synapse models.
The [52]NEST (NEural Simulation Tool) [6] is a mature and easy to use spiking
neural network simulator, that allows users to simulate the activity of large
scale laminar structures with various forms of synaptic plasticity and a range of
neuron and synapse models. It can also be combined with the LFPy tool [7] to
simulate the local field potential generated by a large network. The following
table shows a selection of features present in each of the simulators.
The [53]Neuron simulation environment [8] is a mature software system with a wide
range of features including support for very detailed modelling of cellular
processes. It also supports electrical stimulation through its extracellular
mechanism, and LFP simulation through the LFPy tool [9] and supports a wide
variety of synapse types including short term plasticity and spike timing
dependent plasticity. It provides a graphical user interface to many of its
features.
VERTEX is specifically designed for modelling multi-layered structures such as
the neocortex, and it provides built-in support for modelling LFP generation,
synaptic plasticity, and electric field stimulation. VERTEX is particularly
suited to simulating electrical stimulation and LFP recording in laminar
structures, as it provides built in support to recording the LFP and modeling
electric field stimulation.
Feature BRIAN NEST NEURON VERTEX
User language Python Python Python/hoc Matlab
Multiple compartments Yes Yes Yes Yes
LFP calculation No With LFPy With LFPy Yes
Electric field stimulation No No Yes Yes
Parallelisation Yes Yes Yes Yes
Long term plasticity Yes Yes Yes Yes
Short term plasticity Yes Yes Yes Yes
Acknowledgements
Development of this software was supported by BBSRC (BB/F016980/1), EPSRC
(EP/E002331/1, EP/G03950X/1, EP/K026992/1, NS/A000026/1), and the Wellcome Trust
(102037).
References
1. Tomsett, R. J., Ainsworth, M., Thiele, A., Sanayei, M., Chen, X., Gieselmann,
M. A., ... Kaiser, M. (2015). Virtual Electrode Recording Tool for EXtracellular
potentials (VERTEX): comparing multi-electrode recordings from simulated and
biological mammalian cortical tissue. Brain Struct Funct 220(4), 2333-2353.
[54]https://doi.org/10.1007/s00429-014-0793-x
2. Thornton, C., Hutchings, F., Kaiser, M. (2019). The Virtual Electrode
Recording Tool for EXtracellular Potentials (VERTEX) Version 2.0: Modelling in
vitro electrical stimulation of brain tissue [version 1; peer review: 2
approved]. Wellcome Open Res 4, 20.
[55]https://doi.org/10.12688/wellcomeopenres.15058.1
3. Tsodyks, M., Pawelzik, K., Markram, H. (1998). Neural networks with dynamic
synapses. Neural Computation 10(4), 821-835.
[56]https://doi.org/10.1162/089976698300017502
4. Abbott, L. F., Varela, J. A., Sen, K., Nelson, S. B. (1997). Synaptic
depression and cortical gain control. Science 275(5297), 221-224.
[57]https://doi.org/10.1126/science.275.5297.221
5. Goodman, D. F., & Brette, R. (2008). Brian: a simulator for spiking neural
networks in python. Frontiers in [58]Neuroinformatics, 2, 5.
6. Plesser H.E., Diesmann M., Gewaltig M.O., Morrison A. (2019) NEST: The Neural
Simulation Tool. In: Jaeger D., Jung R. (eds) Encyclopedia of Computational
Neuroscience. Springer, New York, NY.
[59]https://doi.org/10.1007/978-1-4614-7320-6_258-6
7. Hagen, E., Dahmen, D., Stavrinou, M.L., Lindén, H., Tetzlaff, T., van Albada,
S.J., Grün, S., Diesmann, M., Einevoll, G.T. (2016). Hybrid Scheme for Modeling
Local Field Potentials from Point-Neuron Networks. Cerebral Cortex 26, 4461-4496.
[60]https://doi.org/10.1093/cercor/bhw237
8. Hines M.L., Carnevale N.T. (1997). The NEURON simulation environment. Neural
Computation 9(6), 1179-209. [61]https://doi.org/10.1162/neco.1997.9.6.1179
9. Lindén, H., Hagen, E., Leski, S., Norheim, E., Pettersen, K., Einevoll, G.
(2014) LFPy: a tool for biophysical simulation of extracellular potentials
generated by detailed model neurons. Frontiers in Neuroinformatics , Volume 7.
[62]https://doi.org/10.3389/fninf.2013.00041
Internal links
Ted Carnevale (2007) [63]Neuron simulation environment. [64]Scholarpedia,
2(6):1378.
Alain Destexhe and Claude Bedard (2013) [65]Local field potential. Scholarpedia,
8(8):10713.
Marc-Oliver Gewaltig and Markus Diesmann (2007) [66]NEST (NEural Simulation
Tool). Scholarpedia, 2(4):1430.
Dan F. M. Goodman and Romain Brette (2013) [67]Brian simulator. Scholarpedia,
8(1):10883.
Rob Schreiber (2007) [68]MATLAB. Scholarpedia, 2(7):2929.
Frances K. Skinner (2006) Conductance-based models. Scholarpedia, 1(11):1408.
External links
VERTEX website [[69][1]]
VERTEX introduction video [[70][2]]
Dynamic [71]Connectome website [[72][3]]
See also
[73]Brian simulator, [74]Genesis, [75]NEST (NEural Simulation Tool), [76]NEURON
Simulation Environment, [77]XPPAUT
Sponsored by: [78]Thomas Kreuz, Institute for complex systems (ISC), National
research council (CNR), Sesto Fiorentino, Italy
[79]Reviewed by: [80]Thomas Kreuz, Institute for complex systems (ISC), National
research council (CNR), Sesto Fiorentino, Italy
[81]Reviewed by: [82]Prof. Hans E Plesser, Norwegian University of Life Sciences,
Ĺs, Norway
Accepted on: [83]2020-05-21 21:00:00 GMT
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5. http://www.scholarpedia.org/article/VERTEX#mw-head
6. http://www.scholarpedia.org/article/VERTEX#p-search
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21. http://www.scholarpedia.org/article/Periodic_Orbit
22. http://www.scholarpedia.org/article/Brain
23. http://www.scholarpedia.org/article/VERTEX#Motivation
24. http://www.scholarpedia.org/article/VERTEX#Overview
25. http://www.scholarpedia.org/article/VERTEX#What_comprises_a_VERTEX_simulation
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31. http://www.scholarpedia.org/article/VERTEX#Neuron_and_synapse_models
32. http://www.scholarpedia.org/article/VERTEX#Electric_field_stimulation
33. http://www.scholarpedia.org/article/VERTEX#Analysing_simulation_output
34. http://www.scholarpedia.org/article/VERTEX#Performance
35. http://www.scholarpedia.org/article/VERTEX#Potential_Applications
36. http://www.scholarpedia.org/article/VERTEX#Comparison_with_similar_software_tools
37. http://www.scholarpedia.org/article/VERTEX#Acknowledgements
38. http://www.scholarpedia.org/article/VERTEX#References
39. http://www.scholarpedia.org/article/VERTEX#Internal_links
40. http://www.scholarpedia.org/article/VERTEX#External_links
41. http://www.scholarpedia.org/article/VERTEX#See_also
42. http://www.scholarpedia.org/article/Complex_systems
43. http://www.scholarpedia.org/article/Spike-Timing_Dependent_Plasticity
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85. http://www.scholarpedia.org/w/index.php?title=Special:UserLogin&returnto=VERTEX
86. http://www.scholarpedia.org/article/VERTEX
87. http://www.scholarpedia.org/article/Talk:VERTEX
88. http://www.scholarpedia.org/article/VERTEX
89. http://www.scholarpedia.org/w/index.php?title=VERTEX&action=edit
90. http://www.scholarpedia.org/w/index.php?title=VERTEX&action=history
91. http://www.scholarpedia.org/article/Main_Page
92. http://www.scholarpedia.org/article/Scholarpedia:About
93. http://www.scholarpedia.org/article/Special:ProposeArticle
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118. https://twitter.com/scholarpedia
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128. http://www.scholarpedia.org/article/Scholarpedia:Privacy_policy
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133. http://www.scholarpedia.org/article/File:Vertex1.png
134. http://www.scholarpedia.org/article/File:ClassDiagram2.png
135. http://www.scholarpedia.org/article/File:ClassDiagram2.png
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140. http://www.scholarpedia.org/article/VERTEX
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142. http://www.scholarpedia.org/article/Main_Page
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