Dendritic Modelling

Dendrites (from Greek dendron, “tree”) are the branched projections of a neuron that act to conduct the electrical stimulation received from other cells to and from the cell body, or soma, of the neuron from which the dendrites project. Electrical stimulation is transmitted onto dendrites by upstream neurons via synapses which are located at various points throughout the dendritic arbor. Dendrites play a critical role in integrating these synaptic inputs and in determining the extent to which action potentials are produced by the neuron.

80% of all excitatory synapses - at the dendritic spines.

80% of all excitatory synapses - at the dendritic spines.learning and memory, logical computations, pattern matching, amplification of distal synaptic inputs, temporal filtering.

80% of all excitatory synapses - at the dendritic spines.learning and memory, logical computations, pattern matching, amplification of distal synaptic inputs, temporal filtering.

80% of all excitatory synapses - at the dendritic spines.learning and memory, logical computations, pattern matching, amplification of distal synaptic inputs, temporal filtering.

Electrical properties of dendrites

The structure and branching of a neuron's dendrites, as well as the availability and variation in voltage-gated ion conductances, strongly influences how it integrates the input from other neurons, particularly those that input only weakly. This integration is both “temporal” -- involving the summation of stimuli that arrive in rapid succession -- as well as “spatial” -- entailing the aggregation of excitatory and inhibitory inputs from separate branches.

Dendrites were once believed to merely convey stimulation passively, i.e. via diffusive spread of voltage and without the aid of voltage-gated ion channels. Passive cable theory describes how voltage changes at a particular location on a dendrite transmit this electrical signal through a system of converging dendrite segments of different diameters, lengths, and electrical properties.

However, it is important to remember that dendrite membranes are host to a zoo of proteins some of which may help amplify or attenuate synaptic input (like those in the axon). One important feature of dendrites, endowed by their active voltage gated conductances, is their ability to send action potentials back into the dendritic arbor. Known as backpropagating action potentials, these signals depolarize the dendritic arbor and provide a crucial component toward synapse modulation and long-term potentiation.

The dendrite as a “stretched-out” RC circuit

The objective of realistic single cell modeling is to approximate the electrophysiological behavior of biological neurons as closely as possible. Since each type of neuron in the brain is characterized by distinct morphological features, as well as distinct electrophysiological features this requires a case-by-case study -- within the overall famework of cable theory and compartmental modelling.

The theoretical basis of compartmental modeling is given by cable theory, a set of well understood concepts originally introduced by Wilfried Rall. A basic dictum of cable theory is that in passive cylinders signals decay over space and time in a mathematically understood fashion. The concepts of time constant ( ) and space constant ( ) simplify the temporal and spatial description of signal decay.

τ

λ

dendrite segments of different diameters, lengths, and electrical properties. However, it is impor-tant to remember that dendrite membranes are host to a zoo of proteins some of which may helpamplify or attenuate synaptic input (like those in the axon). One important feature of dendrites,endowed by their active voltage gated conductances, is their ability to send action potentials backinto the dendritic arbor. Known as backpropagating action potentials, these signals depolarizethe dendritic arbor and provide a crucial component toward synapse modulation and long-termpotentiation.

Dendritic modelling2

The objective of realistic single cell modelling is to approximate the electrophysiological behaviorof biological neurons as closely as possible. Since each type of neuron in the brain is characterizedby distinct morphological features, as well as distinct electrophysiological features this requires acase-by-case study – within the overall framework of cable theory and compartmental modelling.

The theoretical basis of compartmental modelling is given by cable theory, a set of well understoodconcepts originally introduced by Wilfried Rall. A basic dictum of cable theory is that in passivecylinders signals decay over space and time in a mathematically understood fashion. The conceptsof time constant (τ) and space constant (λ) simplify the temporal and spatial description of signaldecay.

Because passive membrane is assumed, the spatio-temporal distribution of membrane potentialalong the cylinder must obey a partial differential equation, known as the cable equation; thiscan be expressed

∂V

∂T= −V +

∂2V

∂X2

where V = Vm−Vr represents the departure of the membrane potential V (which is the difference,intracellular Vi minus extracellular Ve), from its resting value, Vr. Also,

X =x

λ, λ =

√rm/ri =

√(Rm/Ri)(d/4),

T =t

τm, τm = rmcm = RmCm.

Here ri is the intracellular (core) resistance per unit length of the cylinder, while cm and 1/rm

are the membrane capacity, and membrane conductance, respectively, per unit length of themembrane cylinder; d is the cylinder diameter, while Rm and Cm apply to unit membrane area,and Ri is the volume resistivity of the intracellular medium.

In practice, neurons have finite length processes with closed ends. Steady state voltage decayssignificantly when processes are of length > 1 space constant or so, which is typically the case inlarge neurons.

Importantly, signal decay in cylinders is frequency dependent, and signals with fast rise and falltime (such as action potentials) decay over much shorter distances than steady state potentials.

2from the article by Dieter Jaeger: http://www.brains-minds-media.org/archive/222/

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Membrane decay constant τ. Comparison of normalized charging curves for finite cables ofdifferent electronic lengths. A step of current is injected and the voltage is measured, at x = 0.

Membrane space constant λ. Comparison of voltage decays along finite cables of differentelectronic lengths and with end terminations. Current is injected at x = 0. The solid lines arefor finite cables with sealed ends (no current flows out of either end), the dashed lines are forfinite cables with open ends, and the dotted line is for a semi-infinite cable.

Signal decay can be depicted graphically by “morphoelectronic transforms”. Here it is demon-strated that signal decay is non-symmetrical and that voltage transients generated in the dendritesdecay much more than voltage transients from the soma to the tip of the dendrites. This effectis due to the branching structure of the neuron and the increasing diameter of processes towardthe soma.

Another important feature of signal decay is its dependence on input resistance, which itselfdecreases dramatically during barrages with synaptic input. Thus the shown morphoelectronictransforms would show much more pronounced signal decay yet if the neuron was subjected to asynaptic input barrage.

Beyond the passive structure of a neuron (process diameters, lengths, passive input resistance,membrane capacitance, and axial resistance) neural dynamics are crucially dependent on thepresence of voltage-gated ion channels. The modelling of individual ion channels would be toocomputationally costly, however, and the Hodgkin-Huxley formalism of macroscopic currents (onefor each species of ion channel) is typically employed.

The overall structure of a compartmental model is depicted. The neuron is divided into isopoten-tial compartments (these need to be chosen small enough such that signal decay within the lengthof one compartment is expected to be negligible under all conditions), and each compartment isgiven passive properties: membrane capacitance (Cm), membrane resistance (described by leak

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e−x/λ

dendrite segments of different diameters, lengths, and electrical properties. However, it is impor-tant to remember that dendrite membranes are host to a zoo of proteins some of which may helpamplify or attenuate synaptic input (like those in the axon). One important feature of dendrites,endowed by their active voltage gated conductances, is their ability to send action potentials backinto the dendritic arbor. Known as backpropagating action potentials, these signals depolarizethe dendritic arbor and provide a crucial component toward synapse modulation and long-termpotentiation.

Dendritic modelling2

The objective of realistic single cell modelling is to approximate the electrophysiological behaviorof biological neurons as closely as possible. Since each type of neuron in the brain is characterizedby distinct morphological features, as well as distinct electrophysiological features this requires acase-by-case study – within the overall framework of cable theory and compartmental modelling.

The theoretical basis of compartmental modelling is given by cable theory, a set of well understoodconcepts originally introduced by Wilfried Rall. A basic dictum of cable theory is that in passivecylinders signals decay over space and time in a mathematically understood fashion. The conceptsof time constant (τ) and space constant (λ) simplify the temporal and spatial description of signaldecay.

Because passive membrane is assumed, the spatio-temporal distribution of membrane potentialalong the cylinder must obey a partial differential equation, known as the cable equation; thiscan be expressed

∂V

∂T= −V +

∂2V

∂X2

where V = Vm−Vr represents the departure of the membrane potential V (which is the difference,intracellular Vi minus extracellular Ve), from its resting value, Vr. Also,

X =x

λ, λ =

√rm/ri =

√(Rm/Ri)(d/4),

T =t

τm, τm = rmcm = RmCm.

Here ri is the intracellular (core) resistance per unit length of the cylinder, while cm and 1/rm

are the membrane capacity, and membrane conductance, respectively, per unit length of themembrane cylinder; d is the cylinder diameter, while Rm and Cm apply to unit membrane area,and Ri is the volume resistivity of the intracellular medium.

In practice, neurons have finite length processes with closed ends. Steady state voltage decayssignificantly when processes are of length > 1 space constant or so, which is typically the case inlarge neurons.

Importantly, signal decay in cylinders is frequency dependent, and signals with fast rise and falltime (such as action potentials) decay over much shorter distances than steady state potentials.

2from the article by Dieter Jaeger: http://www.brains-minds-media.org/archive/222/

3

Morphoelectric transforms

Signal decay can be depicted graphically by “morphoelectronic transforms”. Here it is demonstrated that signal decay is non-symmetrical and that voltage transients generated in the dendrites decay much more than voltage transients from the soma to the tip of the dendrites. This effect is due to the branching structure of the neuron and the increasing diameter of processes toward the soma.

Beyond the passive structure of a neuron neural dynamics are crucially dependent on the presence of voltage-gated ion channels. The modeling of individual ion channels would be too computationally costly, however, and the Hodgkin-Huxley formalism of macroscopic currents is employed. In a compartmental model the neuron is divided into isopotential compartments, and each compartment is given both passive and active properties.

Compartmental modelling

The three steps to make a single neuron model are i) create an accurate morphological reconstruction, ii) create an accurate passive model, and iii) match active properties with physiological data.

i) Create an accurate morphological reconstruction

Biocytin fills from slice recordings.

Voltage attenuation along a finite-length cable (L = 1) for current injections (DC to 100 Hz) atx = 0 (i.e., soma) (Rm=50,000 Ω cm2).

conductance and leak reversal potential). Lastly, active conductance properties are inserted, andare described by maximal conductance values (e.g. gK), voltage-dependence (denoted by thearrow), and reversal potential.

The three steps to make a single neuron model are i) create an accurate morphological recon-struction, ii) create an accurate passive model, and iii) match active properties with physiologicaldata.

Create an accurate morphological reconstruction

Creating a model starts with a morphological reconstruction of dye-filled neurons. Typicallybiocytin fills from slice recordings are used. The initial histological processing can already lead todistortions of the true neural morphology due to fixation shrinkage as well as mechanical stresses.Here, the same Purkinje cell is shown before and after cover-slipping. A little bit of squeezeon the cover slip significantly distorts the relative branch lengths. A certain amount of error inmorphological reconstructions due to such distortions as well as incomplete fills of spines anddistal processes should always be expected.

Alternatively grab data from an existing database such as

• Hippocampal Neuronal Morphology Archive athttp://www.compneuro.org/CDROM/nmorph/index/topindex.html.

• Cell Centered Database (CCDB) athttp://ccdb.ucsd.edu/CCDBWebSite/index.html Virtual Neuromorphology electronicdatabase at http://krasnow.gmu.edu/L-Neuron/L-Neuron/database/index.html

• Neocortical microcircuit cell database athttp://microcircuit.epfl.ch/databases/dbCellSearch.asp

Also it is worth knowing about Cvapp – this is a cell viewing, editing and format convertingprogram for morphology files. It can also be used to prepare structures digitized with Neurolucidasoftware for modelling with NEURON. It is available fromhttp://www.compneuro.org/CDROM/nmorph/cellArchive.html.

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ii) Create an accurate passive model

“Morphoelectronic transforms”. A reconstructed CA1 neuron is shown on the left, and themorphoelectronic transforms for the decay of potential fom the soma out the dendrites andfrom the dendrites toward the soma are illustrated on the right. The neuron diagrams on theright are drawn according to a scale that represents the ln of the voltage attenuation ratios (1ln unit represents voltage attenuation of 1/e from the site of current injection). The top middletransform represents the ln (attenuation) of a DC signal applied to the soma as it decays todifferent sites of dendrites. The top right transform represents the ln(attenuation) of a DCsignal applied to different sites in the dendrites as it decays toward the soma. The twotransforms at the bottom right represent the same measurement as those above but resultingfrom the application of a 40 Hz sine wave signal instead of 0 Hz.

Create an accurate passive model

After the morphology description has been imported into the simulation software, the passiveparameters of the cell need to be fit to data. If the morphological reconstruction is less thanideal, the fit parameters may show a shift with respect to the true values to offset erroneousmeasurements on cell surface area and process diameters. For example, if surface area is missing,the specific membrane capacitance may come out higher than the standard value of 1µF cm−2.

Tasks:

• Set Rm, Cm, and Ri correctly for each compartment.

If the assumption that parameters are uniform holds, need to fit 3 values.

Strategy:

• Obtain data from recordings, then optimize Rm, Cm, and Ri to fit data.

To fit passive parameters, recorded neural responses of the same cell type two current injectionpulses need to be available. These recordings need to be undertaken after voltage-gated con-ductances are blocked. A complete block is often difficult, and the recorded passive responsesmay deviate from the true passive response due to several other potential problems as well (suchas due to the electrode serial resistance). Fitting passive responses is done by running currentinjections into the soma of the model cell and optimizing the match of the voltage response withthat of recordings.

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iii) Match active properties with physiological data

Structure of a compartmental model

Biocytin fills from slice recordings.

Match active properties with physiological data

This takes the bulk of the total time in making a good model. The construction of a biologicallyrealistic model is a more hopeful prospect if the availability of voltage-clamp date for the variousvoltage-gated conductances of the cell type under study is good. If such data are not available,one can proceed with default channel kinetics determined in other cell types and fit current-clampdata as well as possible. However, this latter process is unlikely to generate an accurate modelin the end.

Tasks:

• Construct voltage-gated conductances with correct kinetics.

• Incorporate intracellular calcium handling.

• Find needed densities of channels in each compartment.

• Add realistic synaptic input conductances.

Once a model is complete, it can help the scientist to appreciate the 3-dimensional distributionof voltage and currents throughout different activity regimes. Movie files are one helpful tool toobtain this information.

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