# Dendrites

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We now know that the cell membrane contains ion channels that allow specific ions to flow through the lipid bilayer and we treated the neuron cells as small round objects. But Neuron cells are not small round objects. They contain three major areas

• Dendrites
• The Soma (cell body)
• The Axon

The Dendrites are the inputs to the neuron cell. They have extensive arborization and in the case of the Purkinje cells connect to hundreds of thousands of other cells. Normally they are connected to about 10,000 cells. Each connection contributes to the operation of the neuron.

One of the most famous images in neurobiology.

Two Purkinje neurons hand drawn by Santiago Ramon y Cajal

The cell Soma, which is the largest part of the neuron, typically has a diameter of 100 uM but the diameter of the dendrite may be down to 1 uM or less. The dendrites extend 100's of microns but some can extend several meters as in the giraffe brain.

Branching - Dendrites tend to bifurcate repeatedly and create (often several) large and complicated trees. Cerebellar Purkinje cells typically bear one very complicated tree with approximately 400 tips. Cells therefrom cat spinal cord usually have 8-12 trees each with approximately 30 terminal tips.

Diameters - Dendrites are thin tubes of nerve membrane. Near the soma they start with a diameter of few um and fall below 1 um as they successively branch.

Length - Dendritic trees may range from very short (100-200 um) to quite long (1-2 mm) and the total dendritic length may reach 1 cm and more.

Area and Volume - The majority of the brain volume and area is occupied by dendrites. The surface area of a single dendritic tree is in the range of 2,000 to 750,000 Cum.

## Compartments

A detailed multi-compartmental model of a cerebellar Purkinje cell , created with GENESIS simulator. This shows an input spike coming into the dendritic tree and spreading out as it moves down to the soma (the orange ball). The output of the soma is through a single axon - not shown

A neuron cell and its dendrite tree simplified for analysis as compartments:

These dendrite compartments can be thought of Leaky Electrical Cables with capacitance.

## Single Compartment

This picture is a view of the electrical properties of a uniform section of passive dendrite having length l and diameter d.

The ion channels are as described 1.0 The Cell. The current in and out (I axial) is defined by axial resistance to the current and the length of the compartment.

The conducting cytoplasm inside the neuron, the insulating neural membrane, and the liquid (similar to salt water) surrounding the neuron form a cable with a capacitance Cm.

The inner conductor, the cytoplasm, is a poorer conductor than the copper wire used in an undersea cable, and it has an resistance along the length of the cable Ra, the "axial resistance". The membrane in not a perfect insulator due to the ion-conducting channels that pass through it.

The "passive channels" do not vary in conductance, and the "active channels" that have conductances varying with voltage, calcium concentration, or synaptic input. The passive channels account for the membrane resistance Rm and the associated leakage current Ileak. The active channels are represented by the various variable conductances.

## Compartment Electrical Equivalent

The various components of the cell compartment can be thought of as an electrical circuit and can be analyzed as such.

Simplest cable equation is the steady state where you can ignore the current flowing into the membrane capacitance Cm. The membrane voltage is given by $V m = I m R m$ (membrane current times the membrane resistance).

Some Notation

The quantities Rm, Ra, Cm, Vm, in the diagram and equation are given in ohms, farads, or volts, and will depend on the size of the compartment.

Specific units - In order to specify parameters that are independent of the compartment dimensions, specific units are used. For a cylindrical compartment, the membrane resistance is inversely proportional to the area of the cylinder, so we define a specific membrane resistance RM, which has units of ohms·m². The membrane capacitance is proportional to the area, so it is expressed in terms of a specific membrane capacitance CM, with units of farads/.

Compartments are connected to each other through their axial resistances Ra. The axial resistance of a cylindrical compartment is proportional to its length and inversely proportional to its cross-sectional area. Therefore, we define the specific axial resistance RA to have units of ohms·m.

For a piece of dendrite or a compartment of length l and diameter d we then have:

$R m = R M πld$

$R a = 4l R A π d 2$

$C m =πld C M$

These results are useful for determining the length of the model compartment. Sections of dendrite that have a continuous variation of voltage along the length are replaced by a "lumped parameter model" with discrete jumps in membrane potential. By using very many short compartments, the compartmental model can approach the result of the continuous cable equation.

Determining values of CM, RM, and RA

This is done by assuming that the values are the same in all compartments, as they are intrinsic properties of the neural membrane and cytoplasm. CM depends on the intrinsic properties of the thickness and dielectric constant of the membrane, and is usually close to 0.01F/m2 (n this simulator the equivalent value of 1uF/m2 is used).

The membrane time constant for a short uniform section is given by:

$T m = R m C m = R M C M$

## Time Constant

This Dendrite simulator calculates the RC Time Constant of a 1 cm2 patch of cell membrane by applying it to the following formula $V m = V s + ( V o − V s ) e − t / R C$

The time constant is modeled by a passive membrane resistance, Rm, a membrane capacitance, Cm, and a current source.  The membrane capacitance equals 1uF/m2 is, or simply 1 uF.

Where:

• Vm is the compartment membrane voltage at time t
• Vs is the height of the voltage step
• Vo is the voltage on the capacitor (t-1)
• RC is the calculated time constant of the compartment

Note that the Time Constant (RC) is independent of the dimensions, because Rm is inversely proportional to the surface area, and Cm is proportional to the surface area. RC is important to determining the time that it takes for ionic currents to produce changes in the membrane potential.

## Length Constant

The exponential decay of the membrane potential as the current traverses this section of leaky cable is:

Where lambda is the point where the current will have been reduced by 63%

If you were to increase the leakage resistance Rm the slope would become more shallow. Conversely if the axial resistance Raxial were to increase you would get a sharper, faster drop in potential.

## Simulator

The best way to get a feel for the interaction of the various parameters of the dendrite is to run a simulator. Download and install the SETI network dendrite simulator (below) which contains a Step function input and calculates the various results and displays it in an interactive way.

This simulator works like the Genesis simulator referenced in the BoG except that it has only one compartment and no Soma. Those are simulated in final part of this series.

Download Dendrite Simulator (Windows only)

The red image is the step input. It can be adjusted from the spinner on the right side. The blue line is the output of the dendrite after time constant calculation.

### To Use

• Click 'Download Dendrite Simulator' above. Used default installations setup
• Start the application. It will display the step input voltage in red and the dendrite compartment output in blue
• Adjust the vertical position of the oscilloscope traces to your liking using the two small spinners on the upper left
• Click the Membrane Parameters to slide out the panel
• Adjust the Membrane Parameters as necessary. The Rm, Ra, Cm and pA values are calculated based on the specific units and the cell dimensions.
• Adjust the Step input to the desired start, width and amplitude. Set the check box to get amplitude in reference to VRest of the membrane potential -65 mV.