# Dendrites

We now know that the neuron 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 these 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.

This is one of the most famous images in neurobiology.

Two Purkinje neurons hand drawn by Santiago Ramon y Cajal at the turn of the 20th century

The neuron Soma is the physically largest part of the neuron. It typically has a diameter of 100 uM. The diameter of the dendrites 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 from 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.

This is a capture of a detailed multi-compartmental model of a cerebellar Purkinje cell , created with GENESIS simulator. It 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).

## Compartments

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

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

These dendrite compartments can be thought of Leaky Electrical Cables with capacitance. A single compartment is modeled below.

The ion channels are as described 1.0 Cells and Ions. The inner conductor, the cytoplasm, is a much poorer conductor than the copper wire used in an undersea cable. It has an resistance along the length of the cable Ra, the "axial resistance". The conducting cytoplasm inside the cylinder, the insulating neural membrane, and the liquid (similar to salt water) surrounding the neuron form a capacitance Cm.

The membrane is also not a perfect insulator due to the ion-conducting channels that pass through it allowing it to leak. 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.

This sounds like a bit of poor engineering with all the leaking going on but you will see that what you think of as 'bugs' are actually features that allow our brains to work.

## Compartment Electrical Equivalent

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

This drawing allows us to think of the cell in a way that we can reconcile that the inside of the cell is negative with respect to the Outside. Imagine hooking the ground side of your DVM to the Outside of the cell and then probing around at the inside points. You would see negative voltage levels only.

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).

## Dimensions and units of measure

The quantities Rm, Ra, Cm, Vm, in the diagram and equation are given in Kilohm, Microfarad, or millivolts, and will depend on the size of the compartment. Note the use of lower case. This designates them as actual units.

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.

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

$R m = R M π l d$

$R a = 4 l R A π d 2$

$C m = π l d C M$

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. 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 (in this simulator the equivalent value of 1uF/m2 is used).

Time Constant (TM)

The time constant is a value in milliseconds that allows you to predict the percentage of the peak membrane voltage (Vm) at any point in time. The membrane time constant for a short uniform section is given by:

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

This Dendrite simulator calculates value of the membrane voltage at every millisecond point by applying 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.

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 at the last millisecond step
• 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.

The exponential rise and decay of the membrane potential as the current traverses a section of leaky cable is shown in the next figure.

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.

Excited and Depressed Dendrites

Connections from other neurons Axons may be attached to any one of thousands of places along the dendritic tree. The drawing of a compartment of dendrite (above) shows two types of ion channel connections:

• EPSP, Green - If the connected ion channel stimulates the soma, it is called Excitatory Post-Synaptic Potential
• IPSP, Red - If the connection channel depresses the downstream soma from firing it is called an Inhibitory Post-Synaptic Potential .

Each of these are connected through synapse. Leak channels shown in Orange are never connected.

## IPSP Activated

The type of connection (EPSP or IPSP) is set up in the simulator when defining the input to the dendrite compartment. An EPSP connection is shown by a Black connection line. To define a synaptic connection as IPSP, click the right column next to the connection you wish to designate. The word 'yes' will appear, and the connecting line will change to red. A second click will remove the 'yes' and return the connection to EPSP.

This shows Step1 connected to an Inhibitory (IPSP) channel on Dendrite compartment 1. Step 4 is connected to an EPSP channel.

## Simulator

The best way to get a feel for the interaction of the various parameters of the dendrite is to run a simulation. Download and install the NeuronLab Simulator if you have not already (as described in section 3. NeuronLab Simulator).