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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
- 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.
A neuron cell and its dendrite tree simplified for analysis as compartments:
These dendrite compartments can be thought of Leaky Electrical Cables with capacitance.
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 the membrane current times the membrane resistance.
The change in voltage across a small length of the dendrite (one compartment) is equal to
The solution to the voltage drop is therefore
where is the length down the cable and λ is the length constant given by:
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.
As you can see the solution to a complex dendrite arborization is quite complex and therefor numerical simulations are typically used.
Dendrite Compartment Simulator
This Dendrite simulator calculates the RC Time Constant of the dendrite compartment by applying it to the following formula Where:
- Vt is the compartment output voltage at time t
- Vs is the height of the voltage step
- Vo is the initial voltage on the capacitor
- RC is the time constant of the compartment
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 and filtering.
- Start the application. It will display the step input voltage in red and the dendrite compartment output in blue
- Click the slide out panel
- Adjust the Step input to the desired width, amplitude and start/stop time
- Examine the resulting dendrite output wave form
- Notice the delay created at the half power point