Cable Properties III: Design and Analysis of Branching Neurons

Introduction

• In the previous two units, you have focused on understanding some of the key properties of more extended neurons, analyzing their passive properties, and the role of active properties and myelination.

• To analyze more features of the cable properties of neurons, we will explore the concept of safety factor, which determines the reliability of transmission, and then the effect of branching, both on transmitting messages when the branches are active, and on receiving messages (as occur in dendrites) when (as is usually the case) branches are passive.

• Finally, we will ask you to act as a designer, and find the best length for the internode of a myelinated axon.

Cable Properties and Safety Factors

• For an action potential to propagate successfully to its target, it must reliably get from one end of an axon to the other.

• Axons that are narrow in diameter require less current to reach threshold for firing an action potential, and so narrow axons have the advantage of being more reliable when the input current is small or variable.

• On the other hand, as we've seen in the previous problems, axons with wide diameters have greater conduction velocities, and this is often advantageous for the animal.

• One could imagine evolution selecting for an axon that is narrow at the site of current input (allowing for reliable action potential initiation) and which suddenly widens farther down the axon (to maximize conduction velocity).

• What reason might there be for neurons not to use this strategy?

• The key question is whether a change in diameter (in this case, the increase in diameter downstream of the action potential initiation site) will affect the safety factor of the axon, which is defined as the actual current used to charge the membrane relative to the minimum amount necessary to bring it to threshold. Generally, axonal safety factors are very high to ensure rapid and reliable conduction.

• What would happen to the safety factor if the axon diameter expanded along its length too rapidly?

• Here is a simulation that will allow you to answer this question. It simulates a single trunk which terminates in two branches.

• Question 1. To explore the importance of safety factors for neuron function, let us "cut off" one of the branches. To do this, in the parameters listed under the heading Branch 2 properties, change the diameter to $0.001\ \mu\text{m}$, the intracellular resistivity to $10000\ \Omega \text{cm}$, the fast transient sodium conductance to 0, the delayed rectifier potassium conductance to 0, and the branch 2 length to $100\ \mu\text{m}$. Currents that attempt to travel into the branch will encounter a very high resistance over a short length, and will not be able to generate active voltage-gated currents. Run the simulation.
  • What happens to the potential of branch 2? Explain the reason that it moves to the voltage that you observe, based on the remaining properties of that branch.
  • Change the diameter of both the trunk and branch 1 to be 4 $\mu\text{m}$. Run the simulation. Measure and calculate the conduction velocity of the action potential traveling from the trunk to branch 1. Remember to take into account the lengths of the trunk and the branch, and assume that the first electrode is at the beginning of the trunk, and the second electrode is at the end of branch 1. Make sure to measure the peak time of the action potential for the membrane potential of the trunk and the membrane potential of branch 1 (and not the membrane potential at the junction).

• Question 2. Now measure the conduction velocity as you increase the diameter of branch 1 to 8, 12, 16, and 18 $\mu\text{m}$ (keep the trunk diameter at $4\ \mu\text{m}$). Plot the branch 1 diameter versus the conduction velocity for the values 4 through 18. Explain what is happening based on the amount of current needed to charge up branch 1 (which depends on the surface area of the branch) and the axial resistance in branch 1 (which depends on the cross-sectional area of the branch).
  • What happens if the diameter of branch 1 is increased to $21.5\ \mu\text{m}$? Explain this based on the amount of current available from the trunk to charge up branch 1.

• Question 3. Change the diameter of branch 1 to $21.4\ \mu\text{m}$, keeping the same parameters as in the previous question. Run the simulation. What happens? Explain. Now change the diameter of branch 1 to 21.45 $\mu\text{m}$. Run the simulation. What happens? Explain.
  • One hypothesis for the generation of ventricular fibrillation ("sudden death") in the heart is that one region has slowed conduction, and its excitation can re-enter another area, and activate the heart muscle at a time that it should not be activated. Using the results you have just observed, explain what is going on in terms of the refractory period of the excitable heart tissue.

Cable Properties and Branching

• The analysis we have done so far focused solely on a single axon.

• Neurons have very complex branching shapes. Their dendritic trees are given that name because they look like trees - they are very highly branched.

• Have you ever wondered what the advantage is for having highly branched structures?

• Trees are highly branched because this maximizes their ability to expose many different leaves to the sun; if the volume of the tree were enclosed by a single huge photosynthetic organ, the tree would have much less ability to absorb sunlight. Branching is a way to maximize the ratio of surface to volume.

• Similarly, the branching of dendritic trees allows neurons to maximize the number of contacts they make among themselves. In the next units, we will focus on these inputs, the synapses between nerve cells.

• In this unit, we focus on another question: when an input comes into a highly branched structure, how does the branching affect how well it propagates? This matters greatly, because most neurons have a more centralized trigger zone at which they initiate an action potential. If a synapse is just one of tens of thousands of inputs to a nerve cell, how likely is its signal to have an impact on whether the neuron fires an action potential or not? This depends on its electrotonic distance from the trigger zone, which is determined by the number and thickness of the branches between it and the trigger zone.

• Similarly, near their targets, axons often form branches, and distribute their outputs to more than one location. How does branching affect the propagation of an action potential? If an action potential fails to propagate all the way, this is referred to as branch block, and can determine whether the output of the axon is actually effective.

• We can now look at a very simplified dendritic tree, and see how the shapes of the dendrites could affect their ability to influence a nerve cell.

• Question 4. Dendritic trees are complex, highly branched structures. Inputs from other neurons converge on the dendrites, and they perform two kinds of integration of their inputs: temporal and spatial summation. Many dendrites are passive, and thus their passive cable properties are critical for the way in which they integrate inputs.
  • We can use the Branched cable simulation to explore each of these phenomena. Press the Reset button. To explore temporal summation, set both branch 1 length and branch 2 length to 100 $\mu\text{m}$, and set their fast transient sodium conductances and their delayed rectifier potassium conductances to zero (these are found under the Branch 1 Properties and Branch 2 Properties). Under Current Clamp, Trunk, set Number of pulses to zero. Under Current Clamp, Branch 1, set Stimulus current to 0.5 nA, pulse duration to 0.8 ms, and Number of pulses to 1; use the same setting for Current Clamp, Branch 2, except change the Stimulus delay to 5 ms. Run the simulation. What do you observe? Explain.
  • Now set the stimulus delay for Branch 2 to 2 ms. What do you observe? Explain. What do these results imply about the importance of the timing of inputs?

• Question 5. We can also explore spatial summation.
  • Start with the setting for Question 4, but change the Stimulus delay for Branch 2 to be 0.1 ms, so that it occurs at the same time as Branch 1.
  • Change the diameter of branch 1 to 0.5 $\mu\text{m}$, and its length to 466 $\mu\text{m}$. Before running the simulation, predict what the effects of having a smaller diameter and longer length will do to the currents generated by this branch as opposed to branch 2.
  • Run the simulation. What do you observe?
  • Now change the diameter of branch 1 to 1 micron, and run the simulation again. What do you observe? Explain.
  • How does spatial summation allow branches to integrate information?
  • Change the input current for Branch 1 from 0.5 nA to 0 nA. Run the simulation. What do you observe? Explain.
  • Restore the input current for Branch 1 back to 0.5 nA, and change the input current to Branch 2 from 0.5 nA to 0 nA. Run the simulation. What do you observe? Explain.
  • Researchers have claimed that dendritic trees can be used as logic gates, computing functions such as the AND function, where both inputs must be present for the message to be transmitted. Do your results support this claim? Explain.

• One important result of cable theory, due to the work of Wilfred Rall, is that, in some cases, it may be possible to reduce a very complex branching structure into a much simpler equivalent cylinder. If the intracellular resistivity, $R_{i}$, in $\Omega \text{cm}$ and membrane resistance, $R_{m}$, in $\Omega \text{cm}^2$ are identical in all branches (refer to the Cable Theory Parameters and Units page for more information on these), all the branches end with the same terminal condition (i.e., all are sealed, or all are short-circuited), all of the terminal branches end at the same electrotonic distance from the origin in the main branch, and at every branch point, the input resistances are matched, then if $d_{0}$ is the diameter of the part of the cable before the branch, and $d_{1}$ and $d_{2}$ are the diameters of the two branches of the cable, this implies that $$d_{0}^{3/2} = d_{1}^{3/2} + d_{2}^{3/2}. \quad\text{(Equation 7)}$$ This is referred to as the 3/2 rule, and if it applies to branching cables (including multiple branches at a point that satisfy this rule), then the cable theory developed in the previous unit can be exactly applied. In real neurons, there are generally deviations from this rule, so although it may be useful to obtain an initial idea of what it going on in a branching structure, detailed simulations may be essential for understanding how the system actually works.

Designing a Myelinated Neuron

• Here again is the simulation of a myelinated neuron:

• Question 6. In the previous session, we looked at the role of myelination, but we did not determine whether or not the actual length of the myelinated section (the internode) was optimal for increasing the conduction velocity. Design an optimal internode length. If you are more ambitious, see if you can determine a general principle for what this length should be as the diameter of the axon changes.
  • Run the simulation with the default Axon diameter (2 $\mu m$) and Internodal distance (6000 $\mu m$). Find the time of the peak of the action potential in the first node (i.e., the one prior to the internode), and the peak of the action potential in the second node (i.e., the one after the internode). Use the zoom feature (click and drag) to get accurate measurements.
  • Compute the conduction velocity as you did before for this simulation. Make sure that you report the results in meters per second (this will require some conversion factors). Note that since the electrodes are placed at the beginning of the first node and the end of the second node, you need to add 200 $\mu m$ to the total length, so that the total travel distance for the propagating action potential is 6200 $\mu m$.
  • Now systematically shorten the internode length, computing the conduction velocity, and then systematically increase its length, computing the conduction velocity. What Internodal distance maximizes the conduction velocity? Make sure to plot your data.
  • Change the Axon diameters of both the two nodes and the internode to 1 $\mu m$, and repeat your analysis. Repeat your analysis again using diameters of 0.5 $\mu m$. Make sure to plot your data and superimpose your plots.
  • Is there a general pattern? Please describe it as clearly as you can.