Medical Physiology 2005

Problem Set 3:  Action Potentials:

 

Part I:

 

At the axon hillock of a neuron at rest, ENa = +50 mV, EK = -100 mV and Vm = -70 mV.   The resting membrane conductances are gNa = 10 nS, gK = 100 nS (ignore Cl- conductance).  A nearly instantaneous rise in the Na+ conductance elicits an action potential.

 

1.   What is the minimum Na+ conductance that is sufficient to attain threshold?  (Hint:  recall that threshold is attained when the inward [depolarizing] current just exceeds the outward [hyperpolarizing] current.)

 

During the hyperpolarizing after-potential, the K+ conductance rises to 1000 nS, and Vm = ­-90 mV.

 

2.   Can a second action potential be generated during this time (yes/no, why/why not)?

3.   If a second action can be generated, then what Na+ conductance would produce the inward current sufficient to attain threshold?

 

Part II:

 

Much of our understanding of the neural action potential came from a classic series of papers published in 1952 by Alan Hodgkin and Andrew Huxley, from Cambridge University (the work ultimately led to the 1963 Nobel Prize in Physiology or Medicine).  These studies were done prior to the development of glass-capillary microelectrodes, and thus the investigators had to find a preparation that could be studied using gross electrodes (e.g., wires).

 

The investigators chose the squid.  The squid possess a “giant axon” that runs from the head to the mantle; it’s involved in triggering the squid’s escape response—namely, contracting the mantle thereby producing a jet of water that rapidly propels the squid backward.  This axon is approximately 1 mm in diameter (it was originally thought to be a vessel) thereby permitting wires to be snaked down its axis.  The wires were used to control voltage, measure membrane current, etc.  The studies led ultimately to our understanding how the squid’s action potential is generated.  (Note that one should not be confused with the terminology:  the “squid giant axon” is a large-diameter axon from the normal [calamari-size] squid, and not an axon from “the giant squid”—a different rare species that lives at great ocean depths.)

 

The fifth paper of Hodgkin and Huxley’s 1952 series presented a mathematical model (a series of four coupled differential equations), whose solution described the action potentials observed experimentally.  The model is appropriate for invertebrate marine species, whose extracellular solution resembles sea water.  Nevertheless, when adjustments are made for mammalian species (300 mOsm solution as opposed to 1000 mOsm sea water) the model is also surprisingly accurate in describing mammalian action potentials.

 

A web-based Java program that solves the model and generates action potentials can be found at http://pb010.anes.ucla.edu/nerve1.html.  Download the model to a PC or Mac from your web browser.   Read the supplied instructions for running the model.  Your task is to investigate how changing model parameters alters the action potential predicted by the model. 

 

Simulations 4-9 can be done using the “Membrane AP” version of the model (this is a non-propagating action potential occurring at a single region of axon).  You may want to extend the time scale to make the computed action potential better resemble a mammalian action potential (the simulation is done at 6.3ºC, thus rendering the action potential significantly longer than one occurring at 37ºC).

 

4.   What is the effect on the action potential of changing the extracellular Na+ concentration?  Try reducing the extracellular Na+ from the default value down to 200 mM (remember, we’re working with sea water in the model).

5.   What is the effect on the action potential of changing the extracellular K+ concentration?  Try increasing the extracellular K+ from the default value up to 75 mM.

6.   Tetrodotoxin (TTX) binds to the Na+ channel and blocks it.  How does TTX affect the action potential?  Try varying the TTX concentration over the range of 0 to 5 nM.

7.   Pronase is a proteolytic enzyme that specifically cleaves (in a dose-dependent fashion) the inactivation (h) gate; the Na+ channels activate (m gate is functional), but don’t inactivate.  Pronase also does not affect activation of the K+ channel (n gate).  Investigate the effect of pronase on the action potential.  Try varying the pronase concentration from 0 to 200 mg/mL.

8.   The action potentials are initiated by simulating a membrane depolarization resulting from a depolarizing current pulse.  Determine the minimum current-pulse amplitude necessary for generating an action potential.  Also, investigate what happens when polarity is changed from a depolarizing pulse to a hyperpolarizing pulse.

9.   The model permits application of an additional current pulse (pulse 3) at an arbitrary time.  Investigate what happens if a pulse occurs during the relative refractory period (time of elevated potassium conductance).  Specifically, compare the relative pulse amplitude (compared to the initial pulse) needed to initiate a second action potential.

 

The following simulations should be done using the “propagated AP” model.  Note that this simulation is best done on a fast computer (>1 GHz), since it involves solution of a series of partial differential equations (numerically intensive).

 

10. Compute the conduction velocity by noting the times at peak amplitude at electrode positions of 2.5 cm and 5.0 cm.  Remember, the velocity will be slow compared to mammalian axons since (i) the squid axon is unmyelinated, and (ii) the temperature is 6.3ºC.

11. Investigate the relative change in conduction velocity as a function of axon diameter.  Does this follow the predictions of how the length constant (l) varies with diameter?  (Note that when increasing diameter, you’ll also have to increase the initial current-pulse amplitude in order to depolarize to threshold.  [Do you know why this is the case?])