ENGR 093: Biomedical Directed Reading Spring 2004  
Home Hodgkin and Huxley's work Biology History Table of Units References

  Previous Work

The purpose of this directed reading was to create a MatlabTM script that would model the squid action potential experiment performed by Alan Hodgkin and Andrew Huxley in the early 1950s . Examination of several texts showed that none of the texts were fully consistent in the equations and variables used to describe the membrane potential of a space-clamped squid axon. The aim of the directed reading switched to encompass researching and creating a working Matlab script using equations consistent in SI units. The hope of this project is to compile information from various neurological, biological, and engineering texts explaining the Hodgkin-Huxley experiment into one website or paper for the future educational use of students. The website will contain links to the history of the experiment, an interactive mathematical model scripted in Matlab, and a list of all references used in research. We hope to complete these project goals during the spring semester of 2004.

The following is a summary of the work completed in the fall semester of 2003. Five main texts were used ranging from biological, engineering, and mathematical texts. The equivalent cicuit model of an unmyelinated squid axon is illustrated in figure 1. The equations and variables for each text are shown below.

Figure 1. Equivalent circuit model of an unmyelinated squid axon .


In Neuroscience: a Mathematical Primer, by Alywn Scott, sodium current and potassium current are given by



where VNa and VK are the voltages at which the sum of the conduction and diffusion components of sodium and potassium currents cancel, and GNa and GK are membrane conductances for sodium and potassium, respectively, and are given by


Transmembrane voltages are given by



The components of the transmembrane potential are given with respect to a resting potential, VR ~ -65mV. For space-clamped action potentials, the equations governing the membrane voltage dynamics are given by

 


where h, m, and n are the voltage-dependent membrane turn-on and turn-off variables. h, m, and n represent sodium inactivation, sodium activation, and potassium activation in the Hodgkin-Huxley model . The voltage dependences of the coefficients at 6.3 C are given by



in the units of ms-1. At steady state the coefficient values are given by


where t is the rate at which a coefficient variable strives to reach steady state, or the relaxation time of coefficient variable. The parameter values are summarized in the table 1 .

Parameter Mean Range Standard Units
GNa 120 (65-260) 120 mmhos/cm2
GK 34 (26-49) 36 mmhos/cm2
GL 0.26 (0.13-0.5) 0.3 mmhos/cm2
VNa 109 (95-119) 115 mV
VK -11 (9-14) -12 mV
VL 11 (4-22) 10.5995 mV

Table 1. Parameter values from Scott text.

In Mathematical Biology I: An Introduction, by M.D. Murray , sodium current and potassium current are given by



where iL is leakage current due to all other ions and conductances are given by



Variables h, m, and n are bounded by 0 and 1 and determined by the differential equations


The Murray resource fails to give units of measurements for the equations depicted in the reading.

In Modeling and Simulation in Medicine and the Life Sciences, by Frank Hoppensteadt and Charles Peskin , sodium current and potassium current are given by



where v is in mV and conductances are given by



Equilibrium potentials for sodium and potassium are given by



The equations governing the gating variables are given by



The opening and closing rate constants are voltage-dependent and given by



where a and ß have units of ms-1. The parameter values are summarized in the table 2 .

Parameter Units
GNa 120 µA*mV/cm2
GK 36 µA*mV/cm2
GL 0.3 µA*mV/cm2
VNa 45 mV
VK -82 mV
VL -59 mV

Table 2. Parameter values from Hoppensteadt text.

The Hoppensteadt text also includes a computer simulation of the nerve action potential using a Matlab program .
In Biological Engineering, by Herman Schwan , sodium current and potassium current are given by



The components of the transmembrane potential are given with respect to a resting potential, VR = 0V. The equations governing the membrane voltage dynamics are given by



The rate coefficient functions are given by



where a and ß have units of ms-1. The relaxation time and steady-state value for m are given by



The parameter values are summarized in the table 3 .

Parameter Units

Parameter Units
gNa 120 mV/cm2
gK 36 mV/cm2
gL 0.3 mV/cm2
VNa 115 mV
VK -12 mV
VL -10.5989 mV

Table 3. Parameter values from Schwann text.

In Introduction to Biomedical Engineering, by John Enderle, Susan Blanchard, and Joseph Bronzino , sodium current and potassium current are given in mA by



where VNa and VK are the voltages at which the sum of the conduction and diffusion components of sodium and potassium currents cancel and GNa and GK are maximum membrane conductances for sodium and potassium, respectively, and are given by



For space-clamped action potentials, the equations governing the membrane voltage dynamics are given by


The voltage dependences of the coefficients at 6.3 C are given by



where alpha and beta values have units printed as s-1. However, there is a typo in the Enderle text. Equations for alpha and beta values are actually given in units of ms-1. Note that this text did not give an equation for ßn. The text states that V represents the displacement from resting potential and should therefore have a negative value . Further note that the equation for am is written as



but should be written as



The Enderle text also includes a SIMULINK program for an action potential .

A Matlab script was created from the compilation of the equations from various texts in hopes of reproduce the current and conductance graphs and the coefficient and t graphs illustrated in the Enderle text and Schwann text, respectively. Equations were rewritten to give values in SI units. Commented Matlab scripts entitled CalcAlphaBeta.m, CalcTauSS.m, and MatlabBioMed.m are attached in the appendix.

The equations used in our version of the clamped voltage squid axon model were based off those depicted in the Enderle and Schwann texts. Due to inconsistences between the two resources, where Schwann used a voltage, we used a voltage equal in magnitude but opposite sign. Conductance constants for sodium, potassium, and leakage ions are given in units of Vm-2, and a resting potential of -0.07V was used. Voltage constants for sodium, potassium, and leakage ions were based on those given in example problem 3.9 of the Enderle text . Those values were 0.055V, -0.072V, and -0.0494V for sodium, potassium, and leakage ions voltages, respectively. The equations used are listed below.



 

 

 

 

 


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