m, h, and n are the three major variables for this model, and they vary according to the relationship set forth below in the differential equations. These variables are gating variables, and determine the values for the ionic currents, which are
then used to determine the voltage.
2. Input Signal:
Istimulation = 200uA/cm2 current stimulation for 0.05msec
3. Output Signal:
4. Differential equations:
B) Numerical Integration Method:
The 4th order Runge-Kutta method is a numerical integration method used to solve ordinary differential equations. It is a fourth order method with fifth order error. This method calculates slopes at the beginning of the interval, the midpoint of the interval, the midpoint a second time using the first slope to determine the value, and finally at the end of the interval. The weighted average of these four slopes are taken to determine the position of the next point on the curve of interest.
Choose step size, h, for n number
of steps from 0 to N
Calculate slope at beginning
of first step, k1
Calculate slope at midpoint, k2
Calculate slope at midpoint,
Calculate slope at end
Take weighted average of 4
slopes to find first point
Use the previous points, step size,
and previous k values to find next
Repeat until n = N
C) See attached m-file for Matlab code
D) Analysis Results:
1. Steady-state analysis:
Figure D1: Steady state values (m,h,n) of the gating variables as a function of membrane voltage in the range of -100mV to 150mV
2. Time constant analysis:
Figure D2: Time constants (Tau m, Tau h, Tau n) of the gating variables as a function of membrane voltage for the range of -100mV to 150mV
E) Simulation Results:
Figure E1: Plot of the membrane voltage versus time, showing the propagation of an action potential.
Figure E2: Plot of the ionic currents versus time.
Figure E3: Plot of the gating variables as a function of time.
The step size used for this project was h = 0.01. A smaller step size would probably produce more accurate plots, but would also slow down the program. If more computing power was available, it would be possible to have a smaller step size and still run the program reasonably fast. Truncation error is present to some degree due to the finite step size. This error is considered 5th order error, even though this is a 4th order numerical integration. Some data is lost, but the data obtained is still a faithful representation of the actual signal. Round off error is kept low for this program because of the larger step size. If the step size were to be much smaller, the round off error would increase. Numerical instability could be calculated by comparing the step size to α. Based on the resulting figures of this program, the numerical instability seems to be low.
Overall, this project showed an accurate representation of the squid axon model through numerical integration of the Hodgkin Huxley equations. The action potential observed corresponds well to the action potential model learned in class. Therefore, the 4th order Runge-Kutta integration method is found to be valuable in solving differential equations.