Voltage Modeling in Neurons with the Morris-Lecar Equations
John J. Sakon Fall 2003
Abstract
The Morris-Lecar Equations enable neuronal activity to be studied through the use of differential equations.
Introduction
Neurons are cells found in the brain, spinal column and nerves that send informational signals throughout the human (and animal) body. They send these signals by utilizing action potentials created by differences in electrical charge across their membranes. Neurons generally work in two different ways: excitable and bursting. Excitable neurons are inactive until they are acted upon by an outside force while bursting neurons are periodically active. This project will focus on modeling such neurons with the MorrisLecar equations. dv v .01 = I + (2 w(.7 v)) + (.5(.5 v)) + (1.1)(.5)(1 + tanh( ))(1 v) (1) dt .15 dw v .22 v .01 (2) = (cosh( ))(.5)(1 + tanh( ))(1 v) dt .6 .15 (where I is a parameter of injected current into the neuron, w is the percent (%) of open channels permeable to calcium (Ca2+), v is the voltage of the neuron and is a second parameter of injected current
into the neuron)
These equations were created to determine electrical activity within neurons of barnacle muscle fibers. They provide a simplistic model for bursting neurons by analyzing calcium ion flow across the neuronal membrane.
Analysis
The first step in analyzing these equations is to graph equations (1) and (2) and find the nullclines. In order to graph these equations, initial values must be set for I and , the parameters of the equations. We set these values at I =.25 and =.1. The resultant phase plane with nullclines is shown in Figure 1. Next, once we have graphed the nullclines, we can find the equilibrium points of the system by determining where the nullclines cross. From the graph it can be seen that there is one specific equilibrium point at about (.1, .4). Using the equilibrium function of PPlane through Matlab 6.0 we can find detailed information about this point. The actual equilibrium point (where voltage and % channels is at rest) is (.085, .441) with complex eigenvalues of .177+.432(i) and .177-.432(i). Therefore, because there are two complex eigenvalues with a>0 then this equilibrium point is a spiral source.
Figure 1 The next step is to test some solutions of the equations to view how v relates to w over time. ( v , w ) values can be analyzed graphically by finding the solutions on the phase plane. The initial solutions of ( v , w )=(-.5, .1) and (-.5, .2) are shown in Figure 2.
Figure 2
As can be seen in Figure 2 when ( v , w )=(-.5, .1) and (-.5, .2) the solutions are spirals emanating from the equilibrium point. Therefore, as can be seen from the solutions, when the voltage is greater than -.5 the % of open calcium channels steadily increases. However, as the voltage decreases toward 0 the % of open calcium channels falls until it reaches equilibrium. In order to analyze how this system works over time a graph of v and w versus time (t) can be used. Figure 3 graphs v and w versus t for the solution (-.5, .2). This graph shows how v and w repeat over time with a period of about 25-30 seconds. Therefore, at this initial condition the voltage fluctuates slightly over time and the % of open calcium channels fluctuates to match the voltage. The peak % of open channels occurs when the voltage is rising and then steadily declines the until voltage maximizes. As the voltage falls again the % of open channels abruptly falls and then begins to rise again. This indicates that the channels open upon initial influx of voltage and close once the voltage gains are too great.
Figure 3 In order to get an idea of how the solutions work over a larger range of values we can plot multiple solutions along the same voltage value (the same vertical line on the graph). Figure 4 displays multiple solutions when v=-1. As shown, when the voltage is -1 the solutions all spiral towards the equilibrium point. When a lesser % of calcium channels are open (about 0< w <.3) at v=-1 the solutions gently slope toward the equilibrium point, indicating a less severe influx of calcium as the voltage decreases. When a greater % of calcium channels are open (about .3< w <1) at v=-1 the solutions have a much more drastic slope and the calcium lowers quickly as the voltage decreases. This shows how with greater channels open the voltage changes much more quickly, which makes logical sense in such a system.
Figure 4 Now that the system has been analyzed for I =.25 and =.1, the parameters (which denote initial current added to the system) can be changed to get a fuller picture of the Morris-Lecar equations. Figure 5 displays the nullclines, equilibrium point and various solutions when the values are set to I =.25 and =.01. At these conditions the eigenvalues are now real and have values of .390 and .056. Therefore, the equilibrium point is now a source. As can be seen on the graph, the solutions (which all emanate from v=-1, as was done in the last example) at v=-1 all have very gentle slopes no matter how high the % of open calcium channels there are. This is because at a lower initial voltage ( is now 1/10th of the original value) there is only a slight change in the % of channels open as the voltage varies. This makes sense as a neuron without much
Figure 5
Electrical stimulation is not likely going to have as many calcium channels open and any...