Quick start guide for Morris-Lecar model

## Morris-Lecar model†

For example, let us consider the Morris-Lecar neuron model:

and calculate the saddle-node bifurcation of an equilibrium point or the saddle-node bifurcation of a limit cycle in the -parameter plane.

Please also see the references  and  for details.

In the following, we make the following programs, and use these:

1. PP
2. FIX
3. BF

## Creation of Programs †

The following steps are done in a sub-directory. Any name can be used for the sub-directory. First, the following is executed in the MATLAB command window:

`>> BUNKI`

After the execution, a window named Select BUNKI Project Directory will open.

Here the Project to be analyzed is created or selected. At this stage, no Project is generated. Therefore, input a Project name for identification in the Name field and click the OK button. In this case, since the project is the analysis of the Morris-Lecar model, ML is used as the project name.*1

When creating a new Project, you are asked whether it is okay to create a directory (or folder). Please select Yes. SE (system editor) will start.

We now create the programs for analysis.*2

• Description of system
• First, we select what type of system is going to be analyzed. The Morris-Lecar model has two variables and is an autonomous system. Therefore, the Dimension is set to 2，and Autonomous is checked as the system type.
• The equations for the Morris-Lecar model is input into the Equations field. Please follow the following rules.
• State variables : Each state variable should be written as x[i], i = 1,2,...,n . In other words, equations are input using the substitutions V->xN->x.
• Parameters : Parameters are placed between two \$ symbols to identify that the variable is a parameter. For example, items placed between two \$ symbols, such as \$para\$，\$Vm\$，\$alpha\$，etc, are considered parameters. Parameter names can be chosen freely.

For Equilibrium Point

## PP †

Here, we import the initial values of parameters and state variables to draw a phase portrait and to capture a stable equilibrium point.

First, start the PP (phase portrait) program from SE. Select Program -> PP.

The drawing window and the control panel will start. In the first run, the drawing window is the (x,x) plane.*3 Here, there is no need to change the drawing window.

Next, we deal with the control panel. First, we input the values of Initial states and Parameters as the initial condition.*4. The initial points to start the simulations are input (x,x) in the Initial states field. Here, let us start from the origin (0.0) !

Please do not forget to input the parameter values before drawing. When first ran, 1 is entered for all parameters. Parameters are configured as follows.

 parameter value Cm 20 Iext 30 V1 -1.2 V2 18 V3 12 V4 17.4 VK -80 VL -60 VCa 120 gCa 4 gK 8 gl 2 phi 0.0666666667

Next, click the Start button to start simulations. Can you see that the orbit of a solution starting from the origin converges to a single point?

Parameters can be changed in real time. let us change the parameter Iext. By increasing the parameter Iext, a saddle-node bifurcation of the equilibrium point occurs. Here, after the equilibrium point disappears, a stable limit cycle can be observed.

Next, we resume simulations after changing Iext to the initial value 30. We check the check box states flow in the Plot field. In this way, we can check the list of the current values of the state variables in the list box during numerical integration. In this example, the ML system converges to a stable equilibrium point. Therefore, we can see that the same number repeatedly appears in the list box.

Here, by clicking the Clear button, the orbit during the transient response shown in the drawing window can be erased, and the steady state can be observed. Since the system converges to a stable equilibrium point, only one point corresponding to the stable equilibrium point is shown on the drawing window.

Next, let us investigate bifurcations of this equilibrium point. The eigenvalues of the variational equation for the equilibrium point are evaluated when the parameters are changed. For this purpose, the FIX tool can be used. We pass the initial values to use FIX from PP. The important thing is that the values of the state variables and parameters at the steady state must be passed to FIX.

To confirm whether the system reached a steady state, either check the checkbox states flow switch and confirm that the values of the state variables no longer change as described above, or click the Status button a few times and confirm that the information displayed in the list box does not change.

After confirming that there are no more changes, select Export current status from Tools in the menu bar. With this procedure, the values of state variables, the parameter values, and other configuration values at the steady state are preserved and can be passed to the next program (see section on FIX).

## FIX †

Here, the objective is to track the accurate location of an equilibrium point when the parameters are changed and also to monitor the eigenvalues during tracking.

Users must adjust the configurations to run FIX. For simplicity, here let us run FIX as is. Normally, the FIX program is run after the values saved by the PP program (using Export Current status) is imported as the initial values for Newton’s method.*5

First, let us start up the FIX program from the PP control panel. Select Program -> FIX.

When initially starting up the FIX, default values are entered in each field.

Information on the equilibrium point obtained with PP is imported as the initial configuration value. Select Tools ->Import initial point.

After the selection, the information obtained with PP such as the parameter values and the coordinates of the equilibrium point are reflected.

By using PP, we saw that the dynamics of the system changes by increasing the parameter Iext. Therefore, select the parameter to be changed as Iext and configure the step size to increase the parameter.

Click the Start button to start calculations. A new window will open when the calculations start. This window is to monitor the location of the eigenvalues on a Gauss plane. The saddle-node bifurcation of an equilibrium point occurs when one of the eigenvalues becomes "0". We can observe that as the parameter changes, the location of the eigenvalues change, and one eigenvalue on the window approaches "0".

Parameters are continuously changed, and when a bifurcation point is detected, the program stops.*6 In this example, the saddle-node bifurcation is detected because one of the eigenvalues approached 0 and Newton’s method did not converge.

The values of the parameters and the state variables when the program stopped can be used as the initial point for BF, the program to track bifurcation curves. The information when the bifurcation point is detected are exported for BF. When Export BF point is selected here, the status, parameter values, and other necessary configuration values are preserved. The preserved values are passed to the next program (see section on BF).

## BF †

The BF tool is used to obtain a bifurcation curve by changing system parameters.

The user must adjust the configuration to run BF. For simplicity, here let us run BF as is. Normally, the BF program is run after the values of the state variables and parameters at the bifurcation point detected by the FIX program are imported as the initial values for Newton's method.*7

First, let us run the BF program from the FIX main panel. Select Program -> BF.

When starting up BF at the first time, default values are entered in each field.

Various information on the bifurcation point obtained with FIX is imported as the initial value. Select Tools ->Import BF point from main panel.

After the selection, various information such as the type of bifurcation and whether the system has an equilibrium point or a limit cycle are input automatically into each field. To tracke the bifurcation set on the (Iext,V3) plane, check the radio button Iext as the X parameter and V3 as the Y parameter. Furthermore, configure items such as step size to change the parameters and the value to stop the BF when reaching end parameter. To save the calculated results as a data file, input the output file name in Output.

After these configurations, click Start button to start tracking of the bifurcation curve. Then, the Eigenvalues window and Bifurcation diagram window will open as new windows. The Eigenvalues window is used to visualize the change in the root locus of the eigenvalues during calculation of the bifurcation curve. Furthermore, to monitor the change of the root locus, we can visually understand occurrence of a bifurcation point with co-dimension 2. The Bifurcation diagram window is a window to monitor how the saddle-node bifurcation is being tracked in real time. (b): Bifurcation diagram in the (Iext,V3)-plane．The calculation result can be monitored in real time.

For Limit Cycle

We skip configurations and the creations of system, because the steps are the same as that of the equilibrium point. If details are necessary, return to the Top and see Configurations and the Creation of Programs.

## PP †

Here, we import the initial values of parameters and state variables to draw a phase portrait and to capture a stable limit cycle.

First, start the PP (phase portrait) program from SE. Select Program -> PP.

The drawing window and the control panel will start. In the first run, the drawing window is the (x,x) plane.*8 Here, there is no need to change the drawing window.

Next, we deal with the control panel. First, we input the values of Initial states and Parameters as the initial condition.*9 The initial points to start the simulations are input (x,x) in the Initial states field. Here, let us start from the origin (0,0) !

Please do not forget to input the parameter values before drawing. When first ran, 1 is entered for all parameters. Parameters are configured as follows.

 parameter value Cm 20 Iext 80 V1 -1.2 V2 18 V3 12 V4 17.4 VK -80 VL -60 Vca 120 gCa 4 gK 8 gl 2 phi 0.0666666667

Next, click the Start button to start simulations. Can you see that the orbit of a solution starting from the origin converge to a closed curve (limit cycle) ?

Next, let us configure the Poincare section to generate the Poincare map. First, check the check box the Poincare in the Plot field.~

Next, start up the configuration window from menu bar: Setting->Poincare.

In this time, the Poincare section is configured in Fixed value mode. Depending on the properties of the problem, Equilibrium mode may be more appropriate. For details on the configurations of the Poincare section, see PP manual: How to configure the Poincare section.

Here, we fixed the position of the Poincare section at x=-10. Of course, as long as the Poincare section crosses with the limit cycle, configurations such as x=0 or x=20 have no problem.*10 Furthermore, the direction of intersection of the Poincare section and limit cycle is set in the negative direction.*11

After the configuration, points on the Poincare section are shown with red points on the limit cycle.

Parameters can be changed in real time. let us change the parameter Iext. By increasing the parameter Iext, a saddle-node bifurcation of the limit cycle occurs. Here, after the limit cycle disappears, a stable equilibrium point can be observed.

Next, we resume simulations after changing Iext to 100. Then, check the check box states flow in the Plot field. In this way, we can check the current values of the state variables in the list box during numerical integration. In this example, we can see to converge to a stable limit cycle, and the coordinates of the point on the Poincare section converges to one point since the same number repeatedly appears in the list box.

Here, by clicking the Clear button, the orbit during the transient response shown in the drawing window can be erased, and the steady state can be observed. Since the behavior of the system converges to a stable limit cycle, the orbit of the stable limit cycle and the point corresponding to the Poincare map point are shown on the drawing window. Uncheck the check box Orbit, and stop drawing the orbit of the limit cycle. Then only the point of the Poincare map can be shown on the drawing window.

Next, let us investigate bifurcations of this limit cycle. The eigenvalues of the variational equation for the limit cycle are evaluated when the parameters are changed. For this purpose, the FIX tool can be used. We pass the initial values to use FIX from PP. The important thing is that the values of the state variables and parameters at the steady state must be passed to FIX.

To confirm whether the system reached a steady state, either check the check box states flow switch and confirm that the values of the state variables no longer change as described above, or click the Status button a few times and confirm that the information displayed in the list box does not change.

After confirming that there are no more changes, select Export current status from Tools in the menu bar. With this procedure, the values of state variables, the parameter values, and other configuration values at the steady state are preserved and can be passed to the next program (see section on FIX).

## FIX †

Here, the objective is to track the accurate location of a limit cycle when the parameters are changed and also to monitor the eigenvalues during tracking.

Users must adjust the configurations to run FIX. For simplicity, here let us run FIX as is. Normally, the FIX program is run after the values saved by the PP program (using Export Current status) is imported as the initial values for Newton’s method.*12

First, let us start up the FIX program from the PP control panel. Select Program -> FIX.

When initially starting up the FIX, default values are entered in each field.

Information on the limit cycle obtained with PP is imported as the initial configuration value. Select Tools ->Import initial point.

After the selection, the information obtained with PP such as the parameter values, the return time of the limit cycle, and the coordinates of the point of the Poincare map are reflected.

By using PP, we saw that the dynamics of the system changes by increasing the parameter Iext. Therefore, select the parameter to be changed as Iext and configure the step size to increase the parameter.

Click the Start button to start calculations. A new window will open when the calculations start. This window is to monitor the location of the eigenvalues on a Gauss plane. When analyzing bifurcation of a limit cycle, we assume that the limit cycle crosses the Poincare section. Therefore, the effects of configuring the Poincare section appears as the fact that there is always one eigenvalue that is always 1.

The saddle-node bifurcation of a limit cycle occurs when one of the eigenvalues becomes "unity". We can observe that as the parameter changes, the location of the eigenvalues change, and one eigenvalue on the window approaches "unity".

Parameters are continuously changed, and when a bifurcation point is detected, the program stops.*13 In this example, the saddle-node bifurcation is detected because one of the eigenvalues approached 1 and Newton’s method did not converge.

The values of the parameters and the state variables when the program stopped can be used as the initial point for BF, the program to track bifurcation curves. The information when the bifurcation point is detected are exported for BF. When Export BF point is selected here, the status, parameter values, and other necessary configuration values are preserved. The preserved values are passed to the next program (see section on BF).

## BF †

The BF tool is used to obtain a bifurcation curve by changing system parameters.

The user must adjust the configuration to run BF. For simplicity, here let us run BF as is. Normally, the BF program is run after the values of the state variables and parameters at the bifurcation point detected by the FIX program are imported as the initial values for Newton's method.*14~

First, let us run the BF program from the FIX main panel. Select Program -> BF.

When starting up BF at the first time, default values are entered in each field.

Various information on the bifurcation point obtained with FIX is imported as the initial value. Select Tools ->Import BF point from main panel.

After the selection, various information such as the type of bifurcation and the period of the periodic point are input automatically into each field. To tracke the bifurcation set on the (Iext,V3) plane, check the radio button Iext as the X parameter and V3 as the Y parameter. Furthermore, configure items such as step size to change the parameters and the value to stop the BF when reaching end parameter. To save the calculated results as a data file, input the output file name in Output.

After these configurations, click Start button to start tracking of the bifurcation curve. Then, the Eigenvalues window and Bifurcation diagram window will open as new windows. The Eigenvalues window is used to visualize the change in the root locus of the eigenvalues during calculation of the bifurcation curve. Furthermore, to monitor the change of the root locus, we can visually understand occurrence of a bifurcation point with co-dimension 2. The Bifurcation diagram window is a window to monitor how the saddle-node bifurcation is being tracked in real time. (b): Bifurcation diagram in the (Iext,V3)-plane. The calculation result can be monitored in real time.

References

1. C. Morris and H. Lecar, Voltage oscillations in the barnacle giant muscle fiber, Biophys. J. 35 pp.193-213, 1981.
2. J. Rinzel and G.B. Ermentrout, Analysis of neuronal excitability and oscillations, in: C. Koch and I. Segev, 2nd eds., ''Methods in neuronal modeling: From ions to networks'', MIT press, London, pp.251-292, 1999.

*1 Project name can be changed freely. The user should choose a name that is easy to identify.
*2 There is a configuration file for SE as ML.sed in the sample directory. To import this file, it's able to apply several important information to each field in SE. You can create the programs to only click the Create button. Thus, you can skip various configurations for SE. Please refer to see How to import the configuration file for SE.
*3 This configuration can be changed from the setting menu of the Main panel.
*4 There is a configuration file of PP as Eq_point1.ppd in the sample directory. This file is the configuration file saved the values of state variables and parameters to simulate the behavior related to an equilibrium point. To import this file, it's able to simulate the system behavior to only click the Start button. Please refer to see How to import a configuration files for PP. In addition, there is another sample of the configuration file of PP as Eq_point2.ppd in the sample directory. The configuration file differs to the above example. Please enjoy simulations to import their interesting configuration files.
*5 Another method is to import a file describing the values of the parameters and the state variables. There is a file named Eq_point1.fixd under the sample directory where configurations for FIX are saved. This file is a FIX settings file where parameters and initial conditions of the equilibrium point described below are saved. Numerical calculations can start after just clicking the Start Button after loading this file. For directions on how to load the file, see How to import configuration files for FIX. Another configuration file Eq_point2.fixd is also included. This is the initial configuration file for FIX for a stable equilibrium point that can be observed with different parameters than the above equilibrium point. Calculations with this file detects a Hopf bifurcation point. Load the settings file of interest to FIX, and confirm the results that are different from saddle-node bifurcation detection.
*6 Details of this operation can be controlled by using the option of the detection from Settings. For detail, see manual or User's Guides:About FIX.
*7 Another method is to import a file describing parameter and state variable values. There is a file named Eq_point1_SN.bfd under the sample directory where configurations for BF are saved. This file is a BF settings file where parameters and initial conditions to track the saddle-node bifurcation of the equilibrium point described below are saved. The bifurcation curve can be tracked after just clicking the start button after loading this file. For directions on how to load the file, see How to import a configuration files for BF. Another configuration file Eq_point2_Hopf.bfd is also included. The Hopf bifurcation occurs when a parameter is changed, although in this setting, the equilibrium point is different from the above one. This is a configuration file to track this Hopf bifurcation curve. If you are interested, please enjoy tracking bifurcation curves by loading the configuration curves to BF.
*8 This configuration can be changed from the setting menu of the Main panel.
*9 There is a configuration file of PP as file LC_point.ppd in the sample directory. This file is the configuration file saved the values of state variables and parameters to simulate the behavior related to a limit cycle. To import this file, it's able to simulate the system behavior to only click the Start button. Please refer to see How to import a configuration files for PP.
*10 By changing the parameters, a situation that the limit cycle does not cross the Poincare section may occur. In such a case, the Poincare section must be reconfigured.
*11 For details about the direction of the section, see How to change the direction when the limit cycle crosses the Poincare section.
*12 Another method is to import a file describing the values of the parameters and the state variables. There is a file named file LC_point.fixd under the sample directory where configurations for FIX are saved. This file is a FIX settings file where parameters and initial conditions of the limit cycle described below are saved. Numerical calculations can start after just clicking the Start Button after loading this file. For directions on how to load the file, see How to import configuration files for FIX.
*13 Details of this operation can be controlled by using the option of the detection from Settings. For detail, see manual or User's Guides:About FIX.
*14 Another method is to import a file describing parameter and state variable values. There is a file named file LC_point_SN.bfd under the sample directory where configurations for BF are saved. This file is a BF settings file where parameters and initial conditions to track the saddle-node bifurcation of the limit cycle described below are saved. The bifurcation curve can be tracked after just clicking the start button after loading this file. For directions on how to load the file, see How to import a configuration files for BF.

Attach file: eq_img01.gif 313 download

Last-modified: 2009-07-23 (Thu) 20:13:30 (3767d)