Lab 6: Bifurcation diagram: stopping spiking neurons with a single pulse

Transcription

1 Lab 6: Bifurcation diagram: stopping spiking neurons with a single pulse The qualitative behaviors of a dynamical system can change when parameters are changed. For example, a stable fixed-point can become unstable, or vice versa. Moreover, for some dynamical systems it is possible that other types of dynamical behaviors can arise when a fixed-point become unstable, such as a limit cycle oscillation. If the mathematical equations are known then in principle it is possible to summarize the parameter dependence of the dynamics in the form of a bifurcation diagram. A bifurcation diagram provides a road map that can be used in the laboratory to directly link the predictions of a mathematical model to experimental observation. Are the predicted dynamics observed experimentally for the predicted parameter values? Do the dynamics change appropriately as experimentally accessible parameters are changed? Even if a mathematical model has not been developed, a demonstrated qualitative change in dynamics as a parameter is changed can spark the interest of bio-mathematicians to work on the problem. 1 Background In practice it is a challenging task to produce a bifurcation diagram both in mathematical models and in the laboratory. A fundamental problem is that the number of parameters in biological systems can be very large. At present it is not known how to systematically explore the dynamics of a mathematical model using numerical methods when the number of parameters is large. A very important computational package for generating bifurcation diagrams in mathematical models in which the number of parameters is small is called AUTO One of the reasons why bio-mathematicians use XPPAUT is that it affords an easy access to AUTO [1]. AUTO uses a mathematical technique referred to as 1

2 continuation to follow a particular solution (e.g. a fixed-point or a limit cycle) as a parameter changes. For each choice of the parameters the stability of a particular solution branch is determined using a linear stability analysis. As we will see, AUTO provides a number of tools to characterize the bifurcation of fixed-points and limit cycles. AUTO can be tricky to use and can fail in a dramatic fashion for certain problems. However, even though AUTO is over 25 years old it still remains a major tool in the arsenal of an applied mathematician. The experimental determination of a bifurcation diagram can be even more problematic since for a biological system the identify of all of the parameters is not generally known and it may only be possible to alter some parameters, and even then, the range of possible variation may be small. However, as we have seen in lectures, sweeping a parameter, an experimentally attractive idea, can misidentify the location of the point where changes in stability occur. Moreover, as the edges of stability are approached, the identification of the nature of the dynamics becomes difficult due to slowing down phenomena. Nonetheless a number of experimental paradigms have been developed in which it has been possible to at least partially characterize the bifurcation diagram. In this laboratory we illustrate the use of AUTO to characterize the onset of periodic spiking in a neuron. This type of bifurcation is characterized by the replacement of a stable fixed-point with a stable limit cycle and is referred to as a Hopf bifurcation. A surprising prediction of the mathematical analysis is that it is possible to annihilate periodic spiking in a neuron by the application of a single, carefully-timed pulse. We demonstrate this phenomenon using a numerical experiment. This phenomenon was first demonstrated experimentally by R. Guttman and her colleagues using the squid giant axon [3]. Browser use: A user s guide for running AUTO under XPPAUT can readily be located on the Internet bard/xpp/help/xppauto.html In this laboratory we illustrate AUTO using only very simple examples. Readers who wish to extend the use of AUTO to their own research are recommended to read the chapter in Ermentrout s guide to XPPAUT [1]. XPPAUT can be run in Python ( however, we recommend first learning to run AUTO in XPPAUT before taking it on in Python. The ode version of the Hodgkin-Huxley (HH) equation, hh.ode, is the one written by Michael Guevara (McGill University). You can either download this program from our website or from the Internet by typing hh.ode Guevara McGill 2

3 in a web browser. The exercises involving the HH equation described here are modified from the computer exercises developed by Michael Guevara to explore the dynamical properties of the HH and Fitzhugh-Nagumo equations [2]. Students interested in the dynamical behaviors of neurons and excitable systems are recommended to work through the complete set of these exercises. Housekeeping: It is useful to place all of the files related to the HH equation and its analysis in a directory called, for example, HHeqn. 2 Exercise 1: Making one-parameter bifurcation diagrams We illustrate the procedure for obtaining a bifurcation diagram for the saddle node bifurcation discussed in class. The generic equation for this bifurcation is dx dt = µ + x2, (1) where µ is the bifurcation parameter. The steps to determine the bifurcation diagram for (1) are as follows: 1. Write an.ode program to integrate (1). 2. It is very important to ensure that the solution has settled onto either a fixedpoint or a limit cycle before using AUTO to continue the solution as a parameter changes. In the case of (1) we know that a fixed-point solution (x < 0) exists for µ < 0. Pick the initial condition x = 1 and run XPPAUT. The most common cause of failure using AUTO is to use this program when the solution is not already on a branch. With this in mind we extend the integration by clicking on Initialconds Last. This XPPAUT command extends the integration by an amount defined by TO- TAL, from the step at which the previous integration finished. Depending on the differential equation, the choice of initial condition, and the parameter values it may be necessary to click on Initialconds Last several times. 3. The AUTO window (Figure 1) is activated by clicking on Auto which is located under File. The most important functions for our purposes are 3

4 Figure 1: Clicking on File and then AUTO opens up the AUTO window. Parameter, Axes, Numerics, Run and Grab. A complete description of all of the commands in AUTO is provided in the user s guide cited above. The circle in the lower left of Figure 1 summarizes the number of stable and unstable eigenvalues for a given choice of the bifurcation parameter(s): the number of crosses inside the circle corresponds to the number of stable eigenvalues (i.e. eigenvalues with negative real part) and the number outside the circle corresponds to the number of unstable eigenvalues (the number of eigenvalues with positive real part). How many crosses do you see for (1) and are they stable or unstable? How many crosses did you expect to see and why? 4. Click on Parameters. It is possible to declare up to 5 bifurcation parameters in this box. By default, XPPAUT chooses the first five parameters in the *.ode program. In our case there should be just one bifurcation parameter declared. Is this correct? 5. Click on Axes and choose HiLo. Most of this box is already be filled out correctly. We see that the variable X will be plotted on the Y-axis and we are able to choose the range to vary the bifurcation parameter. Note that X min, X max, Y min, Y max refer to the dimensions of the bifurcation diagram and not the variables and thus X min, X max refers to the range over which µ is varied. We suggest that you choose X min = 3 and X max = 1. The values 4

5 Y min, Y max refer to the maximum and minimum values of a periodic orbit. Since (1) does not possess a periodic orbit these values are irrelevant for our purposes (see Exercise 2). Once you have completed the menu press OK. 6. Click on Numerics. For our purposes we can accept all of the default values except those for Par Min and Par Max. Choose these values so that µ is varied from Par Min = -3 to Par Max =1, then push OK. Figure 2: A partial bifurcation diagram computed for (1). We have clicked on Grab and thus we see three special points indicated by crosses. Note that there are actually four crosses but one is not shown. See text for discussion. 7. Click on Run Steady state. A thick black line will appear that extends over the range that we varied µ (Figure 2. The fact that the line is thick means that the fixed-point is stable (a thin black line indicates an unstable fixed-point). If you do not see a line then the most likely causes is that you were not sufficiently close to the stable fixed-point solution when you started. Go back to Step Click on Grab. You will see a cross appear on the thick black line. You can use the right and left arrow keys to move back and forth along this line. As you move along this line in this way what happens to the eigenvalues (see circle in lower left hand corner). Also if you look in the window below the bifurcation diagram you can see more information pertaining to the current 5

6 location of the movable cross in the diagram, such as the value of µ and, in the case of a limit circle, its period. Note that there are also crosses with numbers attached which do not move as you use the arrow keys. The points are special points since it is possible to continue the solution from them. Using the Tab key you can move between the special points. When you are on one of these special points, the bottom window provides information the point type. In our case you will see EP which means end point. If you see MX then this means you screwed up and it is back to step 2 that you go. Figure 3: The complete bifurcation diagram for (1) determined using AUTO. 9. In order to complete the bifurcation diagram it is necessary to continue to the solution from one of the numbered special points. Use the Tab to position the cross on the special point labeled 1. Press Enter. Click on Numerics and change the sign of Ds. Note that when we change the sign of Ds we reverse the starting direction for AUTO. Now click on Run and you should see the bifurcation diagram that we calculated in class (Figure 3). Questions to be answered: 1. Use XPPAUT to determine the bifurcation diagram for the transcritical bi- 6

7 furcation whose generic equation is dx dt = µx x2. 2. Use XPPAUT to determine the bifurcation diagram for the supercritical bifurcation whose generic equation is dx dt = µx x3. 3. Use XPPAUT to determine the bifurcation diagram for the subcritical bifurcation whose generic equation is dx dt = µx + x3. 3 Exercise 2: Bifurcation diagram: Hodgkin-Huxley neuron The determination of the bifurcation diagram for the HH equation by bio-mathematicians made it possible for neuroscientists to test the prediction of this equation in the laboratory using the squid giant axon (for a review see [2]). In this exercise we will AUTO to characterize the onset of regular spiking of the squid giant axon when a sufficient current is injected. The main difference in the use of AUTO for this exercise is that we need to account for the appearance of a limit cycle. Here are the steps we used to make this bifurcation diagram. 1. Download the program hh.ode from the class website. Note that this equation has more parameters that can fit in the shown parameter box shown in XPPAUT. You can maneuver up and down the parameter list by using the,,, buttons located on the parameter box. 2. Run XPPAUT to integrate this equation when the parameter curbias=0. Use the Initialconds Last button several times to ensure that the HH equation has settled onto the initial conditions of v, m, h, n (we obtained, respectively, , , , ). 3. After starting AUTO set the following parameters: 7

8 In the Parameter window set the first parameter to be curbias (the menu will show blockna which is the first parameter declared in hh.ode. In the Axes window click on Hi-Lo. The y-axis should be V and the x-axis (called MainParm) should be curbias. Set X min = 0 and X max = 200 so that curbias ranges from 0 to 200. Set Y min = 80 and Y max = 20 so that the amplitude of the limit cycle oscillation ranges from 80 to 20. In the Numerics menu set Par min = 0 and Par max = 200 (this sets the range over which the bifurcation parameter curbias will be varied. Set Nmax=500 to limit the number of points that will be computed along a given branch of the bifurcation diagram to 500. In addition set Npr=500 and Norm Max =150. Click on Run and then Steady state. You will see a thick line with 4 special points. Click on Grab and use the Tab to move between them. From the bottom screen we see that special points 1 and 4 are end points (EP) and special points 2 and 3 are labeled as Hopf bifurcation points (HB). Note that the line between special points 2 and 3 is thin meaning that the fixed-point that exists for this range of curbias values is unstable. Confirm this by looking at the circle diagrams in the lower left hand corner of the AUTO screen. We can complete the bifurcation diagram by either starting at special point 2 and increasing curbias, or by starting at special point 3 and decreasing curbias. Let s start at special point 2. Use the Tab key to place the cross at this point and then press Enter. Note that the Run menu changes. Click on Periodic and the bifurcation diagram unfolds before your eyes (Figure 4)! XPPAUT (and mathematicians) typically use lines to indicate the stability of fixed-points (remember that thick lines correspond to stable fixed-points and thin lines to unstable ones). Limit cycles are indicate by circles placed which are positioned to indicate the maximum and minimum values of the oscillation: filled circles correspond to stable limit cycle oscillations and open circles to unstable limit cycle oscillations. 8

9 Figure 4: Bifurcation diagram for an HH-neuron determined using AUTO. See text for discussion. Questions to be answered: A bifurcation in which a stable fixed-point is replaced by a stable limit cycle is called a Hopf bifurcation. Specifically the bifurcation that occurs at special point 2 is a subcritical Hopf bifurcation and that which occurs at special point 3 is called a supercritical Hopf bifurcation (more specifically if we consider curbias increasing we have a reverse supercritical Hopf bifurcation). We will discuss these bifurcations in more detail later in the lectures. However, we can interpret the bifurcation diagram to anticipate the properties of these bifurcations. 1. First consider the subcritical Hopf bifurcation that occurs at special point 2. What happens to the stability of the fixed-point at the bifurcation point? Is the fixed-point destroyed at this Hopf bifurcation? Use Zoom to examine the bifurcation diagram close to special point 2. We see that it is possible for a stable fixed-point and a stable cycle to co-exist. This bistability is a characteristic of a subcritical Hopf bifurcation? What does this mean (see also Exercise 3)? Suppose we start with a value of curbias much lower that special point 2 and slowly increase it. How do you expect the amplitude of 9

10 Figure 5: Changes in membrane potential for a periodically spiking HH-neuron. the oscillation to grow: suddenly or gradually? The abrupt onset of a high amplitude oscillation is a characteristic of a subcritical Hopf bifurcation. Now choose a value of curbias on the limit cycle and slowly decrease it. At some point the oscillation will disappear and we will be left with a stable fixed-point. Is the decreasing value of curbias at which the limit cycle disappears the same value of increasing curbias which caused the limit cycle? This observation illustrates the phenomenon of hysteresis. 2. Now consider the supercritical Hopf bifurcation associated with special point 3. What happens to the stability of the fixed-point at the bifurcation point? Is the fixed-point destroyed at this Hopf bifurcation? How does the amplitude of the oscillation grow as the value of curbias is decreased from special point 3: gradually or abruptly? Is this bifurcation associated with bistability? Is this bifurcation associated with hysteresis? 10

11 Figure 6: Stopping a periodically spiking HH-neuron with a single pulse. What stimulation parameters do you think we used? Can you accomplish the same thing with a smaller, briefer and better time pulse? See text for discusison. 4 Exercise 3: Stopping spiking neurons with a single pulse The special point 3 occurs when curbias equals 150µA/cm 2. This is a very high current input for a neuron and likely is not biologically relevant. Thus biological interest centers on the behaviors associated by the subcritical Hopf bifurcation (special point 2). The practical implication of the presence of a subcritical Hopf bifurcation is because of its association with bistability. In particular, a non-spiking state of the neuron co-exists with a periodically spiking state. Suppose we choose curbias close, but smaller, than special point 2. It should be possible to switch between the two stable points by changing the initial conditions. How do we change initial conditions? A simple way is to use a carefully timed electrical pulse. The program hh.ode contains parameters that make it possible to introduce an electrical pulse of width (duration) and height (amplitude) at a specified time (tstart). Figure 5 shows the changes in membrane potential for a period- 11

12 ically spiking neuron. Guess at which point in the spike-to-spike interval where a hyper-polarizing pulse will stop the neuron from periodically spiking (Figure 6). At which point would a depolarizing pulse do the trick? Once the neuron is quiescent what kind of pulse is needed to start it spiking again. To test your predictions set curbias = 13. Set the pulse height to be 9 and the width to be 1.5. Determine a time at which a hyper-polarizing pulse stops the spiking (Figure 6). Can you accomplish this task using an even briefer and and smaller perturbation: sometimes an important goal is to determine the timing for which the smallest and briefest pulse accomplishes the above task. Now try the same exercise using a depolarizing pulse accomplishes the same tasks. DELIVERABLES: Use Lab6_template.tex to prepare the lab assignment. References [1] B. Ermentrout. Simulating, analyzing, and animating dynamical systems: A guide for XPPAUT for researchers and students. SIAM, Philadelphia, [2] M. R. Guevara. Dynamics of excitable cells. In A. Beuter, L. Glass, M. C. Mackey, and M. S. Titcombe, editors, Nonlinear dynamics in physiology and medicine, pages , New York, Springer. [3] R. Guttman, S. Lewis, and J. Rinzel. Control of repetitive firing in squid axon membrane as a model for a neuron oscillator. J. Physiol. (London), 305: ,