Lab 10: Phase resetting oscillators

The response of a dynamical system to brief perturbations underlines the con-cept of the local stability of a fixed-point and consequently that of the bifurcationdiagram. It is natural to wonder about the responses of limit cycle oscillations tosmall perturbations.

Figure 1: Schematic representation of the effect of a single perturbation on aperiodically spiking oscillator. See text for discussion.

1 BackgroundFigure 1 summarizes the key observations for the response of an oscillator to abrief stimulus. The dark vertical lines represent a marker event of the oscillation.Examples of a marker event include the generation of an action potential, or neuralspike (this lab), heel contact with the ground during walking, the onset of mitosisin periodically dividing cells, and so on.

Prior to the introduction of the stimulus, the oscillator produces one neuralspike per period, T . The perturbed inter-spike interval is T ′. We consider thesituation that the rhythm is re-established “quickly” after the perturbation. Thisassumption is equivalent to the observation that the effects of the perturbation dis-appear rapidly on a time scale short compared to the period of the oscillator. Theimportant point is that the re-established rhythm has the same amplitude and inter-spike interval as it did before it was perturbed. The lone exception occurs when

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the single pulses abolishes the rhythm (as we saw in Laboratory 6). Although therhythm is re-established following the perturbation, subsequent spikes producedby the oscillator occur at different times from what would have been observed inthe absence of the perturbation (compare solid and dashed lines in Figure 2. Inother words the effect of the brief perturbation is to simply phase shift the oscilla-tor.

Figure 2: Effect of a single 15nA, 5ms hyper-polarizing square wave pulse ona repetitively spiking motoneuron of the sea slug, Aplysia. A 3nA depolarizingcurrent was injected into the motoneuron to cause it to spike periodically. Thedashed lines show the unperturbed oscillator. Figure reproduced from [2] withpermission.

A single stimulus may lengthen or shorten the inter-spike interval. In Figure 1we show an example in which the perturbation lengthens the cycle, namely T ′ >T . The effects of the perturbation depend on the magnitude of the stimulus andthe time at which it is applied. It is convenient to describe the timing of theperturbation in terms of the phase during the cycle that it arrived, i.e.

φ =t

T. (1)

It should be noted that φ is always a number between 0 and 1 and it is calculatedusing the unperturbed inter-spike interval, T . Thus a more correct way to write(1) is to introduce the concept of the modulus, namely

φ =t

Tmod 1 .

Modulo 1 means the we consider only the remainder when we divide t/T by 1.Hence 1.6 mod 1 = 0.6.

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A plot of the phase shift of the oscillator as a function of the phase at whichthe perturbation was introduced is called the phase resetting curve. Phase resettingcurves are further sub-divided depending on how the phase shift is measured into1) the phase response curve (PRC) and 2) the phase transition curve (PTC).

• The phase response curve (PRC) a plot of the change is phase, ∆φ, as afunction of the phase, φ, at which the perturbation was delivered. As weshowed in lecture

∆φ = 1− T ′

T(2)

• The phase transition curve (PTC) is a plot of the new phase, φ′ versus theold phase, φ. As we showed in lecture

φ′ =

[1 + φ− T ′

T

]mod 1 , (3)

=

[φ− T ′

T

]mod 1 .

Figure 3: Schematic representation of an Arnold tongue diagram. The horizontaldashed line is related to the Devil’s staircase (see text for discussion).

A remarkable observation is that the effects of periodic stimulation of an oscil-lator can be summarized in the form of an Arnold tongue diagram, such as shownin Figure 3. In this diagram stable modes of N : M phase locking occur whereN is the number of cycles of the stimulus and M is the number of cycles of thespontaneous rhythm and where N,M are positive integers. In practice the low-order ratios are most often seen, e.g. 1 : 1, 1 : 2, 3 : 2, 2 : 3 and so on. Foreach N : M , the phase locking patterns are arranged in tongues. As the stimulus

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amplitude (also referred to as the coupling) becomes lower, the region in whicha given N : M phase locking mode is observed decreases. Thus for very lowstimulus amplitudes it can be difficult to maintain stable phase locking becauseof, for example, the effects of noise.

The purpose of this lab is to demonstrate how the PRC and PTC are measuredand the uses of these curves for interpreting the responses of biological oscillatorsto periodic stimulation in the form of an Arnold tongue diagram.

Browser use: The Scholarpedia page by Carmen Canavier on phase resetting

http://www.scholarpedia.org/article/Phase_response_curve

is excellent.

Housekeeping: Some of the laboratory exercises make use of some of theresults we obtained with the HH-equation in Laboratory 6. Thus it might be usefulfor organizational purposes to work in the HHeqns directory. Alternately youmight choose to combine all of the exercises into a new directory called phase.If you choose this option then move into HHeqns and copy hh.ode to phaseusing the command1

cp hh.ode ˜/phase/.

What does the . mean? What does the˜mean?

2 Exercise 1: Phase resetting curvesXPPAUT’s Poincare Max/Min option in XPPAUT can be used to determinethe phase resetting curves when the equation for the oscillator is known (see Sup-plementary Material at end of this lab). However, it is more relevant for thoseinterested in laboratory applications to construct the phase resetting curves pointby point. It is important to note that only three parameters can be measured: φ, T ,and T ′.

First we suggest that you do some house keeping. Move to directory HHeqnand make a second copy of hh.ode by typing the command

cp hh.ode hh_prc.ode

1The purpose of this strategy is to keep an unaltered version of the program so that we can useit at another time.

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Run hh_prc.ode by picking a value of curbias so that the periodic spikingsolution does not co-exist with a stable fixed-point (use the bifurcation diagramfor the HH neuron that you prepared in Lab 6 to determine a suitable value ofcurbias). Click on Initialconds then Last several times to ensure thatyou are on the limit cycle. Using your text editor, e.g. emacs, replace the initialvalues of V, n, h,m with the values that appear in the initial values window. Savehh_prc.ode, exit XPPAUT, and then re-start XPPAUT using the newly editedversion of hh_prc.ode. The advantage of using hh_prc.ode in this way isthat by clicking on Default in the initial conditions menu we ensure that eachtime we run the program we start at the same place on the limit cycle.

It is easier to use XPPAUT to determine the phase resetting curves by choos-ing an inter-spike interval and then varying the timing of the perturbation on thisinterval (see below). For example we took curbias=25 and used the intervalbetween the spike that occurs at 15.92 and 19.84, so that T ≈ 3.92. Pick an am-plitude for the perturbation that is large enough to produce a phase shift. Notethat we measured these times by left clicking on the mouse and reading the (x, y)-coordinates of the cursor below the graph. Use the cursor to check our estimateof T . An even simpler way is to measure the distance between consecutive spikesoff the screen using a ruler. For the purposes of this exercise it does not matter.

Note that T and the magnitude of the perturbation are constant. The timing ofthe perturbation is set in the parameter window. Thus by setting up the programas described above, the only number we need to measure using the mouse cursoris the time of the neural spike that occurs after the perturbation is delivered. Inother words we have reduced our work load.

Pick, for example, ≈ 20 values of the time to introduce the perturbation thatspan the interval [15.92, 19.84]: this means that we increase the time of the pertur-bation in 0.2s jumps. Use your text editor to create a data file (*.tsv) containingtwo columns of data: the first column is the time that the perturbation was deliv-ered and the second column is the length of the perturbed inter-spike interval.

Questions to answer:1. Write a Python program that uses the data you collected with the HH neuron

to calculate the PRC and the PTC.

2. What is the difference between the PRC and the PTC?

Comment: Although the PRC is widely used in the bio-mathematics literature,its use in the laboratory can be problematic. For example it is possible that events

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which occur after the perturbation is delivered are obscured by the perturbationsitself (namely, there can be a stimulus artifact). In these cases it is necessaryto measure the phase difference asymptotically after several periods of the au-tonomous cycle have elapsed. However, an ambiguity arises because phase dif-ferences of −0.05, 0.95, 1.95 are asymptotically the same (remember that phaseis defined modulo the period so that it is a number between 0 and 1). As we willsee in this laboratory an advantage of the PTC is that it can be directly used in acircle map to predict the responses to periodic stimulation (note that the PTC canbe calculated form the PRC) (see below). Thus it has been suggested that the PTCcurve should be used in routine laboratory work [2, 3].

Questions to answer:1. Neurons exhibit two types of phase resetting responses referred to, respec-

tively, as Type 1 and Type 22. Type 1 phase resetting means that phaseis only advanced by using brief depolarizing pulses. Type 2 phase reset-ting means that phase can be either advanced or delayed depending on thetiming of brief depolarizing pulses. What type of phase resetting does aHodgkin-Huxley neuron exhibit?

2. What is a simple mathematical function that approximates the PTC curveyou obtained for the HH neuron?

Comment: The differences in the phase resetting curves for neurons arises be-cause of the different mechanisms involved in spike initiation [1, 4]. Type 1 neu-rons are capable of summing inputs over a broad range of frequencies and henceare referred to as integrators. On the other hand, Type 2 neurons tend to resonateto a preferred frequency and hence are referred to as resonators. Type 1 and 2 neu-rons also display differences in their ability to synchronize with other neurons in aneural network. Type 1 neurons cannot synchronize to purely excitatory synapticinputs, but can synchronize if the inputs are purely inhibitory. In contrast, Type 2neurons can synchronize to both excitatory and inhibitory neurons.

Questions to answer:1. Is a HH neuron a resonator or an integrator?

2. Can a HH neuron synchronize in a neural network with purely excitatoryinputs (YES or NO)?

2Be careful not to confuse Type 1 and Type 2 neurons with Type I and Type II excitability.

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3. Can a HH neuron synchronize in a neural network with purely inhibitoryinputs (YES or NO)?

3 Exercise 2: Arnold tongue diagramsIn the laboratory it is easy to stimulate a neuron periodically. Thus at first sight itwould seem to be straightforward to construct the Arnold tongue diagram. How-ever, this approach is not the most reliable way to determine the Arnold tonguediagram. First, a stable phase locking pattern does not mean that all of the so-lutions within the Arnold tongue look the same (see below). For example, for1 : 1 phase locking solutions can quantitatively differ by the phase in the cyclethat the stimulus is delivered. In other words the Arnold tongue groups solutionsthat are qualitatively the same: 1 : 1 phase locked solutions all have the prop-erty that there is 1 stimulus delivered per 1 cycle of the oscillator. Second, thephase locking properties of oscillators to periodic stimuli is very much a work inprogress. In fact, it is often the case that stable phase locking patterns can be ob-served experimentally for stimulus amplitudes that are larger than those for whicha mathematical theory exists. Moreover, the phenomena that occur between thelow-order Arnold tongues is very complex and only partially explained and ex-plored. For example, it is known that chaotic solutions can exist in these regions.

A reliable way to determine the Arnold tongue diagram for sufficiently smallstimulation amplitudes is to use the PTC. There are three steps in this procedure.

1. Construct circle map: The effect of a stimulus is to generate a new phase;however, its value remains a number between 0 and 1. Thus the dynamicscan be described by the circle map

φn+1 = f(φn) mod 1 ,

where f is a function to be determined3. Specifically the circle map takes

3We have defined the phase as a value between 0 and 1. Over the last century an elegantmathematical theory was developed by Poincare and Arnold to analyze circle maps for situationsin which the amplitude is not too large, and then extended to applications by the work of Winfree,Canavier, Glass, Ermentrout, and Strogatz. The mathematical definition of “not too large” is thatthere must be a 1 : 1 correspondence between φ and φ′. Is this true for the phase resetting curveswe have studied in the lab today? Maps, f(φn, which exhibit this 1 : 1 property are said tobe invertible. Note that if the goal is to compare prediction to observation then an experimentalchallenge is to ensure that the amplitude and duration of the stimulus is in an appropriate range.

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the formφn+1 = g(φ, a) +

tsT

mod 1 , (4)

where, and this is the IMPORTANT POINT, g(φ, a) is the PTC curve, ais the amplitude of the stimulus used to measure the PTC, and ts is thephase at which the stimulus is delivered. Thus by measuring the PTC wemay be able to determine the responses of the oscillator over a range ofappropriately chosen stimulus amplitudes.

2. Calculate the rotation, or winding, number If we have an invertible mapthen for all N : M , where N,M are relatively prime, positive integers (i.e.no common divisor except 1) the solutions of the circle map are periodic.Moreover, for each choice of the initial conditions the solution of the circlemap asymptotically approach a stable N : M phase locking pattern. TheN : M values correspond to the rotation, or winding number, ρ, and can becalculated as

ρ = limt→∞

∑nn=1 φn

n. (5)

In other words, all observed solutions within a given Arnold tongue aresimilar because they have the same ρ. An important practical point is that(5) refers to the cumulated phase and hence we do not take the modulus inevaluating this expression.

3. Identify the N : M pattern using a cobweb diagram If ρ is known thenhow do we figure out the phase locking mode? The answer is make a cob-web diagram using the program cobweb.py located on the website. Acobweb diagram is way to graphically integrate an equation of the form

xn+1 = f(xn) . (6)

For illustration consider the quadratic map,

xn+1 = rxn(1− xn) ,

where r is a constant. Choose r = 3.25 which corresponds to a period-2cycle. Figure 4 illustrates the cobwebbing procedure. Pick an initial valueof x0 on the xn axis. The value of x2 is found by drawing a vertical (green)line from x0 on the xn-axis to the thick line (i.e. 3.25x0(1 − x0), then ahorizontal (red) line to x0 = x1, and then a vertical (green) line (up or downas the case may be) to the line xn = xn+1 , which yields x2, then a horizontal(red) to intersect with xn = xn+1 to obtain x3, and so on.

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Figure 4: Cobweb diagram approach for graphically iterating the quadratic map(thick line) for r = 3.25. The thin line shows xn = xn+1. The alternating vertical(green) and horizontal (red) lines illustrate the iterative process which is describedin the text.

4 Exercise 3: Phase locking modes for standard cir-cle map

We illustrate this procedure by considering the standard circle map

φn+1 = [φn + b+ a sin(2πφn)] mod 1 , (7)

where b is N/M . This map is invertible provided that a < 0.16. Comparing (7)with (4) we have b = ts/T and g(φ, a) = φ+ a sin(2πφn).

Questions to be answered:

1. Use the program devils.py to determine the values of ρ as a function ofb. Setting a = 0.15 results in the Devil’s staircase shown in Figure 5.

2. What is the relationship between Figure 5 and the Arnold tongue diagramshown in Figure 3? The Devil’s staircase is the Arnold’s tongue diagram

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Figure 5: The devil’s staircase, namely a plot of the rotation number, ρ, as afunction of the parameter b.

viewed along one horizontal plane (dashed line) which corresponds to ex-periments done for one value of the stimulus amplitude. The “flat steps” cor-respond to the cross-section of the low-order Arnold tongues for the givenamplitude. By inspecting the figure we see that the most prominent stepsoccur for ρ equal to 0.25, 0.33, 0.5, 0.67, 1.0. If you progressively decreasea you can, in principle, generate the Arnold tongue diagram one slice ata time. Counter-intuitively this is the most reliable way for generating anArnold tongue diagram since trying to accomplish this directly from thetime series is difficult and often can lead to measurement errors related todetermining whether or not the solution has settled onto its asymptotic solu-tion. Run this program for different values of a and use these observationsto sketch the Arnold tongue diagram.

3. Modify the cobweb.py to identify the phase locking pattern that corre-spond to the most prominent steps in the Devil’s staircase (Figure 5). Forexample, for ρ = 2/3, we obtain the 2 : 3 phase locking pattern shown inFigure 6. Why is this a 2 : 3 pattern. The value of m is equal to the numberof steps to go around the closed orbit in Figure 6 for one complete cycle.This is the number of vertical dashed lines and hence we see that m = 3.The value of n is the number of times per complete cycle that the closedorbit hits the branch of the map on the right. We see that n = 2 and hencewe have a 2 : 3 phase locking pattern. Practice identifying phased locked

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Figure 6: Cobweb diagram for (7) for a choice of b corresponds to ρ = 23

withK = 0.15. This solution correspond to a 2 : 3 solution: there are two contactswith the left branch of the circle map for every three contacts of both branches tocompete the cycle.

patterns by trying different values of the parameters. Note that within agiven Arnold tongue the solutions can be quite complex; however, they areall qualitatively described by the same n : m.

4. Notice that in Figure 6 we choose a very small value of a. What happensto the cobweb diagram as a is increased (remember that a ≤ 0.16)? Whathappens to the N : M phase locking pattern?)

5 Exercise 4: Phase locking modes for the periodi-cally spiking HH neuron

In Exercise 1 we measured the PTC for a periodically spiking neuron. Fit theobserved PTC with a continuous mathematical expression, such as a polynomial.For example we choose PTC as

g(φ) = [1 + φ+ k ∗ sin(2πφ+ π)] mod 1 .

In other words we approximated the PTC of the HH neuron as a standard circlemap phase shifted by π. What happens to the Devil’s staircase? Of course ourchoice of function to fit PTC is not very accurate. Can you do better?

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Questions to be answered:

1. Modify devils.py and cobweb.py using your expression that describesthe PTC (remember to first make a copy of this program, give it a new name,and work with the newly created program). The Devil’s staircase can be ob-tained by varying ts over the range [0, T ]. For a value of ts that yields valueof ρ corresponding to a low-order mode locking solution, use your modifiedcobweb.py to identify N : M .

Deliverables: Use Lab10_template.tex to prepare the lab assignment.

6 Supplementary material: Phase resetting curvesusing XPPAUT

XPPAUT contains a very useful function in nUmerics called Poincare Max/Min.This function is based on the concept of a Poincare section that we discussed inlecture. Here we use this function to determine the period of an oscillator and thenits phase resetting properties. The basic idea is that we choose the Poincare sectionto correspond when the variable crosses a threshold value, say 0, in the positivedirection. In other words, in the absence of noise, the time between successivecrossings of the threshold corresponds to the period.

This procedure works best for oscillations characterized by one maxima perperiod. Here we illustrate the method by calculating the phase resetting responsesof the van der Pol oscillator

dx

dt= y ,

dy

dt= −x+ y ∗ (1− x2) .

The program that you will need from the website is vdpphase.ode. Downloadthis program and open it up with your text editor (e.g. emacs).

The first task is to figure out how to make a pulse using XPPAUT. This canbe done by making use of the Heaviside function, heav(arg). This function isequal to 0 if arg1 < 0 and is equal to 1 otherwise. Draw the shape of the pulsethat would be drawn using the command

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pulse(t)=heav(t)*heav(pw-t)

The program vdpphase.ode introduces the pulse using the command

dy/dt = -x + y*(1-xˆ2)+a*pulse(t-tp)

Draw a*pulse(t-delta).The first step is to determine the period, T , of the van der Pol oscillation and

the first step of the procedure is to determine an appropriate initial condition. Inorder to understand how this is done we need to remember the properties of thephase plane and, in particular, that trajectories are traveled in the clockwise direc-tion. Thus the largest value of x is the value to the extreme right of the limit cycle(x is plotted on the x-axis). If we choose this value at the initial condition, thenone full period corresponds to the time taken when x next achieves its maximumvalue.

Set a = 0 and run the program by clicking on Initialcond, then Lastseveral times to allow transients to die out (see Laboratory 6). Click on Data,then on Find. Type in x and then 100. The number 100 is arbitrary. The onlycondition is that it should be larger that you expect that the largest amplitude willbe since it ensures that we find the maximum. This command makes XPPAUTfind the largest value of x closest to 100. Click on Get to load this value an aninitial condition (you can also type this value into vdpprc.ode so that you canby-pass this procedure the next time you use this program).

Now click on nUmerics and then click Poincare map Max/Min toopen up the dialog box. Fill in the parameters as shown (Figure 7). These pa-rameters tell the program (from top to bottom) that the variable to monitor is x,that the threshold (Poincare section is placed at x = 0, that we want to measurecrossings in the upward (1) direction (the downward direction would be −1, andfinally we stop once a crossing has been detected. Click OK.

Now click on Transient and choose 4: this allows the integration to pro-ceed for awhile without looking at and storing values and hence makes the pro-gram faster to run. The choice of 4 is arbitrary; however, it must be a positive num-ber smaller than the period. Exit the nUmerics window by pressing Esc. Nowclick on Initialconds and then Go. When the program has finished click onData, then on Home. You will see that Time has a value of 6.6647 and this valueis that of the period. As above, you can type this value into vdpphase.ode sothat you can by-pass this procedure the next time you use this program.

We are finally ready to calculate phase resetting curves. Note that vdpphase.odeuses an auxiliary function to calculate the phase response curve (PRC). You should

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Figure 7: Filling out the PoincareMax/Win menu.

also define an auxiliary function to calculate the phase transition curve (PTC). Aproblem with XPPAUT is that it does not have a command to calculate the modu-lus. What strategies can you use to overcome this problem? Change the amplitudeof the perturbation to a = 3. Click on Initialconds, then Range to open upthe dialog box shown in Figure 8. Fill it out as shown and then click OK. UseViewaxes to put phase on the x-axis and prc on the y-axis.

Often we wish to compare the PRC’s obtained with different choices of a andpw. Since the memory resources of the computer are limited it is not a good ideato simply plot trajectories without pressing Erase. A better way is to “freeze”the image by clicking on Graphic stuff and then Freeze On Freeze.In this way, the phase resetting curve is frozen onto the graph on the computerscreen, but the data is no longer stored in memory. Thus we free up computerresources for the next calculation.

References[1] B. Ermentrout. Type I membranes, phase resetting curves and synchrony.

Neural Comp., 8:979–1001, 1996.

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Figure 8: Filling out the Range menu.

[2] J. Foss and J. Milton. Multistability in recurrent neural loops arising fromdelay. J. Neurophysiol., 84:975–985, 2000.

[3] L. Glass and M. C. Mackey. From Clocks to Chaos: The rhythms of life.Princeton University Press, Princeton, New Jersey, 1988.

[4] A. L. Hodgkin. The local electric changes associated wirh repetetive actionin a non–modulated axon. J. Physiol., 107:165–181, 1948.

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