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Tunable Oscillations in the Purkinje Neuron

Ze‟ev R. Abrams1,2,3†, Ajithkumar Warrier2†, Yuan Wang2,3, Dirk Trauner4 Xiang Zhang1,2,3*

1. Applied Science & Technology, University of California, Berkeley,

2. NSF Nanoscale Science and Engineering Center, 3112 Etcheverry Hall, University of

California, Berkeley.

3. Material Sciences Division, Lawrence Berkeley National Laboratory, Berkeley

4. Department of Chemistry and Biochemistry, Ludwig-Maximilians-Universität München

(LMU), Butenandtstr. 5–13, 81377 Munich, Germany.

† These authors contributed equally to this work.

*- corresponding author: Xiang@berkeley.edu

Corresponding author:

Correspondence to: xiang@berkeley.edu , Professor Xiang Zhang, NSF Nanoscale Science and

Engineering Center, 3112 Etcheverry Hall, University of California, Berkeley 94720-1740,

Phone: (510) 642-0390, Fax: (510)-643-2311

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ABSTRACT

In this paper, we study the dynamics of slow oscillations in Purkinje neurons in vitro, and derive

a strong association with a forced parametric oscillator model. We demonstrate the precise

rhythmicity of the oscillations in Purkinje neurons, as well as a dynamic tunability of this

oscillation using a photo-switchable compound. We show that this slow oscillation can be

induced in every Purkinje neuron, having periods ranging between 10-25 seconds. Starting from

a Hodgkin-Huxley model, we also demonstrate that this oscillation can be externally modulated,

and that the neurons will return to their intrinsic firing frequency after the forced oscillation is

concluded. These results signify an additional functional role of tunable oscillations within the

cerebellum, as well as a dynamic control of a time scale in the brain in the range of seconds.

I. INTRODUCTION

The Purkinje Neuron (PN) is the largest neuron in the cerebellum, with over 100,000 inputs and

a single output axon [ 1, 2]. Due to its geometry and orientation in the cerebellum, it has been

cited as a possible integrator for the motor control system of the brain [ 2], with many basic

neuroscience and artificial intelligence theories based on its complex neuronal network [ 3, 4].

While most studies of the PN focus on biological sources of memory (plasticity) [ 5, 6], a number

of studies also describe the functionality of the cerebellum in terms of independent oscillators [ 6-

8].

We have previously reinforced this set of theories with an experimental study demonstrating the

intrinsic firing characteristics of the PN [ 9,10]. We identified three frequency bands inherent to

the PN, which we denoted as the Sodium (Na+; >30 Hz), Calcium (Ca2+; 1-10 Hz) and Switching

bands (>1 Hz). This set of frequency bands is distinct from other regions of the brain [ 11- 14],

with the „Switching‟ frequency described and measured for the first time [ 9]. This Switching

frequency operates at lower frequencies than those typically associated with memory and other

cerebellar processes [ 14], however there have been recent in vivo experiments that have

demonstrated similar slow oscillations between 0.039-0.078 Hz [ 15]. Slow oscillations of firing

and quiescence can be defined as an astable mode, as opposed to the known bistable mode in the

PN [ 16- 18], and we have recently described a new theory of PN function using the terminology

of electronic oscillator systems exhibiting astable, bistable and monostable modalities [ 10].

In this paper, we first show that every PN can exhibit this slow form of astable oscillation when

activated using pharmacological compounds in vitro. These slow oscillations are shown to be

precise, with high quality factors of resonance. Next, we modulate the frequency of these

neurons using a unique form of highly specific, photo-switchable compound [ 19]. By doing this,

we show that this frequency pattern acts as a forced-oscillator when externally driven, and that

the oscillations revert back to their initial frequency once the driving force is stopped. Using the

Hodgkin-Huxley neuron model [ 20], we derive a form of parametric oscillator that describes the

slow oscillations observed, as well as their ability to be externally tuned. Finally, we analyze the

parameters of oscillation, and compare them with the existing literature for further understanding

of the gating mechanism in the oscillation behavior of Purkinje neuron cells.

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II. METHODS

For a complete description of animal handling, sample preparation, solutions used and the

optical/patch-clamp setup, please refer to [ 9].

The stimulation of PNs was done by activating kainate receptors using a variety of highly

selective molecular agonists (KRAs). These molecules act only upon kainate receptors without

activating any of the other glutamatergic receptors on the cell, particularly AMPA (α-amino-3-

hydroxyl-5-methyl-4-isoxazole-propionate) receptors, which are the majority of ionotropic

glutamatergic receptors on PNs. Additional pharmacological blockers were used to isolate the

kainate response in the PN. The Photo-Switchable Kainate Receptor Agonist (PSKRA) [ 19] was

used as a traditional KRA when in the dark. The photo-response is described in [ 19], as well as

in the text.

All drugs except the Photo-Switchable Kainate Receptor Agonist (PSKRA) were purchased via

Sigma-Aldrich or Tocris Bioscience. Drugs were applied to the artificial cerebrospinal fluid

(ACSF) reservoir and allowed to perfuse onto the slice using the closed-loop system. Kainate

activation of the PNs was achieved using either highly-specific kainate receptor agonists

(KRAs), or Monosodium Glutamate (MSG, 100 µM) in conjunction with an AMPA receptor

blocker GYKI-52466 (10-20 µM). The KRAs used consisted of the commercially available

(2S,4R)-4-Methylglutamic Acid (SYM-2081, 10-50 µM), a non-selective GluK1/GluK2 agonist

(non-selective for GluR5/6, selective over AMPA receptors), as well as the PSKRA. The

PSKRA was based upon a variant of the commercially available SYM-2081, called LY339434

[ 21], which was designed specifically to be selective towards GluK1 (GluR5) over both GluK2

(GluR6) and AMPA receptors, and was used at 50-100 µM, the ideal concentration as described

in [ 19].

III. RESULTS

A. Induction of Astability

We first demonstrate that every PN, when measured in vitro using a current-clamp setup, will

change its firing pattern to that consisting of slow oscillations of Ca2+ spikes [ 22], nested within

a slow Switching envelope wave. Fig. 1(a) displays the recording of a cell transitioning to this

mode (after the application of KRAs in conjunction with teotrodotoxin (1 µM), which abolishes

the Na+ spikes, and accentuates the underlying Ca2+ spike pattern). The transition to a slow

oscillation mode consisting of Switching and Calcium frequencies occurs after 0.5-2 min, as

shown on the right side of Fig. 1(a). Once induced into this slow oscillation mode, the system

retains its astability, with cells oscillating between firing Ca2+ spikes and quiescence for up to 40

min. The oscillation frequency can be measured from the highest peak in the Fourier Transform

(FFT) of the recording, as shown in Fig. 1(b). Results such as these were obtained in n>50 cells,

with a high yield of greater than 90%, signifying the reproducibility of these results.

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Figure 1: (Color online) Astability induction in a Purkinje neuron.

(a) Signal output recording of a cell transitioning to an astable oscillatory mode after applying a kainate receptor

agonist (PSKRA, 100 µM) in conjunction with an AMPA receptor blocker (GYKI, 10 µM) and Na+ channel blocker

(TTX, 1 µM). Dotted line signifies the drugs‟ arrival at the cell within the bath chamber. Clear oscillations are at the

far right. (b) Fourier transform of the recording before (dashed green) and after (solid blue) the induction of

oscillations. A clear peak at 0.075 Hz can be seen (asterisk).

The frequency of the astable oscillations can be measured by tracking the on/off transitions in

time using a windowing algorithm [ 9], or directly by measuring the Fourier transform using the

FFT algorithm (Matlab). This is shown in Fig. 2 for a cell, already in the astable mode. Both

window-tracking [Fig. 2(b)] and the FFT [Fig. 2(c)] display the clear astable oscillation

frequency of the cells, with the precision of the oscillation measured as the deviation from the

mean in the time domain, or as the Full Width at Half Maximum (FWHM) in the frequency

domain. The transition to clear, precise oscillations would typically occur within 0.5-2 min of the

drugs‟ activation upon the cell, as is seen in the transition region in Fig. 1(a), as well as the initial

firing sequence in Fig. 2(a).

Figure 2: (Color online) Measuring the induced oscillations.

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(a) 9 minute recordings of a cell induced to oscillate via application of glutamate (MSG, 100 µM) and an AMPA

receptor blocker (GYKI, 10 µM). The period of oscillation of bursts can be tracked over time (b), or examining the

low frequency region of the frequency domain (FFT) of the recording (c). The precision of the oscillation frequency

can be measured from the center of the peak in (c), with the Full Width at Half Maximum (FWHM) signifying the

variation over time, or as the standard deviation from the mean period.

B. Tunability via Photo-Switching

Until now we have shown only the natural astability frequencies measured in PNs after

pharmacological activation; presently we show that the PN can act as a forced oscillator, and

thus extend the frequency range available for astability. This can allow the PNs to produce a

wider range of frequencies per cell, which can then be integrated together in other regions of the

cerebellum [ 10, 23]. Using the PSKRA and the appropriate photo-switching wavelengths, shown

in Fig. 3(a), we can gradually toggle the firing of Ca2+ spike bursts in a PN to a given oscillation

period [Fig. 3(b)]. We observed that the firing pattern then matches the period of forced-

modulation. This was done at a wide range of periods (6-30 sec) and DCs (25-75%).

Figure 3. (Color online) Photo-switchable compound modulates the oscillation.

(a) Comparison of the GluK1 selective compound LY339434 and the PSKRA, which has an added azobenzene

moiety, rendering it photo-switchable at 500 and 380 nm. (b) The ultraviolet (UV, 380 nm) and cyan (500 nm) light

turn the bursting on and off, respectively, as a function of the modulation period, here shown for two modulation

patterns with differing periods. The stimulation pattern is noted to the right of each recording, with stimulations at

15 sec cyan, 15 sec UV and 10 sec cyan, 10 sec UV. (c) Photo-modulation followed by induction of oscillation in a

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cell not displaying astability prior to the modulation. The cell continued to oscillate after the photo-switching was

completed.

The tunability of cells using the PSKRA was effective in all cells measured, with the slow

oscillations following the forced photo-modulation. Additionally, in some of the cells that did

not exhibit astability initially when in the presence of the PSKRA in the dark (n=5, resulting in

the less than 100% yield reported above), the photo-modulation still resulted in forced-

oscillation, with the cells continuing to oscillate after the photo-switching was stopped, as

demonstrated in Fig. 3(c).

C. Delayed Recovery of Modulation

The dynamics of the recordings subsequent to the photo-modulation were also studied. Figure

4(a) displays a representative recording from a PN that was initially oscillating at a natural

period of 22 sec (green), and then photo-modulated for 2 min at 5 sec on/ 5 sec off (10 sec

period, dark blue), and subsequently allowed to recover (light blue). The PN faithfully follows

the forced-modulation after 1-2 stimuli when the light is applied, and then slowly recovers back

to the natural oscillation period it had prior to the stimulation, after the photo-stimulation is

stopped. This recovery response of the cell‟s oscillatory modulation is similar to a traditional

forced-oscillator, with an exponential recovery after the forced modulation is stopped. This is

best visualized when plotting out the period and DC over time, as in Figs. 4(b) and (c), which

display both the period-matching during forced-oscillation (dashed blue line) as well as the

exponential recovery (dotted red line).

The time constants for recovery have a distribution among different cells, with a recovery time of

82 ± 48 sec for the period, and 70 ± 41 sec for the DC (n=11 cells. Errors are in SD, displaying

the range of variation among cells). The long recovery time for this process is similar to other

forms of short-term memory in the brain (such as short-term depression and potentiation [ 5]),

allowing the cell to “remember” its forced-modulation for a short duration after the stimulus is

applied, however the cells typically did not retain the forced frequency. The direction of recovery

was generally towards the natural astable period in the cell prior to photo-modulation, with

forced-photo-modulation of the cells done both below and above their natural period [below: 5/5

sec on/off modulation, n=7; above: 10/10 sec on/off modulation, n=9; 15/15 sec on/off

modulation, n=4, with representative cell recordings in Figs. 4(d) and (e), respectively].

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Figure 4: (Color online) Recovery from the photo-modulation follows a forced-oscillator model.

(a) Exemplary recording of a Purkinje neuron showing a natural oscillation (green) that is then photo-modulated at 5

sec on/5 sec off (10 sec period) for 2 min (dark blue), and then allowed to recover from the modulation (light blue).

(b, c) Period and Duty-Cycle tracking of the cell in (a), displaying the forced-modulation (dashed blue line) and

exponential recovery. Dots are color coded as in (a). Red dotted lines are fitted exponential curves for the recovery

segment, with a time constant of τ = 124 sec and τ = 67 sec for the period and duty-cycle, respectively. (d, e)

Representative plotted periods of the natural, modulated and recovering oscillators for two different modulations:

(d), 10/10; (e), 15/15 on/off, in seconds (τ = 97, 40 sec, respectively). Actual recordings appear below each plot.

IV. PARAMETRIC OSCILLATOR MODEL

The bursting oscillations of the PNs are non-dissipating, with a clear frequency signature in the

FFTs. We can simplify the Hodgkin-Huxley equations [ 24- 27] to reach a model similar to the

fundamental parametric oscillator equations to describe this harmonic signal. The generalized

formula for the membrane potential, V, injected current, I, and membrane capacitance, C, is

[ 25, 26]:

ii

VxgIV

)(

i

k

EC

(1)

With each gating variable, xi, being a function of the voltage:

parameters, g, being the conductance, and (V-Ei) term being the driving force per ion channel.

Taking a time derivative of equation 1:

)(/ ]

x

)([

0

v

τ

vxx

, the

)

)

V

i

xgEV

(

dt

xd

gEV

(

xg

IV

C

ii

ki

i

ii

ki

ii

)(

(2)

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Assuming that the injected current (I, if any exists) is constant with time, and that the

conductance coefficients gi, are also time independent, we can neglect the dI/dt and dgi/dt terms,

remaining with:

)

V

i

xgEV

(

dt

xd

gV

C

ii

ki

i

i

)(

(3)

Since we are interested in the slow changing terms only, whose changes with time are of the

order of the oscillation period (10-25 sec), we can isolate these terms by removing them from the

summation:

fast

)

)

iii

ki

i

i

S

SlowSlowSlow

S

Slow

dt

Slow

VxgEV

(

dt

xd

gVxgEV

(

xd

gV

C

)()(

(4)

In equation 4, the Slow gating variable has an exponent of S. We will assume that this exponent

is unity, since the slow action of our experimental results is similar to that of either the

muscarinic gating variable M, or h for the Ih refractory current, both which have an exponent of

unity [ 25, 26]. This assumption will be further justified in the subsequent section.

The rightmost bracketed term in the summation can be taken as the average value for long

periods where the fast acting terms within these brackets change at a rate that is much higher

than the slow oscillations examined here. This is because the time derivative d/dt for long time

durations is defined by changes of the order of 1/ΔtSlow. This is comparable with taking the

average membrane potential during the firing of action potentials, otherwise known as the “up”

state of a bistable system [ 16], such that we are dealing only with a slow waveform that is similar

to a square-wave envelope. With these assumptions, we obtain:

avg

fast

iii

ki

i

iSlowSlowSlowSlowSlowSlowSlow

VxgEV

(

dt

xd

gVxgEx

gVx

gV

C

)

)(

(5)

The middle term: gSlowx’SlowESlow can further be isolated, since it is a term that is dependent upon

the voltage and time, but does not directly include the voltage. Reorganizing the above equation,

we can obtain a generalized 2nd order differential equation:

avg

(6)

fast

iii

ki

i

iSlowSlow SlowSlow SlowSlowSlow

VxgEV

(

dt

xd

gEx

gVx

gVxgV

C

)

)(

Or:

)(tFVx

gVxgV

C

SlowSlow SlowSlow

(7)

Comparing this to the harmonic oscillator with a driven source, F(t) [ 28]:

F(t)kV V

-bV

m

(8)

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Equation 8 is the classic equation for a driven harmonic oscillator with a resonant frequency of

ωo2=k/m and a quality factor of Q=ωom/b, with F(t) being the time-dependent driving input.

Comparing terms, we find that:

SlowSlow

SlowSlow

x

gtV

(

k

xgtV

(

b

Cm

)

,

),

(9)

As can be seen, the parameters of this equation are time/voltage dependent. This makes the

equation a parametric oscillator as opposed to a simple harmonic oscillator. The parameters in

Equation 9 are related to the biological and measureable aspects of each neuron, with the

membrane capacitance directly measureable, and the gating variables measureable using voltage-

clamp experiments to determine the dynamics of the ion channels involved.

If the oscillator is under-damped (Q>1/2, which is equivalent to b<<ωo), it is easy to then

measure the resonance frequency of the Purkinje neuron (2πfo=ωo) as well as the quality factor,

Q, from the Signal-to-Noise-Ratio (SNR). The frequency of an oscillating Purkinje neuron can

be measured directly in the frequency domain, and the SNR can be calculated either through the

time-tracking algorithm, or from the width of the peak in the FFT. The frequency of the damped

oscillator is quite similar to that of the resonance frequency: ω2measured=ωo2(1-1/4Q2), with Q

taken as the SNR. Since most of the oscillating neurons had Q>2 (as will be shown

subsequently), then ωmeasured≈ωo. Therefore, by measuring these two parameters (fo and Q), and

taking a general value of the membrane capacitance of C=1 µF/cm2 [ 26], we can obtain the

parameters k and b.

Equation 7 is the parametric oscillator equation for an oscillating neuron that can be analyzed

using phase-space diagrams [ 27]. Obtaining the van-der-Pol oscillator equation from the

Hodgkin-Huxley equations follows a similar method.

V. DATA ANALYSIS

The period (1/frequency) and duty-cycle (DC) of the astable mode were measured in n=43 cells,

each oscillating for at least 7 min, and are displayed in Figs. 5(a) and (b). Since each cell acts as

an independent oscillator, it is expected to find a range of inherent frequencies. We found that

the average period of the cells was 20 ± 8 sec, and the average DC was 46 ± 8% (± standard

deviation, SD). We again note the similarity between the range of slow oscillations measured

here, and those measured in vivo in tottering mice, which ranged between 12.82 and 25.64 sec

[ 15].

The precision of oscillation over time can be measured by the quality factor of the resonator, Q,

or the SNR. This was measured using the window-tracking algorithm in the time domain, for

each of the cells measured, and displayed in Fig. 5(c) for each individual cell. Of the PSKRA

activated cells, 45% (n=13/29, black circles) had a SNR larger than 10, corresponding to less

than a 10% deviation in period over time, whereas the SYM-2081/MSG activated cells were less

accurate (blue triangles). This may be attributable to wither the selectiveness of the PSKRA, or

the concentration ratios.

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Figure 5: (Color online) Period, duty-cycle and precision of oscillation.

(a, b) Histograms of the average period and duty-cycle (n=43 cells total) showing the central frequency of 20

sec/0.05 Hz, and 46% duty-cycle. (c) Quality factor (Q), or SNR, data for the PSKRA (black circles) and SYM-

2081/MSG (blue triangles) activated cells, as a measure of the precision of oscillation. Each cell is plotted

individually, with the PSKRA activated cells having higher precision values.

Using the formalism of the parametric oscillator above, we can view each PN as an independent

oscillator, each with its own measurable parameters of oscillation. For example, using the cell

shown in Fig. 2, we can isolate the frequency and quality factor directly from the peak in the FFT

in Fig. 2(c): with Q=10.93 and fo=0.044 Hz (T=22.9 sec); these result in values of b=0.025 and

k=0.07. Following equation 9, since we are measuring time averaged values, we can assume that

the gating variables vary sinusoidally (using Floquet analysis), such that the time averaged value

is half the amplitude: <b>=gSlow×<xSlow>=gSlow/2. This results in a value of gSlow~0.05 mS/cm2.

Comparing with existing values in the literature would place this gating channel closer to the h-

current with gIh=0.03 mS/cm2 [ 25] or perhaps the muscarinic channel with gM=0.75 mS/cm2 [ 26].

The relationship between the h-current and bistability has previously been shown [ 16].

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VI. CONCLUSIONS

This work has shown the capability of a PN to act as an astable oscillator with long periods of

oscillation (10-25 sec), as well as the ability to externally tune this frequency for extended

periods of time. This frequency range is notably outside the range typically studied in the brain

[ 11- 14], but matches other in vivo results of the PN [ 15]. The assumed existence of such a timing

functionality of the PN in the cerebellum lies in complete agreement with the conceptual view of

the cerebellum as the feedback control mechanism of the brain [ 1, 6], as well as temporal pattern

generator theories of the cerebellum [ 23]. Using mathematical dynamic systems models and

newly derived optical activation techniques allow us to probe the intrinsic behavior of cells

within a network, thereby enabling us to reverse engineer the neuronal circuitry of the brain at a

higher level.

ACKNOWLEDGMENTS

This research was made with support from the National Science Foundation Nano-Scale Science

and Engineering Center (NSF-NSEC) under award CMMI-0751621. ZRA acknowledges

Government support under and awarded by DoD, Air Force Office of Scientific Research,

National Defense Science and Engineering Graduate (NDSEG) Fellowship, 32 CFR 168a. ZRA

would also like to thank Prof. Harold Lecar for his useful discussion.

References

1. M. Ito, Prog. Neurobiol. 78, 272 (2006).

2. J.C. Eccles, M. Ito and J. Szentágothai, The Cerebellum as a Neuronal Machine

(Springer-Verlag, Berlin, 1967).

3. D. Marr, J. Physiol. London 202, 437 (1967).

4. J.S. Albus, Math Biosci 10, 25 (1971).

5. M. Ito, Physiol. Rev. 81, 1143 (2001).

6. K. Doya, H. Kimura and M. Kawato, IEEE Control Systems Magazine 21, 42 (2001).

7. D. Rokni, R. Llinás and Y. Yarom, Front Neurosci. 2, 192 (2008).

8. J.M. Bower, Front. Cell. Neurosci. 4, 27 (2010).

9. Z.R. Abrams, A. Warrier, D. Trauner and X. Zhang, Front. Neur. Circ. 4, 13 (2010).

10. Z.R. Abrams and X. Zhang, Front. Neur. Circ. 5, 11 (2011).

11. G. Buzsáki and A. Draguhn, Science 304, 1926 (2004).

12. M. Steriade, Neuroscience 137, 1087 (2006).

13. A.K. Roopun, et al., Front. Neurosci. 2, 145 (2008).

14. C.I. De Zeeuw, F.E. Hoebeek and M. Schonewille, Neuron 58, 655 (2008).

15. G. Chen, et al., J. Neurophysiol. 101, 234 (2009).

16. Y. Loewenstein, et al., Nat. Neurosci. 8, 202 (2005).

17. B.E. McKay, et al., J. Neurophysiol. 97, 2590 (2007).

18. M. Yartsev, R. Givon-Mayo, M. Maller and O. Donchin, Front. Sys. Neurosci. 3, 1

(2009).

Page 12

19. M. Volgraf, et al., J. Am. Chem. Soc. 129, 260 (2007).

20. A.L. Hodgkin and A.F. Huxley, J. Physiol. 117, 500 (1952).

21. B. Small, et al., Neuropharm. 37, 1261 (1998).

22. J. Hartmann and A. Konnerth, Cell Calcium 37, 459 (2005).

23. G.A. Jacobson, D. Rokni and Y. Yarom, Trends Neurosci. 31, 617 (2008).

24. C. Morris and H. Lecar, Biophys. J. 35, 193 (1981).

25. F.R. Fernandez, J.D.T. Engbers and R.W. Turner, J. Neurophysiol. 98, 278 (2007).

26. M.A. Kramer, R.D. Traub and N.J. Kopell, Phys. Rev. Lett. 101, 068103 (2008).

27. E.M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and

Bursting (MIT Press, Cambridge, MA, 2006).

28. L. S. Borkowski, Phys. Rev. E 83, 051901 (2011).