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Supplementary Information (SI)
Handedness helps homing in swimming and flying animals
Promode R. Bandyopadhyay, Henry A. Leinhos, & Aren M. Hellum
Autonomous & Defensive Systems Department, Naval Undersea Warfare Center, Newport, RI
02841, USA
Animation
Animation 1: Shows disturbance rejection by a flapping fin driven by analog van der Pol
oscillator circuit13. The fin is given a square pulse of external electrical disturbance. (a) Shows
graph of x-load (N) and y-load (N) with time; shows fin assembly in tow tank. (b) Shows fin roll
versus pitch angle with time. Both show departure from limit cycle due to externally applied
square wave disturbance and subsequent return to the previous limit cycle after the disturbance is
removed. No sensor or separate closed loop controller for error reduction from reference is used.
Animation 2: This shows the sensitivity of the modeled (high-order) force fluctuation maps
( Fx ' Fx ' ) to Strouhal number St in flapping fins subjected to optimized twisting, as evidenced by
their modeling using the Stuart-Landau (SL) equation. The map for nominal experimental St =
0.30 is shown in Fig. 4a. The SL modeling perfect lock-in occurs at St = 0.2999075 and is given
in Fig. 4b. The animation shows SL modeling that produces a small amount of oscillation near
the lock-in at St = 0.3008 because the initial condition in each half of the oscillation is not the
same.
Animation 3: This shows the state map z1-w2 of two inferior-olive neuron oscillators 1 (strong)
and 2 (weak). The ratio of w2/z1 is on the order of 0.001. The weak oscillator effect
congregates at a preferred phase of the strong oscillator. In certain sectional planes, the highfrequency oscillations look deceptively random. The essential nonlinear nature of the weak
oscillator is retained even though the state levels are weak and might be misinterpreted as
“noise.”
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Data and Materials
Part A: Preferential properties of flapping fin instantaneous actuator force vector fields
and of dolphin sonar.
(a)
(b)
(c)
(d)
Fig. S1. Examples of preferential topological properties. (a) Galaxies are flat disks (picture
courtesy of NASA). (b) Instantaneous force vectors (N (x, y, z, t)) from a pair of pectoral fins
flapping at various parameters; note that the pattern is not spherical; it is wider azimuthally and
narrower in elevation (our laboratory measurements7,8,18 on penguin-like realistic (aspect ratio,
fin section, and planform) hinged flapping fins). The fins are hinged at (0, 0, 0). The net fin
force (Fx) acts at one spanwise location (Ravg) at c/3 from the leading edge—the center of pitch
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oscillation. The fins undergo roll and pitch oscillation simultaneously with a 90o phase
difference between them. (c, d) Measurements of dolphin acoustic sensory (transmission and
receiving) response at 120 kHz2,3. Response at other peaks of chirps at 30 and 60 kHz is similar.
Note the 5o and 10o bias in elevation in transmission and receiving presumably due to head
anatomy. The receiving response is wider in azimuth than in elevation as in (a) and (b). The
receiving response in azimuth has a preferential handedness. The operational impact of this last
property is considered in this paper.
[Note on dolphin sonar maps due to Au2,3: The data is due to a stationary animal trained to work
with a bite plate. The dolphin chirps with peaks in the SPL spectra at 30, 60, and 120 kHz. For
the transmitting profile, the SPLs in the peaks measured with a hydrophone are plotted (the
above plots are for 120 kHz). For the receiving profile, in the water there are two sources: one is
emitting noise and the other is a monochromatic projector. The animal is trained to record its
response on the bite plate when the difference between the projector and the noise is discerned.
To clarify, no hydrophone is involved in the plot of the receiving map.]
Experimental data source:
Fig. S1a: NASA.
Fig. S1b: refs. 7, 8, and 18.
Fig. S1c, S1d: refs. 2, 3.
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Part B: Validation of the analog method of solution of the olivo-cerebellar dynamical
system equations
(a)
(b)
(c)
IO Bifurcation April 2010
Steady State Limit Loops at Various '' Values
0.3
w
0.2
0.1
0
-0.1
1
0.03
0.5
0.02
0.01
0
0
-0.5
z
(d)
-0.01
-0.02

(e)
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Fig. S2. Validation of our analog method of solving the nonlinear olivo-cerebellar dynamical
equations (coupled Ca (zi, wi) and Na (ui, vi) oscillators). (a) Photograph of board containing six
independent solvers, three in each of the two columns. (b, c) Comparison of self-referential
phase reset property22 in the analog circuit13 in (b) with the measurements in rodent brain slices22
(c). Initially, the signals have random phase. After the spike is applied (extra-cellular impulse,
the term Iext in the Ca oscillator equation (zi, wi)), the signals synchronize. The inferior-olive
neural behavior is reproduced by our analog circuit. (d, e) Comparison of measurements (d) of
limit cycles of states in the Ca oscillator (zi, wi) in our analog circuit with those from theory (e)33.
The third axis is µ, which is a measure of the strength of the oscillator voltage levels. The analog
circuit in (a) therefore reproduces the olivo-cerebellar dynamics as known experimentally with
animal brains and theoretically. This circuit is used in the results in Fig. 5b.
µ in Figs. S2d and e, and a in Figs. 5b and S913,22,34: In Figs. 5b and S9 (w (a = 0.035), z (a
= 0.015)), the a values are for small and large limit cycles, respectively; a = a*  µ; the
parameter a appears in the cubic nonlinear functions of the u-v and z-w subsystems given in the
main paper; at a critical value of a, the system undergoes Poincare-Andronov-Hopf bifurcation
(it is stable for lower values of a), and at higher values, oscillatory solutions (limit cycles shown
in Figs. S2d and e) appear. Figure 5b shows that, as per our model, the coupling of a strong and
a weak inferior-olive neuron is being used to generate the given bat path in the horizontal plane
in the z and x axes, respectively.
Two other bats (named Frosty and George) have tracks23 similar to those in Fig. 5a—which
are due to Poe—consisting of segments of similar, mainly very long, less-curved and rounded
paths.
Experimental data source:
Fig. S2a, 2b: ref. 13.
Fig. S2c: ref. 22.
Fig. S2e: ref. 34.
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Part C: Modeling of sensor (combined SPL and acceleration) response and comparison
with flapping fin force fluctuation measurements at various wake bifurcating Reynolds
numbers
The relationship between acoustic pressure and the acoustic particle velocity field provided
by the linear Euler equation, as well as the assumptions from the linear wave equation, provides
both the fundamentals of signal propagation and the basis upon which the pressure and velocity
signals can be combined in a coherent hydrophone-accelerometer sensor beamforming
algorithm21:
o
v
 p'.
t
(S1)
Equation (S1) is the linear Euler equation that relates the time dependence of the acoustic
velocity field to the gradient of the pressure for small-amplitude pressure and density
perturbations within a fluid.
Limaçon beamform response: In Figs. S3 through S5, we show the results from the
acoustic sensor response equation given in the main paper for different relative proportions of the
velocity (Wx, Wy) and pressure (Wp) weights.
Figures S3 through S5 show the conditions under which the acoustic combined hydrophoneaccelerometer sensor response would match the acoustic radiation from the flapping fin forces in
the stated regimes of transitional Reynolds numbers.
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(a)
(b)
Fig. S3. (a) Acoustic combined hydrophone-accelerometer sensor response (b, ) for Wx = Wy =
1, Wp = 3/2; (b) fin force fluctuation measurements8 at fin chord Reynolds number Rec = 3,558 ≤
Rec ≤ 7,089, mode A.
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(a)
(b)
Fig. S4. (a) Acoustic combined hydrophone-accelerometer sensor response (b, ) for Wx = Wy =
1, Wp = 1; (b) fin force fluctuation measurements8 at fin chord Reynolds number 17,689 ≤ Rec ≤
26,834, mode B. The force fluctuation maps in the Rec range between 7,089 and 17,689 (mode
A-B) are a mixture of those in Figs. S3b (mode A) and S4b (mode B).
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(a)
(b)
Fig. S5. (a) Acoustic combined hydrophone-accelerometer sensor response (b, ) for Wx = Wy =
1, Wp = ½; (b) fin force fluctuation measurements8 at Rec = 49,121. (The map is stable in the
range 39,923 ≤ Rec ≤ 70,895, mode C; force fluctuation maps in the Rec range between 26,834
and 39,923 (mode B-C) are a mixture of those in Figs. S4b (mode B) and S5b (mode C)).
Experimental data source:
Figs. S3b, S4b, and S5b: ref. 8.
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Part D: Modeling of the motion of a flapping fin platform using bat sonar properties
The target is assumed to be a point. Homing is said to have occurred when the range drops to
less than or equal to half of the platform length. Figure S6 presents background bat sonar results,
which have some validity in dolphins as well. Figure S7 is a graph of the modeling of the fin
flapping frequency taken to be analogous to the bat chirp pulse intensity. Figure S8 shows the
bearing, range, and platform trajectory for static targets to complement the mobile target results
in the main paper (Fig. 6).
(a)
(b)
Fig. S6. Chirp properties during the homing of a bat to a target24.
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Experimental data source:
Figs. S6a, S6b: ref. 24.
Fig. S7. Modeled variation of actuator frequency in a flapping platform as it approaches a target;
this model has been used in the present homing simulations. Thrust is proportional to the square
of the flapping frequency7,18.)
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(a)
(b)
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(c)
Fig. S8. Modeling of flapping fin platform trajectory showing heading and bearing (a), range
(b), and position (c) with time when target is static. The results for even-handedness are
compared with those for 10% and 15% left-handedness. The platform is indicated by a triangle.
The ellipses indicate constant SPL.
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Part E: Phase relationship of a strong olivo-cerebellar oscillator with a weak one where the
state level ratio is weak/strong = 0.001
Also see Animation 3 in the SI.
(a)
(b)
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(c)
Fig. S9. Phase relationship with time between orthogonal states z1 and w2 from two
oscillators—one strong and one weak (see Figs. S2d, e). Although the state is weak in the weak
oscillator, it retains all the properties of the inferior-olive nonlinear oscillator. While (c) looks
noisy and (a) looks clean and “periodic,” (b) clearly shows that the weak oscillator is present at a
certain phase of the large oscillator. For a platform whose states are given by these oscillators,
the weak oscillator will provide a small nudge at a certain phase of the large oscillator.
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References
1.
Hegstrom, R. A., Kondepudi, D. K. The handedness of the universe. Sci. Amer.
262(1), 108-115 (1990).
2.
Cohen, S. Fifty centuries of right-handedness: the historical record. Science 198, 631632 (1977).
3.
Coren, S., Halpern, D. F. Left-handedness: a marker for decreased survival fitness.
Psycho. Bull. 109, 90-106 (1991).
4.
Hori, M. Frequency-dependent natural selection in the handedness of scale-eating
cichlid fish. Science 260, 216-219 (1993).
5.
Au, W. W. L. The Sonar of Dolphins (Springer, New York), p. 106 (1993).
6.
Au, W. W. L., Benoit-Bird, K. Automatic gain control in the echolocation system of
dolphin. Nature 423, 861-863 (2003).
7.
Menozzi, A., Leinhos, H. A., Beal, D. N., Bandyopadhyay, P. R. Open-loop control
of a multi-fin biorobotic underwater vehicle. IEEE J. Oceanic Engrg. 33(2), 59–68
(2008).
8.
Bandyopadhyay, P. R., Beal, D. N., Hrubes, J. D., Mangalam, A. Relationship of roll
and pitch oscillations in a fin flapping at transitional to high Reynolds numbers. J.
Fluid Mech. 702, 298-331 (2012).
9.
Taylor, G. K., Nudds, R. L., Thomas, A. L. R. Flying and swimming animals cruise at
a Strouhal number tuned for high efficiency. Nature Letts. 425, 707-711 (2003).
10.
Triantafyllou, M. S., Triantafyllou, G. S. An efficient swimming machine. Scientific
American 272, 64–70 (1995).
16
10/23/2012
11.
von Kármán, T., Burgers, J. M. General aerodynamic theory: Perfect fluids. Vol. II
(367 p.) of Aerodynamic Theory (ed. W. F. Durand) (Springer Verlag, Leipzig), six
vols., (1934).
12.
Lighthill, J. Mathematical Biofluiddynamics, CBMS-NSF Regional Conf. Series in
Appl. Math., Soc. Indust. Appl. Math., Philadelphia, PA, 17 (1975).
13.
Bandyopadhyay, P. R., Singh, S. N., Thivierge, D. P., Annaswamy, A. M., Leinhos,
H. A., Fredette, A. R., Beal, D. N. Synchronization of animal-inspired multiple fins
in an underwater vehicle using olivo-cerebellar dynamics. IEEE J. Oceanic Engrg.
33(4), 563-578 (2008).
14.
Partridge, B. L. The structure and function of fish schools. Sci. Amer., 246(6), 114123 (1982).
15.
Albarède, P., Monkewitz, P. A model for the formation of oblique shedding and
“Chevron” patterns in cylinder wakes. Phys. Fl. A 4, 744-756 (1992).
16.
Skop, R. A., Balasubramanian, S. A new twist on an old model for vortex-induced
vibrations. J. Fluids Struct. 11, 395-412 (1997).
17.
von Ellenrieder, K. D., Parker, K., Soria, J. Fluid mechanics of flapping wings. Exp.
Thermal and Fluid Science 32, 1578-1589 (2008).
18.
Bandyopadhyay, P. R., Beal, D. N., Menozzi, A. Biorobotic insights into how animals
swim. J. Exp. Biol. 211, 206–214 (2008).
19.
Lentink, D., Dickinson, M. H. Rotational accelerations stabilize leading edge vortices
on revolving fly wings. J. Exp. Biol. 212, 2705-2719 (2009).
20.
Noack, B. R., Ohle, F., Eckelmann, H. On cell formation in vortex streets. J. Fluid
Mech. 227, 293 (1991).
17
10/23/2012
21.
Psaras, S. Masters Thesis, Naval Postgraduate School, Monterey, CA, USA (2008).
22.
Kazantsev, V. B., Nekorkin, V. I., Makarenko, V. I., Llinas, R. Self-referential phase
reset based on inferior olive oscillator dynamics. Proc. Nat. Acad. Sci. 101 (52),
18183–18188 (2004).
23.
Barchi, J. R., Knowles, J. M., Simmons, J. A. Spatial memory and stereotypy of flight
paths by big brown bats in cluttered surroundings. J. Exp. Biol. (2013) (in press).
24.
Hiryu, S., Hagino, T., Riquimaroux, H., Watanabe, Y. Echo-intensity compensation
in echolocating bats (Pipistrellus abramus) during flight measured by a telemetry
microphone. J. Acoust. Soc. Amer. 121, 1749-1757 (2007).
25.
Huerre, P., Monkewitz, P. A. Local and global instabilities in spatially developing
flows. Ann. Rev. Fluid. Mech. 22, 473–537 (1990).
26.
Bates, M. E., Simmons, J. A., Zorikov, T. V. Bats use echo harmonic structure to
distinguish their targets from background clutter. Science 333, 627-630 (2011).
27.
Fletcher, N. H., Thwaites, S. Obliquely truncated simple horns: idealized models for
vertebrate pinnae. Acoustica 65, 194-204 (1988).
28.
Teyke, T. Morphological differences in neuromasts of the blind cave fish Astyanax
hubbsi and the sighted river fish Astyanax mexicanus. Brain Behav Evol.35(1), 23-30
(1990).
29.
Espinasa, L., Rivas-Manzano, P., Pérez, H. E. A new blind cave fish population of
genus astyanax: geography, morphology and behavior. Environ. Biol. Fishes 62, 339344 (2001).
30.
Van Reeuwijk, J., Arts, H. H., Roepman, R. Scrutinizing ciliopathies by unraveling
ciliary interaction networks. Hum. Mol. Genet. 20, 49-57 (2011).
18
10/23/2012
31.
Norris, D. P., Grimes, D. T. Cilia discern left from right. Science 338, 206-207
(2012).
32.
Yoshiba, S. et al. Cilia at the node of mouse embryos sense fluid flow for left-right
determination via Pkd2. Science 338, 226-231 (2012).
33.
Yarusevych, S., Sullivan, P. E., Kawall, J. G. On vortex shedding from an airfoil in
low-Reynolds-number flows. J. Fluid Mech. 632, 245-271 (2009).
34.
Lee, K. W., Singh, S. N. Adaptive global synchrony of inferior olive neurons.
Bioinspiration and Biomimetics 4(3), p. 036003 (2009). Also, K. W. Lee, S. N.
Singh, Bifurcation of orbits and synchrony in inferior olive neurons. J Math. Biol.
(2011); DOI 10.1007/s00285-011-0466-9.
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