Neural mechanism of optimal limb coordination incrustacean swimmingCalvin Zhanga, Robert D. Guya, Brian Mulloneyb, Qinghai Zhangc, and Timothy J. Lewisa,1

Departments of aMathematics and bNeurobiology, Physiology, and Behavior, University of California, Davis, CA 95616; and cDepartment of Mathematics,University of Utah, Salt Lake City, UT 84112

Edited by Eve Marder, Brandeis University, Waltham, MA, and approved July 17, 2014 (received for review December 16, 2013)

A fundamental challenge in neuroscience is to understand howbiologically salient motor behaviors emerge from properties of theunderlying neural circuits. Crayfish, krill, prawns, lobsters, andother long-tailed crustaceans swim by rhythmically moving limbscalled swimmerets. Over the entire biological range of animal sizeand paddling frequency, movements of adjacent swimmeretsmaintain an approximate quarter-period phase difference withthemore posterior limbs leading the cycle. We use a computationalfluid dynamics model to show that this frequency-invariant strokepattern is the most effective and mechanically efficient paddlingrhythm across the full range of biologically relevant Reynoldsnumbers in crustacean swimming. We then show that the organiza-tion of the neural circuit underlying swimmeret coordination providesa robust mechanism for generating this stroke pattern. Specifically,thewave-like limb coordination emerges robustly from a combinationof the half-center structure of the local central pattern generatingcircuits (CPGs) that drive the movements of each limb, the asymmetricnetwork topology of the connections between local CPGs, and thephase response properties of the local CPGs, which we measure ex-perimentally. Thus, the crustacean swimmeret system serves as aconcrete example in which the architecture of a neural circuit leads tooptimal behavior in a robust manner. Furthermore, we consider allpossible connection topologies between local CPGs and show that thenatural connectivity pattern generates the biomechanically optimalstroke pattern most robustly. Given the high metabolic cost ofcrustacean swimming, our results suggest that natural selectionhas pushed the swimmeret neural circuit toward a connectiontopology that produces optimal behavior.

locomotion | coupled oscillators | phase locking | metachronal waves

It is widely believed that neural circuits have evolved to opti-mize behavior that increases reproductive fitness. Despite this

belief, few studies have clearly identified the neural mechanismsproducing optimal behaviors. The complexity of behaviors gen-erally makes it difficult to assess their optimality, and neural circuitsare often too complicated to concretely link neural mechanisms tothe overt behavior. Energy-intensive locomotion such as steadyswimming, walking, and flying provides important model systems forstudying optimality because the goal of the behavior is clear and it islikely to have been optimized for efficiency (1). For example, thekinematics of locomotion has been shown to be optimal in the casesof the undulatory motion of the sandfish lizard and the lamprey(2, 3). On the other hand, the neural circuits underlying locomotionin most organisms are not sufficiently characterized to understandhow they give rise to the optimal motor behavior. Because of thedistinct frequency-invariant stroke pattern and the relative simplicityof the neuronal circuit, limb coordination of long-tailed crustaceansduring steady swimming provides an ideal model system for exam-ining the optimality of motor behavior and its neural underpinnings.During forward swimming, long-tailed crustaceans, like cray-

fish, krill, shrimp, and lobsters, propel themselves through thewater using four or five pairs of abdominal limbs called swim-merets that move rhythmically through cycles of power strokes(PSs) and return strokes (RSs). Although these animals vary insize from 0.5 cm to over 40 cm and beat their swimmerets with

frequencies ranging from about 1 to 10 Hz (4, 5), the strokepattern is invariant: limbs on neighboring abdominal segmentsalways maintain an approximate quarter-period (0.25) phasedifference in a tail-to-head metachronal wave (Fig. 1). Thisphenomenon is known as phase constancy. Measurements of themetabolic cost of krill swimming show that up to 73% of theirdaily energy expenditure is devoted to paddling (6). This and thefacts that the distinct limb coordination is the only stroke patternthat these crustaceans exhibit when the swimmeret system isactive and that this limb coordination is conserved across manyspecies suggest that the stroke pattern is biomechanically opti-mized for swimming. This in turn suggests that evolution haspushed the properties of the underlying neural circuit to producethe distinct phase constant limb coordination.Metachronal waves of motor activity during locomotion are ob-

served in many animals, and the underlying neural circuits havebeen shown to consist of chains of local pattern-generating micro-circuits [i.e., local central pattern generators (CPGs)] (7–10). This isthe case for the neural circuits that control the undulatory motion ofbony fish, amphibians, and lamprey during swimming (9–12) andthe movements of swimmerets in crayfish (Pacifastacus leniusculus)(13). By modeling the neural circuit of the crayfish swimmeretsystem as a chain of generic phase oscillators, previous theoreticalstudies (14–16) showed that the tail-to-head 0.25 phase constantstroke pattern could be achieved if the phase response properties ofthe local CPGs to inputs from ascending and descending inter-CPGconnections satisfy two different constraints. Similar results wereobtained for the lamprey swimming neural circuit (11, 17). Manystudies (e.g., refs. 14, 16, and 18) have addressed aspects of howthese two constraints are satisfied, but it remains unclear how they

Significance

Despite the general belief that neural circuits have evolved tooptimize behavior, few studies have clearly identified theneural mechanisms underlying optimal behavior. The distinctlimb coordination in crustacean swimming and the relativesimplicity of the neural coordinating circuit have allowed us toshow that the interlimb coordination in crustacean swimmingis biomechanically optimal and how the structure of underlyingneural circuit robustly gives rise to this coordination. Thus, weprovide a concrete example of how an optimal behavior arisesfrom the anatomical structure of a neural circuit. Furthermore,our results suggest that the connectivity of the neural circuitunderlying limb coordination during crustacean swimming maybe a consequence of natural selection in favor of more effec-tive and efficient swimming.

Author contributions: C.Z., R.D.G., and T.J.L. designed research; C.Z., R.D.G., B.M., and T.J.L.performed research; C.Z., R.D.G., Q.Z., and T.J.L. contributed new reagents/analytic tools;C.Z., R.D.G., B.M., and T.J.L. analyzed data; and C.Z., R.D.G., and T.J.L. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. Email: tjlewis@ucdavis.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1323208111/-/DCSupplemental.

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are achieved and maintained over a wide range of frequencies.Recent experimental work has elucidated the cellular compositionof the local swimmeret CPGs and the synaptic organization of thecircuit that connects them in the crayfish swimmeret system (13, 19,20), allowing these issues to be directly addressed.In this article, we first use computational fluid dynamics simu-

lations to show that the metachronal multilimb stroke pattern with∼0.25 intersegmental phase differences is the most effective andefficient metachronal stroke pattern across the entire biologicallyrelevant range of body sizes and stroke frequencies in crustaceanswimming. We then show that the half-center structure of the local-CPG circuits (21–23) and the topology of the inter-CPG con-nections in the crayfish swimmeret circuit (19, 20) provide a robustneural mechanism for producing the 0.25 phase-locked metachronalwave. This reduces the previously determined constraints on thephase response properties of the CPGs to a single condition. Fur-thermore, we experimentally measure the phase response proper-ties of the crayfish swimmeret CPG circuit and show that this singlecondition holds. Finally, we consider all possible topologies forconnections between the local CPGs and show that the networktopology present in the crayfish swimmeret circuit generates thebiomechanically optimal stoke pattern most robustly.

The Mechanical Advantage of the Tail-to-Head MetachronalWave of Swimmeret CoordinationMetachronal waves of ciliary beating in microorganisms are thesubject of intense recent study (24–27). These previous worksfound that metachronal waves may allow cilia to propel cellsforward with higher propulsion velocity and efficiency. However,metachronal waves in ciliary beating differ from the metachronalpaddling rhythm in crustacean swimming in many ways: (i)Swimmerets are plate-like paddles, whereas cilia are hair-likeappendages. (ii) The limb angle of swimmerets changes smoothlythrough the cycle with a duty cycle of approximately one-half,whereas cilia use a fast power stroke and a slow sweeping re-covery stroke. (iii) Although both swimmerets and cilia beat ina back-to-front metachronal wave, the phase difference betweenneighboring swimmerets is ∼25%, whereas the phase differencebetween appendages varies from ∼1% to ∼10% depending onthe organism and location of the cilia on the organism. (iv)Evidence suggests that phase locking of cilia occurs throughhydrodynamic forces, whereas neural activity is the primarydeterminant of phase locking of crustacean swimmerets. (v) Thefluid dynamics resulting from cellular level metachronal wavesare characterized by Reynolds numbers (Re) close to 0. The Recharacterizes the relative importance of inertial forces to viscousforces in the flow. For Re ≈ 0, viscous forces dominate and in-ertial forces can be neglected. The Re increases with the size of

the animal (characteristic length scale) and decreases with thestroke period (characteristic time scale). The natural variation ina crustacean’s size and stroke frequency leads to Re ranging fromabout 10 to 1,000 (SI Text, section 1.1), under which both viscousand inertial effects are relevant. Hence, the fluid dynamics of ciliabeating are significantly different from the fluid dynamics of crus-tacean swimming. Relatively few studies have examined meta-chronal limb paddling for the range of Re under which crustaceansoperate (28–30). Recently, a model based on drag forces alonepredicted a slight mechanical advantage of metachronal wave in krillswimming (31). However, this model does not capture the nonlinearinteractions of the forces generated by the multiple limbs arisingfrom the local flow, which have a significant effect on swimmingwhen limbs are close to each other. To capture this essential effect,we build a computational fluid dynamics model and numericallycompute the flow field produced by crayfish’s tail-to-head meta-chronal limb stroke pattern and other hypothetical patterns.Our fluid dynamics model consists of four rigid paddles as

limbs moving with prescribed motion attached to a fixed wallimmersed in a two-dimensional fluid. To approximate the factthat swimmerets are straight and fanned-out during PS and arecurled and folded during RS, we treat swimmerets as imperme-able during PS and permeable during RS (see Materials andMethods, SI Text, section 1, and Figs. S1–S6 for details).We compute the flux of the fluid moving in the tail direction and

take its time average as a measure of the effectiveness of theswimmeret stroke pattern. Fig. 2B depicts the flux of three differentstroke patterns (the natural tail-to-head metachronal wave with 0.25intersegmental phase differences, the in-phase rhythm, and thehead-to-tail metachronal wave with 0.75 intersegmental phase dif-ferences) at an intermediate Reynolds number (Re = 200). Bothmetachronal stroke patterns produce smoother temporal variationin flux compared to the in-phase rhythm because PS and RS areevenly distributed in time among the four limbs. A remarkable ob-servation is that the natural tail-to-head metachronal wave producesa 60% increase in average flux over that of the in-phase rhythm anda 500% increase over that of the head-to-tail metachronal wave.To illustrate this increased effectiveness of the natural tail-to-

head metachronal wave over other stroke patterns, we put infree-flowing tracers in the flow and observe how different strokepatterns lead to different tracer displacement. To allow the ini-tial transient effect to disappear, we first let the swimmerets beatfor five periods. At the end of the fifth period, we put in thepassive tracers uniformly underneath the body. We observe howthe passive tracers move as the swimmerets undergo another fiveperiods of strokes under the three different stroke patterns (Fig.2A and Movies S1–S3). Fig. 2A illustrates that, with the natural0.25 phase-locked tail-to-head metachrony, the majority of thetracers are propelled toward the tail direction. The in-phaserhythm is less effective in driving the tracers toward the tail di-rection compared to the natural tail-to-head metachrony. Withthe 0.75 phase-locked head-to-tail metachrony, it is not clearwhether the tracers are flowing in any particular direction.Overall, among the three stroke patterns, the natural tail-to-headmetachronal wave of swimmeret coordination is the most ef-fective stroke pattern in maximizing flux.The above results are based on simulations for an intermediate

Reynolds number (Re = 200). As the Re changes, the flow char-acteristics change significantly (Movies S1–S3). Nevertheless, therelative advantage of the tail-to-head metachronal wave over othermetachronies is preserved across biologically relevant Reynoldsnumbers. Fig. 2C shows that the natural tail-to-head metachronalwave of swimmeret coordination with 0.25 intersegmental phasedifferences produces the largest average flux among all meta-chronal waves for Re= 50, 200, and 800.As a measure of the efficiency of the stroke pattern, we nor-

malize the average flux by the average power consumption perstroke period (SI Text, section 1.3). The natural tail-to-head

Fig. 1. Forward swimming of a crustacean. Cycles of power strokes (PSs) andreturn strokes (RSs) of the swimmerets provide the thrust for forward swim-ming. Movement of neighboring swimmerets maintains a quarter-periodphase difference with the more posterior swimmeret leading the cycle. Imagecourtesy of Wikimedia Commons/Øystein Paulsen.

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metachronal wave of swimmeret coordination achieves close-to-maximal efficiency among all metachronies over a wide range ofbiologically relevant Reynolds numbers. Fig. 2D shows that theefficiency peaks at 0.125 phase difference closely followed by the0.25 phase difference. Across Reynolds numbers from 50 to 800,the natural tail-to-head metachronal wave is up to 30% moreefficient than the in-phase rhythm and 300% to 550% more ef-ficient than the head-to-tail metachronal wave.Overall, the above results show that the natural 0.25 phase-

locked metachrony leads to near-maximal effectiveness andefficiency over the full span of biologically relevant Reynoldsnumbers in crustacean swimming, i.e., a significant biomechanicaladvantage is preserved despite the large variation in crustaceans’size and swimmeret stroke frequency. The biomechanical opti-mality of the tail-to-head metachronal wave with approximate0.25 intersegmental phase differences that is independent ofswimmeret beat frequencies raises the question: What are theneural mechanisms that maintain this limb coordination in sucha robust manner?

A Robust Neural Mechanism Producing Phase ConstancyThe isolated neural ventral cord of the crayfish, which containsthe neural circuit driving the movement of the swimmerets, dis-plays fictive locomotion (Fig. 3A). That is, it expresses rhythmicneural activity that is analogous to the distinct stroke patternobserved behaviorally (13, 32). This centrally generated rhythm isthe primary determinant of the swimmeret coordination (SI Text,sections 1.2, 2.1, and 3). Experiments on the crayfish neuralventral cord indicate that (i) each swimmeret is innervated by ananatomically separate and functionally independent CPG (22,33), and (ii) these CPGs are connected through ascending anddescending coordinating neurons (19). Thus, the neural circuitdriving metachronal swimmeret movements can be considered asa chain of four pairs of neuronal oscillators.A useful mathematical framework for studying dynamics of

interconnected CPGs is the coupled phase model (17, 34), wherethe state of each CPG is described completely by its phase. If theith CPG is isolated, then its phase θi (0≤ θi < 1) will evolve at itsintrinsic frequency ω, i.e., θi = ðωt+ϕ0

i Þ mod 1, where ϕ0i is the

initial phase of the CPG. If the ith CPG is coupled with the jthCPG, then the rate of change of phase will be sped up or sloweddown due to input from this intersegmental coupling. Themagnitude of the acceleration or deceleration of phase dependson the timing and structure of the input from the jth CPG andthe state-dependent response of the ith CPG. This effect isquantified by the “interaction function,” which is a function of

the phase difference between the two coupled CPGs (18, 35) andis related to the phase response curve (SI Text, section 2.2).Previously, Skinner et al. (15) used the phase model framework

to describe the neural circuit of the crayfish swimmeret system asa chain of four generic oscillators with nearest-neighbor coupling:8>>>>>>><>>>>>>>:

dθ1dt

=ω+Hascðθ2 − θ1Þ;dθidt

=ω+Hascðθi+1 − θiÞ+Hdscðθi−1 − θiÞ;dθ4dt

=ω+Hdscðθ3 − θ4Þ;

i= 2 and 3;

[1]

where Hasc and Hdsc are the interaction functions for ascend-ing and descending connections, respectively. Because the in-teraction functions are functions of the phase differencebetween the CPGs, they are 1-periodic functions. Phase-lockedrhythms correspond to states in which the intersegmental phasedifferences Δθi = θi+1 − θi are constant, i.e., ðdΔθiÞ=dt= 0 fori= 1;   2;   and  3. Metachronal waves correspond to phase-lockedstates where Δθi are equal. Note that the tail-to-head 0.25 phase-locked state requires both Hascð0:25Þ= 0 and Hdscð−0:25Þ= 0. Thequestion remains as to how the structure of the crayfish neuralcircuit maintains these two constraints on the interaction functions,and therefore maintains the optimal stroke pattern, over a broadrange of stroke frequencies. However, recent experimental findingselucidating the structure of the crayfish swimmeret neural circuitallow this question to be addressed.Accumulated anatomical and physiological results on the local

and intersegmental circuitry of the crayfish swimmeret system(13, 19, 20, 23, 36) have revealed the following circuit architec-ture (for a more detailed description of the circuit, see SI Text,section 2.1 and Fig. S7): Each local CPG is composed of a half-center oscillator (HCO) that consists of two mutually inhibitedneurons, a P cell and an R cell, that oscillate in antiphase. The Pcell drives the PS motor neurons, and the R cell drives the RSmotor neurons. Each local HCO has the same frequency andis effectively coupled with its nearest neighbor(s) through in-tersegmental connections diagramed in Fig. 3B. The descendingconnection is effectively excitatory and goes from the P cell of anHCO to the R cell of its more posterior neighbor HCO. The as-cending connection is effectively excitatory and goes from the R cellin an HCO to the R cell of its more anterior neighbor HCO.The ascending and the descending intersegmental connections

Fig. 2. Computational fluid dynamics model ofcrayfish swimmeret paddling. (A) Snapshots ofthe flow field at Reynolds number 200 under thenatural tail-to-head metachrony with 0.25 phasedifference, the hypothetical in-phase rhythm,and the hypothetical head-to-tail metachronywith 0.75 phase difference. Free-flowing tracersare shown as green dots; red denotes positivevorticity; and blue denotes negative vorticity. (B)Flux vs. normalized time for flows in A. (C ) Av-erage flux vs. phase difference. (D) Efficiency vs.phase difference. For B–D, the curves are nor-malized by the time-averaged flux or efficiencyof the in-phase rhythm. For C and D, results forthree Reynolds numbers are provided: 50 (cyancurve with “O”), 200 (black curve with “+”), and800 (magenta curve with “☐”).

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between nearest neighbors have similar strength and dynamics.We will show that the CPGs’ half-center structure and theasymmetric intersegmental connectivity described above providea robust mechanism for generating the 0.25 phase constancyindependent of the intrinsic properties of the CPGs and detailsof the intersegmental coupling.

A Phase Model for Two Coupled HCOs. It is instructive to first considera subnetwork of two coupled HCOs in which each HCO has onlyone incoming connection and one outgoing connection. Because(i) the ascending connection between the HCOs is between Rcells, (ii) the descending connection links a P cell to an R cell, and(iii) the P and R cells in each HCO are in antiphase, thedescending output is effectively phase delayed by a half-periodwith respect to the ascending output. In the phase model, theentire local module is considered to be an oscillatory unit, whosestate is described by its phase θi alone, but the ascending anddescending outputs are in antiphase, i.e., θi and θi + 0:5, re-spectively. Hence, by defining Hðθ2 − θ1Þ=Hascðθ2 − θ1Þ, theinteraction function for the descending connection can be rewritten

as Hdscðθ1 − θ2Þ=Hðθ1 − θ2 + 0:5Þ. Therefore, the dynamics of thetwo HCOs is described by a system of two differential equations:

8>><>>:

dθ1dt

=ω+Hascðθ2 − θ1Þ=ω+Hðθ2 − θ1Þ;dθ2dt

=ω+Hdscðθ1 − θ2Þ=ω+Hðθ1 − θ2 + 0:5Þ:[2]

It follows that the phase difference Δθ= θ2 − θ1 between the HCOsis given by a single differential equation:

dΔθdt

=Hð−Δθ+ 0:5Þ−HðΔθÞ: [3]

Because H has a period of 1, Eq. 3 reveals that Δθ = 0.25 and0.75 are always phase-locked states. We stress that this result isindependent of the frequency of the oscillator and does not relyon tuning any specific biophysical parameters. This implies thatthe phase constancy of the 0.25 phase-locked state arises robustlyfrom the organization of the neuronal circuitry (37).The invariance of the phase-locked state with the 0.25 phase

difference can be understood by considering the timing of the out-put from thehalf-center oscillators relative to the phase atwhich theoscillators receive the input. Suppose thatHCO-2 is phaseadvancedby 0.25 relative toHCO-1. TheR cell ofHCO-1 receives input fromthe R cell of HCO-2 that is phase advanced by 0.25. On the otherhand, theR cell of HCO-2 receives input from the P cell of HCO-1,which is a half-period out of phase with the R cell of HCO-1. Thus,theRcell ofHCO-2 receives input that is effectivelyphaseadvancedby 0.25 (i.e., −0:25+ 0:5). This implies that both oscillators receiveinput at the exact same phases in their cycles. Therefore, they adjusttheir instantaneous frequencies in the exact same way, and HCO-2remains phase advanced by 0.25 relative to HCO-1.

The Full Circuit of Four Coupled HCOs. Although the crayfish’s four-HCO neural circuit is not a trivial generalization of the two-HCO circuit, the half-center structure of the CPG and theasymmetric intersegmental connectivity are still the key organi-zational features of the four-HCO neural circuit that gives riseto an approximate 0.25 phase constancy. From Eq. 1, the dy-namics of the phase differences between each neighboring HCO½ðΔθ1;Δθ2;Δθ3Þ, where Δθi = θi+1 − θi� are described by:8>>>>>>><>>>>>>>:

dΔθ1dt

=HascðΔθ2Þ+Hdscð−Δθ1Þ−HascðΔθ1Þ;dΔθ2dt

=HascðΔθ3Þ+Hdscð−Δθ2Þ−HascðΔθ2Þ−Hdscð−Δθ1Þ;dΔθ3dt

=Hdscð−Δθ3Þ−HascðΔθ3Þ−Hdscð−Δθ2Þ:[4]

As mentioned earlier, ðΔθ1;Δθ2;Δθ3Þ= ð0:25; 0:25; 0:25Þ is aphase-locked state if and only if both Hascð0:25Þ= 0 andHdscð−0:25Þ= 0. Note that these are two independent conditionsthat the crayfish swimmeret neural circuit needs to satisfy. However,because of the half-center structure of the CPGs and the interseg-mental connectivity (Fig. 3B), one of the above two conditions iseliminated. Recall that the descending outputs are effectively phasedelayed by a half-period with respect to the ascending outputs, i.e.,Hascðθ2 − θ1Þ=Hðθ2 − θ1Þ and Hdscðθ1 − θ2Þ=Hðθ1 − θ2 + 0:5Þ.By substituting these expressions into Eq. 4, we find thatðΔθ1;Δθ2;Δθ3Þ= ð0:25; 0:25; 0:25Þ is a phase-locked state if andonly ifHð0:25Þ= 0. That is, because of the special organization ofthe four-HCO neural circuit, the crayfish swimmeret neural cir-cuit only needs to meet one condition to produce the tail-to-head 0.25 phase locking.

Fig. 3. Neuronal circuitry underlying the crayfish swimmeret system. (A)The neural circuit underlying the crayfish swimmeret system consists ofa chain of four CPGs located in the abdominal ganglion A2–A5. Each CPGinnervates motor neurons that drive the power strokes (PSs) and returnstrokes (RSs) of a swimmeret. The dissected neural cord in isolation cangenerate the tail-to-head metachronal wave of neuronal activity as seenin the simultaneous extracellular recordings from the four PS motornerves, PS2–PS5. (B) Schematic diagram of the neural circuit composed ofa chain of four coupled CPGs. Each CPG is modeled as an HCO consistingof two mutually inhibited cells (large circles in diagram) that oscillate inantiphase. Cells in HCOs denoted by iP drive PS motor neurons, and cellsdenoted by iR drive RS motor neurons (i = 1, 2, 3, and 4). The small solidblack circles symbolize inhibitory connections, the small solid trianglessymbolize excitatory connections, and the colors indicate the origin ofeach intersegmental connection. (C ) Experimentally measured inter-action function H for a crayfish swimmeret CPG. (D) The natural networktopology (a1) in the crayfish swimmeret neural circuit and three hypo-thetical network topologies, (a2), (s1), and (s2), in a chain of four HCOswith nearest-neighbor coupling. Only the middle two HCOs are shown.The small open circles symbolize either excitatory or inhibitory con-nections.

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Furthermore, the condition Hð0:25Þ= 0 can be relaxed ifwe allow a small deviation from the exact 0.25 phase locking,i.e., if we consider an approximate 0.25 phase-locked solu-tion ðΔθ1;Δθ2;Δθ3Þ= ð0:25+ «; 0:25; 0:25− «Þ for some small «.From Eq. 4, an approximate 0.25 phase locking exists if andonly if Hð0:25Þ+ Hð0:25− «Þ−Hð0:25+ «Þ= 0. This conditionis equivalent to the condition that «≈Hð0:25Þ=2H′ð0:25Þ is small.For example, if the ratio jHð0:25Þ=H′ð0:25Þj is less than 20%, theresulting phase differences will be between 0.15 and 0.35 approx-imately. In addition, one can show that the tail-to-head 0.25 phaselocking is stable if H has a positive slope at 0.25, i.e., H′ð0:25Þ> 0(SI Text, section 2.3 and Tables S1 and S2). Note that, in the aboveanalysis, we have assumed identical strength for ascending anddescending connections because experimental results have shownthat they are similar in strength and dynamics. In SI Text, section2.4, we further show that this equal strength minimizes the de-viation of the phase-locked state from 0.25.

Experimentally Measured Interaction Function. Our analysis showsthat if each crayfish swimmeret local CPG (the kernel of which isa HCO) satisfies the condition that «=Hð0:25Þ=2H′ð0:25Þ issmall, then the swimmeret neuronal circuitry gives rise to therobust 0.25 phase locking. Indeed, Fig. 3C shows that the ex-perimentally measured interaction function HðθÞ is relativelysmall at 0.25, and has a relatively steep positive slope at thispoint, giving rise to «≈Hð0:25Þ=2H′ð0:25Þ= 0:06. Therefore,both the existence condition (« is small) and the stability con-dition [H′ð0:25Þ> 0] predicted by our mathematical model aresatisfied. It can be further shown that these conditions on theinteraction function arise from the generic phase responseproperties of HCOs (SI Text, section 2.6).

DiscussionOur results demonstrate that the distinct limb coordination in crus-tacean swimming provides the biomechanically optimal stroke pat-tern over a wide range of biologically relevant paddling frequenciesand animal sizes, i.e., Reynolds numbers (Re). Furthermore, therelative simplicity of the crustacean nervous system and recentadvances in the knowledge of the crayfish swimmeret neuronalcircuit allow us to identify how the structure of the circuit robustlygives rise to this stroke pattern. Thus, the swimmeret system of long-tailed crustaceans serves as a concrete example of how the archi-tecture of a neural circuit gives rise to optimal locomotor behavior.Our computational fluid dynamics simulations show that the flow

characteristics are substantially different as the Re changes from 50to 800 (Movies S1–S3). Despite these differences, the distinct tail-to-head metachronal stroke pattern maintains a significant advan-tage over other stroke patterns. Furthermore, the relative advantageof this limb coordination persists over the natural variation of limbspacing (SI Text, section 1.3, and Fig. S6). This suggests that theprimary mechanism that gives the distinct tail-to-head metachronalstroke pattern a significant advantage over other stroke patternsdoes not involve careful timing of limb movements to exploit subtlefluid–structure interactions. Instead, we conjecture that the advan-tage arises from a simple, robust geometric mechanism based on theasymmetric arrangement of the neighboring limbs during PSs andRSs. Because PS generates positive flux whereas RS counteracts theflux generation, an efficient stroke pattern should maximize theeffect of PS while minimizing the effect of RS. As illustrated inFig. 4, under the natural tail-to-head metachrony with 0.25 phasedifference, the volume of fluid enclosed by the neighboring limbsduring PS is much larger than that during RS. This significantasymmetry in the volume of fluid enclosed by the PS and the RSleads to a very effective mechanism for generating positive flux.Therefore, a significantly larger average flux is generated underthe natural 0.25 phase-locked rhythm compared to the case withthe in-phase rhythm, under which the PS and the RS enclose thesame volume of fluid. Similar to the reasoning above, under the

hypothetical 0.75 phase-locked head-to-tail metachrony, the PSencloses a much smaller volume of fluid than the RS does.Hence, the head-to-tail metachrony is an ineffective stroke pat-tern for generating positive flux.Experimental results show that the timing of the swimmeret

movements is strongly correlated with the timing of the bursts ofmuscle activity caused by motor neuron input (38). Furthermore,long-tailed crustaceans exhibit the distinct tail-to-head strokepattern in a variety of swimming modes, including forwardswimming and hovering, in which the limbs experience differenthydrodynamic forces (29). These observations suggest that theintersegmental phase differences between swimmerets result pri-marily from neural input to the muscles rather than from theinteraction between hydrodynamic forces and passive body me-chanics. Neural input to the muscles is shaped by the rhythm in-trinsically generated in the central nervous system and sensoryfeedback. However, while proprioceptive reflexes appear to beable to increase the PS motor drive to compensate for changes inload on the swimmerets (39, 40), extensive experimental evidencesuggests that proprioceptive feedback has little effect on interlimbcoordination (41–43). Thus, sensory feedback, hydrodynamicforces, and limb mechanics are likely to influence the stroke ki-nematics of an individual limb and the animal’s overall swimmingperformance, but the feedforward drive from the central nervoussystem is the primary determinant of coordination between limbs.(See SI Text, section 1.2 and section 3, for details.)The key organizational features of the crayfish neural circuit for

producing the phase constancy with 0.25 intersegmental phasedifferences are the internal half-center structure of the local CPGand the topology of the ascending and descending connectionsbetween the local CPGs. To highlight this fact, let us consider otherhypothetical patterns of intersegmental connection in a chain offour HCO-based CPGs. Fig. 3D shows the four fundamentallydifferent connectivity patterns that can occur between HCOs withonly one ascending and one descending connection. Connectionscheme (a1) is the connectivity that is present in the crayfishswimmeret neural circuit, and (a2), (s1), and (s2) are three hypo-thetical network topologies. Depending on whether the coupling isbetween a P cell and an R cell or between the same cell type, theoverall network topology can be asymmetric or symmetric. It canbe shown (SI Text, section 2.3) that a symmetric network topology(s1) or (s2) can robustly produce the in-phase rhythm and theantiphase rhythm (0.5 phase difference); however, they do notrobustly produce the natural tail-to-head metachronal wave (0.25phase difference). Furthermore, given the response properties ofthe crayfish swimmeret local CPGs (as in Fig. 3C), the asymmetrictopology (a2) can robustly produce the head-to-tail metachronalwave (0.75 phase difference) but not the natural tail-to-headmetachronal wave (0.25 phase difference).The results discussed above can be extended beyond circuits with

single ascending and descending connections between neighboringmodules and equal connection strengths. In fact, it can be shownthat, in the chain of four HCOs, equal ascending and descendingconnection strengths minimize the deviation of the intersegmentalphase difference from 0.25 (SI Text, section 2.4). Furthermore, ifmultiple ascending or descending connections between neighbor-ing HCOs are permitted, then circuit topologies other than (a1) canproduce robust phase locking with phase lags of 0.25, but these con-nectivity schemes must have an analogous structure to the (a1)

Fig. 4. Position of neighboring limbs in the middle of the PS and the RS,respectively, in the natural 0.25 phase-locked rhythm.

13844 | www.pnas.org/cgi/doi/10.1073/pnas.1323208111 Zhang et al.

connection scheme. That is, descending connections must link P andR cells, whereas ascending connections must link R cells to R cellsand/or P cells to P cells, and the connection strengths must be ap-propriately balanced (SIText, section 2.5). Although it is possible thatother connectivity schemes could support the 0.25 tail-to-head met-achronal limb coordination, well-tuned compensatory mechanismswould be required tomaintain this coordination over a wide range ofpaddling frequencies. Thus, the natural asymmetric circuit topology(a1) appears to generate this stroke pattern with the maximal ro-bustness and minimal requirements.HCOs are general, primitive motifs in locomotor CPGs and

often serve as building blocks for networks of CPGs (44, 45).This fact and our finding that the natural asymmetric networktopology in the crayfish swimmeret neural circuit is superior toall other intersegmental connectivity schemes in a chain ofHCOs in robustly producing the biomechanically optimal tail-to-head metachronal stroke pattern suggest that similar neuralcircuits drive coordinated swimmeret movement in all long-tailedcrustaceans. Furthermore, given the large metabolic cost of crus-tacean swimming (6, 46), our results suggest that the asymmetricnetwork topology is the result of natural selection in favor of moreeffective and efficient swimming.

Materials and MethodsIn our fluid model of the swimmeret system, the motion of the limbs drives themotion of the surrounding fluid. The equations that determine themotion of thefluid are Navier–Stokes equations. We use the immersed boundary method, inwhich structures (limbs and body) are represented in a moving, Lagrangian co-ordinate system, whereas fluid variables are represented in a fixed, Eulerian co-ordinated system (47). Because the limbs during PSs and the body areimpermeable, the fluid is forced to move with the prescribed velocity onthese structures. When the limbs become permeable during RSs, the fluid willpermeate throughtheorthogonaldirectionof the limbwithaslipvelocity.SI Text,section 1,providesadetaileddescriptionofourfluidmodel, thebiomechanics ofthe swimmerets, and the numerical method for the fluid model.

To experimentally measure the interaction function H of a crayfishswimmeret local CPG, we effectively isolate local CPGs in the isolated neuralventral cord preparation and subject them to input that mimics the inputfrom neighboring CPGs (19, 20). By varying the phase of the input within thelocal CPG’s cycle and plotting changes in cycle period as a function of phase,we construct the interaction function H plotted in Fig. 3C (SI Text, section2.2; see also figure 2 in ref. 20) (48).

ACKNOWLEDGMENTS. This work was partially supported by the NationalScience Foundation under Grant CRCNS 0905063 (to B.M. and T.J.L.). R.D.G. ispartially supported by the National Science Foundation under Grants DMS-1226386 and DMS-1160438. B.M. is partially supported by the NationalScience Foundation under Grant IOS 1147058.

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