Skip to main content
Log in

The role of long-range coupling in crayfish swimmeret phase-locking

  • Original Article
  • Published:
Biological Cybernetics Aims and scope Submit manuscript

Abstract

During forward swimming, crayfish and other long-tailed crustaceans rhythmically move four pairs of limbs called swimmerets to propel themselves through the water. This behavior is characterized by a particular stroke pattern in which the most posterior limb pair leads the rhythmic cycle and adjacent swimmerets paddle sequentially with a delay of roughly 25% of the period. The neural circuit underlying limb coordination consists of a chain of local modules, each of which controls a pair of limbs. All modules are directly coupled to one another, but the inter-module coupling strengths decrease with the distance of the connection. Prior modeling studies of the swimmeret neural circuit have included only the dominant nearest-neighbor coupling. Here, we investigate the potential modulatory role of long-range connections between modules. Numerical simulations and analytical arguments show that these connections cause decreases in the phase-differences between neighboring modules. Combined with previous results from a computational fluid dynamics model, we posit that this phenomenon might ensure that the resultant limb coordination lies within a range where propulsion is optimal. To further assess the effects of long-range coupling, we modify the model to reflect an experimental preparation where synaptic transmission from a middle module is blocked, and we generate predictions for the phase-locking properties in this system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  • Chevallier S, Ijspeert AJ, Ryczko D, Nagy F, Cabelguen JM (2008) Organisation of the spinal central pattern generators for locomotion in the salamander: biology and modelling. Brain Res Rev 57(1):147–161

    Article  PubMed  Google Scholar 

  • Cohen AH, Ermentrout GB, Kiemel T, Kopell N, Sigvardt KA, Williams TL (1992) Modelling of intersegmental coordination in the lamprey central pattern generator for locomotion. Trends Neurosci 15(11):434–438

    Article  PubMed  CAS  Google Scholar 

  • Daun-Gruhn S (2011) A mathematical modeling study of inter-segmental coordination during stick insect walking. J Comput Neurosci 30(2):255–278

    Article  PubMed  Google Scholar 

  • Fuchs E, Holmes P, Kiemel T, Ayali A (2011) Intersegmental coordination of cockroach locomotion: adaptive control of centrally coupled pattern generator circuits. Front Neural Circuits 4:125

    PubMed  PubMed Central  Google Scholar 

  • Hughes G, Wiersma C (1960) The co-ordination of swimmeret movements in the crayfish, Procambarus clarkii (Girard). J Exp Biol 37(4):657–670

    Google Scholar 

  • Jones S, Kopell N (2006) Local network parameters can affect inter-network phase lags in central pattern generators. J Math Biol 52(1):115–140

    Article  PubMed  CAS  Google Scholar 

  • Jones SR, Mulloney B, Kaper TJ, Kopell N (2003) Coordination of cellular pattern-generating circuits that control limb movements: the sources of stable differences in intersegmental phases. J Neurosci 23(8):3457–3468

    Article  PubMed  CAS  Google Scholar 

  • Kopell N, Zhang W, Ermentrout G (1990) Multiple coupling in chains of oscillators. SIAM J Math Anal 21(4):935–953

    Article  Google Scholar 

  • Laverack M, Macmillan D, Neil D (1976) A comparison of beating parameters in larval and post-larval locomotor systems of the lobster Homarus gammarus (L.). Philos Trans R Soc B Biol Sci 274(929):87–99

    Article  CAS  Google Scholar 

  • Marder E, Bucher D (2001) Central pattern generators and the control of rhythmic movements. Curr Biol 11(23):R986–R996

    Article  PubMed  CAS  Google Scholar 

  • Mulloney B (1997) A test of the excitability-gradient hypothesis in the swimmeret system of crayfish. J Neurosci 17(5):1860–1868

    Article  PubMed  CAS  Google Scholar 

  • Mulloney B, Smarandache C (2010) Fifty years of CPGs: two neuroethological papers that shaped the course of neuroscience. Front Behav Neurosci 4:45

    PubMed  PubMed Central  Google Scholar 

  • Mulloney B, Smarandache-Wellmann C (2012) Neurobiology of the crustacean swimmeret system. Prog Neurobiol 96(2):242–267

    Article  PubMed  PubMed Central  Google Scholar 

  • Mulloney B, Harness PI, Hall WM (2006) Bursts of information: coordinating interneurons encode multiple parameters of a periodic motor pattern. J Neurophysiol 95(2):850–861

    Article  PubMed  Google Scholar 

  • Paul DH, Mulloney B (1986) Intersegmental coordination of swimmeret rhythms in isolated nerve cords of crayfish. J Comp Physiol A 158(2):215–224

    Article  Google Scholar 

  • Perkel DH, Mulloney B (1974) Motor pattern production in reciprocally inhibitory neurons exhibiting postinhibitory rebound. Science 185(4146):181–183

    Article  PubMed  CAS  Google Scholar 

  • Schwemmer M, Lewis TJ (2012) The theory of weakly coupled oscillators. PRCs Neurosci Theory Exp Anal 6:3–31

    Google Scholar 

  • Skinner FK, Kopell N, Mulloney B (1997) How does the crayfish swimmeret system work? Insights from nearest-neighbor coupled oscillator models. J Comput Neurosci 4(2):151–160

    Article  PubMed  CAS  Google Scholar 

  • Smarandache C, Hall WM, Mulloney B (2009) Coordination of rhythmic motor activity by gradients of synaptic strength in a neural circuit that couples modular neural oscillators. J Neurosci 29(29):9351–9360

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  • Smarandache-Wellmann C, Grätsch S (2014) Mechanisms of coordination in distributed neural circuits: encoding coordinating information. J Neurosci 34(16):5627–5639

    Article  PubMed  CAS  Google Scholar 

  • Smarandache-Wellmann C, Weller C, Wright TM, Mulloney B (2013) Five types of nonspiking interneurons in local pattern-generating circuits of the crayfish swimmeret system. J Neurophysiol 110(2):344–357

    Article  PubMed  PubMed Central  Google Scholar 

  • Smarandache-Wellmann C, Weller C, Mulloney B (2014) Mechanisms of coordination in distributed neural circuits: decoding and integration of coordinating information. J Neurosci 34(3):793–803

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  • Somers D, Kopell N (1993) Rapid synchronization through fast threshold modulation. Biol Cybern 68(5):393–407

    Article  PubMed  CAS  Google Scholar 

  • Tschuluun N, Hall WM, Mulloney B (2001) Limb movements during locomotion: tests of a model of an intersegmental coordinating circuit. J Neurosci 21(19):7859–7869

    Article  PubMed  CAS  Google Scholar 

  • Wang XJ, Rinzel J (1992) Alternating and synchronous rhythms in reciprocally inhibitory model neurons. Neural Comput 4(1):84–97

    Article  Google Scholar 

  • Williams TL, Sigvardt KA, Kopell N, Ermentrout GB, Remler MP (1990) Forcing of coupled nonlinear oscillators: studies of intersegmental coordination in the lamprey locomotor central pattern generator. J Neurophysiol 64(3):862–871

    Article  PubMed  CAS  Google Scholar 

  • Zhang C, Lewis TJ (2013) Phase response properties of half-center oscillators. J Comput Neurosci 35(1):55–74

    Article  PubMed  CAS  Google Scholar 

  • Zhang C, Lewis TJ (2016) Robust phase-waves in chains of half-center oscillators. J Math Biol 74(7):1627–1656

    Article  PubMed  Google Scholar 

  • Zhang C, Guy RD, Mulloney B, Zhang Q, Lewis TJ (2014) Neural mechanism of optimal limb coordination in crustacean swimming. Proc Natl Acad Sci 111(38):13840–13845

    Article  PubMed  CAS  Google Scholar 

Download references

Acknowledgements

The authors thank Brian Mulloney for helpful discussions. This work was partially supported by the National Science Foundation under Grant CRCNS 0905063 to TJL, and Grant DMS-0931642 to LES.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Lucy E. Spardy.

Additional information

Communicated by J. Leo van Hemmen.

Appendices

Appendices

A Details of the conductance-based model

In Sect. 2.2, we describe numerical simulations of a chain of half-center oscillators (HCO) in which the dynamics of each HCO are governed by the Wang–Rinzel model. Each HCO contains a “P” and “R” neuron (see Fig. 1b). The membrane potentials and gating variables of the neurons in the jth HCO are governed by

$$\begin{aligned} CV_{jP}'&=-g_{pir}m^3_{\infty }(V)h_{jP}(V_{jP}-V_{pir})-g_L(V_{jP}-V_L) \\&\quad -g_{Syn_I}s(V_{jR},\theta _I)(V_{jP}-V_{Syn_I}) \\ h_{jP}'&=\phi [h_\infty (V_{jP})-h_{jP}]/\tau _h(V_{jP}) \\ CV_{jR}'&=-g_{pir}m^3_{\infty }(V)h_{jR}V_{jR}-V_{pir})-g_L(V_{jR}-V_L) \\&\quad - g_{Syn_I}s(V_{iP},\theta _I)(V_{iR}-V_{Syn_I}) \\&\quad -\sum _{i \in exc(j)} g_{Syn_E}s(V_{i},\theta _E)(V_{jR}-V_{Syn_E}) \\ h_{jR}'&=\phi [h_\infty (V_{jR})-h_j]/\tau _h(V_{jR}) \end{aligned}$$

with \(j \in \{1,2,3,4\}\) and \(m_\infty (V)=\frac{1}{1+e^{(V+65)/7.8}}\), \(h_\infty (V)=\frac{1}{1+e^{(V+81)/11}}\), \(\tau _h(V)=h_\infty (V)e^{(V+162.3)/17.8},\) and \(s(V,\theta )=\frac{1}{1+e^{-(V-\theta )/k_{Syn}}}\) (Wang and Rinzel 1992). The excitatory inputs to neuron jR are indexed by exc(j), respectively, and correspond to the schematic shown in Fig. 1b. Parameters were adapted from (Wang and Rinzel 1992): \(C=1\,\upmu \hbox {F/cm}^2\), \(V_{pir}=120\,\hbox {mV}\), \(g_L=0.1\,\hbox {mS/cm}^2\), \(V_L=-60\,\hbox {mV}\), \(V_{SynI}=-80\,\hbox {mV}\), \(\phi =3\), \(V_{SynE}=0\,\hbox {mV}\), \(g_{Syn_I}=0.2\,\hbox {mS/cm}^2\), \(g_{Syn_E}=0.03\cdot g_{Syn_I}\,\hbox {mS/cm}^2\), \(\theta _I=-44\,\hbox {mV}\), \(k_{Syn}=2\,\hbox {mV}\), \(\theta _{E}=-56\,\hbox {mV}\), \(g_{PiR}=0.3\,\hbox {mS/cm}^2\).

B The effects of the longest-range connections

Here, we consider the effects from the connections between the most posterior and the most anterior oscillators, i.e., the coupling between next-next-nearest-neighbors.

1.1 B.1 Phase model of the complete neural circuitry

We augment the phase model (system (1)) with connections between oscillators 1 and 4 to obtain

$$\begin{aligned} \dot{\theta _1}&=\omega + H_A(\theta _2-\theta _1)+\beta H_A(\theta _3-\theta _1)+ \gamma H_A(\theta _4-\theta _1)\nonumber \\ \dot{\theta _2}&=\omega +H_D(\theta _1-\theta _2)+H_A(\theta _3-\theta _2)+\beta H_A(\theta _4-\theta _2)\nonumber \\ \dot{\theta _3}&=\omega +H_D(\theta _2-\theta _3) +H_A(\theta _4-\theta _3)+\beta H_D(\theta _1-\theta _3)\nonumber \\ \dot{\theta _4}&=\omega +H_D(\theta _3-\theta _4) +\beta H_D(\theta _2-\theta _4) + \gamma H_D(\theta _1-\theta _4). \end{aligned}$$
(13)

Similar to the parameter \(\beta \) that scales the long-range (next-nearest-neighbor) connections, we scale the longest-range connections by the parameter \(\gamma \). Recall that the long-range coupling strength (\(\beta \)) has been shown in experiments to be roughly 30% of the strength between nearest-neighbors, and the longest-range coupling strength (\(\gamma \)) has been shown to be roughly 10% of the strength between neighbors.

By incorporating the topology of the connections into our model as we did in Sect. 2.3.1, we obtain an analogous system that governs the evolution of phase-differences between neighboring HCOs

$$\begin{aligned} \dot{\phi _1}&=H(-\phi _1+0.5)+H(\phi _2)-H(\phi _1) \nonumber \\&\quad +\beta [H(\phi _2+\phi _3)- H(\phi _1+\phi _2)] - \gamma H(\phi _1+\phi _2+\phi _3) \nonumber \\ \dot{\phi _2}&=H(-\phi _2+0.5)+H(\phi _3) -H(-\phi _1+0.5)-H(\phi _2) \nonumber \\&\quad +\beta [H(-\phi _1-\phi _2+0.5)-H(\phi _2+\phi _3)] \nonumber \\ \dot{\phi _3}&=H(-\phi _3+0.5)-H(-\phi _2+0.5)-H(\phi _3) \nonumber \\&\quad +\beta [H(-\phi _2-\phi _3+0.5)- H(-\phi _1-\phi _2+0.5)]\nonumber \\&\quad + \gamma H(-\phi _1-\phi _2-\phi _3+0.5). \end{aligned}$$
(14)

1.2 B.2 The influence of “longest-range” coupling is different than the influence of next-nearest-neighbor coupling

We perform a perturbation analysis as in Sect. 2.3.4, and look for steady-state phase-differences for system (14) of the form

$$\begin{aligned} \phi _j=\phi _0 + \epsilon \phi _j^* +\mathcal {O}(\epsilon ^2). \end{aligned}$$

As before, we assume that \(\epsilon \) scales the shift term \(\delta \), the long-range coupling strength \(\beta \), and now the longest-range coupling strength \(\gamma \), i.e., \(\gamma =\tilde{\gamma }\epsilon \). With these assumptions, the leading order equations for system (14) remain system (3), and therefore, \(\phi _j=0.25 + \epsilon \phi _j^* +\mathcal {O}(\epsilon ^2)\). Including the longest-range connections augments the \(\mathcal {O}(\epsilon )\) system with two additional \(\gamma \) dependent terms

$$\begin{aligned} 0&=-2\phi _1^*\tilde{H}'(0.25)+\phi _2^*\tilde{H}'(0.25)+\tilde{\delta }\tilde{H}'(0.25) -\tilde{\gamma } \tilde{H}(0.75)\\ 0&= \phi _1^*\tilde{H}'(0.25)-2\phi _2^*\tilde{H}'(0.25) + \phi _3^*\tilde{H}'(0.25) + \tilde{\beta }(\tilde{H}(0)-\tilde{H}(0.5)) \\ 0&=\phi _2^*\tilde{H}'(0.25) - 2 \phi _3^*\tilde{H}'(0.25) -\tilde{\delta }\tilde{H}'(0.25) +\tilde{\gamma } \tilde{H}(0.75). \end{aligned}$$

The \( \phi _j^*\) now satisfy the linear system

$$\begin{aligned}&\left( \begin{array}{ccc} -2 &{}1&{} 0\\ 1&{} -2&{} 1\\ 0&{} 1&{} -2 \end{array} \right) \left( \begin{array}{c} \phi _1^* \\ \phi _2^* \\ \phi _3^* \end{array} \right) =\tilde{\delta } \overbrace{\left( \begin{array}{c} -1 \\ 0 \\ 1 \end{array} \right) }^\text {shift term}\\&\qquad +\,\overbrace{\frac{\tilde{\beta }}{\tilde{H}'(0.25)} \left( \begin{array}{c} 0 \\ \tilde{H}(0.5)-\tilde{H}(0) \\ 0 \end{array} \right) }^\text {long-range coupling term} +\overbrace{ \frac{\tilde{\gamma } \tilde{H}(0.75)}{\tilde{H}'(0.25)}\left( \begin{array}{c} 1 \\ 0\\ -1 \end{array} \right) }^\text {longest-range coupling}, \end{aligned}$$

which implies

$$\begin{aligned} \left( \begin{array}{c} \epsilon \phi _1^* \\ \epsilon \phi _2^* \\ \epsilon \phi _3^* \end{array} \right)&=\overbrace{\frac{\delta }{2}\left( \begin{array}{c} 1 \\ 0 \\ -1 \end{array} \right) }^\text {shift term} +\overbrace{\frac{\beta (\tilde{H}(0)-\tilde{H}(0.5))}{\tilde{H}'(0.25)} \left( \begin{array}{c} \frac{1}{2} \\ 1 \\ \frac{1}{2} \end{array} \right) }^\text {long-range coupling term} \\&\ \ \ \ +\overbrace{ \frac{\gamma \tilde{H}(0.75)}{2\tilde{H}'(0.25)}\left( \begin{array}{c} -1 \\ 0 \\ 1 \end{array} \right) .}^\text {longest-range coupling} \end{aligned}$$

This equation is identical to Eq. (5), except for an additional term that quantifies the effects of the longest-range coupling

$$\begin{aligned} \phi ^*_\mathrm{llr}=\gamma \frac{\tilde{H}(0.75)}{2\tilde{H}'(0.25)} \left( \begin{array}{c} -1 \\ 0 \\ 1 \end{array} \right) . \end{aligned}$$

The longest-range coupling modulates the most anterior and posterior phase-differences by equal but opposite amounts. The direction of the modulation depends on the sign of the interaction function at 0.75. Note that for the HCOs under consideration, \(\tilde{H}'(0.25)>0\). The longest-range (\(\gamma \)) term and the H function shift (\(\delta \)) term have similar formulas but differ only by a scalar multiple; thus, the long-range coupling could either exacerbate the edge effects due to the shift of the H function, or counteract them. For all of the H functions we consider here, including the experimentally measured H function in Fig. 3a, the \(\gamma \) term exacerbates the edge effects caused by the \(\delta \) term, leading to an increased gradient in phase-differences between oscillators.

C Next-nearest-neighbor connections in longer chains

Along with the crayfish, some five-limbed crustaceans exhibit the approximately 25% phase lag during forward swimming; therefore, we extend our analysis to consider longer finite chains of HCOs with nearest-neighbor and next-nearest-neighbor coupling. We show that for an asymmetrically coupled chain of arbitrary length, next-nearest coupling tends to decrease phase-difference between oscillators. For a chain of arbitrary length \(n\ge 5\),

$$\begin{aligned} \dot{\theta }_{1}= & {} \omega + H_{A}(\theta _2-\theta _1)+\beta H_{A} (\theta _3-\theta _1) \\= & {} \omega + H(\phi _1)+\beta H(\phi _1+\phi _2)\\ \dot{\theta }_{2}= & {} \omega +H(\phi _1)+\beta H(\phi _1+\phi _2) \\= & {} \omega + H(-\phi _1+0.5)+H(\phi _2)+\beta H(\phi _2+\phi _3) \\ \dot{\theta }_{k}= & {} \omega +H_{D}(\theta _{k-1}-\theta _k)+H_{A}(\theta _{k+1}-\theta _k) \\&+\beta H_{D} (\theta _{k-2}-\theta _k)+\beta H_{A}(\theta _{k+2}-\theta _k) \\= & {} \omega + H(-\phi _{k-1}+0.5)+H(\phi _k) \\&+\beta H(-\phi _{k-2}-\phi _{k-1}+0.5)+\beta H(\phi _k+\phi _{k+1}) \\ \dot{\theta }_{n-1}= & {} \omega +H_{D}(\theta _{n-2}-\theta _{n-1})+H_{A}(\theta _{n}-\theta _{n-1}) \\&+\beta H_{D} (\theta _{n-3}-\theta _{n-1}) \\= & {} \omega +H(-\phi _{n-2}+0.5)+H(\phi _{n-1}) \\&+\beta H(-\phi _{n-2}-\phi _{n-3}+0.5)\\ \dot{\theta }_{n}= & {} \omega +H_{D}(\theta _{n-1}-\theta _n)+\beta H_{D} (\theta _{n-2}-\theta _n) \\= & {} \omega +H(-\phi _{n-1}+0.5)+\beta H (-\phi _{n-2}-\phi _{n-1}+0.5) \end{aligned}$$

This gives us the system of phase-differences

$$\begin{aligned} \dot{\phi }_{1}= & {} H(-\phi _1+0.5)+H(\phi _2)-H(\phi _1) \\&+\beta H(\phi _2+\phi _3)-\beta H(\phi _1+\phi _2)\\ \dot{\phi }_{2}= & {} H(-\phi _{2}+0.5)+H(\phi _3)-H(-\phi _1+0.5) \\&-H(\phi _2)+\beta H(-\phi _{1}-\phi _{2}+0.5) \\&+\beta H(\phi _3+\phi _{4})-\beta H(\phi _2+\phi _3)\\ \dot{\phi }_{k}= & {} H(-\phi _{k}+0.5)+H(\phi _{k+1})\\&-H(-\phi _{k-1}+0.5)-H(\phi _k)\\&+\beta H(-\phi _{k-1}-\phi _{k}+0.5)+\beta H(\phi _{k+1}+\phi _{k+2}) \\&-\beta H(-\phi _{k-2}-\phi _{k-1}+0.5)-\beta H(\phi _k+\phi _{k+1})\\ \dot{\phi }_{n-2}= & {} H(-\phi _{n-2}+0.5)+H(\phi _{n-1})-H(-\phi _{n-3}+0.5)\\&-H(\phi _{n-2})+\beta H(-\phi _{n-2}-\phi _{n-3}+0.5) \\&-\beta H(-\phi _{n-4}-\phi _{n-3}+0.5)-\beta H(\phi _{n-2}+\phi _{n-1})\\ \dot{\phi }_{n-1}= & {} H(-\phi _{n-1}+0.5) -H(-\phi _{n-2}+0.5)\\&-H(\phi _{n-1})+\beta H (-\phi _{n-2}-\phi _{n-1}+0.5)\\&-\beta H(-\phi _{n-2}-\phi _{n-3}+0.5) \end{aligned}$$

Using a perturbation argument with assumptions as in the main text, the \(\mathcal {O}(1)\) system yields \(\phi _0=0.25\). Here, the contribution from the long-range coupling satisfies

$$\begin{aligned} A \phi _\mathrm{lr}^*=\frac{\tilde{\beta }}{\tilde{H}'(0.25)}\left( \begin{array}{c} 0 \\ -\tilde{H}(0) \\ 0 \\ ...\\ 0\\ \tilde{H}(0.5)\\ 0 \end{array} \right) \end{aligned}$$

where A is an \((n-1)\) by \((n-1)\) tridiagonal Topelitz matrix, with \(-\,2\) on the main diagonal and ones along the off diagonals. Independent of n, the inverse of this matrix has all nonpositive components, causing \(\phi _\mathrm{lr}^* \le 0\), as claimed.

D Effects of long-range coupling on stability of phase-locking

The 25% phase-locked state in the phase model of the chain of HCOs (system (2)) inherits its stability from the leading order system in which there are no effects of long-range coupling (\(\beta =\epsilon \tilde{\beta }=0)\) and the preferred phase is 0.25 (\(\delta =\epsilon \tilde{\delta }=0\)). That is, it is stable if \(\tilde{H}'(0.25)\). Here, we assess the modulatory influence of long-range coupling and shifts in preferred phase on the stability of the \(\sim 25\%\) phase-locked state by determining their \(\mathcal {O}(\epsilon )\) effects on the eigenvalues of the system linearized around the phase-locked state.

By computing the Jacobian of the system, evaluating it at \(\phi _k=0.25\), and substituting in the solution form found in Eq. 5, we obtain (to \(\mathcal {O}(\epsilon )\))

$$\begin{aligned} J&=J_0+\epsilon J_1 \\&=\tilde{H}'(0.25)\left( \begin{array}{ccc}-2&{}\quad 1&{}\quad 0 \\ 1&{}\quad -2&{}\quad 1 \\ 0&{}\quad 1&{}\quad -2 \end{array} \right) \\&\quad + \tilde{\delta }\epsilon \tilde{H}''(0.25)\left( \begin{array}{ccc} -2&{}\quad 1&{}\quad 0 \\ \frac{1}{2}&{}\quad -2&{}\quad \frac{1}{2} \\ 0&{}\quad 1&{}\quad -2 \end{array} \right) \\&\quad +\tilde{\beta }\epsilon \frac{\tilde{H}(0)-\tilde{H}(0.5)}{\tilde{H}'(0.25)}\tilde{H}''(0.25)\left( \begin{array}{ccc} 0&{}\quad 1&{}\quad 0 \\ -\frac{1}{2}&{}\quad 0&{}\quad \frac{1}{2} \\ 0&{}\quad -1&{}\quad 0 \end{array} \right) \\&\quad +\tilde{\beta }\epsilon \tilde{H}'(0.5) \left( \begin{array}{ccc} -1&{}\quad 0&{}\quad 1 \\ 0&{}\quad -1 &{}\quad -1 \\ 0&{}\quad 0&{}\quad 0 \end{array} \right) \\&\quad +\tilde{\beta }\epsilon \tilde{H}'(0) \left( \begin{array}{ccc} 0&{}\quad 0&{}\quad 0 \\ -1&{}\quad -1 &{}\quad 0 \\ 1&{}\quad 0&{}\quad -1 \end{array} \right) \end{aligned}$$

We seek the eigenvalues \(\lambda \) of this Jacobian, that is \(J\varPhi =\lambda \varPhi \). We take the eigenvalues and eigenvectors of J to be of the form \(\lambda = \lambda _0+\epsilon \lambda _1\) and \(\varPhi =\varPhi _0+\epsilon \varPhi _1\) and note that \(\lambda _1=<J_1\varPhi _0,\varPhi _0>\). Assuming that \(\tilde{H}'(0.25)\), the eigenvalues of \(J_0\) are all negative. Therefore, to assess stability of the \(\sim 25\%\) phase-locked state, we need only consider the smallest eigenvalue, which is \(\lambda _0=\tilde{H}'(0.25)(-2+\sqrt{2})\) and has a corresponding eigenvector \(\varPhi _0= -\frac{1}{2} ( 1, \sqrt{2} ,1)^\mathrm{T}\). A simple calculation shows that

$$\begin{aligned} \lambda _1= & {} \tilde{\delta }\tilde{H}''(0.25) \frac{-8+3\sqrt{2}}{4}+\tilde{\beta }(\tilde{H'}(0.5)\\&+\tilde{H'}(0))\frac{-2-\sqrt{2}}{4}. \end{aligned}$$

Both of the numerical fractions take values close to \(-\,1\). Furthermore, for the idealized interaction functions that we have considered here, \(\tilde{H}''(0.25), \tilde{H'}(0.5)\), and \(\tilde{H'}(0)\) are small. This implies that long-range coupling and shifts in preferred phase have negligible (\(\mathcal {O}(\epsilon ^2)\)) effects on stability.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Spardy, L.E., Lewis, T.J. The role of long-range coupling in crayfish swimmeret phase-locking. Biol Cybern 112, 305–321 (2018). https://doi.org/10.1007/s00422-018-0752-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00422-018-0752-3

Keywords

Navigation