Order parameter for bursting polyrhythms in multifunctional central patterngenerators

Jeremy Wojcik, Robert Clewley, and Andrey ShilnikovNeuroscience Institute and Department of Mathematics and Statistics,

Georgia State University, Atlanta, GA 30303, USA(Dated: April 7, 2011)

We examine multistability of several coexisting bursting patterns in a central pattern generatornetwork composed of three Hodgkin-Huxley type cells coupled reciprocally by inhibitory synapses.We establish that control of switching between bursting polyrhythms and their bifurcations aredetermined by the temporal characteristics, such as the duty cycle, of networked interneurons and thecoupling strength asymmetry. A computationally effective approach for the reduction of dynamicsof the 9D network to 2D Poincaré return mappings for phase lags between the interneurons ispresented.

PACS numbers: 05.45.Xt, 87.19.L-

A central pattern generator (CPG) is a neural micro-circuit of cells whose synergetic interactions, withoutsensory input, rhythmically produce patterns of burst-ing that determine motor behaviors such as heartbeat,respiration, and locomotion in animals and humans [1].While a dedicated CPG generates a single pattern ro-bustly, a multifunctional CPG can flexibly produce dis-tinct rhythms, such as temporarily distinct swimmingversus crawling, and alternation of blood circulation pat-terns in leeches [2, 3]. Switching between such rhythmsis attributed to input-dependent switching between at-tractors of the CPG.

Here we analyze multistability and transformations ofrhythmic patterns in a 9D model of a CPG motif (Fig. 1)comprised of 3 endogenously bursting leech heart in-terneurons [4] coupled (reciprocally) by fast inhibitorychemical synapses [5]. Our use of fast threshold modu-lation [6] implies that the post-synaptic current is zero(resp., maximized) when the voltage of a driving cell isbelow (resp., above) the synaptic threshold. This is aninherently bi-directionally asymmetric form of coupling.Many anatomically and physiologically diverse CPG cir-cuits involve a 3-cell motif [7], including the spiny lobsterpyloric network, the Tritonia swim circuit, and the Lym-naea respiratory CPGs [8]. We show how rhythms ofthe multistable motif are selected by changing the rel-ative timing of bursts by physiologically plausible per-turbations. We also demonstrate how the set of possiblerhythmic outcomes can be controlled by varying the dutycycle of bursts, and by varying the coupling around thering.

We propose a novel computational tool for detailed ex-amination of polyrhythmic bursting in biophysical CPGmodels with coupling asymmetries and arbitrary cou-pling strength. The tool reduces the problem of stabilityand existence of bursting rhythms in large networks tothe bifurcation analysis of fixed points (FP) of Poincaréreturn mappings for phase lags. Our approach is basedon delayed release of cells from a suppressed state, and

V1

20 mVΘth

−50 mV

V2

20 mV

−50 mV

Δφ21

(n+1)Δφ

21

(n)

V3

20 mV

−50 mV

Δφ31

(n+1)Δφ

31

(n)

(B)

(A)

2s

2

3

1

FIG. 1. (Color online) (A) 3-cell motif with asymmetric clock-wise versus counter-clockwise connection strengths. (B) Volt-age traces: the phase (mod 1) of reference cell 1 (blue) isreset when V1 reaches threshold Θth = −40 mV. The timedelays between the burst onset in the reference cell 1 and theburst onset in cells 2 (green) and 3 (red), normalized overthe recurrent time of cell 1, define a sequence of phase lags{∆ϕ

(n)21 ,∆ϕ

(n)31 }.

complements the phase resetting techniques allowing forfor thorough exploration of network dynamics with spik-ing cells [9]. It demonstrates that more general inhibitorynetworks possess stable bursting patterns that do not oc-cur in similar 3-oscillator motifs with gap-junction cou-pling, which is bi-directionally symmetric [10].

The duty cycle (DC) of bursting, being the fraction ofthe burst period in which cells spike, is turned out to bean order parameter that regulates motif synchronizationproperties [5, 11]. The DC is sensitive to fluctuationsof most cell’s intrinsic parameters, and is affected by ap-plied and synaptic currents [4, 5]. We treat DC implicitlyas a bifurcation parameter that defines short (DC∼20%),medium (DC∼50%), and long (DC∼80%) bursting mo-tifs. DC is controlled by an intrinsic parameter of theinterneuron that shifts the activation of the potassiumcurrent in the leech heart interneuron.

In this study we consider an adequately “weakly” cou-pled motif using the nominal value of a maximal con-ductance gsyn = 5 × 10−4 nS. This coupling strength

2

V1

V2

7 s

V3

(i) (ii) (iii) (iv) (v)

FIG. 2. (Color online) 5 polyrhythms in the medium bursting motif at gsyn = 5 × 10−3 (increased to secure short transientsfor the purpose of illustration). Inhibitory pulses (horizontal bars) suppress the targeted cells, thus causing switching betweenco-existing rhythms: (1⊥{2∥3}) in episode (i), traveling waves (1≺2≺3) in (ii) and (1≺3≺2) in (iii), followed by (2⊥{1∥3})led by cell 2 in (iv). Having released cells 1 and 2 simultaneously, cell 3 leads the motif in the (3⊥{1∥2}) rhythm in the lastepisode.

guaranties relatively slow convergence of transients tophase-locked states of the motif, and hence permits usto visualize “smooth” trajectories that expose in detailthe structure of phase-lag return maps, qualitatively re-sembling time-continuous planer vector fields in this case.The findings obtained for this case prepare the basis forunderstanding more complex patterns in strongly cou-pled, non-homogeneous motifs with the same technique.The asymmetry of the motif is governed by another bi-furcation parameter g�, which weakens (resp., enforces)counter-clockwise (resp., clockwise) coupling strengthsgccc = gsyn(1 ∓ g�), 0 ≥ g� ≤ 1, from the nominal valuegsyn.

As the period of network oscillations can fluctuate intime, we define delays between the onset of bursting incell 2 (green) and cell 3 (red) relative to that in the thereference cell 1 (blue) at the instances the voltages Viincrease through a threshold of Θth = −40 mV (Fig. 1).Initial delays are controlled by the timely release of cells 2and 3 from inhibition (Fig. 2). The subsequent delaysnormalized over the period of the cell 1 define a sequenceor forward trajectory

{∆ϕ

(n)21 , ∆ϕ

(n)31

}of phase-lag re-

turn maps on a torus [0, 1) × [0, 1) with ∆ϕi1 mod 1.The maps are tabulated on a 40 × 40 (or more) grid ofinitial points. The iterates are connected in Figs. 3–5 topreserve order. We then study the dynamical propertiesof the maps, locate fixed points and evaluate their stabil-ity, detect periodic and heteroclinic orbits, and identifythe underlying bifurcations as the parameters, DC andg�, are varied.

Fig. 3A shows the (∆ϕ31,∆ϕ21) phase-lag map for theS3-symmetric, medium bursting motif. The map has fivestable FPs corresponding to the coexisting phase-lockedrhythms: the (red) FP at (∆ϕ31 ≈ 1

2 ,∆ϕ21 ≈ 0), the(green) FP (0, 12 ), the (blue) FP ( 12 ,

12 ), and the (black)

FP ( 23 ,13 ) and the (purple) FP ( 13 ,

23 ). The attraction

basins of the FPs are divided by separatrices (incomingand outgoing sets) of six saddles (small grey dots in thephase diagrams). The neighborhood of (0,0) has a com-plex structure at high magnification (not shown), but iseffectively repelling. Locations of the FP do not corre-spond to exact fractions due to overlap and interactionbetween bursts and a slight ambiguity in the measure-ment and definition of momentary phases.

The coordinates of the FPs determine the phase lockedstates within bursting rhythms of the motif that are de-noted symbolically as follows: (1≺2≺3) and (1≺3≺2)for clockwise and counter-clockwise traveling waves ofbursting (resp.) around the ring (episodes (ii) and (iii)of the voltage trace in Fig. 2) corresponding to the FPslocated near ( 13 ,

23 ) and ( 23 ,

13 ), respectively.

Besides these rhythms (which stability is unknown apriori) in a symmetric motif, the three other FPs corre-spond to the rhythms where one cell fires in anti-phase(∆ϕ ∼ 1

2 , or ⊥) against two cells bursting simultaneouslyin-phase (∆ϕ = 0, or ∥). The stable FP at ( 12 ,

12 ) corre-

sponds to the (1⊥{2∥3})-pattern (episode (i) of Fig. 2);FP (0, 12 ) corresponds to (2 ⊥ {1 ∥ 3})-pattern (episode(iv)); and FP (12 , 0) corresponds to (3⊥{1 ∥ 2})-pattern(episode (v)). We note that such anti-phase patterns arerecorded at switches between peristaltic and synchronousrhythms in the leech heartbeat CPG [3] and in the Tri-tonia CPG during escape swimming [8].

The selection of rhythmic outcome in a multifunctionalmotif depends on the initial phase distributions of thecells. Evaluation of the FP attraction basins in Fig. 3Asuggests that when the cells are simultaneously releasedfrom external inhibition, e.g. with initial phases reset,the medium bursting motif can generate one of five ro-bust rhythms with nearly equal odds. The geometry ofthe phase-lag map also informs how to switch the motifto a specific rhythm by perturbing it in a certain phasedirection, i.e. advancing or delaying cells. A biophys-

3

(A)

(B)

FIG. 3. (Color online) (A) Phase-lag map for the symmetric,medium bursting motif showing five stable FPs: (red) dotat ∼ ( 1

2, 0), (green) (0, 1

2), (blue) ( 1

2, 12), (black) ( 2

3, 13) and

(purple) ( 13, 23), corresponding to the anti-phase (3⊥{1∥2}),

(2 ⊥ {1 ∥ 3}), (1 ⊥ {2 ∥ 3}) bursts, and traveling clockwise(1 ≺ 2 ≺ 3) and counter-clockwise (1 ≺ 3 ≺ 2) waves; theattraction basins are divided by “separatrices” (stable sets)of six saddles (small grey dots). Arrows indicate the forwarditerate direction in the phase plane. (B) Asymmetric motifat g� = 0.154 near the saddle-node bifurcations annihilatingthree stable FPs for anti-phase bursting rhythms.

(A)

(B)

FIG. 4. (Color online) (A) Phase-lag map for the symmet-ric, short bursting motif showing only 3 stable FPs (blue dotat ( 1

2, 12), red dot at ( 1

2, 0), and green dot at (0, 1

2)) corre-

sponding to anti-phase rhythms where one cell bursts followedby synchronized bursts in the other two cells. Unstable FPsat ( 2

3, 13) and ( 1

3, 23) exclude the clockwise (1 ≺ 2 ≺ 3) and

counter-clockwise (1≺ 3≺ 2) traveling waves from the reper-toire of the short bursting motif. (B) Map for the asymmet-ric motif (g� = 0.2) depicting a stable invariant curve near athree-saddle connection around the FP at ( 2

3, 13).

4

ically plausible way to control switching is to apply anappropriately-timed hyperpolarizing pulse to temporar-ily suppress a targeted cell(s). Fig. 2 demonstrates theapproach for the symmetric, medium bursting motif.

Comparison of the maps for the symmetric motifs incases of medium (Figs. 3, 5B), short (Fig. 4) and long(Fig. 5A) bursting demonstrates that DC is a physiolog-ically plausible parameter that determines the dominantobservable rhythms. I.e., short bursting makes it impos-sible for both clockwise (1≺2≺3) and counter-clockwise(1≺ 3≺ 2) traveling wave patterns to occur in the sym-metric motif as the corresponding FPs are unstable ini-tially. In contrast, the symmetric long bursting motifis unlikely to generate anti-phase rhythms, as the cor-responding FPs have narrow attraction basins, dividedequally between the FPs corresponding to the travelingwaves.

We examine next the bifurcations of FPs of thephase-lag map as the motif becomes Z3-rotationally(a)symmetric by increasing g�, which weakens counter-clockwise and enforces clockwise-directed synapses. Thelimit g� → 1 makes the motif unidirectionally coupledwith only the (1≺ 2≺ 3) rhythm observable. Here, theFP at ( 23 ,

13 ) expands its attraction basin over the en-

tire phase range. Intermediate stages of structural trans-formation of the phase-lag map for the medium burst-ing motif are demonstrated in Figs. 3 and 5B. First,as g� increases to 0.154, the three saddles move awayfrom the FP at ( 23 ,

13 ), thereby increasing its attraction

basin, and approach the stable FPs corresponding toanti-phase bursting rhythms. Meanwhile, the other threesaddles move towards the FP at ( 13 ,

23 ) corresponding to

the (1 ≺ 3 ≺ 2) rhythm, narrowing its basin. As g� isincreased further, the attractors and saddles in the bot-tom right of the unit square merge and vanish throughsaddle-node (SN) bifurcations. Increasing g� makes theFP ( 23 ,

13 ) for the (1≺ 2≺ 3) rhythm globally dominant

(Fig. 5B) after the three nearby saddles collapse onto theFP at (13 ,

23 ). Bifurcations in the long bursting motif are

similar, except that the SN bifurcation occurs at smallerg� values.

The bifurcation sequence in the short bursting motifis qualitatively different: the SN bifurcations occur at ahigher degree of asymmetry (g� ≈ 0.48), delayed by an-other bifurcation that makes the clockwise traveling pat-tern the global attractor of the motif. To become stable,the corresponding FP at ( 23 ,

13 ) undergoes a secondary su-

percritical Andronov-Hopf or torus bifurcation. Fig. 4Bdepicts the map at g� = 0.2 showing a stable invariantcurve near the heteroclinic connections between the threesaddles around the given FP. The invariant curve is as-sociated with the appearance of slow phase jitters withinthe (1≺2≺3) rhythm in voltage traces. Once it collapsesonto the FP at (23 ,

13 ) the asymmetric motif gains a new

robust (1≺2≺3) rhythm, making four possible burstingoutcomes.

(A)

(B)

FIG. 5. (Color online) (A) Phase-lag map for the symmetric,long bursting motif revealing two equally dominant attractors:( 23, 13) and ( 1

3, 23) for (1 ≺ 2 ≺ 3) and (1 ≺ 3 ≺ 2) traveling

rhythms. (B) Map for the asymmetric (g� = 0.3) mediumbursting motif depicting two persistent attractors: the one forthe clockwise (1≺ 2≺ 3) rhythms prevails over the attractorfor the counter-clockwise (1≺3≺2) rhythm. Further increasein g� leads to the only observable (1≺ 2≺ 3) rhythm, afterFP at ( 1

3, 23) merges with 3 nearby saddles.

5

The stability of the FPs in the phase-lag maps is defacto proof of the observability of the matching rhythmicoutcomes generated by a motif, symmetric or not. Whilethe existence of some rhythms, like generic (1 ≺ 2 ≺ 3)and (1 ≺ 3 ≺ 2), in a 3-cell motif can hypothetically bededuced using symmetry arguments, the robustness andobservability of the rhythms must be justified by accuratecomputational verification, as their stability strongly cor-relates with the temporal properties of the bursting cells.Using the proposed computational technique for the re-duction of dynamics of the 9D 3-cell motif to the analysisof the equationless 2D mappings for the phase lags be-tween the bursting cells, we have demonstrated that a re-ciprocally inhibitory (non)homogeneous network can bemultistable, i.e. can generate several distinct polyrhyth-mic bursting patterns. It is shown for the first time thatthe observable rhythms of the 3-cell motif are determinednot only by its (a)symmetry, but the duty cycle servingthe role of the order parameter for bursting networks.This knowledge of the existence, stability and possiblebifurcations of polyrhythms in this 9D motif composed ofthe interneuron models is vital for derivations of reduced,phemenologically accurate phase-models for nonhomoge-neous biological CPGs with mixed, inhibitory and exci-tatory synapses.

We thank W. Kristan, A. Neiman, P. Ashwin, C. Laingand R. Lin for valuable suggestions, and M. Brooks fortaking part in the pilot phase of the project. We acknowl-edge support from NSF Grants CISE/CCF-0829742 (toR.C.), DMS-1009591 and RFFI Grant No. 08-01-00083(to A.S.), and the GSU Brains & Behavior program.

[1] A. Selverston (ed), Model Neural Networks and Behavior,(Springer, 1985); F. Skinner, N. Kopell, E. Marder, Com-put. Neurosci. 1, 69 (1994); E. Marder, R. L. Calabrese,Physiol. Rev. 76, 687 (1996); N. Kopell, G. B. Ermen-trout, PNAS USA 101, 15482 (2004).

[2] K. L. Briggman, W. B. Kristan, Annu. Rev. Neurosci.31 (2008).

[3] W. B. Kristan, R. L. Calabrese, W. O. Friesen, Prog.Neurobiol. 76, 279 (2005); A. Weaver, R. Roffman,B. Norris, R. Calabrese, Frontiers in Behav. Neurosci.4(38), 1 (2010).

[4] A. Shilnikov and G. Cymbalyuk, Phys. Rev. Lett. 94,048101 (2005).

[5] A. Shilnikov, R. Gordon, I. Belykh, Chaos 18, 037120(2008).

[6] N. Kopell, D. Somers, Biol. Cybern. 68, 5 (1993).[7] R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan, D.

Chklovskii, U. Alon, Science 298 (2002); M. I. Rabi-novich, P. Varona, A. Selverston, and H. D. Abarbanel,Rev. Modern Phys. 78, 4 (2006); O. Sporns and R. Köt-ter, PLoS Biol. 2, 11 (2004).

[8] W.N. Frost and PS Katz, PNAS 93, 422 (1996); A. G. M.Bulloch and N. I. Sayed, Trends Neurosci. 15, 11 (1992);P. S. Katz and S. L. Hooper, in Invertebrate Neuroscience

G. North and R. Greenspan (eds), (Cold Spring Harbor)(2007).

[9] G. B. Ermentrout, N. Kopell, PNAS USA 95 (1998); C.C. Canavier, D. A. Baxter, J. W. Clark, J. H. Byrne.,Biol. Cybern. 80, 87 (1999).

[10] P. Ashwin, G. P. King, J. W. Swift, Nonlinearity 3,3 (1990). C. Baesens, J. Guckenheimer, S. Kim , R.MacKay, Physica D 49, 387, (1991).

[11] I. Belykh and A. Shilnikov, Phys. Rev. Lett. 101, 078102(2008).

[12] R. H. Clewley, W. E. Sherwood, M. D. LaMar, and J. M.Guckenheimer, http://pydstool.sourceforge.net (2006).

APPENDIX

A reduced model of the leech heart interneuron is givenby the following set of three nonlinear coupled differentialequations:

CdV

dt= −INa − IK2 − IL − Iapp − Isyn,

IL = gL (V − EL), IK2 = gK2m2K2(V − EK),

INa = gNam3Na hNa (V − ENa), mNa = m∞

Na(V ),

τNadhNa

dt= h∞Na(V )− h, τK2

dmK2

dt= m∞

K2(V )−mK2.

Here, C = 0.5nF is the membrane capacitance; V is themembrane potential in V; INa is the sodium current withslow inactivation hNa and fast activation mNa; IK2 is theslow persistent potassium current with activation mK2;IL is the leak current and Iapp = 0.006 nA is an ap-plied current. The values of maximal conductances areset as gK2 = 30nS, gNa = 200nS and gL = 8nS. The re-versal potentials are ENa = 0.045V, EK = −0.07V andEL = −0.046V. The time constants of gating variablesare τK2 = 0.9 sec and τNa = 0.0405 sec. The steady statevalues, h∞Na(V ), m∞

Na(V ), m∞K2(V ), of the of gating vari-

ables are given by the following Boltzmann equations:

h∞Na(V ) = [1 + exp(500(V + 0.325))]−1

m∞Na(V ) = [1 + exp(−150(V + 0.0305))]−1

m∞K2(V ) = [1 + exp (−83(V + 0.018 + Vshift

K2 ))]−1.(1)

The synaptic current is modeled through the fast thresh-old modulation paradigm as follows:

Isyn =

n∑j=1

gsyn(1∓ g�)(Einhsyn − Vi)]Γ(Vj −Θsyn). (2)

The reversal potential Einhsyn = −0.0625V of the postsy-

naptic cell is set to be smaller than it voltage Vi(t) tomake the synapse inhibitory. The synaptic coupling func-tion is modeled by the sigmoidal function Γ(Vj) = 1/[1+exp{−1000(Vj − Θsyn)}]. The threshold Θsyn = −0.03Vis chosen in the presynaptic cell so that every spike withina burst reaches the former. This implies that the synapticcurrent, Isyn, from the presynaptic j-th cell toward thepostsynaptic i-th cell is initiated as soon as the former

6

cell becomes active after its membrane potential, Vj(t),exceeds the synaptic threshold.

The intrinsic bifurcation parameter VshiftK2 of the model

is a deviation from V1/2 = 0.018 V corresponding tothe half-activated potassium channel at m∞

K2 = 1/2. Inthe model (1), decreasing Vshift

K2 delays the activation ofmK2. The bursting range of the bifurcation parameter ofthe given interneuron model is [−0.024235, −0.01862]V;for smaller values of Vshift

K2 the model enters the tonicspiking mode, or becomes hyperpolarizedly quiescentat the upper values. By varying Vshift

K2 (or alternativelyIapp) we can change the duty cycle of bursting (ratio[burst duration]/[burst period]) from 1 through 0. Inthis study, we use the values of V shift

K2 = −0.01895V,V shift

K2 = −0.021V and V shiftK2 = −0.0225V, corresponding

to short (∼20%), medium (∼50%), and long (∼80%)bursting, respectively. Further details about the bifur-cation underlying the transition between bursting andtonic spiking through the blue sky catastrophe, and theregulation of temporal characteristics of the burstingactivity in this leech heart interneuron model can befound in [10].

Numerical methodsAll numerical simulations and phase analysis were per-

formed using the PyDSTool dynamical systems softwareenvironment (version 0.88) [12] The software is freelyavailable. Specific instructions and auxiliary files for thenetwork (1) will be provided upon request.

The algorithm for constructing the 2D phase mappingsfrom the 9D network model is based on the observationthat two solutions x(t) = ψ(x0; t) and x(t) = ψ(x0; t+τ)of an individual model can be considered as the samesolution passing through the initial x0 on the T-periodicbursting orbit at different initial times. By releasing thesolutions from x0 at different delays τ , ( 0 < τ < T ) wecan generate a dense set of initial points which present

a good first order approximation for the solutions of thenetwork (1) in the case of sufficiently weak coupling.

Each sequence of phase lags {∆ϕ(n)31 ,∆ϕ(n)21 } plotted in

Figs. 3–5 begins from an initial lag (∆ϕ(0)31 ,∆ϕ

(0)21 ), which

is the difference in phases measured relative to the re-currence time of cell 1 every time its voltage increasesto a threshold Θth = −0.040 V. Θth marks the begin-ning of the spiking segment of a burst. As that recur-rence time is unknown a priori due to interactions of thecells, we estimate it, up to first order, as a fraction ofthe period Tsynch of the synchronous solution by select-ing guess values (∆ϕ⋆31,∆ϕ⋆21). The synchronous solutioncorresponds to ∆ϕ31 = ∆ϕ21 = 0. By identifying t = 0at the moment when V1 = Θth with ϕ = 0, we can pa-rameterize this solution by time (0 ≤ t < Tsynch) or bythe phase lag (0 ≤ ∆ϕ < 1). For weak coupling andsmall lags, the recurrence time is close to Tsynch, and(∆ϕ⋆31,∆ϕ

⋆21) ≈ (∆ϕ

(0)31 ,∆ϕ

(0)21 ). We use the following al-

gorithm to distribute the true initial lags uniformly on a40× 40 square grid covering the phase portrait.

We initialize the state of cell 1 at t = 0 from the point(V 0, n0, h0) of the synchronous solution when V1 = Θth.Then, we create the initial phase-lagged state in thesimulation by suppressing cells 2 and 3 for durationst = ∆ϕ021Tsynch and ∆ϕ031Tsynch, respectively. On re-lease, cells 2 and 3 are initialized from the initial point(V 0, n0, h0). We begin recording the sequence of phaselags between cells 2 and 3 and the reference cell 1 on thesecond cycle after coupling has adjusted the network pe-riod away from Tsynch. In the case of stronger coupling(increased asymmetry via g�) where the gap betweenTsynch and the first recurrence time for cell 1 widens, weretroactively adjust initial phases using a “shooting” al-gorithm to make the initial phase lags sufficiently closeto uniformly distributed positions on the square grid.