Pacemaker and network mechanisms of rhythm generation: Cooperation and competition

ARTICLE IN PRESS

Journal of Theoretical Biology 253 (2008) 452– 461

Contents lists available at ScienceDirect

Journal of Theoretical Biology

0022-51

doi:10.1

� Corr

E-m

T.Nowo

(A.I. Sel

journal homepage: www.elsevier.com/locate/yjtbi

Pacemaker and network mechanisms of rhythm generation:Cooperation and competition

Mikhail V. Ivanchenko a, Thomas Nowotny b,�, Allen I. Selverston c, Mikhail I. Rabinovich c

a Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UKb Centre for Computational Neuroscience and Robotics, University of Sussex, Falmer, Brighton BN1 9QJ, UKc Institute for Nonlinear Science, University of California San Diego, La Jolla, CA 92093-0402, USA

a r t i c l e i n f o

Article history:

Received 17 September 2007

Received in revised form

16 April 2008

Accepted 16 April 2008Available online 26 April 2008

Keywords:

Oscillatory networks

Bursting neurons

Bistability

CPGs

Brain motifs

93/$ - see front matter & 2008 Elsevier Ltd. A

016/j.jtbi.2008.04.016

esponding author. Tel.: +441273 678593; fax

ail addresses: ivanchen@maths.leeds.ac.uk (M

tny@sussex.ac.uk (Thomas Nowotny), aselver

verston), mrabinovich@ucsd.edu (M.I. Rabino

a b s t r a c t

The origin of rhythmic activity in brain circuits and CPG-like motor networks is still not fully

understood. The main unsolved questions are (i) What are the respective roles of intrinsic bursting and

network based dynamics in systems of coupled heterogeneous, intrinsically complex, even chaotic,

neurons? (ii) What are the mechanisms underlying the coexistence of robustness and flexibility in the

observed rhythmic spatio-temporal patterns? One common view is that particular bursting neurons

provide the rhythmogenic component while the connections between different neurons are responsible

for the regularisation and synchronisation of groups of neurons and for specific phase relationships in

multi-phasic patterns. We have examined the spatio-temporal rhythmic patterns in computer-

simulated motif networks of H–H neurons connected by slow inhibitory synapses with a non-

symmetric pattern of coupling strengths. We demonstrate that the interplay between intrinsic and

network dynamics features either cooperation or competition, depending on three basic control

parameters identified in our model: the shape of intrinsic bursts, the strength of the coupling and its

degree of asymmetry. The cooperation of intrinsic dynamics and network mechanisms is shown to

correlate with bistability, i.e., the coexistence of two different attractors in the phase space of the system

corresponding to different rhythmic spatio-temporal patterns. Conversely, if the network mechanism of

rhythmogenesis dominates, monostability is observed with a typical pattern of winnerless competition

between neurons. We analyse bifurcations between the two regimes and demonstrate how they provide

robustness and flexibility to the network performance.

& 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Oscillatory neural circuits and, in particular, central patterngenerators (GPGs) must contain both basic rhythmogenic andpattern forming mechanisms. A question fundamental to suchnetworks is how the specific intrinsic activity of individualneurons can be constrained by synaptic dynamics and networkorganisation and how the network output emerges from thecooperative activity of its components. Over the last threedecades, neurophysiologists have proposed two different mechan-isms for the generation of low-frequency oscillations: rhythmo-genesis by one or more pacemaker neurons and rhythmgeneration by network mechanisms. Pacemaker mechanisms arebased on a single neuron, or synchronised neural group, that

ll rights reserved.

: +441273 877873.

.V. Ivanchenko),

ston@ucsd.edu

vich).

generates bursting activity as a result of intrinsic cellular proper-ties, while the rest of the network orchestrates the phaserelationships among different principal cells (or cell groups) togenerate a specific spatio-temporal pattern (Rabinovich et al.,2006; Selverston and Miller, 1980; Yuste et al., 2005; Buzsaki,2006; Kopell, 2005; Borgers et al., 2005). In network-basedrhythm generation the spatio-temporal dynamics results fromthe excitatory and inhibitory synaptic connections betweenneurons, which do not have intrinsic rhythmic activity. In thiscase networks of tonically spiking neurons are able to generaterhythmic bursts as a result of a cooperative, synapse-mediated,modulational instability (Nowotny and Rabinovich, 2007).

Little is known about the dynamical principles that govern theinterplay between intrinsic bursting and network mechanismswhen both are found in a single system. There is growingexperimental evidence suggesting that in some neural systems,like the respiratory CPG of mammals, the rhythm is generated by acombined pacemaker-network mechanism (Johnson et al., 2007;Rybak et al., 2004, 2007; Calabrese, 1998; Sohal et al., 2006), inthis case operating on a distributed population of coupled,

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M.V. Ivanchenko et al. / Journal of Theoretical Biology 253 (2008) 452–461 453

bursting pacemaker neurons. It has been suggested that suchhybrid mechanisms of rhythmogenesis are potentially morerobust, more reliable and more flexible than mechanisms basedon pacemakers or network topology alone. They may, therefore, bemore frequently at work in neuronal systems than previouslyassumed, which motivated us to examine their fundamentaldynamical properties. Based on a simple model network (motif,see e.g., Sporns and Kotter, 2004; Milo et al., 2002; Zhigulin, 2004)of three conditionally bursting Hodgkin–Huxley (H–H) typeneurons with reciprocal inhibitory coupling (Fig. 1a), we elucidatesome of these properties. Our model network was inspired by thestomatogastric system of lobsters and we use the term conditionalto signify that the biological bursting requires the presence ofappropriate neuromodulators. In the model this corresponds tothe presence of appropriate de- or hyperpolarisation of thesomatic compartment. Our model neurons are similar to thelateral pyloric (LP) neurons in the lobster Panulirus interruptus,which normally burst irregularly when isolated. The modelneurons also mimic real LP neurons in that their depolarisationcauses them to burst at higher frequencies and with longer burstsand hyperpolarisation slows the burst frequency down anddecreases the burst duration. Furthermore, similar to the graded

Motif Network

5 s

20 m

V

−6 nA

−1 nA

50 ms

−100−80

−60

−40

−20

Vs,

Vs,

3 [m

V]

Fig. 1. (a) Architecture of the simulated network: three neurons are reciprocally coupled

D phase portrait of two examples of network-generated bursting in counterclockwise

neurons, intrinsic currents IDC ¼ 3:0 nA and synaptic coupling g ¼ 1:0mS (black/blue) a

corresponding neurons. (c) Examples of the intrinsic bursting dynamics of isolated

hyperpolarisation IDC ¼ �6:0 nA (lower trace). (d) Basic characteristics of intrinsic burs

IDC . Panels from top to bottom: frequency of bursts f b , number of spikes per burst nsp ,

synapses in the lobster stomatogastric CPG, the model synapsesare slow enough to implement an interaction between burstsrather than responding to single spikes.

We analysed the drastic changes (bifurcations) in the networkactivity which occur when the strength of intrinsic pacemakeractivity is altered by passage of DC current and/or the strength ofthe synaptic connection changes.

We found in our model that the hybrid pacemaker-networkmechanism has modes of cooperation or of competition betweenthe cellular and network dynamics and demonstrates bothrobustness (stability of the produced patterns) and flexibility(the ability to produce different patterns in response to appro-priate stimuli) concurrently. The consequence of cooperation orcompetition is the bistability or monostability of spatio-temporalbursting patterns, and the decisive factors for either cooperationor competition are the parameters underlying intrinsic bursting.The other basic control parameter to switch between cooperationand competition is the asymmetry of coupling and surprisinglywe found that bistability only disappears for strong asymmetry,which suggests that weak background synaptic interactions couldqualitatively change the overall network dynamics and give rise tonew bursting patterns and therefore cannot be neglected a priori.

−10 −8 −6 −4 −2 00

5

10

T b/T

s

−10 −8 −6 −4 −2 00

50

n sp

−10 −8 −6 −4 −2 00

0.2

0.4

f b [H

z]

IDC [nA]

−80−60

−40−20

−50

0Vs, 1 [m

V] 2 [mV]

1

3

2

through strong counterclockwise and weaker clockwise inhibitory synapses. (b) 3-

order f1;2;3g. The three axes are the somatic membrane potentials of the three

nd g ¼ 10:0mS (grey/red). The circled numbers indicate regions of bursting in the

neurons for moderate hyperpolarisation IDC ¼ �1:0 nA (upper trace) and strong

ting of isolated neurons in dependence upon the hyperpolarising DC input current

and ratio of bursting period Tb to burst duration Ts .

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M.V. Ivanchenko et al. / Journal of Theoretical Biology 253 (2008) 452–461454

2. Methods

2.1. Neuron model

The two compartment neuron model used in our study wasbuilt based on voltage clamp recordings and extensive fits to datafrom isolated lobster (Panulirus interruptus) LP neurons, whichwill be discussed elsewhere (Nowotny, T., Levi, R., Selverston A.I.,Probing the dynamics of identified neurons with a data-drivenmodelling approach, under review). The LP neuron plays animportant role in the dynamics of the pyloric CPG of the lobsterand is known to be a conditional burster with a wide range ofdynamics from tonic spiking to irregular spiking, chaotic bursting,and regular bursting for polarisations ranging from depolarisationto increasing levels of hyperpolarisation. The LP neuron model isparticularly well-suited for our purposes because it replicates thiswide range of dynamics.

The membrane potential of axon and soma compartment, Vaxon

and Vsoma, respectively, are described by

dVaxon

dt¼

1

Cað�INa � IKd � IM � Ileak;a þ IVV Þ (1)

dVsoma

dt¼

1

Csð�ICa � IKCa � IA � Ih � Ileak;s

� IVV þ IDC þ Ioffset þ IsynÞ (2)

IDC is an injected current into the cell soma, offset by Ioffset ¼ �2nA. Isyn denotes the total incoming synaptic current. All ioniccurrents except for the ones mentioned explicitly below aregiven by

Ix ¼ gxmphqðV � VxÞ (3)

where gx is the maximal conductance of the current, V is themembrane potential of either the axon compartment (INa, IKd, andIM) or the soma compartment (all other currents), and Vx is thereversal potential of the current. The activation and inactivationvariables are governed by

dm

dt¼ ðamð1�mÞ � bmmÞ (4)

dh

dt¼ ðahð1� hÞ � bhhÞ (5)

for INa and IKd, and

dmx

dt¼ ðm1xðVÞ �mxÞkmx (6)

dhx

dt¼ ðh1xðVÞ � hxÞkhx (7)

for the other currents, denoted by x ¼ CaT, CaS, KCa, A, h, and M.The calcium current is formulated in the Goldman–Hodgkin–Katzformalism

ICa ¼ ðgCaT mCaT hCaT þ gCaSmCaSÞ

�

½Ca� expVsoma

RT=F

� �� ½Ca�out

expVsoma

RT=F

� �� 1:0

PCaVsoma (8)

The IA current has two different components of the inactivationvariable,

IA ¼ gAm3AðahA1 þ ð1� aÞhA2ÞðVsoma � VAÞ (9)

a ¼1

1þ expVsoma � VaA

saA

� � (10)

The leak currents are given by

Ileak ¼ gleakðV � VleakÞ (11)

with gleak ¼ gleak;s; gleak;a and V ¼ Vsoma;Vaxon for the soma andaxon leak currents, respectively. The calcium concentration isdescribed by a first-order kinetic equation

d½Ca�

dt¼ �cICaICa � kCað½Ca� � ½Ca�0Þ (12)

Finally, the coupling between compartments is ohmic, i.e.,

IVV ¼ gVV ðVsoma � VaxonÞ (13)

The activation and inactivation functions am, bm, ah, bh, and m1,h1, km, kh are described in Table 1. The parameters aresummarised in Table 2.

2.2. Synapse model

We constructed a three neuron motif with reciprocal inhibitionbetween neurons (Fig. 1). The synapses were described by astandard synapse model with Ohm’s law for the dependence onthe post-synaptic membrane potential and a non-linear depen-dence of the activation variable S on the pre-synaptic potential.The synaptic current is given by (Pinto et al., 2001)

Ipost;presyn ¼ gpost;preSðtÞðVsyn � Vs;postÞ (14)

where gpost;pre is the maximal conductance of the synapse and S isgoverned by

tsdS

dt¼

S1 � SðtÞ

1� S1(15)

S1 ¼ cothVa;pre � Vth

Vs(16)

where ts ¼ 40 ms, Vth ¼ �40 mV;Vs ¼ 10 mV;Vsyn ¼ �80 mV. Theslow synaptic time scale of ts ¼ 40 ms reflects that neuronsinteract on the basis of their bursting dynamics rather than bysingle spikes in this system.

The maximal conductances of the synapses (and hence thecoupling) were chosen to be asymmetric, gcw ¼ 1=2g clockwiseand gccw ¼ 2g counterclockwise, the base conductance g being acontrol parameter. Note that symmetric connections amongreciprocally inhibitory three cell networks cannot producesequential activity (Rabinovich et al., 2006).

3. Results

Our results are divided into four parts: (1) a brief review of theperformance of the single neuron model, (2) the contribution ofthe three cell network to rhythmogenesis, (3) bifurcations ofactivity following changes in intrinsic bursting and/or synapticstrength, and (4) the robustness and flexibility of the networkmodel.

3.1. Isolated neuron dynamics

The isolated model neurons, like biological LP neurons, burstirregularly in control conditions, i.e., without current injection,IDC ¼ 0 nA. When depolarised beyond IDC � 1:0 nA they ceasebursting and commence tonic spiking. In the bursting regime, wecharacterised the neuron dynamics by measuring the averagenumber of spikes per burst nsp, the period of bursting Tb (orfrequency of bursting f b), and the average ratio between theperiod of bursting and the duration Ts of a burst, Tb=Ts, which wewill further refer to as width factor (WF). We defined bursts by

ARTICLE IN PRESS

Table 1Activation and inactivation functions for INa, IKd , ICa , IKCa , IA , Ih , and IM

p q am bm ah bh

INa 3 10:32

Vaxon þ 52

1� exp �Vaxon þ 52

4

� � 0:28Vaxon þ 25

expVaxon þ 25

5

� �� 1

0:128 exp �Vaxon þ 48

18

� �4

exp �Vaxon þ 25

5

� �þ 1

IKd 4 00:32

Vaxon þ 50

1� exp �Vaxon þ 50

5

� � 0:5 exp �Vaxon þ 55

40

� �

p q m1 h1 km kh

ICaT 1 1 1

1þ expVsoma � VmCaT

smCaT

� � 1

1þ expVsoma � VhCaT

shCaT

� � kmCaT khCaT

1þ expVsoma � VkhCaT

skhCaT

� �

ICaS 1 0 1

1þ expVsoma � VmCaS

smCaS

� � kmCaS

IKCa 1 1 ½Ca�

cmKCa þ ½Ca�

chKCa1

chKCa2 þ ½Ca�kmKCa khKCa

�1

1þ expVsoma � ðVmKCa1 � f � ½Ca�Þ

smKCa1

� �

�1

1þ expVsoma � ðVmKCa2 � f � ½Ca�Þ

smKCa2

� �

IA 3 1;1 1

1þ expVsoma � VmA

smA

� � hA1 and hA2: kmA hA1: khA1,

1

1þ expVsoma � VhA

shA

� � hA2 :khA2

1þ expVsoma � VkhA2

skhA2

� �

Ih 1 0 1

1þ expVsoma � Vmh

smh

� � kmh

� 1þ expVsoma � Vkmh

skmh

� �� �

IM 1 0 1

1þ expVaxon � VmM

smM

� � kmM

1þ expVaxon � VkmM

skmM

� �

M.V. Ivanchenko et al. / Journal of Theoretical Biology 253 (2008) 452–461 455

assuming that they are separated by interburst intervals of at least0.8 s (last to first spike). The listed characteristics depend on thehyperpolarisation level IDC . At a moderate hyperpolarisation level,IDC ¼ �1:0 nA, the average number of spikes per burst wasnsp ¼ 39, and the WF is Tb=Ts � 1:67 (Fig. 1c). (Note that if WF isless than 2 then the duration of bursts exceeds the duration of thesubsequent refractory period. We will call such bursts long.) Forstrong hyperpolarisation at IDC ¼ �6:0 nA, on the other hand,bursts consisted of only nsp ¼ 21 spikes and were short with theWF of Tb=Ts � 5:65 (Fig. 1c). These characteristic changes in thesingle neuron dynamics are summarised in Fig. 1d. Note that spiketrains are modulated with a frequency of about 2 Hz within bursts.(The origin of this faster oscillation in the LP neuron and in themodel is not clear but is presumably related to the interactionbetween the Ca current and the (slower) Ca dynamics.) Such amodulation is seen in the case of long bursts and also appearswhen the network dynamics stretches short bursts. In the lattercase bifurcations to this oscillatory degree of freedom cause jumpsin the average number of spikes per burst and spike train duration(see below).

3.2. Network dynamics

We studied a motif of three neurons with reciprocal inhibitoryconnections of unequal strength. The structure of the modelled

circuit represents a canonical small circuit that is simple enoughto be thoroughly analysed while close enough to known circuits tobe biologically relevant. Burst generation in this circuit is based ontwo separate mechanisms.

Intrinsic bursting: In this case the synaptic connectionsdetermine the relative phases and detailed timings of the burstingactivity of the pacemaker-type neurons. Pacemaker neurons candrive single phase rhythms such as heartbeat by themselves andin synaptically connected clusters they can produce multiphaserhythms as seen in the pyloric CPG in lobster. It has generally beenassumed that pacemaker neurons lend a degree of robustness torhythmic systems and they have indeed been found in almost allsmall circuits that have been described.

Network bursting: Here, the mechanism of rhythmogenesis is amodulational instability of the network (also known as winnerlesscompetition, WLC). The constituent neurons of the circuit are tonicspikers and the sequential bursting emerges due to synapticinteractions for sufficiently asymmetric connections between theneurons of the motif (Rabinovich et al., 2001; Afraimovich et al.,2004a, b). The sequence of bursting is uniquely determined by thenetwork structure: in our case the chosen asymmetry of theinhibitory synapses will usually select a clockwise sequence ofbursts; a reversal of the asymmetry would produce an oppositetemporal sequence by default. This mechanism does not depend onthe details of the dynamics of the constituent neurons other thanthat they are intrinsically active (Nowotny and Rabinovich, 2007).

ARTICLE IN PRESS

Table 2Parameters of the LP neuron model used in the simulations of the three neuron motif network

ICa VmCaT smCaT VhCaT shCaT kmCaT khCaT VkhCaT

15 mV �9.8 mV �40 mV 3.2 mV 45 s�1 20 s�1 �15 mV

skhCaT VmCaS smCaS kmCaS

�10 mV 29.13 mV �4.431 mV 60:86 s�1

IKCa cmKCa VmKCa1 smKCa1 VmKCa2 smKCa2 chKCa1 chKCa2

2:5mM 0 mV �23 mV �16 mV �5 mV 0:7mM 0:6mM

kmKCa khKCa f

600 s�1 35 s�1 0:6 mV=mM

IA VmA smA VhA shA kmA khA1 khA2

�12 mV �26 mV �62 mV 6 mV 140 s�1 50 s�1 3:6 s�1

VkhA2 skhA2 VaA saA

�40 mV �12 mV 7 mV �15 mV

Ih Vmh smh Vkmh skmh kmh

�70 mV 8 mV �110 mV �21.6 mV 0:33 s�1

VmM smM kmM VkmM skmM

IM �26.99 mV �5.957 mV 0:1387 s�1 �60.58 mV �13.26 mV

Reversal potentials VNa VKd VKCa VA Vh VM Vleak

50 mV �72 mV �72 mV �72 mV �20 mV �80 mV �50 mV

gCaT gCaS gKCa gA gh gleak;a gleak;s

Conductances 9:182mS 5:254mS 122:4mS 58:94mS 0:9338mS 0:08209mS 0:05078 mS

gM gNa gKd gVV

21:37mS 584:6m 116:9mS 0:3285mS

RT=F PCa ½Ca�out ½Ca�0 Ca cICa kCa

Other 11.49 mV 1:56 1=mM 15120mM 0:02mM 9.387 nF 505 M=A s 17:3 s�1

Vshift Cs

�9.343 mV 4.447 nF

M.V. Ivanchenko et al. / Journal of Theoretical Biology 253 (2008) 452–461456

The dynamical image of the bursting dynamics for thisnetwork-induced bursting is an attractor (limit cycle or strangeattractor) in the vicinity of the heteroclinic manifold, whichconsists of saddle limit cycles and separatrix manifolds connect-ing them. The saddle limit cycles correspond to spiking in one ofthe neurons while the others are silent. The sequential visits of thesystem to the neighbourhoods of the saddle cycles is the origin ofa periodic switching dynamics between silence and spiking in allneurons, only one of them being active at any time (Fig. 1b). Thestrength of the asymmetric inhibition between neurons deter-mines the proximity of the attractor to the saddle cycles and thefrequency of the network dynamics (Rabinovich et al., 2006). Thestronger the inhibition, the closer the attractor is to these saddles(note better localisation for g ¼ 10:0mS in Fig. 1b) and the longerthe period of sequential dynamics.

If the neurons can exhibit both intrinsic bursting and tonicspiking activity, network-induced bursting and pacemaker-typebursting dynamics may coexist and interact.

Numerical experiments reveal at least two types of dynamicsresulting from the interaction of intrinsic and network mechan-isms. In these experiments we assigned the values of injectedcurrent I1;2;3

DC ¼ IDC to the neurons and changed the conductivity ofall synapses g 2 ½0;20�mS while keeping the ratio of their strengthsconstant.

In the hyperpolarised regime, where the individual neuronshave short bursts (IDC ¼ �6:0 nA, Tb=Ts � 5:65) we observedspontaneous bistability of clockwise f1;2;3g and counterclockwisef3;2;1g bursting sequences (Fig. 2). The characteristics of the twoburst sequences differed. f3;2;1g bursting was slower andexhibited a larger number of spikes per burst compared to the

f1;2;3g sequence and compared to the isolated neuron dynamics.The prolonged bursting is likely due to the post-inhibitoryrebound (PIR) in the LP neuron model, which is typically mediatedby the Ih current. Because the more strongly inhibited neuronbursts next in the f3;2;1g bursting sequence, the effect of PIR isstronger in the f3;2;1gsequence than in the f1;2;3g sequence,where release from inhibition is more gradual (Fig. 2d).

The qualitative explanation for the existence of the additionalstable f3;2;1g is that in this hyperpolarised regime the intrinsicbursts are much shorter (Fig. 1d) than the circuit’s bursting periodand, therefore, can terminate spontaneously before being cut offby synaptic inhibition (for both f1;2;3g and f3;2;1g sequence,see Fig. 2d). The spontaneous burst termination can triggera strong PIR in the other neurons, in particular in the moststrongly inhibited neuron. In the fine balance between gradualrelease and sudden release from inhibition either the morestrongly inhibited (counterclockwise) neuron bursts earlier,leading to the f3;2;1g sequence or the less inhibited neuron does,leading to f1;2;3g. Both solutions are stable and which one occursdepends solely on initial conditions. At first surprising, the f3;2;1gbursting sequence persists for strong synaptic conductances dueto the PIR properties, which allow the more strongly inhibitedneurons to rebound faster even (or especially) for very stronginhibition.

In summary, in the hyperpolarised regime, the network andintrinsic burst generation interact (cooperate) in forming thenetwork oscillations. Bursting is dominantly characterised by PIRand spontaneous burst termination and, interestingly, it is thecooperation of network and intrinsic bursting mechanisms thatleads to multi-stability.

ARTICLE IN PRESS

more inhibitedescapes first

less inhibitedescapes first

3

2

1

2

3

1

0 2 4 6 8 100

0.2

0.4f b

[s−

1 ]

0 2 4 6 8 100

50

100

150

n sp

20 m

V20

mV

2 s

2 s

g [µ S]

Fig. 2. Network dynamics of the three cell motif. (a) Burst frequency and average number of spikes per burst during different modes of sequential activity. For short

intrinsic bursts in all cells, two stable sequences (f1;2;3g (light grey/blue) and f3;2;1g (dark grey/red) are observed. For long intrinsic bursts, IDC ¼ �1:0 nA, only the f1;2;3g

sequence exists (black/green). The steps in the grey curves are caused by the 2 Hz modulation of the plateau potential of long bursts which is intrinsic to the neuron model.

(b) and (c) show example trajectories on the 3D subspace of somatic membrane potentials for moderate coupling, g ¼ 1:0m (b) and strong coupling, g ¼ 10:0mS (c). Both

examples illustrate the bistability for short intrinsic bursts, IDC ¼ �6:0 nA, (f1;2;3g sequence—light grey/blue, f3;2;1g sequence—dark grey/red). The unique f1;2;3g-type

attractor for long intrinsic bursts, IDC ¼ �1:0 nA is shown in black/green. (d) Example time series data for the f1;2;3g (top) and f3;2;1g sequence (bottom). Note how the less

inhibited neuron escapes from inhibition first in the former and the more inhibited neuron escapes first in the latter.

M.V. Ivanchenko et al. / Journal of Theoretical Biology 253 (2008) 452–461 457

When intrinsic bursts are long, induced by small hyperpolar-isation, IDC ¼ �1:0 nA, only the f1;2;3g sequence is generated(Fig. 2). With increasing synaptic strength, we observed that thebursts of individual neurons first undergo restructuring: theybecome shorter and more frequent (Fig. 2a, black/green). Then, atg � 0:4mS the frequency of bursting becomes maximal and thenumber of spikes per burst minimal. The frequency of bursting isapproximately twice and the number of spikes about half thevalue of intrinsic bursts (Fig. 1c and d). Subsequently, for evenstronger inhibition, the burst frequency of the circuit declinesagain.

To understand this phenomenon one has to take into accountthat for these parameters of the neuron the intrinsic burstduration (i.e., the time duration of the spike train) substantiallyexceeds the refractory period and the constituent neurons actessentially like tonically spiking neurons. The long bursts ofintrinsic activity overlap in time leading to competition betweenneurons. Depending upon the asymmetry of coupling and relativephasing of bursts one of the competing neurons can be forced toterminate its burst earlier, which explains the observed restruc-turing. At stronger levels of inhibition ðg � 0:5mSÞ the intrinsicburst properties of each neuron are overcome and spike trains areformed that fill in the refractory stages of the other neuronsTb � Ts;1 þ Ts;2 þ Ts;3. The absolute duration of these bursts

increases with strengthening PIR at larger g. This inherentlynetwork-type WLC dynamics of competing neurons and trajec-tories converging solely to a f1;2;3g-sequence is the onlyobserved activity pattern for moderate hyperpolarisation ofneurons. Bistability of sequences was never observed under anyinitial conditions (randomly picked or corresponding to a f3;2;1gobserved for different IDC values). Like in the classic WLC withtonic neurons, for small hyperpolarisation values there is onlycompetition between neurons.

Summarising the quantitative characteristics of bursting wehave found that in all cases, the increase of inhibitory synapticconductances led to slower bursting dynamics (all frequency plotsdecrease with g in Fig. 2a) and bursts with more spikes per burst(nsp monotonically increases with g in Fig. 2a).

3.3. Bifurcations

The key message of the previous subsection is that the strengthand asymmetry of synaptic connections and the burst duration ofthe constituent neurons are the main control parameters thatdetermine the network dynamics. While we have traced theinteraction of the different rhythmogenic mechanisms by scalingthe strength of synapses, we did not yet fully address the impact

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M.V. Ivanchenko et al. / Journal of Theoretical Biology 253 (2008) 452–461458

of the two other control parameters: the intrinsic burstingproperties, i.e., specifically the burst duration, and the networkarchitecture, i.e., the asymmetry of the coupling. Changing each ofthese characteristics we expect to also observe bifurcations frommono- to bistability (or vice versa). We already know that, e.g., fora coupling strength g ¼ 20mS, the transition must take place inthe interval IDC 2 ½�6:0;�1:0�nA. We also know it must occur onchanging the balance between the coupling in the clockwise andcounterclockwise directions because if the coupling is moderatelyasymmetric we have observed that both bursting sequences arestable while if the coupling is zero in one direction only oneunique sequence can exist.

To identify bifurcations towards bistability, we kept strong,asymmetric inhibition ðg ¼ 20mSÞ, took two sets of initialconditions corresponding to a f1;2;3g and a f3;2;1g sequenceand decreased the amplitude of the hyperpolarising IDC in smallsteps (leading to a gradual decrease in intrinsic burstingfrequency of the constituent neurons). At each decrease of IDC

we used the final value of the variables as the initial value for thesubsequent run. The cessation of bistability with decreasinghyperpolarisation level of the neurons is a two-step process.

The f3;2;1g sequence first becomes imperfect through inter-mittency. Occasionally two neurons start bursting simultaneously,the one being out-of-order resuming silence after a few spikes.These events result in an increase of the bursting frequency and adecrease in the average number of spikes in a burst. The onset ofthese intermittent events was observed at IDC;1 � �2:6 nA (corre-sponding to the WF of Tb=Ts � 2:6). These imperfections becomemore frequent as the amplitude of hyperpolarisation wasdecreased further (Fig. 3a). The reason for intermittent disruptionsof the f3;2;1g sequence is a competitive interaction of overlappingbursts whenever the WF is less than 3. The less-inhibited, out-of-order neuron competes with the longer and more stronglyinhibited neuron and can manage short bursting (typically, onespike) from time to time (depending upon the relative phases ofthe bursts).

At IDC;2 � �2:1 nA, where the WF was Tb=Ts � 2:25, thebistability disappeared completely. The bifurcation point can beunderstood qualitatively as the point where the duration of burstsequals the duration of hyperpolarisation such that both silentneurons are almost equally ready to fire at the end of the burst ofthe third neuron. With less hyperpolarisation the less inhibitedneuron gains the advantage, and the f1;2;3g sequence arises. The

−3.5 −3 −2.5 −2120140160180200

n sp

−3.5 −3 −2.5 −20.04

0.05

0.06

f b [H

z]

IDC [nA]

Fig. 3. The bifurcation from bistability to a unique attractor can be observed with resp

intrinsic bursting as control parameter (as determined by current injection). The conduct

clockwise and counterclockwise connection strength as the parameter. Here, we used

sequences, triangles f3;2;1g sequences. The differently filled triangles in (a) correspon

intermittency route to monostability implies a soft transition,with a continuous change of the basic characteristics. However,the basin of attraction of the imperfect f3;2;1g sequence quicklydiminishes and essentially becomes unattainable at IDC � �2:1 nA,while intrinsic bursting reaches Tb=Ts � 2 at a lower level ofhyperpolarisation ðIDC � �1:5 nAÞ. In practice, one observes adiscontinuous jump from f3;2;1g tof1;2;3g, which is a property ofa hard bifurcation.

It should be noted that a complex multistable regime exists forneurons with intrinsic bursts of intermediate width(IDC ¼ �3:0 nA; Tb=Ts � 3:0). For weak inhibition go2:0mS, up totwo additional sequences were observed. They are period-2f1;2;3g2 with alternation of longer and shorter spike trains in allneurons (data not shown). Their appearance can be explained bythe increasing number of options in ordering bursts with minimalrestructuring, when near the critical ratio Tb=Ts � 3:0. Concur-rently, for strong inhibition (g46:1mS) the bistability of f1;2;3gand f3;2;1g can be observed.

The other principal parameter controlling bistability, asdiscussed above, is the asymmetry of coupling. To determine thethreshold ratio between weighted maximal synaptic conduc-tances in the clockwise gcw and counterclockwise gccw directions,when one of the stable sequences becomes non-observable, wegradually decreased gcw to scale down from gcw=gccw ¼ 0:25 (as inall other simulations) at initial coupling strengths g ¼ 3:0 and6:0mS. We found very similar estimates of 0.11 and 0.14 in thesetwo cases, respectively (Fig. 3b). This result demonstrates the roleof relatively weak synaptic connections and allows us tohypothesise that even background interactions in neural motifsmay substantially contribute to qualitatively new dynamicalfeatures and should by no means be neglected a priori.

3.4. Structural robustness and flexibility

The diversity of biological neurons raises the problem ofrobustness of the observed phenomena. If neurons differ slightlyin their biophysical properties how are systems able to performequivalent operations that are robust and reliable? On the otherhand, what makes these systems flexible, enabling them toprovide specific and distinct responses to inputs?

Assume that the injected DC current has the same value for allneurons, IDC ¼ �1:0 nA, but some of their parameters are different,

0 0.05 0.1 0.15 0.2 0.2520

40

60

80

gcw/gccw

n sp

0 0.05 0.1 0.15 0.2 0.250.5

1

1.5

2 x 10−3

f b [H

z]

ect to different control parameters. (a) bifurcation to a unique attractor with the

ance parameter is fixed to g ¼ 20:0mS. (b) bifurcation with the asymmetry between

g ¼ 3:0mS—open symbols and g ¼ 6:0mS—closed symbols. Circles mark f1;2;3g

d to the data obtained from the three individual neurons.

ARTICLE IN PRESS

10−10

0.2

0.4

0.6

0.8

f b [

s−1 ]

20 m

V

20 m

V

2 s 2 s

g [µ S]

100

Fig. 4. (a) Frequency of bursting for non-identical neurons (light grey) and for two bursting and one spiking neuron (I1;2DC ¼ �1:0 nA; I3

DC ¼ 2:0 nA) (dark grey). The data for

three identical neurons are shown in black for comparison. The different symbols correspond to the data from each of the three constituent neurons. (b) Neuronal activity in

the uncoupled system with two bursters and one tonic neuron. (c) The circuit activity for two bursters and one tonic neuron at g ¼ 0:07mS.

0 5 10 15 200

5

10

15

20

0 5 10 15 200

2

4

6

8

10

20 m

V

20 m

V

2 s2 s

IBI,

ISI [

s]

g [µ S]

IBI,

ISI [

s]

g [µ S]

Fig. 5. The inter-burst intervals strongly depend on the maximal conductance of inhibitory synapses. This dependence can be step-like (a) for short intrinsic bursts,

I ¼ �6:0 nA, or smooth on the large scale (b) for long intrinsic bursts, I ¼ �1:0 nA (both cases correspond to the f1;2;3g-sequence). The quantity on the y-axis are intervals

between successive spikes which can be either ISIs within bursts (dense cloud on the x-axis) or IBIs (monotonically rising cloud above the x-axis). (c) Example of the neuron

activity at g ¼ 3:5mS before a step in (a). (d) Example of the neuron activity at g ¼ 3:8mS after the same step in (a). Note the addition of a whole ‘‘sub-burst’’ in the bursts in

(d) as compared to (c).

M.V. Ivanchenko et al. / Journal of Theoretical Biology 253 (2008) 452–461 459

e.g., the somatic capacitances (or equivalently the maximalconductances of all somatic ion channels) Cs;1 ¼ 4Cs; Cs;2 ¼

2Cs; Cs;3 ¼ Cs and leakage conductances gleak;s;1 ¼ gleak;s; gleak;s;2 ¼

2gleak;s; gleak;s;3 ¼ 4gleak;s. Then bursts in autonomous neurons havethe frequencies f b;1 � 0:25; f b;2 � 0:38; f b;3 � 0:83, the averagenumbers of spikes per burst nsp;1 � 42:6; nsp;2 � 27:7; nsp;3 � 12,

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and the WF Tb;1=Ts;1 � 1:75; Tb;2=Ts;2 � 2:0; Tb;3=Ts;3 � 3:2. Simu-lations showed that quite small inhibition g � 0:3mS was enoughto synchronise neurons and form a WLC pattern, the frequencyalmost coinciding with that of identical neurons (Fig. 4a).

Moreover, the network behaviour remained robust even if oneneuron was intrinsically spiking (Fig. 4a–c). Again, at g � 0:3mSpacemakers induce bursts through network interactions, at g �

0:35mS their frequencies get synchronised and approach thereference value of three identical neurons with increasingstrength of coupling. The interaction of two neurons with shortintrinsic bursts and one spiking neuron (I1;2

DC ¼ �1:0 nA; I3DC ¼

2:0 nA) also demonstrated the induction of bursts in the spikingneuron and a persistence of bistability of sequences even thoughthe spiking neuron could not contribute to the pacemaker-drivenrhythm generation.

While the rhythmogenesis is, as we demonstrated, very robust,it is at the same time very flexible. Synaptic plasticity changingthe asymmetry of coupling and the DC-input changing the WF ofintrinsic bursts are (as we have shown above) able to make thenetwork switch between the regimes of mono- and bistability.Furthermore, over the studied range of synaptic conductance, theperiod of bursting (presented by the ISIs41 s) could increase up to8 times if bursts were short (IDC ¼ �6:0 nA) and 18 times whenbursts were long (IDC ¼ �1:0 nA), see Fig. 5. The steps in Fig. 5aappear because the bifurcations of increasing the number ofspikes per burst involve the slow oscillation degree of freedom(modulating the active phase of bursts with the frequency about2 Hz) discussed in Section 3.1. The bifurcation of adding oneperiod of intra-burst oscillation corresponds to adding on theorder of 10 spikes to a burst. An example is shown in Fig. 5b and cfor IDC ¼ �6:0 nA. The period-2 slow modulation at g ¼ 3:5mSbifurcates to period-3 one at g ¼ 3:8mS and, accordingly, theaverage number of spikes per burst jumps from nsp ¼ 32 to 45.

4. Discussion

Based on a dynamical analysis of a specific neuron motifconsisting of three bursting neurons that are asymmetricallycoupled through slow inhibition, we addressed the followingquestions: 1. What are the characteristics of the hybrid pace-maker-network mechanisms of bursting in this elementary net-work? 2. What combination of neural circuit parameters, e.g.,synaptic strengths or length of bursts, control the qualitativechanges of the rhythmic patterns? 3. Do the concurrent mechan-isms provide robustness and flexibility to the motif network, i.e.,the ability to generate specific but reproducable responses underthe action of an informational signal?

To gain insights into the first and second questions we usednumerical bifurcation analysis of the network dynamics (see Figs.2 and 3). We showed that long intrinsic bursts underlie acompetitive interaction of neurons that essentially restructuresbursts, when inhibition strengthens. This interaction yields aunique bursting pattern, with characteristics that sensitivelydepend upon the level of inhibition (as with a pure networkmechanism). Short intrinsic bursts allow cooperative networkoscillations, in which inhibitory interactions promote activity byPIR, in addition to the competition scenario. This leads tobistability of bursting patterns. Two parameters control theemergence of bistability: the asymmetry of reciprocal inhibitionand the length of intrinsic bursts.

Because of the interaction of intrinsic and synaptic dynamicsboth types of bursting patterns are highly robust, i.e., even if oneof the neurons is changed to a tonic spiking regime, the other twoneurons make it resume bursting and produce almost the samenetwork patterns as if all neurons are identical. The hybrid

pacemaker-network bursting does show a well-pronouncedflexibility as well. The level of DC-input to the cells (interpretedas their intrinsic excitability) and synaptic plasticity can make thecircuit switch between bistability and monostability, and mod-ulating the overall synaptic strength can change the character-istics of bursting on a large scale.

Rhythmogenesis based on inhibition has been studied exten-sively in the past (see e.g., Traub et al., 2000 for a review).However, the existing theoretical work was often focused onnetworks with symmetric reciprocal inhibition. In this work it hasbeen well-established that, depending on synaptic (like synaptictimescale, synaptic depression) and intrinsic (like frequencyadaptation) characteristics the inhibition between two neuronscan produce synchronous or antiphase activity (Calabrese, 1995;Wang and Rinzel, 1992; Skinner et al., 1994; Terman et al., 1997).Concurrent gap-junction coupling is, furthermore, known tocomplement burst generation through inhibition and its synchro-nisation (in-phase or in anti-phase) (Skinner et al., 1999; Mancillaet al., 2007). These ideas have shown a vast potential formodelling coordination, memory and decision-making tasks likeartificial CPG design (Lewis et al., 2005) or the two-intervaldiscrimination problem (Machens et al., 2005). Generalisation totwo competing neural populations in the context of explainingbinocular rivalry yields essentially similar phenomena (Shpiro etal., 2006). The emphasis was typically on the case of symmetricalcoupling, because asymmetry seemed to add little qualitativechange to the dynamics, if any. Another approach is analysingsymmetry breaking bifurcations and arising stable heterocliniccycles (the image of alternating neural activity) in networks withring topology (Buono et al., 1997; Goel and Ermentrout, 2002). Inthis case the symmetry of coupling strengths was the key demandfor the analytical treatment to be fulfilled. A step towardsunderstanding the principal role of asymmetry in the rhythmo-genesis in multi-cell motifs was made in a series of papers byRabinovich et al. (2001), Afraimovich (2004a, b) and Nowotny andRabinovich (2007). They revealed the basics of the origin ofsequential bursting activity in inhibitory spiking neural ensem-bles (the minimal motif had three neurons). These oscillationswere shown to bifurcate from stable heteroclinic sequences in thephase space of the network. Moreover, it was proved thatsequences are uniquely determined by the given asymmetry ofthe network.

Here, we have presented the extension of these ideas to thecase of intrinsically bursting neurons to analyse the interplaybetween intrinsic and network mechanisms of rhythmic activity.While our results are subject to certain assumptions (the type ofmotif, relation between synaptic and neural timescales), wegained significant insights into the very general problem of hybridintrinsic-network bursting mechanisms in neural ensembles, andmarked exciting directions for further theoretical studies as wellas making predictions for experimental research.

Acknowledgement

This work was partly supported by NIH, Grant R01 NS050945.

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