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Journal of Computational Neuroscience 12, 5–25, 2002

c ? 2002 Kluwer Academic Publishers. Manufactured in The Netherlands.

Ghostbursting: A Novel Neuronal Burst Mechanism

BRENT DOIRON, CARLO LAING AND ANDR´E LONGTIN

Physics Department, University of Ottawa, 150 Louis Pasteur, Ottawa, Ontario, Canada K1N 6N5

bdoiron@physics.uottawa.ca

LEONARD MALER

Department of Cellular and Molecular Medicine, University of Ottawa, 451 Smyth Road, Ottawa,

Canada K1H 8M5

Received June 8, 2001; Revised October 17, 2001; Accepted November 1, 2001

Action Editor: John Rinzel

Abstract.

toproducehigh-frequencyburstdischargewithconstantdepolarizingcurrent(Turneretal.,1994).Wepresentatwo-

compartment model of an ELL pyramidal cell that produces burst discharges similar to those seen in experiments.

The burst mechanism involves a slowly changing interaction between the somatic and dendritic action potentials.

Burst termination occurs when the trajectory of the system is reinjected in phase space near the “ghost” of a saddle-

node bifurcation of fixed points. The burst trajectory reinjection is studied using quasi-static bifurcation theory,

that shows a period doubling transition in the fast subsystem as the cause of burst termination. As the applied

depolarization is increased, the model exhibits first resting, then tonic firing, and finally chaotic bursting behavior,

in contrast with many other burst models. The transition between tonic firing and burst firing is due to a saddle-node

bifurcation of limit cycles. Analysis of this bifurcation shows that the route to chaos in these neurons is type I

intermittency, and we present experimental analysis of ELL pyramidal cell burst trains that support this model

prediction. By varying parameters in a way that changes the positions of both saddle-node bifurcations in parameter

space, we produce a wide gallery of burst patterns, which span a significant range of burst time scales.

Pyramidal cells in the electrosensory lateral line lobe (ELL) of weakly electric fish have been observed

Keywords:

bursting, electric fish, compartmental model, backpropagation, pyramidal cell

1. Introduction

Burst discharge of action potentials is a distinct and

complexclassofneuronbehavior(Connorsetal.,1982;

McCormick et al., 1985; Connors and Gutnick, 1990).

Burst responses show a large range of time scales and

temporal patterns of activity. Many electrophysiologi-

cal studies of cortical neurons have identified cells that

intrinsically burst at low frequencies (<20 Hz) (Bland

and Colom, 1993; Steriade et al., 1993; Franceschetti

et al., 1995). However, recent work in numerous sys-

tems has now identified the existence of “chattering”

cells that show burst patterns in the high-frequency

γ range (>20 Hz) (Turner et al., 1994; Par´ e et al.,

1998; Gray and McCormick, 1996; Steriade et al.,

1998;LemonandTurner,2000;Brumburgetal.,2000).

Also,thespecificinterspikeinterval(ISI)patternwithin

the active phase of bursting varies considerably across

burstcelltypes.Certainburstingcellsshowalengthen-

ing of ISIs as a burst evolves (e.g., pancreatic-β cells,

Sherman et al., 1990), others a parabolic trend in the

ISI pattern (e.g., Aplysia R15 neuron, Adams, 1985),

and yet others show no change in the ISI during a burst

(e.g.,thalamicreticularneuron,PinaultandDeschˆ enes,

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Doiron et al.

1992). This diversity of specific time scales and ISI

patterns suggests that numerous distinct burst mecha-

nismsexist.Knowledgeoftheburstmechanismsallows

one to predict how the burst output may be modified

or halted completely in response to stimuli. This may

have consequences for the information content of the

cell’s output (Lisman, 1997).

Pyramidal cells in the electrosensory lateral line

lobe (ELL) of the weakly electric fish Apteronotus lep-

torhynchus have been shown to produce both tonic fir-

ing and γ frequency sustained burst patterns of action

potential discharge (Turner et al., 1994; Turner and

Maler, 1999; Lemon and Turner, 2000). These sec-

ondary sensory neurons are responsible for transmit-

ting information from populations of electroreceptor

afferentsthatconnecttotheirbasalbushes(seeBerman

and Maler, 1999, and references therein). In vivo

recordings from ELL pyramidal cells have indicated

that their bursts are correlated with certain relevant

stimulus features, suggesting the possible importance

of ELL bursts for feature detection (Gabianni et al.,

1996; Metzner et al., 1998; Gabianni and Metzner,

1999).Thus,boththeproximityofELLpyramidalcells

to the sensory periphery and the known relevance of

their bursts to signal detection suggest that studies of

ELLburstingmayprovidenovelresultsconcerningthe

role of burst output in sensory processing.

Previous in vitro and in vivo experiments have fo-

cused both on specifying the mechanism for burst dis-

charge of ELL pyramidal cells and showing methods

for the modulation of burst output (Turner et al., 1994,

1996; Turner and Maler, 1999; Lemon and Turner,

2000; Bastian and Nguyenkim, 2001; Rashid et al.,

2001). Lemon and Turner (2000) have shown that a

frequency dependent or “conditional” action potential

backpropagationalongtheproximalapicaldendriteun-

derliesboththeevolutionandterminationofELLburst

output. Recently, through the construction and analy-

sis of a detailed multicompartmental model of an ELL

pyramidal cell, we have reproduced burst discharges

similartothoseseeninexperiment.Thismodelallowed

us to make strong predictions about the characteristics

of the various ionic channels that could underlie the

burst mechanism (Doiron et al., 2001b). However, a

deeperunderstandingofthedynamicsoftheELLburst

mechanism could not be achieved due to the high di-

mensionality(312compartmentsand10ioniccurrents)

of the model system.

The analysis of bursting neurons using dynami-

cal systems and bifurcation theory is well established

(Rinzel, 1987; Rinzel and Ermentrout, 1989; Wang

and Rinzel, 1995; Bertram et al., 1995; Hoppensteadt

and Izhikevich, 1997; Izhikevich, 2000; Golubitsky

et al., 2001). These studies have reduced complex

neural behavior to flows of low-dimensional nonlin-

ear dynamical systems. In the same spirit, we present

hereatwo-compartmentreductionofourdetailedionic

model of an ELL pyramidal cell (Doiron et al., 2001a,

2001b). The reduced model, referred to as the ghost-

burster (this term is explained in the text), produces

burst discharges similar to both the full model and

in vitro recordings of bursting ELL pyramidal cells.

This analysis supports our previous predictions on

the sufficient ionic and morphological requirements of

the ELL pyramidal cell burst mechanism. In addition

to this, the low dimension of this model allows for

a detailed dynamical systems treatment of the burst

mechanism.

When applied depolarization is treated as a bifur-

cation parameter, the model cell shows three distinct

dynamical behaviors: resting with low-intensity depo-

larizing current, tonic firing at intermediate levels, and

chaoticburstdischargeathighlevelsofdepolarization.

This is contrary to other burst mechanisms that show

burstdischargeforlowlevelsofdepolarizationandthen

transition to tonic firing as applied current is increased

(Hayashi and Ishizuka, 1992; Gray and McCormick,

1996;Steriadeetal.,1998;Wang,1999).Bothofthebi-

furcations separating the three dynamical behaviors of

the ghostburster are shown to be saddle-node bifurca-

tions of either fixed points (quiescent to tonic firing) or

limitcycles(tonicfiringtobursting).Treatingourburst

model as a fast-slow burster (Rinzel, 1987; Rinzel and

Ermentrout,1989;WangandRinzel,1995;Izhikevich,

2000) and using quasi-static bifurcation analysis, we

show that the burst termination is linked to a transi-

tion from period-one to period-two firing in the fast

subsystem, causing the burst trajectory to be reinjected

nearthe“ghost”ofthesaddle-nodebifurcationoffixed

points.Thetimespentnearthesaddle-nodedetermines

the interburst interval length.

This concept of burst discharge is quite different

from the two-bifurcation analysis used to understand

most other burst models (Rinzel, 1987; Rinzel and

Ermentrout, 1989; Wang and Rinzel, 1995; de Vries,

1998; Izhikevich, 2000; Golubitsky et al.,2001). Fur-

ther analysis predicts that the route to chaos in transi-

tioningfromtonictochaoticburstfiringisthroughtype

I intermittency (Pomeau and Manneville, 1980). Com-

parisons of both model and experimental ELL burst

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Ghostbursting: A Novel Neuronal Burst Mechanism7

recordings support this prediction. Furthermore, by

changing the relative position of the two-saddle-node

bifurcations in a two-parameter bifurcation set, the

time scales of both the burst and interburst period can

be chosen independently, allowing for wide variations

in possible burst outputs.

2. Methods

2.1. ELL Pyramidal Cell Bursting

Figure 1A shows in vitro recordings from the soma

of a bursting ELL pyramidal cell with a constant de-

polarizing input. The bursts comprise a sequence of

action potentials, which appear on top of a slow depo-

larizationofthesubthresholdmembranepotential.The

depolarization causes the interspike intervals (ISIs) to

decrease as the burst evolves. The ISI decrease culmi-

nates in a high-frequency spike doublet that triggers a

relatively large after-hyperpolarization (AHP) labeled

a burst-AHP (bAHP). The bAHP causes a long ISI that

separates the train of action potentials into bursts, two

of which are shown in Fig. 1A. The full characteriza-

tionoftheburstsequencehasbeenpresentedinLemon

and Turner (2000).

Immunohistochemicalstudiesoftheapicaldendrites

of ELL pyramidal cells have indicated a patched dis-

tribution of sodium channels along the first ∼200 µm

of the apical dendrite (Turner et al., 1994). Figure 1B

illustrates schematically such a Na+channel distribu-

tion over the dendrite. The active dendritic Na+allows

for action potential backpropagation along the apical

dendrite,producingadendriticspikeresponse(Fig.1B;

Turner et al., 1994). Na+or Ca2+mediated action po-

tential backpropagation has been observed in several

other central neurons (Turner et al., 1994; Stuart and

Sakmann, 1994; for a review of active dendrites, see

Stuartetal.,1997)andhasbeenmodeledinmanystud-

ies(Traubetal.,1994;Mainenetal.,1995;Vetteretal.,

2001;Doironetal., 2001b).Actionpotentialbackprop-

agationproducesasomaticdepolarizingafter-potential

(DAP) after the somatic spike, as shown in Fig. 1B.

The DAP is the result of a dendritic reflection of the

somaticactionpotential.Thisrequiresbothalongden-

dritic action potential half-width as compared to that

of a somatic action potential and a large somatic hy-

perpolarization succeeding an action potential. These

twofeaturesallowforpassiveelectrotoniccurrentflow

fromthedendritetothesomasubsequenttothesomatic

spike, yielding a DAP.

Figure 1.

In vitro recording of burst discharge from the soma of an ELL pyra-

midal cell with constant applied depolarizing current. Two bursts of

actionpotentialsareshown,eachexhibitingagrowingdepolarization

as the burst evolves, causing the ISI to decrease; the burst ends with

a high-frequency doublet ISI. The doublet triggers a sharp removal

of the depolarization, uncovering a prominent AHP, labeled a burst-

AHP. B: Active Na+conductances are distributed along the soma

and proximal apical dendrite of ELL pyramidal cells (left). Na+re-

gionsareindicatedwithverticalbarstotheleftoftheschematic.Note

that the distribution of dendritic Na+is punctuate, giving regions of

high Na+concentration (often referred to as “hot spots”) separated

by regions of passive dendrite. The active dendritic regions allow for

backpropagation of a somatic action potential through a dendritic

action potential response, as seen from ELL recordings from both

the soma and proximal (∼150 µm) dendrite (right). Somatic action

potentialrectificationbyK+currentsandthebroaderactionpotential

in the dendrite allow for electrotonic conduction of the dendritic ac-

tion potential to the soma, resulting in a DAP at the soma (inset). We

thank R.W. Turner for generously providing his data for the figure.

ELL burst discharge and dendritic backpropagation. A:

Recent work has shown the necessity of spike back-

propagation in ELL pyramidal cells for burst dis-

charge (Turner et al., 1994; Turner and Maler, 1999;

Lemon and Turner, 2000). These studies blocked spike

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Doiron et al.

backpropagation by locally applying tetrodoxin (TTX,

aNa+channelblocker)toapicaldendritesofELLpyra-

midal cells, after which all bursting ceased and only

tonic firing persisted. Our previous modeling study

(Doiron et al., 2001b) reproduced this result, since

when active Na+conductances were removed from

all dendritic compartments, similar results were ob-

tained. However, in that study we modeled the prox-

imal apical dendrite with 10 compartments, five of

which contained active spiking Na+channels. The

large number of variables in such a model is incom-

patible with the objectives of the present study. In light

of this and following previous modeling studies in-

volving action-potential backpropagation (Pinsky and

Rinzel, 1994; Bressloff, 1995; Mainen and Sejnowski,

1996;L´ anskyandRodriguez,1999;Wang,1999;Booth

and Bose, 2001), we investigate a two-compartment

model of an ELL pyramidal cell, where one compart-

ment represents the somatic region, and the second

the entire proximal apical dendrite. Note that a two-

compartment treatment of dendritic action potential

backpropagation is a simplification of the cable equa-

tion(KeenerandSneyd,1998).However,inconsidera-

tionofthegoalsofthepresentstudy,whichrequireonly

DAP production, the two-compartment assumption is

sufficient.

2.2.Two-Compartment Model

Aschematicofourtwo-compartmentmodelofanELL

pyramidal cell is shown in Fig. 2, together with the ac-

tive inward and outward currents that determine the

compartment membrane potentials. Both the soma and

dendrite contain fast inward Na+currents, INa,s and

INa,d,andoutwarddelayedrectifying(Dr)K+currents,

respectively IDr,sand IDr,d. These currents are neces-

sary to reproduce somatic action potentials and proper

spikebackpropagationthatyieldssomaticDAPs.Inad-

dition, both the soma and dendrite contain passive leak

currents Ileak. The membrane potentials Vs(somatic)

and Vd (dendritic) are determined through a modi-

fied Hodgkin/Huxley (1952) treatment of each com-

partment. The coupling between the compartments is

assumed to be through simple electrotonic diffusion

giving currents from soma to dendrite Is/d, or vice-

versa Id/s. In total, the dynamical system comprises

six nonlinear differential equations, Eqs. (1) through

(6); henceforth, we refer to Eqs. (1) through (6) as the

ghostburster model, and the justification for the name

is presented in the Results section.

Figure 2.

an ELL pyramidal cell. The ionic currents that influence both the so-

matic and dendritic compartment potentials are indicated. Arrows

that point into the compartment represent inward Na+currents,

whereasarrowspointingoutwardrepresentK+currents(thespecific

currents are introduced in the text). The compartments are joined

through an axial resistance, 1/gc, allowing current to be passed be-

tween the somatic and dendritic compartments.

Schematic of two-compartment model representation of

Soma:

dVs

dt

= IS+gNa,s·m2

+gDr,s·n2

+gleak·(Vl−Vs)

=n∞,s(Vs) − ns

τn,s

Dendrite:

dVd

dt

+gDr,d·n2

+

dhd

dt

dnd

dt

dpd

dt

∞,s(Vs)·(1−ns) · (VNa−Vs)

s·(VK−Vs) +gc

κ·(Vd−Vs)

(1)

dns

dt

(2)

=gNa,d·m2

∞,d(Vd)·hd· (VNa−Vd)

d· pd·(VK−Vd)

(1 − κ)·(Vs−Vd) + gleak·(Vl−Vd)

=h∞,d(Vd) − hd

τh,d

=n∞,d(Vd) − nd

τn,d

=p∞,d(Vd) − pd

τp,d

gc

(3)

(4)

(5)

(6)

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Ghostbursting: A Novel Neuronal Burst Mechanism9

Table 1.

parameters introduced in Eqs. (1) through (6). Each ionic current

(INa,s; IDr,s; INa,d; IDr,d) is modeled by a maximal conductance

gmax(in units of mS/cm2), sigmoidal activation and possibly in-

activation, infinite conductance curves involving both V1/2and k

parameters m∞,s(Vs) =

stant τ (in units of ms). Double entries x/y correspond to channels

with both activation (x) and inactivation (y), respectively. If the

activation time constant value is N/A, then the channel activation

tracks the membrane potential instantaneously. Other parameters

values are gc = 1, κ = 0.4, VNa=40 mV, VK = −88.5 mV,

Vleak= −70 mV, gleak= 0.18, and Cm= 1 µF/cm2. These values

compare in magnitude to those of other two-compartment models

(Pinsky and Rinzel, 1994; Mainen and Sejnowski, 1996).

Model parameter values. The values correspond to the

1

1+e−(Vs−V1/2)/k, and a channel time con-

Current

gmax

V1/2

K

τ

INa,s(m∞,s(Vs))

IDr,s(ns(Vs))

INa,d(m∞,d(Vd)/hd(Vd))

IDr,d(nd(Vd)/pd(Vd))

55

−40

−40

3N/A

2030.39

5

−40/−52

−40/−65

5/−5

5/−6

N/A/1

15 0.9/5

Table 1 lists the values of all channel parameters

used in the simulations. The soma is modeled with two

variables(seeEqs.(1)and(2)).Thereductionfromthe

classic four dimensional Hodgkin-Huxley model is ac-

complishedbyslaving INa,sactivation,m∞,s,toVs(i.e.,

the Na+activation mstracks Vsinstantaneously), and

modeling its inactivation, hs, through IDr,sactivation,

ns(we set hs≡ 1−ns). This second approximation is

aresultofobservinginourlargecompartmentalmodel

(Doiron et al., 2001b) that hs+ ns≈ 1 during spiking

behavior.Bothoftheseapproximationshavebeenused

invariousothermodelsofspikingneurons(Keenerand

Sneyd, 1998). The dendrite is modeled with four vari-

ables(seeEqs.(3)through(6)).Similartothetreatment

of INa,s,weslave INa,dactivationm∞,dto Vdbutmodel

its inactivation with a separate dynamical variable hd.

Lemon and Turner (2000) have shown that the refrac-

tory period of dendritic action potentials is larger than

that of somatic in ELL pyramidal neurons. This result

has previously been shown to be necessary for burst

termination (Doiron et al., 2001b). To model the dif-

ferential somatic/dendritic refractory period we have

chosen τh,dto be longer than τn,s(similar to our large

compartmental model, Doiron et al., 2001b). This re-

sult has not been directly verified through immunohis-

tochemical experiments of ELL pyramidal cells Na+

channels; thus, at present, this remains an assumption

in our model.

The crucial element for the success of our model in

reproducing bursts is the treatment of IDr,d. Dendritic

recordings from bursting ELL pyramidal cells show

a slow, frequency-dependent broadening of the action

potential width as a burst evolves (Lemon and Turner,

2000). Such a cumulative increase in action potential

width has been observed in other experimental prepa-

rations and has been linked to a slow inactivation of

rectifier-likeK+channels(Aldrichetal.,1979;Maand

Koester, 1996; Shao et al., 1999). In light of this, our

previous study (Doiron et al., 2001b) modeled the den-

dritic K+responsible for spike rectification with both

activation and inactivation variables. When the time

constant governing the inactivation was relatively long

(5 ms) compared with the time constants of the spik-

ing currents (∼1 ms), the model produced a burst dis-

chargecomparabletoELLpyramidalcellburstrecord-

ings. Doiron et al. (2001b) also considered other po-

tential burst mechanisms, including slow activation of

persistent sodium; however, only slow inactivation of

dendritic K+produced burst results comparable to ex-

periment. At this time there is no direct evidence for

a cumulative inactivation of dendritic K+channels in

ELL pyramidal cells, and these results remain a model

assumption. However, preliminary work suggests that

the shaw-like AptKv3.3 channels may express such a

slow inactivation (R.W. Turner personal communica-

tion); these channels have been shown to be highly ex-

pressed in the apical dendrites of ELL pyramidal cells

(Rashid et al., 2001). In the present work, our dynam-

ical system also models dendritic K+current IDr,d, as

havingbothactivationndandslowinactivation pdvari-

ables (see Eqs. (3), (5), and (6)). Slow inactivation of

K+channels, although not a mechanism in contempo-

raryburstmodels,wasproposedbyCarpenter(1979)in

the early stages of mathematical treatment of bursting

in excitable cells. We do not implement a similar slow

inactivation of somatic Dr,s since somatic spikes ob-

served in bursting ELL pyramidal cells do not exhibit

broadening as the burst evolves (Lemon and Turner,

2000).

Thesomatic-dendriticinteractionismodeledassim-

ple electrotonic diffusion with coupling coefficient gc

and is scaled by the ratio of somatic-to-total surface

area κ. This form of coupling has been used in pre-

vious two-compartment neural models (Mainen and

Sejnowski, 1996; Wang, 1999; Kepecs and Wang,

2000; Booth and Bose, 2001). ISrepresents either an

applied or synaptic current flowing into the somatic

compartment.Inthepresentstudy ISisalwaysconstant

in time and is used as a bifurcation parameter. Physi-

ological justification for the parameter values given in

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Doiron et al.

Table 1 is presented in detail in Doiron et al. (2001b).

Equations (1) through (6) are integrated by a fourth-

order Runge-Kutta algorithm with a fixed time step of

?t = 5 × 10−6s.

3. Results

3.1. Model Performance

Figure 3A and B shows simulation time series of Vs

and pd, respectively, for the ghostburster with constant

depolarizationof IS= 9.Weseearepetitivebursttrain

similartothatshowninFig.1A.Figure4comparesthe

time series of Vsand Vdfor the ghostburster (bottom

row) during a single burst to both a somatic and den-

dritic burst from ELL pyramidal cell recordings (top

row) and the large compartmental model presented in

Doironetal.(2001b)(middlerow).Allburstsequences

areproducedwithconstantsomaticdepolarization.The

somatic bursts all show the same characteristic growth

in depolarization (DAP growth) and consequent de-

creases in ISI leading to the high-frequency doublet.

Thedendriticburstsallshowthatadendriticspikefail-

ure is associated with both doublet spiking and burst

termination.ThesomaticAHPsinthesimulationofthe

ghostburster do not show a gradual depolarization dur-

Figure 3.

tial Vsduring burst output. B: Dendritic IDr,dinactivation variable

pdduring the same burst simulation as in A. Note the cumulative

(slow) inactivation as the burst evolves and the rapid recovery from

inactivation during the interburst period.

Model bursting. A: Time series of the somatic poten-

ing the burst, as do both the AHPs in the ELL pyrami-

dal cell recordings and the large compartmental model

simulations. This is a minor discrepancy, which is not

relevant for the understanding of the burst mechanism.

The mechanism involved in the burst sequences

shown in Figs. 3 and 4 has been explained in de-

tail (although not from a dynamical systems point of

view) in past experimental and computational studies

(Lemon and Turner, 2000; Doiron et al., 2001b). We

give a short overview of this explanation. Action po-

tential backpropagation is the process of a somatic ac-

tion potential actively propagating along the dendrite

due to activation of dendritic Na+channels. Rapid hy-

perpolarization of the somatic membrane, mediated by

somatic potassium activation ns, allows electrotonic

diffusion of the dendritic action potential, creating a

DAPinthesomaticcompartment.However,withrepet-

itive spiking the dendritic action potentials, shown by

Vd, broaden in width and show a baseline summation

(Fig. 4). This is due to the slow inactivation of IDr,d,

mediatedby pd,asshowninFig.3B.Thisfurtherdrives

electrotonic diffusion of the dendritic action potential

back to the soma; consequently, the DAP at the soma

grows, producing an increased somatic depolarization

as the burst evolves. This results in decreasing somatic

ISIs,asexperimentallyobservedduringELLburstout-

put. This positive feedback loop between the soma and

dendrite finally produces a high-frequency spike dou-

blet (Fig. 4).

Doublet ISIs are within the refractory period of den-

dritic spikes but not that of somatic spikes (Lemon and

Turner, 2000). This causes the backpropagation of the

second somatic spike in the doublet to fail, due to lack

of recovery of INa,dfrom its inactivation, as shown in

the dendritic recordings (Fig. 4). This backpropaga-

tion failure removes any DAP at the soma, uncovering

a large bAHP, and thus terminates the burst. This cre-

ates a long ISI, the interburst period, which allows pd

and hdto recover, in preparation for the next burst (see

Fig. 3B).

3.2. Bifurcation Analysis

In the following sections we use dynamical systems

theory to explore various aspects of the ghostburster

equations (Eqs. (1) through (6)). An introduction to

someoftheconceptsweusecanbefound,forexample,

in Strogatz (1994). An alternative explanation of the

burst mechanism, given in physiological terms, was

presented in Doiron et al. (2001b).

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Ghostbursting: A Novel Neuronal Burst Mechanism11

Figure 4.

full multicompartmental model simulations (middle row; simulation presented in Doiron et al., 2001b), and reduced two-compartment model

simulations (bottom row; Eqs. (1) through (6)). All bursts are produced by applying constant depolarization to the soma (0.3 nA top; 0.6

nA middle; Is = 9, bottom). The columns show both somatic and dendritic responses for each row. The reduced-model somatic spike train

reproduces both the in vitro data and full-model simulation spike trains by showing the growth of DAPs and reduction in ISI as the burst evolves.

All somatic bursts are terminated with a large bAHP, which is connected to the dendritic spike failure.

Model performance. A single burst is obtained from ELL pyramidal cell recordings (top row; data donated by R.W. Turner),

Figure 5A gives the bifurcation diagram of hd as

computed from the ghostburster with IStreated as the

bifurcationparameter.Wechose ISsincethisisbothan

experimentally and physiologically relevant parameter

to vary. Three distinct dynamical behaviors are ob-

served. For IS< IS1two fixed points exist—one sta-

ble, representing the resting state, and one unstable

saddle. When IS= IS1, the stable and unstable fixed

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Doiron et al.

Figure 5.

hdas the representative dynamic variable and plot hdon the vertical axis. For IS< IS1a stable fixed point (solid line) and a saddle (dashed line)

coexist.Asaddle-nodebifurcationoffixedpoints(SNFP)occursat IS= IS1.For IS1< IS< IS2stable(filledcircles)andunstable(opencircles)

limit cycles coexist, the maximum and minimum of which are plotted. A saddle-node bifurcation of limit cycles (SNLC) occurs at IS= IS2. For

IS> IS2a chaotic attractor exists; we show this by plotting the maximum and minimum of hdfor all ISIs that occur in a 1 s simulation for fixed

IS. A reverse-period doubling cascade out of chaos is observed for large IS. The software package AUTO (Doedel, 1981) was used to construct

the leftmost part of the diagram. B: Instantaneous frequency (1/ISI) is plotted for ISsimulations of the ghostburster model for each increment

in IS. The transitions from rest to tonic firing and tonic firing to chaotic bursting are clear. C: The maximum Lyapunov exponent λ as a function

of IS.

A: Bifurcation diagram of the ghostburster equations (Eqs. (1) through (6)) as a function of the bifurcation parameter IS. We choose

points coalesce in a saddle-node bifurcation of fixed

points on an invariant circle, after which a stable limit

cycle exists. This is characteristic of Class I spike ex-

citability (Ermentrout, 1996), of which the canonical

model is the well-studied θ neuron (Hoppensteadt and

Izhikevich, 1997). For IS1< IS< IS2the stable limit

cycle coexists with an unstable limit cycle. Both limit

cyclescoalesceat IS= IS2inasaddle-nodebifurcation

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Ghostbursting: A Novel Neuronal Burst Mechanism13

of limit cycles. For IS> IS2the model dynamics, lack-

ing any stable periodic limit cycle, evolve on a chaotic

attractor giving bursting solutions as shown in Figs. 3

and 4 (lower panel). As ISincreases further a period

doublingcascadeoutofchaosisobserved,andaperiod-

two solution exists for high IS. The importance of both

ofthesaddle-nodebifurcationswillbeexploredinlater

sections.

Figure 5B shows the observed spike discharge fre-

quencies f (≡1/ISI) from the ghostburster as ISis var-

ied over the same range as in Fig. 5A. The rest state

IS< IS1admits no firing, indicated by setting f =0.

For IS1< IS< IS2the stable-limit cycle attractor pro-

ducesrepetitivespikedischargegivingasinglenonzero

fvalueforeachvalueof IS. f becomesarbitrarilysmall

as ISapproaches IS1from above due to the infinite-

period bifurcation at IS1. However, for IS> IS2 the

attractor produces a varied ISI pattern, as shown in

Figs. 3 and 4. This involves a range of observed f val-

ues for a given fixed IS, ranging from ∼100 Hz in the

interburst interval to almost 700 Hz at the doublet fir-

ing. The burst regime, IS> IS2does admit windows

of periodic behavior. A particularly large window of

IS∈ (13.13,13.73) shows a stable period six solution

that undergoes a period doubling cascade into chaos as

ISis decreased. Finally, the period doubling cascade

out of chaos for IS? IS2is evident.

Figure 5C shows the most positive Lyapunov expo-

nent λ of the ghostburster as a function of IS. We see

that λ < 0 for IS < IS1because the only attractor is

a stable fixed point. For IS1< IS< IS2, λ = 0 because

the attractor is a stable limit cycle. Of particular inter-

est is that λ is positive for a range of ISgreater than

IS2, indicating that the bursting is chaotic. The win-

dows of periodic behavior within the chaotic bursting

are indicated by λ being zero (e.g., the large window

for IS∈(13.13,13.73)).For IS>17.65,λ = 0because

the ghostburster undergoes a period doubling cascade

out of chaos, resulting in a stable period two solution.

Figure 6 is a two parameter bifurcation set showing

curves for both the saddle-node bifurcation of fixed

points (SNFP) and of limit cycles (SNLC). The pa-

rameters are the applied current IS, already studied in

Fig. 5, and gDr,d, which controls the influence of the

slowdynamicalvariable pd(seeEq.(3)).Itisnaturalto

choose gDr,das the second bifurcation parameter since

the burst mechanism involves dendritic backpropaga-

tion, which IDr,d regulates, and gDr,d can be experi-

mentally adjusted by focal application of K+channel

blockers to the apical dendrites of ELL pyramidal cells

Figure 6.

furcations of fixed points (SNFP) and limit cycles (SNLC) bifurca-

tions were tracked, using AUTO (Doedel, 1981) in the (IS, gDr,d)

subspace of parameter space. The curves partition the space into

quiescence, tonic firing, and chaotic bursting regimes.

Two-parameter bifurcation set. Both the saddle-node bi-

(Rashid et al., 2001). A vertical line in Fig. 6 corre-

sponds to a bifurcation diagram similar to that pre-

sented in Fig. 5A. The diagram in Fig. 5A corresponds

to the rightmost value of gDr,din Fig. 6 (gDr,d=15).

The intersection of the curves SNFP and SNLC with

anyverticallinegivesthevalues IS1and IS2forthatpar-

ticularvalueofgDr,d.Thus,thecurvesSNFPandSNLC

partitionparameterspaceintoregionscorrespondingto

quiescence, tonic firing, and chaotic bursting solutions

of the ghostburster equations, as indicated in Fig. 6.

The curves intersect at a codimension-two bifurcation

point corresponding to simultaneous fixed-point and

limit-cycle saddle-node bifurcations. The curve to the

left of the intersection point corresponds to the codi-

mensiononeSNFPcurve;thereisnostableperiod-one

limit cycle corresponding to tonic firing in this region.

Figure6demonstratesthatitispossibletomake IS1and

IS2arbitrarily close, by choosing gDr,dappropriately.

This property is of use later in the study.

3.3. The Burst Mechanism: Reconstructing

the Burst Attractor

The dynamical system described by the ghostburster

equations possesses two separate time scales. The time

constants governing the active ionic channels ns, hd,

and nd, are all ∼1 ms, and the half width of the spike

Page 10

14

Doiron et al.

response of the membrane potentials Vs and Vd are

∼0.5 ms and 1.1 ms, respectively. However, the time

scaleof pdischaracterizedbyτp,d,whichisafactorof

five times larger than any of the other time scales. Pre-

viousstudiesofotherburstmodelshaveprofitedfroma

similarcoexistenceofatleasttwotimescalesofactivity

during bursting (Rinzel, 1987; Rinzel and Ermentrout,

1989; Wang and Rinzel, 1995; Bertram et al., 1995; de

Vries,1998;Izhikevich,2000;Golubitskyetal.,2001).

Thisallowedforaseparationofthefulldynamicalsys-

temintotwosmallersubsystems,onefastandoneslow.

Wealsotreatourburstmodelasafast-slowburster.The

natural variable separation is to group Vs, ns, Vd, hd,

and ndinto a fast subsystem, denoted by the vector x,

while the slow subsystem consists solely of pd. This

gives the simplified notation of our model,

dx

dt= f (x, pd)

dpd

dt

(7)

=pd,∞(x) − pd

τp,d

,

(8)

where f (x, pd) represents the right-hand side of

Eqs. (1) through (5) and Eq. (8) is simply Eq. (6)

restated.

Since pd changes on a slower time scale than x,

we approximate pd as constant and use pd as a bi-

furcation parameter of the fast subsystem (quasi-static

Figure 7.

fast-subsystem x. The maxima in the dendritic voltage (dVd

switch to two values, corresponding to the values taken during each ISI of a period-two solution. B: Time series of the dendritic voltage, Vd(t),

while pd=0.13> pd1.Thefastsubsystemfollowsaperiod-onesolution.C:Timeseriesofthedendriticvoltage, Vd(t),while pd= 0.08 < pd1.

The fast subsystem follows a period-two solution. A constant value of IS=9 > IS1is chosen for all simulations in A, B, and C.

A: Quasistatic bifurcation diagram. pdis fixed as a bifurcation parameter, while Vdis chosen as a representative variable from the

dt=0 andd2Vd

dt2 <0) are plotted for each value of pd. At pd= pd1, the maxima of Vd

approximation;see,e.g.,HoppensteadtandIzhikevich,

1997).Wenotethatwith pdconstantthefastsubsystem

(7) cannot produce bursting comparable to that seen

from ELL pyramidal cells. Bursting requires the slow

variabletomodulateDAPgrowthdynamically(Doiron

et al., 2001b). Treating pdas a bifurcation parameter

will show how changes in pdproduce the character-

istics of ELL bursting through the bifurcation struc-

ture of the fast subsystem. Since pddirectly affects the

fast subsystem only through the dynamics of Vd(see

Eq. (3)), we choose Vdas a representative variable of

the fast subsystem x.

Figure 7A shows the local maxima of Vdon a peri-

odic orbit as a function of pd, while the fast subsystem

isdrivenwith IS=9> IS2.Atacriticalvalueof pd,la-

beled pd1, the fast subsystem goes through a transition

from a period-one to a period-two limit cycle. This

is shown by only one maximum in Vd for pd> pd1,

whereastherearetwomaximafor pd< pd1.Figure7B

shows a time series of Vd(t) following the period-one

limitcyclewhen pd=0.13> pd1,whileFig.7Cshows

the period-two limit cycle when pd=0.08< pd1. The

second dendritic action potential in the period-two or-

bit (Fig. 7C) is of reduced amplitude; this corresponds

to the dendritic failure observed in the full dynamical

system(Eqs. (7) and(8)) when pdislow (seeright col-

umnofFig.4).ThebifurcationdiagraminFig.7Amay

be thought of as a “burst shell” in a projection of phase

Page 11

Ghostbursting: A Novel Neuronal Burst Mechanism15

Figure 8.

burster equations with IS=9> IS2. Four bursts are shown with the

corresponding time-stamped spikes given above for reference. A

slow burst oscillation in pd(t) is observed. It is evident that the dis-

crete function ˜ pd(solid circles) tracks the burst oscillation in pd(t).

˜ pd shows a monotonic decrease throughout the burst until the in-

terburst interval, at which point ˜ pdis reinjected to a higher value.

The horizontal lines are the values pd1, corresponding to the period

doubling transition, and pd2, corresponding to the crossing of the

nullcline curve with the ?Vd? curve. The pd(t) reinjection occurs

after pd(t)< pd2asexplainedinthetextandinFig.9A. ˜ pdhasbeen

translated downward to lie on top of the pd(t) time series. This is

required because Eq. (10) uses a unweighted average of Vd, given in

Eq. (9). This produces a ˜ pdseries that occurs at higher values than

pd(t) because Eqs. (9) and (10) ignore the low-pass characteristics

of Eq. (6). However, only the shape of ˜ pdis of interest, and this is

not affected by the downward translation.

pd(t) and ˜ pd computed from integration of the ghost-

space. The full burst dynamics will evolve on the burst

shell as pdis modulated slowly by the fast subsystem.

Wethereforenextaddressthedynamicsof pd(t)during

the burst trajectory in the fast subsystem.

OninspectionofFig.3Bitisclearthatthereexisttwo

oscillations in pd(t)—one fast oscillation occurring on

the time scale of spikes and the other on a much longer

time scale tracking the bursts. Figure 8 shows pd(t)

during a burst solution of the full dynamical system. It

isclearthatthefastspikeoscillationsin pd(t)aredriven

by the instantaneous value of Vd(t). This is due to τp

being small enough to allow pd(t) to be affected by

the spiking in the fast subsystem. In addition, there is

a general decrease in pd(t) as the burst evolves and

a sharp increase in pd(t) after the doublet ISI. The

increase reinjects pd(t) to a higher value allowing the

burst oscillation to begin again. The period of a burst

oscillationencompassesseveralspikesandthuscannot

be analyzed in terms of the instantaneous dynamics of

the fast subsystem.

Due to the separation of time scales and the fact that

dpd

dtdepends only on Vd(Eq. (6)), we expect that the

burst oscillation depends on the average of Vdbetween

consecutive spikes, defined as

?Vd? =

1

ti+1− ti

?ti+1

ti

Vd(t)dt,

(9)

where ti is the time of the ith spike. We construct a

discrete function ˜ pd

˜ pd= pd,∞(?Vd?),

(10)

where pd,∞(·) is the infinite conductance curve as in

Eq. (6). Figure 8 shows a sequence of ˜ pdvalues con-

structed by using ?Vd? from the burst solution of the

full dynamical system. This sequence is plotted (solid

circles) on top of the full pd(t) dynamics during the

burst train. It is evident that the time sequence of ˜ pd

is of the same shape as the burst oscillation in pd(t).

This is evidence that the slow burst oscillation can be

analyzed by considering ?Vd? .

WenowcompletetheburstshellbyaddingtoFig.7A

the nullcline for pd(from Eq. (6)) as well as ?Vd? com-

puted for the stable periodic solutions of the fast sub-

system. This is shown in Fig. 9A. Note that as pdde-

creases through pd1, ?Vd? decreases by ∼10 mV. This

is due to the dendritic spike failure and subsequent

long ISI occurring when pd< pd1, both contributing

to lower Vdon average (see Fig. 7C). The pdnullcline

and?Vd?curvescrossat pd= pd2< pd1.Sincewehave

shown that the burst oscillation is sensitive to ?Vd?, the

crossing corresponds to ?dpd

to positive (see Fig. 9D).

A saddle-node bifurcation of fixed points occurs at

pd= p∗

furcation is similar to the saddle-node bifurcation of

fixed points in Fig. 5A, where ISis the bifurcation pa-

rameter. This is expected, since pd is the coefficient

to a hyperpolarizing ionic current (see Eq. (3)); hence

an increase in pdis equivalent to a decrease in depo-

larizing IS. Because of the saddle-node bifurcation at

pd= p∗

as

√

|pd−p∗

1983).

dt? changing from negative

dfor some p∗

d> pd1(data not shown). This bi-

d,theperiodoftheperiod-onelimitcyclescales

1

d|for pdnear p∗

d(GuckenheimerandHolmes,

Page 12

16

Doiron et al.

Figure 9.

nullcline is inverted so as to give Vd,∞(pd)=V1/2,p−kpln(

at a fixed pd. Note the sharp decline in ?Vd? for pdbelow pd1. B: The diagram in A is replotted with the labels removed. A single directed

burst trajectory projected in the (Vd, pd) plane obtained by integrating the full dynamical system (Eqs. (1) through (6)) is plotted on top of the

burst shell. C: All observed discharge frequencies of the fast subsystem are plotted as a function of pd. At pd= pd1a stable period-one firing

pattern of ∼200 Hz changes to a period-two solution with one ISI being ∼(700 Hz)−1and the other ∼(100 Hz)−1. The inverse of the ISIs of

the single burst shown in Fig. 9B are plotted as well. The ISIs are numbered from 1 (the first ISI) through to 5 (doublet ISI) and 6 (interburst

interval). D: The average of the derivative of pd, ?dpd

has ?dpd

A: The bifurcation diagram of Fig. 6A is replotted along with the pdnullcline pd,∞(Vd) (dashed line labeled N). Note that the pd

1

1−pd). We plot the average of Vdover a whole period of Vd, ?Vd? (solid line),

dt?, is plotted for each ISI in the single burst shown in Fig. 8B. Only the long interburst ISI

dt?<0. A constant value of IS= 9 > IS1is chosen for all simulations in A, B, C, and D.

dt? > 0; all other ISIs have ?dpd

With the burst shell now fully constructed (Fig. 9A),

weplacethefullburstdynamics(Eqs.(7)and(8))onto

the shell. This is shown in Fig. 9B. The directed tra-

jectory is the full six-dimensional burst trajectory pro-

jected into the Vd− pdsubspace. As the burst evolves,

pd(t) decreases from spike to spike in the burst. This

causes the frequency of spike discharge to increase

due to the gradual shift away from the saddle-node

bifurcation of fixed points at pd= p∗

pd(t)< pd1, the spike dynamics shift from period-one

spiking to period-two spiking. This first produces a

high-frequency spike doublet, which is then followed

by a dendritic potential of reduced amplitude, caus-

ing ?Vd? to decrease. When pd(t)< pd2, ?dpd

(see Fig. 9D), and pd(t) increases and is reinjected to

a higher value. The reinjection toward the “ghost” of

d. However, once

dt? > 0

the saddle-node bifurcation of fixed points at pd= p∗

causestheISI(theinterburstinterval)tobelong,since

thevelocitythroughphasespaceislowerinthisregion.

Figure 9C shows the burst trajectory in the fre-

quency domain. The period doubling is evident at

pd= pd1since two distinct frequencies are observed

for pd< pd1,correspondingtoaperiod-twosolutionof

thefastsubsystem,whereasfor pd> pd1onlyaperiod-

one solution is found. As pdis reduced in the period-

one regime (pd> pd1), the frequency of the limit cycle

increases,duetothereducedeffectofthehyperpolariz-

ing current IDr,d. We superimpose the ISIs of the burst

trajectory shown in Fig. 9B on the frequency bifurca-

tion diagram in Fig. 9C. The sequence begins with a

longISI(numbered1)withsubsequentISIsdecreasing,

culminating with the short doublet ISI (numbered 5).

d

Page 13

Ghostbursting: A Novel Neuronal Burst Mechanism17

The reinjection of pdnear p∗

ISI (numbered 6). The reinjection causes this next ISI

to be long; it separates the action potentials into bursts.

Figure 9D shows the average of the derivative of pd,

?dpd

shown in Fig. 9B and C. Notice that ?dpd

tive and decreases as the burst evolves. This is because

theISIlengthreducesastheburstevolves,allowingthe

burst trajectory to spend less time in the region where

dpd

dt>0.However,alargefractionofthebursttrajectory

during the interburst ISI (6) occurs in the region where

dpd

dt> 0. Hence, the average ?dpd

for the interburst interval, producing the reinjection of

pd(t) to higher values.

Izhikevich (2000) has labeled the burst mechanisms

according to the bifurcations in the fast subsystem that

occur in the transition from quiescence to limit cycle

and vice versa. Even though there is never a true “qui-

escent” period during the burst-phase trajectory, the

interburst interval for our model is determined by the

approach to an infinite period bifurcation. This phe-

nomenon is often labeled as sensing the “ghost” of a

bifurcation(Strogatz,1994),andwenaturallylabelthe

burst mechanism as ghostbursting.

The ghostburster system exhibits bursting, for some

range of IS, only for 2<τp<110 ms, with all other

parameters as given in Table 1. The lower bound of τp

is due to the fact that the inactivation of IDr,dmust be

cumulative for there to be a reduction of the ISIs as

the burst evolves. This requires a τplarger than that

of the ionic channels responsible for spike production

(<1 ms). The upper bound on τpis also expected since

significant removal of pdinactivation during the inter-

burst interval is necessary for another burst to occur.

Toolargeavalueofτpwillnotallowsufficientrecovery

of IDr,dfrom inactivation, and therefore bursting will

not occur.

doccurs during the next

dt? =

1

ti+1−ti

?ti+1

ti

(dpd

dt)dt,duringeachISIintheburst

dt? is nega-

dt? is greater than zero

3.4. The Interburst Interval

By varying ISit is possible to set the interburst inter-

val, TIB, to be different lengths. This is because after

thedendritehasfailed(removingtheDAPatthesoma),

the time required to produce an action potential in the

somatic compartment (which is TIB) is dictated almost

solelyby IS.Thespikeexcitabilityofthesomaticcom-

partment is Type I (Ermentrout, 1996), as evident from

the saddle-node bifurcation of fixed points at IS= IS1.

As a consequence TIB is determined from the well-

known scaling law associated with saddle-node bifur-

Figure 10.

IS− IS1. The averaging was performed on 100 bursts produced by

the ghostburster equations at a specific IS. gDr,dwas set to 12.14.

?TIB? shows a similar functional form to that described by Eq. (11).

The dips in ?TIB? are discussed in the text.

Interburst interval. ?TIB? is plotted as a function of

cations on a circle (Guckenheimer and Holmes, 1983),

TIB∼

1

√IS− IS1

.

(11)

Figure 10 shows the average interburst interval, ?TIB?,

as a function of IS− IS1 for the ghostburster with

gDr,d=12.14.Thisvalueof gDr,dsets IS1and IS2close

tooneanother(seeFig.6),allowingthesystemtoburst

with values of ISclose to IS1. It is necessary to form

an average due to the chaotic nature of burst solutions.

Nevertheless, ?TIB? increases as ISapproaches IS1, as

suggestedbyEq.(11).Alinearregressionfitof1/?TB?2

against IS− IS1givesacorrelationcoefficientof0.845,

further verifying that Eq. (11) holds. Figure 10 also

shows downward dips in ?TIB? that occur more fre-

quently as IS− IS1goes to zero. Time series of bursts

with IScorresponding to the dips in ?TIB? show scat-

tered bursts with short interburst intervals that deviate

from Eq. (11), amongst bursts with longer interburst

intervals, which fit the trend described by Eq. (11).

These scattered small values of TIBreduce ?TIB? for

these particular values of IS. These dips contribute to

the deviation of the linear correlation coefficient cited

abovefrom1.Wedonotstudythedipsfurthersincethe

behavior has yet to be observed experimentally. How-

ever, experimental measurements of ELL pyramidal

cell-burst period do indeed show a lengthening of the

period as the applied current is reduced (R.W. Turner,

personalcommunication).Thiscorrespondstothegen-

eral trend shown in Fig. 10. Equation (11) and Fig. 10

Page 14

18

Doiron et al.

show that by choosing the model parameters properly

it is possible to regulate the effect of the ghost of the

saddle-node bifurcation of fixed points on the burst

solutions. We will show later how this property yields

greatdiversityoftimescalesofpossibleburstsolutions

of the ghostburster model.

3.5.The Burst Interval: Intermittency

Regions of chaotic and periodic behavior exist in

many burst models (Chay and Rinzel, 1985; Terman,

1991, 1992; Hayashi and Ishizuka, 1992; Wang, 1993;

Komendantov and Kononenko, 1996). The results of

Fig. 5 show that periodic spiking and chaotic bursting

are also two distinct dynamical behaviors of the ghost-

burster. Moreover, the bifurcation parameter we have

used to move between both dynamical regimes is the

applied current IS, which mimics an average synaptic

input to the cell. This indicates that changing the mag-

nitude of input to the cell may cause a transition from

periodic spiking to chaotic bursting. In ELL pyramidal

cells a transition from tonic firing to highly variable

burstinghasbeenobservedasapplieddepolarizingcur-

rentisincreased(LemonandTurner,2000;Bastianand

Nguyenkim,2001;DoironandTurner,unpublishedre-

sults). It remains to be shown that the experimentally

observed bursting is indeed chaotic; preliminary re-

sults suggest that such an analysis is difficult due to

nonstationarity in the data (Doiron and Turner, unpub-

lished observations). Nonetheless, understanding the

transitions or routes to chaos in the model separating

tonic and chaotic burst regimes is necessary not only

foracompletedescriptionofthedynamicsofthemodel

but also for characterizing the input-output relation of

bursting ELL pyramidal cells.

Figure 5A shows that the transition from periodic

spiking to chaotic bursting occurs at IS= IS2when a

stable limit cycle collides with an unstable limit cycle

in a saddle-node bifurcation of limit cycles. Since we

are analyzing spiking behavior on both sides of the bi-

furcation,itisnaturaltoconsidertheISIreturnmapfor

ISnear IS2. We choose ISslightly larger than IS2and

plot in Fig. 11A the ISI return map for a single burst

sequence from the ghostburster (for IS1< IS< IS2the

return map is a single point). We have labeled the re-

gions of interest in the figure and explain each region

in order: (1) The burst begins here. (2) The ISI se-

quence approaches the diagonal. This produces a clus-

tering of points corresponding to the pseudo-periodic

behavior observed in the center of the burst. We refer

Figure 11.

burst sequence with IS=6.587 and gDr,d=13 is shown (for these

parameters IS1= 5.736 and IS2= 6.5775). The diagonal is plotted

as well (dashed line). The labels (1) through (5) are explained in the

text. B: The ISI return map for a single-burst sequence with IS= 9

and gDr,d=15 as in Fig. 3. C: The ISI return map for a single-

burst recording from an ELL pyramidal cell (data courtesy of R.W.

Turner). Compare with the model burst sequence in B.

Burst intermittency. A: The ISI return map for a single-

Page 15

Ghostbursting: A Novel Neuronal Burst Mechanism 19

to this region of the map as a trapping region. (3) The

ISI sequence leaves the trapping region with a down-

ward trend. (4) The interburst interval involves a sharp

transition from small ISI to large ISI. (5) The ISI se-

quence returns to the trapping region and another burst

begins.

The above description indicates that the route to

chaosisTypeIintermittency(MannevilleandPomeau,

1980; Guckenheimer and Holmes, 1983). Intermit-

tency involves seemingly periodic behavior separated

by brief excursions in phase space. The clustering of

points in the ISI return map in the trapping region of

Fig. 11A (labeled 2) is a manifestation of this appar-

ent periodic firing. A trapping region is a characteristic

feature of Type I intermittency and corresponds to a

saddle-node bifurcation of fixed points in the return

map (which is the saddle-node bifurcation of limit cy-

cles in the continuous system), occurring specifically

at IS= IS2for the ghostburster equations. The escape

and return to the trapping region (regions 3, 4, 5 in

Fig. 11A) are the brief excursions. These events corre-

spondtotheperioddoublingtransitionandthecrossof

the ?Vd? curve and pdnullcline, in the fast subsystem,

as explained in Fig. 9. Figure 11B shows the ISI return

map for a model burst of seven spikes and Fig. 11C

the same map for a seven-spike burst recording from

an ELL pyramidal cell. Both maps show the qualita-

tive structure similar to in Fig. 11A, including a clear

escape from and reinjection into a trapping region near

the diagonal. Wang (1993) has also observed Type I

intermittency in the Hindmarsh-Rose model.

Since intermittent behavior is connected to a saddle-

node bifurcation, the time spent in the trapping region

TB, corresponding to the burst period (the duration of

the spikes in the cluster making up the burst), has a

well-defined scaling law:

TB∼

1

√IS− IS2

.

(12)

Similar to Fig. 10 we consider the average of the burst

period ?TB? because of the chaotic nature of the burst-

ing. Figure 12 shows that ?TB? asymptotes to infinity

as ISapproaches IS2. Linear regression fits to 1/?TB?2

against IS− IS2give a correlation coefficient of 0.886.

TheseresultsvalidateEq.(12)fortheghostbursterburst

sequences. Again the deviation in the correlation coef-

ficient from 1 is caused by slight dips in ?TB?, similar

to the dips observed in ?TIB? (Fig. 10). By choosing

the quantity IS− IS2we can obtain bursts with spike

numbers comparable to experiment.

Figure12.

averagingwasperformedon100burstsproducedbytheghostburster

equationsataspecific IS.gDr,dwassetto12.14.?TB?showsasimilar

functional form to that described by Eq. (12).

Burstinterval?TB?plottedasafunctionof IS−IS2.The

3.6.Gallery of Bursts

Equations (11) and (12) give the inverse square-root

scaling relations of TBand TIB, respectively. These re-

sults showed that TBis determined by IS− IS2and TIB

by IS− IS1. Using this fact and the ability to vary the

difference between IS2and IS1(see Fig. 6) we can pro-

duce a wide array of burst patterns with differing time

scales.

Figure 13A reproduces the (IS, gDr,d) bifurcation

set shown in Fig. 6. The letters B through F mark

(IS, gDr,d) parameters used to produce the spike trains

shown in the associated panels B through F of Fig. 13.

Figure 13B uses (IS, gDr,d) values such that the ghost-

burster is in the tonic firing regime. The burst trains

shown in Figs. 3 and 4 correspond to (IS, gDr,d) values

in the burst regime of Fig. 6, which are not close to

either of the SNFP or SNLC curves. An example of a

burst train with such a parameter choice is shown in

Fig. 13C. However, if we approach the SNLC curve

but remain distant from the SNFP curve, we can in-

crease TBby one order of magnitude yet keep TIBthe

same. The burst train in Fig. 13D shows an example of

this. If we choose ISand gDr,dto be close to both the

SNFP and SNLC curves, we can now increase TIBas

well (Fig. 13E). The interburst period TIBhas now also

increased dramatically from that shown in Figs. 13C

and D.

Finally, for IS and gDr,d values to the left of the

codimension two bifurcation point, burst sequences

show only a period-two solution (Fig. 13F). The burst