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
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
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
bursting, electric fish, compartmental model, backpropagation, pyramidal cell
Burst discharge of action potentials is a distinct and
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.,
the active phase of bursting varies considerably across
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
Doiron et al.
1992). This diversity of specific time scales and ISI
patterns suggests that numerous distinct burst mecha-
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
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,
to the sensory periphery and the known relevance of
their bursts to signal detection suggest that studies of
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
output. Recently, through the construction and analy-
sis of a detailed multicompartmental model of an ELL
pyramidal cell, we have reproduced burst discharges
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
mechanism could not be achieved due to the high di-
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
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
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
This is contrary to other burst mechanisms that show
transition to tonic firing as applied current is increased
(Hayashi and Ishizuka, 1992; Gray and McCormick,
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
model as a fast-slow burster (Rinzel, 1987; Rinzel and
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
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-
I intermittency (Pomeau and Manneville, 1980). Com-
parisons of both model and experimental ELL burst
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.
2.1. 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-
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-
and Turner (2000).
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
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
(DAP) after the somatic spike, as shown in Fig. 1B.
The DAP is the result of a dendritic reflection of the
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
spike, yielding a DAP.
In vitro recording of burst discharge from the soma of an ELL pyra-
midal cell with constant applied depolarizing current. Two bursts of
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-
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
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
Doiron et al.
backpropagation by locally applying tetrodoxin (TTX,
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,
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-
DAP production, the two-compartment assumption is
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
respectively IDr,sand IDr,d. These currents are neces-
sary to reproduce somatic action potentials and proper
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.
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,
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
=n∞,s(Vs) − ns
∞,s(Vs)·(1−ns) · (VNa−Vs)
(1 − κ)·(Vs−Vd) + gleak·(Vl−Vd)
=h∞,d(Vd) − hd
=n∞,d(Vd) − nd
=p∞,d(Vd) − pd
Ghostbursting: A Novel Neuronal Burst Mechanism9
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+e−(Vs−V1/2)/k, and a channel time con-
Table 1 lists the values of all channel parameters
used in the simulations. The soma is modeled with two
classic four dimensional Hodgkin-Huxley model is ac-
the Na+activation mstracks Vsinstantaneously), and
modeling its inactivation, hs, through IDr,sactivation,
ns(we set hs≡ 1−ns). This second approximation is
(Doiron et al., 2001b) that hs+ ns≈ 1 during spiking
Sneyd, 1998). The dendrite is modeled with four vari-
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
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-
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
ables (see Eqs. (3), (5), and (6)). Slow inactivation of
K+channels, although not a mechanism in contempo-
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,
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
in time and is used as a bifurcation parameter. Physi-
ological justification for the parameter values given in
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.
3.1. Model Performance
Figure 3A and B shows simulation time series of Vs
and pd, respectively, for the ghostburster with constant
depolarizationof IS= 9.Weseearepetitivebursttrain
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
somatic bursts all show the same characteristic growth
in depolarization (DAP growth) and consequent de-
creases in ISI leading to the high-frequency doublet.
ure is associated with both doublet spiking and burst
ghostburster do not show a gradual depolarization dur-
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
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,
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
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
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
in Strogatz (1994). An alternative explanation of the
burst mechanism, given in physiological terms, was
presented in Doiron et al. (2001b).
Ghostbursting: A Novel Neuronal Burst Mechanism11
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
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
Doiron et al.
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
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
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
two solution exists for high IS. The importance of both
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-
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
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
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-
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-
limit cycle corresponding to tonic firing in this region.
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
Doiron et al.
response of the membrane potentials Vs and Vd are
∼0.5 ms and 1.1 ms, respectively. However, the time
five times larger than any of the other time scales. Pre-
during bursting (Rinzel, 1987; Rinzel and Ermentrout,
1989; Wang and Rinzel, 1995; Bertram et al., 1995; de
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,
dt= f (x, pd)
=pd,∞(x) − pd
where f (x, pd) represents the right-hand side of
Eqs. (1) through (5) and Eq. (8) is simply Eq. (6)
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
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
dt2 <0) are plotted for each value of pd. At pd= pd1, the maxima of Vd
(7) cannot produce bursting comparable to that seen
from ELL pyramidal cells. Bursting requires the slow
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-
be thought of as a “burst shell” in a projection of phase
Ghostbursting: A Novel Neuronal Burst Mechanism15
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.
the burst trajectory in the fast subsystem.
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
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
be analyzed in terms of the instantaneous dynamics of
the fast subsystem.
Due to the separation of time scales and the fact that
dtdepends only on Vd(Eq. (6)), we expect that the
burst oscillation depends on the average of Vdbetween
consecutive spikes, defined as
where ti is the time of the ith spike. We construct a
discrete function ˜ pd
˜ pd= pd,∞(?Vd?),
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? .
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
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
dt? changing from negative
dfor some p∗
d> pd1(data not shown). This bi-
d|for pdnear p∗
Doiron et al.
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
A: The bifurcation diagram of Fig. 6A is replotted along with the pdnullcline pd,∞(Vd) (dashed line labeled N). Note that the pd
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),
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∗
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
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
culminating with the short doublet ISI (numbered 5).
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,
shown in Fig. 9B and C. Notice that ?dpd
tive and decreases as the burst evolves. This is because
burst trajectory to spend less time in the region where
during the interburst ISI (6) occurs in the region where
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
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.
of IDr,dfrom inactivation, and therefore bursting will
doccurs during the next
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
the time required to produce an action potential in the
somatic compartment (which is TIB) is dictated almost
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-
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),
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
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
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
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,
eral trend shown in Fig. 10. Equation (11) and Fig. 10
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
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
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
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-
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
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-
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
The above description indicates that the route to
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-
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:
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.
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.
functional form to that described by Eq. (12).
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
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
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
Doiron et al.
figure correspond to the (IS, gDr,d) parameter values used to produce panels B through F, respectively. Examples of the interburst period TIBand
burst period TBfor each burst train are indicated (except for the tonic solution shown in B). The exact ISand gDr,dvalues used to produce each
spike train are as follows: B: IS=6.5, gDr,d=14. C: IS=7.7, gDr,d=13. D: IS=7.6, gDr,d=14. E: IS=5.748, gDr,d=12.14. F: IS= 5.75,
gDr,d=11. The vertical mV scale bar in C applies to all panels; however, each panel has its own horizontal time-scale bar.
Burst gallery. A: Reproduction of the two-parameter bifurcation set shown in Fig. 6. The letters B through F marked inside the
Ghostbursting: A Novel Neuronal Burst Mechanism21
sequences are no longer chaotic. This is to be expected
since there no longer is a saddle-node bifurcation of
limit cycles, which gave rise to the intermittency route
to chaos in the ghostburster equations (Fig. 11). Ap-
proaching the SNFP curve allows for a large TIB, but
when gDr,dis small occurs because gDr,dis the coeffi-
cient to the hyperpolarizing K+current in the dendrite
(Eq. (3)) and as such controls the effect of the DAP at
the soma. For gDr,dto the left of the codimension, two-
bifurcation point, the first somatic spike in the doublet
produces a DAP of sufficient strength to cause the sec-
of the dendrite. Thus dendritic failure occurs after the
first reflection, and the burst contains only two spikes.
4.1.Ghostbursting: A Novel Burst Mechanism
We have introduced a two-compartment model
of bursting ELL pyramidal cells—the ghostburster.
The model is a significant reduction of a large
multicompartmental ionic model of these cells
(Doiron et al., 2001b). The large model was moti-
vated by the conditional backpropagation burst mech-
anism that has been experimentally characterized in
ELL pyramidal cells (Lemon and Turner, 2000). The
results of Lemon and Turner (2000) and Doiron et al.
(2001b) suggest that the ionic requirements necessary
and sufficient to support bursting as observed in the
the refractory period of dendritic action potentials that
is longer than that of the somatic potentials, and (3)
slow inactivation of a dendritic K+channel involved in
repolarization. The fact that the ghostburster was de-
succeeds in producing burst discharge comparable to
experiment suggests that these three requirements cap-
ture the essential basis of the burst mechanism used in
ELL pyramidal cells.
The simplicity of the ghostburster, as compared to
the large compartmental model, has allowed us to
understand, from a dynamical systems perspective,
the mechanism involved in this type of bursting. The
ghostburster was analyzed using a separation of the
full dynamical system into fast and slow subspaces
(Eqs. (7) and (8)), similar to the analysis of many other
burst models (Rinzel, 1987; Rinzel and Ermentrout,
1989; Wang and Rinzel, 1995; Bertram et al., 1995;
Hoppensteadt and Izhikevich, 1997; de Vries, 1998;
slow dynamical variable pdas a bifurcation parameter
with respect to the fast subsystem allowed us to con-
struct a burst shell on which the full burst dynamics
evolve. The shell shows that a transition from a period-
one limit cycle to a period-two limit cycle occurs in
the dynamics of the fast subsystem as pdis reduced.
The period-two limit cycle causes a sharp reduction in
?Vd? since the second spike of the limit cycle is of re-
duced amplitude, due to dendritic refractoriness. The
reduction in ?Vd? causes the ?Vd?(pd) curve to cross
the pdnullcline, and pd(t)grows during the second ISI
of the period-two orbit. The growth in pd(t) reinjects
pd(t) near a saddle-node bifurcation of fixed points
occurring at high pd. This passage near the ghost of
the saddle-node bifurcation causes the ISI to be long,
separating the action potentials into bursts.
sification of bursters from a combinatorial point of
view. This has been successful in producing a large
number of new fast-slow bursting mechanisms. One of
biophysically plausible model of bursting corticotroph
cells of the pituitary (Shorten and Wall, 2000). In con-
trast, Golubitsky et al. (2001) (extending the work of
bifurcations. Both these methods have used the im-
plicit assumption that burst initiation and termination
involve bifurcations from quiescence (or subthreshold
oscillation) to limit cycle and vice-versa. However, our
burst mechanism does not appear in any of the above
classifications. This is because the trajectories in the
a limit cycle and are never in true quiescence, corre-
sponding to a stable fixed point. The period of the limit
cycle changes dynamically because the slow subsys-
tem is oscillating, forcing the fast system to sometimes
pass near the ghost of an infinite period bifurcation.
Furthermore, in the ghostburster, burst termination is
connected with a bifurcation from a period-one to a
period-two limit cycle in the fast subsystem. This is a
limit cycle to a stable fixed point in the fast subsystem
Doiron et al.
classifying burst phenomena through the bifurcations
from quiescence to a period-one limit cycle and vice-
versa in the fast subsystem of a dynamical bursting
model has had much success, our work requires an ex-
tension of the classification of bursting to include an
alternative definition of quiescence and a burst attrac-
limit cycles with no stable fixed points.
Rinzel (1987) shows that burst mechanisms with a
one-dimensional slow subsystem require bistability in
the fast subsystem to exhibit bursting. The slow sub-
Fig. 9 shows that the fast subsystem x is not bistable.
This would seem to be a contradiction; however, recall
that as τpapproaches values that are similar to other
bursting mechanisms, bursting is not observed. Thus
our results do not contradict Rinzel’s previous study
and yet do support a separate mechanism entirely. This
illustrates a key distinction between the ghostburster
slow variable has an upper bound in the ghostburster.
The fast and slow timescales are sufficiently separate
to allow us to successfully study the burst mechanism
using a quasistatic approximation. Thus ghostburst-
to ghostbursting, which involve a slow-passage phe-
nomena (requiring saddle-node or homoclinic bifurca-
tions), may exist, placing the ghostburster as only one
in a family of new burst mechanisms.
tonic firing and bursting behavior. Both Terman (1991,
1992) and Wang (1993) have also identified thresholds
and a modified version of the Morris-Lecar equations,
respectively. Both of these models exhibited a homo-
clinic orbit in the fast subsystem as the spiking phase
of a burst terminated. As a result, the bifurcations from
continuous spiking to bursting in the full dynamics
were complicated. Wang observed a crisis bifurcation
showed that a series of bifurcations occurs during the
transition, which could be shown to exhibit dynamics
similar to the Smale horseshoe map (Guckenheimer
and Holmes, 1983). The saddle-node bifurcation of
limit cycles that separates the two regimes in the
ghostburster model is a great deal simpler than ei-
ther of these bifurcations. However, Wang has shown
that an intermittent route to chaos is also observed
in the Hindmarsh-Rose model as continuous spiking
transitions into bursting, much like the ghostburster
The fact that the transition from tonic firing to burst-
ing in the ghostburster system occurs as depolariza-
tion is increased is in contrast to both experimental
and modeling results of other bursting cells (Terman,
1992; Hayashi and Ishizuka, 1992; Wang, 1993; Gray
and McCormick, 1996; Steriade et al., 1998; Wang,
1999). However, since many experimental and model-
ing results, separate from ELL, show burst threshold
behavior, the concept of burst excitability may have
broader implications. To expand, the saddle-node bi-
furcation of limit cycles marking burst threshold can
be compared to the saddle-node bifurcation of fixed
points, which is connected to the spike excitability of
Type I membranes (Ermentrout, 1996; Hoppensteadt
and Izhikevich, 1997). The functional implication of a
burst threshold have yet to be fully understood; how-
ever, recent work suggests that it may have important
implications for both the signaling of inputs (Eguia
et al., 2000) and dividing cell response into stimulus
4.2.Predictions for Bursting in the ELL
An integral part of the burst mechanism in ELL pyra-
drite through action potential backpropagation. One
potential function of backpropagation is thought to
be retrograde signaling to dendritic synapses (Häusser
et al., 2000). Further, a recent experimental study has
shown that the coincidence of action potential back-
propagation and EPSPs produces a significant ampli-
fication in membrane potential depolarization (Stuart
and Häusser, 2001). These results may have conse-
putation. Our results (and those of others; see Häusser
et al., 2000, for a review) imply that backpropagation
can also determine action potential patterning.
As mentioned above, the ghostburster exhibits a
threshold separating tonic firing and bursting as de-
polarization is increased. Similar behavior has been
observed in both in vitro and in vivo experimental
2000; Bastian and Nguyenkim, 2001) and in our full
compartmental model simulations (data not shown).
A reduction of burst threshold was observed in ELL
pyramidal cells when TEA (K+channel blocker) was
Ghostbursting: A Novel Neuronal Burst Mechanism23
with this observation, since dendritic TEA application
is equivalent to a reduction in gDr,dconductance in our
model. Figure 6 shows that as gDr,dis reduced, burst
threshold is lowered.
Bursts, as opposed to individual spikes, have been
suggested to be a fundamental unit of information
(Lisman, 1997). In fact, Gabbiani et al. (1996) have
correlated bursts from ELL pyramidal cells with fea-
results, it is possible that the time scale of burst-
ing, T(=TB+ TIB), could be tuned to sensory input;
hence, the ability of a bursting cell to alter T may im-
prove its coding efficiency. A natural method to al-
ter T would be to change the time constants, τ, that
determine the slow process of the burst mechanism
(Giannakopoulous et al., 2000). Nevertheless, to
achieve an order of magnitude change in T requires a
bursting CA3 pyramidal cell, that the precise timing of
inhibitory synaptic potentials can change the burst pe-
rate and temporal coding of hippocampal place cells.
However, the ghostburster shows that both TBand TIB
can be changed by an order of magnitude but with only
small changes in either depolarizing input and/or den-
dritic K+conductances (see Fig. 13). Small changes
in ISare conceivable through realistic modulations of
trolocation and electrocommunication in weakly elec-
tric fish (Heiligenburg, 1991). Changes in gDr,d can
further occur through the phosphorylation of dendritic
to be abundant over the whole dendritic tree of ELL
bursting mechanism may offer ELL pyramidal cells a
viable method by which to optimize sensory coding
with regulated burst output. Further studies quantify-
ing the information-theoretic relevance of bursting are
required to confirm these speculations.
We conclude our study with a concrete prediction.
of ELL pyramidal cells can be significantly decreased
as either depolarizing current (IS) is increased or den-
dritic K+conductance (gDr,d) is decreased by a small
amount. This prediction can be easily verified by ex-
perimentally measuring T in bursting ELL pyramidal
cells for (1) step changes in ISand (2) before and af-
ter TEA application to the apical dendrites, which will
sistent sodium and somatic K+, in particular) may also
be used to create similar bifurcation sets as in Fig. 13.
We would like to thank our colleague Ray W. Turner
for the generous use of his data and fruitful discus-
sions. Valuable insight on the analysis of our model
was provided by John Lewis, Kashayar Pakdaman,
Eugene Izhikevich, Gerda DeVries, Maurice Chacron,
erating grants from NSERC (B.D., A.L.), the OPREA
(C.L.), and CIHR (L.M.).
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