1ERATO,
Schweighofer, Nicolas,
Kenji Doya, and
Mitsuo Kawato.
Electrophysiological Properties of Inferior Olive Neurons: A
Compartmental Model.
J. Neurophysiol. 82: 804-817, 1999.
As a step in exploring the functions of
the inferior olive, we constructed a biophysical model of the olivary
neurons to examine their unique electrophysiological properties. The
model consists of two compartments to represent the known distribution
of ionic currents across the cell membrane, as well as the dendritic
location of the gap junctions and synaptic inputs. The somatic
compartment includes a low-threshold calcium current
(ICa_l), an anomalous inward rectifier
current (Ih), a sodium current
(INa), and a delayed rectifier potassium
current (IK_dr). The dendritic compartment contains a high-threshold calcium current
(ICa_h), a calcium-dependent potassium
current (IK_Ca), and a current flowing into
other cells through electrical coupling
(Ic). First, kinetic parameters for these
currents were set according to previously reported experimental data.
Next, the remaining free parameters were determined to account for both
static and spiking properties of single olivary neurons in vitro. We
then performed a series of simulated pharmacological experiments using
bifurcation analysis and extensive two-parameter searches. Consistent
with previous studies, we quantitatively demonstrated the major role of
ICa_l in spiking excitability. In addition,
Ih had an important modulatory role in the
spike generation and period of oscillations, as previously suggested by
Bal and McCormick. Finally, we investigated the role of electrical coupling in two coupled spiking cells. Depending on the coupling strength, the hyperpolarization level, and the
ICa_l and Ih
modulation, the coupled cells had four different synchronization modes:
the cells could be in-phase, phase-shifted, or anti-phase or could exhibit a complex desynchronized spiking mode. Hence these simulation results support the counterintuitive hypothesis that electrical coupling can desynchronize coupled inferior olive cells.
Cerebellar Purkinje cells, the sole output of the
cerebellar cortex, receive two major inputs: the climbing fiber input,
which originates from the inferior olive (IO), and the granule cell input, which relays information from the mossy fibers. Each Purkinje cell is innervated by one climbing fiber, and a single IO spike generates a single "complex spike" in a Purkinje cell. In contrast, a Purkinje cell receives IO neurons exhibit a number of unique electrophysiological properties.
First, IO cells respond to both injection of depolarizing currents and
release from hyperpolarization by generating, respectively, dendritic
and somatic spikes (Llinás and Yarom 1981a Several studies have described the membrane currents of IO neurons and
their influence on cell responses (Bal and McCormick 1997 The existence of electrical coupling between IO cell dendrites make
voltage-clamp recording of individual IO cells arduous (Manor
1995 First, we further explored how the modulation of individual current
conductances leads to different asymptotic electrical behaviors in
single olivary neurons in vitro. Specifically, we quantitatively tested
the influence of the low threshold calcium current
ICa_l and the h current
Ih on single IO cell firing patterns. Consistent with previous experimental findings, we quantitatively demonstrated the major role of ICa_l in
spiking excitability. In addition, in line with the study of Bal
and McCormick (1997) Then we examined the role of electrical coupling on the spiking
behavior of IO cells. Because IO cells were discovered to be
extensively coupled by gap junctions (Llinás et al.
1974 General cell model
The electrotonic properties of the two-compartment model (Fig.
1A) were determined by two
morphological parameters (Pinsky and Rinzel 1994
ABSTRACT
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
INTRODUCTION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
200,000 granule cell inputs, which generate a high-frequency discharge ("simple spikes"). Because IO neurons fire at a very low frequency (at most 2 or 3 spikes within a second), there are comparatively few complex spikes. Although it is well known
that simple spikes are involved in motor control, the role of complex
spikes is more controversial (De Zeeuw et al. 1998
; Simpson et al. 1996
). The view that climbing fiber
inputs function as detectors of control errors is well supported (e.g.,
Kitazawa et al. 1998
; Kobayashi et al.
1998
). However, it is still unclear whether climbing fibers
exert a real-time influence on the targets of the Purkinje cells,
and/or exert a short-term modulatory action on simple spike patterns,
or induce long-lasting changes in the potency of granule cell-Purkinje
cell synapses.
).
Second, in vitro IO cells can exhibit 4- to 8-Hz rhythmic activity
(Bal and McCormick 1997
; Benardo and Foster
1986
; Llinás and Yarom 1986
). This
rhythmic activity can either take the form of subthreshold sinusoid-like oscillations or rhythmic generation of sodium spikes. Furthermore these oscillations can occur either spontaneously [in
~10% of the cells recorded by Llinás and Yarom
(1986)
; but Benardo and Foster (1986)
reported a
larger proportion of oscillating cells] or after harmaline or
serotonin administration (Sugihara et al. 1995
). Third,
the cells are coupled by electrotonic gap junctions located on the
dendrites (Llinás et al. 1974
; Sotelo et
al. 1974
). Through the combined actions of the coupling between cells and the underlying rhythmicity of individual neurons, neighboring IO cells have been shown to fire with some rhythmicity in brain slices
and acute preparations (Llinás and Yarom 1986
;
Sasaki et al. 1989
). Yarom (1991)
suggested that the IO is a network of damped oscillators that, when
coupled, can generate sustained oscillations. However, the oscillatory
behavior of IO cells in vivo is still under investigation because
apparently random, rather than periodic, firing has been observed in
behaving animals (Keating and Thach 1995
). Thus it is
probable that synaptic and modulatory inputs influence the oscillating
behavior of single as well as group of coupled cells. Besides the very
strong excitatory dendritic inputs (e.g., De Zeeuw et al.
1998
), IO cells receive two types of inhibitory inputs: the
first affects the excitability of the cell, whereas the second
modulates the coupling strength between cells (Lang et al.
1996
). Moreover, a strong serotonergic input to the IO from the
midbrain raphe has been identified (Weiss and Pellet
1982
).
; Benardo and Foster 1986
; Bleasel
and Pettigrew 1994
; Llinás and Yarom 1981b
,
1986
; Manor 1995
; Manor et al.
1997
; Ruigrok and Voogd 1985
; Yarom and
Llinás 1987
). IO neurons exhibit two types of spatially
distinct calcium currents: a low-threshold current
(ICa_l) located in the soma and a
high-threshold current (ICa_h) located
in the dendrites (Llinás and Yarom 1981a
,b
). IO
cells respond to both release from hyperpolarization and injection of
depolarizing currents via these two calcium currents. The low-threshold calcium current has a window of conductance around the resting membrane
potential, which causes the cell to be excited in response to
hyperpolarizing current pulses. The high-threshold calcium current is
noninactivating, which results in a prolonged plateau potential in
response to depolarizing dendritic input. The calcium influx in the
dendrites activates a calcium-dependent potassium current
(IK_Ca), which abruptly terminates the
plateau potential after ~30 ms (Llinás and Yarom
1981a
). The inactivation of
IK_Ca has a very long time constant
(several hundred milliseconds), resulting in a long
afterhyperpolarization (AHP). The AHP deinactivates the low-threshold
calcium conductance and triggers a postinhibitory rebound. Furthermore,
hyperpolarization activates an anomalous rectifying current
(Bleasel and Pettigrew 1994
; Yarom and
Llinás 1987
). Bal and McCormick (1997)
recently argued that the h current (Ih), which originally was described
in thalamocortical relay neurons (Huguenard and McCormick
1992
), may account for many IO cell properties. They proposed
that Ih, which is activated at hyperpolarized
potentials, contributes to the presence, amplitude, and frequency of
the subthreshold oscillations. Finally, like most other neurons, IO
cells can generate somatic sodium spikes (with the sodium current
INa) that are terminated by a delayed rectifier potassium current (IK_dr).
However, a single spike is generated during the plateau potential,
indicating a strong refractoriness of the sodium current
(Llinás and Yarom 1981a
).
). It is thus difficult to analyze the contribution of
individual currents to the response of olivary neurons.
Conductance-based models of neurons, however, if sufficiently
constrained by biological data, may provide a means to test different
hypotheses by systematically changing their parameters (e.g., maximal
conductances of ionic channels). In the present study, we extended the
single compartment model developed by Manor et al.
(1997)
, which included only
ICa_l and a leak current, and built a
two-compartment model of IO neurons, where the first compartment
represented the soma and the second compartment represented the lumped
dendrites. The dendritic compartment permits us to account for the
known dendritic location of the high-threshold calcium current, the gap
junctions, and the synaptic inputs. We constrained the parameters of
our model with experimental data whenever possible and reproduced many
experimentally observed findings.
, we found that
Ih had an important modulatory role in
the spike generation and period of oscillations.
; Sotelo et al. 1974
), researchers have been
looking for the functional roles of the electrical coupling. In most
neurophysiological studies, electrical coupling is thought to be
responsible for the synchronization of groups of neurons. However, in
several mathematical studies of simple coupled neuron models,
anti-phase as well as irregular spiking have been observed
(Abarbanel et al. 1996
; Kawato et al. 1979
; Sherman and Rinzel 1992
). Thus, although
counterintuitive, electrical coupling theoretically could desynchronize
rather than synchronize coupled cells. In the present study, we found
that two coupled IO cells could have different spiking modes depending on the coupling strength, the hyperpolarization level, and the modulation of ICa_l and
Ih currents: the two cells could be
in-phase, phase-shifted, or anti-phase or could exhibit a complex,
desynchronized spiking mode. We discuss the robustness and significance
of the desynchronization for the in vivo functioning of the IO in light of the known effect of serotonin on IO neurons.
METHODS
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
): the
ratio of the somatic area to total surface area (p) and the
electrotonic coupling conductance between the two compartments
(gint). For each compartment, the
membrane potential (V) changed according to
Cm is the membrane capacitance,
Iapp is the applied current (common to
both compartments) and
accounts for the ionic currents, the current flowing out into the other
compartment, and the current flowing through the gap junctions
(currents in µA/cm2).
View larger version (20K):
[in a new window]
Fig. 1.
Model of inferior olive (IO) neurons. A: 2-compartment
biophysical model of an IO cell. Somatic ionic currents include a
low-threshold calcium inward current
(ICa_l), an anomalous inward rectifier
current (Ih), the Hodgkin-Huxley type inward
sodium (INa) and outward delayed rectifier
potassium (IK_dr) currents, the current
flowing out into the dendritic compartment
(Ids), and a leakage current
(Ils). Dendritic currents include a
high-threshold inward calcium current
(ICa_h), an outward calcium-dependent
potassium current (IK_Ca), the current
flowing into the somatic compartment (Isd),
and a leakage current (Ild). Because the
dendritic potassium Current IK_Ca is
calcium-dependent, Ca2+ concentration dynamics is included
in the model. gint is the coupling
conductance between the two compartments. B: 2 cells
electrotonically coupled by a single hypothetical gap junction with
maximal coupling conductance gc.
Electrotonic coupling current between the cells is given by
Ic.
Somatic currents
The somatic ionic currents included
ICa_l,
Ih, an inward sodium current
(INa) and a delayed rectifier outward
potassium current (IK_dr), as well as
the current flowing out into the dendrites (Ids) and the leakage current
(Ils).1
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
Dendritic currents
The dendritic ionic currents include
ICa_h,
IK_Ca, the current flowing into the
somatic compartment (Isd), a leakage
current (Ild), and the current flowing
into other cells through electrical coupling
(Ic). Thus the dendritic currents are
given by
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
Modeling a "standard" cell
A neural model can have a wide variety of responses when large
parameter ranges are considered (Rinzel and Ermentrout
1989). Our model of IO neurons, although simplified, contains a
large number of parameters. Consequently whenever possible,
physiologically plausible ranges of parameters are first taken into
account (see Table 1). Moreover we use
typical values for sodium, potassium, and calcium reversal potentials
and membrane capacitance: VNa = 55 mV,
VK =
75 mV, and
VCa = 120 mV and
Cm = 1 µF/cm2,
respectively. We took the reversal potential of the h current to be
Vh =
43 mV, as reported by
Huguenard and McCormick (1992)
. The remaining unknown
parameters were investigated using the simulation software package
XPPAUT (Ermentrout 1997
) to integrate the equations of
the system. We developed a "standard" cell that possessed most of
the known electrophysiological properties of IO cells by taking a
priori ranges of parameters into account when available.
|
Sensitivity analysis
We tracked the stationary and periodic solutions of the system
with a change in a parameter (such as the input current) using the
bifurcation analysis software AUTO (Doedel 1981). AUTO
also allowed us to compute the frequency of oscillations, detect the bifurcation of solutions, and track the bifurcation points (e.g., the
onset of oscillations) in two-dimensional parameter space. Because it
was impossible to track the bifurcation of periodic orbits with this
software (e.g., points for which 2 periodic solutions, such as
subthreshold oscillations and spontaneous spiking, coalesce), we
developed a custom simulator based on the integration simulation package CVODE (Cohen and Hindmarsh 1994
). Thus we could
explore spiking behaviors when one or two parameters were varied. By
numerically integrating the model equations with different parameter
values, we uncovered several asymptotic cell response domains:
return to rest (R) after a period of transients, spontaneous
subthreshold oscillations (SO), spontaneous somatic spiking (SS),
spontaneous dendritic spiking (DS). In some cases, the cell could stay
in different symptotic states depending on the initial conditions: the
cell was then described bistable, and in this case the domains were
overlapping.3
For instance, when the SO domain overlapped with SS, the cell behavior
was denoted to be SS/SO bistable. Note that in the sensitivity analysis
graphs, we plotted both the results obtained with AUTO (for the domain
of oscillatory behavior emerging from rest) and the results obtained
with our simulator (for the other domains).
Analyzing the responses of two coupled cells
We examined the behavior of two coupled cells when
gc varied (Fig. 1B). To
describe the extent of synchronization between two identical cells that
receive identical inputs, we computed the average "distance"
D(, gc), in millivolts,
between the somatic membrane potentials (Abarbanel et al.
1996
). This measure is essentially the root mean square
difference between the somatic membrane potentials of the two cells,
Vs1 and
Vs2, with a time shift
. The time
shift between the activities of the neurons is necessary to "align" the neurons' responses. The average distance is mathematically defined
by
![]() |
![]() |
RESULTS |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Standard cell
Using the following parameter values (conductances in
mS/cm2), soma: gNa = 70, gK_dr = 18, gCA_l = 1.0, and gh = 1.5; dendrite: gCa_h = 4.0 and
gK_Ca = 35; leak:
gls = gld = 0.015 and
vl = 10 mV; cell morphology:
gint = 0.13 and p = 0.20, data from our standard cell model were consistent with many
reported experimental findings from actual IO neurons.
STEADY-STATE PROPERTIES.
The membrane potential with no input current
(Iapp = 0 µA/cm2) was 57 mV, whereas the input
resistance derived from the voltage-current curve was 36 M
. When
Iapp =
5
µA/cm2, the potential was
80.3 mV and the
input resistance was 14 M
, reflecting the effect of the h current.
When Iapp = +5
µA/cm2, the membrane potential was
46 mV and
the input resistance was only 10 M
because of the effect of the
delayed rectifier current. These steady-state values are in agreement
with published experimental data (see for instance Table 1 in
Llinás and Yarom 1981a
; Fig. 1A in Yarom
and Llinás 1987
; Manor 1995
). Note that 1 µA/cm2 corresponds to 0.1 nA for a total cell
surface of 10,000 µm2.
DEPOLARIZING CURRENT INPUT.
We injected depolarizing step currents of 50-ms duration, beginning at
t = 200 ms in the cell at rest (Fig.
2). When the injected current intensity
was >3 µA/cm2, the cell first responded with a
somatic sodium spike immediately followed by a dendritic calcium
plateau potential (Fig. 2A), which lasted for ~35 ms. The
calcium plateau potential, due to activation of
ICa_h, created a large calcium ion
influx in the dendrites, which in turn activated
IK_Ca (Fig. 2C). This
eventually terminated the plateau potential and created a large,
long-lasting AHP ~13 mV below the resting membrane potential. At the
offset of the dendritic spike,
ICa_h,
ICa_l, and
Ih are all completely inactivated (as
are INa and
IK_dr). Thus the only nonzero current
is IK_Ca. Because this current is
driven by relatively slow dendritic calcium dynamics and possesses a
long inactivation time constant, the AHP is dominated by
IK_Ca. Note that even though the
conductance of IK_Ca is high, the
amplitude of IK_Ca is small during the
AHP because the membrane potential is close to the potassium reversal potential. The AHP terminated sharply after ~300 ms due to rebound excitation, which is consistent with published data (see Fig. 7A in Llinás and Yarom 1981b). The time
courses of the somatic currents (Fig. 2B) revealed that the
AHP termination is initially due to the activation of
Ih, which increases the membrane
potential. Then when the soma is sufficiently depolarized,
ICa_l starts to be activated, which
results in the rebound of excitation.
|
HYPERPOLARIZING CURRENT INPUT.
We then studied the response of the standard cell model to a
hyperpolarizing current pulse (t = 500 ms,
duration = 100 ms, intensity = 1.5
µA/cm2) superimposed on different tonic
hyperpolarizing currents (Iapp). When
Iapp = 0 µA/cm2, the activation of the low-threshold
calcium conductance, as well as the recovery of
Ih, during the hyperpolarizing step
created a rebound sufficient to generate a sodium spike (Fig.
3A). This is consistent with
previous experimental data (Bal and McCormick 1997
; see
their Fig. 2 for instance). ICa_l then
became inactivated and ICa_h became
activated to a moderate degree. The calcium influx in the dendritic
compartment resulted in the IK_Ca
activation, creating an AHP. The cycle began again, and another spike
was generated. After the second spike, however,
ICa_l was no longer sufficiently
activated to bring the membrane potential above threshold and only a
few more subthreshold oscillations followed. Thus the standard cell was
a damped oscillator, as reported by Lampl and Yarom
(1997)
. When a tonic bias hyperpolarizing current
Iapp =
0.5
µA/cm2 was maintained before the current pulse,
the cell was SO (Fig. 3B). The cell fired a burst of six
spikes after the pulse. When the bias was
Iapp =
1.0
µA/cm2 (Fig. 3C), the cell generated
two sodium spikes in response to the current pulse. Finally, when the
bias was Iapp =
1.5
µA/cm2 (Fig. 3D), only one spike was
generated and the oscillations were rapidly damped.
|
Parameter sensitivity analysis
BIFURCATION ANALYSIS.
The steady-state cell responses at different hyperpolarized membrane
potentials can be predicted from a bifurcation diagram plotting the
minimum and maximum values of the somatic membrane potential for
Iapp. The cell had a stable steady
state for all Iapp values except
between 1.17 µA/cm2 and
0.37
µA/cm2 (Fig. 4;
1.17 and
0.37 corresponded to the 2 Hopf bifurcation points marked
a and b). Within this range, the cell exhibited subthreshold
oscillations that reached a maximum amplitude of 9.8 mV for
Iapp =
0.85
µA/cm2, and the frequency of these subthreshold
oscillations was 5-7 Hz. These values are similar to those recorded in
real IO neurons (Benardo and Foster 1986
; Lampl
and Yarom 1997
; Llinás and Yarom 1986
).
|
LOW-THRESHOLD CALCIUM CURRENT.
Because the low-threshold calcium current has a central role in IO
oscillatory behavior, we examined the variation of
gCa_l in our model. We first analyzed
the behavior of a cell with gCa_l = 1.2 mS/cm2 and compared it to our standard cell
(gCa_l = 1.0 mS/cm2). The relatively small increase in
gCa_l resulted in a very different bifurcation diagram (Fig. 5). The cell
had multiple stable behaviors, specifically, SS, SO, and R (an example
of SS behavior can be seen in Fig. 7D). The spontaneous
somatic spiking was between the points marked a and b in the figure,
which correspond to a large range of input currents (1.48-0.19
µA/cm2). Between the points b and c
(Iapp between
0.23 and 0.19 µA/cm2), the cell had two stable states, either
SS or R. This bistability has been observed previously (Bal and
McCormick 1997
- see their Fig. 1B for instance).
Between the points c and d, i.e., when Iapp was between
0.73 and
0.23
µA/cm2, the stable behaviors were either
subthreshold oscillations or spontaneous somatic spiking. For the
domain delimited by the points d and e
(Iapp between
1.30 and
0.73
µA/cm2), the only stable state was spontaneous
somatic spiking. Finally, the points e and a
(Iapp was
1.47 and
1.30
µA/cm2) delimit another zone of bistability,
with either subthreshold oscillations or spontaneous
spiking.4
The frequencies of the spiking oscillations ranged from
3.8 to 9.4 Hz, which is similar to that previously recorded in IO
neurons (4-8 Hz) (Bal and McCormick 1997
). The
frequencies decreased with stronger hyperpolarization, a result also
consistent with previous observations (see Fig. 2A in
Bal and McCormick 1997
).
|
|
h CURRENT.
Previous in vitro experiments have shown that
gh is reduced with increasing
concentrations of bath-applied Cs+ (Bal
and McCormick 1997). To model the effects of
Cs+, gh was
decreased from its standard value 1.5 mS/cm2
(Fig. 7A) to 1.0, 0.5, and 0.0 mS/cm2 (Fig. 7, B-D), while a tonic
current of
0.5 µA/cm2 was applied (note that
a hyperpolarizing current pulse of
1.5 µA/cm2
was injected at 500 ms for 100 ms to show bistability). The frequency decreased as the conductance value became smaller: it was ~8 Hz with
gh = 1.5 mS/cm2,
and it decreased to 6.2 Hz when Ih was
fully suppressed (frequency analysis conducted with AUTO, data not
shown). Similar results have been found in vitro (Bal and
McCormick 1997
; see their Fig. 7D). Note also that
for gh = 0.5 mS/cm2 and
gh = 0.0 mS/cm2, the
hyperpolarizing pulse generates a dendritic spike. In these cases, as
discussed by Yarom and Llinás (1987)
, somatic and
dendritic compartments are effectively more strongly coupled because of higher input resistance: the sodium spike can invade the dendritic compartment enough to generate a dendritic spike.
|
Behavior of two coupled IO cells
RESPONSES TO A CURRENT PULSE WITH DIFFERENT TONIC CURRENTS.
To investigate the effect of coupling on the oscillating behaviors of
two cells, we examined the membrane potential of cell 1 as a
function of both the tonic input current injected in cell 2 (Iapp2), and the coupling conductance
gc (with
Iapp1 = 0 µA/cm2). The cells were "standard cells,"
except that we set gCa_l = 1.2 mS/cm2 to increase the excitability of the cells.
The two-parameter diagram (Fig.
8A) shows that for
gc < 0.18 mS/cm2, there were two ranges of
Iapp2 with
subthreshold oscillations. In Fig. 8A, to the right of the
nearly vertical line (around Iapp2 = 0.2 µA/cm2), and the area under the curved
line, cell 1 was in a stable-steady state. In other regions,
the cell oscillated.
|
RESPONSES TO CURRENT PULSES IN SIMULATED HARMALINE APPLICATION
CONDITIONS.
The drug harmaline is known to induce very robust somatic spiking in IO
cells via three pharmacological effects (Llinás and Yarom
1986): it increases the voltage sensitivity of the conductance of the low-threshold calcium current, it hyperpolarizes the cell, and
it reduces the anomalous rectification. In simulations, we approximated
the increase in voltage sensitivity of the
ICa_l conductance by increasing
gCa_l, the hyperpolarizing effect by injecting a tonic hyperpolarizing current in the soma, and the reduction in Ih by decreasing
gh. Thus we set
gCa_l = 1.2 mS/cm2, Iapp1 = Iapp2 =
0.8
µm/cm2, and gh = 0.7 mS/cm2 (also we further increased
gNa to 80 mS/cm2
to ensure robust
spiking).5
The cells exhibited strong and robust spontaneous somatic
spiking with this set of parameters.
|
MODULATION OF BOTH COUPLING STRENGTH AND IONIC CURRENTS IN SIMULATED HARMALINE APPLICATION. Here we show that, in addition to the coupling conductance, the maximal conductance values of both ICa_l and Ih have key roles in the appearance of complex, desynchronized firing patterns. In Fig. 10A, the distance D between the two cells is plotted as a function of both gCa_l and gc. The contour D = 2 mV approximately corresponds to the appearance of a significant phase shift. The contour D = 8 mV corresponds to the induction of desynchronizing dendritic spikes by the current flowing through the gap junction (as in Fig. 9A).6 Very weak coupling (gc < 0.003 mS/cm2) had no effect on the phase relationship of the spiking cells. For a large range of intermediate coupling values (gc between 0.003 and 0.15 mS/cm2), spiking was complex. Only very large coupling conductances (gc > 0.2 mS/cm2) brought the two cells into the in-phase mode. Because these large values are probably not biologically plausible (see DISCUSSION), complex firing behavior occurred for almost the whole physiological range of gc. The range of gCa_l for which firing is complex was 1.15-1.4 mS/cm2. This range should be compared with the spontaneous somatic spiking domain: if gCa_l was <0.9 mS/cm2 only subthreshold oscillations were observed, and if gCa_l was >1.6 mS/cm2, only spontaneous dendritic spiking were observed. Thus desynchronization occurred for more than a third of gCa_l values for which somatic spikes were observed.
|
![]() |
DISCUSSION |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
We propose a novel two-compartment biophysical model of IO neurons that reproduced many reported experimental results. The modeled neurons exhibited multiple firing patterns that depended on the level of depolarization or hyperpolarization, on the relative influence of the ionic currents ICa_l and Ih, and on the strength of the electrotonic coupling.
Comparison with related modeling studies
Two-compartment models similar to the one described here were
shown to reproduce successfully many cellular responses of several neurons which contain active dendrites (Booth et al.
1997; Mainen and Sejnowski 1996
; Pinsky
and Rinzel 1994
; Rush and Rinzel 1994
). The
second compartment, corresponding to the lumped dendrites, allowed us
to segregate the soma from the dendritic location of the high-threshold
calcium current, the gap junctions, and the synaptic inputs. The
morphological parameters gint and
p were critical to the behaviors of the two-compartment
model. When the coupling conductance between the two compartment
gint was set to large values (in
effect lumping the somatic and dendritic compartments into 1 isopotential compartment), simulations showed that it was impossible to
generate somatic spikes: the sodium spikes always triggered dendritic
spikes. We found that, for proper functioning of the model
gint must be set to <0.2
mS/cm2. Moreover, because it is reasonable to
assume that intracellular coupling
(gint) is stronger than intercellular
coupling (gc), we have
gc < gint < 0.2 mS/cm2 (see Table 2
for plausible ranges of parameters inferred from simulations). In the
present model, p = 0.2 was used for the ratio of the
somatic area to the total surface area. In two compartment models, it
is difficult, however, to relate directly p to the actual
cell morphology due to the attenuation of the current flowing from
distal dendrites to the soma (Pinsky and Rinzel 1994
).
Thus p should be larger than the actual morphological ratio
(0.1-0.2) (Manor 1995
).
|
In the present study, the calcium dynamics and the kinetics of the
ICa_h and
IK_Ca currents were borrowed from
Traub et al.'s (1991) CA3 hippocampal pyramidal neuron
model. These kinetics lead to satisfactory performance of our IO model
because most of the results of the present study are in agreement with
experimental observations. It is possible, however, that the very low
firing frequency of IO neurons in vivo is due to a longer lasting AHP than in the model (~350 ms when no tonic current is injected). When
we used the following calcium dynamics equation (instead of Traub et
al.'s)
![]() |
To our knowledge, the model developed by Manor (1995)
and Manor et al. (1997)
is the only other published
model of IO neurons. Their study focused on the subthreshold
oscillations, so their model contained only
ICa_l and a leakage current (because
these 2 currents are sufficient to generate subthreshold oscillations). These authors notably showed that when two nonidentical cells were
coupled, oscillations could be generated even though neither of the
individual cells was a spontaneous oscillator. For example, the two
cells shown in Fig. 8B did not oscillate when they were not
coupled because the first cell was too depolarized and the second cell
was too hyperpolarized. When coupled (with
gc = 0.05 mS/cm2), however, the two cells oscillated at the
same frequency, but at different baselines and with a small phase
shift. Moreover, if one cell started spiking, the oscillations in the
other cell increased in amplitude.
Our model significantly extends the Manor et al. model, however,
because we incorporated sodium spikes, dendritic spikes, and AHP. Thus
we could investigate the conditions of IO spike generation when input
current, electrical coupling strength, and the conductances of both
ICa_l and
Ih were varied. We further showed that
a two-cell network model can exhibit anti-phase spiking (as previously
found in simple models of coupled neurons) (e.g., Abarbanel et
al. 1996; Kawato et al. 1979
; Sherman and
Rinzel 1992
) and complex, desynchronized firing patterns (also
observed in the model of Abarbanel et al. 1996
). In the
present model, the crucial and novel element is the triggering of
dendritic spikes by electrical coupling leading to desynchronization.
Our results suggest that this behavior is robust because it can be
attained for a large range of coupling and ionic conductances.
Finally, our two-compartment model is complex enough to reproduce the
essential properties of IO cells, such as the generation of dendritic
and somatic spikes, but computationally simple enough to be later
integrated into models of larger networks. Moreover, realistic, time
varying inputs could be provided to the IO network by incorporating it
into a complete sensorimotor neural system (see Schweighofer et
al. 1998).
Ionic current modulationrhythmicity in single inferior olive cell
As previously described in experiments (Lampl and Yarom
1988; Llinás and Yarom 1986
) and in the
model of Manor et al. (1997)
, simulations showed that
the low-threshold calcium current had a central role in generating
rhythmic behavior. If the maximal conductance of
ICa_l was below a threshold value,
then spontaneous subthreshold oscillations did not occur regardless of
the input current. Small changes in
gCa_l, around the standard cell's
parameter had dramatic effects on the cell excitability pattern (R, SO, or SS) as shown in Figs. 5 and 6. Depending on the brain slice, or even
depending on the cell impaled, IO neurons recorded in vitro exhibit a
variety of responses to the same stimulus, and some cells are more
excitable than others. Our results show that even a relatively narrow
distribution of gCa_l in real IO
neurons can explain the diversity of responses encountered. Figures 4 and 5 suggest two new experiments to test further the effect of the
low-threshold calcium current in single cells, i.e., cells grown in
culture or isolated (this only will be feasible when specific gap
junctions blockers become available). In these experiments, both the
low-threshold calcium conductance and the injected tonic current are
controlled independently. In the first experiment (Fig. 4), starting
from rest at the depolarized level, a very slow continuous decrease in
the injected tonic current first would produce subthreshold
oscillations of increasing amplitudes, then oscillations of decreasing
amplitudes, then rest again. In a second experiment (Fig. 5), if the
maximal conductance gCa_l is increased (by harmaline for instance), the sequence of responses to decreasing injected current then would be: subthreshold oscillations of increasing amplitudes at slightly hyperpolarized membrane potential, then spontaneous somatic spiking, and finally rest again for large hyperpolarizing currents.
Our results indicate that the anomalous rectifier current
Ih plays a major role in controlling
the frequency of the oscillatory behavior and the generation of IO
somatic spikes. This is consistent with the observations of Bal
and McCormick (1997). ICa_l
and Ih play antagonistic roles: while
an increase in gCa_l makes the cell
more excitable (Fig. 6), a decrease in
gh has similar effects (Fig.
7E). The effect of the anomalous rectifier could be tested by controlling simultaneously the conductance of
Ih (with serotonin and/or Tris
artificial cerebrospinal fluid) (see Bleasel and Pettigrew 1994
) and the membrane potentials. For example, an experiment designed to follow a path at a fixed membrane potential similar to
A
B
C
D in Fig. 7 could be performed.
In the present model, the frequency of all oscillations are voltage
dependent. In agreements with our model, the results of Llinás and Yarom (1986) and of Bal and
McCormick (1997)
see their Fig. 2
seem to indicate that
somatic spiking oscillations are voltage dependent. However, the
spontaneous subthreshold oscillations, in slices where they occur, are
voltage independent (Lampl and Yarom 1997
;
Llinás and Yarom 1986
). Part of the discrepancy
between our results and these experimental results may be due to
network oscillations as opposed to cellular oscillations. Lampl
and Yarom (1997)
showed that the pattern of subthreshold
oscillations is network specific rather than cell specific. Thus
injection of current in one cell does not change its oscillatory
frequency. Only when an experimental method is available to isolate a
single IO cell, will it be possible to analyze the voltage dependency of subthreshold oscillations in single cells, as predicted here.
Influence of coupling and current modulationsynchrony and
desynchrony of inferior olive cells
Even though gap junction coupling generally is thought to bring the membrane potential of the two coupled cells closer to each other, we found that, counterintuitively, increasing coupling strength can desynchronize the two spiking cells. If a somatic spike occurred in one cell, a dendritic spike in a coupled cell could be triggered by the current flowing through the gap junctions. Thus under certain conditions, it appears that electrical coupling may induce effects that are similar to those induced by chemical synapses, that is, they may trigger dendritic spikes. In addition, the long AHP after the dendritic spikes increased the complexity of the firing pattern. Desynchronization is robust because there were broad ranges of electrotonic coupling strengths and maximal conductances, gCa_l and gh where desynchronization was observed. Moreover, we verified that desynchronization does not critically depend on the voltage dependency of the coupling conductance: Linear electrical coupling conductances also can desynchronize the cells (however, because of the stronger overall effect of linear coupling, the range of gc where desynchronization was observed is smaller). Note that the occurrence of complex firing behavior by a reduction of gh can be understood by the fact that a reduction of Ih decreases the electrotonic distance between the soma and the dendrites, which, in turn, increases the influence of a somatic spike in one cell on the dendritic compartment of another cell.
It might appear puzzling that because a somatic spike in the leading cell (or cell 1) triggers a dendritic spike in the lagging cell (cell 2), electrical coupling fails to trigger a dendritic spike in cell 1 as well. However, a close inspection of the currents flowing into and out of the two dendritic compartments reveals the following. The cell 1's somatic spike propagates in the dendrite and in turn slightly raises the membrane potential of cell 2. If this happens simultaneously with a rise in the dendritic membrane potential of cell 2 due to the propagation of cell 2's somatic spike, the combined effect is sufficient to trigger a dendritic ICa_h spike in cell 2. This, in turn, creates a relatively large current flow through the electrical coupling. However, the somatic spike of cell 1 has already activated the ICa_h current to some extent (but not suffiently to generate a spike), which in turns lead to some activation of IK_Ca. This later current counterbalances the depolarizing effect of the coupling current: As a consequence, no dendritic spike is generated in cell 1.
To test the effect of coupling on the synchronization, we propose the
following experiment: two electrodes could be inserted into two
neighboring olivary cells after harmaline application. Because
inhibitory inputs from the cerebellar nuclei can decrease significantly
the gap junction coupling strength (Lang et al. 1996),
excitation or inhibition of deep nuclear cells that project to this
region of the IO could be used to control the coupling conductance. We
predict that, depending on the electrical coupling strength, the two
recorded cells could be in-phase, phase-shifted, anti-phase, or desynchronized.
Our results imply that neuromodulatory inputs to the IO that induce
relatively small changes in ionic conductances may bring IO cells into
spontaneous rhythmic firing modes and allow neighboring IO cells to be
desynchronized. Indeed, serotonin, released by the raphe nucleus, has
been shown to have effects very similar to those of harmaline on IO
cellular behavior (even though its mode of actions may be different)
(Sugihara et al. 1995). Thus it is possible that our
results under simulated harmaline application apply to serotonin
neuromodulation. First, because only a slight increase in
gCa_l allows spontaneous rhythmic
somatic spiking (at least at slightly hyperpolarized membrane
potentials), IO cells could be exquisitely sensitive to serotonin: by
increasing the voltage sensitivity of
ICa_l, serotonin may allow the control of repetitive spiking. Second, because anomalous rectification is
reduced by serotonin (Sugihara et al. 1995
), it is
possible that Ih plays a central role,
along with ICa_l, in controlling whether the mode of IO cell firing is rhythmic or not in vivo. Moreover, as shown in Fig. 10, both an increase in
ICa_l strength and a decrease in
Ih strength would make the
desynchronization process more likely. This prediction could be tested
experimentally by recording two neighboring IO cells and by
simultaneously controlling the serotonin level.
![]() |
APPENDIX |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
The kinetics equation for the activation and inactivation
variables h, n, k, l, q, and r is
![]() |
INa
Adapted from Rush and Rinzel (1994).
![]() |
![]() |
![]() |
![]() |
![]() |
IK_dr
Adapted from Rush and Rinzel (1994).
![]() |
![]() |
![]() |
ICa_l
![]() |
![]() |
Ih
Adapted from Huguenard and McCormick (1992).
![]() |
![]() |
ICa_h
Adapted from Traub et al. (1991).
![]() |
![]() |
IK_Ca
Adapted from Traub et al. (1991).
![]() |
![]() |
![]() |
![]() |
Ic
Adapted from Moreno et al. (1994).
![]() |
![]() |
ACKNOWLEDGMENTS |
---|
The authors thank the anonymous referees for valuable comments and criticisms.
XPPAUT3.0 is available from http://www.pitt.edu/~phase. AUTO is available from http://indy.cs.concordia.ca/auto/. CVODE is available from http://www.netlib.org/ode/cvode.tar.gz.
![]() |
FOOTNOTES |
---|
Address for reprint requests: N. Schweighofer, ERATO, Kawato Dynamic Brain Project, 2-2, Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-0288, Japan.
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
1
Yarom and Llinás (1987) demonstrated that
the anomalous rectifying current consists of two parts: a
time-dependent component and an instantaneous component. Due to the
lack of a precise current model, we did not include the instantaneous
component of the anomalous rectifying current in the present model.
2 The electrotonic coupling conductance gint between the somatic and dendritic compartments mathematically represents the inverse of the cell's cytoplasmic resistivity. The electrical coupling conductance gc between the cells reflects the strength of the intercellular electrical coupling. Although these two types of coupling are very different physiologically, in two-compartment models they are represented similarly, by the coupling conductances gint and gc. However, to account for the cell integrity, the value of the former must be much higher than the value of the latter.
3
To detect these cases, current pulses of 21 different intensities (from 8 µA/cm2 to + 8 µA/cm2, for 50 ms) were injected into the cell's soma.
We considered the cell to be bi-stable if at least one current
injection could generate sustained spiking (when the cell was not
spontaneously spiking before the current injection) or stop spiking
(when the cell was spontaneously spiking before the current injection). Note that this method introduces a resolution limit of 0.8 µA/cm2, which corresponds to 0.08 nA only for a total
cell surface of 10,000 µm2. It is therefore possible that
we missed some small ranges of bi-stability. However, these very small
bi-stability ranges would have negligible physiological consequences.
4
For a very small range, 1.477 < Iapp <
1.470, the stable states were the resting
state and somatic spiking.
5
Harmaline shifts the inactivation curve of the
low threshold calcium currents to the right (Llinás and
Yarom 1986). In the present model, we chose to simulate the
modulatory action by adjusting the maximal conductances only, in order
not to increase the parameter search space. We did however run
preliminary simulations which showed that shifting the inactivation
function of ICa_l to the right by 5 mV also reproduced
robust somatic spiking.
6 This value might appear small compared with spike amplitudes. However, since the maximal firing rate is low (around 3-4 spikes per s) and since a sodium spike lasts only a very short time (in the order of 2-3 ms), the two desynchronized cells have similar, hyperpolarized, membrane potentials most of the time. During these relatively long-lasting common hyperpolarized phases, the instantaneous distance is small; this explains the small values of the distance, which was temporally averaged over long duration, compared with the spike amplitudes.
Received 10 December 1998; accepted in final form 12 April 1999.
![]() |
REFERENCES |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|