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J Neurophysiol (May 1, 2003). 10.1152/jn.00845.2002
Submitted on Submitted 24 September 2002; accepted in final form 20 December
2002
1Volen Center for Complex Systems, Brandeis University, Waltham, Massachusetts 02454; 2Instituto de Neurociencias, Universidad Miguel Hernández-Consejo Superior Investigaciones Científicas, 03550 San Juan de Alicante, Spain; and 3Section of Neurobiology, Yale University School of Medicine, New Haven, Connecticut 96510
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ABSTRACT |
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Compte, Albert, Maria V. Sanchez-Vives, David A. McCormick, and Xiao-Jing Wang. Cellular and Network Mechanisms of Slow Oscillatory Activity (<1 Hz) and Wave Propagations in a Cortical Network Model. J. Neurophysiol. 89: 2707-2725, 2003. Slow oscillatory activity (<1 Hz) is observed in vivo in the cortex during slow-wave sleep or under anesthesia and in vitro when the bath solution is chosen to more closely mimic cerebrospinal fluid. Here we present a biophysical network model for the slow oscillations observed in vitro that reproduces the single neuron behaviors and collective network firing patterns in control as well as under pharmacological manipulations. The membrane potential of a neuron oscillates slowly (at <1 Hz) between a down state and an up state; the up state is maintained by strong recurrent excitation balanced by inhibition, and the transition to the down state is due to a slow adaptation current (Na+-dependent K+ current). Consistent with in vivo data, the input resistance of a model neuron, on average, is the largest at the end of the down state and the smallest during the initial phase of the up state. An activity wave is initiated by spontaneous spike discharges in a minority of neurons, and propagates across the network at a speed of 3-8 mm/s in control and 20-50 mm/s with inhibition block. Our work suggests that long-range excitatory patchy connections contribute significantly to this wave propagation. Finally, we show with this model that various known physiological effects of neuromodulation can switch the network to tonic firing, thus simulating a transition to the waking state.
Cortical oscillatory activity as measured by
electroencephalogram (EEG) is a clear signature of the general state of
the brain. The waking state and the rapid-eye-movement (REM) phase of
sleep are characterized by low-amplitude fast oscillations (Gray
et al. 1989
; Steriade et al. 1996
) of a
generally low spatiotemporal coherence (Destexhe et al.
1999
). In contrast, during slow-wave sleep and anesthesia, the
brain shows pronounced oscillatory activity at a variety of frequencies
often with remarkable long range synchrony (Bullock and McClune
1989
; Destexhe et al. 1999
; Steriade et
al. 1993a
, 1996
). During slow-wave sleep, low-frequency (<1 Hz)
oscillations are visible both in the EEG, and in extracellular and
intracellular recordings (Achermann and Borbély
1997
; Lampl et al. 1999
; Steriade et al.
1993b
,c
,
1996
; Stern et al.
1997
). Lesion studies have shown that this type of rhythmic
activity originates in the cortex and is then reflected in subcortical
structures (Amzica and Steriade 1995
; Steriade et
al. 1993c
). Intracellular recordings in vivo showed that the
slow oscillation is mediated by two phases: a period in which nearly
all cell types within the cerebral cortex are depolarized and generate
action potentials at a low rate (the so-called up state) interdigitated
with a period of hyperpolarization and relative inactivity (the down
state). The transition from the up to down states has been proposed to
occur either in response to synaptic "fatigue" or depression
(Contreras et al. 1996
) or to the build-up of
activity-dependent K+ conductances (Sanchez-Vives
and McCormick 2000
). A gradual increase in input resistance of
pyramidal cells during the up state in vivo has been taken to indicate
a steady decrease of a specific ionic conductance, suggesting a
stronger role of depression of excitatory synapses over the activation
of K+ conductances in the transition from the up to the
down state (Contreras et al. 1996
; Timofeev et
al. 2000b
).
Recently, spontaneous activity similar to the slow oscillations (<1
Hz) recorded in vivo has been described in an in vitro slice
preparation of cerebral cortex when maintained in an ionically modified
artificial cerebral spinal fluid (ACSF) solution that mimics ionic
concentrations in situ more closely than the solutions traditionally
used for cortical slice preparations (Sanchez-Vives and
McCormick 2000
). This helped to identify candidate cellular and
circuit mechanisms underlying the generation of slow oscillations and
wave propagation in ferret visual cortex slices. For example, pharmacological manipulations show that this activity depends on
excitatory transmission through AMPA and
N-methyl-D-aspartate (NMDA) receptors,
suggesting a critical role of recurrent excitatory connections. Open
issues remain in relation to the two main aspects of slowly oscillating
cortical activity: the membrane potential sudden transition between and
up state and a down state and the propagation of activity across the
cortical network. Evidence suggests that the transition from the up
state to down state is induced by the opening of a K+
conductance (Sanchez-Vives and McCormick 2000
), whose
time course has led to the hypothesis that it is a slow
Na+-dependent K+ conductance
gKNa known to exist in these neurons
(Sanchez-Vives et al. 2000
). This raises the question of
how an increase of gKNa in a pyramidal cell
could be compatible with an increase of its input resistance observed
during the course of an up state. Another intriguing aspect of
these membrane fluctuations is the sharpness of the transitions between
the up and the down states, where the relative contribution of
intrinsic and network mechanisms remains to be established. On the
other hand, as observed experimentally, an up state episode consists of
barrages of synaptic activity that initiate earlier in infragranular
laminae and occur in all cortical layers. This reverberatory network
activity propagates across the slice at ~10 mm/s and is followed by a
silent period of 2-4 s. Blocking GABAA-mediated inhibition
results in epileptiform discharges that propagate along the slice at
~100 mm/s. The ability of a cortical network to sustain two
propagation velocities has been studied mathematically by Golomb
and Ermentrout (2001)
for a simple model where each cell is an
integrate-and-fire neuron and is allowed to fire only one spike. It
remains to be examined how these two wave propagation modes can be
realized in a biophysically realistic network model of
conductance-based neurons.
The combined results obtained from the in vivo and in vitro preparations provide a framework to build a physiologically realistic network model of the slow oscillation in a cortical slice. We present here this biologically realistic network model, and we use it to address the aforementioned questions about the rhythmogenesis and wave propagation. We then speculate about how our model could relate to slow oscillations during natural slow-wave sleep and activity in the waking state in vivo.
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METHODS |
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The network model consists of a population of 1,024 pyramidal cells and 256 interneurons equidistantly distributed on a line and interconnected through biologically plausible synaptic dynamics. Some of the intrinsic parameters of the cells are randomly distributed, so that the populations are heterogeneous. This and the random connectivity (determined by the synaptic probability distributions; see Fig. 2A) are the only sources of noise in the network.
Model neurons
Especially in vivo, intracellular voltage records show clear
transitions between two well-defined stable membrane potentials (Cowan and Wilson 1994
; Stern et al.
1997
). It has been argued that intrinsic channels may shape the
neuronal membrane potential via an inward rectifier K+
channel (IAR) and a slowly inactivating
K+ channel activated by depolarization
(IKS) (Nisenbaum et al. 1994
; Wilson 1992
; Wilson and Kawaguchi 1996
).
In our model pyramidal neurons, we include these and other channels
found in cortical pyramidal cells.
Our model pyramidal cells have a somatic and a dendritic compartment
(Pinsky and Rinzel 1994
). The spiking currents,
INa and IK, are located
in the soma, together with a leak current IL, a
fast A-type K+ current IA, a
non-inactivating slow K+ current
IKS, and a Na+-dependent
K+ current IKNa. The dendrite
contains a high-threshold Ca2+ current
ICa, a Ca2+-dependent
K+-current IKCa, a non-inactivating
(persistent) Na+ current
INaP, and an inward rectifier (activated by
hyperpolarization) non-inactivating K+ current
IAR. The dynamical equations for the somatic
voltage (Vs) and the dendritic voltage
(Vd) are
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, standard deviation indicates the degree to which this parameter is randomly varied from cell to cell).
Isyn,s and Isyn,d are the
synaptic currents impinging on the soma and dendrites, respectively. In
our simulations, all excitatory synapses target the dendritic
compartment and all inhibitory synapses are localized on the somatic
compartment of postsynaptic pyramidal neurons.
Interneurons are modeled with just Hodgkin-Huxley spiking currents,
INa and IK, and a leak
current IL in their single compartment
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Ion channel kinetics and conductances
All ion channels are modeled following the Hodgkin-Huxley
formalism, with gating variables x governed by the
first-order kinetics equation dx/dt =
[
x(V)(1
x)
x(V)x] =
[x
(V)
x]/
x(V).
being the
temperature factor (
= 1 unless otherwise indicated).
For pyramidal cells, the sodium and potassium spiking currents are
modeled following (Wang 1998
) with slight variations.
The sodium current INa = gNam
VNa) has a maximum conductance of
gNa = 50 mS/cm2, its rapid
activation variable is replaced by its steady-state m
=
m/(
m +
m) with
m = 0.1(V + 33)/[1
exp(
(V + 33)/10)] and
m = 4 exp(
(V + 53.7)/12) and the inactivation variable has
h = 0.07 exp(
(V + 50)/10) and
h = 1/[1 + exp(
(V + 20)/10)]. The temperature factor is
= 4. The delayed rectifier
IK = gKn4(V
VK) has a maximal conductance
gK = 10.5 mS/cm2 and its
inactivation kinetics are set by
n = 0.01(V + 34)/[1
exp(
(V + 34)/10)] and
n = 0.125 exp[
(V + 44)/25], with
= 4. The leakage current
IL = gL(V
VL) is a passive channel with conductance
gL = 0.0667 ± 0.0067 mS/cm2 (Gaussian-distributed in the population, mean ± SD given). The fast A-type K+-channel is as in
Golomb and Amitai (1997)
; IA = gAm
VK) has its fast activation variable replaced
by its steady-state m
= 1/[1 + exp(
(V + 50)/20)] and the inactivation variable is
governed by h
= 1/[1 + exp((V + 80)/6)] and
h = 15 ms.
Its maximal conductance is gA = 1 mS/cm2. The non-inactivating K+-channel is
modeled as in (Wang 1999a
) but with no inactivation variable: IKS = gKSm(V
VK). It has a maximal conductance
gKS = 0.576 mS/cm2 and an
activation controlled by m
= 1/[1 + exp(
(V + 34)/6.5)] and
m = 8/[exp(
(V + 55)/30) + exp((V + 55)/30)]. The remaining currents are modeled with instantaneous
activation because their activation is sufficiently fast and removing
these additional variables significantly reduces the time required to
perform our network simulations.
The persistent sodium channel INaP = gNaPm
VNa) has maximal conductance
gNaP = 0.0686 mS/cm2, it
activates instantaneously according to m
= 1/[1 + exp(
(V + 55.7)/7.7)] and it does not
inactivate. It is borrowed with parameter modification from
(Fleidervish et al. 1996
). The inwardly rectifying
K+ channel was modeled as in (Stern et al.
1997
; Spain et al. 1987
) and adjusting the
parameters: IAR = gARh
(V
VK) activates instantaneously below a
low-lying threshold following h
= 1/[1 + exp((V + 75)/4)] and it has a maximal
conductance gAR = 0.0257 mS/cm2. The high-threshold Ca2+-channel
ICa = gCam
VCa) has gCa = 0.43 mS/cm2 and is instantaneously activated at very
depolarized voltages, thus making it effectively a very transient
current. The voltage dependency is given by
m
= 1/[1 + exp(
(V + 20)/9)]. The concentration of intracellular
calcium, [Ca2+], follows first-order kinetics as
d[Ca2+]/dt = 
CaAdICa
[Ca2+]/
Ca with
Ca = 0.005 µM/(nA · ms) and
Ca = 150 ms. The
Ca2+-dependent K+ channel
IKCa = gKCa[Ca2+]/([Ca2+]+KD)(V
VK) (with KD = 30 µM) activates instantaneously in the presence of intracellular
calcium [Ca2+], and it has a maximal conductance
gKCa = 0.57 mS/cm2. All the
mechanisms involving intracellular calcium are taken from Wang
(1998)
. As for the intracellular sodium concentration [Na+], its dynamics are somewhat more involved because
they incorporate a Na-K pump (Li et al. 1996
):
d[Na+]/dt = 
Na(AsINa + AdINaP)
Rpump{[Na+]3/([Na+]3 + 153)
[Na+]

Na = 0.01 mM/(nA
· ms), Rpump = 0.018 mM/ms, and
[Na+]eq = 9.5 mM. The
Na2+-dependent K+ channel
IKNa = gKNaw
([Na+])(V
VK) has a conductance
gKNa = 1.33 mS/cm2 and
w
([Na+]) = 0.37/[1 + (38.7/[Na+])3.5] (Bischoff et al.
1998
). The kinetics for [Na+] and
IKNa are taken from (Liu 1999
;
Wang et al. 2002
). For all these channels, the reversal
potentials used are VL =
60.95 ± 0.3 mV, VNa = 55 mV,
VK =
100 mV, and
VCa = 120 mV.
For the last three figures, a slight modification in the implementation
of the IKNa current was introduced because we
realized that IKNa was continuously contributing
a sizable [Na+]-independent, voltage-independent
hyperpolarizing current that was confounded with the leakage current.
Because for those figures we were interested in changing independently
gKNa and gL, we opted to
subtract the constant part from IKNa:
IKNa = gKNa
(w
([Na+])
w
([Na+]eq))(V
VK) and change the leakage properties to
compensate for this: gL = 0.07 mS/cm2 and VL =
62.8 mV.
For interneurons, the model was taken from Wang and
Buzsáki (1996)
. The sodium current
INa = gNam
VNa) has a maximal conductance
gNa = 35 mS/cm2, and its rapid
activation is replaced by its steady-state value m
=
m/(
m +
m) with
m = 0.5(V + 35)/[1
exp(
(V + 35)/10)] and
m = 20 exp(
(V + 60)/18). The inactivation gating variable is controlled by
h = 0.35 exp(
(V + 58)/20)
and
h = 5/[1 + exp(
(V + 28)/10)]. The delayed rectifier IK = gKn4(V
VK) has gK = 9 mS/cm2 and it activates with kinetics given by
n = 0.05(V + 34)/[1
exp(
(V + 34)/10)] and
n = 0.625 exp(
(V + 44)/80). The leakage current
IL = gL(V
VL) is a passive channel with conductance gL = 0.1025 ± 0.0025 mS/cm2. The reversal potentials are
VL =
63.8 ± 0.15 mV,
VNa = 55 mV, and
VK =
90 mV.
Model pyramidal neurons set according to these parameters fire at an average of 22 Hz when they are injected a depolarizing current of 0.25 nA for 0.5 s. The firing pattern corresponds to a regular spiking neuron with some adaptation, no bursting pattern was ever observed. In contrast, a model interneuron fires at ~75 Hz when equally stimulated and has the firing pattern of a fast spiking neuron.
Model synapses
Kinetics of synaptic currents is modeled as in (Wang
1999b
): a postsynaptic current Isyn = gsyns(V
Vsyn) enters the postsynaptic neuron when the
presynaptic neuron's action potential activates the gating variable
s(t) following ds/dt =
f(Vpre)
s/
,
with f(Vpre) = 1/[1 + exp(
(Vpre
20)/2)]. For AMPAR-mediated
synaptic transmission,
= 3.48,
= 2 ms, and
Vsyn = 0; while for inhibitory synaptic
transmission
= 1,
= 10 ms, and
Vsyn =
70 mV. In the case of
NMDAR-mediated synaptic transmission, the gating variable follows a
second-order kinetic scheme: ds/dt =
(1
s)
s/
, dx/dt =
xf(Vpre)
x/
x (
= 0.5,
= 100 ms,
x = 3.48,
x = 2 ms, Vsyn = 0) so that the ensuing
excitatory postsynaptic current (EPSC) has a slower rise phase and
saturates at high presynaptic firing rates.
Unless specified otherwise, the synaptic conductances' maximal
strengths are set to the following values: pyramidal to pyramidal: g







Cortical microcircuit connectivity
The neurons in the network are sparsely connected to each other
through a fixed number of connections that are set at the beginning of
the simulation. Neurons make 20 ± 5 (SD) contacts to their
postsynaptic partners (multiple contacts onto the same target, but no
autapses, are allowed). For each pair of neurons separated by a
distance x in the network (see Fig. 2A), the
probability that they are connected in each direction is decided by a
Gaussian probability distribution centered at 0 and with a prescribed
standard deviation
: P(x) = exp(
x2/2
2)/
= 250 µm for excitatory connections so
that the typical size of a patch of connections coming from a single
pyramidal neuron is 500 µm. This number is consistent with anatomical
(Rockland 1985
) data for local connections within ferret
visual cortex. Anatomical studies have also shown that excitatory
horizontal connections in cortex extend further away creating a
periodic patchy pattern (Gilbert and Wiesel 1983
; Rockland 1985
). In some simulations, we include this by
using a probability of connection given by
P(x) = [exp(
x2/2
2) + s exp(
(x
d)2/2
2) + s
exp(
(x + d)2/2
2)]/(1 + 2s)/
= 125 µm, except for simulations in Fig. 12, where
= 500 µm is used. Anatomical and physiological
data indicate that axonal arbors from inhibitory (basket) cells vary considerably, ranging from narrow to widespread (Crook et
al. 1998
). Here we mostly work with the narrow inhibition
architecture but we also briefly explore the case of broader inhibition
(Fig. 12).
Robustness of the model
The model network that we present here has proven its robustness
to parameter modification in a variety of tests. The model is robust to
connectivity sparsity and randomness and to neuronal inhomogeneity.
Furthermore, a certain amount of randomness and heterogeneity seems to
confer more stability to smooth wave propagation. Also, intrinsic
neuronal properties can be varied substantially without changing the
essential propagation and oscillation properties. Using a network of
integrate-and-fire neurons, instead of Hodgkin-Huxley neurons, shows
similar network dynamics (see van Vreeswijk and Hansel
2001
; Wang 1999b
for non-traveling slow
oscillations), but the model cannot reproduce experimental
intracellular data quantitatively. However, as indicated by
Goldman et al. (2001)
in a different context, there are
some intrinsic parameters that do affect the collective network
dynamics importantly. The reversal leak potential, for instance,
controls very finely the excitability of the neurons, and its mean
value across the population has very marked effects on the frequency of
the slow oscillations. However, the standard deviation with which the
reversal leak potential is distributed across the neuronal population
does not have such an important effect on the activity characteristics.
In particular, increasing threefold the reversal leak potential
standard deviation augments the slow oscillation frequency by just 50%
(from 0.27 to 0.4 Hz), the wave velocity 40% (from 5 to 7 mm/s), and
the overall firing frequency 60% (from 1.1 to 1.8 Hz). Instead, with the same manipulation in a network with blocked excitatory synaptic transmission, the average spontaneous rate of firing of the network increases more than fourfold (from 0.06 to 0.26 Hz) and the number of
spontaneously active neurons more than doubles (from 12 to 30%). This
implies that the model is robust with respect to the exact fraction of
neurons spontaneously active in the absence of synaptic excitation, but
it is more sensitive to the overall excitability level of the network.
As for neuron number, our simulations show that doubling or halving the
number of neurons in our model network does not change either the
oscillation frequency, or the wave propagation velocity, or the average
firing rate. This is so because the connectivity of the model is
unaffected by the number of neurons. Any given neuron connects to an
average of 20 postsynaptic cells independently of the size of the network.
Numerical methods
The model was implemented in a C++ code and simulated using a forth-order Runge-Kutta method with a time step of 0.06 ms.
Experimental methods
Experimental data depicted in Fig. 1 was collected
extracellularly in prefrontal cortex slices of the ferret. Details of
the methods can be found in Sanchez-Vives and McCormick,
2000
.
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RESULTS |
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Excitatory synaptic block reveals spontaneous neuronal firing
Extracellular multiple unit recordings in layer V revealed the
basic characteristics of the slow oscillation in vitro, including the
recurrence of synchronized bursts of activity in neighboring neurons,
and the presence of spontaneous activity between up states (Fig. 1,
B-D). The rate of firing of
this multiple unit activity typically decreased following an up state
but increased prior to the onset of the next up state (Fig. 1,
B-D). At least some of this spontaneous activity was not
dependent upon fast glutamatergic excitation because it survived block
of AMPA and NMDA receptors with bath application of
6-cyano-7-nitroquinoxaline-2,3-dione (CNQX; 20 µM) and
DL-2-amino-5-phosphonovaleric acid (DL-APV; 50 µM; n = 6 slices). Block of glutamatergic excitatory
postsynaptic potentials (EPSPs) resulted in both a block of the up
state (see Sanchez-Vives and McCormick 2000
) and a
significant reduction in spontaneous activity during the down state.
However, in 4 of 10 recorded layer V sites in CNQX/APV, significant
spontaneous activity remained (Fig. 1, B-D), suggesting
that at least some layer V pyramidal cells discharge spontaneously
through intrinsic membrane mechanisms (e.g. see Wang and
McCormick 1992
). We used these and other (Sanchez-Vives
and McCormick 2000
) features of the slow oscillation as guide
lines in the generation of our model of this activity.
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Slow oscillation and wave propagation of network activity
In accordance with the experimental observations of
Sanchez-Vives and McCormick (2000)
, neurons in our model
show spontaneous activity as repetitive episodes of low-rate neuronal
firing, separated by long-lasting silences of ~2.5 s (Fig.
2). The oscillation frequency is thus
~0.4 Hz. Activity patterns are organized spatially as synchronous
waves that propagate from one site to neighboring sites, thus
recruiting the whole network at each firing episode of the slow
oscillation (Fig. 2B). Typically, the most active part of
the network initiates the discharge at each cycle (in Fig.
2B, 3rd row from bottom), but the initiation site is not unique and varies from simulation trial to trial.
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The membrane potential of modeled pyramidal neurons, as also seen in experiments, undergoes transitions between more depolarized states with higher spiking activity (up states) and more hyperpolarized states with virtually no spike discharges (down states; Fig. 2C). Slow oscillation of the membrane potential occurs in parallel with waxing and waning of the intracellular sodium concentration: [Na+]i accumulates slowly due to spike-triggered sodium influx during the up state and decays by an extrusion process to the extracellular medium during the down state (Fig. 2, C and D). We will show that the slow [Na+]i dynamics is critical to the generation of slow oscillations (see following text).
Because neurons are not identical in the heterogeneous network model, the membrane potential of pyramidal cells shows quantitatively different firing patterns. Some (highly excitable) cells show spike firing prior to the onset of an up state, and a relatively small after-hyperpolarization in the down state (Fig. 2, C and D, top). Other less excitable cells do not show spiking during the down state and have a pronounced and slowly recovering afterhyperpolarization following the firing discharge in the up state (Fig. 2, C and D, middle). And finally, a subpopulation of (the least excitable) pyramidal neurons exhibit clearly defined voltage up states and down states separated by ~10 mV (Fig. 2, C and D, bottom). These differences arise from the random initialization of intrinsic properties for each cell, and, therefore, other neurons show intermediate behaviors between these three characteristic examples.
Propagating discharges can also be evoked by local external
stimulation, for example by a brief depolarizing current injection to a
subpopulation of 50 neurons (indicated by arrows in Fig. 3). Colliding waves usually merge and
extinguish (3rd and 4th waves in Fig. 3A). The wave nature
can be more clearly revealed when spontaneous activity is absent if all
pyramidal cells are slightly hyperpolarized so that they are unable to
trigger any network event by themselves. In this case, briefly
stimulating by current injection a restricted area of the network
triggers a discharge episode that travels across the network as a wave front (Fig. 3B, left) and merges with waves traveling in the
opposite direction (Fig. 3B, right). Furthermore,
stimulation immediately after an evoked discharge is unable to recruit
the network for a renewed propagation (Fig. 3B, left, 2nd
stimulation arrow). Experiments on the slice have yielded very similar
results: external stimulation during the hyperpolarized phase of the
oscillation generated a wave that propagated across the slice
(Sanchez-Vives and McCormick 2000
), and the slice was
refractory immediately following one of these network events.
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Model pyramidal neurons and interneurons fire in phase during the slow oscillation (Fig. 3, A and B). When pyramidal cells and interneurons at a given spatial location are closely examined, we found that interneurons typically discharge first in response to the arrival of the wave front (blue rim around activity in B, rightward shift in peak of histogram in C, and shift in average activity of pyramidal cells and interneurons in D). Pyramidal neurons have an average maximal rate of ~10 Hz during the network discharge while interneurons fire at ~20 Hz (Fig. 3D).
Change of input resistance during the slow oscillation
In Fig. 5 are plotted the time courses of synaptic and intrinsic
ionic conductances in a pyramidal neuron during the slow oscillation.
In particular, the input resistance of pyramidal neurons (here computed
as the inverse of all open conductances across the membrane, see Fig.
4) is the highest during the down state,
reaching its maximal value right before the onset of the discharge
episode. The input resistance is at its minimum during the initial
phase of the up state and increases gradually in the course of the up
state (Fig. 5B). This is in agreement with experimental recordings during the occurrence of the slow oscillation in cortical cells of the anesthetized cat in vivo (Contreras et al.
1996
). On the basis of this observation, Contreras et
al. (1996)
suggested that a K+ conductance
contributing to spike frequency adaptation cannot be responsible for
the termination of the up state because this would lead to a gradual
reduction, not increase, of the input resistance as the
spike discharge progressed. This argument assumes that the
K+ conductance dominates the input resistance. In our
model, the mechanism terminating the up state is the activation of a
Na+-activated K+ conductance
gKNa (see following text). However, the input
resistance is determined by the sum of all conductances and is
dominated by the synaptic conductances rather than by
gKNa. During an up state, the slow activation of
gKNa produces spike-frequency adaptation; reduced neural firing leads to a decrease of recurrent synaptic conductances and other intrinsic ion conductances, hence an overall increase of the input resistance (Fig. 5, C and
D).
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Balance between synaptic excitation and inhibition
It is apparent in Fig. 5C that the excitatory and
inhibitory synaptic conductances onto pyramidal neurons closely follow
each other during the up state. This is shown more clearly in Fig. 6A, where the two synaptic
conductances are not plotted against time, but against each other for
seven different neurons (middle), or averaged across 32 neurons equally spaced in the network (right). This plot
yields a linear relationship, which means that the excitatory and
inhibitory synaptic conductances increase and decrease in time in such
a way that the ratio of the two
gexc:ginh remains fixed.
This ratio varies from neuron to neuron, ranging from 1:1 to 1:5. On
average, inhibitory synaptic conductances are typically four times
larger than excitatory synaptic conductances,
gexc:ginh
4. Assuming that the average potential in the up state is about
V
=
55 mV, the driving force for the excitatory
current (with a reversal potential of Vexc = 0 mV) is (Vexc
V
)
55 mV, whereas that of the inhibitory
current (with a reversal potential Vinh =
70 mV) is (
V
Vinh)
15 mV. Hence, the driving force for excitation is approximately
four times larger than for inhibition. Therefore, an excitatory
conductance four times smaller than the inhibitory conductance will
yield comparable excitatory and inhibitory postsynaptic currents. In
other words, there is an approximate balance between synaptic
excitation and inhibition that is preserved over time throughout the up
state, when pyramidal neural firing is sustained at relatively low
rates (10 Hz).
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To establish whether this excitation-inhibition balance is the result of a particularly well-tuned parameter choice or is an emergent property of the network, we drew the same graphs for network simulations where one synaptic conductance had been modified (Fig. 6, B and C). We observe that in all cases, the approximately linear relationship between excitatory and inhibitory conductances remains, but the proportionality ratio varies. It also becomes clear that the relationship is not exactly linear but follows a closed cycle, with excitatory conductances above (below) the average linear relationship during the down-up (up-down) transition (see Fig. 6C, right).
The exact ratio of excitatory to inhibitory conductances is likely to
depend on other parameters like VL. Indeed, in a
much simplified model including only leak and synaptic currents, the membrane equation is
CmdV/dt =
gL(V
VL)
gE(V
VE)
gI(V
VI). Assume that VE = 0, VL = VI =
75 mV, and gL = 20 nS. The steady state
of the voltage is given by Vss = (gL + gI)VL/(gL + gI + gE). Then,
the membrane potential could stabilize around its threshold even with
an excitation-inhibition ratio of
gE:gI = 1(Vss ~
56 mV, if
gE = gI = 10 nS). However, the issue of whether such a model could maintain its
stability throughout a recurrently generated up state does not seem
trivial because synaptic conductance increases proportional to the
network activity could lead to reverberatory instability.
Mechanisms of the slow oscillation in the model
The basic mechanism for the emergence of the oscillatory activity in the network model is the interplay between neuronal spontaneous firing amplified by recurrent excitation, and a negative feedback due to slow activity-dependent K+ currents. These positive and negative feedback processes operate along the lines illustrated in Fig 7: because the reversal potential of the leakage current is distributed randomly (see METHODS), some neurons are spontaneously active at very low rates (0.6 ± 0.2 Hz). Occasionally, a sufficient number of closely adjacent spontaneously active neurons fire together, thus triggering a cascade of recurrent excitation that locally brings the network into the firing regime of the oscillation (up state). At that point, activity-dependent K+ currents (notably the Na+-dependent K+ current) start accumulating in active pyramidal neurons, reducing their excitability. The decremental excitability of pyramidal neurons eventually makes the network recurrence unable to sustain the firing state, and the local network reverts to the silent state via a slow afterhyperpolarization. The decay time of the currents responsible for this afterhyperpolarization sets the time scale for the reappearance of spontaneous firing and determines the periodicity of the oscillatory cycle.
|
According to this scheme, recurrent excitation should be responsible for quickly bringing neurons into the firing state in a collective manner, and the activity-dependent slow IKNa eventually leads to a transition back to the down state. More specifically, we hypothesized that recurrent synaptic excitation produces a network bistability with an active up state and an inactive down state and that the slow kinetics of IKNa drives the network to switch back and forth between these two states (see Fig. 8). We tested this prediction by substituting the time-varying IKNa current with a constant hyperpolarizing current of varying intensities simultaneously in all model neurons. We found that the network indeed exhibits two stable dynamic states: the silent state with large hyperpolarization and the persistent firing state with low hyperpolarization. There is a range of current intensity over which these two states coexist (bistability; Fig. 8, A and B). In the control network simulation, the role of the injected hyperpolarizing current of Fig. 8 is fulfilled by the time-varying IKNa, whose slow dynamics causes the neurons to cyclically trace this hysteresis loop (perimeter of shaded area in Fig. 8B): as IKNa progressively builds up in the firing state (upper solid line in Fig. 8B), pyramidal neurons experience an increasing hyperpolarizing current. Eventually, point A in Fig. 8B is reached and neurons in the network collectively and sharply fall into the silent state because recurrent excitation is no longer sufficient to sustain their firing. Then neurons remain silent while IKNa recovers slowly and the network regains the level of excitability (point B in Fig. 8B) where the silent state becomes unstable again and a collective sharp transition to the up state is generated. Thus the cycle goes on indefinitely, driven by the slow kinetics of IKNa. Transitions between up and down states are sharp because of the sudden loss of network stability at a given degree of neuronal excitability.
|
To further test the importance of feedback excitation to the bistability phenomenon, we reduced the maximum conductance of the recurrent excitatory synapses to pyramidal neurons by 20%. The result is a loss of the bistable range (Fig. 8C). Now the network is either silent or, upon progressive depolarization of pyramidal neurons, it becomes spontaneously active. There is, however, no range of input currents for which both a quiescent state and a persistent firing state are simultaneously stable.
The mechanism for adaptation-induced network oscillations has been
mathematically analyzed by van Vreeswijk and Hansel
(2001)
(see also Fuhrmann et al. 2002
;
Wang 1999b
).
Pharmacological manipulations
To compare with the experimental results of Sanchez-Vives
and McCormick (2000)
, we explored the effect of various
synaptic receptor blockers on our slowly oscillating cortical network
(Fig. 9). Blocking AMPAR-mediated
transmission abolishes this rhythmic activity completely, and a
subpopulation of excitable neurons fire spontaneously without any
apparent network amplification. Similarly, the slow oscillation
disappears under NMDARs blockade but this disruption is not as dramatic
as when AMPARs are blocked, and occasional bursts of activity are able
to recruit neighboring neurons into a network event via AMPAR-mediated
transmission. On the other hand, blocking GABAARs results
in a more exuberant discharge that propagates at a much faster velocity
across the cortical network. Also, the time intervals between burst
episodes are significantly prolonged, i.e. the oscillation period is
longer. All these behaviors are very similar to the results of
Sanchez-Vives and McCormick (2000)
after bath
application of various synaptic receptor blockers in the slice
preparation (compare the Fig. 9, left and right).
|
Velocity of wave propagation
Figure 9 shows that the network can sustain two different types of
propagating waves depending on whether intracortical inhibition is
functional (control case) or the network is disinhibited
(GABAAR block). Notably, these two waves have very
different propagation speeds, as in slice experiments (Golomb
and Amitai 1997
; Sanchez-Vives and McCormick
2000
; Wu et al. 2001
). Both in the model and in the experiment, the propagation of the disinhibited wave is almost an
order of magnitude faster than that of the wave in control conditions.
The velocity of the disinhibited wave is largely determined by the
efficacy of AMPAR-mediated excitatory synaptic transmission (Fig.
10A), consistent with
previous analytical and simulation results (Ermentrout
1998a
; Golomb and Amitai 1997
). The dependence of the propagation speed on the strength of feedback inhibition is
plotted in Fig. 10B. Either an increase of the inhibitory
conductance onto excitatory cells or of the excitatory conductances
onto inhibitory cells gradually decreases the wave propagation speed,
according to a smooth sigmoid function (fitted curve).
|
Although the model reproduces roughly a 10-fold increase in the wave
speed with inhibition blockade, a closer examination revealed that the
absolute wave speeds in the model are significantly off the
experimentally measured values. In control conditions, the velocity is
<3 mm/s, whereas in the experiments it was ~10 mm/s; with inhibition
blockade, the wave propagation is <20 mm/s, when the experiment
yielded a value ~80 mm/s. We have explored possible solutions to this
discrepancy. Figure 10A shows that with sufficiently strong
E-to-E coupling, a wave velocity of ~100 mm/s could be achieved in
the disinhibited network. This suggests that one could first increase
the strength of recurrent AMPA-mediated excitatory synapses to achieve
the desired velocity in the disinhibited network; then, by gradually
increasing the inhibitory feedback (as in Fig. 10B), slower
propagation at a desired velocity could be obtained. However, this is
not the case (Fig. 10C): with much stronger excitatory
feedback (4-fold E-to-E AMPA conductances), increasing inhibition in a
network does not lead to the slower propagation mode observed in the
experiment. Instead, when the feedback inhibition is above a threshold
value, the wave phenomenon disappears and the network shows a spatially
uniform (unstructured) tonic firing state (shaded areas in Fig.
10C). Below this threshold but near it (inhibitory synapses
are still very strong) a complex activity pattern emerges (see
illustrative rastergram in Fig. 10D), in which activity
propagates at two different velocities. The rastergram in a small time
window reveals that activity spreads quickly, as in the disinhibited
case ("fast wave" indicated in Fig. 10D). Examination at
a longer timescale reveals also a slower propagation, which is
nevertheless not a smooth wavefront ("slow propagation
mode" indicated in Fig. 10D). The global network pattern is clearly not comparable to the one observed in the slice experiment of Sanchez-Vives and McCormick (2000)
.
Another possibility is to increase the spatial extent (the
"footprint"
EE) of excitatory connectivity because
the wave velocity increases proportionally with
EE
(Ermentrout 1998b
). However, to achieve the
experimentally comparable wave velocities, one would have to increase
EE of the E-to-E coupling by three- to fourfold, from
250 µm (see METHODS) to ~2 mm, which would be
inconsistent with the anatomical and physiological data. We examined
the more plausible scenario that weak but long-range patchy horizontal connections could increase the propagation speeds significantly. This
pattern of connection is prominent in the mammalian cortex (Gilbert and Wiesel 1983
; Rockland 1985
),
and it has been observed in ferret visual cortex in both supragranular
and infragranular layers (Rockland 1985
). Typically,
horseradish peroxidase injection in a restricted area of cortex results
in orthograde striped staining of intracortical connections with stripe
width of 250 µm and center-to-center distance 0.5-0.7 mm
(Rockland 1985
). We model this kind of connectivity as
depicted in Fig. 11A
(right). When we use this type of connectivity in our model,
discharges propagate much faster in both conditions (Fig. 11A,
right compare with left for non-patchy connectivity). For the wave in the disinhibited network, the velocity increased to
>50 mm/s; whereas for the network with functional inhibition, the
propagating wave has a speed close to 10 mm/s. On the other hand, the
inclusion of long-range excitatory connections does not lead to a
significant increase of the pyramidal neural activity, the firing rates
are comparable to the situation without patchy excitatory connections.
Both velocities depend markedly on the spatial extent and strength of
the patchy horizontal connections (Fig. 11,C and
D). Figure 11C shows how the center-to-center
distance for the patchy horizontal connections influences the
propagation speeds. The more separated the patches are spatially, the
faster the wave propagates both in the control and disinhibited
conditions. The firing rates of pyramidal cells decreased slightly as
the waves became faster. In Fig. 11D, we show the dependency
of the wave velocities on the strength of the lateral patches. The
stronger the patched connections, the faster the wave. In this case,
the firing rates grew slightly with the increase in excitatory patch strength.
|
Relative spatial range of excitatory and inhibitory projections
A still unresolved issue in functional cortical architecture is
whether cortical inhibition has a larger or smaller spatial range than
intracortical excitation. The spatial extent of a specific (e.g.
interneuron-to-pyramid) connection depends on the convolution of the
axonal projection from the presynaptic cells and the dendritic extent
of the postsynaptic cells. Judging from the anatomical estimates of the
axonal and dendritic spreads of cortical interneurons and pyramidal
neurons (Lund and Wu 1997
), one could argue that inhibition from most interneurons is likely to act more locally than
excitation. However, there are some classes of interneurons (e.g. a
subset of cortical basket cells) with far-reaching axons (Buzás et al. 2001
; Lund and Wu
1997
) that could be functionally very powerful. Indeed,
functional monosynaptic long-range inhibitory connections have been
recorded electrophysiologically (Crook et al. 1998
). We
used our model to address whether one or the other scenario is more
appropriate to reproduce the slow oscillatory activity observed in the
cortex. To do that, we re-ran the network simulations with a broadened
spatial width (footprint) of the inhibitory connection probability
(from
I = 125 µm to
I = 500 µm), while keeping the footprint for excitation at
E = 250 µm. We still find very robust slow
oscillations; the wave propagation velocity is ~5 mm/s (not shown).
Is there a way to distinguish these two cases
(
I/
E > or < 1)
experimentally? We focus on the temporal relationship between excitatory and inhibitory events as a possible discriminative test
(Fig. 12). In both cases
(
I = 125, 500 µm), interneurons tend to fire
ahead of nearby pyramidal cells at the onset of the propagating
discharge (Fig. 12, A and B). The relative timing
of spike discharges in pyramidal cells and interneurons is therefore not a useful test for the functional significance of long-range inhibition. If the spread of the inhibitory axons is greater than the
excitatory ones, then inhibitory synaptic conductances increase first
in pyramidal neurons as the up state propagates through the network
(Fig. 12, C and D). This is also shown in Fig.
12G, where simultaneous PSCs onto a pyramidal cell are shown
under voltage clamp. If the spread of inhibitory connections is
narrower than excitation (left), then the EPSCs and
inhibitory PSCs (IPSCs) arrive to the pyramidal cells at approximately
the same time. This occurs because, locally, interneurons fire in
advance of excitatory cells. If the inhibitory connections are broader
than excitation, then IPSCs precede the arrival of EPSCs
(right). On the other hand, without voltage-clamp, during
network oscillations, the synaptic currents that the neuron integrates
in the course of the slow oscillation depend on the time-varying
postsynaptic membrane potential Isyn = gsyn(V
Vsyn). Figure 12, E and F,
shows that this excitatory and inhibitory currents show similar time courses in both cases, with EPSCs slightly leading ahead of IPSCs. This
is because initially the voltage is close to the inhibitory reversal
potential, inhibition is purely shunting, and IPSCs become significant
only after the cell is depolarized by EPSCs. As a result, PSPs measured
in current clamp cannot be used to discriminate between long- and
short-range inhibition unless the neuron is significantly depolarized
away from the reversal potential of IPSPs.
|
These data suggest that the timing of EPSPs and IPSPs in intracellular recordings in slices may give clues as to the functional spread of inhibition and excitation in cortical networks. Recent experimental observations with intracellular recordings in layer V pyramidal cells during the generation of the slow oscillation in vitro demonstrate that the synaptic barrages are initially weighted toward excitation with a balance of excitation and inhibition being achieved over ~50-100 ms (Y. Shu, A. Hasenstaub, and D. A. McCormick, unpublished observations). In the view of the current model results, this would suggest that, functionally, the spread of inhibition has a narrower extent than that of excitation in the cortex.
Relation to in vivo slow oscillations under anesthesia
Slow oscillations observed in vivo can have distinct features
depending on the particular experimental protocol. In contrast to the
in vitro condition, up states can be longer than down states (Contreras et al. 1996
; Cowan and Wilson
1994
; Massimini and Amzica 2001
; Stern et
al. 1997
; Timofeev et al. 2001
), but in other
cases the opposite occurs (Steriade et al.
1993b
,c
). In
some cases, regular and robust slow oscillations are observed
(Paré et al. 1998
; Steriade et al.
1993b
), whereas in other instances transitions between up and
down states are irregular and stochastic (Cowan and Wilson
1994
; Lampl et al. 1999
; Paré et
al. 1998
; Stern et al. 1997
). These marked
differences between spontaneous cortical activity in vivo as seen by
different laboratories may be due to the particular anesthetics
protocol (kind and depth of anesthetics) used (Lampl et al.
1999
). Since anesthetic agents often affect intrinsic neuronal
excitability and synaptic transmission (Nicoll et al.
1990
; Schulz and Macdonald 1981
), we explored
how a change in GABA or NMDA-mediated transmission or in the neuronal
baseline excitability may affect spontaneous activity in our model
network (Fig. 13). Neuronal intrinsic
excitability was varied by modifying the leakage membrane permeability
to K+ ions (in the model this affects both the leak
conductance gL and the leak potential
VL). As leakage K+ currents are
progressively reduced, we observe that a sample neuron in the model
goes from very stereotyped pronounced slow oscillations with short up
states and long down states (Fig. 13C) to fluctuations
between a down state and an up state of more irregular durations (Fig.
13, center) and eventually to long sojourns in the up state and only
occasional and brief incursions into the down state (Fig.
13D). Notice that Fig. 13C resembles the very
rhythmic recordings by Steriade et al.
(1993b
,c
); the
center panel in Fig. 13 is reminiscent of the irregular traces in
Cowan and Wilson (1994)
and in Stern et al.
(1997)
; and Fig. 13D simulates closely the data
dominated by tonic-firing of Lampl et al. (1999)
,
Contreras et al. (1996)
, and Timofeev et al.
(2001)
. Synaptic modifications associated with anesthetic
action on GABA channels could also induce changes in the pattern of
network activity: enhanced inhibition made the slow oscillation more
irregular (Fig. 13B) while reduced inhibition turned it more
regular and faster (Fig. 13A). In contrast, NMDAR-mediated
transmission modulation did not produce significant change in the
oscillatory pattern of the network (Fig. 13 E and F). It may thus appear that the combined action of
barbiturates via reduced neuronal intrinsic excitability and via the
augmentation of GABA responses could compensate each other and have
little effect on the pattern of slowly oscillating activity. Our
simulations, however, show that reduced intrinsic excitability
determines the overall activity pattern even when GABA responses are
simultaneously enhanced (data not shown).
|
Therefore our model suggests that anesthetics-dependent changes in the network excitability, most notably through neuronal intrinsic excitability, may explain the various slow oscillating patterns, such as the durations of the up and down states and the statistics of transitions between the two states observed under different anesthesia protocols in vivo.
Transition to the tonic-firing state of vigilance
In unanesthetized cats, local field potential during quiet sleep
shows slow oscillations (Timofeev et al. 2001
); whereas
the waking state is characterized by tonic-firing of spikes
(Evarts 1964
; Hubel 1959
; Steriade
et al. 1974
) and fast oscillations in the local field
potentials (Destexhe et al. 1999
; Steriade et al.
1996
; Timofeev et al. 2001
). We used our network
model, calibrated to reproduce the slice results, to simulate the
transition from the slow wave sleep to the waking state with neuronal
activity patterns observed from unanesthetized animals. It is well
established that the transition from sleep to wakefulness depends
critically on the activation of ascendent activating systems
(especially cholinergic, noradrenergic, and serotoninergic), notably
through an increased cholinergic input to the cerebral cortex
(McCormick 1992
; Steriade and McCarley
1990
). We focused on manipulations that have been linked to
neuromodulatory action by acetylcholine (see DISCUSSION):
reduction of passive, activity- and voltage-dependent K+
conductances, and reduced excitatory transmission (via presynaptic inhibition, see DISCUSSION).
We have considered various manipulations, each in isolation, to see
whether any of them was in itself sufficient to simulate the transition
to waking in terms of voltage traces and input resistance. We found
that when Na+-dependent K+ conductance alone
was significantly reduced, the network reverted to activity with less
marked periodicity, longer up states and shorter down states (see Fig.
14A, middle). Further
reduction of this adaptation current led the network to a tonic firing
mode with no global spatiotemporal coherence (but still some weak
faster oscillations are to be seen in the local field potential at ca. 2-3 Hz), the voltage traces are thus reminiscent of those
characteristic of the "waking state" (compare to data in
Steriade et al. 2001
; Timofeev et al.
2001
). Other aspects, however, indicate that adaptation current
reduction is not the only ingredient of neuromodulatory action
important for the "waking up" of the circuit. In particular, the
overall firing rate of the network increases in the process (from 1 Hz
in the top and 5 Hz in the middle panel, to 9 Hz in the
bottom of Fig. 14A), and the input resistance of
the neuron is significantly reduced. This is in contrast to
experimental observations (Steriade et al. 2001
) that
indicate that the overall firing rate of neurons remains approximately
the same between the natural slow oscillation and the waking state,
even though during the up state of the slow oscillation the apparent
input resistance is lower.
|
On the other hand, the reduction of leakage K+ permeability (reduced gL and increased VL) in pyramidal neurons also affects the transition from a slowly oscillating network into tonically firing neurons (Fig. 14B). With this mechanism, as also observed when reducing IKNa, the overall firing rate also increases considerably (from 1 Hz in the top and 4 Hz in the middle to 8 Hz in the bottom), and the input resistance decreases (presumably due to enhanced synaptic conductance), contrary to the experimental evidence. The last mechanism that we examined is the reduction of intracortical excitation to pyramidal neurons (not shown). The effect is also to transform a slowly oscillating network into tonic activity, the difference with the previous mechanisms is that this tonic firing is now extremely sparse at very low rates (0.2 Hz) and does not look comparable to intracellular recordings during the waking state. Trivially, however, input resistance increases because of the practical absence of synaptic activity.
When these effects of the neuromodulatory action are combined in the
network model (Fig. 15), the reduction
of recurrent excitatory synaptic conductances leads to an increase in
the input resistance, whereas the decreased excitation is compensated
by the reduction of K+ currents so that the firing rate in
the tonic state remains comparable to that in the up state of the slow
oscillation mode. Therefore a transition from the slow oscillation
state characteristic of natural sleep to the tonic firing state of the
waking state is realized, with neural activity and input resistance
changes consistent with the experimental observations (Steriade
et al. 2001
).
|
| |
DISCUSSION |
|---|
|
|
|---|
Cellular and network mechanisms of the slow (<1 Hz) oscillation
In this paper, we present a model that reproduces the slow
rhythmic activity (<1 Hz) observed in vitro by Sanchez-Vives
and McCormick (2000)
and in vivo by Steriade et al.
(1993b)
. The network oscillates between an up state of
sustained but low-rate (~10 Hz) neural activity and a down state of
membrane hyperpolarization. The onset and maintenance of the active up
state is due to an amplification of spontaneous activity by powerful
recurrent excitation, whereas the transition back to the down state is
controlled by a slow negative feedback, the Na+-dependent
K+ current IKNa. We provide
experimental evidence that when excitatory glutamatergic transmissions
mediated by AMPA and NMDA receptors were blocked, a subset of layer 5 neurons still displayed significant spontaneous activities. Our model
suggests that such spike discharges of single neurons are sufficient to
trigger waves of activity that propagate across the cortical tissue.
This initiation mechanism is in contrast to that of spike-independent
spontaneous synaptic transmission proposed by Timofeev et al.
(2000a)
.
Our results are in support of the hypothesis that the up state is
sustained by local reverberatory circuits in the cortex (Contreras et al. 1996
; Metherate and Ashe
1993
; Sanchez-Vives and McCormick 2000
). More
specifically, our model proposes that this oscillatory activity is the
signature of an intrinsic bistability between the up and down state and
that IKNa induces slow switchings between these
two states. One testable prediction of this scenario is that if
IKNa could be suppressed, for example by
neuromodulators such as acetylcholine or norepinephrine, the local
cortical network would become bistable, where both the up and down
states would be dynamically stable. In that case, transient stimulation
would set the network into a self-sustained (up) firing state, whereas transient hyperpolarization would revert it back to the silent (down)
state. Moreover, this bistability would depend on strong excitatory
reverberation and should disappear with reduced recurrent excitation.
Finally, in the presence of IKNa, the cortical
slice is predicted to cycle around the bistability loop as depicted in
Fig. 8 during the slow oscillation.
The ability of a cortical module to sustain several stable states of
firing has been proposed as a possible mechanism for working memory in
association cortical circuits, especially the prefrontal cortex
(Compte et al. 2000
; Wang 1999b
,
2001
). Our findings hint at the
potential existence of a bistable dynamics in the primary visual cortex
that could be made available operationally by reducing slow adaptation
currents or enhancing after-depolarizing currents via neuromodulators
(McCormick 1992
). We emphasize that such a bistability
has not yet been demonstrated experimentally and thus represents a
testable prediction of our model. It is also important to note two
major differences between the kind of bistability discussed here and
that required for working memory. First, for a working memory circuit,
a persistent up state should be stimulus-selective and involve only a
subset of neurons (a neural assembly). By contrast, the up state during
slow oscillation is global and recruits the entire network. Second,
bistability of a working memory network would be between a resting
state of spontaneous activity (at a few Hz) and a persistent active
state with elevated firing rates (20-50 Hz). On the other hand, the bistability associated with slow oscillation is between a down state of membrane hyperpolarization and an up state of relative low
firing rates (at <10 Hz). As the network is transformed from the slow
oscillation mode to the tonic-firing mode, the persistent up state is
similar to the spontaneous state of wakefulness (at
10 Hz)
(Destexhe et al. 1999
; Steriade et al.
1996
), rather than higher-rate mnemonic activity states.
Input resistance and neuronal versus synaptic adaptation
Contreras et al. (1996)
found that during the slow
oscillation, the input resistance is at its lowest of the cycle at the beginning of the depolarized state and then it increases until it
reaches its maximum at the end of the hyperpolarized phase of the
oscillation. This was interpreted as evidence that the primary cause
for the sudden transition to the silent down state was the depression
or "disfacilitation" of an excitatory synaptic current as opposed
to the accumulation of an activity-dependent hyperpolarizing current.
The short-term synaptic depression mechanism has been implemented in a
network model of the slow oscillations as they occur in deafferented
cortical slabs (Bazhenov et al. 2002
; Timofeev et
al. 2000a
). Presumably, that model is also endowed with an
intrinsic bistability, and the network cycles around the hysteresis
loop by the slow dynamics of synaptic depression in a similar manner to
that described in Tabak et al. (2000)
.
We have built our model based on the experimental evidence that a
Na+-dependent K+ current plays a critical role
in the slow oscillation in vitro (Sanchez-Vives and McCormick
2000
). Our model shows that IKNa can be
responsible for the termination of the up state firing, whereas the
input resistance has the same time course as observed experimentally.
This is because an increase in IKNa results in decreased firing and reduced synaptic and voltage-dependent
conductances. Whether the input resistance increases or decreases will
then depend on the particular balance between the opening of the
adaptation conductances and the closing of synaptic conductances
(resulting from reduced activity in the network). Our model suggests
that the increase of input resistance in the course of the up state is
indeed primarily due to a decrease of the synaptic conductances, but
this decrease could arise as an indirect consequence of intrinsic spike-frequency adaptation rather than a direct synaptic depression. Moreover, the inhibitory conductance gI shows a
much larger change than the excitatory conductance
gE, hence dominates the overall input resistance.
Interestingly, during slow oscillations, we found that
gE and gI co-vary in such
a way that the ratio
gI/gE remains roughly fixed in time, so that the two synaptic currents (conductance times the
driving force) are about the same. This dynamic balance of synaptic
excitation and inhibition, as well as intrinsic potassium currents
(such as the low-threshold IKS)
(Nisenbaum et al. 1994
; Wilson 1992
),
contribute to the control of low firing rates (
10 Hz) in the up state
and avoid the generation of runaway excitation.
Propagation of neural activity and the wave velocity
A number of previous theoretical works have been devoted to the
understanding of the velocity of wave propagation in a one-population of pyramidal neurons, corresponding to slice preparations with inhibition blockade (Bressloff 2000
; Ermentrout
1998b
; Osan et al. 2002
). In particular, this
topic was experimentally surveyed and theoretically modeled by
Golomb and Amitai (1997)
. They found that discharges
propagated as traveling pulses in the disinhibited slice, propagation
was possible only above a threshold value for the AMPAR conductance
(g

More recently, Golomb and Ermentrout (2001
,
2002
) extended the
analysis to a model with two (excitatory and inhibitory) neural populations of leaky integrate-and-fire neurons. They found that two
wave propagation modes existed with very different speeds, and they
predicted that in the slow wave inhibitory neuron firing would precede
the pyramidal cell discharge. This is precisely what our more detailed
model shows, thus confirming their prediction. Interestingly, within
the range of parameters that we examined, interneurons always fire in
advance of pyramidal cells at the onset of an up state, regardless of
whether the inhibitory projection is narrower or broader than
excitation. This prediction could be checked experimentally by
comparing the firing times of intracellularly recorded pairs of a
pyramidal cell and an interneuron at the same recording site or by
comparing the discharge of pyramidal cells and interneurons at the same
site to a common reference, such as multiple unit or local field
potential activity. Golomb and Ermentrout (2001)
also
found that the network could be in a regime where both modes of
propagation occurred, depending on the way in which the network was
stimulated. We did not explicitly find this situation in our network.
However, it is conceivable that this behavior could be observed with
the appropriate parameter modification.
In Golomb and Ermentrout (2001
, 2002
), the propagation of a wavefront was
considered with the constraint that each cell is allowed to fire only
one spike. By contrast, our model reproduces both the wave propagation
of the onset of a long-lasting up active state and the slow oscillation
between the up state and down state. We found that with anatomically
estimated spatial ranges of synaptic connections, it is difficult to
reproduce quantitatively the measured velocities of wave propagation.
Typically the wave speeds in the model are three- and fourfold lower
than the experimental data. It is not straightforward to obtain larger
wave speeds by increasing excitatory synaptic conductances because the
latter lead to complex firing patterns and high neural firing rates in
the up state incompatible with the observations. On the other hand, by
introducing weak long-range intracortical excitation, these connections
are effective at recruiting neurons during the firing phase of the
oscillation in such a way that they mediate faster discharge
propagation. Our model predicts a strong relationship between the wave
velocity observed physiologically and the inter-patch distance of
long-range horizontal connections measured anatomically, which can be
tested experimentally. The recruitment of long-range connections during cortical spontaneous activity has also been suggested to operate in
anesthetized preparations in vivo (Tsodyks et al. 1999
).
In our model, whether the inhibitory connections are narrower or broader than the excitatory ones is not crucial for the generation of slow oscillations. In either case, the model can generate slow propagating waves. These two scenarios of intracortical connectivity cannot be distinguished based upon the relative timing of activity in nearby inhibitory and excitatory neurons but only by examining in voltage-clamp the relative timing of EPSCs and IPSCs that arrive in pyramidal cells in the transition to the up state. We predict that inhibition should arrive earlier than excitation in pyramidal cells if long-range inhibition is a significant ingredient of intracortical circuitry. If the cell is at or near the reversal potential for this inhibition, then this should appear as an initial increase in membrane conductance (shunt) without significant change in membrane potential.
Slow oscillations in vitro and in vivo
The spontaneous cortical activity recorded in in vivo preparations
under urethan or ketamine-xylazine anesthesia is not always univocally
identified as slow oscillations. In some instances, membrane voltage
transitions between an up state and a down state can be quite
unpredictable, and they are then classified as a fluctuating rather
than an oscillating pattern (Lampl et al. 1999
; Stern et al. 1997
). The issue of the coherency of the
activity of cortical neurons is also a matter of debate since some
authors find very good correspondence between neuronal up states and
EEG waves (Paré et al. 1998
; Steriade et
al. 1993c
), but other experiments show poor coherence between
individual neurons and EEG (Lampl et al. 1999
). An
obvious question is then how to reconcile all these various
observations with the hypothesis that they are manifestations of one
single phenomenon. Our network simulations provide a hint as to how
these types of behavior could be related to each other. We found that
an irregularly oscillating network (Fig. 13B) turns into a
more periodic oscillation at a lower frequency (Fig. 13A) when the pyramidal neuron intrinsic excitability is reduced (by increasing permeability to K+ ions through leakage
channels). Also the inverse manipulations make the neuron spend more
time in the up state and less in the down state (as in Fig.
13C). One of the anesthetic agents typically used in in vivo
experiments is xylazine, which increases K+ conductances
through
2 noradrenergic receptors (Nicoll et al. 1990
). For anesthetic effective concentrations, barbiturates
also have a hyperpolarizing effect through a GABA-mimetic action
(opening Cl
channels) that occurs in parallel to an
augmentation of GABA responses (Schulz and Macdonald
1981
). Our model suggests that these variations in excitability
and connectivity may explain the variations in the pattern of activity
generated in cortical networks between the in vitro slice, anesthetized
and naturally sleeping animal.
Transition to the tonic firing state of vigilance
In unanesthetized animals, the membrane potential of cortical
pyramidal cells is characterized by transitions between an up and a
down state in slow-wave sleep and a tonically depolarized state with
firing at low rates during waking (when the animal's head is kept
rigid) (Steriade et al. 1996
, 2001
; Timofeev et al. 2001
). The average
firing rate over long periods of time of a neuron is relatively similar
over the sleep-waking cycle, and the input resistance of pyramidal
cells in the waking state is as high as it is for the down state of the
slow oscillation (whereas it is significantly lower for the up state of
the slow oscillation) (Steriade et al. 2001
).
Several neuromodulators are involved in the regulation of the brain's
state of vigilance, including acetylcholine, norepinephrine, and
serotonin (McCormick 1992
; Steriade et al.
1997
). Cholinergic modulation in cortical neurons is known to
reduce several K+ conductances: leak, A-type, M-type, and
Ca2+- and Na+-dependent (Constanti and
Sim 1987
; Foehring et al. 1989
; McCormick 1992
; Schwindt et al. 1989
). Another known
effect of acetylcholine is the reduction via a presynaptic mechanism of
EPSPs both in hippocampus (Hasselmo and Fehlau 2001
;
Seeger and Alzheimer 2001
; Valentino and
Dingledine 1981
) and in neocortex (Kimura and Baughman 1997
; Kimura et al. 1999
; Tsodyks and
Markram 1997
). Kimura et al. (1999)
have
estimated that ACh suppresses
50% of intracortical synaptic
excitation. On the other hand, serotonin also reduces these adaptation
currents while diminishing the size of unitary EPSPs (McCormick
1992
). These neuromodulators are likely to be at low levels in
the in vitro preparation, and in vivo their concentration depends
greatly on the state of vigilance of the brain.
We have performed manipulations to simulate some of the actions of
neuromodulators with our model of a slowly oscillating cortical slice
(see Figs. 15 and 14). We show that changing individual parameters of
the model does not reproduce the most salient neurophysiological differences between the slow-wave sleep and the waking states (Fig.
14). However, concomitant manipulations of adaptation currents, leakage
K+ currents, and recurrent excitation transformed a
regularly oscillating network into a more irregular oscillating
behavior (Fig. 15), closely resembling the slow oscillations observed
during natural slow-wave sleep in unanesthetized cats (Steriade
et al. 1996
; Timofeev et al. 2001
). Enhancement
of the neuromodulator effects eventually brings the network into a
tonic discharge state with no large-scale spatio-temporal coherence
(but notice local coherent inhomogeneities in the multiunit record),
reminiscent of typical cortical activity in the awake state
(Destexhe et al. 1999
; Steriade et al.
1996
). This effect has been recently observed also in a
thalamocortical network model of slow oscillations (Bazhenov et
al. 2002
). Unlike the in vivo observations, though, we find
that in Fig. 15B the oscillations still propagate slowly
across the network, whereas experiments have shown rapid
synchronization of episodes across very long distances (Destexhe
et al. 1999
). A possibility is that the integrity of the
corticothalamic feedback system has a synchronizing action on the
cortically generated slow oscillation (Amzica and Steriade
1995
; Contreras and Steriade 1997
). However, the
recently reported network model of Bazhenov et al.
(2002)
still shows slow wave propagation even in the presence
of thalamocortical influences.
Our simulations were not meant to explore the myriad of known effects
of neurotransmitters in the cerebral cortex. Specifically, the ability
of acetylcholine and other neurotransmitters to enhance calcium-activated cation currents or to modulate activity
differentially in subpopulations of cortical interneurons
(Kawaguchi 1997
, 1998
) may be relevant in the natural transition from the
sleep to waking and remain to be explored.
In summary, our model proposes that a given cortical microcircuit is more reverberatory during sleep than during waking states. Slow oscillations are a manifestation of both strong feedback synaptic excitation and neuronal adaptation during quiet sleep. A concomitant reduction by neuromodulators of recurrent excitatory synaptic transmission and adaptation currents leads to the transition to the tonic-firing state of wakefulness.
| |
ACKNOWLEDGMENTS |
|---|
We thank C. Wilson and E. Stern for helpful discussions.
This work was supported by the National Institutes of Health (NIH), the Alfred P. Sloan Foundation, and the W. M. Keck Foundation to X.-J. Wang and A. Compte; by grants GV00-138-3 (Generalitat Valenciana) and PM98-0102-CO2-01 (Dirección General de Investigación, Spain) to M. V. Sanchez-Vives; and by the NIH to D. A. McCormick.
| |
FOOTNOTES |
|---|
Address for reprint requests: A. Compte, Instituto de Neurosciencias, Universidad Miguel Hernández-CSIC, 03550 San Juan de Alicante, Spain (E-mail: acompte{at}umh.es).
| |
REFERENCES |
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