1Institute of Neurobiology, University of Amsterdam, 1098 SM Amsterdam, The Netherlands; and 2Department of Cell Biology, Duke University Medical Center, Durham, North Carolina 27710
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ABSTRACT |
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Kager, H., W. J. Wadman, and G. G. Somjen. Simulated Seizures and Spreading Depression in a Neuron Model Incorporating Interstitial Space and Ion Concentrations. J. Neurophysiol. 84: 495-512, 2000. Sustained inward currents in neuronal membranes underlie tonic-clonic seizure discharges and spreading depression (SD). It is not known whether these currents flow through abnormally operating physiological ion channels or through pathological pathways that are not normally present. We have now used the NEURON simulating environment of Hines, Moore, and Carnevale to model seizure discharges and SD. The geometry and electrotonic properties of the model neuron conformed to a hippocampal pyramidal cell. Voltage-controlled transient and persistent sodium currents (INa,T and INa,P), potassium currents (IK,DR and IK,A), and N-methyl-D-aspartate (NMDA) receptor-controlled currents (INMDA), were inserted in the appropriate regions of the model cell. The neuron was surrounded by an interstitial space where extracellular potassium and sodium concentration ([K+]o and [Na+]o) could rise or fall. Changes in intra- and extracellular ion concentrations and the resulting shifts in the driving force for ionic currents were continuously computed based on the amount of current flowing through the membrane. A Na-K exchange pump operated to restore ion balances. In addition, extracellular potassium concentration, [K+]o, was also controlled by a "glial" uptake function. Parameters were chosen to resemble experimental data. As long as [K+]o was kept within limits by the activity of the Na-K pump and the "glial" uptake, a depolarizing current pulse applied to the cell soma evoked repetitive firing that ceased when the stimulating current stopped. If, however, [K+]o was allowed to rise, then a brief pulse provoked firing that outlasted the stimulus. At the termination of such a burst, the cell hyperpolarized and then slowly depolarized and another burst erupted without outside intervention. Such "clonic" bursting could continue indefinitely maintained by an interplay of the rise and fall of potassium and sodium concentrations with membrane currents and threshold levels. SD-like depolarization could be produced in two ways, 1) by a dendritic NMDA-controlled current. Glutamate was assumed to be released in response to rising [K+]o. And 2) by the persistent (i.e., slowly inactivating) Na-current, INa,P. When both INMDA and INa,P were present, the two acted synergistically. We conclude that epileptiform neuronal behavior and SD-like depolarization can be generated by the feedback of ion currents that change ion concentrations, which, in turn, influence ion currents and membrane potentials. The normal stability of brain function must depend on the efficient control of ion activities, especially that of [K+]o.
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INTRODUCTION |
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The electrophysiology of epileptiform seizures
has been studied since the early 1930s and the phenomenon of spreading
depression (SD) was discovered by Leão in the 1940s
(Leão 1944). The extra- and intracellular
electrical signs and the ion concentration changes accompanying these
processes have been described in some detail (Bure
et al.
1974
; Heinemann et al. 1978
; Marshall
1959
; Nicholson 1984
; Somjen et al.
1986
), yet the biophysical mechanisms underlying these
phenomena are not completely understood. During the tonic phase of an
epileptic seizure, large numbers of neurons are firing, the discharge
being driven by steady depolarization. Gloor et al. (1961
,
1964
) have shown that in the hippocampal formation, tonic
discharge is accompanied by a negative shift of the extracellular potential, Vo, that is limited to the
cell body layers and is usually accompanied by a positive shift of
potential in layers containing mostly neuron dendrites and neuroglia.
Unlike in neocortex and spinal cord, in the hippocampal formation, the
contribution of glia to extracellular potential shifts is negligible
(reviewed by Somjen 1995
). We have confirmed the
distribution of Vo shifts during
seizures, and we have also shown that the seizure-related rise in
extracellular potassium concentration,
[K+]o, is maximal in cell
body layers (Somjen and Giacchino 1985
; Somjen et
al. 1985
). Current source density analysis then revealed a
current sink limited to cell soma layers that persisted as long as the
seizure continued. By contrast, during SD, current source density maps
show very large inward currents in the zones of dendritic trees
(Wadman et al. 1992
). Unlike during seizures, during SD, the membrane potential measured in neuron somata is reduced to between
0 and
20 mV and firing ceases. The cell input resistance measured
with sharp electrodes in current clamp drops to less than 12% of its
normal value (Czéh et al. 1993
;
Müller and Somjen 1998
, 2000
; Schwartzkroin
1984
; Snow et al. 1983
).
In spite of the obvious differences between seizures and SD, there are
also important similarities. Both seizures and SD can be triggered by
similar insults, and both are inhibited by similar physical or
pharmacological interventions, such as cooling, hypertonicity, acidosis, and certain depressant drugs (Bure et al.
1974
; Marshall 1959
). An event that begins with
an epileptiform discharge can sometimes terminate in SD (Somjen
and Aitken 1984
), and a seizure that starts in a focus can then
spread over a large area with a velocity resembling that of SD
(Bure
et al. 1974
; Van Harreveld and Stamm
1953
). Finally, intense, persistent inward currents characterize both processes, albeit differently distributed over the
neuron surface (Wadman et al. 1992
).
Neither for tonic-clonic seizures nor for SD have the channels been
identified that drive the persistent depolarization. Hypothetically the
depolarizing current could be generated by the abnormal operation of
one or more of the known physiological membrane channels or it could
involve ion flow through pathways that are not normally present.
Pharmacological blockade of some of the known ion channels delays SD,
shortens its duration, and reduces the amplitude of the associated
depolarization, but none could completely prevent it
(Bure et al. 1984
;
Hernández-Cáceres et al. 1987
;
Herreras and Somjen 1993a
; Marrannes et al.
1993
). This failure seemed to favor the idea that SD is caused
by the opening of a pathological pathway for ion flow that is not
normally present in the membrane. Recently, however,
Müller and Somjen (1998)
found that simultaneous blockade of all known major inward currents did prevent hypoxia-induced SD-like depolarization. This observation could mean that the
depolarization is generated by the cooperative action of several
channels, and this could explain why blocking any one of the channels
can slow down the process or curtail its intensity but not stop it.
To test this hypothesis, we now used computer simulation to examine whether, under the appropriate conditions, the activation of physiological channels could produce SD-like depolarization. The results of the computations suggest that this is indeed possible. In the course of the trials, we discovered that the model could also behave in ways resembling discharges that are recorded from neurons during some forms of epileptic seizures. The key to both classes of pathological behavior appears to be positive feedback loops in which ion currents produce ion concentration changes, which, in turn, profoundly alter ion currents.
An abstract of some of these results is in Somjen et al.
(2000).
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METHODS |
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All simulations were run in the NEURON modeling environment
designed by Hines, Moore, and Carnevale for simulating electrical behavior of branched neuronal structures (Hines and Carnevale 1997).
Morphology
Several series of experiments were run on reduced neurons, which
consisted either of a single somatic compartment or a soma with
sparsely branched dendritic loads attached. However, we illustrate in
this report our findings using a model cell with morphology based on
reconstruction of a hippocampal CA1 neuron of a young adult rat. This
is cell n408 from the Duke-Southampton Archive of Neuronal
Morphology (Cannon et al. 1998; Pyapali et al.
1998
) (Fig. 1A and
Table 1). This neuron was represented in
201 electrically coupled compartments. For simplification we have
assumed membrane parameters that were identical in all dendritic
compartments but distinct from those of the somatic compartment. In
text Figs. 3-10, we illustrate the variables that were recorded from
the somatic compartment, a proximal dendritic compartment and a distal
dendritic compartment (see Fig. 1, d2 and d14). The electrical
parameters were in part similar to those of the hippocampal neuron
model of Traub et al. (1994)
, and in part they were
based on data obtained from hippocampal preparations in our
laboratories.
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Passive electrical properties
The model did not incorporate the axon nor dendritic spines.
Spatial discretization of the numerical compartments was chosen so that
no compartment was longer than 0.2 electrotonic lengths (Rall et
al. 1992). With a specific membrane capacitance
Cm of 0.75 µF/cm2, the total membrane capacity of the cell
became 212 pF. The axial resistivity was set to 100
*cm. Resting
sodium and potassium permeabilities were then used to define the input
resistance and resting membrane potential. With a sodium leak
conductance of 2 * 10
5
S/cm2 and a potassium leak conductance of 7 *
10
5
S/cm2, we obtained an input resistance of 100 M
and a membrane resting potential near to
70 mV. In the following
simulations we added a fixed leak of 20 *
10
5
S/cm2 with a reversal potential set to
70 mV to
stabilize the membrane potential and reduce the input resistance to
between 50 and 100 M
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Active membrane conductances
The voltage-dependent sodium and potassium conductances were
simulated using the classical Hodgkin and Huxley kinetic description (Hille 1992; Hodgkin and Huxley 1952
).
The expressions used for the rate constants that describe the
voltage-dependent transition of the first order m and
h gate are based on a model of hippocampal pyramidal cells
described by Traub et al. (1994)
. The
Goldman-Hodgkin-Katz equation (GHK) was used to describe the
current-voltage relation for each ionic current as a function of
absolute membrane potential V
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Transient Na current, INa,T
The fast transient sodium current,
INa,T was inserted only in the somatic
compartment. We used the description given by Traub et al.
(1994), but based on our own observations (Vreugdenhil et al. 1998
), we shifted the activation function 5 mV in
depolarizing direction. This slightly increased the threshold for
action potentials and reduced the "window" current (Fig.
2, A and B). The
following equation relates the membrane current to the gating
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Persistent Na current, INa,P
A persistent sodium current was inserted into the somatic and
the dendritic compartments. The voltage dependence and kinetics that
describe this sodium current were mainly based on data obtained from
hippocampal preparations. In dissociated hippocampal CA1 neurons
INa,P activates between 60 and
70
mV, and it is maximal between
20 and
40 mV in different cells
(Somjen, unpublished data, similar to those published by French
et al. 1990
; Hammarström and Gauge 1998
).
We used a kinetic scheme (see also Fig. 2, A and
B)
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Delayed rectifier K current IK,DR
The noninactivating potassium current was inserted in all
compartments and obeyed the kinetic scheme of
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Transient K current IK,A
The fast transient potassium current
IA was implemented in all compartments
with a kinetic scheme that obeyed
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NMDA receptor mediated current, INMDA
We reasoned that global glutamate-dependent depolarization can
be modeled as NMDA receptor activation. Since there is evidence that
glutamate is released from cells when
[K+]o is elevated
(Crowder et al. 1987; Fujikawa et al.
1996
) and high
[K+]o also enhances NMDA
receptor activation (Poolos and Kocsis 1990
), the
conductance, gNMDA, was made a
function of [K+]o as well
as of voltage (Hestrin et al. 1990
) (see Fig.
2C). Given the long-lasting phenomena that we study and the
fast desensitization rate of the AMPA receptor, we concentrated our
efforts on depolarizations mediated by the NMDA receptor. It was only
inserted in the dendritic compartments and was implemented including
its voltage-dependent Mg2+ block (Traub et
al. 1994
). The current was carried by both,
Na+ and K+, giving it an
apparent reversal potential around
10 mV (Fig. 2C)
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The kinetics of the receptor consisted of a fixed activation time
constant of 2 ms and a fixed desensitization of 2,000 ms so that we get
for the activation
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Ion accumulation
In our model, the membrane currents are carried by ions, and
this has been taken into account as an actual change in ion
concentration. This made it necessary to define an extracellular space
as the interstitial volume fraction, ISVF, which was taken to be a
fixed fraction of the intracellular space (15%) based on published
data (see DISCUSSION) (Mazel et al. 1998;
McBain et al. 1990
). For simplicity, we did not
calculate the lateral diffusion between adjacent compartments.
All transmembrane sodium and potassium currents were integrated and
converted into chemical units to continuously calculate intra- and
extracellular ion concentrations. In all compartments instantaneous
diffusion equilibrium was assumed
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Active pumping of Na and K ions
An essential feature of our model is the accumulation and
depletion of ions in the intra- and extracellular space (see following text). To balance these effects, we implemented an active pump that was
able to restore the balance for sodium and potassium ions. It is
stimulated by extracellular [K+] and
intracellular [Na+]. We assume instantaneous
kinetics, which, however, due to the integrating nature of the ion
concentrations, will always be slow. The pump will contribute an
electrogenic factor because it exchanges 2 K+ for
3 Na+. Its rate is determined by the
concentrations according to the following relation (Läuger
1991)
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Control of extracellular K accumulation
Potassium accumulation in the interstitial volume was controlled
by a first-order buffering scheme that simulated an effective glial
potassium uptake system. It had a fixed backward rate constant (k1) and a forward rate constant
(k2) that was potassium dependent. In
an extracellular volume fraction of 15% of the intracellular volume,
the total capacity of the potassium buffer was set to have a
concentration of 500 mM
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Numerical considerations
The actual simulation and numerical integration was performed in
NEURON using a higher-order variable time step integration procedure
(Hines and Carnevale 1997). The model was numerically stable. In many situations, we repeated the simulation with a forced
smaller time step to check whether numerical stability or accuracy was
affecting the results. The most important aspect was calculation of the
membrane voltage in the extended dendritic tree treated with cable
equations (further described by Hines and Carnevale
1997
). An additional but essential complication was that we
also simulated the effects of ionic currents on extra- and
intracellular ion concentrations. Under conditions of the repetitive
firing, seizures and SD large ionic shifts occur that have to be taken
into account when driving forces for the ionic currents are calculated.
They were therefore continuously evaluated using the accumulated ion
concentrations. Local diffusion in the neighborhood of ionic channels
or flux limitations through the channels were not taken into account.
By adjusting the densities of the leaks and fine tuning the "pump,"
the resting potential was initially always set to near 70 mV. Cell
impedance and membrane time constant were kept within the physiological
range. When intracellular stimulation was performed, we mimicked the
experimental situation with a sharp electrode in the somatic
compartment that allowed the injection of current, ignoring possible
shunt effects. Input impedance was determined through the same current
injection configuration, by injecting a small hyperpolarizing current
pulse and measuring the resulting voltage step.
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RESULTS |
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"Resting" properties and normal excitability of the model
With the initial ion concentrations set to
[K+]o/[K+]i = 3.5/133.5 and
[Na+]o/[Na+]i = 140/10, the equilibrium potentials of these two ions was at
"rest" EK = 97 mV and
ENa = +71 mV. Before starting a trial, the leak conductances and the pump maximal turnover rate were set to
achieve at the soma a stable membrane potential,
Vm, stable ion concentrations and
input resistance, Rin, conforming to
experimental data. Thus the resting Vm
was between
69 and
71 mV in different trials, and, in the absence
of stimulation, it did not change by more than 0.5-1 mV in 80 s
of simulated time. The resting Rin varied between 40 and 105 M
in different simulations, in the great
majority between 40 and 60 M
, values well within the range measured
with "sharp" electrodes in CA1 pyramidal neurons
(Müller and Somjen 1998
, 2000
;
Schwartzkroin and Mueller 1987
).
The response of the model neuron to stimulation when
[K+]o was well controlled
is illustrated in Fig. 3. A depolarizing
current of 0.1 nA was applied for 200 ms to the cell soma. As long as the current flowed, the model generated a series of action potentials that stopped shortly after cessation of the pulse. At first each action
potential was followed by a hyperpolarizing afterpotential, generated
by the voltage-gated potassium conductances
gK,A and gK,DR. During each action potential,
the cell lost K+ to the extracellular fluid and
gained Na+ at the expense of
[Na+]o, resulting in
step-wise shifts of EK and
ENa. After several spikes,
EK reached the "foot" of the
spikes so that there was no more driving force remaining for the
generation of the afterhyperpolarizations, and eventually
Vm was forced to a more positive level
in the interspike intervals. The disappearance of the hyperpolarizing
afterpotential under the influence of K+
accumulating in interstitial space is akin to the
Frankenhaeuser-Hodgkin effect seen during repetitive stimulation of the
squid giant axon (Frankenhaeuser and Hodgkin 1956).
After termination of the stimulating current,
Vm briefly dipped below
EK, forced into hyperpolarization by
the electrogenic Na-K ion pump, which was strongly activated by the
rises in [Na+]i and
[K+]o. After the end of
the stimulus-induced firing it took about 17 s for
Vm,
ENa, and
EK to be restored to their resting
values by the Na-K pump.
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Afterdischarge and intermittent ("clonic") burst behavior
The behavior of the model changed dramatically when the maximal turnover capacity of the Na-K pump was weakened by 44%. In this state, when the same current pulse was applied to the soma as in Fig. 3, the spike train did not stop at the end of stimulus but continued for several hundreds of milliseconds (Fig. 4, A and B). Firing continued because the continued elevation of [K+]o kept Vm depolarized above the threshold for firing. The firing stopped when [Na+]i increased and [Na+]o decreased so much that the declining electrochemical gradient for Na+ and the lingering inactivation of gNa raised the threshold for firing beyond reach. Thereafter the model continued to generate bursts of action potentials at regular intervals without any additional outside intervention (Fig. 4C). Following each burst, the electrogenic effect of the Na-K pump drove Vm more negative than EK. The electrical load imposed on the soma by the dendrites (Fig. 4D) also aided hyperpolarization of the soma. In the absence of fast sodium channels, the dendrites did not generate action potentials of their own, but the electrotonic coupling with the soma produced the spikelets of the Vm tracing seen in Fig. 4D (see Fig. 1, for the location of dendritic segment d2). Because of the slow recovery of EK, the membrane potential in the soma could not return to its rest level after completing a spike burst but began to slowly depolarize again. Since the pump current is a joint function of [K+]o and of [Na+]i, as the ion levels shifted toward their normal value, the pump current gradually decreased. As the pump's action weakened, the control of Vm, was once more taken over by the leak currents, as shown by the crossing of Vm over EK (Fig. 4C). Now because ENa recovered faster than EK, Vm was still depolarized. At this time, gNa,T inactivation had been sufficiently removed, and the driving force for INa was restored in order for another burst of action potentials to be triggered. So the cycle repeated itself.
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SD-like depolarization
Figures 57 illustrate various features of one example of a
simulated SD-like event. To generate this SD-like depolarization, gNa,P was increased in the entire
neuron membrane. Furthermore the dendritic tree has been equipped with
a conductance whose properties resemble those of NMDA
receptor-controlled channels. To activate
INMDA, it was assumed that NMDA
receptor activation is enhanced and that glutamate is released into the
interstitial space from glial cells and axon terminals whenever these
structures are depolarized by rising
[K+]o (Crowder et
al. 1987
; Fujikawa et al. 1996
; Poolos
and Kocsis 1990
).
In the simulation of Fig. 5, a
depolarizing current of 0.2 nA was applied for 0.5 s to the soma
of the model. The cell fired at a high rate, and firing continued well
beyond the end of the stimulation (Fig. 5A). During the
afterdischarge, the firing frequency increased while the amplitude of
the spikes decreased and the level of
Vm in the inter-spike intervals became
more and more positive. Slightly more than 0.5 s after the end of
the stimulating pulse, the membrane settled into a depolarized and
inactivated state. In the ensuing seconds,
Vm continued to move from around 40
to about
20 mV and then started to repolarize slowly due to an ever
reduced ENa (Fig. 5B).
During and after the depolarizing pulse,
EK rose and approached
Vm, but it caught up with the latter only during the inactivated state. In other simulated SD-like depolarizations, EK did not always
come to equal Vm, but the two variables always approached and tended to move along parallel trajectories during the depolarization but suddenly parted company at
the moment of Vm repolarization. How
near Vm and
EK became during an SD-like event
depended on the relative values chosen for ion conductances and pump
turnover capacity. In the example of Fig. 5, approximately 25 s
after the start of the trial, Vm suddenly repolarized and overshot to a hyperpolarized level and then it
gradually returned toward its resting voltage, while
EK now slowly descended to eventually
reach a level considerably more negative than at rest (Fig.
5B). Undershoots of voltage and [K+]o are well-known
features of tracings of SD in live brain tissue (Hansen and
Zeuthen 1981
; Heinemann and Lux 1975
).
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The changes in ion concentrations are illustrated in Fig. 5,
C and D. These were computed assuming that ISVF
is 15% of cell volume (for a critique see DISCUSSION).
[K+]o rose to almost 30 mM during the SD-like depolarization, which is well within the range
seen in experiments.
[Na+]o dipped to about 30 mM, which is lower than usual in real life but not excessively so.
Hansen and Zeuthen (1981) reported an average of 60 mM,
Kraig and Nicholson (1978)
reported 57 mM during SD, and
we found recently 61 ± 16 (StD) mM during hypoxic SD-like depolarization (Müller and Somjen, 2000
).
[K+]o started to recover
already during the depolarized phase, but [Na+]o continued to
decrease and [Na+]i to
increase while the depolarized state lasted due to the continued flow
of Na+ current (Fig. 7, A and
B). In this period, the slowly inactivating voltage-gated
conductances gNa,P and
gK,DR were both high, but while
Na+ ions were still experiencing a considerable
driving force, there remained no driving force for
K+ ions. Figure 6, A and
B, illustrates the courses of
Vm,
ENa, and
EK in the proximal dendrites (Fig. 1,
dendrite 2) during the same trial as Fig. 5. As before, only
attenuated, electrotonically conducted spikelets were recorded here
because no INa,T existed in this
segment (Fig. 6A). After a short delay following the spike discharge, Vm in the dendrites
depolarized rapidly to a short-lived summit, and then it began to
repolarize slowly (Fig. 6B). Abrupt hyperpolarization
occurred in the dendrites at the same time as in the soma. The time
courses of Vm in the soma and in the
proximal and distal dendrites are compared in Fig. 6, C and
D. A delay of 0.28 s between the steep depolarizations
of distal and proximal dendrites is evident in Fig. 6A. With
the distance between the two segments being 238 µm, this works out to
an apparent propagation velocity of 0.84 µm/ms. The initial courses
of Vm in the dendritic segments d2 and
d14 are remarkably similar, suggesting all-or-none type responses that
were propagated slowly along the dendrites. Vm in segment d14 repolarized in two
steps, the first one after about 10 s, probably due to the
activity of the d14 segment's membrane itself, while the second
hyperpolarizing effect was apparently imposed on this segment
electrotonically from its "upstream" neighbors. Similar
discontinuities have been seen in other SD simulations.
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Figure 7 illustrates the currents that
generated the voltages shown in Figs. 5 and 6. Following the burst of
action potentials, Vm remained
depolarized in the soma by the combined effect of the window current of
INa,T plus the activation of
INa,P (Fig. 7, A and
B). The window current subsided as
Vm depolarized into the voltage range
where inactivation (h) became more effective (Fig.
5A), and then it increased again somewhat as
Vm repolarized into the range where
the window is most open (Fig. 7B; compare also Fig. 2,
A and B). Thus
INa,P and the window of
INa,T were taking turns in keeping the
membrane depolarized, and their joint action was shaping the course of
Vm. The window closed suddenly when
Vm repolarized beyond 50 mV. Even
though both currents, INa,P and the
window current, are small, they could keep the membrane depolarized
because of the minimal opposition by
IK, which was minimal in the absence
of an electrochemical gradient for K+ ions. In
fact, for a short while IK was flowing
inward instead of outward, reversing course ever so slightly, (Fig.
7B). In other successful simulations of SD, such a reversal
did not always occur, and it is therefore not a requirement for the
generation of SD. In such cases, it was enough if the outwardly
directed IK became small.
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In the dendrites, the sustained SD-like depolarization was the result of cooperative activation of INa,P and INMDA (Fig. 7, C and D). INa,P was activated first, due to the initial depolarization conducted electrotonically from the soma. Then INMDA followed when [K+]o began to rise. A positive feedback evolved as INMDA released K+ ions raising [K+]o, which then reinforced INMDA so that its increase accelerated, driving Vm in the dendrites to a depolarized summit well before the soma (compare Figs. 7C and 6, C and D). Rapid change in dendrites is also aided by the greater surface to volume ratio. After reaching its peak, INMDA subsided due to the desensitization of the receptor. Meanwhile, however, INa,P has increased sufficiently to maintain depolarization (Fig. 7D).
During most of the SD-like event the only force opposing
depolarization was the net pump current, and it was the pump current alone that hyperpolarized the cell after the collapse of the inward currents (Fig. 7, B and D). It may be surprising
that the very small net outward pump current could generate the very
large and sudden hyperpolarizing shift. This was possible because of
the rebounding membrane resistance, which suddenly became high once all
active ion channels had shut down. The input resistance of the soma of
the model of Fig. 7 was 50 M at rest, it dropped to 0.67 M
at the
peak of the depolarization and shortly after cessation of impulse
firing. At the end of the depolarized phase it was 6.6 M
and during
the maximally hyperpolarized state again 51 M
. SD was produced in
numerous simulations under varying conditions. While the details
varied, the essential features were similar in all cases.
Critical ignition point of the SD process
The tracings of Fig. 8 were produced by the model in the same state as for Figs. 5-7 except that the threshold at which the glial uptake function began to operate was set at 8 mM [K+]o for Fig. 8, while it was 10 mM for the simulation of Fig. 5-7. With the glial uptake starting at a lower level, [K+]o could not rise above 7.9 mM during the fixed period of stimulation (Fig. 8B). With [K+]o remaining at a low ceiling, the conditions for maintained depolarization and afterdischarge were absent, INa,P, the INa,T window current, and INMDA were not activated. The stimulating current induced repetitive firing but no afterdischarge, and the SD-like all-or-none response was not triggered (Fig. 8A).
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Component actors of the SD-like process
The simulation illustrated in Figs. 5-7 was produced by the
cooperation of a number of currents. We asked whether fewer of these currents could produce an SD-like response. In the simulation shown in
Fig. 9
INa,P and both K currents were
inserted in soma and dendrites, but
INa,T and
INMDA were absent. Stimulation by a
depolarizing current of 0.3 nA for 0.5 s produced an SD-like response in soma as well as in the dendrites (Fig. 9, A-C).
As shown in Fig. 9B, the time delay among the three
compartments was quite short. Depolarization began in the soma, the
site of the stimulating current (Fig. 9B), but the summit of
depolarization was reached first in the distal dendrites followed by
the proximal dendrites and finally in the soma (Fig. 9C).
The maximal amplitude of the depolarization was, however, largest in
the soma, smaller in the proximal dendrites, and smaller yet distally
(Fig. 9C) even though the initial courses of
Vm were very similar (Fig. 9B). As also seen in the case of Fig. 6D, in Fig.
9C, in the distal dendrites,
Vm repolarized before the other
segments, and then it was pulled into hyperpolarization by the
neighboring segments several seconds later. In this case, the input
resistance at rest was 43 M, at the height of depolarization, it
dropped to 0.96 M
, toward the end of the SD-like event it was 4.1 M
, and immediately after repolarization, in the hyperpolarized state
it was 46 M
.
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Figure 10 illustrates a trial in which the cell soma was endowed only with IK,DR and IK,A, while the dendritic tree had the two K currents and also an NMDA receptor-controlled current. The soma was depolarized by a 2-nA current for 2 s. As expected, there was no active response to this very strong stimulus in the soma, yet the dendrites did generate an SD-like response (Fig. 10, A-D). When there was an "active" SD-process in the soma (Figs. 5-7 and 9), it tended to prolong the depolarization of the dendrites. In the absence of SD in the soma, the dendrites repolarized in less than 8 s (Fig. 10B). The cell soma, which had no inward current, was nevertheless forced to remain partially depolarized by electrotonus by the dendritic response (Fig. 10, A and B). The courses of Vm in three of the model's compartments are compared in Fig. 10A. The shape of the curve depicting Vm in the distal dendrite (d14) is almost identical to that in the proximal dendrite (d2), but it is delayed by almost 1 s (Fig. 10, A and B). The propagation velocity in this case was only 0.28 µm/ms, again suggesting an all-or-none type response propagating along the cable structures of the dendritic tree.
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Other pathophysiological behaviors of the model
When INa,T and the two K currents
were in place in the soma, but INa,P
was small, INMDA was absent, and the
Na-K pump and "K-uptake" functions were set at low capacity, strong
depolarizing pulses often produced high-frequency firing that ended in
a partially depolarized and inactivated state either already during the
flow of stimulating current or following a few seconds of
afterdischarge. In these cases, the membrane potential seemed fixed
between 35 and
50 mV, within the window of the
gNa,T. Eventually the cell started to
repolarize, and, as it did so, it sometimes resumed firing for a short
period before returning to its resting state. This sequence resembled
intracellular recordings sometimes obtained from neurons during tonic
seizures (e.g., Fig. 10 of Somjen et al. 1985
).
In a simulated cell soma without dendrites, surrounded by the usual interstitial space and endowed with the Hodgkin-Huxley style conductances but no INa,P or INMDA, strong stimuli provoked repetitive firing at a constant frequency that could indefinitely outlast the stimulating pulse. The continuous firing was maintained by elevated [K+]o, which fluctuated with each spike around a stable mean level. We mention this behavior for the sake of completeness even though it has no equivalent counterpart in real life.
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DISCUSSION |
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The results of these simulations warrant several general conclusions. First is that single neurons are capable of producing the self-sustaining pathophysiological processes typical of at least some types of epileptiform seizures as well as those of Leão's SD. Second is that there is no need to postulate novel membrane conductances but rather that these processes can be generated by the abnormal activity of "physiological" ion channels. Third, the key to the generation of prolonged seizure discharge and SD is the positive feedback based on ion currents causing ion concentration changes, and ion concentration changes altering ion currents. And, as a corollary, is that the normal stability of brain function rests on the ability of cerebral tissue to keep ion distributions within well-regulated limits.
It follows, that the ability to generate self-sustaining inward current is not specific to any single ion channel type. To produce such a current, a channel must have the following properties: it must conduct inward current, it must be either voltage or [K+]o dependent (or both), and the inward current must (secondarily) induce the release of K+ ions into a restricted extracellular space. A seizure will erupt if the membrane potential is forced to remain above firing threshold but below inactivation level. SD-like suspended excitability requires an inward current with inactivation (or desensitization) that is slow or absent. SD will result if the membrane potential moves into the range where the spike-generating currents are inactivated.
The basic idea is not new. Years ago Grafstein (1956)
proposed a potassium hypothesis for SD and Hodgkin used diffusion
theory to show how the movement of K+ in
interstitial space could explain the propagation of SD; Hodgkin's unpublished derivation was incorporated in a communication by Grafstein (1963)
. Later Green (1964)
and
Fertziger and Ranck (1970)
proposed that positive
feedback mediated by elevated
[K+]o had a key role in
the initiation and maintenance of epileptiform seizures. As an
alternative to Grafstein's (1956)
potassium hypothesis, Van Harreveld and Fifková (1970)
proposed that the
agent of positive feedback is glutamate. Like K+
ions, glutamate is released from cells during SD and it can induce SD.
Later Van Harreveld modified his position and proposed a dual hypothesis in which both K+ and glutamate play a
part (Van Harreveld 1978
). The limited knowledge of the
biophysics of central neurons and the absence of computational techniques prevented critical testing of these early propositions at
the time of their publication. Our simulations lend credence to Van
Harreveld's dual hypothesis.
It has been known for some time that drugs blocking either
glutamate-controlled or voltage-gated Na currents can delay and curtail
but not prevent SD and SD-like hypoxic depolarization (Hernández-Cáceres et al. 1987;
Herreras and Somjen 1993a
,b
; Marrannes et al.
1988
). More recently we found that a pharmacological cocktail
inhibiting all known major voltage-gated and receptor-controlled Na+ and Ca2+ currents
succeeds in suppressing hypoxic SD where individual ingredients of the
cocktail failed (Müller and Somjen 1998
). In the
model neuron, it was sufficient to have either a voltage-gated persistent Na conductance or an NMDA receptor-controlled conductance to
generate the SD-like depolarization. In real life and in the absence of
blocking drugs, glutamate-induced and voltage-gated inward currents
probably cooperate in generating SD. This explains why blocking either
the one or the other will reduce but not eliminate SD, while
suppressing all inward currents simultaneously does succeed in
preventing SD. More recently M. Müller and G. G. Somjen (unpublished data) tested whether blocking glutamate receptors and
voltage-gated Na+ currents, but leaving
Ca2+ currents active, prevents hypoxic SD in
hippocampal slices. In this condition, withdrawing oxygen still
provoked SD in three of six slices and in five of eight pyramidal
neurons, albeit only after an unusually long delay and at a greatly
elevated threshold level of
[K+]o. On the other hand,
replacing all but 25 mM of the Na+ in the bathing
solution by N-methyl-D-glucamine
(NMDG+) completely but reversibly prevented the
occurrence of hypoxic SD even though none of the membrane channels was
blocked, indicating that Ca2+ alone cannot
mediate SD. It follows that, besides the glutamate- and
voltage-controlled channels, there is some additional pathway for
Na+ influx that can mediate SD-like
depolarization even if in blunted form. A possible candidate may be the
acetylcholine and Ca2+ dependent cation current
described by Fraser and MacVicar (1996)
(see also
Kawasaki et al. 1999
).
The potassium hypothesis of seizure generation proposed by
Fertziger and Ranck (1970) was disputed by
Heinemann et al. (1978)
, who could not find a fixed
threshold of [K+]o at
which seizures would erupt. Yet they emphasized that once a seizure
started, elevated [K+]o
could shape its progress. Recently Borck and Jefferys
(1999)
proposed that elevated
[K+]o causes the
transition from interictal to ictal discharge. In the simulations
presented here, seizures had to be triggered by a depolarizing
stimulating current; elevated
[K+]o maintained the
discharge, it did not start it. Seizures could be provoked by the
initial stimulus provided that the regulation of
[K+]o was less than
optimal. This role of high
[K+]o in the simulation
is entirely compatible with the observed behavior of
[K+]o during seizures
(Fertziger and Ranck 1970
; Heinemann et al. 1978
; Somjen and Giacchino 1985
). Also in
agreement with measurements in live brain tissue are the membrane
hyperpolarization and the undershoot of
[K+]o below its rest
level, mediated in the simulations by the Na-K pump after the
termination of seizure discharges and of SD-like depolarization
(Heinemann and Gutnick 1979
; Heinemann et al.
1978
). These authors already emphasized the role of
electrogenic ion transport in the termination of seizure discharges. In
our simulations, postexcitation undershoot of the membrane potential
preceded the recovery and undershoot of potassium levels both in the
"normal" state (Fig. 3) and in the inter-burst intervals during
seizures (Fig. 4, B and C). As a result, for
short periods, Vm was more negative
than EK. There are no published
simultaneous measurements of K+ concentrations
and neuron membrane potentials during seizures in live brain tissue to
verify this feature, but extracellular recordings are compatible with
it because following spike trains, seizures or SD,
Vo returns to baseline or undershoots
baseline well before
[K+]o does (Borck
and Jefferys 1999
; Hansen and Lauritzen 1984
;
Heinemann and Gutnick 1979
; Heinemann and Lux
1975
; Heinemann et al. 1978
; Nicholson
1984
; Somjen and Aitken 1984
; Somjen and
Giacchino 1985
).
While computer simulations cannot positively prove a point, they can
decide whether a theoretical explanation of a process is feasible and
consistent (for a detailed discussion, see Borg-Graham 1999). The model presented here had many but by no means all
the properties of a real neuron. Epileptic attacks do, of course, require an intact organism whose brain, muscles, and viscera produce the clinical picture. Single neurons are, however, the constituent units of the system, and understanding their properties is essential for understanding the operation of the organism. Our model did not
produce a complete tonic-clonic seizure sequence only prolonged afterdischarges followed by unceasingly recurring "clonic"
discharges. It may be that grand mal attacks can only be
generated by cell populations connected in a network of excitatory and
inhibitory connections. Moreover these simulations cannot speak to the
mechanism by which SD or Jacksonian seizures propagate only to the
mechanism by which these processes evolve once they have been ignited.
The propagation velocities estimated from dendritic recordings (Figs. 6C and 10A) are by far faster than the 2-5
mm/min with which SD spreads in the brain. In the model neuron, the SD
wave was conducted along the dendritic cables, while the customary
measurements in brain tissue concern movement from cell to cell, and
this cannot be imitated in a model consisting of a single neuron.
Other, earlier computer simulations interpreted the propagation of SD
in terms of a diffusion-reaction process but did not address the
specific membrane currents generating the depolarization (Grafstein 1963; Tuckwell and Miura
1978
). Reggia and Montgomery (1996)
and
Revett et al. (1998)
attempted to relate simplified theoretical models of SD to the pathophysiology of clinical conditions. In a recent abstract, Shapiro (1999)
outlined a model
testing the role of gap junctions in the propagation of SD.
In our model, the SD-like depolarization evolved faster in the
dendrites than in the soma. In an analysis of high
K+-induced SD in rat hippocampus in situ, the
extracellular voltage shifts (Vo)
almost always started in stratum radiatum before st. pyramidale
(Herreras and Somjen 1993b
). The typical, biphasic, "inverted saddle" shape of the
Vo suggested at least two
mechanisms contributing to the process. This impression was reinforced
when the NMDA antagonist compound CPP apparently blocked the second, larger component of the dual-peaked
Vo. The analysis of multiple recordings of
Vo (Herreras
and Somjen 1993a
,b
) and the simulations presented here are
compatible with the idea of at least two generators of the SD process,
only one of which is NMDA dependent.
The geometry of the model was that of a real neuron, as published in
the Duke-Southampton Archive of Neuronal Morphology (Cannon et
al. 1998) with values for membrane capacitance and cytoplasmic resistivity conforming to known physiological parameters (see METHODS). The input conductance of cells was within the
range encountered in CA1 pyramidal neurons in real life (e.g.,
Brown et al. 1981
; Fujimura et al. 1997
;
Fujiwara et al. 1987
; Müller and Somjen
1998
, 2000
; Schwartzkroin and Mueller 1987
). The
input conductances dropped during the height of SD to low values, as expected from published recordings (Czéh et al.
1993
; Müller and Somjen 1998
, 2000
;
Snow et al. 1983
). The changes in ion concentrations and
voltages that evolved during these simulations were also within the
range of those measured in live brain tissue (Hansen and Zeuthen 1981
; Kraig and Nicholson 1978
;
Müller and Somjen 2000
). The all-or-none
character of the simulated SD-like depolarization is similar to real
SD. The ignition point for SD appears to have been reached when
[K+]o exceeded 8 mM
(Figs. 8B and 5C), close to the 9 mM
[K+]o at which hypoxic
SD-like depolarization took off in hippocampal slices
(Müller and Somjen 2000
).
While the behavior of the model was in many respects life-like, it did
not emulate all aspects of real neurons. The differences can be
ascribed to the numerous features that were missing from the model. The
only Na+ conductance in the dendrites of the
model was the gNa,P. There is much
evidence for the existence of fast or intermediately inactivating Na+ channels in dendritic membranes (Magee
and Johnston 1995). These could play a role in initiating
pathological discharges, but they do not seem to be essential for the
process. Perhaps more important is the absence of calcium currents.
Leaving calcium from our computations for the sake of simplicity was
justified because seizure-like events and SD are known to occur in
brain slices bathed in low-calcium solutions (Albrecht and
Heinemann 1989
; Basarsky et al. 1998
, 1999
; Dudek et al. 1990
;
Young and Somjen 1992
). Glutamate is released into
interstitial space during SD and other pathological conditions by both
Ca-dependent and -independent processes (Basarsky et al.
1999
; Benveniste et al. 1984
; Drejer et
al. 1985
; Szerb 1991
; Van Harreveld and
Fifková 1970
). Nonetheless, Ca currents and
calcium-dependent K currents undoubtedly do modulate real seizures and
SD. For example, the firing frequency of the model during simulated
seizures was unusually high probably because of the absence of
IK(Ca), which contributes to the slow
hyperpolarizing afterpotentials. Another important omission from the
model is the absence of chloride and of other anions. According to
Nicholson (1984)
, the cerebellar cortex generates SD
only if "conditioned", and one of the methods of conditioning is to
replace Cl
in the extracellular fluid by
certain other anions. In isolated chick retina, Do Carmo and
Martins-Ferreira (1984)
found that replacing
Cl
by isethionate depressed the
Vo shift associated with SD. Recently Müller (2000)
confirmed this for hypoxic SD in
hippocampal tissue slices using methylsulfate as the
Cl
substitute. Intracellular recording from
neurons revealed, however, that while hypoxic SD-like depolarization of
neurons was greatly delayed when
[Cl
]o was lowered, the
ultimate level of the depolarization was not changed. It therefore
appears that Cl
flux modulates SD but is not an
essential ingredient for its generation. A further omission from the
model was any consideration given to charge balance.
Also missing, yet important, is the effect of cell swelling. Cells
swell during seizures and during SD and SD-like hypoxic depolarization
(Dietzel and Heinemann 1986; Hansen and Olsen
1980
; Jing et al. 1994
;
Pérez-Pinzón et al. 1995
), restricting
interstitial space and hence amplifying extracellular ion concentration
changes. The size of the interstitial space chosen for the simulation, 15% of the cell volume, was close to the 12-13% of the total tissue volume reported by McBain et al. (1990)
. A larger
figure, about 20%, was recently suggested by Mazel et al.
(1998)
. During SD, however, the interstitial volume fraction
shrinks to around 5% (Jing et al. 1994
). In an earlier
and much simpler version of the model that consisted of a cell soma
with sparsely branched apical dendrites, interstitial space was made to
vary inversely and cell volume to vary directly with
Na+ uptake. The result was a dramatic decrease of
the interstitial space during simulated SD. Restriction of interstitial
space amplifies extracellular concentration changes and therefore
accelerates the evolution of SD but does not qualitatively alter the
process. Cell swelling probably participates in SD in yet another way
by causing the release of glutamate through stretch-activated anion channels (Basarsky et al. 1999
). Since elevated
K+ causes swelling, this is one more pathway
cooperating in the positive feedback.
Another deficiency was the absence of lateral diffusion. Thus even
though cell "compartments" were electrically coupled, for the
computation of ion fluxes each was treated in isolation from its
neighbors. Ion diffusion would smooth the sudden transitions in space
and time. The next improvement of the model should take into account
cell swelling as well as the missing ions, Ca2+
and Cl, charge balance, and spatial diffusion.
These considerations will increase the complexity of computation by an
order of magnitude. In the model, the distributions of
K+ and Na+ were controlled
by the Na-K pump. Conforming to its known properties (Läuger 1991
), the pump transported three
Na+ ions against two K+
ions. In addition, increases in
[K+]o were also limited
by a glial uptake. The glial buffer hypothesis has appeared in the
literature in several versions, but none is quantitative (reviewed by
Somjen 1987
). We represented the glial uptake in the
form of a buffer without implying that it necessarily represents the
actual mechanism. In intact brain, extracellular ion concentrations are
also regulated by capillary endothelium that is capable of transporting
not only K+ but also Na+
ions as well as other solutes between interstitial fluid and capillary blood.
French et al. (1990) first reported the existence of a
persistent Na+ current in dissociated hippocampal
neurons. Mittmann et al. (1997)
confirmed its presence
in the dendritic tree of central neurons. It appears that
INa,P of hippocampal neurons is
greatly enhanced during hypoxia (Hammarström and Gauge
1998
). It has also been known since some time that raising
[K+]o had such an effect
in skeletal and heart muscle (Barchi 1995
; Hoffman et al. 1995
). Enhancement of
INa,P by high
[K+]o has now been
confirmed also in isolated hippocampal neurons (Somjen
2000
) and in neurons in hippocampal slices (M. Müller and
G. G. Somjen, unpublished results). More surprisingly,
INa,P was even more strongly
potentiated when neurons filled with fluorescent calcium-indicator dyes
were illuminated. These various methods of potentiating
INa,P may have a "final common
path" that has yet to be discovered. The enhancement mediated by high
[K+]o and by hypoxia
could amplify the positive feedback leading to SD and to hypoxic
SD-like depolarization.
We conclude that seizure discharges and SD can be generated in healthy brain tissue if the normal regulatory mechanisms that keep ion concentrations within limits are overwhelmed. Spontaneous seizures, i.e., clinical epilepsy, can result from inherently enhanced excitatory processes, defective synaptic inhibition, or failure of ion concentration regulation, three defects that could drive ion concentrations into the pathological range, and perhaps also from abnormal sensitivity (reduced threshold) of neurons to ion concentration change.
Finally we offer this speculation: could ordinary, everyday muscle cramp be the SD of skeletal muscle? Its sudden onset, all-or-none, autonomous, hard-to-influence course and its all too slow release are provocatively reminiscent of the course taken by SD of brain. Certainly potassium is abundantly available to serve as the agent of positive feedback in muscle and, as the cramp stops local blood flow, K+ could accumulate in the interstitium.
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ACKNOWLEDGMENTS |
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This work was supported by National Institute of Neurological Disorders and Stroke Grant NS-18670.
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FOOTNOTES |
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Address for reprint requests: G. Somjen, Dept. of Cell Biology, Duke University Medical Center, Box 3709, Durham, NC 27710 (E-mail: g.somjen{at}cellbio.duke.edu).
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.
Received 1 November 1999; accepted in final form 20 March 2000.
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REFERENCES |
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