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, V_o, 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 V_o 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).

	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.

View larger version (26K): [in this window] [in a new window]   Fig. 1. A: the geometry of the model neuron. , the regions from which recordings are illustrated in Figs. 3-10: dendrite (2) is the second, dendrite (14) is the 14th dendritic segment. The computations were based on the responses of the entire neuron. The model is based on a CA1 pyramidal cell, published in the Duke-Southampton Archive of Neuronal Morphology (Cannon et al. 1998). B: schema of the main constituents of the model: currents, fluxes and concentrations. The drawing in B is not to scale.

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 C_m 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 * 10⁵ S/cm² and a potassium leak conductance of 7 * 10⁵ 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 * 10⁵ 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

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
with
where P₀ is the membrane permeability, F is Faraday's constant, R is the gas constant, T is the absolute temperature, and z is the valence of the current carrying ion. The temperature was set to 37°C. The generalized formula for each active current can now be written as
where "gates" (m, h) describe the dependence of the current on the activation and/or inactivation gates. Five active membrane currents were incorporated.

Transient Na current, I_Na,T

The fast transient sodium current, I_Na,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
where g_Na,T was the conductance at maximal activation, and it was usually set to about 100 * 10⁵ S/cm². The forward () and the backward () rate constants describe the transition between the closed and open state of the first-order activation gate, m
and
as well as for the inactivation gate h
and
To illustrate the properties of some of the simulated currents, Fig. 2, A and B, shows sodium currents computed in voltage-clamp mode in a neuron soma. Figure 2B (inset) shows the transient and persistent sodium currents, I_Na,T and I_Na,P and their sum, evoked by a voltage step from 90 to 10 mV. Similarly to "real life" whole cell recordings, repolarization elicited an inward tail current due to the slow closing of the I_Na,P channels. In Fig. 2A, the activation and steady-state inactivation (m³ and h) curves are plotted. Of note is the presence of a window current between the membrane potentials of 50 and 20 mV, in which region activation begins but steady-state inactivation is not yet complete (see also Magee and Johnston 1995; Sah et al. 1988). This enables I_Na,T to flow continuously, influencing the behavior of the model in depolarized states. In Fig. 2B, inset, the window current is visible as the continued flow of I_Na,T beyond the termination of the main current surge. Its importance will become clear in the results.

View larger version (27K): [in this window] [in a new window]   Fig. 2. Properties of some of the simulated currents. A: activation and inactivation functions: m3 INa,T: activation of the transient Na+ current; m2 INa,P: activation of the persistent Na+ current; hinf: steady-state inactivation of the transient Na+ current. Note that the curves for hinf and m3 overlap, creating a voltage domain or "window" where INa,T is partially activated but inactivation is as yet incomplete. B: current-voltage (I-V) curves of the transient and persistent sodium currents and the window current of INa,T. The left ordinate axis refers to INa,T, the right axis to INa,P and the "window" current. Inset: INa,T, INa,P, and their sum, evoked by depolarizing step from 90 to 20 mV. (note the window current as continued flow of INa,T). C: I-V curves of the glutamate-induced N-methyl-D-aspartate (NMDA) current that is dependent on voltage (through Mg2+ block) as well as, indirectly, on extracellular potassium (concentrations indicated at right side of figure). D: properties of the glial potassium buffering system. Abscissa: flux of K+ out of the neuron; ordinate: extracellular potassium concentration resulting from the flux. Channel currents and pump current were ignored for this illustration. The kinetics of the buffering were fast enough to assume virtually instantaneous equilibrium.

Persistent Na current, I_Na,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 I_Na,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)
The value for the maximal conductance, g_Na,P was taken to be much smaller than the one for g_Na,T and equal to 2 * 10⁵ S/cm². Because little information is available about the activation kinetics we simplified it to
with
Inactivation is extremely slow (10⁶ times slower than that of I_Na,T), and it was modeled similar to the fast transient Na current giving
and

Delayed rectifier K current I_K,DR

The noninactivating potassium current was inserted in all compartments and obeyed the kinetic scheme of
Using a density for g_K,DR of 100 * 10⁵ S/cm² and the following voltage dependent rate constants
and

Transient K current I_K,A

The fast transient potassium current I_A was implemented in all compartments with a kinetic scheme that obeyed
and a value of g_K,A of 10 * 10⁵ S/cm². The activation kinetics was implemented as
and
and for the inactivation gate
and

NMDA receptor mediated current, I_NMDA

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, g_NMDA, 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)
which is apart from the driving force equal to
with [Mg²⁺]₀ set to 1.2 mM/l and g_NMDA equal to 10 * 10⁵ S/cm².

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
with
where Kha is 13.5 mM/l and Ksa is 1.42 mM/l, and for desensitization
with
where Khi is 6.75 mM/l and Ksi is 0.71 mM/l.

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
where F is the Faraday constant, Surface is the surface area related to the ion densities included, and Vol_intra is the intracellular volume of the compartment under study. Since the ions have to flow into the accompanying extracellular compartment they also change the concentration there
where we assume a fixed relation between the intracellular volume and the interstitial space
(see also DISCUSSION). "Resting" ion concentrations were set to (in mM/l): [Na⁺]_o, 130; [Na⁺]_i, 10; [K⁺]_o, 3; [K⁺]_i, 130.

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)
using Km_K of 3.5 mM/l and Km_Na of 10 mM/l, contributing to the ion flux with

The maximal flux (converted to electrical units), I_max, generated by the pump was adjusted to obtain a steady state under resting conditions just compensation the ion leaks. The usual value of I_max was 0.013 mA/cm².

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
with rate constants
and
where [K⁺] is the potassium concentration in the interstitial space, and [Buffer] and [Kbuffer] are, respectively, the lumped free and bound buffer in the same volume. This function describes extracellular buffering phenomenologically. The speed of the buffering was dependent on the extracellular K⁺ concentration, but despite the relative low rate constants, it appeared in all situations that we encountered to achieve a virtually instantaneous equilibrium. The kinetic aspects of the K⁺ buffering system were not investigated in further detail. In Fig. 2D we illustrate this properties by plotting the final K⁺ concentration that is reached in the extracellular space when a certain amount of K⁺ is entered, assuming in this figure that no other mechanisms are present. At low concentrations relatively little K⁺ is buffered, but at around 10 mM/l the buffer capacity is quite high, creating a relative ceiling corresponding to the so-called "ceiling" of [K⁺]_o (Heinemann and Lux 1977). It takes a large amount of K⁺ before the buffer is saturated and thereafter fast increase can occur again, ultimately raising K⁺ up to levels where the driving force becomes the limiting factor (around 30-60 mM/l). We have limited our model calculations by ignoring diffusion of K⁺ (and other ions) among the compartments of the extracellular space. This probably exaggerates the duration of extracellular ionic gradients in comparison with the real situation, although lateral diffusion through the limited interstitial volume will be unimportant compared with diffusion/uptake at locations farther away from the cell membrane.

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.

	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" E_K = 97 mV and E_Na = +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, V_m, stable ion concentrations and input resistance, R_in, conforming to experimental data. Thus the resting V_m 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 R_in 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 g_K,A and g_K,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 E_K and E_Na. After several spikes, E_K reached the "foot" of the spikes so that there was no more driving force remaining for the generation of the afterhyperpolarizations, and eventually V_m 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, V_m briefly dipped below E_K, 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 V_m, E_Na, and E_K to be restored to their resting values by the Na-K pump.

View larger version (30K): [in this window] [in a new window]   Fig. 3. The normal response of the model in its stable state. The tracings were obtained from the "soma" segment of the model neuron shown in Fig. 1. Black trace: the membrane potential (Vm); red line: the equilibrium potential computed for Na+ (ENa); blue line: the equilibrium for K+ (EK). The horizontal bar indicates the application of a 0.1-nA depolarizing pulse to the soma. A and B represent the same event on 2 different time scales.

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 V_m 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 g_Na 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 V_m more negative than E_K. 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 V_m tracing seen in Fig. 4D (see Fig. 1, for the location of dendritic segment d2). Because of the slow recovery of E_K, 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 V_m, was once more taken over by the leak currents, as shown by the crossing of V_m over E_K (Fig. 4C). Now because E_Na recovered faster than E_K, V_m was still depolarized. At this time, g_Na,T inactivation had been sufficiently removed, and the driving force for I_Na was restored in order for another burst of action potentials to be triggered. So the cycle repeated itself.

View larger version (48K): [in this window] [in a new window]   Fig. 4. Prolonged afterdischarge followed by recurrent (clonic) seizure discharges generated by the model neuron. The 6 panels show different aspects of the same event. In each graph, the horizontal bar indicates the single 0.1-nA depolarizing pulse applied to the soma at the beginning of the simulation. A-C: the membrane potential and the equilibrium potentials for Na+ and K+ in the cell soma, plotted on 3 different time scales. D: the same 3 variables in the proximal dendrites (dendritic segment 2, see Fig. 1). E: sodium and potassium currents recorded in the soma. F: transport of Na+ (red line) and of K+ (blue line) in the soma by the Na-K exchange pump, expressed in electric units (pA/µm2). The green line represents the net pump current, which is electrogenic.

SD-like depolarization

Figures 57 illustrate various features of one example of a simulated SD-like event. To generate this SD-like depolarization, g_Na,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 I_NMDA, 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 V_m 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, V_m continued to move from around 40 to about 20 mV and then started to repolarize slowly due to an ever reduced E_Na (Fig. 5B). During and after the depolarizing pulse, E_K rose and approached V_m, but it caught up with the latter only during the inactivated state. In other simulated SD-like depolarizations, E_K did not always come to equal V_m, but the two variables always approached and tended to move along parallel trajectories during the depolarization but suddenly parted company at the moment of V_m repolarization. How near V_m and E_K 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, V_m suddenly repolarized and overshot to a hyperpolarized level and then it gradually returned toward its resting voltage, while E_K 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).

View larger version (44K): [in this window] [in a new window]   Fig. 5. Seizure discharge followed by spreading depression (SD)-like depolarization, recorded in the cell soma. All 4 panels derived from the cell soma during the same event. The horizontal bars represent a depolarizing pulse of 0.2 nA, 500 ms, applied to the cell soma. A and B: Vm, ENa, and EK represented similarly to Figs. 3 and 4, A-C. C and D: the changes in concentrations of Na+ and K+ in the interstitial space and in cytosol ([Na+]o and [K+]o, respectively, [Na+]i and [K+]i).

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 g_Na,P and g_K,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 V_m, E_Na, and E_K 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 I_Na,T existed in this segment (Fig. 6A). After a short delay following the spike discharge, V_m 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 V_m 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 V_m in the dendritic segments d2 and d14 are remarkably similar, suggesting all-or-none type responses that were propagated slowly along the dendrites. V_m 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.

View larger version (44K): [in this window] [in a new window]   Fig. 6. Voltages recorded in different segments of the model neuron during the same SD-like event as in Fig. 5. A and B: Vm, ENa, and EK in the proximal dendrites (d2), on 2 different time scales. C and D: superimposed tracings of the membrane potentials recorded in soma, in proximal, and in distal dendrites (d2 and d14).

Figure 7 illustrates the currents that generated the voltages shown in Figs. 5 and 6. Following the burst of action potentials, V_m remained depolarized in the soma by the combined effect of the window current of I_Na,T plus the activation of I_Na,P (Fig. 7, A and B). The window current subsided as V_m depolarized into the voltage range where inactivation (h) became more effective (Fig. 5A), and then it increased again somewhat as V_m repolarized into the range where the window is most open (Fig. 7B; compare also Fig. 2, A and B). Thus I_Na,P and the window of I_Na,T were taking turns in keeping the membrane depolarized, and their joint action was shaping the course of V_m. The window closed suddenly when V_m repolarized beyond 50 mV. Even though both currents, I_Na,P and the window current, are small, they could keep the membrane depolarized because of the minimal opposition by I_K, which was minimal in the absence of an electrochemical gradient for K⁺ ions. In fact, for a short while I_K 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 I_K became small.

View larger version (36K): [in this window] [in a new window]   Fig. 7. Ion currents recorded during the same SD-like event as Figs. 5 and 6. A and B: the transient and persistent sodium currents (INa,T and INa,P), the total potassium current (IK,DR+IK,A) and the net pump current in the soma, plotted on 2 different sets of abscissal and ordinal scales. C and D: similar tracings for the proximal dendrites, d2. INMDA represents the NMDA receptor-controlled current.

In the dendrites, the sustained SD-like depolarization was the result of cooperative activation of I_Na,P and I_NMDA (Fig. 7, C and D). I_Na,P was activated first, due to the initial depolarization conducted electrotonically from the soma. Then I_NMDA followed when [K⁺]_o began to rise. A positive feedback evolved as I_NMDA released K⁺ ions raising [K⁺]_o, which then reinforced I_NMDA so that its increase accelerated, driving V_m 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, I_NMDA subsided due to the desensitization of the receptor. Meanwhile, however, I_Na,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, I_Na,P, the I_Na,T window current, and I_NMDA 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).

View larger version (24K): [in this window] [in a new window]   Fig. 8. Restoration of a stable state. All the parameters as well as the stimulating current were the same as for the SD-like event of Figs. 5-7, except that the threshold for the "glial uptake" function was lowered from 10 to 8 mM [K+]o. Compare especially the traces of [K+]o in B of with Fig. 5C.

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 I_Na,P and both K currents were inserted in soma and dendrites, but I_Na,T and I_NMDA 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 V_m were very similar (Fig. 9B). As also seen in the case of Fig. 6D, in Fig. 9C, in the distal dendrites, V_m 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.

View larger version (41K): [in this window] [in a new window]   Fig. 9. SD-like depolarization can be produced when the persistent sodium current is the only inward current. For this simulation, INa,T and INMDA have been deleted. The stimulus was 0.3 nA for 500 ms. A: membrane potential and ion equilibrium potentials recorded in the cell soma. B and C: membrane voltages in three segments of the model neuron superimposed on different time scales. D: the persistent sodium current, the summed potassium currents, and the net pump current recorded in the cell soma.

Figure 10 illustrates a trial in which the cell soma was endowed only with I_K,DR and I_K,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 V_m in three of the model's compartments are compared in Fig. 10A. The shape of the curve depicting V_m 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.

View larger version (39K): [in this window] [in a new window]   Fig. 10. SD-like depolarization generated in the dendrites by NMDA receptor-controlled current (INa,P) by itself. INa,T and INa,P have been deleted. The stimulus was 2 nA applied for 2 s. A and B: the membrane potentials in 3 cell segments superimposed on 2 different time scales. C and D: membrane potential and ion equilibrium potentials in the proximal dendritic segments (d2). E and F: ion currents in the proximal dendritic segments (d2).

Other pathophysiological behaviors of the model

When I_Na,T and the two K currents were in place in the soma, but I_Na,P was small, I_NMDA 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 g_Na,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 I_Na,P or I_NMDA, 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.