Role of potassium lateral diffusion in non-synaptic epilepsy: A computational study

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Abstract

An increase of extracellular potassium ion concentration can result in neuronal hyperexcitability, and thus contribute to non-synaptic epileptiform activity. It has been shown that potassium lateral diffusion alone is sufficient for synchronization in the low-calcium epilepsy in-vitro model. However, it is not yet known whether the lateral diffusion can, by itself, induce seizure activity. We hypothesize that spontaneous sustained neuronal activity can be generated by potassium coupling between neurons. To test this hypothesis, neuronal simulations with 2-cell or 4-cell models were used. Each model neuron was embedded in a bath of K+ and surrounded by interstitial space. Interstitial potassium concentration was regulated by both K+-pump and glial buffer mechanisms. Simulations performed with two coupled neurons with parameter values within physiological range show that, without chemical and electrical synapses, potassium lateral diffusion alone can generate and synchronize zero-Ca2+ non-synaptic epileptiform activity. Simulations performed with a network of four zero-Ca2+ CA1 pyramidal neurons modeled in zero-calcium conditions also show that spontaneous sustained activity can propagate by potassium lateral diffusion alone with a velocity of ∼0.93 mm/sec. This diffusion model used for the simulations is based on physiological parameters, is robust for various kinetics, and is able to reproduce both the spontaneous triplet bursting of non-synaptic activity and speed of propagation in low-Ca2+ non-synaptic epilepsy experiments. These simulations suggest that potassium lateral diffusion can play an important role in the synchronization and generation on non-synaptic epilepsy.

Introduction

Fluctuation of ion concentration can modulate neuronal excitability. In particular, an elevation of extracellular potassium ion concentration ([K+]o) has been shown to be responsible for neuronal hyperexcitability (Fertziger and Ranck, 1970; Jefferys, 1995). An increase of [K+]o diminishes the driving force, defined as a difference between membrane potential and potassium Nernst potential. This reduction lowers outward potassium current, resulting in relative depolarization. Frankenhaeuser and Hodgkin initially showed that potassium ions released from neuronal firing accumulate into a restricted space around squid giant axon membranes (Frankenhaeuser and Hodgkin, 1956). Considering the very small interstitial space around pyramidal neurons in the hippocampus, Green has suggested that K+ can accumulate in the space when neurons fire, which, in turn, increases membrane excitability and causes seizure activity (Green, 1964). In 1970, Fertziger and Ranck have proposed ‘potassium accumulation hypothesis’, whereby a rise in [K+]o increases neuronal excitability and firing rate causing further rise of [K+]o. Potassium accumulation effect has been observed not only in hippocampal pyramidal cells but also in Purkinje cells from the guinea-pig (Hounsgaard and Nicholson, 1983). This accumulation hypothesis has been suggested as a possible mechanism for seizure generation. Physiological evidence suggests that potassium accumulation only modulates the frequency of oscillations outside the hippocampus (Towers et al., 2002). Therefore, the hippocampus is found to be the most important structure for seizure generation by potassium accumulation effect. Yet, the ability of an increase in [K+]o in the interstitial space to initiate seizure activity has not been determined. Due to a lack of threshold value of [K+]o and an increase of [K+]o after the onset of neuronal firing, some investigators concluded that [K+]ois an influential factor in the course of seizure activity but not involved in neither initiation nor termination of seizure activity (Fisher et al., 1976; Heinemann et al., 1977; Lux, 1974; Moody et al., 1974; Kager et al., 2000; Somjen, 2004). However, a number of experiments supporting the potassium hypothesis have shown that [K+]o accumulation could initiate seizure activity (Izquierdo et al., 1970; O’Connor and Lewis, 1974; Zuckermann and Glaser, 1968). This is due to the fact that neurons sharing the same extracellular space (ECS) were depolarized by the diffusion of high [K+]o. Recent simulations performed with a network model have also shown that an increase in [K+]o in the interstitial space was sufficient for not only evoking a transition from normal oscillations to paroxysmal oscillations but also maintaining the activity through the network (Bazhenov et al., 2004). The effects of changes of extracellular potassium concentrations on the dependence of stimulus-induced synaptic potentials have been also studied by Rausche et al. (1990). The results show that an increase of [K+]o from 5 to 8 mM suppressed the rise of excitatory postsynaptic potentials, as well as the amplitude of population spikes.

The role of non-synaptic mechanisms in the generation of epileptiform activity has been investigated in a number of experiments and simulations. Specifically, the fluctuation of [K+]o has been proposed as a major factor for non-synaptic epileptogenesis in low-Ca2+ experiments (Konnerth et al., 1986; Yaari et al., 1986). In particular, the authors have suggested that spatial potassium dispersion can cause excitation of neurons lateral to initial focus underlying seizure-like events (SLE) activity in low-Ca2+ medium. Unlike lateral diffusion (diffusion between two neurons), excitation by dispersion is caused by shared depolarization of neurons in the same potassium bath. Nelken and Yaari (1987) have proposed that seizure activity in low-Ca2+ solution (Jefferys and Haas, 1982; Taylor and Dudek, 1982; Yaari et al., 1983) could be explained solely by the potassium hypothesis from both experimental and simulation studies. In a similar experimental condition (blocked synaptic transmission), Lebovitz has quantitatively analysed paroxysmal depolarization (PD) mediated by extracelluar K+ diffusion with a HH type model (assuming that synaptic transmission was blocked) (Lebovitz, 1996). The author reported that one “subthreshold” action potential (AP) signal was sufficient to generate “autogenic” potassium mediated PD, supporting the “potassium regenerative hypothesis” (Fertizger and Ranck, 1970) and similar to self-regenerating activity (Kager et al., 2000; Bazhenov et al., 2004).

Several studies point to the crucial role of lateral diffusion of potassium ions in zero-calcium epileptiform activity. A theoretical analysis by Lebovitz (1996) showed that a “suprathreshold” AP and a close distance between primary and nearby neuronal membrane were sufficient to generate “cooperative” PD, resulting from a reverberation of the AP from boundary of adjacent membrane to primary neuronal membrane. In-vitro low-Ca2+ experiments (Lian et al., 2001) showed that epileptiform activity was synchronized across a mechanical cut directly implicating potassium lateral diffusion in the synchronization of neuronal activity.

In general, both the spread (or propagation) and the synchronization of epileptiform activity can be assumed to take place after the generation of seizure activity. In addition, experimental studies (Konnerth et al., 1986; Yaari et al., 1986; Lian et al., 2001) showed that potassium lateral diffusion plays a critical role in the synchronization and propagation of epileptiform activity in low-Ca2+ medium. Therefore, we hypothesize in this model study that spontaneous neuronal activity can be generated by coupling between neurons through potassium ion lateral diffusion. To test this hypothesis, a 2-cell model (modified Hodgkin–Huxley neurons) and a 4-cell model (modified zero-Ca2+ CA1 pyramidal neurons) were used to study directly the effect of potassium lateral diffusion on the synchronization and generation of neuronal activity.

Section snippets

Two-neuron diffusion coupled model

A three compartment model for diffusion was used (Fig. 1): one compartment for the cell and two for ECS including interstitial space and the bath. Two cells were embedded in the potassium bathing solution ([K+]bath) and placed side by side. Each cell was surrounded by interstitial space or K+ shell. One of the cells was excited by current injection (Istim). During neuronal firing, potassium currents (IK) contributed to the release of K+ from the cell into the interstitial space where K+ ions

Sustained activity generated by potassium lateral diffusion

To study the role of lateral diffusion in the generation of potassium-induced activity, simulations with a two-cell model with HH kinetics with and without lateral diffusion were performed. A current pulse (1 nA, 840 ms at threshold) was injected in one of the two cells. The stimulation current amplitude of 1 nA was chosen to be larger than the minimal threshold amplitude (0.99 nA).

Activity was initially generated in cell-1, but as [K+]o increased during the stimulation, the cell stopped firing due

Conditions for the potassium lateral diffusion-induced activity

Simulations of neuron models lacking both synaptic transmission and other types of non-synaptic mechanisms such as field effect and gap junction show that a mutual (Table 3, Table 4, Table 5) interaction between two neurons via potassium lateral diffusion induces and synchronizes neuronal activity. In a recent simulation study of a single pyramidal neuron, Kager et al. (2000) and Somjen (2004) proposed a composite mechanism of self-regenerating activity: positive feedback between ion currents

Conclusion

The results of simulations showed that: (1) lateral diffusion of potassium ions alone is necessary and sufficient to generate and synchronize sustained neuronal activity in a simple 2-cell model system, (2) there is a bifurcation (from transient to sustained activity) produced by changing parameters, τss and rv, and (3) synchronization level decreased as the lateral diffusion time constant increased or as the space constant increased. The simulations performed with a zero-Ca2+ CA1 pyramidal

Acknowledgments

We thank M.A. Jaber for helpful discussions and E.J. Durand for proof reading of the manuscript. The financial support of this work was provided by NIH grant NS40785-03.

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