Journal of Neurophysiology

Bursting in Inhibitory Interneuronal Networks: A Role for Gap-Junctional Coupling

F. K. Skinner, L. Zhang, J. L. Perez Velazquez, P. L. Carlen

Abstract

Bursting in inhibitory interneuronal networks: a role for gap-junctional coupling. Much work now emphasizes the concept that interneuronal networks play critical roles in generating synchronized, oscillatory behavior. Experimental work has shown that functional inhibitory networks alone can produce synchronized activity, and theoretical work has demonstrated how synchrony could occur in mutually inhibitory networks. Even though gap junctions are known to exist between interneurons, their role is far from clear. We present a mechanism by which synchronized bursting can be produced in a minimal network of mutually inhibitory and gap-junctionally coupled neurons. The bursting relies on the presence of persistent sodium and slowly inactivating potassium currents in the individual neurons.Both GABAA inhibitory currents and gap-junctional coupling are required for stable bursting behavior to be obtained. Typically, the role of gap-junctional coupling is focused on synchronization mechanisms. However, these results suggest that a possible role of gap-junctional coupling may lie in thegeneration and stabilization of bursting oscillatory behavior.

INTRODUCTION

Recent work indicates that interneurons are part of an extensive inhibitory network which play an essential role in molding the synchronous rhythmic output of principal cells (Buhl et al. 1994; Cobb et al. 1995; Freund and Buzsáki 1996). Besides synapsing onto principal cells, these GABAergic interneurons are heavily interconnected synaptically (Acsády et al. 1996; Hájos et al. 1996). Furthermore, gap junctions (Benardo 1997;Gulyás et al. 1996; Katsumaru et al. 1988; Kosaka 1983; Kosaka and Hama 1985) and dye coupling (Michelson and Wong 1994;Strata et al. 1997; Zhang et al. 1998) have been found between interneurons. Theoretical models have been instrumental in explaining that mutually inhibitory neural networks can produce synchronized behavior when the inhibition is slow relative to the neuronal firing rate (van Vreeswijk et al. 1994;Wang and Rinzel 1993; White et al. 1998).Buzsáki and Chrobak (1995) postulate that oscillating inhibitory networks (in basal forebrain, hippocampus, neocortex, and thalamus) may provide the precise temporal structure necessary for ensembles of neurons to perform specific functions.

Recently, slow (≤1 Hz) oscillatory behavior has been demonstrated in in vitro hippocampal and cortical rat brain slice preparations after blockade of Cs+-sensitive currents (Zhang et al. 1998). At the principal cell level, it has been shown that these oscillations are synaptically mediated because they are abolished in the presence of TTX and attenuated in high external Mg+. Pharmacological and ionic properties indicate that inhibitory GABAA synapses are responsible for the oscillatory behavior, and ionotropic glutamate receptors are not required. At the interneuronal level, the addition of Cs+ also can invoke rhythmic discharges with electrophysiological and pharmacological properties similar to the principal cells. Moreover, in paired recordings, correlations are present between the rhythmic discharges or bursting oscillations in the interneuron and the voltage-clamped oscillations in the principal cell. In addition, the rhythmic behavior at both the principal and interneuronal levels disappear after treatments that block gap junctions. These experimental observations lead to the suggestion that an interneuronal network that acts reciprocally via electrical and GABAergic synapses plays a role in the observed oscillatory behavior.

These experiments are intriguing because they show that interneuronal bursting oscillations cannot be maintained if eitherinhibitory coupling or gap-junctional coupling is blocked. Clearly this is a network phenomenon, but how are these oscillations generated and why would both types of coupling be necessary? Motivated by these questions, we use a computational approach to examine these observations. We develop a minimal biophysical model of a two-cell neural network which suggests why both mutual inhibition and gap-junctional coupling are required.

METHODS

Experimental

Transverse brain slices (400–500 μm) were obtained from Wistar rats (13- to 52-day old) and maintained in an oxygenated (5% CO2-95% O2) artificial cerebrospinal fluid (ACSF) medium with (in mM) 125 NaCl, 2.5 KCl, 1.25 NaH2PO4, 2 CaCl2, 1,8 MgSO4, 26 NaHCO3, and 10 glucose. The pH of the ACSF was 7.4. Whole cell patch-clamp recordings were done in a submerged chamber at 32–33°C. The patch pipette solution (intracellular) contained (in mM) 150 K-gluconate, 0.1 Na-EGTA, and 5 HEPES.

Computational

Each interneuron consists of a single compartment cell with a persistent sodium current, I Nap, a slowly inactivating potassium current, I D, and Hodgkin-Huxley (HH) sodium and potassium currentsCdVdt=Iext[gL(VVL)+gKDab(VVK)+gNapp(VVNa)+gNam3h(VVNa)+gKn4(VVK)+Icoupl] Equation 1 dadt=(aa)/τa Equation 2 dbdt=(bb)/τb Equation 3 dhdt=φ(αh(1h)βhh) Equation 4 dndt=φ(αn(1n)βnn) Equation 5where C = 1 μF/cm2 is the capacitance, I ext (μA/cm2) is the external or imposed current, V is the voltage, tis time, and I coupl is the coupling current (see following text). Using a HH formalism and adopting parameters used byWang (1993) to describe cortical cells, the leak current is described by its conductance g L = 0.1 mS/cm2 and its reversal potential,V L = −60 mV. g Na = 52 mS/cm2 is the maximal sodium conductance, andV Na = 55 mV is the sodium channel reversal potential. The sodium channel activation is assumed to be fast enough so that its steady-state value, m , can be used, i.e., m = αm/(αm + βm), where αm = −0.1(V + 30)/ {exp[−0.1(V + 30)] − 1} and βm= 4 exp(−(V + 55)/18); h is the sodium channel inactivation and αh = 0.07 exp[−(V + 44)/20]; βh = 1/{exp[−0.1(V + 14)] + 1}. For the delayed rectifier potassium channel,g K = 20 mS/cm2 is the maximal conductance, V K = −90 mV is the reversal potential, and n is the activation where αn = −0.01(V + 34)/{exp[−0.1(V + 34)] − 1}; βn = 0.125 exp[−(V + 44)/80]; φ = 28.57.

I Nap is based on data from French et al. (1990): g Nap is the maximal conductance and the activation kinetics are assumed to be fast so that they can be described by their steady-state behavior: p = 1/{1 + exp[−(V + 51)/5]}. I Dis based on data from Storm (1988):g KD is the maximal conductance, a is the activation and b is the inactivation. A sigmoidal form is used to describe the steady-state activation and inactivation curves: a = 1/{1 + exp[−(V + 55)/]5} and b = 1/ {1 + exp[(V + 85)/6]}. The steady-state activation and inactivation curves are illustrated in Fig.1.

Fig. 1.

Steady-state activation and inactivation curves. A: — (- - -), a (b ), the steady-state activation (inactivation) of the slowly inactivating potassium current,I D. Notice that this current is only active at a hyperpolarized level. B: —,p , the steady-state activation of the persistent sodium current, I Nap.

Two identical neurons are coupled together to form a neural network. As described by following equations, nonrectifying gap junctions and mutual inhibitory synapses are used in the network architecture. The coupling current for each neuron is given byIcoupl=gsyns(VVsyn)+gelec(VVpre) Equation 6wheredsdt=αT(Vpre)(1s)βs Equation 7 T(Vpre)=1/(1+exp((VpreVthresh)/2) Equation 8 g syn and g elec are the maximal synaptic and electrical conductances, respectively, andV pre is the presynaptic membrane potential. The synaptic activation, s, is assumed to follow first-order kinetics where the normalized transmitter concentration T is assumed to be a sigmoid function of V pre. The channel opening, α = 12 ms−1, assumes a fast rise of the synaptic current and β, the inverse of the decay time constant of the synaptic current is set to 0.1 ms−1 (as done by others forGABA A synapses) (e.g., see Wang and Buzsáki 1996). V thresh = −10 mV so that transmitter release occurs only when the presynaptic cell produces a spike. The synaptic reversal potential,V syn, is set to −75 mV.

The resulting set of 12 differential equations is solved with a 10 μs time step using CVODE, an industrial-strength stiff differential equation solver included as part of the XPPAUT software developed by Bard Ermentrout and available via anonymous ftp fromftp.math. pitt.edu/pub/bardware.

RESULTS

Experimental observations

Coherent oscillations in hippocampal and cortical principal neurons of rat brain slices are obtained after perfusion with 3–5 mM CsCl. Field recordings and dual principal cell recordings indicate that these oscillations occur in a population of neurons, and these oscillations do not require activation of ionotropic glutamate receptors to occur (Zhang et al. 1998). An example of a voltage-clamped CA1 pyramidal neuron is shown in Fig.2. In the presence of Cs+, this neuron displays slow oscillations. In this voltage-clamped mode, the “peaks” of the oscillation thus correspond to the maximal outward current or where the cell receives the most inhibition.

Fig. 2.

Cs+-induced slow oscillations in a CA1 pyramidal neuron of a rat brain slice preparation. CA1 neuron was voltage-clamped at −60 mV and recordings were made before (A), during perfusion of 5 mM CsCl for 7 min (B), and after wash (C).

As shown in Fig. 3, Cs+ also can induce rhythmic bursts in hippocampal interneurons. These Cs+-induced rhythms or bursting oscillations are dependent on GABAA neurotransmission (Fig. 3 A) and are arrested by manipulations that block gap junctions (Fig.3 B). In the absence of Cs+, the interneurons do not burst but fire randomly with a membrane potential of −54.4 ± 3.3 (SD) mV and an input resistance of 266 ± 182 MΩ. It is quite possible that the coherent rhythms observed in the principal cell population (Fig. 2) are due to the innervation of a synchronized GABAergic interneuronal population onto the principal cells (Zhang et al. 1998). To suggest how and why a network of mutually inhibitory and gap-junctionally coupled interneurons could produce synchronized bursting oscillations (as in Fig. 3), we have used a computational approach.

Fig. 3.

Slow rhythmic activity in hippocampal interneurons. A: recordings from an oriens/alveus (O/A) interneuron in the presence of 5 mM CsCl and 1.5 mM kynurenic acid, a general ionotropic glutamate receptor antagonist (top), and after adding 10 μM bicuculline methiodide, a GABAA receptor antagonist, to the perfusate (bottom). B: Cs+-induced oscillations in an O/A interneuron before (top), during application of 0.1 mM octanol, a gap junction uncoupler (Perez Velazquez et al. 1994) (middle), and after wash (bottom).

Minimal model

To help understand and explain the experimental observations described earlier, we developed a minimal two-cell network model. As described in methods, each cell in the network contains HH-like spiking currents, a slowly inactivating potassium current, and a persistent sodium current. Depending on the level of injected current, individual, model neurons are silent or fire tonically. They do not burst. For example, the situation where an individual neuron fires tonically is illustrated in Fig.4 A. For bursting on the time scale of seconds (as observed experimentally), an underlying slow process with such kinetics is required. The inactivation ofI D has such kinetics. To produce the bursting, oscillatory behavior, the outward I D must be counterbalanced by some inward current. However,I D is active at a hyperpolarized level (see Fig.1) relative to the spike threshold (−30 mV; i.e., where the HH inward sodium current would begin to be active), andI Nap is too small at these hyperpolarized levels to bring the cell up to spike threshold and fire a spike to initiate a burst. Therefore an individual cell does not exhibit slow, bursting behavior in our model: At more hyperpolarized levels, the inward currents are not active to counterbalance the outwardI D; at more depolarized levels,I D with its appropriate kinetics is not active.

Fig. 4.

Hyperpolarization produces bursting. A: individual neuron tonically fires. I ext = 1.7 μA/cm2; g elec =g syn = 0 mS/cm2.B: model network also fires tonically. Output from 1 cell is shown; the other cell fires in antiphase.I ext = 1.7 μA/cm2;g elec = 0.1, g syn= 0.1 mS/cm2. C–E: with hyperpolarizing current, (i.e., I ext < 1.7), injected into each cell of the network in B (to simulate the effect of Cs+), the network exhibits bursting oscillations. Again, output from 1 cell is shown; the other cell is in phase or synchronized. g elec = 0.1, g syn = 0.1 mS/cm2.C: I ext = 1.4 μA/cm2; D: I ext= 1.2 μA/cm2; E:I ext = 1.0 μA/cm2. Other parameter values in (A–E) areg KD = 20, g Nap = 0.1 mS/cm2; τa = 5, τb = 1,500 ms. In each plot, the dotted line represents −50 mV as indicated.

If two cells (as in Fig. 4 A) are coupled with mutual inhibition and gap junctions (as described in methods), the model network also does not produce bursting oscillations (Fig.4 B). This represents the interneuronal situation without the addition of Cs+. Experimentally, the bursting oscillations are induced with the addition of Cs+, which blocks inward rectifier currents, leading to hyperpolarization (Maccaferri and McBain 1996). The addition of Cs+ is modeled as a hyperpolarization to each cell. With hyperpolarization, the model neural network produces stable, synchronized, bursting behavior (Fig.4, C–E). The bursts are synchronized so each cell simultaneously feels some average inhibition from the other and the gap-junctional current is negligible. Simulations (not shown here) in which hyperpolarization-activated inward currents are blocked were performed, and bursting also could be obtained in this way.

Bursts obtained via hyperpolarization of the network are shown in Fig.4. More specifically, if a hyperpolarizing current is injected into each cell of the network such that the inactivation ofI D is sufficiently removed, then the tonically firing network (Fig. 4 B) changes its behavior to bursting oscillations (Fig. 4 C). The frequency of these bursts is affected by the level of injected current each cell receives (Fig. 4,C–E).

Both forms of coupling are required for these bursts to be obtained. This is shown in Fig. 5 where either the inhibitory coupling (Fig. 5 A) or the gap-junctional coupling (Fig. 5 B) is removed. If the inhibitory coupling is removed, the bursts are lost immediately, and the two cells eventually move to a state where they fire tonically in antiphase with each other (Fig.5 A, small graph). If the gap-junctional coupling is removed, the bursts are eventually lost because they are no longer stable (see following text). The system moves to a state where one cell is tonically firing and the other is hyperpolarized, being kept down by the inhibition (as shown in the small graph of Fig. 5 B). How and why do these bursts occur?

Fig. 5.

Both inhibitory synaptic and gap-junctional coupling are required for bursting. A: without inhibitory coupling, the bursts are lost. At the vertical dashed line, g syn is set to 0. Before removal of inhibition, the 2 cells are bursting in phase. After inhibition is removed, the cells move to a state where they fire tonically in antiphase with each other. This is shown in the smaller plot which is an expanded version of the last part of the larger plot. The two cells are distinguished by solid and dashed lines.B: without gap-junctional coupling, the bursts are eventually lost. At the vertical dashed line,g elec is set to 0. After some time, the synchronized bursts are lost, and 1 cell fires tonically and the other is hyperpolarized. This is illustrated in the smaller plot, which is an expanded version of the last part of the larger plot; 1 cell fires tonically (solid line) and the other cell is hyperpolarized (dashed line). Parameters as in Fig. 4 D. In each plot, the dotted line represents −50 mV.

Model mechanism

The bursting oscillations in the aforementioned model occuronly if I D actively participates in the behavior. This means that I D must be sufficiently deinactivated in both neurons. IfI D is not in a sufficiently deinactivated range for either neuron (for example, by depolarizing 1 of the cells for a long enough time so that it is no longer deinactivated), then bursting oscillations would no longer occur.

The underlying mechanism of the bursting behavior in the model can be described as follows: inhibition via the synaptic current is required to hyperpolarize the postsynaptic cell so that the inactivation ofI D can be removed sufficiently. HenceI D increases and the bursts eventually terminate because at the same time that I D is increasing,I Nap is not. This can be seen in Figs.6 C and7 if one considers hypothetical outlining envelopes of the currents (I Nap andI D) during the burst. The cells can no longer be brought up to spike threshold by I Nap and fire, and so the burst ends. Once firing stops, inhibition stops so that the cells are no longer taken to the more hyperpolarized values. NowI Nap increases more thanI D during the interburst interval (as seen in Fig. 6 C) so that a burst can eventually be initiated, i.e., the first spike fires. The cycle repeats. Figure 6 Billustrates that b increases during the burst, i.e., the inactivation is removed as the burst progresses. Because overall (i.e., considering envelopes of the voltage and current),I D increases (see Figs. 6 C and 7) and there is an overall hyperpolarization (see Figs. 6 A and 7) during the burst, the increasing I D is due to the increasing removal of inactivation. Note that even though the variation in b is not large, it is enough of a variation to generate the slow bursting behavior. The voltage and currents are shown on an expanded time scale in Fig. 7.

Fig. 6.

Bursting requires the participation of the slowly inactivating potassium current, I D. A: voltage output from 1 neuron (the other cell is in phase).B: b, the inactivation ofI D for the neuron depicted in A. C: I Nap (bottom curve, inward current, negative) andI D (top curve, outward current, positive) for the neuron in A. Note that during the burst, i.e., when the neuron is firing, there is a net increase in the outward current (I D) because the inactivation variable is increasing during this time. During the burst, the outward I D increases more relative toI Nap, eventually leading to the burst termination, and during the interburst interval, the inwardI Nap increases more relative toI D eventually leading to the burst initiation. Parameters are as in Fig. 4 D.

Fig. 7.

Burst and underlying currents. Voltage and currents from Fig. 6 are shown on an expanded timescale to illustrate the role of the slowly inactivating potassium current, I D. One burst of the voltage output in Fig. 6 A is shown (top) with the corresponding currents from Fig.6 C (bottom). Note the hyperpolarization during the burst due to the net increase of outward current (I D) during this time.

The particular parameter values affect the regime in which it is possible to obtain bursts, but without both types of coupling, and sufficient deinactivation of I D in both cells, it was not possible to obtain these bursting oscillations within the parameter regimes investigated. With the appropriate balance of currents and operational voltages as given by the described mechanism earlier, the model will exhibit robust bursting behavior. For example, if g Nap is increased by ∼50%, the slow bursting behavior will not occur because nowI D is not large enough to overcomeI Nap to terminate the bursts, and fast spiking will continue without any slow bursting behavior. However, if at the same time, g KD also is increased by ∼50% (all other parameters remaining the same), slow bursting will again be present. Because the experimental data shows that both types of coupling are required to obtain this bursting behavior (Fig. 3) and this is also true in the model, it is possible that the model with its given parameters capture the essential details of the mechanism underlying the experimental observations.

As described in the preceding text, without the inhibition, the neurons would not be able to enter a regime in which I Dis sufficiently deinactivated so that it can participate in the behavior and produce bursting oscillations. If there was no gap-junctional coupling present, the two cells eventually would diverge and remain stable with one cell tonically firing and the other suppressed (as shown in Fig. 5 B). What role are the gap junctions playing?

Role of the gap-junctional coupling

Without gap junctions, stable bursting behavior cannot be generated or maintained. We already showed in Fig. 5 B that bursting is lost when gap junctional coupling is removed. In Fig.8, we illustrate that the presence of gap junctions is key in allowing the system to recover from perturbations. For the three cases shown in Fig. 8, the system is started from the same initial conditions, but the gap-junctional coupling is set to 0 (Fig. 8 A), 0.1 (Fig. 8 B), or 0.2 (Fig.8 C) mS/cm2. These initial conditions are such that I D would actively participate in the behavior so that bursting would occur. A perturbation is applied to one of the cells in each of the cases. When there is no gap-junctional coupling (Fig. 8 A), no bursts are obtained and one cell goes to a tonically firing state and the other to a hyperpolarized state, being inhibited by the other cell. This is expected because we already know from Fig. 5 B that bursts cannot be maintained without gap-junctional coupling. With some gap-junctional coupling (Fig.8 B), bursts are generated and temporarily maintained. They are eventually lost, not being able to recover from the perturbation. However, with enough gap-junctional coupling (Fig. 8 C), the system can recover from the perturbation, generate bursts, and stably maintain them.

Fig. 8.

Gap junctions are needed to maintain the bursting behavior. Using the parameters in Fig. 4 D, simulations are run and bursting oscillations are present. In this state, the gap junctional coupling is adjusted to 0 (A), 0.1 (B), or 0.2 mS/cm2 (C). At the arrow, a perturbation (50-ms injected current of −1 μA/cm2) is delivered to 1 of the neurons. Note that without any gap-junctional coupling, bursts cannot be recovered after the perturbation (A). Depending on the strength of the gap-junctional coupling, bursts are maintained temporarily (B) or stably (C). Output of the 2 cells are differentiated by solid and dashed lines. If the output in B is continued for a longer period of time, it would arrive at a similar output shown in the final state of A, with 1 cell tonically firing and the other hyperpolarized. An expanded version of this final state would be similar to the small plot in Fig. 5 B. However, in C, the output of the 2 cells burst stably in phase and remain like this for longer runs.

The size of the gap junctional conductance does not play a critical role in the frequency of the bursts. However, the gap junctional conductance affects how long it takes to and whether the system can achieve or lose stable, bursting behavior. For example, with all other parameters the same as in Fig. 4 D, if the gap-junctional conductance was doubled (to 0.2 mS/cm2), the frequency of the bursts still would be 0.9 Hz. However, ifg elec is now set to zero (as in Fig.5 B), it would take about twice as long for the bursts to be lost (not shown).

Therefore, the role of the gap-junctional coupling in the model lies in stabilizing the bursts. Even though it is possible to have bursts without gap-junctional coupling (as in Fig. 5 B afterg elec = 0), they are not stable and are eventually lost.

Decreased synaptic conductance

To determine whether the model reasonably captures the behavior of the real system, the synaptic conductance is manipulated. The strength of the synaptic conductance is decreased by using very low doses of bicuculline, a GABAA antagonist. This is shown in Fig.9 for recordings in the rat brain slice at the principal cell level. What is observed from the voltage-clamp recordings is that there is a significant increase in the amplitude of the oscillations as well as a decrease in the frequency. These are intricate manipulations and at present, the more difficult recordings at the interneuronal level with low doses of bicuculline have not been obtained. Even though we do not know what proportion of the synaptic conductance is blocked by these low doses of bicuculline, it is clear that there is a large increase in current amplitude when these inhibitory conductances are partially blocked. As assumed earlier, if the interneuronal population is responsible for this slow rhythmic behavior, then the larger amplitude excursion in current suggests that more inhibition is being received by the principal cells from the interneuronal cell population.

Fig. 9.

Low doses of bicuculline increase the amplitude of the Cs+-induced slow oscillations in CA1 pyramidal neurons. CA1 neuron was voltage-clamped at −45 mV, and recordings were made after application of 5 mM CsCl for ∼10 min (A), after adding 10 nM BMI after 3 min (B), and after 8 min (C). Notice the increase in the amplitude of the slow oscillations induced by the low concentration of bicuculline which partially blocks the GABAA synapses.

Figure 10 shows what happens in the model if the synaptic conductance is partially blocked. The burst duration increases so that there are more spikes per cycle, and the period increases (Fig. 10 B) relative to the control (Fig.10 A). Given the description of the model mechanism above, it is understandable why this occurs. If there is less inhibition between interneuronal cells, the postsynaptic cell is less hyperpolarized, the inactivation, b, is smaller and it takes longer forI D to build up and overcomeI Nap to terminate the burst (via preventing the cell from reaching spike threshold). Therefore, the cells can fire for a longer period of time, hence the increase in burst duration. Therefore, decreased synaptic conductance translates into less inhibition between interneurons. Thus, the bursts are prolonged. This in turn translates into more inhibition from the interneurons onto the principal cells. At the least, the model behavior is consistent with the experimental results. The fact that the interneurons produce more spikes per cycle means that there would be increased inhibition onto the principal cell population, and an increased current amplitude excursion in the voltage clamp mode would be observed, as it is.

Fig. 10.

Decreasing the synaptic conductance increases the burst duration. In phase bursting oscillations, voltage output of the network wheng syn = 0.1 mS/cm2(A). Parameters as in Fig. 4 D. B: voltage output of the network when the synapses are blocked partially, i.e.,g syn = 0.07 mS/cm2. Parameters as in A except for g syn. Notice the increase in the burst duration leading to a decrease in frequency with this decreased synaptic inhibitory conductance.

DISCUSSION

Gap junctions and bursting

In accordance with experimental observations, stable bursts are obtained in a model network of mutually inhibited and gap-junctionally coupled interneurons. To obtain these bursts, both mutual inhibition and gap-junctional coupling are required between interneurons that have persistent sodium and slowly inactivating potassium currents, of which there is evidence for their presence in interneurons (Chikwendu and McBain 1996; Klink and Alonso 1993; McBain 1994; Zhang and McBain 1995). The molecular nature of I Dis unknown as present, but it has been found that slowly inactivating potassium currents are encoded by the Kv3.1 gene known to be present in GABAergic interneurons (see Du et al. 1996; Lenz et al. 1994; Massengill et al. 1997;Sekirnjak et al. 1997). The bursts are synchronized with both spikes and bursts in phase. The gap-junctional coupling in the model is needed to stabilize the bursting behavior. Without it, stable bursts cannot be generated or maintained (Figs. 5 B and 8).

Bursting, a more involved firing pattern, is expressed by many neurons. Mathematically, bursting requires more than a two variable model, and on the basis of their characteristics, different types of bursting have been named (Bertram et al. 1995; Rinzel and Ermentrout 1998). The bursts exhibited by our model resemble an “elliptic burster” type, but because it is a network-generated bursting pattern, it is not as straightforward to characterize and analyze (Bertram et al. 1995). Because of the large number of variables in the model system (12), complex dynamics such as quasiperiodicity and chaos can emerge. Our model network does exhibit other stable solutions, but the particular solution of bursting oscillations emerges only when I D is deinactivated sufficiently so that it actively participates in the network output.

Theoretical work (van Vreeswijk et al. 1994; Wang and Rinzel 1993; White et al. 1998) has demonstrated that synchronized behavior can be obtained in mutually inhibited neural networks. In White et al.’s (1998)recent work, networks of mutually inhibitory spiking neurons, in which the individual cells had different properties, were investigated. It was found that the spike synchrony was quite fragile in the presence of mild heterogeneity (intrinsic spike rates <5% different). Moreover, when synchrony was lost, it was lost either in a tonic (out of phase firing) or a phasic (1 cell suppresses the other) fashion depending on the ratio of spike period and synaptic decay time constant. Based on their analyses, they suggest that the fragile synchrony in such heterogeneous networks could be remedied by the presence of gap junctions. The bursting behavior presented in this paper is an emergent network phenomenon because individual model cells do not burst, and their synchronization is a result of mutual inhibition and gap junctions. When gap-junctional coupling is blocked, synchrony is lost (Fig. 5 B) in a phasic fashion using the terminology ofWhite et al. (1998). The ability of our model system to exhibit bursts critically depends on the active participation ofI D, which in turn can be seen to affect the spike frequency within the burst (see Fig. 6). Taken together with the detailed analyses by White and colleagues (1998) on mutually inhibitory spiking neurons, the spike frequency within a burst is an important consideration for assessing the level of gap-junctional coupling required for stable bursting behavior.

The correspondence between model and experiment is not expected to be exact because the experimental data reflects a large population of interneurons and the model data reflects a two-cell network. However, the model does produce oscillations that look similar to the experimental data (i.e., tonically firing bursts with silent periods, compare Figs. 3 and 4) and are in the appropriate frequency range. Work in progress will examine the mathematical details of the dynamical processes.

The most direct prediction that this model makes is that withoutI D, bursts will not occur. Specifically, the interplay of I Nap and I Dis critical for the observed bursting behavior, suggesting that these currents should be investigated further. Another prediction is that one would expect to see some interneurons tonically firing and others silent when gap-junctional coupling is blocked (Fig. 5 B). The particular interneuron shown in Fig. 3 B fires tonically when gap-junctional coupling is interrupted. Furthermore, the strength of the gap-junctional coupling should not affect the frequency of the bursts, and weaker synaptic conductances should lead to an increased number of spikes per cycle in the interneuronal population (Fig. 10). Although straightforward in the model, such experiments have to be designed carefully to examine these predictions. For example, although low doses of 4-aminopyridine are known to blockI D, such manipulations also affect synaptic coupling (Avoli et al.1996; Storm 1988). Techniques such as the dynamic clamp (Sharp et al. 1993) may be useful in this regard. To investigate coupling manipulations, several simultaneous recordings from multiple interneurons are required.

Related models

It has been shown that bursting can be obtained with slowly inactivating potassium and persistent sodium currents in individual neurons (Wang 1993), but in that case the potassium activation/inactivation kinetics occurred at more depolarized potentials. In our model, bursting does not occur in the individual neurons because the activation/inactivation ofI D occurs at more hyperpolarized values. Inhibition is required to remove the inactivation and so allow bursting to occur. The mechanism described here is also different from that found in Sherman and Rinzel’s (1992) model. In their work, only gap-junctional coupling was present between the two model cells, and they found that the gap-junctional coupling could convert spiking cells into bursting cells where the bursts were in phase but the spikes within the burst were antiphase.

The work of Manor et al. (1997) shows that oscillations can emerge from electrically coupled cells with heterogeneous channel densities even when the individual cells do not oscillate. Their study is similar to ours in that they hypothesize that gap-junctional coupling is responsible for generating oscillatory behavior. However, it is different from our work in that their focus was the inferior olive cells with low-amplitude, sinusoidal-like subthreshold oscillations, and no synaptic inhibitory coupling was present.

In a detailed model of 128 51-compartment interneurons, Traub (1995) showed that synchronized bursts could be obtained with dendritic gap junctions in which the dendrites contained active conductances. However, unlike our model, his model included GABAB synapses and was focused on synchronized bursts in the absence of GABAA and ionotropic receptors. Recent experimental and modeling work indicate that at the principal cell level, gap-junctional coupling is responsible for synchronizing high-frequency (150–200 Hz) oscillations (Draguhn et al. 1998).

Concluding remarks

It is important to stress that the model presented here is a minimal model. For example, it does not include dendrites that could certainly play a role in the bursting behavior (e.g., see Traub 1995). Active dendrite properties pose another level of complexity (Mainen and Sejnowski 1998) that require additional interneuronal data not currently available. Future work involves a consideration of larger, heterogeneous networks of multicompartment neurons in the context of the mechanism described in this work. Given the difficulty in recording from multiple interneurons of which there are several types (Freund and Buzsáki 1996), mechanistic insights from mathematical models can be very instructive. Therefore although minimal, our model perhaps captures essential components underlying the observed bursting behavior.

The importance of interneuronal gap junctions has been suggested in previous studies to primarily play a synchronizing role (Benardo 1997; Michelson and Wong 1994). However, using a computational model, we have shown that the gap-junctional coupling is needed to generate and stabilize the bursting behavior. Using in vitro models, it has been suggested that rhythmic seizure-like activities are suppressed by manipulations that block gap-junctional communication (Han et al. 1996; McDonald et al. 1997;Perez Velazquez et al. 1994). In addition to metabotropic receptors (Whittington et al. 1995), it may be that gap junctions are needed to generate oscillatory behavior.

Acknowledgments

F. K. Skinner thanks N. Kopell and J. White for discussions leading to a better understanding of the model mechanism.

This work was supported by the Medical Research Council of Canada, the Natural Sciences and Engineering Research Council of Canada, the Toronto Hospital Research Institute, and Defence and Civil Institute of Environmental Medicine Grant W7711–8–7451.

Footnotes

  • Address for reprint requests: F. K. Skinner, Playfair Neuroscience Unit, The Toronto Hospital, Western Division, 399 Bathurst St., MP12–303, Toronto, Ontario, M5T 2S8, Canada.

  • 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.

REFERENCES

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