NMDA and GABA_B (KIR) Conductances: The “Perfect Couple” for Bistability

Abstract

Networks that produce persistent firing in response to novel input patterns are thought to be important in working memory and other information storage functions. One possible mechanism for maintaining persistent firing is dendritic voltage bistability in which the depolarized state depends on the voltage dependence of the NMDA conductance at recurrent synapses. In previous models, the hyperpolarized state is dependent on voltage-independent conductances, including GABA_A. The interplay of these conductances leads to bistability, but its robustness is limited by the fact that the conductance ratio must be within a narrow range. The GABA_B component of inhibitory transmission was not considered in previous analyses. Here, we show that the voltage dependence of the inwardly rectifying potassium (KIR) conductance activated by GABA_B receptors adds substantial robustness to network simulations of bistability and the persistent firing that it underlies. The hyperpolarized state is robust because, at hyperpolarized potentials, the GABA_B/KIR conductance is high and the NMDA conductance is low; the depolarized state is robust because, at depolarized potentials, the NMDA conductance is high and the GABA_B/KIR conductance is low. Our results suggest that this complementary voltage dependence of GABA_B/KIR and NMDA conductances makes them a “perfect couple” for producing voltage bistability.

Introduction

Persistent neuronal firing underlies information storage in functions such as working memory, motor control, and spatial navigation (for review, see Goldman-Rakic, 1995; Durstewitz et al., 2000; Wang, 2001; Major and Tank, 2004). A proposed cellular mechanism underlying persistent activity is membrane potential bistability. Brief depolarizing stimuli can switch a bistable cell from a hyperpolarized state, in which the firing rate is low, to a depolarized state, in which the firing rate is high (Hounsgaard and Kiehn, 1989; Major et al., 2008). Such bistability has been observed in motoneurons and cortical and hippocampal pyramidal cells (Lee and Heckman, 1998; Schiller et al., 2000; Wei et al., 2001).

According to one hypothesis (Lisman et al., 1998), bistability depends on the voltage dependence of the NMDA receptor (NMDAR), a class of glutamate-activated channels. This voltage dependence results from block of the channel pore by extracellular Mg²⁺ (Nowak et al., 1984). This block is relieved by depolarization, leading to peak inward (depolarizing) current at elevated membrane potentials (Fig. 1A,B). Working memory networks based on this mechanism can store patterns of input by the selective persistent firing of the subset of cells excited by a brief stimulus. Even with identical recurrent synapses, current selectively flows into active (depolarized) cells through their NMDARs, thereby maintaining activity in these cells. Feedback inhibition through GABA_ARs keeps other cells in the network from spuriously activating. In contrast to attractor network models of persistent firing that can only store previously learned patterns, networks with NMDAR-dependent bistability can store novel patterns (Durstewitz et al., 2000). Such models of NMDAR-dependent bistability have been extended to maintain graded patterns of activity (multistability) (Koulakov et al., 2002; Goldman et al., 2003). Consistent with a working memory function of NMDARs, antagonists of these channels interfere with delay period persistent firing (Shima and Tanji, 1998) and behaviorally measured working memory (Adler et al., 1998).

Figure 1.

Voltage dependence of NMDA, GABA_A, and GABA_B/KIR currents and the robustness of the bistability that they create. A, B, Voltage dependence of the NMDA and GABA_B-activated KIR conductances (A) and corresponding currents I = g(V − E_reversal) for maximal conductances of 1 μS (B). Also shown in B is the GABA_A I–V curve for g_GABAA = 0.1 μS. C, Combining the NMDA conductance with a GABA_A conductance can produce voltage bistability. Stable points are noted by solid circles. Solid curve, I–V curve for 5 μS GABA_A and 18 μS NMDA maximal conductances exhibits two stable membrane potentials. Dashed curves, ±5% change in GABA_A conductance destroys bistability. D, Adding a 40 μS GABA_B/KIR maximal conductance to 5 μS GABA_A and 20 μS NMDA maximal conductances produces more robust bistability that is maintained through the same changes in GABA_A that destroyed bistability in C. E, The range of combinations of GABA_A and NMDA conductances that give rise to bistability without GABA_B and with a fixed AMPA conductance of 0.04 mS/cm² (as shown by Lisman et al., 1998). F, Range of bistable combinations of GABA_A and NMDA is greatly expanded when a GABA_B/KIR conductance is added (here illustrated for a maximal GABA_B/KIR conductance of 51.2 mS/cm²). G, With a more typical 2:1 NMDA/AMPA ratio, there are no bistable combinations without GABA_B/KIR. H, The GABA_B/KIR conductance allows bistability in dendrites with a 2:1 NMDA/AMPA maximal conductance ratio.

One limitation of this mechanism of producing bistability is that it requires a rather precise balancing of GABA_A and NMDA conductances. Another limitation is that recurrent AMPA conductance must be very small (a 75:1 NMDA/AMPA conductance ratio was used by Lisman et al., 1998). Although NMDA-only (“silent”) synapses exist, they are rare later in development (Kerchner and Nicoll, 2008). Here, we consider how these limitations might be overcome. Synaptic inhibition is due to GABA_A and GABA_B receptors, but previous models of bistability considered only the former. In this study, we examine the effects of including the GABA_BR, a G-protein-coupled receptor with a variety of downstream targets, including a G-protein-activated inwardly rectifying potassium (KIR) channel (GIRK) (Kohl and Paulsen, 2010). Like the NMDAR, the KIR channel has a Mg²⁺ block. However, because the channel is blocked by intracellular Mg²⁺ rather than extracellular Mg²⁺, the block of the KIR channel is reduced as the cell hyperpolarizes (Yamada et al., 1998). As a result, the GABA_B/KIR conductance increases with hyperpolarization, enhancing outward (hyperpolarizing) current at hyperpolarized potentials (Fig. 1A,B). The potential utility of GABA_BRs in bistability has been noted (Shoemaker, 2011), but the efficacy of their cooperation with NMDARs in network persistent activity has not been analyzed. Here, we show that cooperation of GABA_B/KIR and NMDA conductances greatly enhances the robustness of bistability in working memory networks.

Materials and Methods

Network architecture.

The networks used in this study contained N = 400 neurons, of which N_p = 320 were excitatory and N_I = 80 were inhibitory. There were 320 incoming axons, each of which synapsed onto one excitatory cell and all inhibitory cells (feedforward inhibition was not strictly necessary, as successful simulations were run without it). The axons representing the pattern to be remembered fired a train of action potentials that was generated by a Poisson process of mean rate 200 Hz for the first 100 ms. Otherwise, incoming axons did not fire at all. The size of the activated population ranged from 10% to 75% of the total population. All excitatory neurons synapsed onto all other excitatory neurons with equal synaptic weights and onto all inhibitory neurons with equal synaptic weights. All inhibitory neurons synapsed onto all excitatory neurons with equal synaptic weights.

Neuron model.

The model neurons used in this study were Hodgkin-Huxley-type conductance-based neurons, modified from Lisman et al. (1998). The excitatory neurons had two compartments: a dendrite with voltage V_d and a soma with voltage V_s. Separating the spike-generating conductances from the bistable synaptic compartment allows bistability to be maintained during the large somatic voltage fluctuations associated with action potential generation. The equations for the integration of voltage are as follows:

where C_m = 1 μF/cm² and g_c = 0.1 mS/cm².

The leak current I_L = g_L(V − E_L), where g_L = 0.1 mS/cm², and E_L = −80 mV.

The spike-generating currents I_Na and I_K are described by the Hodgkin-Huxley formalism as in the work of Lisman et al. (1998), giving a threshold of ∼−45 mV and an action potential duration of ∼1 ms. Inhibitory interneurons were as described by Wang and Buzsáki (1996).

The synaptic currents I_syn,e = I_NMDA + I_AMPA and I_syn,i = I_GABA, where for AMPA and GABA_A, $I_x=g_x{\displaystyle \sum _i{s}_i\left(V_d-E_x\right)}$ (all maximal synaptic conductances g_x and reversal potentials E_x are in Table 1). $\frac{ds}{dt}=k_f\frac{1}{1+\exp \left(-V_{{\text{s}}_{\text{pre}}}/2\right)}\left(1-\text{s}\right)-k_{r}s$, where k_f = 12 ms⁻¹ for both AMPA and GABA_A, k_r = 1 ms⁻¹ for AMPA, and k_r = 0.1 ms⁻¹ for GABA_A.

Table 1.

Ranges of values of maximal synaptic conductances used in simulations

Synaptic conductance	Maximal g per synapse (mS/cm2)		Reversal potential
	Onto p-cells	Onto I-cells
AMPA	0.04/Np*	0.5/Np*	0 mV
NMDA	1–99.7/Np	0.3/Np	0 mV
External input	2	0.25/Np	0 mV
GABAA	0.1–3.2/NI	0	−70 mV
GABAB/KIR	0–102.4/NI	0	−90 mV

*The values of maximal AMPA conductances in the table do not apply where noted that AMPA scaled with NMDA. In simulations where AMPA scaled with NMDA, g_AMPA = g_NMDA/2 for pyramidal-to-pyramidal connections and g_AMPA = g_NMDA/16 for pyramidal-to-interneuron connections.

The NMDA current $I_{\text{NMDA}}=g_{\text{NMDA}}{\displaystyle \sum _i{s}_{\text{i}}\frac{V_d-E_{\text{NMDA}}}{1+0.15\exp \left(-0.08\left(V_d-E_{\text{NMDA}}\right)\right)}}$, giving a half-activation voltage of −23.7 mV. The synaptic gating variable s has second-order kinetics conferred by $\frac{ds}{dt}$ = α_sx(1 − s) − β_ss, where $\frac{ds}{dt}=\frac{{\alpha }_x}{1+\exp \left(-V_{s_{\text{pre}}}/s\right)}\left(1-x\right)-{\beta }_{x}x$, and α_x = 10, β_x = 0.5, α_s = 0.1, and β_s = 0.01 ms⁻¹, giving an NMDA time to peak of ∼10 ms and decay time constant of ∼100 ms. α_s was changed from 1 as described by Lisman et al. (1998) to reflect the lack of saturation of the NMDA conductance with single spikes (Popescu et al., 2004).

The intrinsic and GABA_B-activated KIR current $I_{{\text{GABA}}_{\text{B}}/\text{KIR}}=g_{{\text{GABA}}_{\text{B}}/\text{KIR}}\left(0.25+0.75{\displaystyle \sum _i{s}_i}\right)\frac{V_d-E_{{\text{GABA}}_{\text{B}}/\text{KIR}}}{1+\exp \left(0.1\left(V_d-E_{{\text{GABA}}_{\text{B}}/\text{KIR}}+10\right)\right)}$ (see Yamada et al., 1998, p. 742). The factor of $\left(0.25+0.75{\displaystyle \sum _i{s}_i}\right)$ makes 25% of the maximal KIR conductance constitutively active and 75% activated through GABA_BR activation of G-protein, the cooperative dynamics of which (Thomson and Destexhe, 1999) is given by $s=\frac{G^4}{G^4+K_d}$ and four coupled first-order rate equations: the normalized G-protein concentration G changes as $\frac{dG}{dt}$ = K₃R − K₄G, the fraction of active GABA_BRs R changes as $\frac{dR}{dt}$ = K₁T(1 − R) − K₂R, extracellular GABA concentration T in mm changes as $\frac{dT}{dt}$ = − k₁T(B_m − B) + k₋₁B − (1/τ_D)T and is incremented by 1 mm with each spike, and bound GABA transporter B changes as $\frac{dB}{dt}$ = k₁T(B_m − B) − (k₋₁ + k₂)B. K_d = 17.83, K₁ = 0.18 (mm*ms)⁻¹, K₂ = 0.0096 ms⁻¹, K₃ = 0.19 ms⁻¹, K₄ = 0.060 ms⁻¹, k₁ = 30 (mm*ms)⁻¹, k₋₁ = 0.1 ms⁻¹, k₂ = 0.02 ms⁻¹, and the maximum binding capacity of transporter B_m = 1 mm. The only change from Thomson and Destexhe (1999) is that for each synapse, 1 mm GABA was released with each spike (instead of 3 mm) and decayed with a time constant τ_D = 10 ms instead of diffusion being calculated by the gradient of transmitter. These equations give a latency of GABA_B-activated KIR current of ∼10 ms, time to peak of ∼50 ms, and a decay time constant of ∼80 ms as described by Thomson and Destexhe (1999). A slow time constant for GABA removal was chosen because much of the GABA_B response is mediated by extrasynaptic receptors (Kohl and Paulsen, 2010) and because the time constant for GABA decline at extracellular sites appears to be much slower than at synaptic sites. However, we found that results similar to those in Figure 2 could be obtained using properties characteristic of synaptic GABA (peak 3 mm; decay time constant 0.5 ms) if the decreased GABA lifetime was compensated for by raising interneuron firing rates through moderate increases in AMPA-mediated excitation onto inhibitory neurons. Thus, the model performance was robust across a broad range of assumptions about the dynamics of GABA diffusion and reuptake.

Figure 2.

Robustness of persistent firing in network simulations is increased by including GABA_B/KIR. A, Network architecture. A subset of excitatory cells (large triangular somas with rectangular dendrites) receive external synaptic input for 100 ms. The dendrites of these cells are depolarized (red), leading to somatic firing (red), while the dendrites of the other cells are hyperpolarized (blue). Recurrent excitatory axons cause both NMDA and AMPA current to pass at their synapses onto depolarized dendrites (small red triangles), while little NMDA current passes at synapses onto hyperpolarized dendrites (yellow). Inhibitory axons cause both GABA_A and GABA_B/KIR current to pass at their synapses onto hyperpolarized dendrites (small red circles), while little GABA_B/KIR current passes at synapses onto depolarized dendrites (yellow). B, Top, Initially stimulated cells (black bars on y-axis) maintain their firing after external stimulation ends, without activity spreading to other cells (black points at time of each spike). Bottom, Neuron 200, which is in the activated subset, continues to fire after the external input ceases, while neuron 300 barely moves away from its resting potential despite receiving the same presynaptic activation as neuron 200 after the first 100 ms. Maximal conductances: NMDA, 7; AMPA, 3.5; GABA_B/KIR, 50; GABA_A, 0.7 mS/cm²; 160 excitatory cells initially stimulated. The voltage dependence of the GABA_B/KIR conductance causes the very large nominal GABA_B/KIR to GABA_A conductance ratio despite their reasonable effective conductance ratio (see Materials and Methods). C–G, The simulation was run repeatedly, varying the size of the initially stimulated population and maximal GABA_A, GABA_B/KIR, and NMDA conductances (see Materials and Methods). The number of successful pattern sizes of 14 tested pattern sizes for a given combination of conductances is shown. C, Combinations of maximal GABA_A and NMDA conductances that lead to selective persistent activity for simulations based on the original model (Lisman et al., 1998) (no GABA_B/KIR, negligible AMPA). D, Adding the GABA_B/KIR conductance increases the robustness of the persistent firing. E, If the NMDA/AMPA maximal conductance ratio was set at a more typical value of 2, no simulations produced selective persistent activity without GABA_B/KIR. F, With the addition of sufficient GABA_B/KIR conductance, selective persistent activity did occur. G, The proportion of simulations over the tested range of parameters that exhibited selective persistent activity increases with the addition of the GABA_B/KIR conductance. Each bar represents an average over an entire 2-D plot of the type illustrated in C–F. For example, C is quantified in the leftmost bar of the left bar graph and F is quantified in the penultimate bar of the right bar graph.

The external synaptic input I_syn,ext = s_input g_input(V − E_syn), where g_input is the maximal conductance given in Table 1 and the synaptic activation s_input is the fraction of that conductance that is activated. s_input increased by 50% of the as-yet-unactivated conductance (i.e., 0.5(1 − s_input)) with each spike and decayed exponentially with a time constant of 2 ms.

Noise was implemented as independent excitatory and inhibitory conductance noise rand drawn at each time step from a uniform distribution between ±0.05/$\sqrt{dt}$ mS/cm², where dt is in milliseconds. I_noise = rand*(V − E_AMPA) + rand*(V − E_GABAA). Noise was generated independently for each compartment of each cell.

Simulations.

Simulations with the above model were run over all combinations of the following parameters: pattern sizes P were spaced linearly according to the formula P = (0.1 + 0.05i)N for integers i from 0 to 13. Maximal conductances were chosen to give approximately even spacing on a logarithmic scale, as follows: g_NMDA = (4/3)ⁱ/10 mS/cm², rounded to the nearest tenth, for i from 8 to 24; g_GABAA = 0.1, 0.13, and (4/3)ⁱ/10 mS/cm², rounded to the nearest tenth, for i from 3 to 12. g_GABAB = 0 and 2ⁱ/10 mS/cm², for i from 0 to 10. The initial dendritic and somatic voltages for each neuron were chosen independently from a uniform distribution over the interval −80 to −60 mV.

To check the plausibility of the relative magnitudes of the GABA_A and GABA_B conductances used, we compared simulations of GABA_A and GABA_B IPSP magnitudes to data presented by Benardo (1994). Their Figures 6D and 7B show that the change in IPSP magnitude per 10 mV change in somatic membrane potential was ∼2 mV for GABA_B and ∼1.7 mV for GABA_A, giving a ratio r of ∼1.2. We checked whether our model neurons gave similar results for the conductance values used in Figure 2B (which is near the center of the parameter space giving successful simulations). To do this, we simulated the IPSP amplitudes measured at the soma for different somatic membrane potentials and found r of ∼1.3, in good agreement with experimental values.

The criterion for successful maintenance of selective persistent activity was that the firing rate of the externally stimulated subset be >50 Hz during the last 50 ms of the 250 ms simulation, while the unstimulated population must have a firing rate of <10 Hz. The firing rates clustered bimodally above 90 Hz and below 5 Hz, so the precise cutoffs did not substantially affect the results.

Simulations were written in C++. Numerical integration was performed using Euler's method and dt = 0.025 ms. Code is available upon request.

Results

Previous work that did not consider GABA_B (Lisman et al., 1998) showed analytically that the sum of NMDA and GABA_A currents (Fig. 1B) can produce an “N-shaped” I–V (current–voltage) curve that crosses zero current at three voltages (Fig. 1C, solid curve). The crossings that have positive slope are stable: small perturbations from these voltages result in corrective current returning the voltage to its original level. As can be seen in the I–V curve, there are two zero crossings with this property, thus demonstrating bistability. However, the bistability is not very robust. Because the peaks and troughs of the I–V curve are relatively shallow, small changes in GABA_A conductance can cause loss of one of the stable points. As shown in Figure 1C, ±5% change in GABA_A is sufficient to destroy bistability. When GABA_B/KIR is added (Fig. 1D), the sharpness of the N-shaped I–V curve is increased so that GABA_A can change over a much larger range without loss of bistability. A more systematic exploration of the robustness produced by GABA_B/KIR is shown in Figure 1E–H. The I–V curve was analyzed for bistability over a wide range of combinations of GABA_A and NMDA conductances. The thinness of the sliver of combinations leading to bistability in E indicates that synaptic conductances have to be precisely tuned under a model with no GABA_B/KIR conductance. Including a GABA_B/KIR conductance (Fig. 1F) greatly increases the range of combinations of GABA_A and NMDA conductances that give bistable I–V curves. When the model is constrained to a more typical 2:1 NMDA/AMPA ratio (see Myme et al., 2003, their Table 1, for review), no combinations of NMDA and GABA_A alone lead to bistability (Fig. 1G). However, when GABA_B/KIR is included, there is a large region of GABA_A and NMDA conductance combinations that lead to bistability (Fig. 1H). Furthermore, the GABA_B/KIR conductance does not have to be precisely tuned with respect to the NMDA conductance (data not shown). Of course, if NMDA conductance is greatly decreased and/or GABA_B/KIR greatly increased, the dendrite will be unable to maintain the UP state, and likewise if NMDA conductance is greatly increased and/or GABA_B/KIR greatly decreased, the dendrite will be unable to maintain the DOWN state. However, overall, inclusion of the GABA_B/KIR conductance leads to robust bistability of I–V curves that is tolerant to large changes in either conductance.

We next examined the effects of including the GABA_B-activated KIR conductance in network simulations of working memory. The network (Fig. 2A) included 320 excitatory neurons, connected all-to-all, and 80 inhibitory interneurons that each provide global feedback inhibition. Excitatory neurons had a dendritic compartment containing the synaptic conductances connected to a somatic compartment containing the spike-generating conductances. The inhibitory neurons had a single compartment. Each excitatory neuron received external input from a single external axon; external inputs also projected nonselectively to all inhibitory neurons, although the presence of external input to inhibitory neurons was not found to be critical to the results presented below. The pattern to be stored in working memory was determined by the subset of these axons that were briefly active (200 Hz for 100 ms). KIR channels can be activated by receptors or can be constitutively active. In our simulations, 25% of the maximal KIR conductance was constitutively active and the other 75% was activated by GABA_BRs through G-proteins (similar results hold for different ratios, data not shown). The relative magnitudes of GABA_A and GABA_B/KIR conductances were chosen to be physiologically plausible (see Materials and Methods). If operating correctly, this network should selectively maintain activity in the excitatory cells that received external input. This selective maintenance should occur despite the fact that all cells receive identical recurrent presynaptic activity. As shown in Figure 2B, the externally activated excitatory neurons (red trace, example neuron) flip into the stable depolarized state, while excitatory neurons not receiving external input (blue trace) stably remain in the hyperpolarized state.

To understand quantitatively how persistent firing is affected by the GABA_B/KIR conductance, we tested the maintenance of persistent firing in the network simulation for a broad set of combinations of maximal NMDA and GABA_A conductances, sizes of the input pattern to be remembered, and maximal GABA_B/KIR conductance (for ranges and criterion of success, see Materials and Methods).

When we ran the simulations with negligible AMPA (as described by Lisman et al., 1998), the range of acceptable conductances and pattern sizes increased dramatically when a GABA_B/KIR conductance was included (Fig. 2C,D, quantified in G). Without GABA_B/KIR, the range of input pattern sizes that could be selectively maintained in memory for a given combination of NMDA and GABA_A conductances occupied only ∼10% of the total network size. When GABA_B/KIR was added, the size of the input pattern could be varied over the entire range that we inspected (10%–75% of the network). We next ran simulations in which the relative AMPA component was larger (2:1 NMDA/AMPA maximal conductance ratio), as in Figure 1G,H. Under these conditions, persistent activity occurred when the GABA_B/KIR conductance was >∼10 mS/cm² (Fig. 2E,F, quantified in G, left). Without GABA_B/KIR, none of the simulations maintained selective persistent activity. In summary, our simulations show that combining GABA_B/KIR with the NMDA conductance greatly enhances the robustness of bistability and persistent firing dependent on it.

Discussion

Here, we showed that the GABA_B-activated and intrinsic KIR conductances can work together with the NMDA conductance to produce robust bistability. In particular, we showed that bistability is maintained over much larger ranges of conductances and input pattern sizes when GABA_B/KIR is present than when it is not. We also showed that, with GABA_B/KIR, selective persistent firing can be maintained even when there is a significant AMPA component of transmission at recurrent synapses.

Networks based on the principles that we have described may be of special utility in the brain because they can store novel patterns and thus do not depend on previous learning. This stands in contrast to traditional attractor-based models of persistent firing, in which recurrent excitation is mediated by synapses whose strength has already been selectively altered by learning (Hopfield and Herz, 1995; Wang, 2001). In attractor networks, a memory trace is maintained because cells that represent a stored pattern have strong mutual synapses, while their synapses onto other cells are weak. Such networks have the disadvantage that they cannot store novel patterns. On the other hand, because they have attractor dynamics, the network can reconstruct and stably maintain patterns that are presented in a corrupted form. In our model, all synaptic weights are equal; the effective strength of connections between the neurons is determined only by the level of depolarization in the postsynaptic dendrites. This allows the network to maintain the firing of a pattern that it has never seen before, but its fidelity to the presented pattern prevents it from correcting errors. One function of such networks may be to produce the brief periods of repetitive firing that are necessary to drive attractor formation in downstream networks.

Although bistable behavior that is dependent on NMDARs has been observed in brain slices (Schiller et al., 2000; Wei et al., 2001), we predict that it is more common in vivo. This is because interneurons in vivo are typically more active than in slice preparations, leading to enhanced GABA_B activation. Thus, if the bistability based on GABA_B is to be investigated in slices, methods will have to be developed to activate these receptors in a manner similar to that expected in vivo (Kohl and Paulsen, 2010).

The NMDA and GABA_B/KIR conductances appear to be the perfect couple for producing bistability. Most notably, the two conductances have a complementary voltage dependence: in the depolarized state, the NMDA conductance is “on,” whereas the KIR channel activated by GABA_B is “off”; the converse is true in the hyperpolarized state. Furthermore, bistability requires that the conductance ratio of GABA_B/KIR to NMDA stays within a certain range. The fact that both conductances have long time constants of decay helps to smooth out fluctuations and thereby maintain a fixed ratio. Another way in which they are well suited for balancing each other is that both conductances scale with network activity. This enhances robustness compared with the situation in which NMDA current scales with activity, but the KIR conductance is fixed (Gruber et al., 2003). Some limits on perfect balancing of the two conductances should, however, be noted. Constitutively active KIR (Doupnik et al., 1995) or KIR channels activated by neuromodulators (Yamada et al., 1998) may also be beneficial for maintaining persistent firing, for example by providing a baseline level of KIR current while additional GABA_B/KIR conductance slowly activates. This means that the activity-dependent scaling of NMDA and GABA_B/KIR will not be exact. Furthermore, it is unclear exactly how well the time course of the two conductances match, as the time course of GABA_B can vary widely over brain regions (Thomson and Destexhe, 1999); measurement of the two conductances under the same conditions would be useful for more detailed analysis of this issue.

It had generally been assumed that the functions of GABA_B were based on its slow time course (e.g., theta oscillations; Wallenstein and Hasselmo, 1997). Our work suggests that the strong voltage dependence of the GABA_B/KIR conductance is also important, specifically that this voltage dependence couples with that of the NMDA conductance to produce a robust form of membrane voltage bistability. Such bistability may be important in various forms of working memory. Previous work has shown that NMDAR antagonists interfere with working memory (Adler et al., 1998). It will thus be important to test whether GABA_B antagonists or interneuron hypofunction also produce deficits in working memory. With this in mind, the GABA_B/KIR conductance should be taken into account in further work on bistability and working memory.