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1Department of Complexity Science and Engineering, Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa; 2Exploratory Research for Advanced Technology Aihara Complexity Modelling Project, Japan Science and Technology Agency, Tokyo; and 3Institute of Industrial Science, The University of Tokyo, Tokyo, Japan
Submitted 21 September 2004; accepted in final form 27 January 2005
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ABSTRACT |
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INTRODUCTION |
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-aminobutyric acid (GABA), well known as an inhibitory neurotransmitter in the vertebrate central nervous system, could play an excitatory role in a sense that GABAergic inputs can facilitate action potential generation in cooperation with glutamatergic inputs, depending on their timing and location in the mature mammalian neocortex (Gulledge and Stuart 2003
). This excitatory action of GABA is attributed to the reversal potential of GABAA-receptor–associated chloride channels, which was recently revealed to be considerably higher (by about 10 mV) than the resting membrane potential (Gulledge and Stuart 2003). Because this reversal potential is still lower than the firing threshold, GABAergic inputs can have bidirectional actions: those temporally or spatially separated from glutamatergic inputs facilitate action potential generation, whereas nonseparated inputs inhibit it. Such behavior might agree with the notion of discriminative inhibition, originally thought to occur in the hippocampus (Andersen et al. 1980). However, because such input-specific discriminative inhibition at the level of single synaptic inputs can be implemented even more faithfully through the traditional purely shunting inhibition with the GABAA reversal potential equal to the resting potential, the exact meaning of the depolarized value of the GABAA reversal potential, and that of the facilitatory action of GABA, are still elusive.
Shunting inhibition has often been studied in relation to the neuronal input–output relationship. Originally, the GABAA-associated shunting conductance was considered to naturally implement the divisive inhibition on the neuronal input–output relationship observed in simple cells in primate primary visual cortex (Carandini and Heeger 1994). However, using both a simple integrate-and-fire neuron model and a detailed multicompartment model, Holt and Koch (1997) showed that shunting inhibition has a subtractive rather than a divisive effect on the firing rate. Recently, Mitchell and Silver (2003) showed that GABAergic shunting inhibition actually decreases the firing rate in a divisive manner if in vivo–like highly fluctuating glutamatergic and/or GABAergic inputs are considered instead of tonic ones in the cerebellar granule cells. In another context, Chance et al. (2002) showed that fluctuating inhibition can implement gain modulation when it is balanced with fluctuating excitation. In all these studies, the GABAA reversal potential was set to be equal to, or slightly lower than, the resting potential in the theoretical models and the simulations, as well as in the experiments using the dynamic clamp (conductance injection) technique (Robinson and Kawai 1993; Sharp et al. 1993). How these results would change if the GABAA reversal potential is more depolarized has not yet been examined either in experiments or in models.
In this paper, we examine the effects of the depolarized GABAA reversal potential on the relationship between the input conductances and the output firing rate. We use a simple conductance-based 2-dimensional (2D) model of neocortical neurons (Wilson 1999a, b), which permits us to dynamically understand the effects of the GABAergic conductance on the phase plane. At first, we show that experimental results with respect to the depolarized GABAA reversal potential (Gulledge and Stuart 2003) are quantitatively reproduced by this neuron model. Then, we examine the effect of the tonic GABAergic conductance on the input–output relationship, that is, the relationship between the tonic glutamatergic conductance and the output firing rate. We show that the depolarizing GABAergic conductance up to a certain amount does not affect the input–output relationship, whereas the greater GABAergic conductance prevents all spike generation; thus, the depolarizing GABAergic conductance does not have a continuous inhibitory effect, but behaves instead in an ON–OFF-like manner. We explain the mechanism of this feature on the phase plane in comparison with the case of purely shunting inhibition. Next, we consider the effect of highly fluctuating inputs instead of tonic ones. We represent the fluctuating input conductances as a summation of the unitary conductances that are generated by a Poisson process. When there is no coincidence between glutamatergic and GABAergic inputs, a wide range of depolarizing GABAergic inputs does not affect the input–output relationship, indicating that they have little effect on the firing rate of a neuron driven by uncorrelated glutamatergic inputs. As the degree of coincidence between glutamatergic and GABAergic inputs increases, the strength of GABAergic inputs required to modulate the firing rate decreases. Therefore there is a particular strength range of depolarizing GABAergic inputs that allows the inputs to selectively modulate the firing rate only if a large quantity of glutamatergic driving inputs are coincident with them. Based on these findings, we propose a possibility that the depolarized GABAA reversal potential works as a sophisticated device for the discriminative firing rate modulation so that depolarizing GABAergic inputs modulate the firing rate only when the neuron is driven by the glutamatergic inputs that are considerably correlated with the GABAergic inputs. This main prediction on the effects of fluctuating depolarizing GABAergic inputs still holds, at least to some extent, if we use the integrate-and-fire neuron model instead of Wilson's 2D neocortical neuron model, although there is a certain difference between the results for these two models, especially in the tonic conductance case, as we will discuss.
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METHODS |
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We use the single-compartment model of a neocortical neuron proposed by Wilson (Wilson 1999a, b)
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 | (2) |
(ms) is the time constant of the inactivation variable. This neuron model has its roots in the early works where Hodgkin–Huxley-type equations (Hodgkin and Huxley 1952) with 4 or more variables were reduced to 2D ones (Hindmarsh and Rose 1982; Rinzel 1985). The key feature of this Wilson's model is that the existence of the A-current causes the shape of f(V) to curve as a quadratic function, enabling the neuron firing rate to vary over a wide range, starting from 0 Hz [called type I or class I excitability (Hodgkin 1948)] as observed in cortical pyramidal cells. The precise forms of functions gNa(V) and f(V), as well as the values of other parameters, were determined by Wilson (1999b) as gNa(V) = 0.1781 + 4.758 x 10–3V + 3.38 x 10–5V2; f(V) = 1.29 x 10–2V + 0.79 + 3.3 x 10–4(V + 38)2; C = 1 µF/cm2; as = 10 (10–10 m2)1 (that is, equal to 1000 µm2); ENa = 48 mV; EK = –95 mV; = 5.6 ms; and cK = 260 nS. Isyn (pA) in Eq. 1 represents the following current through synaptic channels
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Wilson's two-compartment model
To study the effects of the somato-dendritic location of GABAergic inputs, we use a 2-compartment version of Wilson's model (Wilson 1999a)
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 | (7) |
associated with the relative density of the voltage-dependent channels (see below). Vs and Vd (mV) are the respective membrane potentials of the axo-somatic compartment and the dendritic compartment; Rs and Rd are the respective inactivation variables; as and ad (10–10 m2) are the respective areas; is a ratio of the density of voltage-dependent sodium and potassium channels in the dendritic compartment compared to that in the axo-somatic compartment; gsd (nS) is the conductance between the axo-somatic compartment and the dendritic compartment; Is and Id (pA) represent the current through synaptic channels (described as Eq. 3) into the axo-somatic compartment and the dendritic compartment, respectively. Parameter values are set as follows: as = ad = 10 (10–10 m2); = 0.05 (Wilson 1999a); gsd = 5 nS, which approximately correspond to the conditions that intracellular resistivity (the total resistance across a 1-cm cube of intracellular cytoplasm) is 300 
· cm (Major et al. 1993), the diameter of the dendrite is 2 µm, and the distance between the axo-somatic compartment and the dendritic compartment is 200 µm. Resting potential, steady-state action potential threshold, and reversal potentials
The key experimental result by Gulledge and Stuart (2003) is that the reversal potential of the chloride channels associated with GABAA synapses is considerably higher than the resting potential. They report that the resting potential was –79 ± 1 mV and the GABAA reversal potential at a soma was –68 ± 2 mV. They also report that the steady-state action potential threshold was –63 ± 1 mV. To find an appropriate value for the GABAA reversal potential in Wilson's neuron model, we have to compute its resting potential and the steady-state firing threshold. To obtain these, we consider the steady-state current–voltage relationship, whose minimal and middle zero crossing points respectively correspond to the resting potential and the steady-state action potential threshold (Koch et al. 1995). As a result, the resting potential of Wilson's model is –75.4 mV, and the steady-state action potential threshold is –58.2 mV. In this way, we obtain a range between the resting potential and the steady-state firing threshold in Wilson's model that is similar to the value for the neocortical pyramidal cells recorded in Gulledge–Stuart experiments: 17.2 mV in Wilson's model and 16 mV in the Gulledge–Stuart experiments, although both values of the resting potential and the steady-state firing threshold are about 4–5 mV higher in Wilson's model. Thus we set the value of the GABAA reversal potential of Wilson's model at –64 mV so that it is nearly the same proportion as in Gulledge–Stuart experiments; the difference between the resting potential and the GABAA reversal potential is 11.4 mV in the model and 11 mV in the Gulledge–Stuart experiment, and the difference between the GABAA reversal potential and the steady-state firing threshold is 5.8 mV in the model and 5 mV in the Gulledge–Stuart experiment. In the following, we refer to this value (EGABA = –64 mV) of the GABAA reversal potential in Wilson's model as the depolarized GABAA reversal potential. For comparison, we will also examine the case with the GABAA reversal potential that is equal to the resting potential of Wilson's model (i.e., –75 mV). We refer to this value (EGABA = –75 mV) of the GABAA reversal potential as the purely shunting GABAA reversal potential. The reversal potential of the non-NMDA glutamate receptor-channel is set to EGlu = 0 mV throughout this paper. We confirm that our main results basically do not depend on the precise values of these parameters.
Synaptic conductances
When we try to reproduce the results of the Gulledge–Stuart experiments, we assume that the time courses of synaptic conductances, gGlu(t) and gGABA(t), are respectively represented by a double and a triple exponential, as in their experiments
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Glu and GABA are the maximum conductances of glutamatergic and GABAergic synaptic inputs; Glu_rise and Glu_decay are the rising and decaying time constants of glutamatergic synapses; GABA_rise, GABA_decay1, and GABA_decay2 are the rising, the first, and the second decaying time constants of GABAergic synapses; a1 and a2 are the amplitudes of the first and the second exponentially decaying terms of GABAergic synapses; c and c' are the normalizing factors so that the maximum values of the blankets is equal to 1. All the parameter values are set to equal those in Gulledge–Stuart experiments and simulations: Glu_rise = 0.3 ms; Glu_decay = 3 ms; GABA_rise = 0.5 ms; GABA_decay1 = 3.2 ms; GABA_decay2 = 12.3 ms; a1 = 1; a2 = 2.2. In this setting, a glutamatergic synaptic input cannot evoke an action potential if its maximum conductance (Glu) is 17 nS, which corresponds to the maximum excitatory current of about 12 nA, but can do so if Glu is 17.5 nS. We use these values as the subthreshold and the suprathreshold input, respectively, when we try to reproduce the experimental results of Gulledge and Stuart (2003). Nullclines and phase plain analysis
A nullcline is the set of the variable values where the velocity of the state point of one variable is equal to 0. The nullclines of Wilson's model with tonic conductances are defined as the 2 curves obtained from Eq. 1, where Isyn is substituted by Eq. 3, and from Eq. 2 by setting each differential coefficient equal to 0 as follows
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), and therefore that the neuron does not generate action potentials but stays at the resting state if without stimulus, as shown in Fig. 1C. On the other hand, Fig. 1B shows the V-nullcline (the red curve) and the R-nullcline (the blue curve) of Wilson's model when the glutamatergic conductance is added (gGlu = 0.5 µS and gGABA = 0). In this case, there is only one equilibrium point (indicated by the green square) that is unstable, and thus there is no stable equilibrium point. Instead, there is a stable oscillatory solution (indicated by green dashed curve), which represents the repetitive firing of the neuron as shown in Fig. 1D.
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Bifurcations occur when the system meets a singularity. The set of the parameter values where a bifurcation occurs is referred to as a bifurcation set, or a bifurcation curve when there are 2 parameters. Wilson's model with tonic conductances is a 2D autonomous dynamical system with 2 parameters. It is abstractly represented as follows
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, 
) represent time derivatives. At an equilibrium point satisfying
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). Repeating this process for a series of gGlu values produces (gGlu, gGABA) value sets on the bifurcation curve, which is drawn later in Figs. 3E and Fig. 7.
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The in vivo input conductances in neocortical neurons are known to fluctuate greatly. For simplicity, we represent such a fluctuated conductance as a summation of the unitary conductances generated according to a Poisson process. Specifically, we assume that a unitary conductance, which is described as the following difference of the 2 exponential functions with the rising time constant rise = 1 ms and the decaying time constant decay = 10 ms; is generated according to a Poisson process.
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. Instead, the firing irregularity is well accounted for through the assumption that synaptic inputs are clustered rather than temporally independent; that is, spikes from up to tens or hundreds of presynaptic neurons arrive almost synchronously (Stevens and Zador 1998). If that is the case, setting the rate of the Poisson process in our model equal to that of the presynaptic spike arrivals, which may be several thousands per second (Hz) or more, would result in too many regular input conductances to generate a spike train as irregular as that observed in vivo. Therefore, we set the Poisson rate much lower than the spike arrival rate, specifically to 50 Hz. Under this context, the unitary conductance (Eq. 17) generated at each Poisson event does not represent the unitary conductance at a single synapse—instead it represents a summation of several tens or hundreds of nearly synchronized synaptic inputs of the same type, that is, either glutamatergic or GABAergic. In this sense, we refer to the unitary conductance (Eq. 17) as the input event. The strength, or the maximum amplitude, of each input event (
in Eq. 17) is proportional to the number of presynaptic neurons whose spikes are included in the event. Integrate-and-fire neuron model
The integrate-and-fire neuron model consists of 2 elements: one equation describing the temporal evolution of the membrane potential below the firing threshold and one condition determining the generation of action potentials. We describe the subthreshold dynamics of the membrane potential as follows
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m is the membrane time constant, which is a product of the membrane capacitance and the membrane resistance; and Isyn is the current through synaptic channels represented by Eq. 3. Spike generation is described as thresholding and resetting of the membrane potential in the above equation. Whenever the membrane potential V reaches a value Vth, which is called a firing threshold, we presume that an action potential is generated at that instance, and V is reset to another value Vreset, which is called a reset potential. Afterward, the membrane potential V is kept constant at V = Vreset during an absolutely refractory period abs, and then evolves again according to Eq. 18. The resting potential and the firing threshold are set equal to those of Wilson's model: Vrest = –75 mV and Vth = –58 mV. The reset potential is set to Vreset = –75 mV. We have confirmed that changing the reset potential to Vreset = –69 mV does not largely alter the overall properties. Other parameters are set as m = 20 ms and abs = 2 ms. When the glutamatergic and GABAergic conductances are tonic—i.e., gGlu(t) and gGABA(t) in Eq. 3 take constant values gGlu and gGABA, respectively—combining Eq. 18 with Eq. 3 produces a one-dimensional linear differential equation on the membrane potential V. Subsequently, the condition under which the neuron generates action potentials can be calculated exactly as follows
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m and c2 = (Vrest + gGluEGlu + gGABAEGABA)/m. Although this expression is too complicated to provide any particular insights, it is clear that the firing rate is a continuous function of gGlu and gGABA, and so there should be no discontinuous jumps of the firing rate on the gGlu–gGABA parameter plane.
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All simulations were calculated by MATLAB (The MathWorks, Natick, MA).
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RESULTS |
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We tried to reproduce the experimental results regarding the effects of a somatically applied depolarizing GABAergic synaptic input on spike generation (Gulledge and Stuart 2003) using the conductance-based model of a neocortical neuron with a depolarized GABAA reversal potential that is about 11 mV higher than the resting potential (see METHODS for details). In this setting, a glutamatergic synaptic input cannot evoke an action potential if its maximum conductance (Glu in Eq. 8) is 17 nS (Fig. 2B), which corresponds to the maximum excitatory current of about 12 nA, but it can do so if Glu is 17.5 nS (Fig. 2D). If a GABAergic input evokes an action potential in cooperation with the subthreshold (Glu = 17 nS) glutamatergic input, it is called a facilitatory action. On the other hand, if a GABAergic input inhibits spike generation by the suprathreshold (Glu = 17.5 nS) glutamatergic input, it is called an inhibitory action. For example, a GABAergic input that is of the same strength as the subthreshold glutamatergic input and precedes it by 8 ms evokes an action potential (Fig. 2C), and thus has a facilitatory action. On the other hand, another GABAergic input that is coincident with the suprathreshold glutamatergic input prevents spike generation (Fig. 2E), and thus has an inhibitory action. We systematically examined which timing and strength conditions are necessary for depolarizing GABAergic inputs so that they have facilitatory or inhibitory actions. The results are summarized in Fig. 2K. When the strength of the glutamatergic input is fixed at Glu = 17 nS (subthreshold) and the strength of the GABAergic input is set to the same value, the depolarizing GABAergic input has a facilitatory action if it arrives about 5.8 ms or more before the glutamatergic input (see the black dotted line in Fig. 2K). This critical timing of 5.8 ms closely matches the results of Gulledge and Stuart (5.5 ± 0.6 ms in Fig. 5E of Gulledge and Stuart 2003). On the other hand, when the strength of the glutamatergic input is fixed as Glu = 17.5 nS (suprathreshold) and the strength of the GABAergic input is set to the same value, the depolarizing GABAergic input has an inhibitory action if it arrives about 4.5 ms or less before the glutamatergic input, which also agrees with the experimental results. Therefore, Wilson's model with the depolarized GABAA reversal potential can reproduce the results of Gulledge and Stuart (2003) regarding the timing dependency of the effects of a somatically applied depolarizing GABAergic input quantitatively as well as qualitatively. Actually, our simulation results suggest that as the magnitude of the GABAergic input increases, the temporal borderline between a facilitatory action and an inhibitory one shifts toward the earlier times as shown in Fig. 2K. If the maximum conductance of the GABAergic input is about 5 times higher than that of the glutamatergic input (i.e., GABA = 90 nS), the temporal borderline is about 12 ms before the onset of the glutamatergic input.
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was to examine effects of dendritically applied depolarizing GABAergic inputs on spike generation by somatically applied glutamatergic inputs. Because Wilson's model we have used above is a single-compartment model, it cannot reproduce the results regarding the somato-dendritic difference. To further verify the representational ability of Wilson's model, we examined whether the extension of Wilson's model to 2 compartments (Wilson 1999a) could reproduce the experimental results regarding dendritically applied GABAergic inputs as well or not. The 2-compartment version of Wilson's model consists of the axo-somatic compartment representing the axon hillock and initial segment of the neuron where the densities of the voltage-dependent sodium and potassium channels are so high that action potentials are generated, and the dendritic compartment representing a portion of the dendrite located about 200 µm from the soma where the densities of the active channels are about 20 times lower than those in the axon hillock and initial segment (see METHODS for details). In this extended setting, a glutamatergic synaptic input to the axo-somatic compartment cannot evoke an action potential if its maximum conductance (Glu in Eq. 8) is 18.5 nS (Fig. 2F), which corresponds to the maximum excitatory current of about 13 nA, but it can do so if Glu is 19 nS (Fig. 2I). We examined whether a dendritically applied GABAergic input facilitates, or inversely prevents, spike generation by a somatically applied glutamatergic input. For example, a dendritically applied GABAergic input that is of the same strength as the somatically applied subthreshold glutamatergic input and precedes it by 8 ms evokes an action potential (Fig. 2G), and thus has a facilitatory action. Moreover, another dendritically applied GABAergic input that coincides with the somatically applied subthreshold glutamatergic input also evokes an action potential (Fig. 2H), and thus also has a facilitatory action. Meanwhile, a dendritically applied GABAergic input that coincides with the somatically applied suprathreshold glutamatergic input does not prevent spike generation (Fig. 2J), and thus does not have an inhibitory action. We systematically examined which timing and strength conditions are necessary for somatically or dendritically applied depolarizing GABAergic inputs to have facilitatory or inhibitory effects on somatically applied glutamatergic inputs. Figure 2, L and M show the result for somatically applied GABAergic inputs and that for dendritically applied ones, respectively. Figure 2L resembles the result obtained from the single-compartment model in Fig. 2K, and again reproduces well the experimental results of Gulledge and Stuart (2003); however, the result for the 2-compartment model (Fig. 2L) has a slightly larger "inhibitory action" region (indicated by blue in the figure) than that for the single-compartment model in Fig. 2K, possibly because the total resistivity is reduced because of the dendritic compartment. Also, Fig. 2M indicates that dendritically applied GABAergic inputs facilitate spike generation unless they arrive 1.2 ms or more later than somatically applied glutamatergic inputs, and that dendritically applied GABAergic inputs have no inhibitory actions at all. These results closely match those of Gulledge and Stuart (Fig. 5B of Gulledge and Stuart 2003). Therefore, Wilson's 2-compartment model is able to quantitatively reproduce most of the main findings of Gulledge and Stuart (2003) regarding the effects of somatically and dendritically applied GABAergic inputs on spike generation by somatically applied glutamatergic inputs. In the following, we will return to use the single-compartment model for the sake of simplicity to examine the effects of GABAergic inputs on the neuronal input–output relationship. This choice might be justified if we restrict our discussion to the somatically applied GABAergic inputs, presumably from fast spiking cells that preferentially innervate somata or perisomatic regions. We discuss more about this issue under DISCUSSION. Input–output relationship with tonic conductances
At first, we examine how a neuronal firing rate changes depending on tonic (i.e., not time-varying) glutamatergic and GABAergic conductances. These tonic conductances might be regarded as a model of the input conductance produced by asynchronous synaptic inputs with a high frequency. In addition, they also provide a basis for understanding the case with fluctuating inputs that we describe later. We consider Wilson's neuron model with tonic synaptic conductances represented by 2 time-independent parameters gGlu (the glutamatergic conductance) and gGABA (the GABAergic conductance). When the GABAergic conductance is absent (gGABA = 0 nS), the increase of the glutamatergic conductance causes the firing rate to monotonically increase as shown in Fig. 3A. In the following, we examine the effects of GABAergic conductance on this input–output relationship. Although there have been many studies regarding the effect of the purely shunting GABAergic conductance, whose reversal potential is equal to the resting potential, on the input–output relationship (Holt and Koch 1997; Mitchell and Silver 2003), we reexamine it here for comparison with the depolarizing case. Figure 3B illustrates the sample spike trains of Wilson's model with a fixed glutamatergic conductance (gGlu = 5 nS) and 9 different values of the purely shunting (left column) or depolarizing (right column) GABAergic conductance (gGABA = 0, 5, 10, 15, ... , 40 nS).
As the purely shunting GABAergic conductance increases from gGABA = 0 nS to gGABA = 30 nS, the input–output relationship curve gradually shifts to the right as shown in the left panel of Fig. 3C: 7 black solid curves indicate the input–output relations corresponding to gGABA = 0, 5, 10, 15, ... , 30 nS, from the left to the right. This result is in agreement with the previous work, which showed that the purely shunting GABAergic conductance has a subtractive effect on the firing rate (Holt and Koch 1997). On the other hand, as the depolarizing GABAergic conductance, whose reversal potential is about 11 mV higher than the resting potential, increases at the same rate, the input–output relationship does not shift to the right; it barely changes for this range (gGABA = 0–30 nS) of the GABAergic conductance as shown in the right panel of Fig. 3C: 7 overlapped black solid curves indicate the input–output relations corresponding to gGABA = 0, 5, 10, 15, ... , 30 nS. A further increase of the depolarizing GABAergic conductance then drastically causes the input–output curve to drop to the baseline: 3 blue dotted curves in the right panel of Fig. 3C indicate the input–output relations corresponding to gGABA = 35, 40, 45 nS from the left to the right. The difference in the effect between the purely shunting GABAergic conductance and the depolarizing one is also apparent in the figure of spike trains (see Fig. 3B). The depolarizing GABAergic conductance up to 35 nS hardly changes the firing rate, although it looks like slightly decreasing the spike amplitudes, whereas that of 40 nS completely stops the firing. Between gGABA = 35 nS and gGABA = 40 nS, there is a critical amount of the conductance where the repetitive firing disappears.
Figure 3D shows the dependency of the firing rate on the glutamatergic and the GABAergic conductances. In the purely shunting case (Fig. 3D, left), the contour lines (boundaries between different colors) with positive slopes run almost parallel, indicating the subtractive effect of such GABAergic conductances. On the other hand, in the depolarizing case (Fig. 3D, right), the contour lines run almost vertically up to a critical height of the GABAergic conductance, indicating that the input–output relationship does not greatly change in this region. Figure 3F shows the relationships between the GABAergic conductance and the firing rate when the glutamatergic conductance is fixed at gGlu = 5 nS. It is clearly shown that the purely shunting GABAergic conductance has a continuous inhibitory effect on the firing rate (see Fig. 3F, left), whereas the depolarizing GABAergic conductance does not have such a continuous effect, instead having a somewhat ON–OFF-like effect (see Fig. 3F, right).
Mechanism of the different properties of the purely shunting and the depolarizing GABAergic inhibition
To explore how the effect of the GABAergic conductance depends on its reversal potential, we examine the nullclines of the model on the V–R phase plane. Nullclines are the sets of the variables where the velocity of the state point of one direction is equal to 0; the V-directional velocity is 0 on the V-nullcline, whereas the R-directional velocity is 0 on the R-nullcline. Thereby, the nullclines divide the phase plane qualitatively to subregions according to the direction of the velocity vector, and thus are convenient to visualize how the state point moves on the phase plain (see METHODS for details). First, let us examine the effect of the glutamatergic conductance without the GABAergic conductance. The left panels of Fig. 4A show the V-nullcline (the red solid curve) and the R-nullcline (the blue solid curve), whereas the right panels of Fig. 4A show the difference between these 2 nullclines (the black solid curve) of Wilson's model with various amount of the glutamatergic conductance (gGlu = 0, 3.2, 5, and 10 nS from top to bottom). As the glutamatergic conductance increases, the shape of the V-nullcline changes so that the part of it in the left side of EGlu (0 mV) moves upward (Fig. 4A, left, from top to bottom) because of the middle term [–gGlu (V – EGlu)] of the numerator in Eq. 10. In due course, the leftmost intersection (the stable equilibrium point) and the middle one (the saddle point) coalesce at a critical value about 3.2 nS of gGlu (see the left-second panel of Fig. 4A), which is called a saddle-node bifurcation (see METHODS for details). At this point, a stable oscillatory solution representing repetitive firing of action potentials emerges, a part of which is indicated by a green dashed curve in the left-second panel of Fig. 4A. Further increase of the glutamatergic conductance makes the part of the V-nullcline to the left of EGlu (0 mV) moves upward further (third and fourth panels of Fig. 4A, left).
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), squeezed between the 2 nullclines. Because the nullcline is a place where the velocity of one direction is equal to 0, the velocities of the state point of both directions are very small in such a narrow path between the 2 nullclines where the state point is close to both of the nullclines. Therefore, such a narrow path is a rate-determining region for the oscillatory solution that passes through it. Specifically, the period of the oscillatory solution is almost inversely correlated with the minimum width of the narrow path. Immediately after the bifurcation point (the left-second panel of Fig. 4A), the minimum width of the narrow path is infinitely small, so that the period of the oscillatory solution is infinitely long. That is, the firing rate of the neuron begins from 0 Hz at the bifurcation point, which is a feature of type I (class I) neurons (Ermentrout 1996; Fujii and Tsuda 2004; Hodgkin 1948; Wilson 1999a, b). As the glutamatergic conductance increases, the part of the V-nullcline to the left of EGlu (0 mV) moves upward so that the minimum width of the narrow path increases (the left-third panel of Fig. 4A). Thereby the period of the oscillatory solution becomes shorter and shorter, or in other words, the firing rate of the neuron monotonically increases. When the glutamatergic conductance is further increased (the left bottom panel of Fig. 4A), the part of the V-nullcline to the left of EGlu (0 mV) moves upward further, resulting in a dissolution of the rate-determining narrow path. Now, let us examine the effect of the GABAergic conductance when the glutamatergic conductance is fixed at gGlu = 5 nS. The left panels of Fig. 4, B and C show the V-nullcline (the red solid curve) and the R-nullcline (the blue solid curve), whereas the right panels of Fig. 4, B and C show the difference between these 2 nullclines (the black solid curves) of Wilson's model with a fixed amount of the glutamatergic conductance (gGlu = 5 nS) and various amounts of the GABAergic conductance (gGABA = 10, 14, 30, and 45 nS from top to bottom). For comparison, the V-nullcline when the GABAergic conductance is absent (gGlu = 5, gGABA = 0) is also shown by the red dotted curves in the left panels in Fig. 4, B and C. Figure 4B shows the purely shunting GABA case (i.e., EGABA = –75 mV), whereas Fig. 4C shows the depolarizing GABA case (i.e. EGABA = –64 mV). In both Fig. 4B and Fig. 4C, increasing the GABAergic conductance changes the shape of the V-nullcline so that the part of it in the left side of the GABAA reversal potential (EGABA) moves upward, whereas the right part moves downward because of the last term [–gGABA(V – EGABA)] of the numerator in Eq. 10. On the other hand, the shape of the R-nullcline does not change. First, let us consider the case with purely shunting inhibition in Fig. 4B. In this case, the purely shunting GABAA reversal potential EGABA that is indicated by the square on the horizontal axis in Fig. 4B is equal to the resting potential that is located in the left side of the narrowest point of the narrow path between the 2 nullclines, through which the oscillatory solution passes. Thereby, as the GABAergic conductance increases, the minimum width of the narrow path, which is the rate-determining region of the oscillatory solution as described above, becomes narrower (Fig. 4B, top to the second panel). As a result, the firing frequency of the neuron decreases continuously. In due course, the V-nullcline makes contact with the R-nullcline (the second panel of Fig. 4B), where the saddle-node bifurcation (see METHODS for details) occurs so that the firing ceases. In this way, the purely shunting GABAergic conductance stops repetitive firing by a saddle-node bifurcation. Next, consider the case with the depolarizing GABAergic conductance in Fig. 4C. In this case, the depolarized GABAA reversal potential EGABA that is indicated by the cross on the horizontal axis in Fig. 4C is about 11 mV higher than the resting potential, and thus is located just around the narrowest point of the narrow path between the 2 nullclines. Thereby, even if the GABAergic conductance increases, the minimum width of the narrow path hardly changes (see Fig. 4C, top to second panel), so the firing frequency of the neuron does not greatly changes. Instead, the intersection of the 2 nullclines, which is an unstable equilibrium point, suddenly becomes stable at a critical point (bottom of Fig. 4C), because of the change of the vector field around that point, and thus the firing suddenly ceases. Such a change in the stability of the equilibrium point is called the Hopf bifurcation (see METHODS for details). In this way, the depolarizing GABAergic conductance stops repetitive firing by a Hopf bifurcation.
Thus, the difference between the effect of the purely shunting GABAergic conductance and that of the depolarizing GABAergic conductance observed in the simulation described above, that the purely shunting GABAergic conductance has a continuous inhibitory effect on the firing rate whereas the depolarizing GABAergic conductance does not have such a continuous effect but has a somewhat ON–OFF-like effect (Fig. 3F), can be understood as the difference in the types of associated bifurcations. Because a local bifurcation is characterized as a point where the Jacobian matrix of system's equations meets a special condition, a set of parameters at which the bifurcation occurs, that is, a bifurcation set, as well as the type of bifurcations can be obtained by numerically solving the associated algebraic equations (see METHODS for details). Figure 3E shows the bifurcation sets and their types obtained through this method for the purely shunting (left) or the depolarizing (right) GABA cases. The solid curves represent saddle-node bifurcation curves, whereas the dashed curves represent Hopf bifurcation curves. In the purely shunting GABA case, the region in which the neuron keeps firing (the shadowed area in Fig. 3E, left) is bounded mainly by the saddle-node bifurcation curve. On the other hand, in the depolarizing GABA case, the firing region (the shadowed area in Fig. 3E, right) is bounded by the saddle-node bifurcation curve from the left, but is bounded by the Hopf bifurcation curve from the above. Note that bifurcation sets associated with an appearance or a disappearance of a pair of unstable equilibrium points, which are not related to the neuronal firing, are also shown in Fig. 3E (indicated by dotted lines).
Input–output relationship with highly fluctuating conductances
To consider effects of highly fluctuating synaptic conductances in vivo, we represent the synaptic conductance as a summation of the unitary conductances. We refer to such unitary conductances as input events and generate them according to a Poisson process (see METHODS for details). A neuron receiving such highly fluctuating glutamatergic inputs generates an irregular sequence of action potentials, as shown in Fig. 5A. As the strength of each glutamatergic input event increases, while their rate remains unchanged, the firing rate monotonically increases. Figure 5B shows the relationship between the glutamatergic input strength, which is measured by mean conductance (
Glu), and the output firing rate. In the following, we examine how this input–output relationship is affected by the strength of GABAergic inputs, which is also measured by its mean conductance GABA. At first, we assume that GABAergic inputs are statistically independent of glutamatergic inputs, and thus can be represented as a summation of GABAergic input events whose occurrence depends on a Poisson process that is independent from the one producing the glutamatergic input events. The case where there is a correlation between glutamatergic input events and GABAergic ones will be examined in the next section. Figure 5C illustrates sample spike trains with a fixed strength of glutamatergic inputs (Glu = 5 nS) and 9 different strengths of purely shunting (left column) or depolarizing (right column) GABAergic inputs (GABA = 0, 10, 20, 30, ... , 80 nS).
The left panel of Fig. 5D shows the effects of purely shunting GABAergic inputs on the input–output relationship. The 11 black solid curves indicate the input–output relations corresponding to GABA = 0, 5, 10, 15, ... , 50 nS from the top to the bottom. As the strength of GABAergic inputs increases from GABA = 0 nS to GABA = 50 nS, the slope of the input–output curve decreases but the onset point (the intercept on the abscissa axis) does not change. Therefore, this effect of GABAergic inputs can be called a gain modulation or a divisive inhibition, and it is consistent with previously reported experimental and theoretical results (Mitchell and Silver 2003). On the other hand, the right panel of Fig. 5D shows the effects of depolarizing GABAergic inputs varying over the same range (GABA = 0–50 nS). The 11 overlapped black solid curves indicate the input–output relations corresponding to GABA = 0, 5, 10, 15, ... , 50 nS. The depolarizing GABAergic inputs hardly affect the input–output relationship over the range of Glu = 0–5 nS, although a weak gain modulation is observed when glutamatergic inputs are stronger (Glu > 5 nS).
Figure 5E shows the dependency of the firing rate on the strengths of the glutamatergic and the GABAergic inputs. For the purely shunting case (Fig. 5E, left), the contour lines are not parallel, contrary to the case with tonic conductances (Fig. 3D, left), indicating that the effect is divisive rather than subtractive. On the other hand, for the depolarizing case (Fig. 5E, right), the contour lines extend rather longitudinally on the left half of the parameter plane (Glu < 5 nS), beyond the bifurcation curve in the case with tonic conductances (Fig. 3E, right). This indicates that a wide range of depolarizing GABAergic inputs hardly modulate the firing rate. Figure 5F shows the relationships between the strength of the GABAergic inputs and the firing rate when the strength of the glutamatergic inputs is fixed at Glu = 5 nS. It is shown that increasing the strength of purely shunting GABAergic inputs continuously decreases the firing rate, quickly at first and then more slowly (Fig. 5F, left). On the other hand, increasing the strength of depolarizing GABAergic inputs only gently decreases the firing rate (Fig. 5F, right). Actually, depolarizing GABAergic inputs that are 8 times stronger (GABA = 40 nS) than the glutamatergic inputs decrease the firing rate by only about 10%, and even stronger GABAergic inputs decrease the firing rate only gradually. Further increase of the strength of the depolarizing GABAergic inputs beyond the range indicated in Fig. 5F does not induce a sudden stop of firing, contrary to the tonic conductance case, but only decreases the firing rate moderately (data not shown).
Effects of coincidence between glutamatergic and GABAergic input events
So far we have assumed that glutamatergic input events and GABAergic ones are statistically independent; that is, they are generated by 2 independent Poisson processes. For cortical pyramidal neurons in vivo, however, this may not always be the case. Because there are common inputs by shared connections in feedforward networks and/or recurrent connections in feedback networks, glutamatergic input events and GABAergic ones may well be temporally correlated according to circumstances. In other words, under a certain situation, some portion of the glutamatergic input events and GABAergic ones may be coincident, or more generally share some temporal relations. Because such temporal correlation may crucially affect the effects of GABAergic inputs on the input–output relationship, we examine it in the following. Specifically, we change the degree of coincidence between glutamatergic input events and GABAergic ones, and examine its effects for both the purely shunting and the depolarizing GABA cases. An example of the glutamatergic conductance and the GABAergic conductance with 60% coincidence and the resulting spike trains are shown in Fig. 6A.
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Glu = 5 nS and 9 different strengths of purely shunting (left column) or depolarizing (right column) GABAergic inputs (GABA = 0, 10, 20, 30, ... , 80 nS) that are completely coincident with the glutamatergic inputs. The 7 black solid curves in the left panel of Fig. 6C indicate the input–output relations corresponding to GABA = 0, 5, 10, 15, ... , 30 nS from left to right. It is shown that purely shunting GABAergic inputs that are completely coincident with glutamatergic inputs do not have divisive effects on the input–output relationship, contrary to the case without any coincidence (Fig. 5D, left), but have subtractive effects as in the tonic conductance case (Fig. 3C, left). Moreover, in the case of complete coincidence, depolarizing GABAergic inputs do have considerable inhibitory effects on the input–output relationship, as shown in the right panel of Fig. 6C. The 11 black solid curves in the right panel of Fig. 6C indicate the input–output relations corresponding to GABA = 0, 5, 10, 15, ... , 50 nS from left to right. It is shown that increasing the strength of depolarizing GABAergic inputs that are completely coincident with glutamatergic inputs shifts the input–output curve to the right and slightly decreases its slope, indicating that these GABAergic inputs have mainly subtractive effects on the input–output relationship of a neuron driven by coincident glutamatergic inputs. Figure 6D shows how the dependency of the firing rate on the input strengths varies according to the degree of coincidence between glutamatergic input events and GABAergic ones [from 0% coincidence (top) to 100% coincidence (bottom)]. As the degree of coincidence increases, the effect of purely shunting GABAergic inputs gradually changes from divisive to subtractive. On the other hand, as the degree of coincidence increases, the inhibitory effect of the depolarizing GABAergic inputs becomes more and more conspicuous. Figure 6E shows the relationship between the strength of the GABAergic inputs having various degrees of coincidence with the glutamatergic inputs (0–100%) and the firing rate when the strength of the glutamatergic conductance is fixed at Glu = 5 nS for the purely shunting GABA case (Fig. 6E, left) and the depolarizing GABA case (Fig. 6E, right). Although the purely shunting GABAergic inputs with arbitrary degree of coincidence have considerable inhibitory effects, whereas its type (divisive or subtractive) depends on the degree of coincidence, the depolarizing GABAergic inputs have apparent inhibitory effect only when the degree of coincidence is large.
The dependency of the effects of fluctuating GABAergic inputs on the degree of coincidence can be partly understood on the parameter plane of the case with tonic conductances. Figure 7 shows simultaneous probability distributions of the instantaneous amounts of the glutamatergic conductance and the GABAergic conductance when Glu = 5 nS and GABA = 50 nS and the degree of coincidence between glutamatergic input events and GABAergic ones ranges from 0% (top) to 100% (bottom). When there is no coincidence between glutamatergic and GABAergic input events (Fig. 7, top), their instantaneous amounts have no correlation. Thus, the probability distribution of the conductances is concentrated near both axes. As the degree of coincidence increases, the probability distribution becomes concentrated close to the line connecting the mean-conductance point (Glu = 5 nS and GABA = 50 nS, indicated by the white asterisks) and the coordinate origin. In the extreme case, when glutamatergic input events and GABAergic ones are perfectly coincident (Fig. 7, bottom), all the probability rides on this line. The white solid curves and the white dashed curves in Fig. 7 show the bifurcation curves corresponding to the purely shunting case (Fig. 3E, left) or the depolarizing case (Fig. 3E, right) with the tonic conductance, respectively. Thus the areas under these bifurcation curves represent parameter ranges that produce repetitive firing of action potentials. When there is no coincidence between glutamatergic and GABAergic input events (Fig. 7, top), a certain portion of the probability near the horizontal axis is located below both of the bifurcation curves. This implies that the neuron can fire on some epochs even if it is receiving such strong GABAergic inputs as this example with GABA = 50 nS, and this is consistent with the results that GABAergic inputs independent of glutamatergic inputs never prevent neuronal firing thoroughly (Fig. 5D). On the other hand, when glutamatergic input events and GABAergic ones are coincident (Fig. 7, bottom), most of the distribution is located above both of the bifurcation curves, so the neuron can hardly generate spikes. This is consistent with the results that GABAergic inputs that are coincident with glutamatergic inputs would prevent neuronal firing almost completely (Fig. 6C).
Comparison with the integrate-and-fire neuron model
Many studies investigated the effects of GABAergic inputs on the neuronal input–output relationship using the integrate-and-fire neuron model (Chance et al. 2002; Holt and Koch 1997; Mitchell and Silver 2003; Salinas and Sejnowski 2000). To examine whether our main results hold for the integrate-and-fire model as well, we compared the properties of Wilson's model with those of the integrate-and-fire model (see METHODS for details). Figure 8 shows the results for the integrate-and-fire model corresponding to Figs. 3, 5, and 6 for Wilson's model.
First, we consider the tonic conductance case (top rows in Fig. 8, A–C). Although the purely shunting GABAergic conductance provides a substantial level of subtractive inhibition (left panel in top row of Fig. 8A), the effect of the depolarizing GABAergic conductance is more restricted (see the corresponding right panel). These results are similar to those for Wilson's model in Fig. 3C in the sense that the depolarizing GABAergic conductance has less effect on the input–output relationship than the purely shunting GABAergic conductance. However, for the effect of the depolarizing GABAergic conductance, there is a certain difference between Wilson's model and the integrate-and-fire model. Although the depolarizing GABAergic conductance that is larger than a certain amount prevents all firing over a wide range of the glutamatergic conductance in Wilson's model (see dotted lines in Fig. 3C, right), even a large amount of the depolarizing GABAergic conductance merely causes subtractive inhibition at an almost constant rate of shift in the integrate-and-fire model (see dotted lines in right panel of top row in Fig. 8A). This difference reflects the different shapes of the boundaries between 2 regions, one region where the neuron fires repetitively and the other where the neuron is in the resting state. Whereas the boundary consists of 2 distinct curves, a nearly vertical saddle-node bifurcation curve and a nearly horizontal Hopf bifurcation curve, in Wilson's model (see Figs. 3, D and E, right), it consists of a single straight line in the integrate-and-fire model (see right panel of top row in Fig. 8B). Indeed, the boundary for the integrate-and-fire model is the straight line represented by Eqs. 19 and 20.
Next, let us turn to the fluctuating conductance cases (second and bottom rows in Fig. 8, A–C). When there is no correlation between glutamatergic inputs and GABAergic ones, the integrate-and-fire model predicts that purely shunting GABAergic inputs cause divisive inhibition, whereas the depolarizing ones have little effect on the firing rate (second row in Fig. 8A). Note that these results are similar to those from Wilson's model in Fig. 5D. When GABAergic inputs are strongly correlated with glutamatergic inputs, on the other hand, the integrate-and-fire model predicts that purely shunting GABAergic inputs produce not divisive but subtractive inhibition (see left panel in bottom row of Fig. 8A). Note again that this is also consistent with Wilson's model (see Fig. 6C). Finally, for fluctuating depolarizing GABAergic inputs that are strongly correlated with glutamatergic inputs, the integrate-and-fire model predicts that they have an inhibitory effect (see right panel in bottom row of Fig. 8A). This result is similar to that from Wilson's model (Fig. 6C) in the sense that increasing the coincidence between glutamatergic inputs and GABAergic ones makes depolarizing GABAergic inputs more effective. However, the ways depolarizing GABAergic inputs modulate the firing rate seem to be somewhat different in the 2 models: although these inputs shift the input–output curve to the right (subtractive inhibition) and slightly decrease the slope (divisive inhibition) for Wilson's model as shown in Fig. 6C, for the integrate-and-fire model they shift the onset point of the input–output curve to the right but have smaller effects when the glutamatergic inputs are strong, resulting in a steeper slope.
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; Holt and Koch 1997; Mitchell and Silver 2003): a tonic GABAergic conductance has a subtractive effect, whereas a highly fluctuating conductance causes gain modulation. The primary novelty of this study lies in our analysis of the depolarizing GABA case. First, we have analyzed the case with tonic conductances. A tonic depolarizing GABAergic conductance causes almost ON–OFF or binary, rather than continuous, response: when the amount of the glutamatergic conductance is fixed, there is a critical amount of the depolarizing GABAergic conductance (see Fig. 3F, right); under this critical amount, the depolarizing GABAergic conductance hardly affects the firing rate, whereas above this critical amount the neuron ceases firing completely. Such a switching effect of the depolarizing GABAergic conductance is potentially useful for some kinds of neural computation, and is expected to be examined by experiments. Next, we have considered highly fluctuating inputs, and found that a wide range of highly fluctuating depolarizing GABAergic inputs have little effect on the firing rate of a neuron driven by glutamatergic inputs that are uncorrelated with them (see Fig. 5D, right). We have then examined the influence of the degree of coincidence between glutamatergic input events and GABAergic ones, and found that increasing the degree of coincidence makes depolarizing GABAergic inputs effective (ree Fig. 6E, right). These results imply that a certain range of depolarizing GABAergic inputs enables discriminative modulation of the firing rate; that is, they modulate the firing rate only if many of the driving glutamatergic inputs are coincident with them. The depolarized GABAA reversal potential causes both facilitatory and inhibitory effects on the spike generation, depending on the precise temporal relations to other glutamatergic inputs at a single synaptic input level, as shown by Gulledge and Stuart (2003). Such a dual action can provide a rich computational ability, and is potentially useful in terms of the precise timing-based temporal spike coding. On the other hand, from the aspect of the firing-rate coding, it is not the purely shunting GABAA reversal potential but the depolarized GABAA reversal potential that is suitable for the input-specific discriminative firing rate modulation, as we have described above. Concrete functional meanings of the depolarized GABAA reversal potential in specific systems are expected to be examined in the future. In the following, we discuss the GABAA reversal potential, the input fluctuation, the existence of correlation between glutamatergic and GABAergic inputs, and several issues about the use of neuron models, including the comparison of the results for Wilson's model with those for the integrate-and-fire model. GABAA reversal potential
The reversal potential of GABAA receptor–associated chloride channels has long been considered to be nearly equal to the resting potential. A novel experimental technique enabled Gulledge and Stuart to obtain the surprising results regarding the depolarized GABAA reversal potential (Gulledge and Stuart 2003). They used the gramicidin-perforated patch-clamp method that allows one to perform intracellular recordings without disturbing anion concentrations. Although the influence of other conditions, such as pH, may still prevent recording of the precise value of the GABAA reversal potential, there have been several reports supporting Gulledge–Stuart results (Grande et al. 2004; Wirth and Luscher 2004). There are also many reports on the depolarized GABAA reversal potential in other contexts. For example, it is well known that the GABAA reversal potential is depolarizing in the early developmental stages because of the composition of the chloride ion transporters. Another example includes the findings associated with tetanus stimulation (Kaila et al. 1997; Staley et al. 1995), in which the gradient of the chloride concentration collapses. Furthermore, in the striatum spiny neurons, the GABAA reversal potential is higher than the membrane potential at the so-called down state (Wilson and Kawaguchi 1996). Because we deal with the depolarizing value of the GABAA reversal potential, but do not refer to its origin in this paper, our results may be applicable to situations in which the GABAA reversal potential is depolarizing, regardless of its causes. It should be also noted that there have been several theoretical studies (Fujii et al. 2004; Vogel 2001) regarding possible functions of the depolarizing GABA in more abstract forms than our study.
Fluctuating inputs
As described in METHODS, we assume that the fluctuating glutamatergic or the GABAergic conductance is represented by a summation of unitary conductances, which we call input events, generated by Poisson processes with a relatively low (50 Hz) rate. This representation is inspired by the previous studies examining the nature of the in vivo input conductance (Stevens and Zador 1998). Actually, we tried various forms for the representation of the fluctuating conductance: Poisson trains with various rates and Ornstein–Uhlenbeck processes with various variances. Fluctuating inputs generated by a Poisson process with a higher rate (e.g., 1,000 Hz) are much more regular than those generated by input events with a lower rate, as pointed out by Softky and Koch (1993), and their effect on the input–output relationship is more similar to that in the tonic conductance case (results not shown). Prominent features shown in our simulations with fluctuating inputs—divisive inhibition in the purely shunting GABA case and correlation-dependent inhibition in the depolarizing GABA case—are conspicuous only when the generating Poisson rate is low so that the fluctuation is sufficiently large. These features are also obtained by using Ornstein–Uhlenbeck processes, but only when the variance is very large, such as a half of the mean or more (results not shown). Recently, Mitchell and Silver (2003) showed that GABAergic inputs can have a divisive effect on the input–output relationship if the glutamatergic and/or GABAergic inputs are fluctuating rather than tonic. They represent the fluctuating conductance as a summation of unitary conductances generated by a Poisson process whose rate is low [i.e., several tens per second (Hz)] to fit the situation of the cerebellar granule cells. Compared to the cerebellar granule cells, neocortical pyramidal cells usually receive much more frequent synaptic inputs. If these inputs are statistically independent, the total input conductance becomes rather regular, and the effect on the input–output relationship might be approximated by that of the tonic conductance. On the other hand, if there is considerable temporal correlation between synaptic inputs such as the synchronized arrival of many spikes (Stevens and Zador 1998), the effective rate of the inputs decreases so that the total input conductance becomes highly fluctuating as in the situation of the cerebellar granule cells, which is the standpoint of our description of the fluctuating inputs. Although our simulating fluctuating inputs have a similarity to those of Mitchell and Silver (2003) in the meaning that both are generated by Poisson processes with low frequencies like several tens per second (Hz), there is a difference in the way of changing the strengths of the fluctuating inputs: Mitchell and Silver (2003) changed the input frequency but not the amplitude of each input, whereas we have changed the amplitude of each input event but kept the frequency constant. Because individual input event in our description of the fluctuating inputs does not represent the unitary conductance at a single synapse but instead represents a summation of several tens or hundreds of synchronized synaptic inputs of the same type, that is either glutamatergic or GABAergic (see METHODS for details), the amplitude of each input event is considered to be proportional to a number of presynaptic spikes that are included in the event. Therefore, our manipulation on the amplitude of each input event corresponds to changing the sizes of a presynaptic synchronously firing neuronal cluster.
Correlation between glutamatergic and GABAergic inputs
Consideration of the temporal correlation between glutamatergic inputs and GABAergic ones is also a major feature of our study in addition to consideration of the depolarized GABAA reversal potential. In general, different neocortical neurons receive common inputs from identical cells, possibly by glutamatergic neurons or GABAergic interneurons. Furthermore, there are massive recurrent connections in the neocortex. Both of these are possible sources of correlation between glutamatergic inputs and GABAergic ones, and such correlation would substantially affect the neuronal input–output relationship. However, although there have been numerous studies regarding the effects of input correlation, including those dealing with correlation between glutamatergic and GABAergic inputs (e.g., Mikula and Niebur 2004; Salinas and Sejnowski 2000), few of the previous studies have examined the effects of the glutamate–GABA correlation and the strength of GABAergic inputs simultaneously. In our study, the correlations within glutamatergic or GABAergic inputs are implicitly assumed to be fixed values by setting the frequency of each input event to the fixed value (50 Hz). Thereby we have examined the effects of the correlation between glutamatergic and GABAergic inputs and the strengths of GABAergic inputs. In addition to our results concerning the depolarizing GABA case on which we have focused and discussed above, we have also found that increasing the degree of coincidence between glutamatergic input events and GABAergic ones switches the action of purely shunting GABAergic inputs from divisive inhibition to subtractive inhibition (Fig. 6, C, left and D, left). This result, which was also supported by the integrate-and-fire model (the left panels in the second and bottom rows of Fig. 8A), may be important for understanding the function of purely shunting GABAergic inputs.
Although in this paper we have examined only the case where glutamatergic inputs and GABAergic ones are coincident with no time delay, the temporal correlation between those 2 types of the inputs would generally accompany a certain time delay. As the other extreme case, it is also possible that glutamatergic inputs and GABAergic ones are anticorrelated. In that case, depolarizing GABAergic inputs might excite the neuron rather than inhibit it. The time delay between glutamatergic inputs and GABAergic ones is considered to depend on the local circuitry around the neuron, and thus the effects of GABAergic inputs could be different according to circumstances.
Wilson's model versus the integrate-and-fire model
We have examined whether the characteristics obtained by Wilson's 2D neocortical neuron model also hold for the integrate-and-fire neuron model, which has been widely used in the studies on the neuronal input–output relationship (Chance et al. 2002; Holt and Koch 1997; Mitchell and Silver 2003; Salinas and Sejnowski 2000). As described in RESULTS, when we consider the purely shunting GABA case where the GABAA reversal potential is equal to the resting potential, the results are almost the same for these 2 models. When we consider the depolarized GABAA reversal potential, on the other hand, the characteristics of these 2 neuron models differ according to conditions. When we consider the tonic conductances, there is a certain qualitative difference between them. Specifically, the boundary between the firing region and the nonfiring region in the parameter space is upward convex in Wilson's model (right panels of Fig. 3, D and E), but is a straight line in the integrate-and-fire model (right panel of the top row in Fig. 8B). This discrepancy could be considered to result from the fact that the representation ability of the integrate-and-fire model is restrictive compared with 2D models like Wilson's model. Although whether such a curving boundary predicted by Wilson's model really appears in actual neocortical neurons remains to be experimentally explored, if it was true, it cannot be described by the integrate-and-fire model. Contrary to the tonic conductance case, the results for the 2 models are rather similar in the highly fluctuating input case, except for some difference seen in the coincident GABAergic and glutamatergic inputs situation. This observation might be consistent with a notion that the integrate-and-fire model with conductance-based synapses may provide a good representation of the large fluctuating synaptic conductance (Meffin et al. 2004).
Issues about using a 2D model
Although Wilson's model as well as other 2D models seems to have a richer representation ability, backed with biophysical relevance, than the integrate-and-fire model, it is still too simple compared with more elaborative models, or of course, with real neurons. First, Wilson's model, like any 2D models, consists of a single compartment and thus cannot deal with phenomena strongly related to the somato-dendritic properties. We have succeeded in reproducing the experimental results of Gulledge and Stuart (2003) concerning the effects of dendritically applied GABAergic inputs using a 2-compartment version of Wilson's model, which in total is a 4-dimensional model. However, we have used the single-compartment model rather than this 2-compartment model to examine the input–output relationship, which is the main theme of this paper. This was partly because of its simplicity, but could be further justified as a model of the effects of somatically targeted GABAergic neurons like the fast spiking cells (Buhl et al. 1997; Tamas et al. 1997).
Next, we have explained the difference in the effects of the tonic GABAergic conductance between the purely shunting case and the depolarizing case using the V-nullcline and the R-nullcline on the 2D phase plane. The R-nullcline, however, is itself a product of the reduction, which represents the approximate activation curve for the potassium current including both the noninactivating Hodgkin–Huxley type and the transient so-called A-type. In models with more than 2 dimensions as well as in real neurons, a wider variety of bifurcations can occur than in 2D models, so that whether a Hopf bifurcation really occurs and stops firing is unclear and may be depend on types and properties of the channels that they posses. A special concern relating to this point is the wave form of the membrane potential in Wilson's model in the tonic conductance case. In the simulation in the tonic conductance case (see Fig. 3B), the amplitude of the membrane potential oscillation changes continuously from a large spikelike size at gGABA = 35 to zero at gGABA = 40 with the increase of the depolrazing GABAergic conductance. Because the relevant Hopf bifurcation is supercritical, the oscillation with a small or intermediate size is observed near the critical value of gGABA that stops the firing between gGABA = 35 and gGABA = 40 (not shown). Such small or intermediate oscillations, although limited to a relatively small region in the parameter space, may be inappropriate for a neuron model because the amplitudes of real action potentials are known to be usually rather stereotypical. This point is considered to be a drawback of using the 2D Wilson's model. More elaborate high-dimensional models would generate oscillatory solutions with stereotypical action potentials through the subcritical Hopf bifurcation, or other bifurcations. Wilson's model also sometimes generates wave forms whose amplitudes seem to be insufficient for those of stereotypical action potentials but are apparently larger than those of subthreshold oscillations in the fluctuating input cases as appeared in Figs. 5C and 6B. This point is expected to be addressed using a more detailed model as well as a real cortical neuron in future studies.
Finally, Wilson's model does not include inactivation of sodium channels, which is included in the original Hodgkin–Huxley equations (Hodgkin and Huxley 1952) as well as in real neocortical neurons. Because a transient (Hodgkin–Huxley type) sodium current with an inactivation process is inactivated by prolonged depolarization, it can be expected that lengthy subthreshold depolarization caused by depolarizing GABAergic inputs would suppress spiking rather than facilitate it. However, in the experiments of Gulledge and Stuart, temporally preceding depolarizing GABAergic inputs did facilitate spike generation. Possibly, the depolarization caused by depolarizing GABAergic inputs was not large enough to induce considerable inactivation of the transient sodium current. The absence of an inactivation variable in Wilson's model might be a good representation of the effect of noninactivating or slow-inactivating sodium current (INaP) (Crill 1996), as pointed out by Wilson (1999b), which could help sustain a small depolarizing postsynaptic potential generated by depolarizing GABAergic inputs.
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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1 Although the area itself is not defined explicitly in the original paper (Wilson 1999), it can be calculated from other defined values. 
Address for reprint requests and other correspondence: K. Morita, Institute of Industrial Science, The University of Tokyo, Ce601, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan (E-mail: morita{at}sat.t.u-tokyo.ac.jp)
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