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Possible Effects of Depolarizing GABA_A Conductance on the Neuronal Input–Output Relationship: A Modeling Study

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

	ABSTRACT

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
Recent in vitro experiments revealed that the GABA_A reversal potential is about 10 mV higher than the resting potential in mature mammalian neocortical pyramidal cells; thus GABAergic inputs could have facilitatory, rather than inhibitory, effects on action potential generation under certain conditions. However, how the relationship between excitatory input conductances and the output firing rate is modulated by such depolarizing GABAergic inputs under in vivo circumstances has not yet been understood. We examine herewith the input–output relationship in a simple conductance-based model of cortical neurons with the depolarized GABA_A reversal potential, and show that a tonic depolarizing GABAergic conductance up to a certain amount does not change the relationship between a tonic glutamatergic driving conductance and the output firing rate, whereas a higher GABAergic conductance prevents spike generation. When the tonic glutamatergic and GABAergic conductances are replaced by in vivo–like highly fluctuating inputs, on the other hand, the effect of depolarizing GABAergic inputs on the input–output relationship critically depends on the degree of coincidence between glutamatergic input events and GABAergic ones. Although a wide range of depolarizing GABAergic inputs hardly changes the firing rate of a neuron driven by noncoincident glutamatergic inputs, a certain range of these inputs considerably decreases the firing rate if a large number of driving glutamatergic inputs are coincident with them. These results raise the possibility that the depolarized GABA_A reversal potential is not a paradoxical mystery, but is instead a sophisticated device for discriminative firing rate modulation.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
Recent electrophysiological experiments using gramicidin-perforated patch-clamp techniques have revealed that -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 GABA_A-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 GABA_A reversal potential equal to the resting potential, the exact meaning of the depolarized value of the GABA_A 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 GABA_A-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 GABA_A 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 GABA_A 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 GABA_A 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 GABA_A 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 GABA_A 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.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
Wilson's model

We use the single-compartment model of a neocortical neuron proposed by Wilson (Wilson 1999a, b)

(1)

(2)

Here, V (mV) is the membrane potential (inside against outside); R is the inactivation variable that qualitatively represents the conductance of the potassium channel; c_K (nS) is a certain constant; a_s (10^–10 m²) is the area of the axon hillock and initial segment; C (µF/cm²) is the membrane capacitance per unit area; E_Na (mV) and E_K (mV) are the respective reversal potentials of sodium and potassium channels; g_Na(V) (nS) is the steady-state voltage dependency of the sodium conductance; f(V) is the voltage dependency of the potassium channels, including both the delayed-rectifier channel and the A-current channel; and (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 g_Na(V) and f(V), as well as the values of other parameters, were determined by Wilson (1999b) as g_Na(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; a_s = 10 (10^–10 m²)¹ (that is, equal to 1000 µm²); E_Na = 48 mV; E_K = –95 mV; = 5.6 ms; and c_K = 260 nS. I_syn (pA) in Eq. 1 represents the following current through synaptic channels

(3)

where E_Glu and E_GABA (mV) represent the reversal potentials of the channels coupled with the non-NMDA (N-methyl-D-aspartate) glutamate receptors and the GABA_A receptors, respectively. g_Glu(t) and g_GABA(t) (nS) represent the corresponding total time-dependent synaptic conductances. When we consider tonic conductances, g_Glu(t) and g_GABA(t) are assumed to be time-independent parameters g_Glu and g_GABA (nS), respectively. The NMDA receptors and the GABA_B receptors are not considered in this paper. In the simulation, we count spikes by regarding that Wilson's model generates a spike when the membrane potential passes through the value V = –30 mV upward. We have confirmed that even if this value is increased to V = –10 mV, overall results do not largely change.

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)

(4)

(5)

(6)

(7)

Equations 4 and 5 describe the dynamics of a compartment representing the axon hillock and initial segment, which we refer to as the axo-somatic compartment, and they are essentially the same as for Wilson's single-compartment model (Eqs. 1 and 2) except that they include the last term to represent coupling with a dendritic compartment. Equations 6 and 7 describe the dynamics of the dendritic compartment, and they are the same as those for the axo-somatic compartment except that they have an extra parameter associated with the relative density of the voltage-dependent channels (see below). V_s and V_d (mV) are the respective membrane potentials of the axo-somatic compartment and the dendritic compartment; R_s and R_d are the respective inactivation variables; a_s and a_d (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; g_sd (nS) is the conductance between the axo-somatic compartment and the dendritic compartment; I_s and I_d (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: a_s = a_d = 10 (10^–10 m²); = 0.05 (Wilson 1999a); g_sd = 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 GABA_A synapses is considerably higher than the resting potential. They report that the resting potential was –79 ± 1 mV and the GABA_A 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 GABA_A 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 GABA_A 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 GABA_A reversal potential is 11.4 mV in the model and 11 mV in the Gulledge–Stuart experiment, and the difference between the GABA_A 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 (E_GABA = –64 mV) of the GABA_A reversal potential in Wilson's model as the depolarized GABA_A reversal potential. For comparison, we will also examine the case with the GABA_A reversal potential that is equal to the resting potential of Wilson's model (i.e., –75 mV). We refer to this value (E_GABA = –75 mV) of the GABA_A reversal potential as the purely shunting GABA_A reversal potential. The reversal potential of the non-NMDA glutamate receptor-channel is set to E_Glu = 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, g_Glu(t) and g_GABA(t), are respectively represented by a double and a triple exponential, as in their experiments

(8)

(9)

Here, _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 I_syn is substituted by Eq. 3, and from Eq. 2 by setting each differential coefficient equal to 0 as follows

(10)

(11)

The first of these curves is the V-nullcline where the V-directional velocity of the state point is equal to 0, and the second is the R-nullcline where the R-directional velocity is equal to 0. Intersections of these 2 nullclines are the equilibrium points. Because these nullclines divide the phase plane qualitatively according to the direction of the velocity vector, they are convenient to visualize how the state point moves on the phase plane. For example, Fig. 1A shows the V-nullcline (the red curve) and the R-nullcline (the blue curve) of Wilson's model when the input is absent (i.e., g_Glu = g_GABA = 0). The direction of the velocity vector of a state point, or the vector field, in each region divided by these 2 nullclines is indicated by each arrow. In this case, all the velocity vectors around the leftmost intersection of the 2 nullclines (indicated by the green circle), which is one of the equilibrium points, face to this equilibrium point. In fact, this equilibrium point is asymptotically stable, in the sense that any trajectories except unstable equilibrium points corresponding to the other 2 intersections will approach it (Wilson 1999b), 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 (g_Glu = 0.5 µS and g_GABA = 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.

View larger version (12K): [in this window] [in a new window]   FIG. 1. Phase plane of Wilson's model. Horizontal axis represents the membrane potential [V(mV)], whereas the vertical axis represents the inactivation variable (R). Black squares and black crosses on the horizontal axis represent the purely shunting and depolarized GABAA reversal potentials, respectively. A: case without the input conductance, gGlu = gGABA = 0 (nS). B: case with the tonic glutamatergic conductance, gGlu = 5, gGABA = 0 (nS). Red curves indicate the V-nullcline, whereas blue curves indicate the R-nullcline. Arrows indicate direction of the velocity vector of a state point, or the vector field, in each region divided by these 2 nullclines. Green circle in A represents the stable equilibrium point corresponding to the resting state of the neuron. Green square in B represents the unstable equilibrium point and the green dashed curve in B represents the stable oscillatory solution corresponding to the repetitive firing of the neuron. C and D: time evolution of the membrane potential of Wilson's model. Horizontal axis represents the time (ms), whereas the vertical axis represents the membrane potential [V(mV)]. C: same case as A without the input conductance, gGlu = gGABA = 0 (nS). D: same case as B with the tonic glutamatergic conductance, gGlu = 5, gGABA = 0 (nS).

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

(12)

where V (the membrane potential) and R (the inactivation variable) are the state variables, whereas g_Glu and g_GABA are the bifurcation parameters. Dots on the letters (, ) represent time derivatives. At an equilibrium point satisfying

(13)

(14)

a saddle-node bifurcation accompanied with an appearance or a disappearance of a pair of stable and unstable equilibrium points occurs if

(15)

is satisfied, whereas a Hopf bifurcation accompanied with a genesis of an oscillatory solution, which is called a limit cycle, occurs if

has pure imaginary eigenvalues, that is

(16)

is satisfied. The simultaneous equations, consisting of Eqs. 13–15, define the saddle-node bifurcation set, whereas those of Eqs. 13, 14, and 16 define the Hopf-bifurcation set. If a value is assigned to g_Glu, the simultaneous equations can be numerically solved by the Newton method so as to obtain corresponding values of g_GABA (Kawakami 1984). Repeating this process for a series of g_Glu values produces (g_Glu, g_GABA) value sets on the bifurcation curve, which is drawn later in Figs. 3E and Fig. 7.

View larger version (58K): [in this window] [in a new window]   FIG. 3. Effects of the tonic GABAergic conductance on the neuronal input–output relationship. A: relationship between the tonic glutamatergic conductance (horizontal axis) and the neuronal firing rate (vertical axis, unit: spikes/s) when the GABAergic conductance is absent. B: 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). C: effects of the tonic GABAergic conductance on the input–output relationship. Seven black solid curves in the left panel represent the relations corresponding to 7 different values of the purely shunting GABAergic conductance (gGABA = 0, 5, 10, 15, ... , 30 nS) from left to right. Overlapped black solid curves in the right panel represent the relations corresponding to 7 different values of the depolarizing GABAergic conductance (gGABA = 0, 5, 10, 15, ... , 30 nS). Three blue dotted curves in the right panel represent the relations with a larger depolarizing GABAergic conductance (gGABA = 35, 40, 45 nS) from left to right. D: dependency of the firing rates on the tonic glutamatergic (horizontal axis) and the tonic GABAergic (vertical axis) conductances. E: bifurcation curves on the gGlu–gGABA parameter plane. Solid curves indicate the saddle-node bifurcation sets, whereas dashed curves indicate the Hopf bifurcation sets. Shadowed areas indicate the regions in which the neuron keeps repetitive firing. Dotted curves indicate the saddle-node bifurcation curves that are associated with an appearance or a disappearance of a pair of unstable equilibrium points that are not related to the neuronal firing. F: dependency of the firing rates on the GABAergic conductance at a fixed glutamatergic conductance (gGlu = 5 nS). Left panels in B–F correspond to the purely shunting GABA case, whereas the right panel correspond to the depolarizing GABA case.

View larger version (28K): [in this window] [in a new window]   FIG. 7. Probability distributions of the instantaneous values of glutamatergic (horizontal axis, unit: nS) and GABAergic (vertical axis, unit: nS) conductances of the fluctuating inputs generated by Poisson processes (see METHODS for details) with the mean conductance Glu = 5 nS and GABA = 50 nS. Degree of coincidence between the glutamatergic input events and the GABAergic ones ranged from 0% (top) to 100% (bottom). White solid curves and the white dashed curves represent the bifurcation curves for the purely shunting GABA case and the depolarizing GABA case, respectively. Areas under these bifurcation curves represent parameter ranges that produce repetitive firing of action potentials for the tonic conductance.

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.

(17)

To determine a plausible value for the rate of the Poisson process, we have made considerations and assumptions as follows. Although a neocortical neuron typically has thousands of glutamatergic and GABAergic synaptic sites, the high irregularity of the neuronal firing cannot be explained if the input at each of these thousands of synapses is independent of the other inputs, as shown by Stevens and Zador (1998). 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

(18)

where V is the membrane potential (inside against outside); V_rest is the resting potential; _m is the membrane time constant, which is a product of the membrane capacitance and the membrane resistance; and I_syn 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 V_th, which is called a firing threshold, we presume that an action potential is generated at that instance, and V is reset to another value V_reset, which is called a reset potential. Afterward, the membrane potential V is kept constant at V = V_reset 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: V_rest = –75 mV and V_th = –58 mV. The reset potential is set to V_reset = –75 mV. We have confirmed that changing the reset potential to V_reset = –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., g_Glu(t) and g_GABA(t) in Eq. 3 take constant values g_Glu and g_GABA, 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

(19)

For the purely shunting (E_GABA = –75 mV) or the depolarized (E_GABA = –64 mV) GABA_A reversal potential, this condition becomes

(20)

respectively. Note that the above inequalities represent straight lines on the g_Glu–g_GABA parameter plane, and also that the slope of that line is about 3 times steeper for the depolarized GABA_A reversal potential than for the purely shunting one, as demonstrated in the top 2 panels of Fig. 8B. The firing rate of the neuron can also be calculated exactly as follows

(21)

where c₁ = (1 + g_Glu + g_GABA)/_m and c₂ = (V_rest + g_GluE_Glu + g_GABAE_GABA)/_m. Although this expression is too complicated to provide any particular insights, it is clear that the firing rate is a continuous function of g_Glu and g_GABA, and so there should be no discontinuous jumps of the firing rate on the g_Glu–g_GABA parameter plane.

View larger version (50K): [in this window] [in a new window]   FIG. 8. Response properties of the integrate-and-fire model. Pictures in the left columns of A, B, and C show the purely shunting GABA case, whereas those in the right columns show the depolarizing GABA case. A: effects of GABAergic inputs on the input–output relationship. Pictures in the top rows show the tonic conductance case, whereas those in the second and third rows show the fluctuating input cases. Although the second row shows the case where glutamatergic inputs and GABAergic inputs are independent (0 % coincidence), the third row shows the case where glutamatergic inputs and GABAergic ones are completely synchronized (100 % coincidence). Five black solid curves in the 2 pictures in the top row correspond to 5 different amounts of the GABAergic input (gGABA = 0, 0.5, 1, 1.5, 2) increasing from left to right. Four blue dotted curves in the right picture in the top row correspond to further increase of the depolarizing GABAergic conductance (gGABA = 2.5, 3, 3.5, 4) from left to right. Eleven black solid curves in the second and third rows correspond to 11 different strengths of GABAergic inputs (GABA = 0, 0.5, 1, 1.5, ... , 5) increasing from left to right (only the 7 curves are within the range of the lower left picture). B: dependency of the firing rate on the mean conductance of glutamatergic (horizontal axis) and GABAergic (vertical axis) inputs. Top row shows the tonic conductance case, whereas the second and third rows show the fluctuating input cases when the degree of coincidence between glutamatergic and GABAergic inputs is 0% or 100%, respectively. C: top row shows the dependency of the firing rate on the tonic GABAergic conductance at a fixed tonic glutamatergic conductance (gGlu = 0.5). Second row shows the dependency of the firing rate on the mean strength of GABAergic inputs at a fixed strength (Glu = 0.5) of glutamatergic inputs with various degrees of coincidence between them. Six black solid curves correspond to 6 different degrees of coincidence (0, 20, 40, ... , 100 %) increasing from top to bottom.

All simulations were calculated by MATLAB (The MathWorks, Natick, MA).

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
Quantitative reproduction of the experimental results using the conductance-based neuron model

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 GABA_A 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 GABA_A 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.

View larger version (30K): [in this window] [in a new window]   FIG. 2. Quantitative reproduction of the experimental results obtained by Gulledge and Stuart (2003) with Wilson's single-compartment neuron model (B–E, K) or Wilson's 2-compartment model (F–J, L–M). A: time evolution of the synaptic conductances of a glutamatergic input (Eq. 8) (top) and a GABAergic input (Eq. 9) (middle). Time evolution of a unitary conductance (Eq. 17) used for fluctuating conductances is also shown (bottom). B: time evolution of a subthreshold glutamatergic input (top) and a resulting membrane potential (bottom). C: a depolarizing GABAergic input (blue curve in top panel) 8 ms before a subthreshold glutamatergic input (red curve) facilitates spike generation (bottom). D: time evolution of a suprathreshold glutamatergic input (top) and a resulting action potential (bottom). E: a depolarizing GABAergic input (blue curve in top) coincident with a suprathreshold glutamatergic input (red curve) prevents spike generation (bottom). F: time evolution of a somatically applied subthreshold glutamatergic input (top) and the resulting membrane potential of the dendritic compartment (middle) and that of the axo-somatic compartment (bottom). G: a dendritically applied depolarizing GABAergic input (blue curve in top) 8 ms before a somatically applied subthreshold glutamatergic input (red curve) facilitates spike generation at the axo-somatic compartment (bottom). A back-propagating action potential at the dendritic compartment is shown in the middle. H: a dendritically applied depolarizing GABAergic input (blue curve in top) coincident with a somatically applied subthreshold glutamatergic input (red curve) also facilitates spike generation (bottom). I: time evolution of a somatically applied suprathreshold glutamatergic input (top) and the resulting membrane potential of the dendritic compartment (middle) and the action potential in the axo-somatic compartment (bottom). J: a dendritically applied depolarizing GABAergic input (blue curve in top) coincident with a somatically applied suprathreshold glutamatergic input (red curve) does not prevent spike generation (bottom). K: facilitatory or inhibitory action of a depolarizing GABAergic synaptic input on the action potential generation in the case of Wilson's single-compartment model. Horizontal axis shows the timing of a GABAergic input relative to a glutamatergic input (ms). Vertical axis represents the maximum conductance of a GABAergic input [GABA (nS)]. Red indicates the region where a GABAergic input has a facilitatory action; that is, where it evokes an action potential in cooperation with a subthreshold (Glu = 17 nS) glutamatergic input. Blue indicates the region where a GABAergic input has an inhibitory action; that is, it prevents the action potential generation by a suprathreshold (Glu = 17.5 nS) glutamatergic input. Green indicates the region where a GABAergic input has both facilitatory and inhibitory actions, whereas white indicates the region where a GABAergic input has no action in the above sense. Horizontal black solid line indicates the maximum conductance equal to that of the suprathreshold glutamatergic input (GABA = 17.5 nS), whereas the black dotted line indicates the maximum conductance equal to that of the subthreshold glutamatergic input (GABA = 17 nS). L: facilitatory or inhibitory action of a somatically applied depolarizing GABAergic synaptic input on the action potential generation by a somatically applied glutamatergic input using Wilson's 2-compartment model. M: facilitatory action of a dendritically applied depolarizing GABAergic synaptic input on the action potential generation by a somatically applied glutamatergic input using Wilson's 2-compartment model. Note that a dendritically applied depolarizing GABAergic input does not have any region with an inhibitory action.

View larger version (62K): [in this window] [in a new window]   FIG. 5. Effects of the highly fluctuating GABAergic inputs on the neuronal input–output relationship when there is no temporal correlation between glutamatergic inputs and GABAergic ones. A: an example of the fluctuating glutamatergic conductance generated by summing the unitary conductances produced by a 50-Hz Poisson process (top), and the spike train generated by this input conductance (bottom). B: relationship between the mean conductance of glutamatergic inputs Glu (nS) (horizontal axis) and the neuronal firing rate (vertical axis, unit: spikes/s) when GABAergic inputs are absent. C: 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) D: effects of fluctuating GABAergic inputs, which are independent of glutamatergic inputs, on the input–output relationship. Eleven black solid curves in the left panel represent the relations corresponding to 11 different strengths of the purely shunting GABAergic inputs (GABA = 0, 5, 10, 15, ... , 50 nS) from top to bottom. Overlapped black solid curves in the right panel represent the relations corresponding to 11 different strengths of the depolarizing GABAergic inputs (GABA = 0, 5, 10, 15, ... , 50 nS), E: dependency of the firing rates on the mean conductances of glutamatergic (horizontal axis) and GABAergic (vertical axis) inputs. F: dependency of the firing rates on the mean conductance of GABAergic inputs at a fixed mean conductance (Glu = 5 nS) of glutamatergic inputs. Left panels in C–F correspond to the purely shunting GABA case, whereas the right panels correspond to the depolarizing GABA case.

_Glu

_Glu

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 g_Glu (the glutamatergic conductance) and g_GABA (the GABAergic conductance). When the GABAergic conductance is absent (g_GABA = 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 (g_Glu = 5 nS) and 9 different values of the purely shunting (left column) or depolarizing (right column) GABAergic conductance (g_GABA = 0, 5, 10, 15, ... , 40 nS).

As the purely shunting GABAergic conductance increases from g_GABA = 0 nS to g_GABA = 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 g_GABA = 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 (g_GABA = 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 g_GABA = 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 g_GABA = 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 g_GABA = 35 nS and g_GABA = 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 g_Glu = 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 (g_Glu = 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 E_Glu (0 mV) moves upward (Fig. 4A, left, from top to bottom) because of the middle term [–g_Glu (V – E_Glu)] 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 g_Glu (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 E_Glu (0 mV) moves upward further (third and fourth panels of Fig. 4A, left).

View larger version (32K): [in this window] [in a new window]   FIG. 4. Mechanism of the different properties of the purely shunting and the depolarizing GABAergic inhibition on the phase plane. Left columns in A–C: V-nullcline (red solid curves) and the R-nullcline (blue solid curves) of Wilson's neuron model. Horizontal axis represents the membrane potential [V (mV)], and the vertical axis represents the inactivation variable (R). Right columns in A–C: black solid curves indicate the difference between the V-nullcline and the R-nullcline plotted against the membrane potential [V (mV)]. A: changes of the V-nullcline (red solid curves of left columns) and the difference of the 2 nullclines (black solid curves of right columns) along with the increase of the glutamatergic conductance without the GABAergic conductance. gGlu = 0, 3.2, 5, and 10 nS from top to bottom. B, C: changes of the V-nullcline (red solid curves of left columns) and the difference of the 2 nullclines (black solid curves of right columns) accompanying an increase in the purely shunting (B) or the depolarizing (C) GABAergic conductance with a fixed amount of the glutamatergic conductance (gGlu = 5 nS). gGABA = 10, 14, 30, and 45 nS from top to bottom. For comparison, the V-nullcline in the case without the GABAergic conductance is also shown by the red dotted curves in left columns in B and C. Difference of the 2 nullclines in the case without the GABAergic conductance is also shown by the black dotted curves in right columns in B and C. Squares on the horizontal axis in A and B and the crosses on the horizontal axis in A, and C represent the purely shunting and the depolarized GABAA reversal potentials, respectively. Green circles and green dashed curves represent stable equilibrium points and stable oscillatory solutions, respectively.

Now, let us examine the effect of the GABAergic conductance when the glutamatergic conductance is fixed at g_Glu = 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 (g_Glu = 5 nS) and various amounts of the GABAergic conductance (g_GABA = 10, 14, 30, and 45 nS from top to bottom). For comparison, the V-nullcline when the GABAergic conductance is absent (g_Glu = 5, g_GABA = 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., E_GABA = –75 mV), whereas Fig. 4C shows the depolarizing GABA case (i.e. E_GABA = –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 GABA_A reversal potential (E_GABA) moves upward, whereas the right part moves downward because of the last term [–g_GABA(V – E_GABA)] 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 GABA_A reversal potential E_GABA 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 GABA_A reversal potential E_GABA 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 th