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The Journal of Neuroscience, April 1, 1999, 19(7):2765-2779
Network Oscillations Generated by Balancing Graded Asymmetric Reciprocal Inhibition in Passive Neurons
Yair Manor1, Farzan Nadim1, Steven Epstein2, 3, Jason Ritt3, Eve Marder1, and Nancy Kopell3 1 Volen Center, Brandeis University, Waltham, Massachusetts 02454, 2 Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York 12180, and 3 Department of Mathematics and Center for BioDynamics, Boston University, Boston, Massachusetts 02215
ABSTRACT
TOP
ABSTRACT
INTRODUCTION
We describe a novel mechanism by which network oscillations can arise from reciprocal inhibitory connections between two entirely passive neurons. The model was inspired by the activation of the gastric mill rhythm in the crab stomatogastric ganglion by the modulatory commissural ganglion neuron 1 (MCN1), but it is studied here in general terms. One model neuron has a linear current-voltage (I-V) curve with a low (L) resting potential, and the second model neuron has a linear current-voltage curve with a high (H) resting potential. The inhibitory connections between them are graded. There is an extrinsic modulatory excitatory input to the L neuron, and the L neuron presynaptically inhibits the modulatory neuron. Activation of the extrinsic modulatory neuron elicits stable network oscillations in which the L and H neurons are active in alternation. The oscillations arise because the graded reciprocal synapses create the equivalent of a negative-slope conductance region in the I-V curves for the cells. Geometrical methods are used to analyze the properties of and the mechanism underlying these network oscillations.
Key words: neural oscillators; central pattern generators; crustaceans; coupled oscillators; phase plane analysis, mathematical model
INTRODUCTION
The mechanisms by which reciprocal inhibition among neurons give rise to network oscillations have been studied extensively both experimentally and theoretically (Marder and Calabrese, 1996
; Stein et al., 1997
There has been a significant amount of theoretical work on half-center oscillators in which the neurons are essentially identical and the connections symmetric (Perkel and Mulloney, 1974
At the center of the MCN1-activated gastric mill rhythm (Coleman et al., 1995
To understand better the operation of this circuit, Nadim et al. (1998)
MATERIALS AND METHODS In this paper we describe a reduced version of the LG/Int1/MCN1 network that retains the essential features of the compartmental model of Nadim et al. (1998)
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Figure 1. Simplifying a compartmental model of the MCN1-elicited gastric mill rhythm. A, Schematic representation showing the compartmental model (left) and voltage traces of the model Int1 and the LG neuron when the model MCN1 is stimulated at 15 Hz. [Adapted from Nadim et al. (1998)
To simplify the circuit, we model LG and Int1 as two passive neurons, one neuron with a low (L) resting membrane potential (
60 mV) and the other with high (H) resting membrane potential (+10 mV). The modulatory neuron provides a slow excitation (s) to L. This slow excitation is controlled by the membrane potential of L via presynaptic inhibition. This circuit and the resulting oscillation are shown in Figure 1B. The mechanism that we describe depends only on graded, not spike-mediated, synaptic transmission. Therefore we consider only the slow envelopes of the LG and Int1 oscillations shown in Figure 1A, not the fast spiking activity of these cells. The electrical coupling between MCN1 and LG that helps sustain the LG burst (Coleman et al., 1995
Our model is a three-dimensional dynamical system. The three variables are VL, the membrane potential of L, VH, the membrane potential of H, and s, the strength of the excitatory input to L. The effect of AB is added to the circuit as a periodic input (P), and the consequences are discussed. Figure 1C shows a schematic drawing of the circuit and the voltage traces when P is added.
Equations describing the reciprocally inhibitory pair. We first consider the subcircuit formed by L and H, leaving out the M excitation and the periodic input from P. The membrane potentials of L and H are given by first-order differential equations, each with a unique equilibrium point. For simplicity, the membrane capacitances of the two cells are set to 1. The equations of the coupled L/H circuit have the form:
(1)
The parameters of the reciprocally inhibitory synapses are EH
(2) L and EL
H
(3)
These functions are plotted in Figure 2B,C. The functions fH (VH) and fL (VL) represent the intrinsic properties of H and L, respectively. We assume that the intrinsic dynamics of H and L are purely passive:
(4)
(5)
where gleak,L, gleak,H, Eleak,L, and Eleak,H are the conductances and reversal potentials of the leak currents in L and H, respectively. In the biological MCN1-elicited gastric mill circuit, without the slow excitation, L rests at a low potential whereas H fires tonically with a high baseline potential (Coleman et al., 1995
Setting the right-hand side of Equation 1 to 0 and solving for VL, we obtain the following formula for the L nullcline:
(6) L is a sigmoidal function of VH: when VH is low, mH
NH is a sigmoidal function of VL.
(7) Adding the slow modulatory excitation. We now introduce the slow chemical excitation (s) of L. In our model, this excitation is completely controlled by the voltage of L: when VL is above some threshold VT, the excitation decays, and when VL is below VT, the excitation grows. The slow excitation s is therefore governed by the following equation:
where
(8) r,
where
(9)
Adding the fast input from P. When P is added to the circuit, the network is described by Equations 8, 9 and:
(10)
where the conductance of the P to H synapse gP
(11)
We will sometimes use NH, NL, or ÑH to denote the graphs of the respective formulae in the VL-VH space.
(12) The physiological interpretation of nullclines in the VL-VH phase plane. In the absence of periodic input P, and for a fixed value of s, the VL-VH phase plane is used to describe the relationship between the membrane potentials of H and L at any time. The L and H nullclines are the sets of points in the phase plane where dVL/dt and dVH/dt, respectively, are zero. Another view, more intuitive to some physiologists, is that the H nullcline at any value of VL is where VH would settle if L were voltage-clamped at that value of VL. When H is at a membrane potential higher than the H nullcline, VH decays toward NH. VH rises toward NH when H is at a membrane potential lower than the H nullcline. NH thus divides the VL-VH phase plane into two parts: above NH, where dVH/dt < 0, and below NH, where dVH/dt > 0. Similarly, the L nullcline at any value of VH is where VL would settle if H were voltage-clamped at that VH value. NL divides the VL-VH phase plane into two parts: to the left of NL, where dVL/dt > 0, and to the right of NL, where dVH/dt < 0.
The intersections of the two nullclines are equilibrium points at which both dVL/dt and dVH/dt are zero. An equilibrium point may be stable
after any local perturbation, the membrane potential of the perturbed cell will return toward the same equilibrium point; or the equilibrium point may be unstable
By solving the differential equations for VH and VL we obtain VH(t) and VL(t). The trajectory in the VH-VL phase plane is obtained by plotting the (VH, VL) values for all times t. For fixed s, the nullclines NH and NL are fixed curves in the VH-VL plane. In the full system (given by Equations 2, 8, and 9), s changes slowly, producing a family of curves NL that change slowly (NH is independent of s). Thus, the intersections of these curves also change slowly in time, creating "quasi-static" equilibrium points.
Construction of the current-voltage curves of the reciprocally inhibitory pair. One can get additional information from the current-voltage (I-V) relationships of the neurons H and L. We first describe the derivation of the I-V curve for L. For simplicity, we consider the case without the fast periodic input. The total current IL flowing into L is given by
The negative of the right-hand side of this expression gives IL as a function of VL and is used to plot the family of I-V curves of L, dependent on s. Using similar arguments, we can derive an expression for the family of I-V curves of H. In this case, we use Equation 10 instead of Equation 9.
All numerical simulations were performed with the software XPPAUT by B. Ermentrout (available at ftp://ftp.math.pitt.edu/pub/bardware).
Model parameters. The simulations for Figures 1B, 2, 5, 7, and 8 were performed using the following parameter values:
RESULTS Figure 2 illustrates oscillations that result from reciprocal inhibition between two passive neurons with different resting membrane potentials, provided that the L cell receives a slow excitation. Figure 2A shows the voltage traces of the two cells, L and H, together with s (the slow excitation of L), mL
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Figure 2. Slow modulatory excitation can balance an asymmetric half-center to produce oscillations. A, The top two traces are the voltages of the two cells H and L. Also shown are the slow modulatory excitation s to L, the activation of the L to H inhibition mL
We first describe how graded reciprocal inhibition gives rise to a state equivalent to excitability. Two complementary methods are used. The first method uses the I-V curves in the pair of neurons, because these are more familiar to many electrophysiologists. Subsequently we use phase plane analysis, because this is a mathematically compact formalism to elucidate the mechanism of oscillation in this system.
Reciprocally inhibitory graded synapses produce excitability: I-V curves
In this section we use the I-V curves of the pair of neurons to show how inhibitory graded synapses between two passive cells can produce a state equivalent to excitability. The inhibitory graded synapses create a region of negative slope conductance in the I-V curves of the two cells. In physiological terms, the negative slope in the I-V curves is tantamount to membrane excitability. For either cell, this region of negative slope conductance can produce regenerative dynamics, provided that the membrane potential is brought into this region. We show that this shift in the membrane potential can be obtained by an excitatory input to one of the cells. We treat the excitatory input s as a parameter: the effects of the reciprocal synapses on the I-V curves at three representative values of s are discussed.
We start by describing the I-V curves of the two passive cells when uncoupled. In Figure 3A the I-V curves of the L neuron and H are plotted for three different values of excitatory input into L (s = 0, 0.5, and 1) when there are no synaptic connections between L and H. The two I-V curves are plotted together. The zero points on the I-V curves are the resting potentials of the cells. Because both cells are passive, the I-V curves are linear. Without the excitatory input s, the resting potential of the L neuron (marked by
) is low, whereas the resting potential of H (marked by
) is high (Fig. 3A, left panel). With a larger excitatory input s, the I-V curve of L becomes steeper, and the resting potential of L becomes larger (Fig. 3A, middle and right panels). Because there is no slow excitation to H, the I-V curve of H is not affected.
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Figure 3. Reciprocal graded inhibition produces a region of negative slope conductance in the I-V curves. To obtain steady-state I-V curves, s was treated as a parameter. The I-V curves of L (solid line) and H (dashed line) are shown for three values of s (0, 0.5, and 1) in the absence (A) and presence (B) of the reciprocal inhibitory synapses. The resting potentials of L (marked by
Figure 3B shows the I-V curves of L and H when the reciprocally inhibitory synaptic connections between these two cells are included. We first describe the case in which there is no excitatory input to L (Fig. 3B, left panel). At rest, H is at a voltage greater than its threshold for transmitter release. When L is at low membrane potentials, H will release transmitter, so the L I-V curve has a steeper slope, reflecting this additional conductance. When L is at a higher membrane potential, where it inhibits H, the H to L synapse is turned off, so the L conductance is identical to its value in Figure 3A (left panel). The transition between the two regions of the I-V curves creates the negative slope conductance. Both reciprocally inhibitory synapses are therefore responsible for producing the cubic shape, or negative slope conductance region, in the I-V curve of L. In contrast with the I-V curve of L, the I-V curve of H is not affected by the reciprocal synapses and remains linear. This linearity comes from the fact that in the absence of excitatory input, L is never at a membrane potential where it can inhibit H. This will be clarified in the discussion of Figure 4B (left panel) below.
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Figure 4. Graded inhibition produces sigmoidal nullclines in the VL-VH phase plane. To plot the L nullcline (NL) and the H nullcline (NH) in the VL-VH phase plane, s was treated as a parameter. NL (solid line) and NH (dashed line) are shown for three values of s (0, 0.5, and 1) in the absence (A) and presence (B) of the reciprocal inhibitory synapses. The resting potentials of L (marked by ) or unstable (
).
With a larger s, the excitatory (inward) current into L shifts the I-V curve of L downward (Fig. 3B, middle and right traces). When VH is low, this excitatory drive may allow L to depolarize enough to activate the L to H synapse. The activation of the L to H synapse generates an inhibitory (outward) synaptic current from L to H that causes the I-V curve of H to shift upward. The L to H synapse affects only a portion of the I-V curve of H, namely the portion where VH is small enough to allow the L to H synapse to be active. Therefore, when s is large enough, the I-V curve of H assumes a cubic shape.
Excitability in terms of nullclines
In Figure 4A, we show the VL-VH phase planes corresponding to the panels of Figure 3A. The nullclines NL and NH are obtained from Equations 7 and 10 by setting the right-hand side to 0 when the maximal conductances of the reciprocal synapses are also set to 0. Because H receives no synaptic input from L, NH is independent of VL and is a horizontal line at the H resting potential. Similarly, NL is independent of VH and is a vertical line. This vertical line is the average of the resting potential of L and the reversal potential (Es) of the excitatory input s, weighted by their respective conductances. When s = 0, the vertical line is at the L resting potential (Fig. 4A, left panel). With larger values of s, the vertical line shifts to the right toward Es (Fig. 4A, middle and right panels). At the intersection of NL and NH, both VL and VH are at steady state. The steady-state points (
In Figure 4B, we plot the H and L nullclines (from Equations 7 and 10) in the VL-VH phase plane, for the cases corresponding to the I-V curves plotted in Figure 3B. We first describe the shape of NH (same in all three panels of Fig. 4B). At any value of VL, NH depends on the H resting potential, the reversal potential of the L to H synapse, and their corresponding conductances. When VL is small, the L to H synapse is off, and NH lies at H resting potential. As VL increases, the L to H synapse activates, and NH gets closer to the reversal potential of the L to H synapse. The sigmoidal shape of NH reflects the shape of the gating function of the L to H synapse. Similarly NL assumes a sigmoidal shape lying between the resting potential of L and the reversal potential of the H to L synapse. NL depends on s, and changes are shown in Figure 4B.
Now consider the case where there is no excitatory input into L (Fig. 4B, left panel). Because the resting potential of L is close to the reversal potential of the H to L synapse, NL spans a small range of VL. As a consequence, at steady state VL is restricted to values for which NH lies near its maximum, the resting potential of H. Hence, for the whole range of VH, the synapse from L to H is off. This explains the linearity of the I-V curve of H in Figure 3B (left panel).
From Equation 10, NL depends on the conductances and reversal potentials of L and the inputs to L. When s becomes larger, the relative "weight" of the reversal potential of the excitatory input increases. Because this reversal potential is more positive than both the L resting potential and the reversal potential of the H to L synapse, NL moves to the right. Moreover, when VH is low (and therefore the inhibitory input from the H to L synapse is small), s has a larger relative contribution, and NL is stretched more to the right.
The intersections of NL and NH represent steady states for a fixed value of s. When s = 0 (no excitation to L), the two nullclines intersect at a high VH and a low VL (Fig. 4B, left panel). When s is at its maximal possible value of 1 (maximal excitation to L), the two nullclines intersect at a low VH and a high VL (Fig. 4B, right panel). The intermediate case s = 0.5 is shown in Figure 4B (middle panel). In this case, the two nullclines intersect at three points. In such a case, the middle intersection (
Dynamics of the L/H/s oscillation when L and H are passive
The oscillation in the L/H/s system comes about because s, gated by the voltage of L, varies slowly. The analysis of the oscillation can be performed using either the I-V curves or the nullclines in the VH-VL phase plane. Although reasoning with the I-V curves is more intuitive to some physiologists, constructing the I-V curves requires an extra step in which one voltage is computed in terms of the other and the current value of s (see Materials and Methods). An advantage of nullclines is that they are directly computable from Equations 2 and 9. Thus, with nullclines, it is easier to be explicit about the effect of changing parameters such as degree of gradation of synapses and ionic conductances. Moreover, the I-V curves describe the behavior of the cells in steady state based on the assumption that VH adjusts instantaneously as VL changes and vice versa. This assumption is only an approximation to the full dynamics of VH and VL captured in the phase-plane analysis. For these reasons, we shall use the phase-plane analysis to describe the oscillations in the full system.
We start by describing one cycle of the model oscillation in the case where L and H have passive intrinsic properties. Figure 5A shows VH, VL, and s in the time domain. The presynaptic inhibition of s by L is all-or-none. The white line denotes the threshold VT for presynaptic inhibition of s by L. The slow excitatory input s grows when VL is below VT and decays otherwise (Equation 8). Figure 5B-G illustrates the phase plane at six representative times during the oscillation. These times are marked in Figure 5A. In each of the panels (B-G), we show the L nullcline (solid curve) and the H nullcline (dashed curve). We also show the complete trajectory of the oscillation (dotted curve; same in all panels) in the VL-VH phase plane. In each of the panels (B-G) the white circle marks the point on the trajectory at the correspondingly labeled time in A. We refer to this point in B-G as the phase point. The gray circle marks the phase point at the next indicated time. A single arrow indicates the motion of the phase point along the trajectory from the white to the gray circle. Double arrows in D and G indicate fast transitions between L plateau and H plateau. The vertical white line denotes the threshold VT.
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Figure 5. Analysis of one cycle of the oscillation using phase planes. A, Top, middle, and bottom traces show the voltage traces of H and L and the modulatory excitation s. The dotted line superimposed on the voltage trace of L is the threshold VT for presynaptic inhibition of the excitatory input. B-G show the VL-VH phase plane at the six representative times marked in A. The corresponding values of s are marked in each panel. The intersection of the L nullcline (solid line) and H nullcline (dashed line) is the quasi-steady state ( ) indicates the movement of the phase point along the trajectory (dotted line; same in B-G). Vertical white line indicates the threshold VT.
In Figure 5B,C, the phase point follows the quasi-steady state, with VH decreasing from a high value and VL increasing from a low value. The slow movement of the phase point corresponds to the H plateau. In panels B-D, VL is below VT and s increases, causing the L nullcline to shift to the right. In D, the L nullcline separates from the H nullcline at the phase point. At this time the quasi-steady state is lost through a saddle-node bifurcation. Immediately after this time, the phase point jumps to the only remaining quasi-steady state (D, bottom right). This is the beginning of the L plateau and the termination of the H plateau. This fast transition occurs because, away from the quasi-steady states, dVL/dt and dVH/dt are large in magnitude. During this transition VL jumps above VT, and L turns the modulatory excitation off. At this time s starts to decay, causing the L nullcline to shift to the left (E-G). The slow movement through the quasi-steady state shown in E corresponds to the movement along the L plateau. In G, the L nullcline separates from the H nullcline, and the bottom right quasi-steady state is lost through a saddle-node bifurcation. The phase point jumps to the top left quasi-steady state. This is the beginning of the H plateau and the termination of the L plateau. During this transition VL falls below VT and turns the modulatory excitation on. The L nullcline configuration returns to that of B, and the cycle repeats.
A necessary condition for sustained oscillations
For the oscillation mechanism described above to work, the slow modulatory excitation s must grow during the H plateau and decay during the L plateau. During the H plateau, VL must be below VT, otherwise s will stop growing. Similarly, during the L plateau, VL must be above VT, otherwise s will stop decaying. Therefore, the threshold VT for presynaptic inhibition of s by L must lie between the two values of VL, just before the onset and just before the termination of the L plateau. This condition can be formulated as a geometrical condition on the nullclines in the VL-VH phase plane.
In the VL-VH phase plane there are two values of s (Fig. 5,D,G) for which the L nullcline is tangent to the H nullcline. These two values of s define the two saddle-node bifurcation points. The two corresponding phase planes are shown again in Figure 6. VBLeft and VBRight (marked by dotted drop lines) are the values of VL at the two bifurcation points. These two values are, respectively, the voltages of L just before the onset and just before the termination of the L plateau. The top left quasi-steady state (Fig. 5D) may be lost only when VBLeft is below VT. Otherwise the voltage of L reaches VT, where s stops growing, and the L plateau does not occur. The bottom right quasi-steady state (Fig. 5G) may be lost only when VBRight is above VT. Otherwise the voltage of L reaches VT where s stops decaying, and the L plateau does not terminate. Therefore, oscillations occur only if VT (indicated by the white line in Fig. 6) is strictly between VBLeft and VBRight. Note that this is the only restriction on VT. As long as this requirement is satisfied, the exact location of VT with respect to other voltage-related factors (such as the synaptic transfer functions) does not affect the oscillations.
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Figure 6. The L nullcline (solid line) and H nullcline (dashed line) are tangent at two distinct values of VL. A, The tangency on the top left branch of the H nullcline defines the saddle-node bifurcation point that corresponds to the onset of the L burst. B, The tangency on the bottom right branch of the H nullcline defines the saddle-node bifurcation point that corresponds to the termination of the L burst. A necessary condition for oscillations is that the threshold VT (white line) for presynaptic inhibition lies strictly between the VL values (VBLeft and VBRight) at these two tangency points.
We now discuss how several parameters affect the generation of oscillations.
The effect of graded synaptic transmission
In this model, the reciprocally inhibitory synapses are graded. The graded nature of these synapses gives rise to the sigmoidal shape of the L and H nullclines (Fig. 4), and the cubic shape of the I-V curves (Fig. 3). As discussed below, the sigmoidal shapes of both nullclines are essential to the existence of oscillations. Below, we discuss the effect of the slope (kH
If the activation curve of the H to L synapse is too steep, oscillations will not occur. Figure 7A shows how the voltage traces of H and L are affected when the H to L synapse is abruptly changed from graded to all-or-none. Shortly after the beginning of an H plateau, at the time indicated by the vertical arrow, the H to L synapse is made all-or-none by changing its activation curve to a step function. As a result, the plateau of H is prolonged, and as it terminates L starts its plateau. The L plateau, however, does not terminate; instead, VL settles at VT and the oscillation dies. Figure 7B-D shows the phase planes at the times indicated in A. When the H to L synapse is not graded, the synapse is "off" below vH
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Figure 7. Oscillations are disrupted when the activation curve of one of the reciprocal inhibitory synapses is too steep. A, Voltage traces show the alternation of activity in L and H. At the time indicated by the vertical arrow, the H to L synapse is made all-or-none by changing its activation curve to a step function. The dotted line denotes the threshold VT for presynaptic inhibition. B-D show the phase planes at the three times marked in A. Solid and dashed curves are the L and H nullclines. The vertical white line indicates the threshold VT.
If the activation curve of the H to L synapse is too shallow, oscillations will also die. Figure 8A shows how the voltage traces of H and L are affected when the slope of the gating function of the H to L synapse is decreased. At the time indicated by the first arrow, the slope of the gating function is reduced fivefold (see Fig. 8 legend for values). The oscillations after this change become fast and small in amplitude. A further twofold reduction of the slope at the time indicated by the second arrow causes the oscillation to die completely. Figure 8B-D shows the phase planes at the times indicated in A. In B and C, the H and L nullclines are shown at the L plateau termination. In the case shown in B, the two bifurcation points are well separated and oscillations occur as described in Figure 5. In C, because the slope of the L nullcline is closer to that of the H nullcline, the two bifurcation points are much closer to each other. Consequently, oscillations still occur, but with small amplitude. In D, the L nullcline is steeper than the H nullcline, and the two nullclines always intersect in only one point. As a result, no bifurcation points exist, and there is no oscillatory solution.
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Figure 8. Oscillations are disrupted when the activation curve of one of the reciprocal inhibitory synapses is too shallow. A, Voltage traces show the alternation of activity in L and H. At the time indicated by the first vertical arrow, the activation curve of the H to L synapse is made fivefold shallower. At the time indicated by the second vertical arrow, the activation curve of the H to L synapse is made less steep by a factor of two. B-D show the phase planes at the three times marked in A. Solid and dashed curves are the L and H nullclines. The vertical white line indicates the threshold VT.
The effect of the strength of the reciprocally inhibitory synapses
The growth of the slow modulatory excitation s is directly opposed by the H to L synapse; therefore, the stronger this synapse, the longer the period. The decay of s interacts with the L to H synapse by determining how long L will inhibit H. Therefore, if the L to H synapse is stronger, s must decay longer before H can escape the inhibition. These effects can be analyzed by examining how changing the strength of the synapses affects the shape of the nullclines.
Increasing the strength of the H to L synapse pulls the left (top) branch of the L nullcline left toward EH
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Figure 9. The effect of strength of the reciprocally inhibitory synapses on the nullclines. A, Increasing the strength of the H to L synapse pulls the top left branch of the L nullcline (solid line) toward the reversal potential (dotted line) of the H to L synapse. B, Increasing the strength of the L to H synapse pulls the bottom right branch of the H nullcline (dashed line) toward the reversal potential (dotted line) of the L to H synapse.
Similarly, increasing the strength of the L to H synapse pulls the bottom (right) branch of the H nullcline down toward EL
The effect of the strength and time constant of the excitatory input s
Decreasing the maximal conductance gs of the excitatory input to L has an effect that is similar to increasing the maximal conductance of the H to L synapse. If gs is too small, the L nullcline does not move enough to the right for the bifurcation and the transition to L plateau to occur. In this case the oscillations are disrupted. If gs is decreased, but not to the extent that the bifurcation is prevented, the period of oscillation will increase, mainly by increasing the H plateau duration. Note that changing the time constants for the growth and decay of s will change the period of oscillation in a linear manner. This linear relation is a simple consequence of the fact that Equation 8 is linear on either side of VT.
The effect of fast periodic inhibition to H
Up to this point we have described the mechanism of oscillation in an asymmetric pair of reciprocally inhibiting neurons, balanced by a slow excitatory modulation. In the model of MCN1-activated gastric mill rhythm that inspired this work (Nadim et al., 1998
The input of P to H is fast, inhibitory and periodic. When P is added to the circuit, the network is described by Equations 8, 9, and 11. Figure 10A shows the H nullclines NH, when the P input is at 0, and ÑH, when the P input is at its peak value. ÑH depends on the conductances and reversal potentials of H and all of its synaptic inputs (Equation 12). When VL is low, the L to H synapse is off and the effect of the P input on ÑH is relatively large. Therefore, the P input lowers the left branch of the H nullcline more than the right branch. This in turn implies that during an L plateau, when the phase point is on the bottom right branch of the H nullcline, the input from P can be effectively ignored.
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Figure 10. The effect of fast periodic inhibition to H on the nullclines. A, The H nullcline (dashed line) is shown in the absence of the inhibition (P) and in the presence of maximal inhibition. This inhibition pulls the top left branch of the H nullcline toward the reversal potential (dotted line) of the P to H synapse. B, The periodic inhibition of H by P swings the H nullcline back and forth between the two limits shown in A. The L nullcline is not affected. At some intermediate value of the P input, the two nullclines become tangent, producing a saddle-node bifurcation point (
We start with an L/H/s network that is not oscillatory when no P input is present. Figure 10B (top H nullcline) shows such an example, where VT lies to the left of the left bifurcation point (
Figure 11A shows VH, VL, and s in the time domain. In B-E, we show the phase planes at the times indicated in A. In each panel, the white circle denotes the position of the phase point along the trajectory. In B, the phase point is at the quasi-steady state (the intersection of the two nullclines). The periodic input from P moves the H nullcline down rapidly, back and forth between the two limits shown as dashed curves. The trajectory follows this nullcline as shown by the dotted curve. As s grows, the L nullcline moves to the right. In C, the top left quasi-steady state is lost during a P input (through a saddle-node bifurcation), and the phase point jumps to the bottom right quasi-steady state. At this time, s starts to decay, and the L nullcline moves to the left (D). In E the bottom right quasi-steady state is lost, and the phase point jumps back to the top left quasi-steady state, completing the cycle. Note that the P input does not contribute to the loss of the bottom right quasi-steady state. In fact, the P input hinders the termination of the L plateau, although to a very small extent.
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Figure 11. Analysis of one cycle of the oscillation in the presence of the periodic P input. A, Top, middle, and bottom traces show the voltage traces of H and L and the modulatory excitation s. B-E show the VL-VH phase plane at the four representative times marked in A. In each panel, the two dashed curves show the H nullcline when the periodic input P is at zero and at its maximum. The solid line is the L nullcline, and
We mentioned earlier that oscillations are disrupted if either of the reciprocal synapses is all-or-none. However, in the presence of the P input, oscillations are still possible when either one of the synapses is all-or-none and the other is graded. We will discuss the case in which the L to H synapse is all-or-none (and the H nullcline is step-like) and the H to L synapse is graded (and the L nullcline is sigmoidal). The discussion of the other case is similar. As discussed in Figure 6, when one of the two synapses is all-or-none, the two saddle-node bifurcations values VBLeft and VBRight are identical. Recall that the decay of s will result in the termination of the L plateau only if VT is to the left of VBRight. In general, the transition to an L plateau onset occurs when the two nullclines separate, causing the top left intersection point to disappear. However, when the L to H synapse is all-or-none, the growth of s is not sufficient to separate the two nullclines, because VT is to the left of VBLeft (because VBLeft = VBRight; see the section entitled A necessary condition for oscillations). In the presence of the P input, the left branch of the H nullcline is periodically shifted downward (as shown in Fig. 10B). This provides an alternative way for separating the two nullclines, allowing an L plateau onset. Note that the growth of s corresponds to accumulating excitatory input in L, whereas the P input corresponds to removing inhibition from L. The L plateau onset in the presence of the P input is caused by this periodic removal of inhibition.
The P input can also be added to an L/H/s network that is already oscillatory, with minimal changes in the description of Figure 11. Moreover, whether or not the L/H/s network is oscillatory, the input from P determines the timing of the transition from H plateau to L plateau. The control of the timing of this transition by P leads to the frequency control mechanism (F. Nadim, S. Epstein, Y. Manor, J. Ritt, E. Marder, and N. Kopell, unpublished observations).
DISCUSSION Symmetric and asymmetric half-centers
Reciprocal inhibition is a common circuit element in the nervous system. Brown (1914)
There have been a number of theoretical studies on the factors that control the behavior of half-center oscillators when the two neurons that form them are identical (Perkel and Mulloney, 1974
Several workers have noted the necessity of "balancing the excitability" of the two sides of the half-center to produce rhythmic alternating bursts. For example, Miller and Selverston (1982b)
