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1Life Sciences Department and Zlotowski Center for Neurosciences, Ben-Gurion University of the Negev, Beer-Sheva, 84105 Israel; 2Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey 07102; 3Courant Institute of Mathematical Sciences, New York University, New York City, New York 10012; 4Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, 07102; and 5Department of Biological Sciences, Rutgers University, Newark, New Jersey 07102
Submitted 28 April 2003; accepted in final form 15 June 2003
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXDISCLOSURESREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXDISCLOSURESREFERENCES |
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; Fries et al. 2001; Gray et al. 1989; Rodriguez et al. 1999), in various sensory states (Laurent et al. 1996; Patel and Baladan 2000; Perez-Orive et al. 2002) and in the production of motor patterns (Grillner et al. 2000; Marder 2000; Nusbaum and Beenhakker 2002). In many cases, such neuronal networks produce rhythmic activity over a wide range of frequencies (Marder and Calabrese 1996). The output of a rhythmic network is defined by the frequency dependence of the activity of individual neurons or groups of neurons. For instance, the activity of different neurons may be separated by a fixed time latency, despite wide changes in frequency (Ahissar et al. 2000; Bartos et al. 1999; Faulkes and Paul 1998). Often, however, rather than a constant latency, neurons maintain a constant phase difference (DiCaprio et al. 1997; Friesen and Pearce 1993; Grillner 1981; O'Keefe and Recce 1993; Young 1975), where the phase difference is the ratio of latency and cycle period. In other cases, when frequency is changed neurons in the network show differences in activity times that range between constant latency and constant phase (Davis 1969; Fischer et al. 2001; Grillner 1981; Hooper 1997a,b; Pearson and Iles 1970).
Nowhere is the importance of phase adjustment more obvious than in rhythmic motor systems, which operate in a wide range of frequencies (Brodin et al. 1985; Marder and Calabrese 1996; Mercier and Wilkens 1984; Skinner and Mulloney 1998) and where mechanical restrictions dictate that the pattern in muscle activity should be adjusted in accordance to speed of movement. In the crab ventilatory system, for instance, phase constancy is necessary to produce a smooth and coordinated movement of the ventilatory pump (DiCaprio et al. 1997). In lamprey and other aquatic organisms propelled by undulatory locomotion, fixed phase lags between consecutive segments guarantee that the rostra-caudal axis of the body forms exactly one wavelength. This ensures the minimization of lateral thrust and the optimization of swimming at all possible speeds (Grillner 1974; Sigvardt 1981).
Despite its importance in central pattern generation, the mechanisms underlying phase constancy across different cycle frequencies are not well understood. Several studies have addressed the problem of phase maintenance in chains of coupled oscillators and have proposed that the mechanism underlying phase constancy is embedded within the circuitry. In Williams (1992), for example, the synapses that make up a unit oscillator are repeated in neighboring segments, albeit with a reduced synaptic strength. In Skinner and Mulloney (1998), in each segment one neuron makes equal excitatory and inhibitory connections to two identical neurons in the previous segment. In both works, even though phase constancy was demonstrated, the essence of the mechanism remains unclear.
In this work, we propose a novel mechanism involving short-term synaptic depression to create phase maintenance between neurons in rhythmic networks. We consider a simple model network consisting of an oscillatory neuron that has an inhibitory synapse onto a follower neuron. The mechanism is based on the simple idea that the latency of firing in the postsynaptic cell is directly affected by the synaptic strength. In a nondepressing synapse, the strength and time course of the synapse are independent of cycle frequency. In contrast, when the synapse is depressing, synaptic strength varies with cycle frequency, allowing postsynaptic latency to vary as well. With the correct choice of synaptic dynamics, we show that it is possible to vary the postsynaptic latency in a way that is approximately proportional to the change in cycle frequency, thereby allowing the phase to be relatively well maintained.
We have intentionally chosen a very simple oscillator-follower model to illustrate our proposed mechanism for phase maintenance. In spite of the simplicity of this model, the explanation of how a depressing synapse contributes to maintain phase is surprisingly complicated. As we will show, this complexity arises because of the interaction between synaptic dynamics and intrinsic properties of the postsynaptic neuron. At certain cycle periods, the latency is determined by the intrinsic properties alone, whereas in other cycle periods, it is mostly affected by the synaptic dynamics. In principle, there can be numerous ways in which cycle period of an oscillatory system can change. We explore in detail three cases where cycle period is varied while preserving the duration of the active state, the duration of the inactive state, or both durations proportionally. Despite their simple nature, these three representative ways of changing the cycle period provide results that are insightful for the general case. Indeed, in all three cases synaptic depression provides a degree of flexibility that can be used to automatically adjust the time difference between the activities of the two neurons, such that their phase difference is approximately maintained. Although a depressing synapse does not produce perfect phase maintenance, we show that in most cases it is superior to a nondepressing synapse in promoting a constant phase difference.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXDISCLOSURESREFERENCES |
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The durations of the active and inactive states of neuron O are defined as TActive and TInactive, respectively. The period P is equal to TActive + TInactive (Fig. 1, bottom). In this study, we consider three ways of changing the period: 1) the duration of the nonactive state of neuron O (TInactive) varies and the duration of the active state of neuron O (TActive) remains constant (TActive = 250 ms in RESULTS). 2) Both TInactive and TActive vary such that the ratio TActive/(TActive + TInactive), defined as the duty cycle (DC), remains constant (DC = 0.3 in RESULTS). 3) TInactive remains constant and TActive varies (TInactive = 750 ms in RESULTS).
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We now describe the model of the follower neuron F and the synapse from O to F. For each case of changing the period, we hereafter refer to the model with the parameter values given below as the "reference model." When noted, the reference model will be changed to assess the dependence of our model on parameters (see RESULTS).
Cellular model
The follower neuron F is modeled with standard current balance equations based on the Morris-Lecar model (Morris and Lecar 1981). The ionic conductances of this neuron include a fast activating conductance with high equilibrium potential (the "calcium" conductance), a slow activating conductance with low equilibrium potential (the "potassium" conductance), and a non-voltage-dependent conductance (the "leak" conductance)
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Ca = 0.3, K = 0.6, and gL = 0.15; the reversal potentials are (in mV) ECa = 100, EK = –70, and EL = –50; the steady-state activation functions are m
(VF) = 0.5 {1 + tanh[(VF–1)/14.5]} for the calcium conductance, and w(VF) = 0.5 {1 + tanh[(VF - 20)/15]} for the potassium conductance; the applied current Iext is 7.5 µA/cm2. The parameter 
F, which determines the speed of the intrinsic dynamics in neuron F, is equal to 150 ms in the constant TActive case and 100 ms in the other two cases. With these parameters, neuron F is a quiescent cell with a high-voltage equilibrium point. Synaptic model
The current balance equation for neuron F includes an additional term that represents the synaptic current from O to F
 | (2) |
syn (in mS/cm2) is: 0.185 in the constant TActive case, 0.22 in the constant duty cycle case, and 0.35 in the constant TInactive case. The gating variable s represents the fraction of open synaptic channels. In general, except for the single time point when neuron O switches from its nonactive to its active state, s decays toward 0. This decay represents the closure of synaptic channels after the onset of transmitter release. When neuron O switches to its active state, s is set to the fraction of available synaptic resources at that time. This fraction is described by a second variable d, which decreases toward 0 when the presynaptic cell O is active (representing the decay of available synaptic resources, i.e., the synaptic depression), and increases toward 1 when the presynaptic cell O is not active (representing the recovery of available synaptic resources).
Figure 1 shows time traces of the postsynaptic voltage VF (top), s (thick, middle), and d (thin, middle). At the onset time of the active state in neuron O, s is set to the value of d. Except for this single time point, s decays to 0 in two steps: 1) during the active state of neuron O (black bars, bottom trace), s decays with a slow time constant 
. This slow decay represents the fatigue of the synapse, for example as a result of desensitization of synaptic receptors. In our study, we assumed that the synaptic fatigue is small. Hence, we set to 25 s such that s remains essentially constant for the duration of the active state. And 2) during the nonactive state of neuron O (between black bars, bottom trace), s decays with a faster time constant 
. This decay represents the closing of synaptic channels as the neurotransmitter is removed from the synaptic cleft. The value of is 1.5 s in the constant TActive case, 0.5 s in the constant duty cycle case, and 0.3 s in the constant TInactive case. Hence, except for the single time point where s is set to the value of d, the dynamics of s are governed by the following equation
 | (3) |
The variable d represents the fraction of available synaptic resources (depression). It evolves independently of s, decreasing toward 0 with time constant 
when neuron O is active and recovering toward 1 with time constant 
when neuron O is nonactive
 | (4) |
= 3 s; = 1.5 s for the constant TActive case and 0.5 s for the other two cases. For a depressing synapse, the peak strength of the synaptic current depends on the peak value of d during the cycle.
To make a synapse nondepressing, we assume that the fraction of available synaptic resources is constant, that is, independent of the period. Thus for a nondepressing synapse the value of d is not determined by Eq. 4 but is fixed to d = 1. We can tune a nondepressing synapse to have dynamics identical to a depressing synapse at a specific value of the period by setting syn in the nondepressing case to the maximum value of g = syns for that specific case of the depressing synapse.
Definitions of key values of the synaptic conductance
To facilitate the understanding of how synaptic conductance determines the onset of activity in neuron F, we define several key values of the synaptic conductance. The synaptic conductance itself, g, is simply the product of the maximal conductance syn and the fraction of open synaptic channels s.
Note that if the synaptic conductance is too large, activity in neuron F is suppressed. Thus a necessary, but not sufficient, condition for neuron F to become active is that the synaptic conductance g is less than some value g* defined here as the critical conductance.
The conductance immediately before the onset of activity in neuron F is defined as the transition conductance gjump. It is important to emphasize that the transition conductance gjump is not identical to the critical conductance g*. For example, if the inhibition is weak (i.e., the maximal synaptic conductance syn is small), the synaptic conductance g may never be sufficiently large to reach the critical conductance g*, but by definition it will reach the value of gjump immediately before the onset of activity in neuron F. On the other hand, when the inhibition is strong (syn is large), at the onset of activity in neuron O, the synaptic conductance g may be larger than the critical conductance g*. If the duration of the nonactive state in neuron O is sufficiently long, s can decay to a sufficiently small value such that the synaptic conductance g reaches the critical conductance g*, and at that time neuron F becomes active. It is only in this latter case that the critical conductance g* and the transition conductance gjump are equal, and the condition g < g* becomes a sufficient condition for the transition to the active state.
We further define gpeak as the synaptic conductance at the onset of activity in neuron O, i.e., at the instant that s is set to the value of d. Because s decays at all other times, gpeak is the peak synaptic conductance reached during a cycle. Depending on the period, gpeak may be greater than or less than the critical conductance g*. In the former case, the synapse is defined as weak; it is strong in the latter case.
Definition of time interval and phase
For the analysis of the effects of the depressing synapse on the firing time of neuron F, we arbitrarily define the onset of firing of a neuron as the time at which the membrane potential of this neuron increases past 0 mV. The period (P) is defined by the interval between consecutive onsets of firing in neuron O. 
t is defined as the time interval between the onset of firing in neuron O and the subsequent onset of firing in neuron F. The phase (
) of firing in neuron F is defined as t/P. Thus the phase ranges between 0 and 1 with both extremes corresponding to the case where neurons O and F start their active states at the same time. Whenever the period is changed, we allow the system to equilibrate before we measure t and .
All numerical simulations were done with the software XPPAUT (Ermentrout 2002).
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXDISCLOSURESREFERENCES |
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t) of follower neurons when the period changes. t is a direct function of the synaptic strength. When the synapse is nondepressing, the synaptic strength is independent of the period. In contrast, with a depressing synapse, the synaptic strength changes as a function of the period. Hence, a depressing and a nondepressing synapse affect the activity time of a follower neuron in different ways. Figure 2 illustrates this fact by comparing the activity of a follower neuron with a depressing and nondepressing synapse, for three different period values. Here, when the synapse was nondepressing, t was fixed for all three periods; with a depressing synapse, t was larger for larger period values. If the increase in t was proportional to the increase in period (P), the phase ( = t/P) would be perfectly maintained. This example shows that under appropriate conditions, the dependence of t on period could provide a mechanism for keeping phase approximately constant, across different periods.
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In general, how t changes with period depends, among other things, on how the period is changed. Of the numerous ways to change the period, we considered three different cases. These cases are shown in Fig. 3 and are labeled the constant TActive case, the constant duty cycle case and the constant TInactive case. In the constant TActive (TInactive) case, we varied the period by modifying only TInactive (TActive). In the constant duty cycle case, both TActive and TInactive were proportionally varied such that the duty cycle remained constant. There are, of course, other ways to vary the period. However, we believe that these three cases are good representatives of the effect of a depressing synapse when the period is arbitrarily changed.
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Our results are divided in three sections. First, we describe how the synaptic conductance changes as a function of period for the constant TActive, constant duty cycle, and constant TInactive cases. This is followed by a detailed explanation of the dependence of t and phase on period in these three cases. Finally, we give a description of how various biophysical model parameters affect the dependency of phase on period.
Dependence of synaptic conductance on cycle period
The oscillatory activity of neuron O dictates an oscillatory time course for the synaptic conductance g (= syns). At the onset of activity in neuron O, the synaptic conductance g is at its peak value gpeak. The synaptic conductance then decays, first with time constant (when neuron O is active) and then with time constant (when neuron O is nonactive; see Fig. 1). Immediately before the onset of activity in neuron F, the synaptic conductance is equal to the transition conductance gjump. t measures the time during which the synaptic conductance decays from gpeak to gjump. Analytical derivations for gpeak and t are provided in the APPENDIX. An analytical expression for gjump depends on the intrinsic properties of neuron F and will not be derived here. However, we will show that for a wide range of period values, gjump is constant.
The primary effect of synaptic depression was to change the value of gpeak. The following equation defines gpeak, regardless of how the period is changed (see APPENDIX)
 | (5) |
Equation 5 shows that gpeak depends not only on the period (= TInactive + TActive), but also on the time constants of recovery () and depression () and the maximal conductance syn.
Figure 4 shows how the peak conductance gpeak and the transition conductance gjump depended on period for the three different cases when the synapse was depressing. The effect of the synapse on the firing time of neuron F depended on the relationship of gpeak and gjump to g*, the critical synaptic conductance below which the synapse was too weak to keep neuron F inactive (see METHODS). The critical conductance g* is constant. In general, two factors can determine the firing time of neuron F: the intrinsic properties of neuron F (determined by F) and the synaptic dynamics (determined by , , , Esyn, and syn). The extent to which the synapse recovered from depression determined which of these two factors affected the firing time of neuron F. For example, if the synapse was recovered enough so that gpeak was larger than the critical conductance g* and the transition conductance gjump was equal to g*, then the firing time of neuron F was completely determined by the synapse. In this case, the necessary condition for firing of neuron F (gjump 
g*) became a sufficient condition. On the other hand, if the synapse was depressed and gpeak was smaller than g*, the firing of neuron F was mainly determined by its intrinsic properties because the synapse was too weak to greatly affect it. In this case, gjump was not fixed: its variation with period resulted from the variation of gpeak with period.
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In Fig. 4A, the membrane potential of neuron F (VF) and the synaptic conductance (g) are shown for the constant TActive case when the period is 500, 1,000, and 2,000 ms. At a period of 500 ms, both the peak conductance gpeak and the transition conductance gjump were less than the critical conductance g* (· · ·). At periods of 1,000 and 2,000 ms, gpeak was larger than the critical conductance g*, whereas gjump was equal to g*. Figure 4B shows the dependencies of gpeak and gjump on the period for a continuous range of period values. Because the period was increased by increasing TInactive, as the period increased, the synapse had more time to recover from depression, and thus gpeak increased (as can also be seen directly from Eq. 5). For a period <500 ms, the synapse was too weak to inhibit neuron F, and neuron F remained at its equilibrium point and displayed no rhythmic activity (not shown). For period values <500 ms, both gpeak and gjump increased with the period. For period values <750 ms, gpeak was smaller than the critical conductance g*: the synapse was weak and the firing of neuron F was mainly affected by the intrinsic properties of neuron F. For period values<750 ms but <1,250 ms, gpeak was larger than the critical conductance g*, but gjump was smaller than g*. In this range, the synapse affected the firing time of neuron F together with the intrinsic properties of neuron F. When the period was <1,250 ms, gjump was equal to the critical conductance g* (a constant value) and the firing of neuron F was completely determined by the synapse.
Figure 4, C and D, shows the corresponding situation for the constant duty cycle case. This case is similar to the constant TActive case in the qualitative dependencies of the peak conductance gpeak and the transition conductance gjump on period. Note that gpeak and gjump increased with the period, and this result was independent of the choice of time constants of recovery and depression (not shown). Here the synapse completely determined the firing time of neuron F (gpeak < g* and gjump = g*) when the period was <2,500 ms. This value is relatively large, compared with the constant TActive case (where the synapse completely determined the firing time of neuron F when the period was <1,250 ms). This is because here the period was increased by increasing both TActive and TInactive. Hence, not only the recovery from depression increased when the period increased but also the extent of depression.
Figure 4, E and F, shows the situation for the TInactive constant case. Here the dependencies of the peak conductance gpeak and the transition conductance gjump on the period were opposite to those found in the previous two cases. Both gpeak and gjump decreased when the period increased. This is because the period was increased by increasing TActive only, and a larger value of TActive produced a greater depression of the synapse and thus a smaller value of gpeak. This can also be seen directly from Eq. 5. In this case, the two conditions for which the synapse totally determined the firing time of neuron F, gpeak < g* and gjump = g*, were obtained for periods smaller than 1,750 ms. For periods <1,750 ms, gjump decreased from the critical conductance g* as the period increased, and the firing time of neuron F was mostly determined by its intrinsic dynamics (the synapse was too depressed).
Analytical description of t
In this section, we show how the values of the peak conductance gpeak and the transition conductance gjump determine t, the time delay between the activity in neuron O and neuron F. Because t is determined by the rates at which the conductance decays from gpeak to gjump, the following equation describes the dependence of t on model parameters for a sufficiently large maximal conductance syn (see also APPENDIX)
 | (6) |
and are decay time constants of the synapse during the inactive and active states of neuron O, respectively. In general, both t and gjump are values that are a priori not known. When the O to F synapse is weak, the firing time of neuron F is largely controlled by its intrinsic properties. Thus at this time the transition conductance gjump is mostly determined by the intrinsic dynamics of neuron F. In this case, Eq. 6 cannot be used to compute t without resolving the dependence of gjump on the dynamics of neuron F. However, with a stronger O to F synapse, gjump becomes equal to the critical conductance g* (a constant value), as seen in Fig. 4, and hence
 | (7) |
t can be analytically calculated. Equations 6 and 7 apply when the synapse is recovered enough to determine the firing time of neuron F. In other words, these equations apply to the constant TActive and duty cycle cases when the period is large and to the constant TInactive case when the period is small. For simplicity, in our model we assumed that is large relative to , such that during the active state of neuron O the synaptic decay is minimal. In this case, Eq. 7 can be approximated by the following equation
 | (8) |
t and phase of firing in neuron F on the period, in each of these three cases.
Dependence of t and phase on period in the constant TActive case
As we saw from Fig. 4, A and B, increasing the period by keeping TActive constant allowed the synapse to recover from depression without changing the extent of depression. In Fig. 5A, we compare the activity of neuron F when the synapse is nondepressing and depressing at period values of 1,000 and 2,000 ms. The top and bottom traces show the membrane potential of neuron F and the synaptic conductance, respectively. For the sake of comparison, we tuned the parameters of the nondepressing synapse such that its effect was identical to the effect of a depressing synapse when the period was 1,000 ms (see METHODS). Thus, at a period of 1,000 ms, the time courses of the synaptic conductance and the membrane potential of F were identical for the depressing and the nondepressing synapses (traces are superimposed, Fig. 5A, left). In both cases, the peak of the synaptic conductance was 120 µS/cm2. At a period of 2,000 ms, in the nondepressing case the strength of the synapse was identical to when the period was 1,000 ms (compare the peak values of the synaptic conductance, horizontal dotted line). Thus t was unchanged. In contrast, in the depressing case, the peak of the synaptic conductance was 155 µS/cm2 because the synapse recovered more from depression. This caused t to increase almost 1.5-fold.
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Figure 5B shows the relationship between the period and t (computed numerically) for the depressing (black) and nondepressing (blue) cases. To compare the two cases, the maximal synaptic conductance (syn) for the nondepressing synapse was chosen such that the phase was equal to 1 when the period was 500 ms (the smallest period value that sustained a rhythm in neuron F when the synapse was depressing). The red line represents an idealized relationship in which phase is perfectly maintained at 0.643. Note that the depressing synapse case remained close to phase constancy for a large range of periods extending from 500 to 1,200 ms, whereas the nondepressing synapse case did not.
Figure 5C plots the phase () as a function of the period (P). The black, blue, and red curves represent the depressing synapse, nondepressing synapse, and idealized constant phases. In the case of a depressing synapse, the curve was cubic shaped with a local minimum at the point (P1, 1) and a local maximum at (P2, 2). In the nondepressing case, t was constant in this case and hence the phase monotonically decreased like 1/P. Consequently, between periods of 500 and 1,500 ms (1-s interval marked by the vertical dotted lines in Fig. 5C), the change in phase for the depressing case (0.063) was much less than that of the nondepressing case (0.668).
In the depressing case, the dependence of phase on period followed a cubic shape. This shape was obtained because t was controlled by different mechanisms in different ranges of period values. With small period values (P < P1), the synapse was largely depressed and hence t was mostly determined by the intrinsic dynamics of neuron F. Because these intrinsic dynamics did not change with period, t was almost constant and thus the phase behaved like 1/P. For P < P1, the synapse increasingly contributed to t because in this range of periods, the synapse increasingly recovered from depression. Between P = P1 and P = P2, the phase increased because t increased more rapidly than the period (Fig. 5C). This was due to the choice of synaptic parameters, in particular the fact that the synaptic decay (governed by ) was much slower than the intrinsic dynamics of neuron F (governed by F). To understand the existence of the local maximum point at (P2, 2), consider the situation when the period was very large. In this case, the synapse maximally recovered and gpeak and gjump approached constant values (the maximal conductance syn and the critical conductance g*, respectively), hence t approached a constant value. Therefore for large periods the phase decreased like 1/P. The increase and then decrease of the phase, at intermediate and then large period values, imply the existence of a local maximum for the phase.
Dependence of t and phase on period in the constant duty cycle case
In contrast to the constant TActive case, when period was increased by keeping the duty cycle constant, both the recovery from depression and the extent of depression increased. Nevertheless, the effect of the depressing synapse in the constant duty cycle case was qualitatively similar to the constant TActive case because the dependence of the peak conductance gpeak and the transition conductance gjump on period was similar in these two cases (Fig. 4, B and D). Figure 6A compares the activity of neuron F when the synapse was nondepressing (dotted traces) and depressing (solid traces) at periods of 1,000 and 2,000 ms. The description of this figure is similar to that of Fig. 5A.
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It is important to emphasize that in the constant duty cycle case the O to F synapse could affect the activity of neuron F in two distinct ways, depending on whether the maximal conductance syn is smaller or larger than the critical conductance g*. This is true whether the synapse is depressing or not. When the synapse is nondepressing, syn < g* implies that neuron F remains inhibited as long as neuron O is active independent of the period. We shall refer to this case as the strong nondepressing synapse. In contrast, when syn < g* (weak nondepressing synapse), neuron F need not remain inhibited for the entire duration of the active state in neuron O (see for example the P = 2,000 ms traces of Fig. 6A). The weak depressing synapse behaves essentially the same way as the weak nondepressing synapse. This is not true for a strong synapse (syn < g*). Thus, when discussing the effect of a depressing synapse in the following text, we treat the case of a strong depressing synapse.
Figure 6, B and C, respectively, shows the dependence of t and phase on period for a range of period values for a depressing synapse (black curve), a strong (green curve), and three weak (blue curves) nondepressing synapses. The idealized constant phase case is represented as red curves. In the case of the depressing synapse, at short and intermediate period values, the dependency of t (and phase) on period was similar to the constant TActive case. However, at large period values, the constant duty cycle case was qualitatively different from the constant TActive case. As the period was increased, t approached a constant value in the constant TActive case, whereas here it continued to increase linearly. To understand why, recall that in both the constant TActive and constant duty cycle cases the activity in neuron F was suppressed during the activity of neuron O. Thus the activity of neuron F started at some time interval after the end of activity in neuron O (t = TActive + 
t, where t is the time interval between the end of activity in neuron O and the beginning of activity in neuron F). At large period values, the synapse was maximally recovered. Hence, t no longer changed as the period continued to increase. Because in the constant duty cycle case TActive increased linearly with period, so did t. From the definition of phase = t/P = (TActive + t)/P, at large period values the term t/P became negligible. Consequently, the phase approached the duty cycle value (DC = TActive/P).
The dependence of t on period was qualitatively different for the weak and strong nondepressing synapses. When the nondepressing synapse was strong (green curves in Fig. 6, B and C), at all period values neuron F started to fire at a fixed time after the end of activity in neuron O. Thus, t was equal to TActive + t and t was constant for all period values because it only depended on , the decay time constant of the synapse (which is independent of the period). Hence, as period increased, t increased linearly and the phase decayed toward the duty cycle DC. When the nondepressing synapse was weak (blue curves in Fig. 6, B and C, with 1 denoting the weakest synapse), the effect of the synapse on t and the phase was qualitatively different for small and large period values. With large period values t was constant, whereas it increased for small period values. The transition point between these two regions moved to larger period values when the maximal conductance syn was increased (from curve 1 to curve 3). To explain the existence of these two regions, recall that with a weak nondepressing synapse, following the onset of inhibition neuron F depolarized according to its intrinsic dynamics. Because the synapse was not strong enough to keep neuron F inactive indefinitely, when the period was large enough neuron F eventually started its activity while neuron O was still active. In this case, t was independent of the period because it only depended on the dynamics of neuron F. Hence, for large period values, t was constant and the phase decayed to 0. On the other hand, when the period was small, the intrinsic dynamics of neuron F did not have sufficient time to enable it to become active before neuron O stopped its activity. Therefore, just as in the case of a strong nondepressing synapse, t was equal to TActive + t. However, in contrast to the strong nondepressing synapse where t was fixed, here t decreased with the period. To explain this, note that t is the time it took neuron F to reach activity after the end of activity in neuron O. As such, when the synapse was weak, t was determined primarily by the intrinsic properties of neuron F. When the period increased, TActive increased and hence during the time that neuron O was active, the membrane potential in neuron F depolarized more. Therefore, at the end of the activity in neuron O the membrane potential of neuron F was closer to its activity threshold, and consequently it took less time for neuron F to become active. Because the decrease of t was small relative to the increase in TActive (since the depolarization of neuron F was relatively slow), overall the sum TActive + t increased as the period increased. Note that the increase in t in this case was due only to the increase in TActive and was completely independent of changes in TInactive. However, the increase in period was due to increases in both TActive and TInactive, and as a result the phase decreased with period.
Note that in the 1-s interval marked by the vertical dotted lines, the change in phase for the depressing synapse (0.149) was smaller than any of the nondepressing cases (strong: 0.467; weak: 1, 0.269; 2, 0.272; 3, 0.214).
Dependence of t and phase on period in the constant TInactive case
In the constant TInactive case, the time during which the synapse depressed, and hence the extent of synaptic depression grew as the period increased. Hence, the synapse was most relevant when the period was small. Figure 7A shows the activity of neuron F when the synapse was nondepressing (dotted lines) and depressing (solid lines) at period values of 1,000 and 2,000 ms. With a depressing synapse, the activity in neuron F began at a later time for a period of 1,000 ms compared with when the period was 2,000 ms. This was due to the fact that the conductance of the depressing synapse was smaller when the period increased (bottom).
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Figure 7, B and C, respectively, show the dependence of t and phase on period for a depressing synapse (black curve), a strong (green curve), and three weak (blue curves) nondepressing synapses. The idealized constant phase cases are shown as red curves. In the depressing case, the dependence of t on period went through a clear transition from being dependent on the properties of the depressing synapse to being dependent on the intrinsic properties of neuron F. The transition occurred at the period value that corresponded to the local maximum of the t versus P curve (at a period value of 1,450 ms). For period values below this transition point the synapse was strong (the peak conductance gpeak was larger than the critical conductance g*) and hence neuron F could switch to its active state only after the end of activity in neuron O. In this regime, t = TActive + t, where t is the time interval from the end of activity in neuron O to the beginning of activity in neuron F. Note that both TActive and t changed as the period was increased but in opposite directions. The time interval t decreased because the synapse became more depressed, while TActive increased linearly with the period (by definition of the constant TInactive case). These two time intervals were determined by completely separate parameters. We chose the reference parameters so that t initially decreased and then increased. For period values above the transition point (1,450 ms), the synapse became too weak (gpeak < g*) to keep neuron F silent while neuron O is active, and hence neuron F started to fire before the end of activity in neuron O. Near the transition point, the synapse still had an effect in delaying the firing of neuron F, but this effect weakened as the period was increased. Hence t decayed. At period values larger than 2,700 ms, the synapse was totally depressed, and the firing time of neuron F was exclusively determined by the intrinsic dynamics of neuron F. As a result, t approached a constant value. The phase curve had a local minimum and a local maximum. The local maximum of the phase curve occurred at the period value of the local maximum in t (Fig. 7B). For larger period values, t decreased to a constant value, and the phase decayed to 0. To explain the occurrence of the local minimum in the plot of phase versus period, recall that for period values below the local maximum there were two competing effects on t. As the period increased, t (and therefore the phase) first decreased. With further increases in the period, t increased rapidly, causing the phase to increase as well. This resulted in a local minimum at low period values.
When the synapse was nondepressing, as for the constant duty cycle case, the dependence of t (and hence of the phase) on the period was qualitatively different if the synapse was strong (syn < g*, green curve) or weak (syn < g*, blue curves). As in Fig. 6B, the weak nondepressing cases are numbered from 1 to 3 with 1 marking the weakest case. When the nondepressing synapse was strong, neuron F started to fire at some fixed time interval after the end of activity in neuron O. Hence t = TActive + t, where TActive increased linearly with the period and t was constant (as in the strong nondepressing constant duty cycle case). Hence t increased. Recall that = TActive/P + t/P. As the period was increased, the first term on the right side of this equation approached 1, whereas the second term approached 0. Thus the phase increased toward 1.
When the synapse was nondepressing and weak, the dependence of t on the period was identical to the constant duty cycle case because the arguments presented for the constant duty cycle case depended only on changes in TActive. In the present case, in the range of period values where t increased with the period, the phase also increased. This increase occurred for the same reasons that the phase increased in the strong nondepressing case. In contrast, for larger period values, where t was constant, the phase decayed to 0.
In contrast to the constant duty cycle and constant TActive cases, in the constant TInactive case a depressing synapse did not show a significantly different variation of phase in the 1-s interval between period values of 500 and 1,500 ms (vertical dotted lines), compared with a strong nondepressing synapse (0.292 and 0.302, respectively). Moreover, the weak nondepressing case gave a smaller variation in phase, compared with the depressing case (1: 0.094, 2: 0.118, 3: 0.213). These results suggest that, in the constant TInactive case, a depressing synapse does not promote phase maintenance better than a nondepressing synapse.
Dependence of the phase-period relationship on intrinsic and synaptic parameters when TActive is constant
In general, the shape of the phase versus period curve was determined by the intrinsic properties of neuron F (the intrinsic time constant F) when the period was small and by the properties of the synapse (the synaptic time constants , , , Esyn and the maximal synaptic conductance syn) when the period was large. Because these parameters were independent, the phase versus period curve could be cubic-like, depending on the choice of these parameters. Moreover, each of these parameters played a distinct role in determining the value and location of the local maximum and minimum points of the phase versus period curve. The effects of these parameters on the phase versus period curve are illustrated in Fig. 8. The effect of increasing the time constant of synaptic depression () was qualitatively equivalent to decreasing the time constant of synaptic recovery (). Hence, we only show the effect of the latter parameter. The effect of increasing Esyn (making it less negative) was tantamount to decreasing the maximal synaptic conductance syn. Hence, we only show the effect of syn.
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In each panel, the thick curve shows the reference model; other curves show variations of one of the studied parameters. The two synaptic parameters, the maximal conductance syn and the time constant of synaptic recovery , both controlled the strength of the synapse directly. The effect of syn on synaptic strength was present across all periods. Increasing the maximal conductance syn caused the phase versus period curve to shift up (Fig. 8A). Decreasing the time constant of synaptic recovery had a similar effect, although this effect was more diminished for sufficiently large period values because at these period values the synapse was mostly recovered from depression (Fig. 8B). In both cases, at any particular period value, strengthening the synapse caused the phase to increase because t increased. This can be explained by considering Eqs. 5 and 6. Indeed, Eq. 6 shows that for sufficiently large period values (when the transition conductance gjump is fixed at the critical conductance g*), t is an increasing function of gpeak. gpeak itself increases when the time constant of synaptic recovery is decreased or the maximal synaptic conductance syn is increased, as can be seen from Eq. 5.
In Fig. 8A, the maximal phase value moved to the left as the maximal synaptic conductance syn increased. Near the maximum, we recall that the phase was determined entirely by the properties of the synapse so that neuron F would only jump to its active state when the synaptic conductance g decayed below the critical conductance g*. Because the synaptic conductance g is equal to ssyn, when the maximal synaptic conductance syn increased, the synaptic conductance g decayed below the critical conductance g* at a lower value of s, i.e., at a smaller period value. Thus the synapse would become more relevant for determining the firing time of neuron F at lower values of the period, causing the phase versus period curve to shift up and to the left. Similar trends occur in Fig. 8B, as the time constant of synaptic recovery decreased. For any fixed period value, smaller values of (faster recovery) caused the synaptic conductance g to increase. Hence, at any fixed period value, the synaptic conductance g decayed below the critical conductance g* at a lower value of s when the synaptic recovery was faster. Again, this happened for smaller period values implying that, as the synaptic recovery was faster, the phase versus period curve shifted up and to the left.
Of the three synaptic parameters analyzed, the time constant of synaptic decay (decay of synaptic transmission when neuron O is not active) is the only one that is not directly related to synaptic depression. Recall that the main effect of depression is to change the peak conductance gpeak, whereas the time constant of synaptic decay affects the time spent between gpeak and the transition conductance gjump, but not gpeak itself. At small period values, where the synapse was weak, had almost no effect on the phase (Fig. 8C). However, for larger periods, was the predominant parameter in determining how long the synaptic conductance g had to decay to release neuron F from the inhibition of neuron O. Thus, at larger period values, changes in had a large effect on t and consequently on the phase. Increasing (slower decay) increased the maximum phase and shifted it to the right. This can be explained using Eqs. 6–8, which show that a simultaneous increase in both the period and shifts the phase versus period curve up and to the right. Because t increased with , for a fixed period value, the phase increased as well, thereby shifting up the curve. Alternatively, if the phase remained fixed, an increase in would result in an increase in t and therefore in period, thereby shifting the curve to the right.
At large period values, the intrinsic parameter F did not have any effect on the phase because the synapse was dominant in determining the firing time of neuron F (Fig. 8D). The effect on phase was restricted to smaller values of the period when the synapse was relatively weak. For such a fixed period value, decreasing F forced neuron F to spend less time in the silent state, thereby decreasing t and therefore the phase.
As seen in Fig. 8, changing any of the parameters shown could remove the local minimum and maximum of the phase versus period curve. An examination of the shown parameter ranges for which the phase versus period curve decreased monotonically revealed that, in these ranges, the effect of the intrinsic properties of neuron F was dominant over the effect of the depressing synapse in determining the phase. This shift in the dominance of the intrinsic versus synaptic effects was produced either through a change in the relative time courses of these two factors (Fig. 8, C and D) or by weakening the synapse (Fig. 8, A and B).
Dependence of the phase-period relationship on intrinsic and synaptic parameters when the duty cycle is constant
Figure 9 shows the dependence of the phase versus period curve on parameters when the duty cycle is constant. In principle, the different parameters affected the phase versus period curve in a way that was similar to the constant TActive case. There were some differences between the two cases that are worth pointing out. In Fig. 9A, as the maximal synaptic conductance syn was decreased, the value of the local minimum of the phase curve decreased and shifted to the right. Recall that in the constant duty cycle case, a larger period means that there is not only a longer time for the synapse to recover but also a longer time for it to depress. This can be seen in Fig. 4, where gpeak < g* at much larger period values for the constant duty cycle case compared with the constant TActive case. Thus at smaller values of the maximal conductance syn, the range of period values for which the intrinsic properties of neuron F determined phase became larger. However, despite the smaller value of syn, as the period was increased to very large values the synapse became sufficiently large and determined phase. A similar effect was seen by increasing the time constant of synaptic recovery (Fig. 9B). In both A and B, the cubic shape of the phase curve persisted when the synapse was weakened because, as the period approached infinity, the phas
) and a low voltage of -50 mV (
). In the constant TActive case (top), P was varied by changing the duration TInactive of the inactive state in neuron O. In the constant DC case (middle), P was varied by changing TActive and TInactive proportionally, such that the duty cycle DC = TActive/(TInactive + TActive) remained constant. In the constant TInactive case (bottom), cycle period was varied by changing TActive, the duration of the active state in neuron O.
