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Life Sciences Department and Zlotowski Center for Neurosciences, Ben-Gurion University of the Negev, Beer-Sheva, Israel
Submitted 4 August 2003; accepted in final form 25 June 2004
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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). Within a CPG, different neurons are active at different times, to ensure the activation of motoneurons and their muscle targets in an optimal sequence. Most CPGs produce rhythms that are highly variable, often severalfold, in the cycle period. Nevertheless, in many of these cases the firing pattern is kept such that the repetitive activity remains coherent despite changes in speed.
A central question in central pattern generation is what mechanisms allow the production of robust patterns of activity in the face of large variations in the cycle period. One possible strategy is that any temporal interval in the rhythm is expanded when the cycle period increases, or contracted when the cycle period decreases, such that the phase (ratio of time interval and cycle period) of this event remains constant when the cycle period changes. Indeed, a number of experimental studies have shown that within a neuronal network some of the neurons fire with an approximately fixed phase difference with respect to each other (DiCaprio et al. 1997; Fischer et al. 2001; Friesen and Pearce 1993; Hill et al. 2003; Hooper 1997a, b).
Maintenance of phase in rhythmic networks, especially in cases where the cycle period can vary severalfold, is not a simple problem. The mechanisms that underlie phase constancy are still unclear. A recent theoretical study proposed that short-term synaptic depression of inhibitory synapses could be instrumental to maintain phase when the cycle period changes (Manor et al. 2003). When an oscillator was coupled by an inhibitory synapse to a follower neuron, if (and only if) the synapse showed depression the 2 neurons fired with little phase variation with respect to each other, in some range of cycle periods. When the rhythm was fast, the synapse was mostly depressed and thus less effective in delaying the follower neuron. With a slower rhythm, the synapse could increasingly recover and delay the follower neuron. As the cycle period changed the delay of the follower neuron was proportionally adjusted such that the 2 cells could fire with an approximately fixed phase difference with respect to each other, independent of cycle period.
The mechanism proposed by Manor et al. (2003) exclusively depended on short-term depression of an inhibitory synapse, and did not take into account other possible sources of delay, such as intrinsic conductances in the follower neuron. A number of experimental studies established that several types of intrinsic conductances, in particular transient potassium current (A-currents), could play an important role in setting the phase of follower neurons in CPG circuits (Harris-Warrick et al. 1995a; Hartline and Gassie 1979; McCormick 1991; Storm 1988). To our knowledge the possibility that intrinsic currents or synaptic depression (not saying a combination of the 2) are involved in the mechanisms underlying phase invariance was not experimentally explored.
Taking a modeling approach, in the present study we examine this question. We explore the dynamical interaction between synaptic depression and a particular type of current, a slowly inactivating form of A-current, in the context of phase maintenance and under the constraints of a complex and realistic CPG. To this end, we chose the pyloric circuit as a representative example. The choice of this well-studied CPG as our working model was motivated by the fact that it includes all the ingredients necessary for the proposed mechanism: the oscillator produces a rhythm that is variable in the cycle period; follower neurons are entrained by inhibitory synapses; the synapses are graded and show short-term depression; most followers are endowed with a slowly inactivating A-current; and some of the neuronal pairs burst with a relatively fixed phase with respect to each other. We now describe the pyloric circuit in greater detail.
The pyloric circuit is involved in the feeding behavior of crustaceans. Depending on the species it consists of 12 to 14 neurons, which together produce a triphasic pattern of firing activity. Although the cycle period of the pyloric rhythm is commonly around 1 s, it can vary between about 0.5 to 5 s, depending on various factors such as temperature or the neuromodulatory environment (Hooper and Marder 1987; Nusbaum and Marder 1989b). The rhythm originates from a pacemaker ensemble of neurons [the anterior burster (AB) and the 2 pyloric dilators (PD)], and is propagated to the other neurons of the network by inhibitory synapses. The 3 pacemaker neurons fire together in the first phase of the rhythm. One class of follower neurons, to which the lateral pyloric (LP) neuron belongs, is active in the second phase. The third class of follower neurons consists of 6–8 pyloric constrictor (PY) neurons, depending on species. In the crab Cancer borealis, these PY neurons are subdivided to early pyloric constrictors (PE) and late pyloric constrictors (PL). The LP, PE, and PL neurons are all directly inhibited by the PD neurons, and are thus silent when the PD neurons are bursting. Earlier work suggested that the follower neurons rebound from synaptic inhibition at different rates, and thus become active at different times because of different densities or kinetics of their A-current (Harris-Warrick et al. 1995a, b; Hartline and Gassie 1979). Indeed, in pyloric neurons several types of A-currents were measured, including a rapid form (IA) and a slowly inactivating form (IAS) (Golowasch and Marder 1992; Turrigiano et al. 1995).
In the first part of this work we constructed a model of the pyloric network, which consisted of 5 representative members of the pyloric circuit and their connections: the AB, PD, LP, PE, and PL neurons. We used a realistic way of modifying the cycle period of the rhythm by incorporating a proctolin-like current in the AB neuron, and modifying its maximal conductance. The proctolin current is elicited by the endogenous neuropeptide proctolin (Freschi 1989; Hooper and Marder 1987; Nusbaum and Marder 1989a, b). It is active in many of the pyloric neurons, in particular the AB neuron (Swensen and Marder 2001). Because of its sharp dependency on voltage, this inward current depolarizes the AB neuron and increases its excitability. Indeed, when the rhythm is slow bath application of proctolin speeds up the rhythm (Hooper and Marder 1987; Swensen and Marder 2001). In our model of the pyloric rhythm we study how changes in the cycle period, induced by different levels of the proctolin current in the AB neuron, affect the delay between the onsets of bursting in the PD and LP neurons. We find that when the dynamics of the PD to LP synapse is such that it cannot, by itself, support phase constancy, nor the dynamics per se of the A-current in the LP neuron, the combination of these 2 components can produce a synergistic interaction that promotes phase maintenance in a wide range of cycle periods.
Although the detailed model of the pyloric network clearly demonstrates this synergistic interaction, the complexity of this model obscures the mechanism and impedes a complete understanding of it. Thus in the second part of this work we reduce the full circuit to a much simpler model, based on the model of Manor et al. (2003). The insights obtained from the reduced model facilitate our understanding of the synergistic interaction between synaptic depression and A-current.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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CELLULAR MODELS.
The model included 5 representative members of the pyloric circuit in the stomatogastric nervous system of the crab Cancer borealis. Figure 1A is a schematic drawing of the model circuit: the anterior burster (AB) and the pyloric dilator (PD), which together form the pacemaker ensemble, were coupled by a bidirectional electrical synapse. The PD neuron formed inhibitory chemical synapses onto 3 follower neurons: the lateral pyloric neuron (LP), the early pyloric constrictor (PE), and the late pyloric constrictor (PL). The LP and PE neurons were reciprocally coupled with inhibitory chemical synapses, and also by a bidirectional electrical synapse. Similar connections were set between the LP and PL neurons. In the biological circuit, LP forms an inhibitory chemical synapse back to the PD neuron (shown with a dotted line and open circle in Fig. 1A). Recent studies suggest that this feedback synapse may play an important role in stabilizing the rhythm (Mamiya and Nadim 2004; Weaver 2003). Other works show that this synapse may not have a significant effect in a nonperturbed rhythm (Prinz et al. 2003). For simplicity, and in view of the controversial role of this synapse in frequency regulation of the pyloric rhythm, we decided not to implement it in our model of the pyloric rhythm.
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): a primary neurite extends from the cell body, branches into secondary neurites and dendrites, and continues as an axon. In such neurons the surface area of the cell body + dendrites is much larger than that of the axon. Thus in our cellular models the membrane capacitance and leak conductance (in absolute terms) was an order of magnitude larger in the somatodendritic compartment, relative to the axonal compartment. The whole system consisted of 47–48 differential equations: 10 for the AB neuron; 9 for the PD neuron; 7 for each of the LP, PE, and PL neurons; and 7–8 for the synapses. Except for the PD to LP chemical synapse, for simplicity all 6 other synapses were modeled as nondepressing, with a single differential equation that determined the time-dependent activation of the synapse. The PD to LP chemical synapse was in some cases modeled as nondepressing (with one differential equation, for activation) and in others as depressing (with an additional differential equation, for depression).
In each cell XX, the voltages in the somatodendritic ("sd") and axonal ("ax") compartments were governed by the following equations
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XX, and gZZXX are time- and voltage-dependent conductances (in nS) of intrinsic conductance ion, chemical synapse from soma of cell YY to soma of cell XX, and electrical synapse from somatodendritic compartment of cell ZZ to somatodendritic compartment of cell XX. Parameters for capacitance C (in pF) were: 2.5 (AB,ax, PD,ax), 10 (AB,sd), 25 (LP,ax, PE,ax, PL,ax), 50 (PD,sd), 100 (PL,sd), 150 (PE,sd), and 500 (LP,sd). The cytoplasmatic conductance was calculated as the sum of 2 serially connected conductances
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sd (in nS) was 0.1 (AB), 0.21 (PD), and 8 (LP, PE, PL); and gsdax (in nS) was 0.5 (AB), 1 (PD), and 50 (LP, PE, PL).
IONIC CONDUCTANCES.
The voltage- and time-dependent conductance gion(V, t) of ionic conductance ion obeyed the following equation
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j. For a voltage-dependent persistent current (no inactivation) p > 0, q = 0, and gion(V, t) = ionmp. For a voltage-dependent transient current, p > 0, q > 0, and gion(V, t) = ionmphq.
Each state variable x was modeled with the following type of equation
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(V) described by
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(V) = 0.5, and kx (in mV) represents the slope (steep when close to 0). In agreement with the model described in Turrigiano et al. (1995), in all our cellular models the time constant of inactivation of the sodium current was described with a bell-shaped function
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, all other times constants of conductance activation/inactivation were described by sigmoid functions of the form
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x,lo and x,hi (in ms) are the time constant values at low (V –) and high (V +) voltages, respectively. Instantaneous variables (i.e., dependent on voltage but not on time) were modeled by setting the time constant value to 0 and setting the variables according to Eq. 5. For example, when a conductance ion had an instantaneous activation and no inactivation, the conductance was calculated as gion = ionm(V). Tables 1– 7 denote the values of parameters for the different ionic conductances, in each compartment.
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C2 = gcoup(VC1 – VC2). Values for gcoup, in nS, were: 2 (LP,sd PE,sd), 6 (LP,sd PL,sd), 2.5 (PD,sd AB,sd, AB,sd PD,sd), 0 (PE,sd LP,sd, PL,sd LP,sd).
CHEMICAL SYNAPSES.
In general the synaptic current from the presynaptic compartment C1 to the postsynaptic compartment C2 was described as: Isyn,C1C2 = gsyn(VC1)(VC2 – Esyn). In all chemical synapses, Esyn = –80 mV. Regarding the conductance gsyn(V), we distinguished between nondepressing and depressing synapses.
NONDEPRESSING SYNAPSES.
Nondepressing chemical inhibitory synapses were set between PD,sd and PE,sd, PD,sd and PL,sd, LP,sd and PE,sd, LP,sd and PL,sd, PL,sd and LP,sd, and sometimes between PD,sd and LP,sd (in other cases the PD,sd and LP,sd synapse was modeled as depressing; see following text). Parameters for these synapses were (using Eqs. 3–5): Esyn (in mV) = –80 (all synapses), syn (in nS) = 0.5 (LP,sd PL,sd), 0.1 (LP,sd PE,sd), 7 (PE,sd LP,sd), 4 (PD,sd PE,sd), 5 (PD,sd PL,sd), 3.2 (PD,sd LP,sd), 2 (PL,sd LP,sd); p = 1 (all synapses), Vm,1/2 (in mV) = –50 (LP,sd PE,sd, LP,sd PL,sd, PE,sd LP,sd, PL,sd LP,sd), –53 (PD,sd LP,sd), –55 (PD,sd PE,sd, PD,sd PL,sd); km (in mV) = 1.25 (PD,sd LP,sd), 1 (LP,sd PE,sd, LP,sd PL,sd, PE,sd LP,sd, PL,sd LP,sd, PD,sd PE,sd, PD,sd PL,sd); m,lo (in ms) = 0 (PD,sd LP,sd, LP,sd PE,sd, LP,sd PL,sd, PL,sd LP,sd, PD,sd PE,sd, PD,sd PL,sd), 50 (PE,sd LP,sd); m,hi (in ms) = 0 (PD,sd LP,sd, LP,sd PE,sd, LP,sd PL,sd, PL,sd LP,sd, PD,sd PE,sd, PD,sd PL,sd), 50 (PE,sd LP,sd).
Note: A value of 0 for a time constant indicates an instantaneous process.
DEPRESSING SYNAPSES.
When the PD to LP was modeled as a depressing synapse, the synaptic conductance was computed similar to an intrinsic conductance with time-dependent activation and inactivation. Parameters for this synapse were (using Eqs. 3–5): p = 1; Vm,1/2 (in mV) = –53; km (in mV) = 1.25; m (in ms) = 1; m,hi (in ms) = 1; q = 1; Vh,1/2 (in mV) = –57; kh (in mV) = –1; h,lo (in ms) = 300; h,hi (in ms) = 1,900. The value of syn was tuned so that it was comparable to the effect of the nondepressing version of the synapse, at a cycle period of 1,000 ms (see RESULTS).
The reduced model
The complexity of the detailed model prevented us from gaining a clear and complete understanding of the mechanism. Thus we decided to dissect the mechanism by reducing this model to a much simplified one. The reduced model consisted of an oscillator (O) coupled by a single inhibitory synapse to a follower F. Both O and F were modeled with Morris-Lecar equations (Morris and Lecar 1981). In O, the selected set of conductances produced spontaneous oscillations. Cell F was endowed with a set of ionic conductances that, in the absence of any input from cell O, yielded a high stable resting membrane potential. The synapse from O to F was modeled with 2 variables: one represented the fraction of open synaptic channels (which decayed between bursts; this decay represented the closure of synaptic channels after the onset of transmitter release) and the other represented the depression state of the synapse (which decreased when cell O was active, and increased when it was nonactive). Equations for the cellular and synaptic variables were as described in Manor et al. (2003), with the exception that an A-like current was introduced in cell F.
In the reduced model, the cycle period was modified by changing the value of the time constant of the recovery variable in cell O when the membrane potential of cell O was low. This computational manipulation changed the cycle period by modifying the duration of time that cell O was not active, while keeping the duration of the active state (high-voltage) fixed.
COMPUTATION OF TIME INTERVAL AND PHASE. Numerical simulations and analysis were done with a MatLab program using the Symulink environment. All simulations were run for 30-s simulation time. In each run we ignored the first 20 s, to eliminate transient effects and allow the system to reach a stationary state. We then divided the last 10 s of the run to n individual cycles. We arbitrarily decided that each cycle starts at the onset time of PD burst (in the detailed model), or the time that the voltage exceeds 0 mV in the O neuron (in the reduced model). This time was defined as the reference time. In any one cycle, all temporal events were measured relative to this time. In each cycle i, we measured the following temporal events for cell XX:
1.) TXX,on, the time of the onset of activity in neuron XX. In the PD, LP, PE, or PL neurons (detailed pyloric model) this time was equal to the time of the first action potential peak in a burst. In all cases, we empirically found that the interspike interval was always <250 ms. Thus in these neurons an action potential was defined as first action potential in a burst if no spike occurred within the 250 ms preceding this action potential. The action potentials in the AB neuron were very small in amplitude and could not be used to define the burst limits. Thus in this case we defined the onset of burst as the time that the voltage of AB became more positive than –55 mV. In the reduced model, the 2 cells O and F showed only slow waves of activity (no fast action potentials). In these cells, the onset of activity was defined as the time at which the voltage of the cell became more positive than 0 mV.
2.) TXX,off, the time of the termination of activity in cell XX. In the PD, LP, PE, or PL neurons (detailed pyloric model) this time was equal to the time of the last action potential peak in a burst. An action potential was defined as last in a burst if no spike occurred within the 250 ms after this action potential. In the case of the AB neuron, the end of the burst was defined as the time at which the membrane potential of AB became more negative than –55 mV. In the simplified model, the end of activity was defined as the time at which the voltage of the cell became more negative than 0 mV.
3.) Burst duration in neuron XX was calculated as TXX,off – TXX,on.
4.) P, the cycle period. This time was equal to the time difference between TXX,on in the current cycle and TXX,on in the subsequent cycle, where XX = PD in the detailed model and XX = O in the simplified model.
5.) The phase of the onset of the burst in cell XX: 
XX,on = TXX,on/P.
6.) The phase of the termination of the burst in cell XX: XX,off = TXX,off/P.
A CRITERION FOR PHASE CONSTANCY.
To compare how well phase was maintained in different cases, we developed the following criterion: we found the phase at a cycle period of P = 1,000 ms, and referred to this phase as the "pivot point." In the plot of phase versus cycle period, we defined a window of ±0.05 around the pivot point, and found the largest continuous range of cycle periods (
P) for which the phase varied within this window only. Larger P values indicate better phase maintenance.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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The parameters of the model were tuned such that the network produced a triphasic rhythm with a cycle frequency (frequency of slow waves) of 1 Hz, which is the typical frequency of the pyloric rhythm. Five top traces in Fig. 1B are voltage traces of the somatodendritic compartments of the AB, PD, LP, PE, and PL neurons. The AB and PD neurons produced simultaneous bursts. The fast action potentials riding on the slow wave of the PD and AB neurons were of small amplitude, reflecting the fact that the site of fast action potentials generation was electrically distant from the somatodendritic compartment. After a brief interval, the LP neuron produced a burst, followed by bursts in the PE and then the PL neurons. There was a brief overlap in the bursting times of the LP neuron and the 2 PY (PE and PL) neurons. The onset of activities in the PE and PL neurons terminated the LP burst. The bursts of PE and PL terminated when PD started the subsequent burst. These activity profiles are characteristic of the intracellular activities that can be recorded in a typical biological pyloric network (Bal et al. 1994; Marder and Calabrese 1996). The bottom trace in Fig. 1B represents the spiking activity in a nerve constituting axons of the 4 motoneurons PD, LP, PE, and PL (the AB neuron is an interneuron and does not send an axon through this nerve). This trace was constructed by first detecting the times of action potential peak in these 4 neurons. Then, for each spike detected, a vertical stroke was added to the trace at the corresponding time. These vertical strokes were scaled differently, according to the identity of the cell: 5, 2, 1.2, and 1 length units for an LP, PD, PE, or PL spike, respectively. This procedure accurately reproduced the type of extracellular activity that is recorded on the biological nerve (lvn), which constitutes the axons of the PD, LP, and PY neurons (Hartline and Gassie 1979).
Increasing a proctolin current in the AB neuron produced a biphasic effect on the cycle period
The cycle period of the rhythm was modified by changing the level of the proctolin current in the AB neuron. We assumed that larger levels (greater concentrations) of proctolin recruit more proctolin channels. Thus we modeled the effect of proctolin concentration by changing the maximal conductance of the proctolin current, proc. In the simulations shown in Fig. 2, the synapse from PD to LP was nondepressing, and there was no A-current in LP (the effects of synaptic depression in the PD to LP synapse and existence of A-current in the LP neuron are studied in later sections). In Fig. 2A are plotted voltage traces of the somatodendritic compartment in the PD neuron, with different proc values in the AB neuron. When proc = 0 (top trace), the cycle period of the rhythm was 1,888 ms, which is within the physiological range of slow pyloric rhythms. At low levels of the proctolin current, increasing the proctolin current sped up the rhythm: compare top trace to 2nd trace (proc = 2.5 nS), and 2nd trace to 3rd trace (proc = 5 nS). At higher levels of proctolin, increasing the proctolin current had an opposite effect: it slowed down the rhythm (compare 3rd trace to 4th trace, proc = 7.5 nS).
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proc = 0 nS, the burst durations were 240 and 197 ms in the AB and PD neuron, respectively. As proc was increased, the burst duration in the AB neuron gradually became longer, as a result of the addition of an inward current that delayed the termination of the AB burst. Because of the electrical coupling between the AB and PD neurons, the PD burst duration gradually became longer as well. The interburst duration in the AB neuron (and, consequently, in the PD neuron) decreased as proc was increased because the addition of inward current in AB facilitated the depolarization of the AB neuron and advanced its activity time. As long as the burst duration, which in general increased, was smaller than the interburst duration, which in general decreased, the cycle period (= burst + interburst) decreased as a function of proc. This occurred for proc <6.1 nS. Above this value, the burst duration exceeded the interburst duration, and the cycle period increased with larger proc values, until the rhythm collapsed when proc was greater than 7.7 nS (the AB and PD neurons became permanently active). We see that, in general, increasing the proctolin current produced a gradual change in the cycle period and the durations of burst and interburst in the AB and PD neurons. This effect is illustrated in Fig. 2C, where PD bursts at 3 different proc values are superimposed. The transition of proc from 1.0 to 1.6 nS demonstrates the gradual effect of the proctolin current: the addition of inward current in the AB neuron prolonged the AB burst and thus delayed the last (7th) spike in the PD burst. Recall that in the PD neuron the burst duration is calculated as the interval from the first to the last action potential in the burst. Thus the duration of the PD burst gradually increased. At several discrete points, increasing the proctolin current in AB produced a stepwise change in the durations of PD burst and interburst
At several discrete proc values the PD burst and interburst durations abruptly changed. These results were obtained when the number of action potentials in the PD burst decremented, or incremented, by one. Note that this effect was observed only for the PD neuron, and not the AB neuron. Because the PD burst duration is calculated as the interval from the first to the last action potential in the burst, the addition or subtraction of a single spike resulted in a step change (about 60 ms) in the durations of the PD burst and interburst. This effect was not observed in the AB neuron because, in this case, the calculation of burst duration did not rely on the times of first and last action potentials in the burst (see METHODS).
The number of action potentials in the PD burst incremented at proc = 4.9 (from 5 to 6 spikes), 6.6 (6 to 7 spikes), 7.12 (7 to 8 spikes), 7.4 (8 to 9 spikes), 7.58 (9 to10 spikes), and 7.7 nS (10 to 11 spikes). In these cases, the increase in inward current prolonged the AB burst and, consequently, delayed the repolarization of the PD burst. Thus the number of action potentials in the PD burst incremented and the PD burst duration abruptly increased (by about 60 ms).
At proc = 1.8 and 4.2 nS, the number of action potentials in the PD burst decreased (from 7 to 6 and from 6 to 5, respectively). A possible explanation is that when proc was increased, the addition of an inward current tended not only to prolong the AB burst, but also to increase the amplitude of the AB burst (as observed in our simulations; not shown). A larger amplitude of the AB burst could increase/speed up the inactivation of calcium currents, thereby acting as a negative feedback and accelerating the rate of repolarization in both AB and PD. This effect would cause each spike in the PD burst to occur incrementally later, and at a lower membrane potential. Let n be the number of spikes in the PD burst at some proc value. As proc is increased, eventually the membrane potential of PD would fall below threshold shortly after the n – 1 spike, and the number of action potentials would suddenly decrement from n to n – 1.
This effect is illustrated in Fig. 2C when proc was increased from 1.6 to 1.8: the addition of inward current in the AB neuron slightly increased the rate of repolarization in the PD burst, as can be seen from the slightly lower membrane potential at the time of the 6th spike in the PD burst. The 7th spike failed because at the time this action potential should have been generated, the membrane potential was already too negative; thus the duration of the PD burst sharply decreased.
Dependency of frequency of the onset and termination times of bursting in the pyloric neurons
Next, we examined the dependency of burst onset and termination as a function of the cycle period (as a result of changing proc) in the PD, LP, PE, and PL neurons (Fig. 3, A–D). In each panel, the dotted line T = P represents the time of the subsequent burst in the PD neuron. Black and gray curves represent times of burst onset and termination, respectively. On any of these curves, each point was obtained with a different proc value. The reference time, relative to which all other times were measured, was the onset of bursting in the PD neuron (TPD,on = 0, lies on the x-axis in Fig. 3A).
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proc, and not the initial conditions of the system variables. On the 2 curves TPD,off and TLP,on proc was incremented from 0 to 6 nS when following the lower branch from the right (P = 1,888 ms) to the left (P = 597 ms). In general TPD,off gradually increased along the lower branch, except at 2 points where it sharply decreased (see Fig. 2). Thus overall the change in TPD,off was relatively small along this branch. Because of the strong inhibitory synapse from PD to LP, as long as the PD neuron was bursting the LP burst was delayed. Consequently TLP,on was vertically shifted relative to TPD,off. Because the synapse was nondepressing, this shift was almost constant and TLP,on was mostly fixed. There was a weak decrease of TLP,on along the lower branch, mainly because of the gradual nature of the PD to LP synapse. When following the upper branch from the left (P = 598 ms) to the right (P = 918 ms), proc was incremented from 6.1 to 7.7 nS. At these large proc values, the increased excitability in the AB neuron delayed the termination of the AB and PD bursts, and TPD,off increased along its upper branch. Note that the slope of the upper branch was parallel to the linear curve T = P, indicating that along this branch the increase in cycle period was a direct result of the increase in PD burst duration. The burst of the LP neuron was delayed accordingly, and TLP,on increased along its upper branch as well.
The end of the LP burst (TLP,off) depended on the activity of the PE (Fig. 3C) and PL (Fig. 3D) neurons. Thus its dependency on cycle period cannot be understood before analyzing how the cycle period affected the onset of burst in the PE and PL neurons. We therefore start by describing the dependency of TPE,on (onset of PE burst) and TPL,on (onset of PL burst) on the cycle period. The PE neuron included a large A-current. After inhibition from the PD neuron, the large A-current in PE strongly decreased the rate of depolarization, and PE started to burst about 250 ms after the onset of LP burst. The PL neuron included an even larger A-current and therefore its burst was further delayed, and TPE,on and TPL,on were vertically shifted with respect to TLP,on and with respect to each other. Because the PD to PE and PL synapses were nondepressing, TPE,on and TPL,on were almost constant. As soon as the PD neuron started to depolarize, it ended the bursts in PE and PL (thus TPE,off and TPL,off were close to the linear line T = P). As proc increased and the cycle period decreased, as seen in Fig. 2, the PD interburst duration approached a minimal value. Because PL and PE could burst only when PD was not active, the burst duration in these 2 cells became shorter. In the PL neuron, between 742 < P < 803 ms the burst shortened down to the duration of a single spike (note the merging of TPL,on and TPL,off in Fig. 3D). At P < 742 ms, the interburst of the PD neuron was too short to allow the PL neuron to produce even a single spike, and the PL neuron stopped firing. Likewise for the PE neuron (Fig. 3C): the burst shortened down to a single spike between 617 < P < 693 ms, and at P < 617 ms the PE neuron stopped firing. TPE,on and TPL,on lacked the wedgelike shape seen for TPD,off and TLP,on because the PE and PL neurons stopped firing at large proc values.
The strong inhibitory synapses from the PE to LP and PL to LP neurons ensured that LP stopped firing slightly after PE started to fire, and slightly before PL started to fire. Thus at low and intermediate values of proc TLP,off (Fig. 3B) was larger than TPE,on and smaller than TPL,on. With large proc values, as explained above the PE and PL neurons stopped firing. Thus in this case the burst of the LP neuron was terminated by the subsequent burst in the PD neuron, and TLP,off was close to the linear curve T = P.
We hereafter focus our attention to the onset of bursting in the LP neuron. In the DISCUSSION, we briefly elaborate how other temporal events may be affected by the cycle period as well, using principles similar to those outlined for the onset of bursting in the LP neuron.
With no A-current in LP, the delaying effect of a depressing synapse was similar to a nondepressing synapse
In this section we study the frequency-dependent contribution of synaptic depression per se to the onset of bursting in the LP neuron. All simulations were done when the LP neuron did not include an A-current. To properly compare between the depressing and nondepressing cases, we tuned the maximal synaptic conductance (gsyn) of the depressing PD to LP synapse such that at a cycle period of 1,000 ms the delay it produced onto the LP burst (TLP,on) was identical to that of the nondepressing case. With a nondepressing synapse, at P = 1,000 ms TLP,on was 330 ms. The maximal conductance gsyn of the nondepressing synapse was 3.1 nS. When the synapse was implemented as a depressing synapse, we found that at P = 1,000 ms a gsyn value of 20 nS produced a delay TLP,on of 330 ms. We therefore chose a maximal conductance of 20 nS for the depressing synapse.
In Fig. 4A we show voltage traces of the PD neuron (top row) and the LP neuron when the PD to LP synapse was nondepressing (middle row, –Dep–A) or depressing (bottom row, +Dep–A), at 4 values of proc: 7.5 nS (P = 762 ms, leftmost column), 6.5 nS (P = 598, 2nd column), 2.7 nS (P = 1,005, 3rd column), and 0 nS (P = 1,888, rightmost column). The length of the 2 gray rectangles on top of the PD traces represents the delay of LP burst when the synapse was nondepressing (top rectangle) or depressing (bottom rectangle). When the synapse was nondepressing, in all 4 cases the depth of inhibition (lowest voltage in LP) was almost identical (compare troughs on middle row traces); the durations of inhibition depended only on the duration of the PD burst and were not significantly different. Thus as expected, the effect of the synapse on the delay of the LP burst was nearly fixed and independent of cycle period. When the synapse showed depression, the depth of hyperpolarization was strongly correlated with proc and cycle period (compare troughs on bottom row traces). The lowest voltages of LP were: –55.0 mV with proc = 7.5 (P = 762 ms); –57.6 mV with proc = 6.5 (P = 598 ms); –63.3 mV with proc = 2.7 (P = 1,005 m); and –65.8 mV with proc = 0 (P = 1,888 ms); With proc values of 0, 2.5, or 5 nS (2nd, 3rd, and rightmost columns), the onset of LP burst was not different whether the synapse was depressing or nondepressing (compare widths of gray rectangles in the 2nd, 3rd, and rightmost columns).
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proc = 7.5 nS (leftmost column) there was a marked difference when the synapse was depressing or nondepressing: With a nondepressing synapse, as the duration of the PD burst increased the LP neuron was inhibited for a longer time, and the LP burst was delayed (note the longer length of the bottom gray rectangle, compared with all other cases). When the synapse was depressing, because of the short interburst in the PD neuron the PD to LP synapse did not recover from inhibition and became weak. The weak synapse was not sufficient to inhibit the LP neuron for as long as PD was active, and LP started to burst before the end of the PD burst (the bursts of LP and PD overlapped in this case). Thus the LP burst was advanced (note the shorter length of the upper gray rectangle, compared with all other cases).
These results are quantitatively presented in Fig. 4B. With proc values <3.4 nS (P < 856 ms), TLP,on was almost identical (up to 1 ms difference) whether the synapse was nondepressing (thin curve, –Dep–A) or depressing (thick curve, +Dep–A): it weakly decreased as proc increased and the cycle period decreased, between 370 ms at P = 1,888 and 320 ms at P = 856 ms. With higher values of proc (>3.3 nS), the dependency of TLP,on on proc and the cycle period became different in the 2 cases: when the synapse was nondepressing synapse, TLP,on increased as proc increased, as can be seen on the top branch of the curve –Dep–A, and also on the bottom branch of this curve for 597 < P < 856 ms. In contrast, when the synapse was depressing synapse, TLP,on decreased as proc increased, as can be seen on the bottom branch of the curve +Dep–A, and also on the top branch of this curve for 597 < P < 856 ms.
As explained in the METHODS, the pivot point was defined as the phase LP,on at P = 1,000 ms. Because of the tuning process of the depressing synapse, by definition the pivot points for the depressing and nondepressing cases were identical and equal to 0.33 (dashed linear curve in Fig. 4B represented a fixed phase of 0.33; this curve crossed the 2 curves, –Dep–A and +Dep–A, at P = 1,000 ms). In Fig. 3C, the width of the gray rectangle represented a phase variation of ±0.05 around the pivot points; the length of this rectangle represented the range P of cycle periods for which phase varied within this rectangle (i.e., ±0.05 around the pivot point). Because, in this range of cycle periods, the phases were identical with a depressing or a nondepressing synapse, the gray rectangles corresponding to the 2 cases were identical (only one rectangle is shown in Fig. 4C). In both the nondepressing and depressing versions of the synapse, P was 1,250 – 838 = 412 ms. This relatively narrow range represented a poor level of phase maintenance.
These results suggest that with no A-current in the follower neuron, and with the kinetics chosen for synaptic depression in this model, a depressing synapse may not be advantageous for phase maintenance.
When the synapse was nondepressing, the delaying effect of an A-current was constant and independent of cycle period
In this section we study the frequency-dependent contribution of an A-current per se to the onset of bursting in the LP neuron. All simulations were done when the synapse from PD to LP was nondepressing. In Fig. 5A, voltage traces of the PD (top traces) and LP neuron without A-current (middle row traces, –Dep–A) and with A-current (bottom row traces, –Dep+A) are shown for 4 values of proc: 7.5 nS (P = 762 ms, leftmost column), 6.5 nS (P = 598, 2nd column), 2.7 nS (P = 1,005, 3rd column), and 0 nS (P = 1,888, rightmost column). Because the synapse from PD to LP was nondepressing, in all 4 cases the depth of synaptic inhibition was similar, whether the LP current included or did not include an A-current (troughs of bottom traces and middle traces, respectively). The addition of A-current slightly delayed the burst of LP in all 4 cases, but this additional delay was too short to be visually distinguishable when comparing the lengths of the gray rectangles in Fig. 5A.
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The onset phase of LP burst was plotted against the cycle period in Fig. 5C. Each of the 2 gray rectangles represented a phase variation of ±0.05 around the corresponding pivot point of 0.33 (without A-current in LP) and 0.34 (with A-current in LP). The ranges P of cycle periods for which phase satisfied the criterion for phase maintenance (i.e., variation within ±0.05 of the pivot point, as illustrated by the gray rectangles) were almost identical with and without A-current: 420 and 412 ms, respectively.
These results demonstrate that when the synapse is not depressing, the existence of A-current in the follower neuron is not advantageous for phase constancy.
When the synapse was depressing, the delaying effect of an A-current was incrementally larger as cycle period increased
In this section we examined whether, when the synapse from LP to PD synapse showed depression, the existence of A-current in the LP neuron promoted phase constancy. All simulations were done when the LP to PD synapse was depressing. In Fig. 6 A, voltage traces of the PD (top traces) and LP neuron without A-current (middle row traces, +Dep–A) and with A-current (bottom row traces, +Dep+A) are shown for 4 values of proc: 7.5 nS (P = 762 ms, leftmost column), 6.5 nS (P = 598, 2nd column), 2.7 nS (P = 1,005, 3rd column), and 0 nS (P = 1,888, rightmost column). When the synapse was nondepressing, in all 4 cases the LP burst occurred at similar times. In contrast, with a depressing synapse the delaying effect of the A-current became highly dependent on the cycle period. Except for the case proc = 7.5 nS (leftmost column), the LP burst was incrementally delayed as the cycle period was increased. At P = 1,888 ms, with an A-current the LP neuron fired a single action potential: here the onset of firing in the LP neuron was delayed for so long that it occurred just slightly before the onset of firing in the PE and PL neurons. As soon as the PL neuron started its burst, it inhibited the LP neuron and prevented it from generating additional spikes.
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proc was incremented from 0 to 6 nS and cycle period decreased from 1,888 to 597 ms, TLP,on decreased from 368 to 309 ms (+Dep–A, thin curve in Fig. 6B). Thus at low and intermediate values of proc, TLP,on weakly decreased as the cycle period decreased because the interburst in the PD neuron became shorter, the synapse had less time to recover from depression, and was less effective in delaying the burst in the LP neuron. However, it is important to emphasize that with no A-current in the LP neuron this effect was subtle relative to the case where LP included an A-current (see following text). With further increment of proc from 6.1 to 7.7 nS (bottom branch +Dep–A curve in Fig. 6B), the cycle period increased from 598 to 918 ms and TLP,on decreased because the synapse became too weak to inhibit the LP neuron, and the LP burst started to burst before the end of the PD burst.
A different picture emerged when the LP neuron was endowed with an A-current. As proc was increased from 0 to 6 nS and the cycle period decreased from 1,888 to 597 ms in the LP neuron, TLP,on decreased from 752 to 314 ms (+Dep+A curve in Fig. 6B, when following the top branch of the curve from right to left). Note that this part of the +Dep+A curve was close to the linear function 0.45P (dashed curve in Fig. 6B), implying that with low and intermediate values of proc, the phase LP was well maintained around 0.45 (see also Fig. 6C). At higher values of proc, when proc was increased from 6.1 to 7.7 nS and the cycle period increased from 598 to 918 ms, TLP,on decreased, again because the synapse became incrementally weak and ineffective in delaying the burst of LP. Focusing on the low and intermediate values of proc (see DISCUSSION), these results demonstrate that the existence of A-current greatly amplified the tendency of TLP,on to increase as the cycle period increased, thereby promoting phase constancy in this regime.
These results are also presented in Fig. 6C, where the onset phase LP,on of the LP burst is plotted against cycle period. Without A-current, when proc was incremented from 0 to 6 nS and cycle period decreased from 1,888 to 597 ms, the phase rose from 0.19 to 0.52 (+Dep–A curve in Fig. 6C). As proc was further increased from 6.1 to 7.7 nS and the cycle period increased from 598 to 918 ms, the phase decreased to 0.21. The pivot point on this curve (i.e., the phase at P = 1,000 ms) was 0.33. The range P of cycle period values for which phase varied by less than ±0.05 around the pivot point (length of bottom gray rectangle in Fig. 5C) was 412 ms. This range represented a poor level of phase maintenance.
In contrast, with an A-current in the LP neuron, the phase was approximately constant and around 0.45) in a wide range of cycle period values (thick curve in Fig. 6C). This occurred with low and intermediate values of proc, but not with proc values >6 nS, where the cycle period increased with proc and the phase decreased (bottom branch of the curve +Dep+A). The pivot point on this phase curve was 0.45. Considering only the low and intermediate values of proc, the range P of cycle period values for which phase varied by less than ±0.05 around the pivot point (upper gray rectangle) was 1,245 ms. This represents a 3.02-fold improvement in the range of cycle periods for which phase was well maintained, according to the criterion defined in METHODS, compared with all other cases (no A-current with and without depression, A-current with no depression).
These results support our hypothesis that when the synapse is depressing, the existence of A-current in the follower neuron promotes phase constancy.
Sensitivity analysis of the kinetics of the A-current in the detailed model
Next, we investigated how the kinetics of the A-current affects the relationship between LP,on and the cycle period P. In Fig. 7, each panel illustrates the effect of a different parameter of the A-current. In each panel, the thick curve represents the phase obtained with the set of canonical parameters outlined in METHODS. Other curves were obtained with a different value of the parameter, as indicated on the graphs. The dotted curve represents the phase obtained with no A-current. The A-current started to affect the phase at the P value at which diverged from this dotted curve.
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A, was 22 nS. With larger or smaller maximal conductance of the A-current, A, the delaying effect of the A-current was stronger or weaker, respectively, and LP,on increased or decreased, respectively (Fig. 7A). These effects were particularly emphasized with small and intermediate values of the proctolin current, where the cycle period was long or moderate and the A-current could sufficiently deinactivate (during the interburst of the LP neuron) and become active. Note that with larger values of A, the LP,on curve was truncated at long P values. In these conditions, the LP burst was excessively delayed such that the PL neuron became active before the onset of burst in the LP neuron. Thus LP,on was not defined in these cases. For example, with A = 26 nS the LP neuron stopped to burst at P > 1,100 ms.
The onset phase of LP burst was also sensitive to the midpoint (V1/2,h) of the inactivation current of the A-current (Fig. 7B). The canonical value of V1/2,h was –65 mV. More depolarized values increased the phase because it allowed A-current to incrementally deinactivate during the interburst of the LP neuron. With more depolarized values of V1/2,h, the LP,on curve was truncated at long P values, again because in these conditions, the LP burst was excessively delayed such that the PL neuron became active before the onset of burst in the LP neuron.
The effect of the slope (kh) of the inactivation curve of the A-current is presented in Fig. 7C. The canonical value of kh was –1.5 mV. Less-negative values (e.g., kh = –0.5 mV) yielded a steeper inactivation curve. At all cycle periods, with the parameters used for this model the lowest membrane potential of LP was more depolarized than the midpoint of the inactivation curve, V1/2,h = –65 mV. For example, when P = 1,000 ms, the lowest membrane potential of LP was –64.2 mV. Therefore at the lowest membrane potential of LP the value of h(V) was smaller when kh was less negative. The deinactivation of the A-current was smaller and thus the A-current was less effective in delaying the LP neuron. The phase was therefore smaller when kh was less negative (opposite effects would have been obtained if the lowest membrane potential of LP was more hyperpolarized than V1/2,h; in that case less-negative values of kh would have increased the phase; see also Fig. 9C). This effect was emphasized at small P values because at short cycle periods the synapse from PD to LP was weaker and the hyperpolarization level of the LP neuron was smaller. When kh was more negative and the slope of the steady-state curve was more shallow, the phase increased, causing the LP neuron to stop bursting at large cycle periods (again, because of the strong inhibitory synapse from PL to LP, which terminated the LP burst). For example, with kh = –2 mV the LP neuron stopped to burst at P > 1,080 ms.
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h) of inactivation of the A-current on phase was examined in Fig. 7D. The canonical value of h was 500 ms. Around this value, the phase was weakly sensitive to differences in h because the interburst duration of the LP neuron was long relative to h. When the time constant was too small (h = 100 ms in Fig. 7D), the A-current rapidly inactivated during the depolarization phase of the LP neuron, and it became ineffective in delaying the LP n
