Abstract
In oscillatory systems, neuronal activity phase is often independent of network frequency. Such phase maintenance requires adjustment of synaptic input with network frequency, a relationship that we explored using the crab, Cancer borealis, pyloric network. The burst phase of pyloric neurons is relatively constant despite a >2-fold variation in network frequency. We used noise input to characterize how input shape influences burst delay of a pyloric neuron, and then used dynamic clamp to examine how burst phase depends on the period, amplitude, duration, and shape of rhythmic synaptic input. Phase constancy across a range of periods required a proportional increase of synaptic duration with period. However, phase maintenance was also promoted by an increase of amplitude and peak phase of synaptic input with period. Mathematical analysis shows how short-term synaptic plasticity can coordinately change amplitude and peak phase to maximize the range of periods over which phase constancy is achieved.
Introduction
Oscillatory neural activity is often organized into different phases across groups of neurons, both in brain rhythms associated with cognitive tasks or behavioral states (Hasselmo et al., 2002; Buzsaki and Wang, 2012; Buzsaki and Tingley, 2018), and in central pattern generating (CPG) circuits that drive rhythmic motor behaviors (Marder and Bucher, 2001; Marder et al., 2005; Grillner, 2006; Bucher et al., 2015; Katz, 2016; Stein, 2018). The functional significance of different phases in the latter is readily apparent, as they for example provide alternating flexion and extension of limb joints, and coordination of movements between joints, limbs, and segments (Krantz and Parks, 2012; Grillner and El Manira, 2015; Kiehn, 2016; Le Gal et al., 2017; Bidaye et al., 2018). A hallmark of many such patterns is that the relative timing between neurons is well maintained over a range of rhythm frequencies (Dicaprio et al., 1997; Hooper, 1997b, a; Wenning et al., 2004; Marder et al., 2005; Grillner, 2006; Mullins et al., 2011; Le Gal et al., 2017). If the latency across different groups of neurons changes proportionally to the rhythm period, phase (latency over period) is invariant, in some cases providing optimal coordination at all speeds (Zhang et al., 2014).
The ability to maintain phase arises from central coordinating mechanisms between circuit elements, as it is present in isolated nervous system preparations, but the underlying cellular and circuit mechanisms are not well understood. Constant phase lags between neighboring segments in the control of swimming can be explained mathematically on the basis of asymmetrically weakly coupled oscillators, but the role of intrinsic and synaptic dynamics within each segment is unknown (Cohen et al., 1992; Skinner and Mulloney, 1998; Grillner, 2006; Mullins et al., 2011; Zhang et al., 2014; Le Gal et al., 2017).
The pyloric circuit of the crustacean stomatogastric ganglion (STG) has inspired a series of experimental and theoretical studies of cellular and synaptic mechanisms underlying phase maintenance. The pyloric circuit generates a triphasic motor pattern with stable phase relationships over a wide range of periods (Eisen and Marder, 1984; Hooper, 1997b, a; Bucher et al., 2005; Goaillard et al., 2009; Tang et al., 2012; Soofi et al., 2014). Follower neurons burst in rebound from inhibition from pacemaker neurons (Marder and Bucher, 2007; Daur et al., 2016), and post-inhibitory rebound delay scales with the period of hyperpolarizing currents (Hooper, 1998). Voltage-gated conductances slow enough for cumulative activation across cycles could promote such phase maintenance (Hooper et al., 2009). Similarly, short-term depression of graded inhibitory synapses is slow enough to accumulate over several pyloric cycles, meaning that effective synaptic strength increases with increasing cycle period (Manor et al., 1997; Nadim and Manor, 2000).
Theoretical studies have shown that short-term synaptic depression, by increasing inhibition strength with cycle period, should promote phase maintenance (Manor et al., 2003; Mouser et al., 2008), particularly in conjunction with inactivating (A-type) potassium currents (Bose et al., 2004; Greenberg and Manor, 2005), which control the rebound delay (Harris-Warrick et al., 1995b; Harris-Warrick et al., 1995a; Kloppenburg et al., 1999). These predictions remain experimentally untested.
Additionally, postsynaptic responses also depend on the actual trajectory of synaptic conductances, which are shaped by presynaptic voltage trajectories and short-term synaptic plasticity (Manor et al., 1997; Mamiya et al., 2003; Zhao et al., 2011; Tseng et al., 2014). If amplitude, duration, and trajectory of synaptic conductance determine rebound delay, phase maintenance necessitates all three of these parameters to change with cycle period in coordination. We used the dynamic clamp technique to exhaustively explore the range of these parameters and understand how the coordinated changes in synaptic dynamics determines the phase of follower neurons in an oscillatory circuit. Our findings are consistent with a mathematical framework that accounts for the frequency dependence of amplitude and peak phase of the synaptic conductance.
Results
Phase maintenance and latency maintenance
The firing of neurons in oscillatory networks is shaped by a periodic synaptic input. The relative firing latency of such neurons is often measured relative to a defined reference time in each cycle of oscillation, and is used to determine the activity phase of the neuron (see, e.g., Belluscio et al., 2012). For example, in a simple network consisting of a bursting oscillatory neuron driving a follower neuron (Fig. 1A1), at a descriptive level, the latency (L) of the follower neuron activity relative to the onset of the oscillator’s burst onset may depend on the oscillation cycle period (P). In response to a change in period (say, to P2), the follower neuron may keep constant latency (L2 = L), or constant phase, i.e., modify its latency proportionally to the change in period (L2 /P2 = L/P; Fig. 1A2). However, in many oscillatory systems, for example the pyloric circuit (Hooper, 1997b, a), the relationship between L and P falls between these two extremes. A sample recording of the bursting activity of the lateral pyloric (LP) neuron in response to controlled changes in the pyloric cycle frequency demonstrates such a relationship (Fig. 1B and 1C).
The burst onset time of the isolated LP neuron depends on the temporal dynamics of its input
The LP neuron does not have intrinsic oscillatory properties, but oscillates due to the synaptic input it receives from the pacemaker anterior burster (AB) and pyloric dilator (PD) neurons, and the follower pyloric constrictor (PY) neurons (Fig. 2A). The burst onset phase of the LP neuron (φLP = Δt / P; Fig. 2A) is shaped by the interaction between synaptic inputs and the neuron’s intrinsic dynamics that influence post-inhibitory rebound. We measured an overall burst onset phase of the LP neuron of φLP =0.34 ± 0.03 (N=9).
As a first-order quantification, we measured how inputs to the LP neuron interact with its intrinsic properties to determine the timing between its bursts, in the absence of network oscillations. To this end, we synaptically isolated the LP neuron (Fig. 2B) and drove it with a noise current input (see Methods). In response to the noise input, the LP neuron produced an irregular pattern of spike times, which included a variety of bursting patterns with different spike numbers (Fig. 2C). We were interested in the characteristics of inputs producing different burst onset latencies. However, unlike a periodic input, noise input does not provide a well-defined reference point to measure the burst onset latency. We therefore categorized bursts with respect to the preceding inter-burst intervals (IBIs) during which no other action potentials occurred. We classified these IBIs in bins (300, 500, 700 and 900 ms) and tagged bursts based on the IBI values (Fig. 2C). We characterized the driving input leading to bursts with specific IBIs by burst-triggered averaging the input current (IBTA; an example shown in Fig. 2D). Our analysis produced a single IBTA for each of the four IBIs in each preparation (N=23). IBTA ‘s of each preparation were first normalized in amplitude by the IBTA amplitude at IBI = 300 ms (Fig. 2E; average shown in Fig. 2F) to examine how peak amplitude (Ipeak) varied with IBI. These data were then normalized in time (Fig. 2G) to examine the effect of IBI on peak phase (Δpeak) and the rise (slopeup) and fall (slopedown) slopes of the input current across preparations. We found that IBI had a significant effect on Ipeak, Δpeak, slopeup and slopedown (all one-way RM-ANOVA on ranks; data included in Figure 2-source data). In particular, larger IBIs corresponded to larger Ipeak values (Fig. 2F-2H; p<0.001, χ2 = 65.87) with smaller (more advanced) Δpeak (Fig. 2I; p<0.001, χ2 = 41.35). The change in Δpeak was due to a decrease in slopeup (p<0.001, χ2 = 65.25), whereas slopedown did not vary as much (Figs. 2J-2K; p=0.002, χ2 = 14.77).
The burst onset phase of the LP neuron oscillation depends on its synaptic input
Injection of noise current revealed that the timing of the LP response is exquisitely sensitive to the duration and amplitude of inputs. In the intact system, the primary determinant of input duration and amplitude is the network period (P), as increasing P increases both presynaptic pacemaker burst duration (Hooper, 1997b, a) and synaptic strength (Manor et al., 1997; Nadim and Manor, 2000). To explore the effect of the duration and strength of the synaptic input, we used dynamic clamp to drive the LP neuron with a realistic synaptic conductance waveform.
We constructed this realistic waveform by measuring the synaptic current input to the LP neuron during ongoing pyloric oscillations (Fig. 3A). These measurements showed the two components of inhibitory synaptic input, those from the pacemaker AB and PD neurons (left arrow) and those from the follower PY neurons (right arrow). In each cycle, the synaptic current always had a single peak, but the amplitude and phase of this peak showed variability across preparations (Fig. 3B, average in blue).
The realistic conductance input was injected periodically with strength gmax (Fig. 3C). For any fixed gmax, φLP decreased as a function of P (Fig. 3D), i.e., the relative onset of the LP burst was advanced in slower rhythms. In contrast to the effect of P, for any given P, φLP increased sublinearly as a function of gmax (Fig. 3E). Fig. 3F combines the simultaneous influence of both parameters on φLP. The results shown in Fig. 3D indicate that the LP neuron intrinsic properties alone do not produce phase constancy. However, level sets of φLP (highlighted for three values in Fig. 3F), indicate that phase could be maintained over a range of P values, if gmax increases as a function of P. This finding was predicted by our previous modeling work, in which we suggested that short-term synaptic depression promotes phase constancy by increasing synaptic strength as a function of P (Manor et al., 2003; Bose et al., 2004). We will further discuss the role of synaptic depression below.
To clarify the results of Fig. 3, it is worth examining the extent of phase maintenance for fixed gmax. An example of this is shown in Fig. 4A (turquoise plots). A comparison of these data with the theoretical cases in which either delay or phase is constant suggests that the LP neuron produces relatively good phase maintenance, at least much better in comparison with constant delay. However, this conclusion is misleading because, in these experiments, the duty cycle of the synaptic input was kept constant. Therefore, most of the phase maintenance is due the fact that the synaptic input keeps perfect phase. In fact, if the reference point measures phase relative to the end –rather than onset– of the PD burst (Fig. 4B), phase maintenance of the LP neuron is barely better than in the constant delay case (Fig. 4A, purple plots). It is therefore clear that phase maintenance by the LP neuron would require the properties of the synaptic input to change as a function of P, a hallmark of short-term synaptic plasticity (Fortune and Rose, 2001; Grande and Spain, 2005). As mentioned above, short-term plasticity such as depression could produce changes in gmax as a function of P. Independently of gmax, the peak time of the synaptic current is another parameter that could change with P and influence the timing of the postsynaptic burst. We therefore proceeded to systematically explore the influence of P, gmax and the synaptic peak time on φLP.
A systematic exploration of synaptic input parameters on the phase of the LP neuron
For a detailed exploration of the influence of the synaptic input on φLP, we approximated the trajectory of the (unitary) synaptic conductance in one cycle by a simple triangle (Fig. 5A), which could be defined by three parameters: duration (Tact), peak time (tpeak) and amplitude (gmax) (Fig. 5B). This simplified triangular synaptic conductance waveform could then be repeated with any period (P) to mimic the realistic synaptic input to the LP neuron. For a given synaptic duration Tact, the peak phase of the synapse can be defined as Δpeak = tpeak / Tact). The parameter Δpeak is known to vary as a function of P (Tseng et al., 2014) and, in a previous study, we found that Δpeak may influence the activity of the postsynaptic neuron, independent of P and gmax (Mamiya and Nadim, 2004). We therefore systematically explored the influence of three parameters of the synaptic input (P, gmax and Δpeak) on φLP.
As with the realistic synaptic waveforms (Fig. 3), we used the dynamic clamp technique to apply the triangular conductance waveform periodically to the synaptically isolated LP neuron. Across different runs within the same experiment, the parameters P, gmax and Δpeak were changed on a grid (see Methods). In addition, all combinations of these three parameter values were run in two conditions in the same experiment, 1: with constant duration, i.e., constant Tact across different P values (C-Dur of 300 ms), and 2: with constant duty cycle, i.e., Tact changing proportionally to P (C-DC of 0.3; Fig. 5C). Using these protocols, we measured the effects of synaptic parameters on φLP (Fig. 5D).
The LP neuron produced burst responses that followed the synaptic input in a 1:1 manner across all values of P that were used (Fig. 6A1). When gmax and Δpeak were kept constant, φLP decreased as a function of P (Fig. 6A2). This decrease was always larger for the C-Dur case than the C-DC case. For both C-DC and C-Dur, this trend was seen across all values of Δpeak and gmax (Fig 6A3). The effect of P on φLP was highly significant for both C-DC (Three-Way ANOVA, p<0.001, F=100.677) and C-Dur (Three-Way ANOVA, p<0.001, F=466.424), indicating that the period and duration of the inhibitory input to the LP neuron had a significant effect on its phase.
Changing gmax produced a large effect on the level of hyperpolarization in the LP neuron, but this usually translated to only a small or modest effect on the time to the first spike following inhibition (Fig. 6B1). Overall, increasing gmax at constant values of P and Δpeak produced a significant but only small to moderate increase in φLP (Three-Way ANOVA, p<0.001, F=10.798). Although increasing gmax produced the same qualitative effect for both the C-DC and C-Dur (e.g., Fig. 6B2), φLP in the C-DC case was restricted to a smaller range (Fig. 5F top vs. bottom panels). Overall, this increase was robust for most values of P and Δpeak (Fig. 6B3).
Increasing Δpeak for a constant value of P and gmax (Fig. 6C1), produced a small but significant increase in φLP (Three-Way ANOVA, p<0.001, F=17.172). This effect was robust for most values of P and gmax, for both C-DC and C-Dur (Fig. 6C2 and 6C3).
These results showed that all three parameters that define the shape of the IPSC influence φLP. Clearly, the strongest effect is the decrease in φLP as a function of P. However, φLP modestly increases as a function of the other two parameters, gmax and Δpeak. This raised the question how gmax and Δpeak would have to change in coordination as a function of P to counteract the effect of P on φLP and achieve phase constancy.
Coordinated changes of gmax and Δpeak produce the largest effect on phase
To explore how gmax and Δpeak might interact to influence φLP, we examined the sensitivity of φLP to these two parameters, individually and in combination, for all values of P in our data (see Methods). Sensitivity of φLP to these two parameters varied across P values, with larger sensitivity at lower values of P (data not shown; Two-Way RM-ANOVA, p<0.001, F=16.054). For simplicity, we averaged the sensitivity values across different P values to obtain an overall measure of the influence of gmax and Δpeak. These results showed that, for the C-DC case, φLP had a positive sensitivity to gmax and a smaller positive sensitivity to Δpeak (Fig. 7A). The sensitivity was largest if the two parameters were varied together (gmax + Δpeak) and smallest if they were varied in opposite directions (gmax - Δpeak; Two-Way RM-ANOVA, p<0.001, F=3.330). Similarly, these sensitivity values were also significantly different for the C-Dur case (Fig. 7B; Two-Way RM-ANOVA, p<0.001, F=2.892), with largest sensitivity for gmax + Δpeak and smallest for gmax - Δpeak.
Level sets of φLP in the P-gmax -φpeak space for C-DC and C-Dur cases
To search for phase constancy across different P values in our dataset, we expressed φLP as a function of the three IPSC parameters, P, gmax and Δpeak : φLP = Φ(P,gmax, Δpeak). Figure 8 shows heat map plots of the function Φ, plotted for the range of values of P and Δpeak and four values of gmax. In these plots, phase constancy can be seen as the set of values in each graph that are isochromatic, indicating the level sets of the function Φ. These level sets are mathematically defined as hypersurfaces on which the function has a constant value: Φ(P,gmax, Δpeak) = φc. For the C-DC case, in each gmax section of the plot, the level sets (e.g. φc = 0.34 denoted in white) spanned a moderate range of P values as Δpeak increased (Fig. 8A1). The span of P values across all four panels indicates the range of cycle periods for which phase constancy could be achieved by varying gmax and Δpeak. This range of P values (spanned by the white curves) was considerably smaller for the C-Dur case (Fig. 8A2).
For any constant phase value φc, these level sets can be expressed as which describes a surface in the 3D space, yielding the P value for which phase can be maintained at φc, for the given values of gmax and Δpeak. The level set indicated by the white curves in panel A for the C-DC case is plotted as a heat map in Fig. 8B1 and can be compared with the same plot for the C-Dur case in Fig. 8B2. The range of colors in each plot (marked next to each panel) indicates the range of P values for which phase can be kept at φc = 0.34. To reveal how this range depends on the desired phase, we measured this range for all values of φc between 0.2 and 0.8 (Figs. 8C1 and 8C2). We found that the LP neuron could not achieve phases below 0.3 in the C-DC case (Fig. 8C1), which is simply because the neuron never fired during the inhibitory synaptic current (which had a duty cycle of 0.3). Furthermore, the range of P values for which the LP phase could be maintained by varying gmax and Δpeak was much larger for C-DC inputs compared to C-Dur Inputs, for all φc values between 0.31 and 0.54.
A model of synaptic dynamics could predict activity onset phase of LP neuron
To gain a better understanding of our experimental results, we considered Equations (7) and (8)—the mathematical description of φLP as a function of P, gmax and Δpeak, for the C-Dur and C-DC cases, respectively, that we derived in the Methods section—repeated here for convenience:
In the C-Dur case, described by Equation (7), the input period has the most significant affect and φLP decays like 1/P. In contrast, in the C-DC case, described by Equation (8), φLP is bounded from below by Δpeak ·DC and thus behaves very differently than in the C-Dur case. In particular, as P increases, φLP approaches Δpeak ·DC for the C-DC case, whereas it approaches 0 in the C-Dur case.
Keeping gmax (respectively, Δpeak) constant in these equations allows us to obtain a relationship between P and Δpeak (respectively, gmax), for which φLP is kept constant at φc. Consider Equations (7) and (8) for fixed values of both φLP (= φc) and gmax. Then these equations reduce to simple functional relationships where Δpeak can be expressed as a function of P. In the C-DC case, for example, evaluating Δpeak from Equation (8) produces
Equation (12) describes how gmax must vary with P for the system to maintain a constant phase φc for any given Δpeak.
Alternatively, Δpeak can be expressed as a function of P. In the C-DC case, evaluating Δpeak from Equation (8) produces
This equation describes how Δpeak must vary with P for the system to maintain a constant phase φc for any given gmax. A comparison of these two cases can be seen in Fig. 9A, where either gmax (green) or Δpeak (blue) is varied, while keeping the other parameter constant, to keep φLP constant at φc =0.34 across different P values. (The red curve is the depressing case, described below.) As the figure shows, phase constancy can be achieved by varying either parameter, but each parameter produces a different range of P across which phase is maintained.
In fact, Equation (13) can be used to calculate the range of P values over which changing Δpeak (from 0 to 1) can maintain a constant phase φc. Solving 0 < Δpeak < 1 using Equation (13) yields
Performing the same procedure in the C-Dur case, we find
If ΔP denotes the range of P values that respectively satisfy Equation (14) or (15), it is straightforward to show that ΔPDC > ΔPDur (compare black and blue curves in Fig. 9B and 9C). To see this, note that the lower bound for each interval is the same, thus we need to show that the upper bound for the C-DC case is larger than that for the C-Dur case. That is,
Equation (16) is true for τs and gmax large enough.
Two additional points are notable from Fig. 9B. First, the lower bound on φLP for which phase constancy can occur (i.e., φc) is smaller in the C-Dur than C-DC case. This is because we have assumed that in the C-DC case the LP neuron cannot fire during inhibition (i.e., until after Δpeak DC). Second, for φc larger than ∼ 0.5, ΔP is larger for the C-Dur case. This occurs because, when φc is sufficiently large, Equation (16) can no longer be satisfied. These findings are consistent with our experimental results described above, indicating that although phase constancy can be achieved when either gmax or Δpeak increases with P, a concomitant increase of both—which could occur for example with a depressing synapse—greatly expands the range of P values for which a constant phase is maintained.
We now consider how short-term depression of the synapse—a property known to exist in the pacemaker to LP synapse (Zhao et al., 2011)—influences phase constancy by changing gmax and Δpeak. We will restrict this section to the C-DC case. A similar derivation can be made for the C-Dur case. As mentioned in the Methods, the effect of synaptic depression on synaptic strength can be obtained by Equation (11) (repeated from the Methods): where smax is the maximum value of sd at the onset of the pacemaker burst:
Note that smax is a monotonically increasing function with values between 0 and 1. Its value approaches 1 as P increases, indicating that the synapse becomes stronger. In this equation, is constant and is chosen so that the non-depressing and depressing conductances match at P = 1 s. As seen in Fig. 9A, when synaptic depression dictates how gmax varies with P as in Equation (11), and Δpeak varies with P and gmax according to Equation (13), the simultaneous changes in gmax and Δpeak (red) greatly increase the range of P values over which φLP is constant.
Returning to Fig. 9B, note that the C-DC case with depression spans a larger range of P values than the non-depressing case. Similarly, in Fig. 9C, the range of P values for which phase can be maintained is larger than the non-depressing case across φLP values, except where φLP is so large that the depressing synapse operates outside its dynamic range.
Discussion
The importance of phase in oscillatory networks
A common feature of oscillatory networks is that the activities of different neuron types are restricted to specific phases of the oscillation cycle. For example, different hippocampal and cortical neurons are active in at least three distinct phases of the gamma rhythm (Hajos et al., 2004; Hasenstaub et al., 2005), and distinct hippocampal neuron types fire at different phases of the theta rhythm and sharp wave-associated ripple episodes (Somogyi and Klausberger, 2005).
Experimental studies quantify the latency of neural activity with respect to a reference time in the cycle, but in most cases, these latencies are normalized and reported as phase. Distinct neuron types can maintain a coherent activity phase, despite wide variations in the network frequency (30-100 Hz for gamma rhythms, 4-7 Hz for theta rhythms, and 120-200 Hz for sharp wave-associated ripple episodes). Phase-specific activity of different neuron types is proposed to be important in rhythm generation (Wang, 2010), and indicates the necessity of precise timing for producing proper circuit output and behavior (Kopell et al., 2011). For example, phase locking of spike patterns to oscillations is important for auditory processing, single cell and network computations and Hebbian learning rules (Kayser et al., 2009; McLelland and Paulsen, 2009; Panzeri et al., 2010). For brain oscillations, phase relationships may provide clues about the underlying circuit connectivity and dynamics, but a behavioral correlate of varying frequencies is not obvious. In contrast, the activity phase of distinct neuron types in rhythmic motor circuits is a tangible readout of the timing of motor neurons and muscle contractions, thus defining phases of movement (Grillner and El Manira, 2015; Kiehn, 2016; Le Gal et al., 2017; Bidaye et al., 2018). Because meaningful behavior depends crucially on proper activity phases, whether neurons maintain their activity phase in face of changes in frequency simply translates to whether the movement pattern changes as it speeds up or slows down.
Determinants of phase
In oscillatory networks, the activity phases of different neuron types depend to different degrees on the precise timing and strength of their synaptic inputs (Oren et al., 2006). Our results from noise current injections showed that the timing of the LP neuron is strongly dependent on the timing of inputs it receives. Dynamic clamp injection of realistic or triangular conductance waveforms with different periods (P) indicated that φLP was largely determined by the duration of the synaptic input. φLP changed substantially with P when inputs had constant duration, but much less when inputs had a constant duty cycle, i.e., when duration scaled with P. However, our experiments also showed that inputs of constant duty cycles alone are insufficient for phase constancy. φLP decreased with P even with a constant duty cycle of inputs, but increased with either synaptic strength (gmax) or peak phase of the synaptic input (Δpeak). The increase in φLP had similar sensitivity to gmax and Δpeak, and therefore a larger sensitivity to a simultaneous increase in both. Consequently, it was possible to keep φLP constant over a wide range of cycle periods by increasing both parameters with P.
The fact that an increase in gmax with P promotes phase constancy is biologically relevant, as short-term depression in pyloric synapses means that synaptic strength indeed increases with P (Manor et al., 1997). Previous modeling studies show that short-term synaptic depression of inhibitory synapses promotes phase constancy (Nadim et al., 2003; Bose et al., 2004), largely because of longer recovery times from depression at larger values of P.
The case is less clear for the finding that an increase of Δpeak with P promotes phase maintenance, as we have previously shown that Δpeak in LP actually decreases with P (Manor et al., 1997; Tseng et al., 2014). On the face of it, this suggests that an increase in Δpeak is not a strategy employed in the intact circuit. However, the caveat is that such results may critically depend on the cause of the change in P, either technically and biologically. While in our current study we varied Δpeak with direct conductance injection into LP, previous results were obtained by changing the waveform and period of the presynaptic pacemaker neurons. When P is changed in an individual preparation by injecting current into or voltage-clamping the pacemakers, phase of follower neurons is not particularly well maintained. An example of this is shown in Fig. 1, where φLP values fall between constant phase and constant duration and, additionally, all pyloric neurons show behavior that falls between constant phase and constant latencies (Hooper, 1997b, a). This may reflect that individuals are not keeping phase particularly well when the only cause of changing P is the presynaptic input. This is supported by the observation that even during normal ongoing pyloric activity, phases change with cycle-to-cycle variability of P in individual preparations (Bucher et al., 2005). However, it does not preclude the possibility that Δpeak plays an important role in stable phase relationships when P differs because of temperature, neuromodulatory conditions, or inter-individual variability (discussed below).
It is noteworthy that a change in the synaptic strength or peak phase with P is not peculiar to graded synapses. The fact that short-term synaptic plasticity can act as a frequency-dependent gain control mechanism is well known for many spike-mediated synaptic connections. In bursting neurons, the presence of a combination of short-term depression and facilitation in the same spike-mediated synaptic interaction could also result in changes in the peak phase of the summated synaptic current as a function of burst frequency and duration, and the intra-burst spike rate (Markram et al., 1998).
Phase relationships in changing temperatures
An interesting case is provided by the observation that phases are remarkably constant when pyloric rhythm frequency is changed with temperature. Tang et al. (2012) report a 4-fold decrease in P of the pyloric rhythm between 7 and 23° C. In this study, none of the pyloric phases changed significantly, and it is worth noting that under conditions of changing temperatures, the relationships between P, gmax, and Δpeak appeared to be fundamentally different from when P is changed at a constant temperature. Presynaptic voltage trajectories scaled with changing P, and Δpeak of postsynaptic currents was independent of P, in contrast to the decrease described at constant temperature (Manor et al., 1997; Tseng et al., 2014). Amplitudes of synaptic potentials did not change with temperature, despite an increase in synaptic current amplitudes with increasing temperature (and associated decrease in P). This is in contrast to the positive relationship between gmax and P that results from synaptic depression at a constant temperature (Manor et al., 1997). Therefore, it appears that the likely substantial effects of temperature on synaptic dynamics and ion channel gating are subject to a set of compensatory adaptations different from when P is changed at constant temperature.
Slow compensatory regulation of phase
Phase maintenance in the face of changing P in an individual animal requires the appropriate short-term dynamics of synaptic and intrinsic neuronal properties. The fact that characteristic (and therefore similar) phase relationships can also be observed under the same experimental conditions across individual preparations is a different conundrum, particularly when P can vary substantially, as is true for brain oscillations (Hajos et al., 2004; Hasenstaub et al., 2005; Somogyi and Klausberger, 2005). Phases show different degrees of variability across individuals in a variety of systems, e.g., leech heartbeat (Wenning et al., 2018), larval crawling in Drosophila (Pulver et al., 2015), and fictive swimming in zebrafish (Masino and Fetcho, 2005), but in all of these cases phases are not correlated with P. In the pyloric rhythm, phases are also variable to a degree across individuals, but not correlated with the mean P, which varies >2-fold (Bucher et al., 2005; Goaillard et al., 2009). This phase constancy occurs despite considerable inter-individual variability in ionic currents, and is considered the ultimate target of slow compensatory regulation, i.e., homeostatic plasticity (Marder and Goaillard, 2006; Ma and LaMotte, 2007; Marder et al., 2014). Slow compensation can also be observed directly when rhythmic activity is disrupted by decentralization, and subsequently recovers to similar phase relationships over the course of many hours (Luther et al., 2003). It is interesting to speculate if our findings about how synaptic parameters must change to keep phase constant would hold across individuals with different mean P. The prediction would be coordinated positive correlations of both gmax and Δpeak with P.
Phase relationships under different neuromodulatory conditions
The flipside of the question how neurons maintain phase is the question how their phase can be changed. In motor systems in particular, changes in phase relationships are functionally important to produce qualitatively different versions of circuit output, for example to produce different gaits in locomotion (Vidal-Gadea et al., 2011; Grillner and El Manira, 2015; Kiehn, 2016). The activity of neural circuits is flexible, and much of this flexibility is provided by modulatory transmitters and hormones which alter synaptic and intrinsic neuronal properties (Brezina, 2010; Harris-Warrick, 2011; Jordan and Slawinska, 2011; Bargmann, 2012; Marder, 2012; Bucher and Marder, 2013; Nadim and Bucher, 2014). The pyloric circuit is sensitive to a plethora of small molecule transmitters and neuropeptides which affect cycle frequency and phase relationships (Marder and Bucher, 2007; Stein, 2009; Daur et al., 2016). With respect to our findings, any given neuromodulator could act presynaptically to alter P, duration, or duty cycle on the one hand, and gmax and Δpeak on the other. In addition, the neuromodulator could affect the postsynaptic neuron’s properties and alter its sensitivity to any of these parameters. Therefore, our findings could not just further our understanding of how phase can be maintained across different rhythm frequencies, but also provide a framework for testing if and how changes in synaptic dynamics may contribute to altering phase relationships under different neuromodulatory conditions.
Materials and Methods
Adult male crabs (Cancer borealis) were acquired from local distributors and maintained in aquaria filled with chilled (10-13°C) artificial sea water until use. Crabs were prepared for dissection by placing them on ice for 30 minutes. The dissection was performed using standard protocols as previously described (Tohidi and Nadim, 2009; Tseng and Nadim, 2010). The STNS, including the four ganglia (esophageal ganglion, two commissural ganglia, and the STG) and their connecting nerves, and the motor nerves arising from the STG, were dissected from the stomach and pinned into a Sylgard (Dow-Corning) lined Petri dish filled with chilled saline. The STG was desheathed, exposing the somata of the neurons for intracellular impalement. Preparations were superfused with chilled (10-13°C) physiological Cancer saline containing: 11 mM KCl, 440 mM NaCl, 13 mM CaCl2 2H2 O, 26 mM MgCl2 6H2 O, 11.2 mM Trizma base, 5.1 mM maleic acid with a pH of 7.4.
Extracellular recordings were obtained from identified motor nerves using stainless steel electrodes, amplified using a differential AC amplifier (A-M Systems, model 1700). One lead was placed inside a petroleum jelly well created to electrically isolate a small section of the nerve, the other right outside of it. For intracellular recordings, glass microelectrodes were prepared using the Flaming-Brown micropipette puller (P97; Sutter Instruments) and filled with 0.6 M K2 SO4 and 20 mM KCl. Microelectrodes used for membrane potential recordings had resistances of 25-30MΩ; those used for current injections had resistances of 15-22 MΩ. Intracellular recordings were performed using Axoclamp 2B and 900A amplifiers (Molecular Devices). Data were acquired using pClamp 10 software (Molecular Devices) and the Netsuite software (Gotham Scientific), sampled at 4-5 kHz and saved on a PC using a Digidata 1332A (Molecular Devices) or a PCI-6070-E data acquisition board (National Instruments).
Individual pyloric neurons were impaled and identified via their membrane potential waveforms, correspondence of spike patterns with extracellular nerve recordings, and interactions with other neurons within the network (Weimann et al., 1991).
Constructing realistic IPSC waveforms
Inhibitory postsynaptic currents (IPSCs) were recorded from LP neurons during the ongoing rhythm using two-electrode voltage clamp and holding the LP neuron at −50mV, far from the IPSC reversal potential of ∼ −80 mV (Fig. 3A). When the LP soma is voltage clamped at −50 mV, the axon (which is electrotonically distant from the soma) can still produce action potentials following the synaptic inhibition from the pacemaker neurons. The onset of the LP neuron action potentials (recorded in the current trace) was used to calculate the mean IPSC for each experiment averaging the IPSCs over 10-20 cycles. The IPSC waveforms were then extracted by normalizing both the amplitude and the duration of the mean IPSC.
Driving the LP neuron with noise current
In these experiments, the preparation was superfused in Cancer saline plus 10−5 M picrotoxin (PTX; Sigma Aldrich) for 30 minutes to block the synaptic currents to the LP neuron. The removal of synaptic inhibition onto LP neurons changed the activity of these neurons from bursting to tonic firing. Then, noise current, generated by the Ornstein-Uhlenbeck process (Lindner), was injected into the isolated LP neurons for 60 minutes using the Scope software (available at http://stg.rutgers.edu/Resources.html, developed in the Nadim laboratory). The baseline of the noise current was adjusted by adding DC current so that it can provide enough inhibition to produce silent periods alternating with bursts of action potentials.
Driving the LP neuron with realistic or triangular IPSC waveforms in dynamic clamp
The dynamic clamp current was injected using the Netclamp software (Netsuite, Gotham Scientific). The synaptically isolated (10−5 M PTX) LP neuron was driven with an artificial synaptic current in dynamic clamp. The synaptic current was given as where the synaptic conductance gsyn was a pre-determined waveform, repeated periodically with period P, and Esyn was the synaptic reversal potential set to −80 mV (Zhao et al., 2011).
Two sets of dynamic clamp experiments were performed on different animals. In one set of experiments, gsyn was set to be a triangular waveform. We measured the effects of four different parameters in these triangle conductance injections (Fig. 1): peak phase (Δpeak), duration (Tact), period (P = time between onsets of dynamic clamp synaptic injections), and maximal conductance (gmax, the peak value of gsyn). This allowed us to explore which combinations of the different parameters influences the LP phase. Five values for P were used: 500, 750, 1000, 1500, and 2000 ms, which cover the typical range of pyloric cycle periods. Three values of gmax were used: 0.1, 0.2 and 0.4 µS, consistent with previous measurements of synaptic conductance (Zhao et al., 2011; Tseng et al., 2014). The value of Δpeak was varied to be 0, 0.25, 0.5, 0.75 or 1. In the same experiment, all runs were done in two conditions: with Tact constant across different P values (C-Dur case with Tact = 300 ms) or with Tact changing proportionally to P (C-DC case with duty cycle DC = Tact /P = 0.3).
In the other set of experiments, gsyn was a realistic IPSC waveform, based on a pre-recorded IPSC in the LP neuron. In these experiments, P was varied to be 500, 750, 1000, 1250, 1500, or 2000 ms by scaling the realistic waveform in the time direction. In these experiments, gmax was set to be 0.1, 0.2, 0.4, 0.6, or 0.8 μS. The LP neuron burst onset delay (Δt) was measured relative to the onset of the pacemaker component of the synaptic input (identified by the kink in the synaptic conductance waveform) in each cycle. The burst phase was calculated as φLP = Δt /P. Phase constancy means that Δt changed proportionally to P. To measure the LP neuron phase with respect to the end of the pacemaker input, this reference used was the point on the synaptic conductance waveform marked by drawing a horizontal line from the kink that identified the onset of the pacemaker input.
Determining relationship between cycle period (P), synaptic strength (gmax) and LP phase (φLP) using the realistic IPSC waveform
We determined how well the mathematical model derived for constant input duty cycles (see Equation 8 below), matched the experimental data obtained with realistic IPSC waveforms. To this end, we fit the model to φLP values measured for all values of gmax and P, using the standard fitting routine ‘fit’ in MATLAB (Mathworks).
Sensitivity of φLP to gmax and Δpeak across all P values
To explore how gmax and Δpeak might interact to influence φLP, we first examined the sensitivity of φLP to these two parameters, individually and in combination, for all values of P in our data. For each cycle period, we computed the mean phase across all of our experiments (N=9) and all values of gmax (0.1, 0.2 and 0.4 µS) and we interpolated the φLP for gmax (0.2 and 0.4) to obtain φLP for 0.3 and Δpeak (0, 0.25, 0.5, 0.75 or 1). This produced a 4 by 5 matrix of all of the values. For each data point in the matrix, we moved along eight different directions (increase both gmax and Δpeak,, increase gmax, increase Δpeak,, decrease both gmax and Δpeak,, decrease gmax, decrease Δpeak, increase gmax and decrease Δpeak, and decrease gmax and increase Δpeak) and calculated the change in phase per unit (normalized) in gmax, Δpeak, or both.
A model of synaptic dynamics
In the derivation of the model, the firing time of the LP neuron was assumed to be completely determined by its synaptic input. This synaptic conductance (gsyn) was assumed to rise and fall with distinct time constants. The following holds over one cycle period and therefore time is reset with period P (t (mod P)): where the time tpeak, corresponding to Δpeak, is tpeak = Δpeak Tact. We assumed that LP neuron remained inactive when gsyn was above a fixed threshold (g*) less than gmax. Because the synaptic input is periodic with period P, we solved for the minimum and maximum values of gsyn in each cycle. The minimum (glo) occurred just before the onset (t = 0) of AB/PD activity, whereas the maximum occurred at the peak synaptic phase Δpeak for the C-Dur case. In the C-DC case, Tact = DC ·P, where DC is the duty cycle (fixed at 0.3 in our experiments).
To calculate g*, we set the value t = 0 so that gsyn (0) = glo (and, by periodicity, gsyn (P) = glo), and solved the first part of Equation (1) where gsyn increases until t = tpeak. This yielded
We then used the second part of Equation (1) to track the decay of gsyn for tpeak < t < P:
Using Equation (3), we calculated the time Δt at which the synaptic conductance gsyn (Δt) = g* as follows:
Solving Equation (4) for Δt yielded
Dividing this equation by P yielded φLP: where gpeak is given by Equation (2). This expression provides a description of the dependence of φLP as a function of P, gmax and Δpeak. To explore the role of the parameters in this relationship, we made a simplifying assumption that the synaptic conductance gsyn (t) rapidly reached its peak (i.e., τr was small), stayed at this value and started to decay at t = tpeak. In this case g(t) = gmax on the interval (0,tpeak) and the value of glo is irrelevant. With this assumption, Equation (5) reduced to
Substituting tpeak = Δpeak ·Tact in Equation (6), gave which we used to describe the LP phase in the C-Dur case. To describe the C-DC case, after substituting tpeak = Δpeak ·DC·P, we obtained
Note that these equations also describe the relationship between φLP with Tact (or DC, in the C-DC case).
Equations (7) and (8) can be used to approximate a range of parameters over which φLP is maintained at a constant value. To do so, we assumed a specific parameter set, say , satisfies for some fixed phase value, . We could now ask whether there are nearby parameters for which phase remains constant, i.e., F remains equal to . The Implicit Function Theorem (Krantz and Parks, 2012) guarantees that this is the case, provided certain derivatives evaluated at are non-zero, which turns out to be true over a large range of parameters. Since the partial derivative with respect to Δpeak of F(P,gmax, Δpeak) at this point is a non-zero constant equal to Tact /P (or DC) in the C-Dur (or C-DC) case, there is a function Δpeak = h(P,gmax) such that for values of P and gmax near . In other words, the Implicit Function Theorem guarantees that small changes in P and gmax can be compensated for by an appropriate choice of Δpeak in order to maintain a constant LP phase. A similar analysis can be done by solving for gmax in terms of P and Δpeak or by solving for P in terms of gmax and Δpeak.
Adding synaptic depression to the model of synaptic dynamics
In a previous modeling study, we explored how the phase of a follower neuron was affected when the inhibitory synapse from an oscillatory neuron to this follower had short-term synaptic depression (Manor et al., 2003). In that study the role of the parameter Δpeak was not considered. It is, however, straightforward to add synaptic depression to Equations (7) and (8).
An ad hoc model of synaptic depression can be made using a single variable sd which will be a periodic function that denotes the extent of depression and takes on values between 0 and 1 (Bose et al., 2004). sd decays during the AB/PD burst (from time 0 to Tact, indicating depression) and then recovers during the inter-burst interval (from Tact to P, indicating recovery). Thus, sd can be described by an equation of the form:
Using periodicity, it is straightforward to show that the maximum value of sd, which occurs at the start of the AB/PD burst, is given by:
Note that smax is a monotonically increasing function with values between 0 and 1. Its value approaches 1 as P increases, indicating that the synapse becomes stronger. For a complete derivation and description, see (Bose et al., 2004). The effect of synaptic depression on synaptic strength can be obtained by setting where smax is given by Equation (10).
Software, analysis and statistics
Data were analyzed using MATLAB scripts to calculate the time of burst onset and the phase. Statistical analysis was performed using Sigmaplot 12.0 (Systat). Significance was evaluated with an α value of 0.05, error bars and error values reported denote standard error of the mean (SEM).
Competing interests
The authors declare no competing financial interests.
Source Data Files
Figure 2-1. File: Figure2_sourcedata.xlsx
This Excel file contains 4 sheets, including all measured attributes of the burst-triggered average current (IBTA) for different IBIs (N=23) as shown in Fig. 2H-2K.
Acknowledgements
We thank Drs. Horacio Rotstein and Eric Fortune for helping with the initial MATLAB scripts in the analysis. This study was supported by NIH MH060605 and NSF DMS1122291.