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 > two 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 shortterm synaptic plasticity can coordinately change amplitude and peak phase to maximize the range of periods over which phase constancy is achieved.
https://doi.org/10.7554/eLife.46911.001Introduction
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; Buzsáki and Wang, 2012; Buzsáki 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 of firing between neurons is well maintained over a range of rhythm frequencies (Dicaprio et al., 1997; Hooper, 1997b; Hooper, 1997a; Wenning et al., 2004; Marder et al., 2005; Grillner, 2006; Mullins et al., 2011; Le Gal et al., 2017). If the latency of firing across different groups of neurons changes proportionally to the rhythm period, phase (latency over period) is invariant, in some cases providing optimal limb coordination at all speeds (Zhang et al., 2014).
The ability of the system to coordinate phases with changes in period 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. For instance, constant phase lags between neighboring segments in the control of swimming in lamprey fish and crayfish 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; Hooper, 1997a; Bucher et al., 2005; Goaillard et al., 2009; Tang et al., 2012; Soofi et al., 2014). Synapses in the pyloric circuit use graded as well as spikemediated transmission (Graubard et al., 1980; HarrisWarrick and Johnson, 2010; Zhao et al., 2011; Rosenbaum and Marder, 2018). Follower neurons burst in rebound from inhibition from pacemaker neurons (Marder and Bucher, 2007; Daur et al., 2016), and postinhibitory rebound delay scales with the period of hyperpolarizing currents (Hooper, 1998). Voltagegated conductances slow enough for cumulative activation across cycles could promote such phase maintenance (Hooper et al., 2009). Similarly, shortterm 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 shortterm 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 (Atype) potassium currents (Bose et al., 2004; Greenberg and Manor, 2005), which control the rebound delay (HarrisWarrick et al., 1995b; HarrisWarrick 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 shortterm 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 dependence of amplitude and peak phase of the synaptic conductance on cycle period.
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, for example Belluscio et al., 2012). For example, in a simple network consisting of a bursting oscillatory neuron driving a follower neuron (Figure 1A1), at a descriptive level, the latency (Δt) 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 P_{2}), the follower neuron may keep constant latency (Δt _{2} = Δt), or constant phase, that is modify its latency proportionally to the change in period (Δt_{2} / P_{2} = Δt/P; Figure 1A2). However, in many oscillatory systems, for example the pyloric circuit (Hooper, 1997b; Hooper, 1997a), the relationship between L and P falls between these two extremes.
We demonstrated this point in the pyloric follower LP neuron using the following protocol. We voltage clamped one of the pacemaker PD neurons and drove this neuron with its own prerecorded waveform, but applied at five different cycle periods (also denoted P). This protocol entrained the pacemaker group at this period, which forced the follower LP neuron to obey the same period (Figure 1B). We then measured the latency (Δt) of the LP burst onset with respect to onset of the PD neuron burst. A plot of the LP latency Δt or phase (Δt/P) for different cycle periods demonstrates the abovementioned finding that the LP neuron activity falls between the two limits of constant phase and constant latency (Figure 1C).
The burst onset time of the 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 (Figure 2A). The burst onset phase of the LP neuron (φ_{LP} = Δt/P; Figure 2A) is shaped by the interaction between synaptic inputs and the neuron’s intrinsic dynamics that influence postinhibitory rebound. We measured an overall burst onset phase of the LP neuron to be φ_{LP}=0.34 ± 0.03 (N = 9).

Figure 2—source data 1
 https://doi.org/10.7554/eLife.46911.004
As a firstorder 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 blocked the synaptic input from the pacemaker AB and follower PY neurons to the LP neuron (Figure 2B) and drove the LP neuron with a noise current input (see Materials and 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 (Figure 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 welldefined reference point to measure the burst onset latency. We categorized bursts with respect to the preceding interburst intervals (IBIs; see Materials and methods), 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 (Figure 2C). We characterized the driving input leading to bursts with specific IBIs by bursttriggered averaging the input current (I_{BTA}; an example shown in Figure 2D). Our analysis produced a single I_{BTA} for each of the four IBIs in each preparation (N = 23). I_{BTA}’s of each preparation were first normalized in amplitude by the (negative) peak value of the I_{BTA} at IBI = 300 ms (Figure 2E; average shown in Figure 2F) to examine how peak amplitude (I_{peak}) varied with IBI. These data were then normalized in time (Figure 2G) to examine the effect of IBI on peak phase (Δ_{peak}) and the rise (slope_{up}) and fall (slope_{down}) slopes of the input current across preparations. We found that IBI had a significant effect on I_{peak}, Δ_{peak}, slope_{up} and slope_{down} (all oneway RMANOVA on ranks; data included in Figure 2—source data 1). In particular, larger IBIs corresponded to larger I_{peak} values (Figure 2F–2H; p<0.001, χ^{2} = 65.87) with smaller (more advanced) Δ_{peak} (Figure 2I; p<0.001, χ^{2} = 41.35). The change in Δ_{peak} was due to a decrease in slope_{up} (p<0.001, χ^{2} = 65.25), whereas slope_{down} did not vary as much (Figure 2J–2K; p=0.002, χ^{2} = 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, as increasing period increases both presynaptic pacemaker burst duration (Hooper, 1997b; Hooper, 1997a) 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 (Figure 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 (Figure 3B, average in blue).
The realistic conductance input was injected periodically with strength g_{max} (Figure 3C). For any fixed g_{max}, φ_{LP} decreased as a function of P (Figure 3D), that is the relative onset of the LP burst was advanced in slower rhythms. In contrast to the effect of P, for any given P, φ_{LP} increased sublinearly as a function of g_{max} (Figure 3E). Figure 3F combines the simultaneous influence of both parameters on φ_{LP}. The results shown in Figure 3D indicate that the LP neuron intrinsic properties alone do not produce phase constancy. However, level sets of φ_{LP} (highlighted for three values in Figure 3F), indicate that phase could be maintained over a range of P values, if g_{max} increases as a function of P. This finding was predicted by our previous modeling work, in which we suggested that shortterm 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 Figure 3, it is worth examining the extent of phase maintenance for fixed g_{max}. An example of this is shown in Figure 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 (Figure 4B), phase maintenance of the LP neuron is barely better than in the constant delay case (Figure 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 shortterm synaptic plasticity (Fortune and Rose, 2001; Grande and Spain, 2005). As mentioned above, shortterm plasticity such as depression could produce changes in g_{max} as a function of P. Independently of g_{max}, 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, g_{max} and the synaptic peak time on φ_{LP}.
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 φ_{LP}, we approximated the trajectory of the (unitary) synaptic conductance in one cycle by a simple triangle (Figure 5A), which could be defined by three parameters: duration (T_{act}), peak time (t_{peak}) and amplitude (g_{max}) (Figure 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 T_{act}, the peak phase of the synapse can be defined as Δ_{peak}=t_{peak}/ _{Tact}). The parameter Δ_{peak} is known to vary as a function of P (Tseng et al., 2014) and, in a previous study, we found that Δ_{peak} may influence the activity of the postsynaptic neuron, independent of P and g_{max} (Mamiya and Nadim, 2004). We therefore systematically explored the influence of three parameters of the synaptic input (P, g_{max} and Δ_{peak}) on φ_{LP}.
As with the realistic synaptic waveforms (Figure 3), we used the dynamic clamp technique to apply the triangular conductance waveform periodically to the LP neuron in the presence of the synaptic blocker picrotoxin. Across different runs within the same experiment, the parameters P, g_{max} and Δ_{peak} were changed on a grid (see Materials and methods). In addition, all combinations of these three parameter values were run in two conditions in the same experiment, 1: with constant duration, that is constant T_{act} across different P values (CDur of 300 ms), and 2: with constant duty cycle, that is T_{act} changing proportionally to P (CDC of 0.3; Figure 5C). Using these protocols, we measured the effects of synaptic parameters on φ_{LP} (Figure 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 (Figure 6A1). When g_{max} and Δ_{peak} were kept constant, φ_{LP} decreased as a function of P (Figure 6A2). This decrease was always larger for the CDur case than the CDC case. For both CDC and CDur, this trend was seen across all values of Δ_{peak} and g_{max} (Figure 6A3). The effect of P on φ_{LP} was highly significant for both CDC (threeway ANOVA, p<0.001, F = 100.677) and CDur (threeway 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 g_{max} 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 (Figure 6B1). Overall, increasing g_{max} at constant values of P and Δ_{peak} produced a significant but only small to moderate increase in φ_{LP} (threeway ANOVA, p<0.001, F = 10.798). Although increasing g_{max} produced the same qualitative effect for both the CDC and CDur (e.g., Figure 6B2), φ_{LP} in the CDC case was restricted to a smaller range (Figure 6B3 top vs. bottom panels). Overall, this increase was robust for most values of P and Δ_{peak} (Figure 6B3).
Increasing Δ_{peak} for a constant value of P and g_{max} (Figure 6C1), produced a small but significant increase in φ_{LP} (threeway ANOVA, p<0.001, F = 17.172). This effect was robust for most values of P and g_{max}, for both CDC and CDur (Figure 6C2 and C3).
These results showed that all three parameters that define the shape of the IPSC influence φ_{LP}. Clearly, the strongest effect is the decrease in φ_{LP} as a function of P. However, φ_{LP} modestly increases as a function of the other two parameters, g_{max} and Δ_{peak}. This raised the question how g_{max} and Δ_{peak} would have to change in coordination as a function of P to counteract the effect of P on φ_{LP} and achieve phase constancy.
Coordinated changes of g_{max} and Δ_{peak} produce the largest effect on phase
To explore how g_{max} and Δ_{peak} might interact to influence φ_{LP}, we examined the sensitivity of φ_{LP} to these two parameters, individually and in combination, for all values of P in our data (see Materials and methods). Sensitivity of φ_{LP} to these two parameters varied across P values, with larger sensitivity at lower values of P (twoway RMANOVA, p<0.001, F = 16.054; data included in Figure 7—source data 1). For simplicity, we averaged the sensitivity values across different P values to obtain an overall measure of the influence of g_{max} and Δ_{peak}. These results showed that, for the CDC case, φ_{LP} had a positive sensitivity to g_{max} and a smaller positive sensitivity to Δ_{peak} (Figure 7A). The sensitivity was largest if the two parameters were varied together (g_{max} + Δ_{peak}) and smallest if they were varied in opposite directions (g_{max}  Δ_{peak}; twoway RMANOVA, p<0.001, F = 3.330). Similarly, these sensitivity values were also significantly different for the CDur case (Figure 7B; twoway RMANOVA, p<0.001, F = 2.892), with largest sensitivity for g_{max} + Δ_{peak} and smallest for g_{max}  Δ_{peak}.

Figure 7—source data 1
 https://doi.org/10.7554/eLife.46911.010
Level sets of φ_{LP} in the Pg_{max}Δ_{peak} space for CDC and CDur cases
To search for phase constancy across different P values in our dataset, we expressed φ_{LP} as a function of the three IPSC parameters, P, g_{max} and Δ_{peak}: ${\phi}_{LP}=\Phi (P,{g}_{\mathrm{max}},{\Delta}_{peak})$. Figure 8 shows heat map plots of the function Φ, plotted for the range of values of P and Δ_{peak} and four values of g_{max}. 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: $\Phi (P,{g}_{\mathrm{max}},{\Delta}_{peak})={\phi}_{c}$. For the CDC case, in each g_{max} section of the plot, the level sets (e.g. φ_{c}=0.34 denoted in white) spanned a moderate range of P values as Δ_{peak} increased (Figure 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 g_{max} and Δ_{peak}. This range of P values (spanned by the white curves) was considerably smaller for the CDur case (Figure 8A2).
For any constant phase value φ_{c}, 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 φ_{c}, for the given values of g_{max} and Δ_{peak}. The level set indicated by the white curves in panel A for the CDC case is plotted as a heat map in Figure 8B1 and can be compared with the same plot for the CDur case in Figure 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 φ_{c}=0.34. To reveal how this range depends on the desired phase, we measured this range for all values of φ_{c} between 0.2 and 0.8 (Figure 8C1 and C2). We found that the LP neuron could not achieve phases below 0.3 in the CDC case (Figure 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 g_{max} and Δ_{peak} was much larger for CDC inputs compared to CDur Inputs, for all φ_{c} values between 0.31 and 0.54.
A model of synaptic dynamics could predict activity onset phase of the LP neuron
To gain a better understanding of our experimental results, we derived a mathematical description of the phase of a follower neuron such as LP, based on the following assumptions: 1, that the firing time of this neuron was completely determined by its synaptic input, 2, that in each cycle the synaptic conductance g_{syn} increased to a maximum value g_{max} for a time interval T_{act} (the active duration of the synapse) and decayed to 0 otherwise, and 3, that the follower neuron remained inactive when g_{syn} was above some threshold g*. The derivation of this model is described in the Materials and methods.
This simple model provided a mathematical description of φ_{LP} as a function of P, g_{max} and ∆_{peak}, for the CDur and CDC cases. In the CDur case (Equation (7)), as P increased, φ_{LP} decayed and approached 0 like 1/P. In contrast, in the CDC case (Equation (8)), φ_{LP} approached its lower limit Δ_{peak}·DC, as P increased, and thus behaved very differently than in the CDur case.
We used these equations to describe g_{max} as a function of P (for any given Δ_{peak}) so that LP maintained a constant phase φ_{c}, (Equation (10) for the CDC case). Alternatively, Δ_{peak} could be given as a function of P (for any given g_{max}, Equation (11) for the CDC case). We used these derivations to compare how phase constancy depends on g_{max} or Δ_{peak} in the CDC case. A comparison of these two cases can be seen in Figure 9A, where either g_{max} (green) or Δ_{peak} (blue) is varied to keep φ_{LP} constant at φ_{c}=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.
These equations and their corresponding counterparts for the CDur case can be used to calculate the range of P values over which changing Δ_{peak} (from 0 to 1) can maintain a constant phase φ_{c}. If ΔP denotes the range of P values for which phase can be constant, it is straightforward to show that ΔP_{DC} > ΔP_{Dur} (compare blue and black curves in Figure 9B and C; see Materials and methods for derivation).
Two additional points are notable in Figure 9C. First, the lower bound on φ_{LP} for which phase constancy can occur is smaller in the CDur (black) than the CDC (blue) case. This is because we have assumed that in the CDC case the LP neuron cannot fire during inhibition and therefore the constant value of DC produces a lower limit for φ_{LP}. Second, for φ_{c} larger than ~0.5, ΔP is larger for the CDur case. This occurs because Equation (12) can no longer be satisfied when φ_{c} is large. That is, with constant duty cycle, it is not possible to produce an arbitrarily large follower neuron phase, but with constant duration, any large phase is attainable if the cycle period is not much larger than the synaptic duration. These findings are consistent with our experimental results described above (see Figure 8).
The pacemaker synaptic input to the LP neuron shows shortterm synaptic depression (Rabbah and Nadim, 2007). 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 shortterm synaptic depression (Manor et al., 2003). In that study the role of the parameter Δ_{peak} was not considered. We now consider how the presence of shortterm synaptic depression influences phase constancy by changing both g_{max} and Δ_{peak}. As stated in the Materials and methods (Equation (16)), the effect of synaptic depression on synaptic strength can be obtained as ${g}_{max}={\overline{g}}_{max}\cdot {s}_{max}\left(P\right)$), where s_{max} is an increasing function whose value approaches one as P increases. This indicates that the synapse becomes stronger due to more recovery from depression at longer cycle periods. When synaptic depression dictates how g_{max} varies with P and Δ_{peak} also varies with P and g_{max} (Equation (11)), the simultaneous changes in g_{max} and Δ_{peak} (red) greatly increase the range of P values over which φ_{LP} is constant (Figure 9A).
Note that the CDC case with shortterm depression spans a larger range of P values than the nondepressing case (Figure 9B). Similarly, the range of P values for which phase can be maintained is larger than the nondepressing case across φ_{LP} values, except where φ_{LP} is so large that the depressing synapse operates outside its dynamic range (Figure 9C). These results are consistent with our experimental results, indicating that although phase constancy can be achieved when either g_{max} or Δ_{peak} 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.
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 (Hájos et al., 2004; Hasenstaub et al., 2005), and distinct hippocampal neuron types fire at different phases of the theta rhythm and sharp waveassociated 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 waveassociated ripple episodes). Phasespecific 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 φ_{LP} was largely determined by the duration of the synaptic input. φ_{LP} changed substantially with P when inputs had constant duration, but much less when inputs had a constant duty cycle, that is when duration scaled with P. However, our experiments also showed that inputs of constant duty cycles alone are insufficient for phase constancy. φ_{LP} decreased with P even with a constant duty cycle of inputs, but increased with either synaptic strength (g_{max}) or peak phase of the synaptic input (Δ_{peak}). The increase in φ_{LP} had similar sensitivity to g_{max} and Δ_{peak}, and therefore a larger sensitivity to a simultaneous increase in both. Consequently, it was possible to keep φ_{LP} constant over a wide range of cycle periods by increasing both parameters with P.
The fact that an increase in g_{max} with P promotes phase constancy is biologically relevant, as shortterm depression in pyloric synapses means that synaptic strength indeed increases with P (Manor et al., 1997). Previous modeling studies show that shortterm 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 finding that an increase of Δ_{peak} with P promotes phase maintenance is somewhat surprising, as we have previously shown that Δ_{peak} 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 Δ_{peak} 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 experimentally or biologically. While in our current study we varied Δ_{peak} 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 voltageclamping the pacemakers, phase of follower neurons is not particularly well maintained. An example of this is shown in Figure 1, where φ_{LP} 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; Hooper, 1997a). This may reflect that neurons 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 cycletocycle variability of P in individual preparations (Bucher et al., 2005). However, it does not preclude the possibility that Δ_{peak} plays an important role in stable phase relationships when P differs because of temperature, neuromodulatory conditions, or interindividual 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 shortterm synaptic plasticity can act as a frequencydependent gain control mechanism is well known for many spikemediated synaptic connections. In bursting neurons, the presence of a combination of shortterm depression and facilitation in the same spikemediated 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 intraburst spike rate (Markram et al., 1998).
The mathematical model in the current study provides mechanistic explanations for several of our experimental findings. First, it can be used to produce a quantitative measure of phase, given the values of g_{max}, Δ_{peak} and P. Thus, these equations can be used to compare the CDC and CDur cases, which match our experimental results. They show that, for most phase values, the CDC case provides a larger range of cycle periods at which phase constancy can occur. Second, these equations provide the activity phase no matter how the pacemaker synaptic input duration changes with cycle period. For instance, our experiments were conducted by changing synaptic input through sampling individual values of the parameter pairs g_{max} and Δ_{peak}, and then calculating the resulting phase. We then used fitting to find level sets of constant phase (Figure 8). In contrast, when we combined our mathematical derivation here with previous results on the role of shortterm synaptic depression (Bose et al., 2004), we could demonstrate how a neuronal circuit can naturally follow a level set of phase (Equation (7), (8), (15), (16)). Moreover, we showed that the combined increase in g_{max} and Δ_{peak} with P produces a larger range of periods for phase constancy than increasing either parameter alone. In short, this mathematical formulation produces a simple quantitative distillation of our experimental results.
In this study, we did not explore the role of the intrinsic properties of the LP neuron on its phase. In separate experiments, we simultaneously measured postinhibitory rebound properties in LP neurons and the levels of voltagegated ionic currents (the transient potassium current I_{A} and the hyperpolarizationactivated inward current I_{h}) that influence rebound spiking. These data were not included in this study for brevity and because they showed that the timing of postinhibitory spiking was relatively stable across preparations. Therefore, we would expect the contribution of intrinsic properties in controlling the timing of the LP neuron burst onset to be relatively small. However, this result does not generalize to all follower neurons. For example, the follower ventral dilator (VD) and PY neurons have a much higher levels of I_{A}, which in turn has a larger effect on the timing of postinhibitory spiking. In a set of computational studies, we addressed the role of I_{A} in determining the burst phase in response to periodic inputs (Zhang et al., 2008; Zhang et al., 2009) and in conjunction with shortterm depression in the synaptic input (Bose et al., 2004). An experimental clarification of the relative contribution of intrinsic properties vs. synaptic input could be done with controlled dynamic clamp synaptic input, such as those used in the current study, injected in PY or VD neurons. Such a data set would fittingly complement the results of the current study to elucidate more general rules in determining the activity phase of neurons in an oscillatory network.
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 fourfold 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, g_{max}, and Δ_{peak} appeared to be fundamentally different from when P is changed at a constant temperature. Presynaptic voltage trajectories scaled with changing P, and Δ_{peak} 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 g_{max} 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.
Variability and slow compensatory regulation of phase
Phase maintenance in the face of changing P in an individual animal requires the appropriate shortterm 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 (Hájos et al., 2004; Hasenstaub et al., 2005; Somogyi and Klausberger, 2005). Phases show different degrees of variability across individuals in a variety of systems, for example 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 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 interindividual variability in ionic currents, and is considered the ultimate target of slow compensatory regulation, that is 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 g_{max} and Δ_{peak} with P.
Synaptic inputs to the LP neuron show considerable variability across preparations (e.g. Figure 3B), which mirrors the variability seen in the levels of voltagegated ionic currents in pyloric neurons (Schulz et al., 2006). We did not address the role and extent of variability in this study, because a proper analysis of variability required us to first establish the mechanisms that give rise to a consistent output, in this case phase constancy. Based on our findings regarding the influence of synaptic parameters on phase, a natural next step is to explore whether the variability of different parameters defining the synaptic input influences variability of phase or, alternatively, whether variability in some synaptic parameters may be irrelevant to phase or restrained by the postsynaptic neuron.
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 (VidalGadea 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; HarrisWarrick, 2011; Jordan and Sławińska, 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). Indeed, extensive research has indicated the role of amine modulation of synaptic strength and neuronal firing phase in the pyloric circuit, and how amine modulation of synaptic and intrinsic firing properties changes firing phases (Johnson et al., 2003; Gruhn et al., 2005; Johnson et al., 2005; Peck et al., 2006; HarrisWarrick and Johnson, 2010; HarrisWarrick, 2011; Kvarta et al., 2012). With respect to our findings, any given neuromodulator could act presynaptically to alter P, duration, or duty cycle on the one hand, and g_{max} and Δ_{peak} 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 min. 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 (DowCorning) 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 (1013°C) physiological Cancer saline containing: 11 mM KCl, 440 mM NaCl, 13 mM CaCl_{2} · 2H_{2}O, 26 mM MgCl_{2} · 6H_{2}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 (AM 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 FlamingBrown micropipette puller (P97; Sutter Instruments) and filled with 0.6 M K_{2}SO_{4} and 20 mM KCl. Microelectrodes used for membrane potential recordings had resistances of 25–30 MΩ; 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 PCI6070E 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 graded IPSC waveforms
Request a detailed protocolInhibitory postsynaptic currents (IPSCs) were recorded from LP neurons during the ongoing rhythm using twoelectrode voltage clamp and holding the LP neuron at −50 mV, far from the IPSC reversal potential of ~ −80 mV (Figure 3A). We refer to the total current measured in the voltageclamped LP neuron during the activity of the PD and PY neurons as a synaptic current for the following reasons: 1, the after blocking the PTXsensitive component of the pacemaker synapses, the LP neuron produces tonic spiking activity (see, for example Figure 2B), and 2, holding the LP neuron at different voltages (e.g. −60 or −110 mV) produces a similarly shaped current, but with a different amplitude or reversed sign (at −110 mV).
When the LP soma is voltage clamped at −50 mV, the axon (which is electrotonically distant from the soma) produced action potentials following the synaptic inhibition from the PY neuron and 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
Request a detailed protocolIn these experiments, the preparation was superfused in Cancer saline plus 10^{5} M picrotoxin (PTX; Sigma Aldrich) for 30 min 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 OrnsteinUhlenbeck (OU) process (Lindner, 2019), 60 min 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. The OU process was defined as
The parameters used for noise injection were τ = 10 to 20 ms, σ = 200 pA and a DC current of −200 to −100 pA. In these experiments, we defined bursts as groups of at least two action potentials with interspike intervals < 300 ms, following a gap of at least 300 ms.
Driving the LP neuron with realistic or triangular IPSC waveforms in dynamic clamp
Request a detailed protocolThe dynamic clamp current was injected using the Netclamp software (Netsuite, Gotham Scientific). We pharmacologically blocked synaptic inputs from the pacemaker AB and follower PY neurons to the LP neuron by superfusing the perparation in Cancer saline plus 10^{5} M picrotoxin (PTX; Sigma Aldrich) for 30 min. This treatment does not block the cholinergic synaptic input from the PD neurons for which no clean pharmacological blocker is known. Although the PD neuron input has some influence on the LP neuron activity, this input only constitutes <20% of the total pacemaker synapse and cannot drive oscillations in the follower LP neuron.
The LP neuron was driven in PTX with an artificial synaptic current in dynamic clamp. The synaptic current was given as
where the synaptic conductance g_{syn} was a predetermined waveform, repeated periodically with period P, and E_{syn} 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, g_{syn} was set to be a triangular waveform. We measured the effects of four different parameters in these triangle conductance injections (Figure 1): peak phase (Δ_{peak}), duration (T_{act}), period (P = time between onsets of dynamic clamp synaptic injections), and maximal conductance (g_{max}, the peak value of g_{syn}). 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 g_{max} 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 Δ_{peak} 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 T_{act} constant across different P values (CDur case with T_{act} = 300 ms) or with T_{act} changing proportionally to P (CDC case with duty cycle DC =T_{act}/P=0.3).
In the other set of experiments, g_{syn} was a realistic IPSC waveform, based on a prerecorded 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, g_{max} 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 φ_{LP} = Δt/P. Phase constancy means that Δt changed proportionally to P. To measure the LP neuron phase with respect to a new reference point, the end of the pacemaker input. This reference point was defined by drawing a horizontal line from the kink on the synaptic waveform that identified the onset of the pacemaker input, and chosing the first intersection point.
Determining relationship between cycle period (P), synaptic strength (g_{max}) and LP phase (φ_{LP}) using the realistic IPSC waveform
Request a detailed protocolWe 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 φ_{LP} values measured for all values of g_{max} and P, using the standard fitting routine 'fit' in MATLAB (Mathworks).
Sensitivity of φ_{LP} to g_{max} and Δ_{peak} across all P values
Request a detailed protocolTo explore how g_{max} and Δ_{peak} may interact to influence φ_{LP}, we examined the sensitivity of φ_{LP} to these two parameters, individually and in combination, for all values of P in our data. For each P, we computed the mean value of φ_{LP} across all experiments, and all values of g_{max} (0.1, 0.2, 0.3 and 0.4 µS) and Δ_{peak} (0, 0.25, 0.5, 0.75 or 1). (The φ_{LP} value for g_{max} = 0.3 µS was obtained in this case by linearly interpolating the values for 0.2 and 0.4 µS.) This produced a 4 by 5 matrix of all values. For each data point in the matrix, we moved along eight directions (+g_{max}, +Δ_{peak}, –g_{max}, –Δ_{peak},+g_{max} and +Δ_{peak}, –g_{max} and –Δ_{peak},+g_{max} and –Δ_{peak},+g_{max} and –Δ_{peak}). Here "+” denotes increasing and “ “denotes decreasing. We then calculated the change in φ_{LP} per unit g_{max} (normalized by 0.4 µS), Δ_{peak}, or both. For example, the sensitivity of φ_{LP} when Δ_{peak} was changed from 0.25 to 0.5 was measured as
Similarly, the sensitivity of φ_{LP} when g_{max} was changed from 0.2 to 0.4 was measured as
These data are provided in Figure 7—source data 1. As the next step, we averaged the sensitivity along each aligned direction: [+g_{max} and –g_{max}]; [+Δ_{peak} and –Δ_{peak}]; [+g_{max} & +Δ_{peak} and –g_{max} & –Δ_{peak}]; [+g_{max} & –Δ_{peak} and +g_{max} & –Δ_{peak}]. This produced the four cardinal directions, shown in Figure 7. Finally, we averaged the sensitivity across all P values.
A model of synaptic dynamics
Request a detailed protocolIn 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 (g_{syn}) 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 t_{peak}, corresponding to ∆_{peak}, is t_{peak} = Δ_{peak} T_{act}. We assumed that LP neuron remained inactive when g_{syn} was above a fixed threshold (g^{∗}) less than g_{max}. Because the synaptic input is periodic with period P, we solved for the minimum and maximum values of g_{syn} in each cycle. The minimum (g_{lo}) occurred just before the onset (t = 0) of AB/PD activity, whereas the maximum occurred at the peak synaptic phase ∆_{peak} for the CDur case. In the CDC case, T_{act} = 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 g_{syn}(0) = g_{lo} (and, by periodicity, g_{syn}(P)=g_{lo}), and solved the first part of Equation (1) where g_{syn} increases until t = t_{peak}. This yielded
We then used the second part of Equation (1) to track the decay of g_{syn} for t_{peak} <t < P:
Using Equation (3) , we calculated the time ∆t at which the synaptic conductance g_{syn}(∆t)=g^{∗} as follows:
Solving Equation (4) for ∆t yielded
Dividing this equation by P yielded φ_{LP}:
where g_{peak} is given by Equation (2) . This expression provides a description of the dependence of φ_{LP} as a function of P, g_{max} and Δ_{peak}. To explore the role of the parameters in this relationship, we made a simplifying assumption that the synaptic conductance g_{syn}(t) rapidly reached its peak (i.e., τ_{r} was small), stayed at this value and started to decay at t = t_{peak}. In this case g(t)=g_{max} on the interval (0,t_{peak}) and the value of g_{lo} is irrelevant. With this assumption, Equation (5) reduced to
Substituting t_{peak} = Δ_{peak}·T_{act} in Equation (6) , gave
which we used to describe the LP phase in the CDur case. To describe the CDC case, after substituting t_{peak} = Δ_{peak}·DC·P, we obtained
Note that these equations also describe the relationship between φ_{LP} with T_{act} in the CDur case, and DC in the CDC case).
Equations (7), (8) and can be used to approximate a range of parameters over which φ_{LP} is maintained at a constant value φ_{c}. To do so, we assumed a specific parameter set, say $\left(\hat{P},{\hat{g}}_{max},{\hat{\mathrm{\Delta}}}_{peak}\right)$, satisfies
for some fixed phase value, φ_{c}. We could now ask whether there are nearby parameters for which phase remains constant, that is F remains equal to φ_{c}. The Implicit Function Theorem (Krantz and Parks, 2012) guarantees that this is the case, provided certain derivatives evaluated at $\left(\hat{P},{\hat{g}}_{max},{\hat{\mathrm{\Delta}}}_{peak}\right)$ are nonzero, which turns out to be true over a large range of parameters. Since the partial derivative with respect to ∆_{peak} of F(P,g_{max},∆_{peak}) at this point is a nonzero constant equal to T_{act}/P (or DC) in the CDur (or CDC) case, there is a function ∆_{peak} = h(P,g_{max}) such that
for values of P and g_{max} near $\left(\hat{P},{\hat{g}}_{max}\right)$. In other words, the Implicit Function Theorem guarantees that small changes in P and g_{max} can be compensated for by an appropriate choice of Δ_{peak} in order to maintain a constant LP phase. A similar analysis can be done by solving for g_{max} in terms of P and Δ_{peak} or by solving for P in terms of g_{max} and ∆_{peak}.
Keeping g_{max} (respectively, Δ_{peak}) constant in these equations allows us to obtain a relationship between P and Δ_{peak} (respectively, g_{max}), for which φ_{LP} is kept constant at φ_{c}. Consider Equations (7), (8) and for fixed values of both φ_{LP} (= φ_{c}) and g_{max}. Then these equations reduce to simple functional relationships where Δ_{peak} can be expressed as a function of P. In the CDC case, for example, evaluating Δ_{peak} from Equation (8) produces
Equation (10) describes how g_{max} must vary with P for the system to maintain a constant phase φ_{c} for any given Δ_{peak}.
Alternatively, Δ_{peak} can be expressed as a function of P. In the CDC case, evaluating Δ_{peak} from Equation (8) produces
Equation (11) can be used to calculate the range of P values over which changing Δ_{peak} (from 0 to 1) can maintain a constant phase φ_{c}. Solving 0 < Δ_{peak} < 1 using Equation (11) yields
Performing the same procedure in the CDur case, we find
The lower limits of the two cases (P_{DC} and P_{Dur}) are the same. The upper limit for P_{DC} is larger than that of P_{Dur} if
If ΔP denotes the range of P values that respectively satisfy Equation (12) or (13), then ΔP_{DC} > ΔP_{Dur} if the inequality given by holds, which it does for true for τ_{s} and g_{max} large enough.
Adding synaptic depression to the model of synaptic dynamics
Request a detailed protocolIn 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 shortterm synaptic depression (Manor et al., 2003). In that study the role of the parameter Δ_{peak} was not considered. It is straightforward to add synaptic depression to Equations (7), (8) and therefore examine how phase is affected if Δ_{peak} increases with P and synaptic strength also changes with P according to the rules of synaptic depression. We will restrict this section to the CDC case. A similar derivation can be made for the CDur case.
An ad hoc model of synaptic depression can be made using a single variable s_{d} which will be a periodic function that denotes the extent of depression and takes on values between 0 and 1 (Bose et al., 2004). s_{d} decays during the AB/PD burst (from time 0 to T_{act}, indicating depression) and then recovers during the interburst interval (from T_{act} to P, indicating recovery). Thus, s_{d} can be described by an equation of the form:
Using periodicity, it is straightforward to show that the maximum value of s_{d}, which occurs at the start of the AB/PD burst, is given by:
Note that s_{max} is a monotonically increasing function with values between 0 and 1. Its value approaches one 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 s_{max} is given by Equation (15).
Software, analysis and statistics
Request a detailed protocolData 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) unless otherwise noted.
Data availability
Source data files have been provided for Figures 2 and 7.
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Decision letter

Ronald L CalabreseSenior and Reviewing Editor; Emory University, United States

Astrid A PrinzReviewer; Emory University, United States
In the interests of transparency, eLife includes the editorial decision letter and accompanying author responses. A lightly edited version of the letter sent to the authors after peer review is shown, indicating the most substantive concerns; minor comments are not usually included.
Thank you for submitting your article "Shortterm synaptic dynamics control the activity phase of neurons in an oscillatory network" for consideration by eLife. Your article has been reviewed by Ronald Calabrese as the Senior Editor and Reviewing Editor, and three reviewers. The following individual involved in review of your submission has agreed to reveal their identity: Astrid A Prinz (Reviewer #1).
The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.
Summary:
This manuscript addresses the role of synaptic input dynamics in controlling the phase of activity of neurons in the oscillatory pyloric motor network of the crustacean stomatogastric nervous system. Through systematic experimental and mathematical, analysis, the paper explores and explains the influence of various synaptic timing and amplitude parameters on activity phasing and phase maintenance, using the LP follower neuron. A major conclusion of the work is that intrinsic cellular response properties are not sufficient to ensure proper cell activity phasing and instead need to be complemented by appropriate synaptic input timing and dynamics to achieve a functional activity pattern. This and other findings should generalize well to other oscillatory neuronal networks.
Essential revisions:
There are a few concerns that must be addressed, however, before acceptance. The expert reviews are provided, but as a guide to orchestrating a revision, we provide the following points.
1) The mathematical model should be better integrated into the overall paper and discussion should include what we learn from the model. One reviewer (#2) was unable to see the value of the model given how peripheral it seemed to the results/discussion and so suggested a network model. Such a model would be beyond the scope of the present paper, but this review underscores how important it is for the authors to show the relevance of the mathematical model to the results.
2) Reviewer #1 was concerned about the treatment of variability in the paper. This concern can be addressed by analyzing, describing, displaying, and especially discussing cycletocycle and animaltoanimal information that should already be present in the data underlying this paper.
3) Reviewer #2 was concerned that there is no mention of whether/how intrinsic properties of the LP neuron might contribute to its phasing.
4) Reviewer #3 was concerned that the authors make clear the predominantly graded nature of synaptic transmission in the pyloric network, and that the authors discuss the work of the HarrisWarrick lab that addresses amine modulation of synaptic strength and neuronal firing phase in the pyloric network, and how amine modulation of synaptic and intrinsic firing properties changes firing phases.
[Editors’ note: The separate reviews follow for reference; for the purposes of publication, the minor comments sent to the authors have not been included.]
Reviewer #1:
This manuscript addresses the role of synaptic input dynamics onto neurons in oscillatory circuits in controlling the phase of activity of those neurons in the ongoing rhythmic circuit activity. Through an experimental, mathematical, and analysis tour de force, the paper comprehensively explores and explains the influence of various synaptic timing and amplitude parameters on activity phasing and phase maintenance, using the LP follower neuron in the crustacean pyloric circuit as a testbed. A major conclusion of the work is that intrinsic cellular response properties are not sufficient to ensure proper cell activity phasing and instead need to be complemented by appropriate synaptic input timing and dynamics to achieve functionally meaningful oscillations. This and other findings should generalize well to other oscillatory neuronal systems.
While some individual findings presented here confirm prior results by the authors and others, the manuscript presents the most comprehensive framework, to date, that organizes a multitude of findings and parameter dependencies of neuronal oscillatory activity into a coherent picture supported by the fruitful combination of experimentation and mathematical analysis. It will likely become a goto publication in the area of cellular and circuit oscillation analysis.
My only major comment concerns the treatment of variability in the paper, and can likely be addressed by analyzing, describing, displaying, and especially discussing cycletocycle and animaltoanimal information that should already be present in the data underlying this paper, rather than requiring additional experiments. I note that with the exception of data points in Figure 2HK and error bars (are these SD or SE?) in Figure 6 and Figure 7, all individual data presented is already in one or several ways normalized or scaled. It is therefore almost impossible for the reader to get a sense how variable rhythmic activity really is between individuals. To derive insights about general operational principles of the pyloric circuit (and oscillatory circuits in general), it makes total sense and provides the clearest answers to average data across animals, as is done in this manuscript. However, as the authors correctly state, proper phasing of neuronal activity in an oscillatory circuit can be of vital importance for physiologically functional performance of a motor system. The level at which achieving functional phase relationships matters most is the individual level, not the population average – in other words, for a crab to properly process its food it matters that its own pyloric circuit produces the right phasing, not whether crab pyloric circuits on average do so. Furthermore, work from the Calabrese lab shows that synaptic properties in an oscillatory circuit can be tuned to postsynaptic neuron properties on an individual basis to ensure that every animal implements a good solution for the problem of achieving proper phasing. All I am suggesting is that the authors provide some more detail about interindividual variability in their text and figures (perhaps by showing some individual raw traces at extreme ends of the spectrum), and, more importantly, that they discuss animaltoanimal variability, its biological significance, and the advantages and disadvantages of their approach of processing their results largely in average form.
I further note that cycletocycle variability is not mentioned, described, quantified, displayed, or discussed in the paper at all. However, in the context of this work I find a discussion of variability at the animaltoanimal level more pertinent than at the cycletocycle level.
Reviewer #2:
This manuscript addresses the question of how phase is maintained within a rhythmically active circuit. The simplest model that could be used to address this question is a twocell network consisting of a pacemaker and a follower neuron. This is well embodied by the PDLP synapse in the stomatogastric system. The authors use a combination of electrophysiology and the dynamicclamp to determine how peak synaptic conductance and the phase of the synaptic input combine to maintain phase across periods. The use of the dynamicclamp allows for a substantive exploration of the parameter space composed of Period, peak synaptic conductance and peak phase interact. The data are compelling and suggest that the combined effect of g_max and peak phase are important for phase constancy even though individually they don't appear to be. I think the sensitivity analysis in their Figure 7 is insightful because it is likely that these types of combinations of parameters are what will be important for phase constancy in other systems in which exhaustive measure of intrinsic properties will be prohibitive. For example, a lot of work has been done to try and "unwrap" the individual components of rhythmic inhibition in the hindlimb locomotor network. A similar analysis in that system could inform the relative timing of excitation and inhibition necessary for rhythmic output.
In terms of substantive concerns, my biggest concern is the use of the mathematical model. As far as an explanatory tool ("to get a better understanding of our experimental results"), it is helpful, but in my opinion, it doesn't add to the overall outcome. Perhaps a rewrite of that section, or a simulation of a network of the PDLP synapse could add to the analysis. Furthermore, the model results are not discussed in any great detail in the discussion, which detracts from their use in the Results section.
The second substantive concern regards the use and numbering of the equations in the Results section and Materials and methods section for the model. They aren't numbered properly, and they are introduced in the Results section sort of midstream from the methods derivation. Perhaps leaving it out of the Results section or referring to just the equations would work better. In alignment with the above concern, this may allow for a more focused explanation of how the parameters in the model explain the experimental results.
The final substantive concern is that there is no mention of intrinsic properties of the LP neuron. Although not asking for a measure of them or an inclusion of new experiments, a statement about how the follower neuron's intrinsic properties might contribute to phase would provide a final context for the current results. Presumably the activity of the muscles innervated by LP also need to be properly phased relative to those innervated by PD, and as such the LP neuron may make a contribution to phase via its integration of the synaptic input. As noted, that remains an untested hypothesis, but it is known that intrinsic properties of hindlimb motor neurons contribute to their output.
Reviewer #3:
This an important and interesting paper that carefully examines the parameters of synaptic transmission that contribute to the phase maintenance of neuronal firing across different neural network periods. The authors use experimental and computational methods to determine the contribution of specific parameters of synaptic transmission to firing phase constancy of neurons in a model central pattern generator network. They systematically manipulate experimentally and in a mathematical model the duration, timing and amplitude of synaptic currents to determine how each parameter works separately and together to maintain firing phase. These results should be of interest to neuroscientists studying small and large ensembles of neural networks that oscillate. This a complicated paper but the results and figures are mostly well explained.
In the Introduction introduce the concept that the neurons studied use graded as well as action potential evoked synaptic inhibition. That will clarify the results.
In the last paragraph of the Discussion section, the importance of change in neuronal phase relationships for proper network function is raised, The paper could note the work of the HarrisWarrick lab that addresses amine modulation of synaptic strength and neuronal firing phase in the pyloric network, and how amine modulation of synaptic and intrinsic firing properties changes firing phases.
https://doi.org/10.7554/eLife.46911.014Author response
Essential revisions:
There are a few concerns that must be addressed, however, before acceptance. The expert reviews are provided, but as a guide to orchestrating a revision, we provide the following points.
1) The mathematical model should be better integrated into the overall paper and discussion should include what we learn from the model. One reviewer (#2) was unable to see the value of the model given how peripheral it seemed to the results/discussion and so suggested a network model. Such a model would be beyond the scope of the present paper, but this review underscores how important it is for the authors to show the relevance of the mathematical model to the results.
The mathematical model sections are now completely revised in the Results section and Materials and methods section and a new section has been added to the Discussion section to address the findings of the model.
2) Reviewer #1 was concerned about the treatment of variability in the paper. This concern can be addressed by analyzing, describing, displaying, and especially discussing cycletocycle and animaltoanimal information that should already be present in the data underlying this paper.
This is a major issue for us too and the subject of a manuscript in preparation. Please see the response to the comment of the reviewer.
3) Reviewer #2 was concerned that there is no mention of whether/how intrinsic properties of the LP neuron might contribute to its phasing.
Please see the response to the comment of the reviewer.
4) Reviewer #3 was concerned that the authors make clear the predominantly graded nature of synaptic transmission in the pyloric network, and that the authors discuss the work of the HarrisWarrick lab that addresses amine modulation of synaptic strength and neuronal firing phase in the pyloric network, and how amine modulation of synaptic and intrinsic firing properties changes firing phases.
Done and done.
Reviewer #1:
[…] My only major comment concerns the treatment of variability in the paper, and can likely be addressed by analyzing, describing, displaying, and especially discussing cycletocycle and animaltoanimal information that should already be present in the data underlying this paper, rather than requiring additional experiments. I note that with the exception of data points in Figure 2HK and error bars (are these SD or SE?) in Figure 6 and Figure 7, all individual data presented is already in one or several ways normalized or scaled. It is therefore almost impossible for the reader to get a sense how variable rhythmic activity really is between individuals. To derive insights about general operational principles of the pyloric circuit (and oscillatory circuits in general), it makes total sense and provides the clearest answers to average data across animals, as is done in this manuscript. However, as the authors correctly state, proper phasing of neuronal activity in an oscillatory circuit can be of vital importance for physiologically functional performance of a motor system. The level at which achieving functional phase relationships matters most is the individual level, not the population average – in other words, for a crab to properly process its food it matters that its own pyloric circuit produces the right phasing, not whether crab pyloric circuits on average do so. Furthermore, work from the Calabrese lab shows that synaptic properties in an oscillatory circuit can be tuned to postsynaptic neuron properties on an individual basis to ensure that every animal implements a good solution for the problem of achieving proper phasing. All I am suggesting is that the authors provide some more detail about interindividual variability in their text and figures (perhaps by showing some individual raw traces at extreme ends of the spectrum), and, more importantly, that they discuss animaltoanimal variability, its biological significance, and the advantages and disadvantages of their approach of processing their results largely in average form.
I further note that cycletocycle variability is not mentioned, described, quantified, displayed, or discussed in the paper at all. However, in the context of this work I find a discussion of variability at the animaltoanimal level more pertinent than at the cycletocycle level.
We thank the reviewer for these comments. We realized that addressing (animaltoanimal) variability within the context of phase constancy requires us to first deal with the synaptic mechanisms that influence phase. Dr. Anwar is lead author on a followup study that focuses only on the issue of variability. In that study, we examine how variability of synaptic parameters reflects on the variability of the LP neuron phase. We are in complete agreement with the findings of the Calabrese lab on this topic but beg the patience of the reviewer on a proper treatment of variability in this context in a follow up study.
We now address the issue of variability in the Discussion section.
Reviewer #2:
[…] In terms of substantive concerns, my biggest concern is the use of the mathematical model. As far as an explanatory tool ("to get a better understanding of our experimental results"), it is helpful, but in my opinion, it doesn't add to the overall outcome. Perhaps a rewrite of that section, or a simulation of a network of the PDLP synapse could add to the analysis. Furthermore, the model results are not discussed in any great detail in the Discussion section, which detracts from their use in the Results section.
The second substantive concern regards the use and numbering of the equations in the Results section and Materials and method sections for the model. They aren't numbered properly, and they are introduced in the Results section sort of midstream from the methods derivation. Perhaps leaving it out of the Results section or referring to just the equations would work better. In alignment with the above concern, this may allow for a more focused explanation of how the parameters in the model explain the experimental results.
The modeling sections of the Results section and Materials and methods section are completely redone and, as the reviewer suggested, all equations were moved to the Materials and methods section.
The final substantive concern is that there is no mention of intrinsic properties of the LP neuron. Although not asking for a measure of them or an inclusion of new experiments, a statement about how the follower neuron's intrinsic properties might contribute to phase would provide a final context for the current results. Presumably the activity of the muscles innervated by LP also need to be properly phased relative to those innervated by PD, and as such the LP neuron may make a contribution to phase via its integration of the synaptic input. As noted, that remains an untested hypothesis, but it is known that intrinsic properties of hindlimb motor neurons contribute to their output.
The reviewer is correct, and this is a topic of a separate study by led Dr. Anwar. We now address this point in the Discussion section.
Reviewer #3:
[…] In the Introduction introduce the concept that the neurons studied use graded as well as action potential evoked synaptic inhibition. That will clarify the results.
Done.
In the last paragraph of the Discussion section, the importance of change in neuronal phase relationships for proper network function is raised, The paper could note the work of the HarrisWarrick lab that addresses amine modulation of synaptic strength and neuronal firing phase in the pyloric network, and how amine modulation of synaptic and intrinsic firing properties changes firing phases.
Added to the last paragraph of the Discussion section.
https://doi.org/10.7554/eLife.46911.015Article and author information
Author details
Funding
National Institutes of Health (MH060605)
 Dirk M Bucher
 Farzan Nadim
National Science Foundation (DMS1122291)
 Amitabha Bose
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
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.
Senior and Reviewing Editor
 Ronald L Calabrese, Emory University, United States
Reviewer
 Astrid A Prinz, Emory University, United States
Publication history
 Received: March 15, 2019
 Accepted: June 8, 2019
 Accepted Manuscript published: June 10, 2019 (version 1)
 Version of Record published: June 24, 2019 (version 2)
Copyright
© 2019, Martinez et al.
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
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