Frequency-Dependent Action of Neuromodulation

The amplitude of I_Proc is sensitive to the slope and direction of voltage ramps

During an ongoing pyloric rhythm, the LP neuron membrane potential expresses slow-wave oscillations in time with the pyloric rhythm (Fig. 1A). We approximated the LP neuron waveform (Fig. 1B) slow-wave depolarization slope at different cycle periods (Fig. 1C) and measured the proctolin-activated current I_Proc with ramp voltage protocols with corresponding slopes (Fig. 1D; Extended Data Fig. 1-1; see Materials and Methods).

With both positive and negative voltage ramps, proctolin elicited a voltage-gated inward current (I_Proc), which had a similar appearance for all ramp slopes (Fig. 2A). However, the maximum magnitude (I_max) of I_Proc was larger on the positive ramp than on the negative ramp. Furthermore, the magnitude of the current was sensitive to the slope of the positive ramp, with larger slopes eliciting larger currents (Fig. 2, raw difference currents of one example experiment in A, corresponding I-V curves with fits in B, quantification in C). For the negative ramp, I_max was mostly independent of slope value, except for the largest slope of 400 mV/s. Additionally, the voltage (V_Imax) at which I_max was attained was more depolarized for larger slopes on the positive ramp, and less so on the negative ramp (Fig. 2C; two-way RM ANOVA results in Extended Data Fig. 2-1). Together, these data suggested that I_Proc includes both a persistent and a transient (inactivating) component. Both components activated with the positive ramp, but the negative ramp elicited mostly the persistent component because, by this time, the other component had inactivated. The presence of these two components would explain (1) the difference between the current amplitudes on the positive and negative ramps; (2) the slope dependence of the current amplitude on the positive ramps; (3) the fact that the largest negative ramp slope also revealed a larger current compared with other negative ramps, likely because of incomplete inactivation of the transient component; and (4) the shift of V_Imax to more depolarized values for larger slopes because with these more depolarized voltages are reached in a shorter amount of time, before complete inactivation of the transient component.

The component of I_Proc measured with negative ramps was (mostly) independent of ramp slope and qualitatively matched the previously described I_MI (Golowasch and Marder, 1992; Swensen and Marder, 2000; Goaillard et al., 2009; Gray and Golowasch, 2016). Therefore, we will refer to this persistent component as I_MI, and to the inactivating (transient) component as I_MI-T. Thus, the negative ramp current consists primarily of I_MI, whereas the positive ramp current consists of I_MI + I_MI-T.

I_Proc has an additional, slow inactivating component

In the ramp measurements (Fig. 2), we allowed a 5 s time interval between ramps, so that the currents measured could recover from any inactivation. Hence, there was little difference between currents in subsequent identical ramp measurements. However, in normal biological conditions, the pyloric rhythm is active continuously and, therefore, I_Proc would be at some steady-state level of inactivation. To measure the steady-state levels of I_MI and I_MI-T, we applied symmetric ramp-and-hold stimuli repetitively for 30 cycles in a separate set of experiments (Fig. 3A). We switched to a ramp-and-hold protocol, expecting that the addition of a depolarized hold interval between the positive and negative ramp would allow for full inactivation of any inactivating component and, therefore, a better separation of I_MI and I_MI-T.

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Figure 3.

Proctolin-activated currents show slow inactivation. A, Proctolin-activated currents (I_Proc) in response to 30 sweeps of ramp-and-hold stimuli with different slopes (color-coded). The gray dots connected by lines depict the average I_Proc for each sweep in this experiment. We refer to I_Proc during the first sweep as initial state, and to the average of the last five sweeps as steady state (gray box). B, First (initial state) and averaged last five (steady state) sweeps from the experiment in A. C, Ratio of I_Proc between steady state and initial state. The slow inactivation of I_Proc is greater at 100 mV/s ramps compared with 200 and 400 mV/s ramps (RM ANOVA; Table 1; results in Extended Data Fig. 3-1). Each dot represents an individual experiment. D, Time constants for slow inactivation. Each dot represents an individual experiment. Time constants were not significantly different between slopes (ANOVA on ranks; Table 1; results in Extended Data Fig. 3-2). Asterisks indicate significance at α = 0.05. n.s. indicates no significant changes.

The repeated depolarization in this longer protocol revealed a clear slow reduction in the amplitude of the total current (Fig. 3; raw difference currents across all sweeps of one experiment in A, initial and steady state sweeps of the same experiment in B, quantification of the levels of slow inactivation in C and of the time course in D). We refer to the response to the first depolarizing ramp-and-hold stimulus in each run as the initial state, and to the average of the last five responses as steady state (Fig. 3A,B). The distinction between the positive and negative ramp currents was present both in the initial state and at steady state (Fig. 3A,B). The first stimulus produced a current that was comparable to the currents we measured with triangular ramps and analyzed in Figure 2, and therefore we do not repeat that analysis here. To quantify the slow change, we measured the average inward current amplitude during each stimulus (Fig. 3A, gray dots connected by lines). A comparison of the initial response amplitude with that at steady state showed a slope-dependent reduction in amplitude for all ramp slopes (Fig. 3C; statistical comparisons in Extended Data Fig. 3-1). Note that this reduction in amplitude was very slow, in the range of several seconds (Fig. 3D) and therefore distinct from the fast inactivation effect that distinguished the positive and negative ramp currents. The time constant of the reduction of the current amplitude at this slow timescale did not depend on the steepness of the ramp slope (ANOVA on ranks; results in Extended Data Fig. 3-2).

We explored the effect of ramp slopes at steady state (Fig. 4, raw difference currents of three steady state cycles of one experiment in A, corresponding I-V curves with fits in B, quantification in C) as in the triangular ramp experiments without a depolarized hold interval. Generally, the steady-state I-V curves showed the same slope and directional dependencies as in the simple ramp currents described in Figure 2 (Fig. 4B,C; two-way RM ANOVA results in Extended Data Fig. 4-1), with one exception: I_max and V_Imax values at steady state on the negative ramp no longer showed slope dependence, probably because this difference was due to incomplete inactivation of the transient component with the triangular ramps.

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Figure 4.

Steady-state levels of the proctolin-activated currents elicited by a periodic ramp-and-hold stimulus depend on ramp slope and direction. A, Proctolin-activated currents (I_Proc) in response to the last three of 30 sweeps of ramp-and-hold stimuli with different slopes (color-coded). Data are from the same experiments as Figure 3A. B, Steady-state I-V curves for different ramp slopes (color-coded) from one experiment (same experiment as in A). Gray lines show raw current recordings, colored lines show logistic fits that were used to smoothe the raw data. C, Quantitative analysis of I_max (left) and V_Imax (right) for different ramp slopes and ramp directions (N = 10). Dots represent data from individual experiments. I_max is sensitive to ramp slope on the + ramp but not to the – ramp (two-way RM ANOVA; Table 1; results in Extended Data Fig. 4-1). Asterisks indicate significant differences between slopes within the same direction, daggers indicate significant differences between directions within the same slope at α = 0.05. n.s. indicates no significant changes.

Together, these findings led to two conclusions. First, I_Proc has a slowly inactivating component, with a time constant of several seconds, that diminishes the total current activated and measured across frequently repeated stimuli until it reaches stable levels at steady state. Second, even at steady state, I_Proc has a persistent component (I_MI) and a component that inactivates rapidly during depolarizing ramps (I_MI-T).

The amplitude of I_Proc is sensitive to the cycle period of the pyloric rhythm

The I-V curves revealed that I_Proc was most sensitive to the depolarizing slope. Because the pyloric rhythm operates over a large range of cycle periods (Bucher et al., 2005), and the shape of the voltage waveform (e.g., rising and falling slopes) is influenced by this period, the size of the inward current activated by proctolin would also be expected to be influenced by the period. We, therefore, measured how the cycle period of the LP waveform influenced the total current activated by proctolin. To do so, we repeatedly played back a prerecorded LP voltage waveform scaled to, and applied at, different cycle periods in the voltage-clamped LP neuron, in control and in the presence of proctolin, and measured I_Proc as the difference current (Fig. 5A). We used periods of 250–2000 ms (4–0.5 Hz), which corresponds to rates of membrane potential rise of ∼100–950 mV/s (see Fig. 1C; Extended Data Fig. 1-1).

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Figure 5.

Slope-sensitivity of proctolin-activated currents is different during ongoing LP activity. Ai, The last three (of 20) sweeps of the proctolin-activated currents I_Proc in response to voltage clamping with a realistic LP waveform. Shown is one experiment at different cycle frequencies (color-coded). Aii, Overlay of the averages of the last five cycles (same experiment as Ai), normalized to time. B, I_max and phase of I_max (ϕ_Imax) are sensitive to cycle frequency (RM ANOVA; Table 1; results in Extended Data Fig. 5-1). Dots represent data from individual experiments. Asterisks indicate significant differences between frequencies at α = 0.05.

Based on our ramp and ramp-and-hold measurements we expected that I_Proc would increase with increasing waveform cycle frequency (or shorter cycle period), which corresponds to an increase in its depolarization slope. There was a small increase in the average I_Proc amplitude as cycle frequency was increased up to 2 Hz (Fig. 5Bi), but this increase was not observed in every experiment (Fig. 5Aii). Surprisingly, and contrary to our expectation from the ramp measurements, the peak I_Proc value (I_max) was significantly smaller for the highest frequency tested (4 Hz; Fig. 5Bi; RM ANOVA results in Extended Data Fig. 5-1). Finally, the phase (peak time relative to the trough potential divided by period) at which I_max was measured at each period got slightly but significantly delayed at the higher cycle frequencies (Fig. 5Bii).

I_MI-T explains the observed frequency dependence of proctolin effects

Our findings so far led to three observations on I_Proc in the LP neuron. (1) Ramps activate a current whose amplitude depends on the steepness and sign of the slope. (2) Realistic waveforms produce a current whose amplitude somewhat increases with increasing cycle frequencies (and thus larger positive slopes) up to 2 Hz but decreases at the highest frequency tested (4 Hz) in contrast to what is predicted from the ramp experiments. (3) The size of the current decreases gradually to a steady state over a timescale of several seconds when stimuli are repeated at frequencies consistent with pyloric cycle frequencies.

To see how these observations could be explained by proctolin activating voltage-dependent currents, we resorted to computational modeling. We first examined the steady-state effects of the ramp-and-hold protocols. We found that the steady-state slope-dependent effects of the data could be explained assuming that proctolin activated two voltage-gated currents (Fig. 6A). This match required that at least one of the two currents (which we have referred to as I_MI-T) be an inactivating current but could not be explained with a single persistent inward current that has been previously described (Golowasch and Marder, 1992; Swensen et al., 2000).

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Figure 6.

A model with only I_MI and I_MI-T adequately captures the fast and slow inactivations. A, Model parameters for I_MI and I_MI-T were tuned to capture the steady-state I_Proc trajectories, i.e., a larger inward current on the positive ramps, and larger inward currents with larger slopes. Ai, Overlay of the biological (red) and model (black) I_Proc trajectories in response to steady-state ramp-and-hold stimuli. Aii, I-V curves separated by positive (purple) and negative (green) ramps similar to those shown in Ai. I-V curves for the positive ramps were obtained after holding the voltage at –80 mV to remove inactivation, and the negative ramps after holding the voltage at +20 mV to maximize inactivation of the transient current. Bi, Time constants for the activation (solid lines) and inactivation (dashed lines) gates of the model I_MI (green) and I_MI-T (pink). Bii, Model response to realistic LP waveform stimulations with different cycle frequencies (based on the biological data from the same preparation as in Ai). The gray shading around V_comm. indicates the voltage range of the ramp and ramp-and-hold stimuli used in Ai. Biii, Model response to the same waveforms as in Bii but with an upscaled amplitude that is similar to the amplitude of the ramp and ramp-and-hold stimuli. C, The slow inactivation of the proctolin-activated current can be mimicked in a computational model by modeling I_MI-T as Ca²⁺ current following the Goldman–Hodgkin–Katz formalism.

However, a simple division of I_Proc into a persistent and an inactivating component did not explain the second observation. As described above, the measurement of I_Proc using realistic LP neuron voltage waveform protocols (slow-wave oscillation range −60 to −20 mV) showed that the amplitude of the current did not consistently follow the same slope dependence as the current using the ramp protocols. Rather, the value of I_Proc decreased at the largest applied cycle frequency of 4 Hz (largest depolarization slope), which was the opposite of what we expected from the ramp experiments. As expected, when we simulated the simplest model that fit the ramp data with the LP voltage waveforms it produced the expected result that the current became consistently larger, not smaller, with increased cycle frequencies, and thus did not replicate the realistic waveform data (not shown). However, we noted that in our experiments the ramps spanned a voltage range (−80 to +20 mV) that was much larger than that of the realistic waveforms (−60 to −20 mV). It was therefore possible that the currents involved have different kinetics in the more restricted voltage range. Because the persistent current I_MI has very fast activation kinetics (Golowasch and Marder, 1992), we focused on the kinetics of the inactivating current I_MI-T. We found that an adjustment of the kinetics of I_MI-T to have slower activation over the voltage range of the realistic waveform but remain fast outside of this range (Fig. 6Bi, pink trace), was sufficient to reproduce the effect of the realistic waveform in that increasing cycle frequency would lead to a smaller I_Proc (Fig. 6Bii). A prediction of this modified model would be that applying the realistic waveform in the larger range of −80 to +20 mV should produce an increase of I_Proc when cycle frequency is increased. Indeed, this was precisely what we observed in the model (Fig. 6Biii).

So far, we focused on describing how the model currents matched the slope-dependent effects of I_Proc at steady state, both for ramps and for realistic waveforms. However, these currents could not reproduce the third observation, that the total current decayed slowly when the ramp-and-hold protocol was applied repeatedly (Fig. 3). This was because both I_MI and I_MI-T had fast activation, and the inactivation time constant of the model I_MI-T was <800 ms throughout the voltage range, which did not allow the current to keep a memory over repeated voltage waveforms. It is possible that the slow decay of I_Proc over several seconds involves yet another slower inactivation component or another slow-inactivating proctolin-activated ionic current altogether. However, there is an alternative and perhaps simpler possibility. Previous studies have suggested that the peptide-activated ion channels (such as those underlying I_Proc) in the STG may have calcium permeability or even be calcium currents (Zhao et al., 2011; Oh et al., 2012; Rodriguez et al., 2013; Gray et al., 2017). If I_MI-T were a calcium current, calcium entry due to this current could change local internal calcium concentrations so that the driving force of the current would be reduced over repetitive depolarizations. To examine this possibility, we modeled I_MI-T as a calcium current using the Goldman–Hodgkin–Katz formalism (see Materials and Methods). Although we did not attempt to optimize the parameters of this model (many of which remain unmeasured), we found that this formalism could sufficiently explain how the current amplitude reduced with repetitive depolarization (Fig. 6C).

Examining the model predictions

To examine the prediction of the model that low-amplitude stimuli produce a different slope-dependent or frequency-dependent effect than high-amplitude stimuli (Fig. 6B), we repeated the ramp-and-hold experiments but reduced the amplitude to match the original realistic waveform voltage range of −60 to −20 mV and scaled the cycle frequencies to match the pyloric frequencies commonly observed in experiments at 12°C (∼0.5−1 Hz; Tang et al., 2012; Haddad and Marder, 2018; Rosenbaum and Marder, 2018; Kushinsky et al., 2019; Fig. 7A). In this voltage range, I_Proc largely lost its slope dependence, both for I_max and V_Imax (Fig. 7B; two-way RM ANOVA results in Extended Data Fig. 7-1), as predicted by the model.

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Figure 7.

Decreasing the amplitude of ramps to that of the realistic waveforms removes the slope dependence of I_MI-T. A, I_Proc measured in LP in response to downscaled ramp-and-hold stimuli with an amplitude similar to the realistic LP wave stimulus. B, Quantification of steady state I_max and V_Imax for the downscaled ramp-and-hold stimuli (N = 6). Dots represent data from individual experiments. There is no significant slope dependence (n.s.; two-way RM ANOVA; Table 1; results in Extended Data Fig. 7-1).

We also tested the possibility that I_MI-T is a calcium current by repeating our ramp protocols with LP in saline containing 200 μm Cd²⁺, which blocks Ca²⁺ channels in the LP neuron (Golowasch and Marder, 1992). In Cd²⁺ saline, the slope dependence was greatly reduced (Fig. 8A–C, raw difference currents of one example in A, example I-V curve with fits of the same experiment in B, quantification in C). Only the largest positive slope of 400 mV/s resulted in a significantly larger I_max compared to the smallest slope of 50 mV/s (Fig. 8C). Additionally, except for the largest slope, I_max and V_Imax of I_MI and I_MI-T were not different in Cd²⁺ saline (Fig. 8C; two-way RM ANOVA results in Extended Data Fig. 8-1). In normal saline, the increase in I_Proc across positive ramp slopes was approximately linear. Therefore, we compared the slopes of linear fits across all positive and negative ramps between normal saline and Cd²⁺ saline. For the positive ramps, this slope was significantly smaller in Cd²⁺, and unchanged for the negative ramps (Fig. 8D; two-way ANOVA results in Extended Data Fig. 8-2), indicating that I_Proc did not depend on depolarization slope in this voltage range in Cd²⁺. We were unable to calculate time constants for the slow inactivation because I_Proc at steady-state was not different from I_Proc at the initial state in Cd²⁺ saline (Fig. 8E; two-way RM ANOVA; results in Extended Data Fig. 8-3). From these experiments we conclude that Ca²⁺ is at least partially involved in I_MI-T, possibly as the charge carrier, and it also appears to be necessary for the slow decay of the current.

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Figure 8.

Proctolin-activated currents are partially blocked by cadmium. Ai, Proctolin-activated currents in Cd²⁺ saline evoked by symmetrical ramp stimulations with four different slopes (color-coded), averaged over the last three of five sweeps of one experiment. Aii, Overlay of the proctolin currents shown in Ai, normalized by time. B, Example I-V curves for I_proc shown in A, 100 mV/s. Gray solid lines show original current recordings, colored solid lines show logistic fits. C, Quantitative analysis of I_max (left) and V_Imax (right) for different ramp slopes and ramp directions (N = 4). Dots represent data from individual experiments. Ramp slope and direction show statistically significant interactions for I_max (two-way RM ANOVA; Table 1; results in Extended Data Fig. 8-1). Asterisks indicate significant differences between slopes within a direction, daggers indicate significant differences between directions within a slope at α = 0.05. n.s. indicates no significant changes. D, Linear fits for I_max. Di, Example showing the fits from a linear regression model to the I_max for different ramp slopes and directions in normal saline and Cd²⁺ saline. Dii, Slopes of linear fits for I_max in normal saline (filled boxes) and Cd²⁺ saline (hatched boxes). Cd²⁺ significantly reduced the slope of I_max for the + ramps (purple), but not the – ramps (green), indicating a reduction of I_MI-T in the presence of Cd²⁺ (two-way ANOVA; Table 1; results in Extended Data Fig. 8-2). E, Example (same preparation as in A) showing the initial (black) and steady-state (gray, average of the last five sweeps) responses to ramp-and-hold stimuli in Cd²⁺ saline. In all four experiments, I_Proc was not significantly different between initial and steady state (two-way RM ANOVA; Table 1; results in Extended Data Fig. 8-3).

The roles of I_MI and I_MI-T in the activity patterns of the LP neuron at different cycle periods

The bursting activity of follower pyloric neurons and, in particular, the LP neuron is exquisitely sensitive to the action of neuromodulation (Marder and Bucher, 2007; Stein, 2009; Harris-Warrick and Johnson, 2010), including proctolin (Nusbaum and Beenhakker, 2002; Daur et al., 2016). It is therefore natural to ask how the two currents I_MI and I_MI-T may differentially influence the LP bursting activity. Because we currently do not have an experimental method of separating the two components of I_Proc during the ongoing activity (no pharmacological blockers of these currents exist), we addressed this question in a computational model of the LP neuron (see Materials and Methods) based on the model developed by Taylor et al. (2009). Because the LP neuron bursts in response to periodic inhibition it receives from the pyloric pacemaker neurons, we drove the model with a periodic inhibitory synaptic input. We then examined the effect of the proctolin modulation on the bursting activity of the model neuron. I_MI and I_MI-T were adjusted to account for the spatial distance to the putative receptor location distal from the soma (see Materials and Methods). Note that the lack of slope dependence with small amplitude stimulations (Fig. 7) could be because of incomplete space clamp. All data shown from these modeling results are at steady state once the transient response of the model neuron has finished.

To see how Ca²⁺ permeability would influence the activity of the model neuron, we drove the LP neuron with synaptic input at a cycle frequency of 1 Hz, a typical average pyloric cycle frequency, and compared the effect of I_Proc = I_MI + I_MI-T at steady state, when the current was assumed to be permeable to Ca²⁺ (I_MI-T(Ca)) and when it was not (Fig. 9Ai, black traces: not Ca²⁺ permeable, purple traces: Ca²⁺ permeable). We found that permeability to Ca²⁺ did not change the burst onset of the model neuron but shortened the burst duration so that the burst terminated before the arrival of the periodic inhibition. Consistent with the shorter burst, the number of spikes and the intra-burst spike frequency were both reduced, but there was no change in the burst structure in that the instantaneous spike frequency (f_inst = 1/interspike interval) decayed during each burst in both cases (Fig. 9Ai). To understand these effects, we compared the ionic currents of the models and found that the changes in the burst duration and spike number were because of the additional activation of the outward Ca²⁺-dependent K⁺ current, I_K(Ca), when the model I_Proc was assumed to be permeable to Ca²⁺ (Fig. 9Aii, I_MI-T(Ca), purple and pink traces), whereas the other ionic currents (which were not influenced by Ca²⁺ entry) were not affected (data not shown).

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Figure 9.

The frequency dependence of I_MI-T shifts the burst phases in a model of the LP neuron. LP model with I_K and I_Na in the axon compartment, leak current in both axon and soma/neurite compartment, and I_A, I_h, I_Ca, I_K(Ca), I_MI, and I_MI-T in the soma/neurite compartment. Calcium permeability is indicated by the subscripted addition of (Ca) to I_MI-T. Ai, Model voltage waveforms and the corresponding instantaneous spike frequencies (f_inst) within a burst with I_MI and I_MI-T where I_MI-T is either contributing to the intracellular Ca²⁺ concentration (purple; I_MI-T(Ca)) or not (black). The model received periodic inhibition (g_syn) at 1 Hz. Aii, Levels of I_K(Ca) when I_MI-T is either contributing to the intracellular Ca²⁺ concentration (purple) or not (black). The larger I_K(Ca) contributes to the earlier burst termination. The model equations and parameters are as described in Materials and Methods. B, Contribution of the different components to the activity phases of the model neuron at different cycle frequencies. Currents were matched so that the model produced the same number of spikes per burst at 1 Hz (gray rectangles in Bi). Bi, Voltage trajectories and instantaneous frequencies within a burst at 0.5, 1, and 1.66 Hz when the model contained I_MI and I_MI-T(Ca) (purple), when the model contained only I_MI but not I_MI-T(Ca) (green), and when the model contained only I_MI-T(Ca) but not I_MI (pink). Bii, Phase plots of the activity at different cycle frequencies. I_MI-T(Ca) had a substantially greater effect on phase than I_MI (Extended Data Fig. 9-1). Gray bars indicate the duration of the inhibitory synaptic current. In this panel, the model parameters for I_MI and I_MI-T were adjusted as follows: I_MI + I_MI-T(Ca): g¯MI=0.227 , g¯MI−T=2.27 ; I_MI only: g¯MI=0.995 , g¯MI−T=0 ; I_MI-T(Ca) only: g¯MI=0 , g¯MI−T=3.63 .

To examine the frequency-dependent effects of the two proctolin-activated currents, I_MI and I_MI-T, we started with the model in Figure 9A (with Ca²⁺ permeability) at the cycle frequency of 1 Hz and tuned the levels of the modulatory currents to have seven spikes per burst. We refer to this as the reference model. We then changed the frequency of the periodic inhibitory synaptic input without changing the duty cycle, shape, or amplitude of this synapse. For simplicity, we only show the activity of the model neuron at two other frequencies, one lower (0.5 Hz) and one higher (1.66 Hz). Decreasing the cycle frequency slightly advanced the onset and end phases of the burst, and increased the number of spikes per burst, but did not affect the burst structure, in that f_inst followed the same decreasing trend within each burst (Fig. 9Bi, purple traces). Consistent with this, increasing the cycle frequency did the opposite. We then examined how proctolin modulation would influence LP activity if proctolin were to activate only I_MI or only I_MI-T. However, simply removing I_MI-T from our reference model resulted in a complete absence of spiking at all three frequencies. Therefore, to do this comparison in the I_MI only case, we first increased the maximal conductance of I_MI so that, for the 1 Hz case, the number of spikes per burst matched the number of spikes of the reference model. Similarly, in the I_MI-T only case, we increased the maximal conductance of I_MI-T to achieve the same. These cases are denoted as “matched.” We then changed the frequency of the periodic input in each case and compared the effects with the reference case.

In the I_MI only case, changing the cycle frequency simply increased or decreased the number of spikes in the burst but the spike structure (f_inst) of the model neuron remained relatively constant for the duration of the burst. In contrast, in the I_MI-T only case, both increasing and decreasing the cycle frequency shortened the duty cycle, but at 1.66 Hz, the burst was delayed in its onset, whereas at 0.5 Hz, it was advanced. In this case, the burst structure was also influenced by the change in cycle frequency. When cycle frequency was decreased, f_inst dropped dramatically within each burst (pink trace at 0.5 Hz), whereas it became somewhat more constant as cycle frequency was increased (pink trace at 1.66 Hz). The changes in the burst onset and end were reflected in the phase diagram of the model neuron (Fig. 9Bii). In the reference model (purple), there was a shift in the burst onset and a smaller shift in the burst end with increased cycle frequency. These shifts were reduced in the I_MI only case (green) and the phases were more similar across cycle frequencies. In contrast, the shifts became more pronounced in the I_MI-T only case (pink) and both the beginning and the end of the burst greatly shifted with increasing cycle frequency. Together, these modeling results indicate the ability of the two modulatory-activated currents to provide different influences on the burst structure, with I_MI producing a consistent burst duration and strength across all cycle frequencies and I_MI-T producing a dynamic change in burst structure across different cycle frequencies.