Abstract
Many oscillatory networks involve neurons that exhibit intrinsic rhythmicity but possess a large variety of voltagegated currents that interact in a complex fashion, making it difficult to determine which factors control frequency. Yet these neurons often have preferred (resonance) frequencies that can be close to the network frequency. Because the preferred frequency results from the dynamics of ionic currents, it can be assumed to depend on parameters that determine the neuron's oscillatory waveform shape. The pyloric network frequency in the crab Cancer borealis is correlated with the preferred frequency of its bursting pacemaker neurons anterior burster and pyloric dilator (PD). We measured the preferred frequency of the PD neuron in voltage clamp, which allows control of the oscillation voltage range and waveforms (sine waves and realistic oscillation waveforms), and showed that (1) the preferred frequency depends on the voltage range of the oscillating voltage waveform; (2) the slope of the waveform near its peak has a strongly negative correlation with the preferred frequency; and (3) correlations between parameters of the PD neuron oscillation waveform and its preferred frequency can be used to predict shifts in the network frequency. As predicted by these results, dynamic clamp shifts of the upper or lower voltage limits of the PD neuron waveform during ongoing oscillations changed the network frequency, consistent with the predictions from the preferred frequency. These results show that the voltage waveform of oscillatory neurons can be predictive of their preferred frequency and thus the network oscillation frequency.
Introduction
Oscillatory neuronal networks usually produce their output in a certain frequency range. The network frequency can be determined by the properties of individual neurons and the interaction between neurons (Marder and Calabrese, 1996) and, in some systems, solely by the oscillation frequency of the pacemaker neurons (Marder and Bucher, 2001). Previous studies have shown that the frequency of a bursting network oscillation can be strongly correlated with the preferred (resonance) frequency of the neurons within the network (Tohidi and Nadim, 2009). An important characteristic of a bursting neuron is its slow oscillation membrane potential, which can be represented as a unitary waveform. We explored the question of how the parameters of this waveform affect the preferred frequency of the neuron and, potentially, the frequency of the network activity.
The crustacean pyloric network generates a triphase rhythmic activity around 1 Hz. Many pyloric neurons, including the pacemaker neurons anterior burster (AB) and pyloric dilator (PD), show a frequencydependent response to oscillatory inputs, resulting in a preferred frequency at which the neuron shows the highest impedance (Tohidi and Nadim, 2009). Here we examine the hypothesis that changes in the PD neuron waveform shape can shift its preferred frequency and, consequently, the frequency of the pyloric network activity as a whole. Thus, by knowing how defining parameters of the PD neuron waveform influence the preferred frequency, we should be able to predict shifts in the pyloric network frequency as a result of physiological inputs or experimental manipulations that reshape the waveform.
The oscillation waveform of the PD neuron is defined by multiple parameters, including the voltage minimum and maximum, duty cycle, peak phase, and the rise and fall slopes. In an ongoing pyloric activity, these parameters are defined by a variety of factors. For instance, synaptic inhibition from the follower lateral pyloric (LP) neuron alters the lower bound of the PD waveform (Thirumalai et al., 2006), whereas voltagegated ionic currents, such as the A current, contribute to the process of rebound by shaping the rising slope of the waveform and its duty cycle (Tierney and HarrisWarrick, 1992; MacLean et al., 2003). Additionally, these parameters are subjected to neuromodulation (Kloppenburg et al., 1999; Peck et al., 2001; Thirumalai et al., 2006).
We demonstrate that changes in the preferred frequency of a neuron can be predicted by examining parameters that determine its voltage waveform. By measuring the preferred frequency with the voltageclamp technique, we find that the voltage range of the oscillation and a few descriptive parameters of the PD neuron waveform, including its rising slope near peak, are strongly correlated with its preferred frequency. Using the dynamicclamp technique, we introduced artificial ionic currents in the PD neuron during its ongoing oscillation and found that changing the waveform parameters shift the network frequency in a manner consistent with the predicted shifts in the preferred frequency. As such, novel changes in the network frequency can be predicted by examining the effect of ionic or synaptic currents on the pacemaker neuron's waveform shape. These results demonstrate the ability to predict shifts in the network frequency in response to changes in certain parameters characterizing the pacemaker neuron waveform, regardless of how these parameters are altered.
Materials and Methods
Experimental protocols.
Adult male crabs (Cancer borealis) were purchased from local distributors and kept in aquaria filled with chilled artificial sea water until use. Before dissection, the crabs were anesthetized by putting them on ice for at least 20 min. The dissection was done using standard protocols as described in previous studies (Tohidi and Nadim, 2009). After dissection, the nervous system, including the commissural ganglia, the esophageal ganglion, the stomatogastric ganglion (STG), the nerves connecting these ganglia, and motor nerves, were pinned down in a 100 mm Petri dish coated with the silicon elastomer Sylgard (Dow Corning). The STG was then desheathed to expose the neurons for impalement. During the experiment, the whole dish was superfused with normal crab saline (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; pH 7.4) maintained at 10–13°C. The PD neurons were identified by matching their intracellular activity with the extracellular action potentials on the corresponding motor nerves.
The electrodes were prepared by using the Flaming–Brown micropipette puller (P97; Sutter Instrument) and filled with 0.6 m K_{2}SO_{4} and 0.02 m KCl. For current injection, the resistance of the electrode was kept at 10–20 MΩ; for membrane potential measurements, the resistance of the electrode was 20–35 MΩ. Extracellular recording from the motor nerves was done using a differential AC amplifier model 1700 (AM Systems) and intracellular recording was done with Axoclamp 2B amplifiers (Molecular Devices).
Measurements of the preferred frequency.
The two PD neurons are electrically coupled to the AB neuron with which they form the pyloric pacemaker kernel. We have previously shown that the measurement of preferred frequency in either of these neuron types is not influenced by the contribution from their electrically coupled partners (Tohidi and Nadim, 2009). However, it should be noted that in all manipulations in the current study, the AB neuron is present and may contribute to some of the observed results. In particular, the AB neuron is essential for the generation of the pyloric network rhythm and therefore the effects of current injections into the PD neuron on the pyloric frequency during the ongoing rhythm most probably involve indirect influences on the AB neuron.
To measure the preferred frequency, after identifying the PD neuron, we used 10^{−7} m tetrodotoxin (TTX; Biotium) to remove the neuromodulatory inputs and thus the endogenous network activity (Raper, 1979; Luther et al., 2003). The preferred frequency was measured in voltage clamp or current clamp using the software Scope (version 7.75). All experiments were done with two electrodes, one used for injecting current and the other for recording the membrane potential. In the currentclamp experiments, the injected current used was a sweepingfrequency sinusoidal (impedance amplitude profile or ZAP) function. A similar ZAP function was used in the voltageclamp experiments to clamp the membrane potential. The ZAP function was calculated as follows: where A is the amplitude of the oscillation and f(t) is a monotonically increasing function that determines the range of the sweeping frequencies (in Hz) and is defined as where F_{max} and F_{min} are the maximum and minimum swept frequencies, respectively; dur is the duration of the sweep; and L is the rate of the exponential rise in frequency (the exponential rise in frequency is used so that lower frequencies are sampled sufficiently). Because time was measured in milliseconds, the scaling factor 10^{−3} was used to convert the frequency units to Hz. All experiments were done with F_{min} = 0.1 Hz, F_{max} = 4 Hz, and dur = 100 s. To avoid transients at the beginning of the oscillations, the ZAP waveform was preceded with three cycles of a sinusoidal waveform injected at the lowest frequency F_{min}, which transitioned smoothly into the ZAP function, bringing the total duration of each sweep to 130 s. The first sinusoidal waveform was phase shifted to start at its minimum value. Henceforth, we use the term ZAP function to refer to the total injected function, including the first three sinusoidal waveforms.
In the currentclamp experiments, the amplitude of the ZAP function current was adjusted to produce a 30 mV difference in membrane potential when the PD neuron was oscillating at 0.1 Hz. To make the results comparable to those from the voltageclamp recordings, when necessary, we also injected a bias DC current to bring the baseline membrane potential to −60 mV.
The voltageclamp experiments were done in twoelectrode voltageclamp mode. In each sweep, the membrane potential was clamped first at a holding value (equal to the minimum value of the ZAP function), followed by the ZAP function. The voltage range of oscillation varied depending on the experiment. The PD neuron produces bursting oscillations at a frequency of ∼1 Hz with a slowwave profile that ranges approximately between −60 and −30 mV. We therefore focused our study around these voltage and frequency ranges (Tohidi and Nadim, 2009). In the experiments in which the effect of the lower bound was examined, the lower bound was shifted by ±10 mV (−70, −60, and −50 mV) while the upper bound was kept at −30 mV. When examining the effect of the upper bound, this bound was shifted by ±4 mV (−34, −30, and −26 mV) while the lower bound was kept at −60 mV. In a subset of experiments, we also did sweeps in which the upper bound was shifted by ±10 mV and the frequency range was between 0.1 and 10 Hz. However, setting the upper bound too high (−20 mV) resulted in a second impedance peak in the higher frequency range (>5 Hz), whereas setting the upper bound too low (−40 mV) often resulted in a shift of the peak to zero frequency (Tohidi and Nadim, 2009). We therefore limited the shift of the upper bound to ±4 mV, which resulted, on average, in a shift of f_{max} that was similar to that seen when the lower bound was shifted by ±10 mV.
The impedance profile was generated with a Matlab (MathWorks) script. For each cycle of the recording, the script measured the frequency and the amplitudes of the changes in voltage and current, and the impedance [Z(f)] was calculated as a ratio of the voltage and current amplitudes as a function of frequency. Alternatively, Z(f) can be measured as the ratio of the Fourier transforms of the voltage and current traces (Hutcheon and Yarom, 2000). The preferred frequency is defined as the frequency at which the impedance power Z(f) is maximal. In Results, below, we refer to the impedance power as impedance.
Dynamic clamp.
The Dynamic Clamp software (version 1.55) was used to produce artificial ionic conductance to shift the voltage range of an oscillating PD neuron during its ongoing network activity. The current was injected into the PD neuron in discontinuous currentclamp mode. In the upperbound shift experiments, the artificial current had a reversal potential at 0 mV and activated when the PD membrane potential was above a given threshold; in the lowerbound shift experiments, the artificial current had a reversal potential at −80 mV and activated when the PD membrane potential was below a given threshold. In each experiment, we used two different thresholds. The lower threshold was set at the voltage at which the inhibitory synapse from the LP neuron activated (∼−55 mV), whereas the higher threshold was always set 10 mV higher than the lower threshold. The dynamic clamp current was assumed to activate with firstorder activation kinetics and no inactivation: The activation variable followed the equations Here, v_{th} is the activation threshold, k was set to −1 mV to produce a sharp activation curve, and the time constant (τ_{m}) was set to a constant value of 1 ms. By adjusting the maximal conductance g_{max} (range, 5–35 nS), we could move the upper bound or the lower bound of the PD neuron membrane potential to different levels. In the upper and lower bound shift experiments, we limited the value of g_{max} so that only the upper or lower bounds, respectively, were affected. This restriction resulted in a smaller possible range of g_{max} values for the low threshold activation inward current and thus a more limited shift in the upper bound when compared with the high threshold activation current.
Generation of the representative realistic PD neuron waveforms.
The representative unitary waveforms were extracted from the experimental recordings using the Readscope software (version 7.75) as described below. Five consecutive cycles of each selected PD neuron recording were analyzed. The voltage trace was lowpass filtered at 10 Hz to reveal the slowwave oscillation. The average membrane potential in the five cycles was used as the voltage threshold to align and average these cycles and produce a single averaged waveform. The averaged waveform was sampled at 1000 points and rescaled to be between 0 and 1. Finally, the waveform was shifted so that the minimum point 0 became the start point of the waveform. These representative unitary waveforms were then used in the waveform analysis and in the experiments. The experiments in which the waveform was used to voltage clamp the PD neuron and measure the corresponding preferred frequency were performed by replacing the sine function in Equation 1 with the unitary waveform using the Scope software.
Waveform analysis.
The waveform analysis was done using Matlab scripts. The waveform duty cycle was defined as the fraction of time that the waveform was above its mean value. We also calculated the portions of waveform >75% or <25% of the peak value. The peak phase is the phase at which the maximum value (1) occurs when the minimum (0) is used as the reference point. Area refers to the area underneath the waveform curve where the duration of the waveform is normalized to 1. The various slopes were calculated from different parts of the waveform after dividing the waveform into a rising and a falling interval. The rising interval is the portion of the waveform from between the start point 0 and the peak point 1 and the falling interval is between the peak and the end point. Each interval was divided into different slope ranges. The rising slopes used were as follows: 0–100, 0–25, 25–50, 50–75, and 75–100%; the falling slopes were as follows: 100–0, 100–75, 75–50, 50–25, and 25–0%. These parameters were not meant to be independent or exhaustive descriptions of the waveforms. These were selected from among a larger set of parameters because they provide a relatively distinct and comprehensive description of the waveforms.
The software tools Scope, Readscope, and Dynamic Clamp were developed in the Nadim laboratory and are available for download at http://stg.rutgers.edu/software.
Statistical analysis.
All statistical analysis was performed in SigmaStat (version 2.03, Aspire Software International). Significance was evaluated by comparing with the α value of 0.05.
Results
A previous study from our lab showed that the pacemaker PD neuron bursting frequency, and therefore the pyloric network frequency (f_{pyloric}), is strongly correlated with the preferred frequency (f_{max}) of the PD neurons (Tohidi and Nadim, 2009). We now explore the parameters of the slowoscillation waveform underlying the bursting activity of PD neurons that affect f_{max} and examined whether these parameters also change f_{pyloric}. Our results are divided into four sets of experiments. First, we show that the preferred frequency measured in voltage clamp is statistically identical to the frequency traditionally measured in current clamp. Voltageclamp measurements allow for the precise determination of the dependence of f_{max} on the voltage range and oscillation waveform shape. Second, we demonstrate that shifts in the upper or lower limits of the PD neuron voltage range result in a shift in f_{max} in the same direction. Third, using the dynamicclamp technique, we add inward and outward voltagedependent currents to shift the upper and lower limits, respectively, of the PD neuron voltage range during its ongoing pyloric oscillation and show that these manipulations typically result in a shift in the f_{pyloric}, similar to that of f_{max}. However, the dependence of f_{pyloric} on the upper bound of the voltage waveform is sensitive to the activation threshold for the dynamicclamp current. Finally, we measure f_{max} using different prerecorded realistic PD neuron waveforms and show that there are correlations between f_{max} and certain parameters that determine the waveform shape.
Measurement of impedance in voltage clamp
The PD neuron exhibits a preferred frequency when its impedance is measured by injecting a ZAP current of fixed amplitude that allows the membrane potential to oscillate between the limits of ∼−60 and ∼−30 mV (Tohidi and Nadim, 2009). These values provide an approximation of the slowwave oscillation of the PD neuron bursting waveform during ongoing pyloric activity (Fig. 1A). Using ZAP current injection is the traditional way of measuring neuronal impedance and demonstrating the presence of resonance (Puil et al., 1986; Hutcheon and Yarom, 2000). However, in the current study, we were interested in how the properties of the PD neuron waveform, such as its voltage range and shape, affect f_{max}. To change the voltage range in a controlled manner, we measured the impedance of the PD neuron in voltage clamp (see Materials and Methods). After the TTX treatment, the PD neuron was voltageclamped with a ZAP function waveform sweeping frequencies between 0.1 and 4 Hz and the voltage range between −60 and −30 mV, and the injected current was recorded (Fig. 1B). The impedance peak in such a measurement corresponds to the frequency at which the injected current is minimal. One important question was whether the impedance measured in voltage clamp would produce the same value of f_{max} as that measured in current clamp. To address this question, we did paired current and voltageclamp impedance measurements of the PD neuron. Under currentclamp conditions, the membrane potential of the PD neuron at the lowest frequency of 0.1 Hz was matched to the voltageclamp range of −60–−30 mV (Fig. 1C). These comparisons showed that the values of f_{max} obtained under the currentclamp or voltageclamp conditions are statistically identical (Student's t test, N = 7, p = 0.742) (Fig. 1D). In all subsequent experiments described in our study, we measured f_{max} in voltage clamp.
The effect of the PD neuron voltage range on f_{max}
During the ongoing pyloric network activity, many factors may affect the upper and lower bounds of the bursting oscillations the PD neuron. These include the inhibitory synaptic input from the follower LP neuron (Thirumalai et al., 2006) (Fig. 1A) as well as the modulatory environment (Marder and Eisen, 1984; Kloppenburg et al., 1999; Thirumalai and Marder, 2002; Goaillard et al., 2004). Thus, we decided to examine possible effects of the waveform voltage range on the f_{max} value in the PD neuron. The waveform voltage range can be altered by shifting the upper bound, the lower bound, or both. We first explored the change in the upper bound while keeping the lower bound at a fixed voltage of −60 mV. As our control value, we used an upper bound voltage of −30 mV and then increased or decreased this upper bound by 4 mV (Fig. 2A1, top). In a preliminary set of experiments, we also examined the effect of changing the upper bound by 10 mV and a wider frequency range (0.1–10 Hz), but decided to use the smaller range for reasons explained in the Materials and Methods section.
An example of the impedance profiles for these three different upper bounds is shown in Figure 2A1. When the upper bound was shifted to a higher value of −26 mV, the average f_{max} increased by ∼27% compared with the control value. Similarly, when the upper bound was shifted to a lower value of −34 mV, the average f_{max} decreased by ∼23% compared with the control. Figure 2A2 summarizes this effect for 15 experiments. showing that changing the upper bound of the voltage range significantly shifts f_{max} in the same direction (N = 15; oneway ANOVA; p < 0.001).
Adjusting only the lower bound of the oscillation had a similar effect on f_{max}. An example of the impedance profiles for three different lower bounds is shown in Figure 2B1. When the lower bound was increased from −60 mV to −50 mV, f_{max} increased by ∼18% compared with the control value. Likewise, a decrease of the lower bound to −70 mV resulted in a shift of f_{max} by ∼30% compared with control (N = 12; oneway ANOVA; p < 0.05) (Fig. 2B2).
To examine which is the more effective way of shifting the preferred frequency, we compared the sensitivity of f_{max} to variations of the upper and lower bounds of the voltage range. We did this by comparing the total change in f_{max} value divided by the voltage change during the upper or lower bound shifts. We found that changing the upper bound of the voltage range was significantly more effective in changing the f_{max} value (upper bound Δf_{max}/ΔV = 0.07 ± 0.044 Hz/mV; lower bound Δf_{max}/ΔV = 0.02 ± 0.017 Hz/mV; p = 0.001, Student's t test).
Finally, we shifted both upper and lower bounds by 4 mV in the same direction. An example of the frequencydependent profiles for these different voltage windows is shown in Figure 2C1 (top). Note that with this manipulation, the amplitude of the membrane potential oscillations remains 30 mV in all three cases. Shifting the voltage range up by 4 mV, f_{max} increased by ∼29% and a downward shift of 4 mV resulted in an ∼25% decrease of f_{max} (N = 9; oneway ANOVA; p < 0.001) (Fig. 2C2). The change in f_{max} due to the upward shift of the voltage range was not significantly different from the change from increasing the upper bound alone (Student's t test, p = 0.876; increase in the upper bound by 4 mV, Δf_{max} = 26.9 ± 25.5%, N = 15; shifting up the voltage range by 4 mV, Δf_{max} = 28.5 ± 18.9%, N = 9). It is possible that the lack of difference is due to the low sensitivity of f_{max} to the lower bound of the oscillation, such that moving the lower bound up by 4 mV in the shift experiment does not produce a significant additional effect. In general, the f_{max} in the PD neuron shifted in the same direction with the voltage range of oscillation; the higher the voltage range, the higher the f_{max}.
The effect of the PD neuron voltage range on the pyloric frequency
The preferred frequency f_{max} of the PD neuron has been shown to be correlated with the pyloric network frequency (Tohidi and Nadim, 2009). Our data showed that when the PD neuron was driven with an oscillatory stimulus, the measured f_{max} was influenced by the upper and lower bounds of the membrane potential oscillations. We therefore posed the following hypothesis: Factors that shift the preferred frequency (f_{max}) in one direction also change the ongoing pyloric frequency (f_{pyloric}) in the same direction. To examine this hypothesis, we first explored whether modifications in the voltage range of the oscillating PD neuron during the ongoing rhythm had an effect on f_{pyloric}. Our prediction was that shifting upward the upper or lower bound of the PD neuron voltage range during the ongoing pyloric oscillation would lead to an increase in f_{pyloric}, consistent with the effects of the voltage range on f_{max}.
To change the upper bound of the voltage in an oscillating PD neuron during an ongoing pyloric activity, we used the dynamicclamp technique to generate an artificial voltagegated inward current (I_{dynamic}) that activated when the membrane potential of the PD neuron was above a predetermined threshold. Because, in different preparations, PD neurons have slightly different voltage ranges in their ongoing oscillation, we needed to decide on a voltage threshold that was tied to some feature of the PD neuron waveform. One prominent feature of the PD neuron oscillation waveform is the IPSPs it receives from the follower LP neuron (Fig. 1A). Thus, we used the voltage at the onset of these IPSPs (referred to as the low threshold, V_{low}) (Fig. 1A, arrow) as a reference voltage and the activation threshold for I_{dynamic} was set to be at this value. Because of the sharp activation threshold of I_{dynamic}, only the upper bound of the PD neuron voltage range was increased whereas the lower bound remained unchanged (Fig. 3A1). By adjusting the maximal conductance value of I_{dynamic}, we were able to push the upper bound of the PD oscillation to different voltage levels. In each run, we injected the artificial voltagegated current into the PD neuron during the ongoing oscillation for 20 s and compared the resulting f_{pyloric} and voltage ranges before and during injection. The traces in which the lower bound of the voltage range was also shifted and were discarded. As predicted, the inclrease in the upper bound of the PD oscillation resulted in higher pyloric network frequencies (N = 6; r = 0.45; p < 0.05) (Fig. 3A2).
Similarly, in separate sets of runs, we changed the lower bound of the voltage range of the oscillating PD neurons by injecting an artificial dynamicclamp outward current that activated when the membrane potential was below a predetermined threshold (also set to V_{low}). In this case, when the artificial current was activated, the lower bound of the PD oscillation was more hyperpolarized, whereas the upper bound remained unchanged (Fig. 3B1). Different values of the maximal conductance of the dynamic clamp current resulted in different lower bound values. Again, as expected, when the lower bound of the oscillation was shifted to more negative values, f_{pyloric} also decreased (N = 6; r = 0.61; p < 0.01) (Fig. 3B2). Overall, these results showed that changing the upper or lower bounds of the voltage range of the oscillating PD neuron shifted the f_{pyloric} in the same direction that it shifted f_{max} in the previous set of experiments, consistent with our abovestated hypothesis.
The decision to set the threshold of the dynamic clamp current to V_{low} was arbitrary. To see whether the value of the threshold voltage affected our results, we also examined the effect of dynamicclamp currents with a different threshold value that was 10 mV higher than the V_{low} (the high threshold, V_{high}). Examples of low versus highthreshold dynamicclamp inwardcurrent injections are shown in Figure 4, A and B. The inward dynamicclamp current with threshold at V_{high} also effectively shifted the upper bound of the PD neuron oscillation to higher values without changing the lower bound. Interestingly, unlike the current injection with threshold at V_{low}, the highthreshold current injection did not change f_{pyloric} despite increasing the upper bound (N = 6; r = 0.06; p = 0.67) (Fig. 4C).
We also examined the effect of the threshold voltage in adjusting the lower bound of the PD oscillation with an outward artificial current. Once again, dynamicclamp current injection could effectively decrease the lower bound of the PD oscillation without changing its upper bound, independent of the voltage threshold. However, in this case, the highthreshold dynamicclamp current acted similarly to the lowthreshold current and decreased f_{pyloric} (N = 6; r = 0.35073; p = 0.01817).
The difference between the effects of the dynamicclamp inward current with low or high thresholds in the upperbound shift experiments indicated that the PD neuron voltage range was not the only factor that affected f_{pyloric}. Because the dynamicclamp current, in addition to the voltage range, also changes the voltage waveform of the PD neuron, it is possible that some other properties of the waveform affected f_{pyloric} in a way that could potentially oppose the shift due to increasing the upper bound. If so, according to our abovestated hypothesis, changes in the PD neuron waveform (that do not affect its voltage range) may also result in shifts of its preferred frequency f_{max}.
The effect of the PD neuron waveform on f_{max}
To explore the effect of the PD neuron waveform properties on f_{max} required waveforms with distinct properties. One possibility was to use artificial waveforms with distinct properties (e.g., different duty cycles, rise and fall slopes, etc.). Instead, we took advantage of the fact that the PD neurons in different preparations produce different waveforms during their natural oscillation. We selected seven PD neuron oscillation waveforms from different preparations and generated a group of representative realistic unitary waveforms from these recordings. To produce the unitary waveforms, each selected PD recording was filtered at 10 Hz to remove the action potentials (Fig. 5A) and the representative realistic waveforms were extracted from each of these filtered recordings (see Materials and Methods) (Fig. 5B, left). These waveforms differed in their duty cycle and slopes of rise and fall as well as other parameters (Table 1; Fig. 5B, right).
Each of the seven waveforms was used as a unitary waveform for generating a ZAP function (see Materials and Methods) and, in subsequent experiments, each PD neuron was voltage clamped with all these ZAP functions (0.1–4 Hz) as well as the standard sinusoidal ZAP. Two examples of the traces recorded from the PD neuron voltage clamped with the realistic waveform ZAP function are shown in Figure 5C. We calculated the f_{max} as described above for all seven representative realistic waveforms and the standard sinusoidal ZAP. In these experiments, we were mainly interested in the effect of the waveform shape on F. A statistical analysis of all values measured indicated that this increase was highly significant (N = 7; oneway ANOVA, p < 0.001).
What features of the PD neuron waveform are correlated with f_{max}?
The representative realistic waveforms have different properties, including their duty cycle, the peak phase, the portions above or below certain threshold, and the slopes in different sections of the waveform. We were interested in which, if any, of these waveform characteristics correlated with the value of f_{max}. For each waveform, we measured values of several characteristic parameters (see Materials and Methods) (Table 1). Surprisingly, most of these parameters showed only a weak correlation with the normalized f_{max} (Fig. 6; Table 2). For example, the duty cycle (r = 0.18, p = 0.17) (Fig. 6A) and the peak phase (r = 0.08, p = 0.58) (Fig. 6B) had almost no correlation with f_{max}. However, a few specific properties did stand out. In particular, the rising slope at 75–100% of the waveform amplitude showed a strong negative correlation (r = −0.58) (Fig. 6J) with the most statistically significant p value (<0.0001). In addition to the 75–100% rising slope, the total waveform area and the portion >75% amplitude of the waveform also showed good correlation with f_{max} (area, r = 0.43, p < 0.05; portion >75%, r = 0.45, p < 0.05) (Fig. 6C,D). Altogether, the analysis of the waveform shape provided additional information on factors other than the upper and lower voltage range that affected the f_{max} value. In particular, when the top portion of the PD oscillation waveform was steeper, the PD neuron will had a smaller f_{max} value.
The PD neuron waveform shape in the ongoing pyloric oscillation is affected by dynamicclamp currents
To examine whether any of the waveform parameters that showed a good correlation with f_{max} were predictive of the changes seen in f_{pyloric} during the dynamicclamp experiments, we measured how these parameters were affected by the dynamicclamp currents. To analyze how the properties of the waveforms were changed by the low or highthreshold dynamicclamp current injection in the upperbound shift experiments, we compared the PD neuron waveforms during the current injection with those before the current injection in the same experiment. This comparison required us to generate representative waveforms for each recording, as described above. Examples of how the dynamicclamp current injection altered the slowoscillation waveform are shown in Figure 7, A (lowthreshold) and B (highthreshold).
We next explored how the shifts in the lower or upper bound of the voltage range during the dynamicclamp experiments affected the three waveform parameters that showed a good correlation with f_{max}: the 75–100% rising slope, the total waveform area, and the portion of the waveform >75% amplitude. The effect of the dynamicclamp currents was measured as the ratio of these values during the current injection to the values before injection in the same experiment and plotted against the shift in the upper bound of the voltage range. A ratio higher or lower than 1 means that the characteristic parameter increased or decreased, respectively, during current injection.
With the lowthreshold dynamicclamp current, the 75–100% rising slope showed a slight decrease as the upper bound of the voltage range increased (r = −0.36, p < 0.05) (Fig. 7C). In contrast, with highthreshold injection, the slope increased as the upper bound was increased (r = 0.39, p < 0.05) (Fig. 7C). The other two waveform parameters examined, total area and portion >75% amplitude, increased with both low and highthreshold dynamicclamp current injections, but the values for the lowthreshold current were higher (Fig. 7D,E). The difference in the effects of the waveform parameters, especially the 75–100% rising slope, between the low and highthreshold experiments could provide a possible explanation for the distinct effects on f_{pyloric} in the upperbound shift experiments.
Discussion
A fundamental characteristic of oscillatory activity is its frequency. Different behavioral needs of the animal often require network oscillations to be active at different frequencies (Tryba et al., 2008); as such, it is important to know what factors determine this frequency. Modeling results show that many ionic currents can shape the preferred frequency of a neuron (Tohidi and Nadim, 2009) and even more currents potentially underlie oscillatory activity (SotoTreviño et al., 2005). It is experimentally difficult, if not impossible, to measure every one of these ionic currents in the same neuron to determine their effects on the oscillation frequency. Measuring the membrane potential waveform, on the other hand, is relatively easy to do. The correlations between the parameters of waveform and the preferred frequency provide a new tool to examine the effects of neuronal parameters on the preferred frequency. Reducing from the complexity of ionic current dynamics to a few parameters of the voltage waveform makes it more feasible to predict shifts in the preferred frequency, which can be predictive of the neuron's ongoing oscillation frequency.
We explored the parameters of the slowoscillation waveform underlying the bursting activity of PD neurons, members of the crab pyloric pacemaker group, that affect their preferred frequency (f_{max}) and examined whether these parameters also change the network frequency (f_{pyloric}). We found correlations between f_{max} and certain parameters that determine the waveform shape, including the lower and upper bounds of the voltage range and the peak rising slope. Shifting the voltage bounds with dynamic clamp during an ongoing oscillation produced shifts in f_{pyloric} similar to those predicted by changes in f_{max}. The dependence of f_{max} on the PD neuron waveform indicates that changes in the network frequency are reflected in corresponding changes in the waveform voltage range and other parameters.
The effect of voltage range on preferred frequency
Traditionally, preferred frequency is measured by injecting a ZAPfunction current into a neuron to record its membrane potential, which results in membrane potential oscillations that vary in voltage range and amplitude (Puil et al., 1986; Hutcheon and Yarom, 2000). We measured the preferred frequency of the PD neuron, with both current and voltageclamp methods, using a range of membrane potential oscillations (−60 to −30 mV) similar to the slowwave of the bursting activity, and found similar values of f_{max} with both methods (Fig. 1). The voltageclamp measurements demonstrated that f_{max} increases when the PD neuron oscillates with a higher voltage range, consistent with the effect of a DCbias current (Tohidi and Nadim, 2009). In other systems, depolarizing the membrane potential may shift the f_{max} to higher (Gutfreund et al., 1995) or lower values (Hutcheon et al., 1996), but it is the presence of such a change that is important, not its direction.
The traditional description of membrane resonance involves the combination of the passive resistive and capacitive properties of the membrane together with at least one voltagedependent current acting as an inductor (Hutcheon and Yarom, 2000). This standard resistor–inductor–capacitor circuit description of membrane resonance describes a linear phenomenon, whereby the voltage response scales with the amplitude of the input current and, therefore, the resonance frequency does not shift. Our results do not support this linear model. The significantly higher sensitivity of f_{max} to the shifts in the upper bound compared with the lower bound provides support for the nonlinearity of this phenomenon. Different ionic currents (e.g., I_{Ca} and I_{h}) are often activated in the upper and lower envelopes of the voltage oscillation, which may affect the value of f_{max} differently (Tohidi and Nadim, 2009). These currents have different activation properties, which may result in different sensitivities to the same amount of membrane potential change.
Conversely, it is important to understand which ionic currents shape the upper and lower limits of the voltage range. In a PD model neuron, increasing the conductance of I_{h} moves the baseline of the membrane potential to a more depolarized level and increases f_{max} (Tohidi and Nadim, 2009). Consistent with these results, our experiments show that an increase (i.e., depolarization) in the lower bound of the voltage range results in a higher f_{max}. Another factor affecting the membrane potential in pyloric neurons is neuromodulation. Some modulators, such as dopamine, hyperpolarize the PD neuron whereas others, such as octopamine and pilocarpine, depolarize it (Marder and Eisen, 1984; HarrisWarrick et al., 1998; Kloppenburg et al., 1999; Goaillard et al., 2004). Most neuromodulators have an effect on the pyloric network frequency, but the influence of neuromodulators on the preferred frequency of pyloric neurons remains to be explored. The inhibitory LPtoPD synapse is the only chemical feedback synapse from pyloric follower neurons to the pacemakers (Manor et al., 1997). The strength of the LPtoPD synapse can be changed by neuromodulators, resulting in smaller or larger IPSPs and therefore moving the lower bound of the oscillation voltage range to a different value (Ayali et al., 1998; Thirumalai et al., 2006). Based on our results, we predict that this inhibitory synapse also has a role in controlling the frequency. However, previous studies have shown that blocking or enhancing this synapse (with neuromodulators) can have little effect on the pyloric network frequency (Nadim et al., 1999; Mamiya and Nadim, 2004; Thirumalai et al., 2006). One explanation is that the IPSP generated by the LP neuron under these situations is not large enough to produce an observable shift when it is removed or enhanced during the oscillation, particularly because much of the LPinduced IPSPs in the PD neuron are transient (Fig. 1A) and the dependence of f_{max} on the lower bound of the voltage range is significantly less sensitive than the upper bound (see Results). Alternatively, it is possible that other changes in the waveform shape of the PD neuron counteract the effect of the shift in the lower bound, resulting in a limited change in both f_{max} and f_{pyloric}.
Waveform shape and the determination of the network frequency
The relationship between the parameters of waveform and the dynamics of ionic currents is bidirectional. For example, overexpression of I_{h} in the PD neuron changes its bursting waveform by increasing the duty cycle and number of action potentials per burst (Zhang et al., 2003), whereas the Acurrent delays the postinhibitory rebound (Tierney and HarrisWarrick, 1992), resulting in different delay times before the burst in pyloric constrictor neurons (Hooper, 1998; Zhang et al., 2008). The shape of the waveform also affects the dynamics of the voltagegated ionic and synaptic currents. In leech heart interneurons, for example, voltage waveforms with different rise slopes produce different amounts of I_{Ca} and I_{h} (Olsen and Calabrese, 1996). When the waveform is altered, the resulting changes in the dynamics of ionic currents in turn change f_{max}. Similarly, distinct realistic voltage waveforms in the pyloric network can result in different synaptic outputs, presumably due to their effect on the kinetics of presynaptic Ca^{2+} currents (Mamiya et al., 2003; Rabbah and Nadim, 2007).
Because f_{max} results from the dynamics of ionic currents, it can be assumed that it also depends on parameters that affect the neuron's waveform shape. The seven realistic PD neuron waveforms, when applied in the same voltage range, each produced a somewhat different value of f_{max}. We ordered these waveforms according to their effect on f_{max} and examined how different waveform parameters correlated with these f_{max} values. Our goal was not to find independent parameters but to focus on those that provided intuitive distinctions between the waveforms. Of the 15 parameters examined, several were correlated with f_{max}, but one, the 75–100% rising slope, showed the highest significance value and largest (negative) correlation.
We therefore focused on two important waveform characteristics that affected f_{max}: the voltage range of oscillation and the 75–100% slope. We asked whether a combination of these two parameters explains the effect of the dynamicclamp current on f_{pyloric}. During ongoing oscillations, the dynamicclamp outward current shifts the lower bound of oscillations and results in a lower value of f_{pyloric}, as predicted by the effect of the lower bound on f_{max} (Fig. 3B), independent of its threshold. The dynamicclamp inward current also affected f_{pyloric} by shifting the upper bound of oscillations, but this effect was only present with a low but not highthreshold current. Similar shifts in the upper bound of oscillations may occur in the biological setting as actions of low or highthreshold activated inward currents, both of which have been shown to be activated in pyloric neurons by neuromodulators. For example, the modulatoractivated inward current I_{MI}, which is activated by a variety of peptides, has a low activation threshold (Golowasch and Marder, 1992; Zhao et al., 2010), whereas the calcium currents, which are subject to modulation by dopamine, have a much higher activation threshold (Johnson et al., 2003).
An examination of the PD waveform shape showed that the high (but not low) threshold current significantly increased the 75–100% slope of the waveform. Whereas the increase in the upper bound will push f_{max} to a higher value, the increase in the 75–100% slope will decrease it. The combined effect of these two factors could produce little or no shift in the preferred frequency and, as a result, the network frequency, which is correlated with f_{max}, would not change. Two other waveform parameters, total area and the portion >75%, that correlated positively with f_{max} increased with changes in the upper bound in both the low and highthreshold current experiments. However, the values for the lowthreshold experiments were higher, predicting a higher f_{max}, which was consistent with the higher value of f_{pyloric} measured in these experiments.
Conclusions
In other oscillatory systems, the network frequency is also similar to the preferred frequency of the neuron within the network (Leung and Yu, 1998; D'Angelo et al., 2001), although few studies have shown a direct correlation (Erchova et al., 2004; Schreiber et al., 2004). If the preferred frequency can be predictive of ongoing oscillations, it may be possible to extend our methodology to understand how parameters that shape the preferred frequency affect the network oscillation in many other systems. It should be noted, however, that the extent to which the parameters of one type of oscillatory neuron shape the network frequency may become more or less influential as a result of parallel changes in other neuron types in the network.
Footnotes

This work was supported by National Institutes of Health Grant MH60605. We gratefully acknowledge Jorge Golowasch and Isabel Soffer for their critical comments on the manuscript.
 Correspondence should be addressed to Farzan Nadim, Department of Mathematical Sciences, New Jersey Institute of Technology, 323 Martin Luther King Boulevard, Newark, NJ 07102. farzan{at}njit.edu