## Abstract

Oscillatory networks often include neurons with membrane potential resonance, exhibiting a peak in the voltage amplitude as a function of current input at a nonzero (resonance) frequency (*f*_{res}). Although *f*_{res} has been correlated to the network frequency (*f*_{net}) in a variety of systems, a causal relationship between the two has not been established. We examine the hypothesis that combinations of biophysical parameters that shift *f*_{res}, without changing other attributes of the impedance profile, also shift *f*_{net} in the same direction. We test this hypothesis, computationally and experimentally, in an electrically coupled network consisting of intrinsic oscillator (O) and resonator (R) neurons. We use a two-cell model of such a network to show that increasing *f*_{res} of R directly increases *f*_{net} and that this effect becomes more prominent if the amplitude of resonance is increased. Notably, the effect of *f*_{res} on *f*_{net} is independent of the parameters that define the oscillator or the combination of parameters in R that produce the shift in *f*_{res}, as long as this combination produces the same impedance vs. frequency relationship. We use the dynamic clamp technique to experimentally verify the model predictions by connecting a model resonator to the pacemaker pyloric dilator neurons of the crab *Cancer borealis* pyloric network using electrical synapses and show that the pyloric network frequency can be shifted by changing *f*_{res} in the resonator. Our results provide compelling evidence that *f*_{res} and resonance amplitude strongly influence *f _{net}*, and therefore, modulators may target these attributes to modify rhythmic activity.

- resonance
- oscillations
- stomatogastric
- computational modeling
- dynamic clamp

## NEW & NOTEWORTHY

In many oscillatory systems, the membrane potential resonance frequency (f._{res}) of some network neuron types is correlated with the network frequency. With the use of theoretical and experimental methods, we show, for the first time, a causal relationship between the f_{res}of a resonator neuron and the frequency of an electrically coupled network. This relationship is independent of individual parameters of the neurons and only depends on the impedance of the resonator as a function of input frequency

many behaviors and sensory states are associated with rhythmic activity in the central nervous system, generating interest in mechanisms governing oscillations in neuronal networks. A number of such mechanisms have been identified for the generation of rhythmic oscillations. An exploration of these mechanisms points to a variety of interacting factors that influence the network frequency (*f*_{net}), including the intrinsic properties of the individual oscillators and the network connectivity. This raises the question of whether such factors, independent of their origin, interact in a coherent manner. Recent work has shown that oscillatory network activity can be correlated with membrane potential resonance of participating neurons, a property that arises through interactions of multiple ionic currents. Membrane potential resonance is the ability of neurons to produce a maximal voltage response to oscillatory inputs at a preferred nonzero input frequency [resonance frequency (*f*_{res})], a property found both in neurons that are intrinsically oscillatory, i.e., produce stable subthreshold or slow-wave membrane potential oscillations, and in neurons that are not (Hutcheon and Yarom 2000; Richardson et al. 2003).

The *f*_{res} of various neuron types has been shown to fall in the frequency range of the oscillatory networks in which they are embedded (Leung and Yu 1998; Moca et al. 2014; Tohidi and Nadim 2009; Wu et al. 2001). A prominent example is the theta oscillations observed in the hippocampus, neocortex, and medial entorhinal cortex (Buzsaki 2002; Buzsaki and Draguhn 2004; Colgin 2013). Hippocampal pyramidal cells, medial entorhinal cortex layer II stellate cells, and neocortical cells exhibit theta *f*_{res} at theta frequencies generated by various combinations of ionic channels (Engel et al. 2008; Erchova et al. 2004; Gastrein et al. 2011; Hu et al. 2002, 2009; Hutcheon et al. 1996; Leung and Yu 1998; Pike et al. 2000). However, to our knowledge, a causal relationship between the *f*_{res} and *f*_{net} has not been established. The goal of this paper is to address this issue both theoretically and experimentally in a simple neuronal network consisting of electrically (gap junction) coupled neurons, where one of the neurons is a resonator, and the other is an intrinsic oscillator and generates the network oscillations.

Resonance arises as a consequence of the interaction between voltage-gated ionic currents and passive properties and manifests itself as a peak in the impedance profile (the impedance amplitude *Z* plotted vs. the input frequency *f*) (Hutcheon et al. 1996; Hutcheon and Yarom 2000). Different combinations of ionic currents can give rise to similar impedance profiles. For example, a resonant peak in the impedance profile can arise either through the actions of a slow inward current, such as the h current (*I*_{h}), or a slow outward current, such as the potassium M current (Gutfreund et al. 1995; Haas and White 2002; Rotstein and Nadim 2014). It would therefore be informative to test whether the influence of resonator neurons on network rhythms is due to the properties of these ion channels or due to the properties of the impedance profile, independent of which ion channels give rise to this profile. To this end, we characterize the resonant neurons in terms of *f*_{res}, which depends on one or more combinations of the biophysical model parameters. The resonator we use in this study exhibits membrane potential resonance but not necessarily persistent oscillations (Rotstein and Nadim 2014) or spiking resonance [a detailed comparison between subthreshold and spiking resonance can be found in Beatty et al. (2015); see also Stark et al. (2013)].

A number of electrically coupled rhythmic networks consist of both neurons that produce subthreshold or bursting oscillations (when isolated) and nonoscillators. A subset of both neuron types can be resonators. In the inferior olive, synchronous rhythmic activity with a frequency of 5–10 Hz arises from electrical coupling among the constituent neurons (Kitazawa and Wolpert 2005; Leznik and Llinas 2005) that have a range of *f*_{res} similar to the *f*_{net} (Lampl and Yarom 1997), but the neurons involved are not all oscillatory (Manor et al. 1997). A similar oscillatory activity arises through the cerebellar cortex Golgi neurons, which are extensively connected through electrical coupling (Dugue et al. 2009). Oscillations in central pattern generator networks also often involve electrical coupling among pacemaker neurons and neurons that are not intrinsically oscillatory. The crustacean pyloric network, for example, produces oscillations of ∼1 Hz, generated by a pacemaker group of strongly, electrically coupled neurons that show resonance and whose *f*_{res} is strongly correlated to the *f*_{net} (Tohidi and Nadim 2009).

To address rigorously the question of the influence of resonance on network oscillations through electrical coupling, the following would be necessary: to *1*) vary the *f*_{res} and other attributes that determine the shape of the impedance profile, independently of the biophysical parameters that produce them; *2*) produce the same impedance profiles using different parameter sets, and *3*) explore the influence of resonance on a number of different oscillatory mechanisms. We explore this problem using a two-cell model network consisting of an oscillator neuron O coupled to a resonator neuron R. For the oscillator, we use a Morris-Lecar model with an additional *I*_{h} (Gutierrez et al. 2013).

In biophysical model resonator neurons, there is no simple dependence of *f*_{res} or other attributes of the impedance profile on the model parameters. However, in a recent theoretical study, we showed that the attributes of the impedance profile, such as *f*_{res}, maximum impedance (*Z*_{max}), resonance amplitude [*Q*_{Z} = *Z*_{max} − *Z*_{0}], half-width of the impedance profile, etc., can be calculated as a function of the model parameters for a linear resonator (Rotstein and Nadim 2014). We then used these calculations to construct level sets in the model parameter space for each of the important attributes that characterizes the impedance profile; that is, we found how combinations of the model biophysical parameters can be covaried to keep any single attribute unchanged. Given this knowledge, we can use a linear resonator R in the current study to control the resonance properties precisely and to change them along these level sets.

With the use of the two-cell O–R electrically coupled network, we explore the influence of *f*_{res} and *Q*_{Z} of the model neuron R, as well as the electrical coupling strength on the *f*_{net}, and examine the hypothesis that shifts in *f*_{res} result in a similar shift in the *f*_{net} and that the effect of *f*_{res} on the *f*_{net} becomes larger if *Q*_{Z} is larger. Finally, we examine the predictions of this model experimentally in the crab pyloric network using the dynamic clamp technique (Sharp et al. 1993). To our knowledge, our results show for the first time a causal relationship between *f*_{res} of neurons and the frequency of the network in which they are embedded.

## MATERIALS AND METHODS

#### Model neurons.

We examined the influence of *f*_{res} on oscillation frequency through electrical coupling in a two-cell model network consisting of an oscillator O and a linear resonator R. The oscillator O was modeled as a Morris-Lecar neuron, modified by adding an *I*_{h} (Gutierrez et al. 2013), as described by the following equations
(1)

The current balance equation includes a leak current, persistent calcium current with instantaneous activation, noninactivating potassium current, and *I*_{h}, which are modeled using the Hodgkin-Huxley formalism.

*I*_{leak} = *g*_{leak}(*V*_{O} − *E*_{leak})

*I*_{Ca} = *g*_{Ca}*m*_{∞}(*V*_{O})(*V*_{O} − *E*_{Ca})

*I*_{K} = *g*_{K}*w*(*V*_{O} − *E*_{k})

*I*_{h} = *g*_{h}*H*(*V*_{O} − *E*_{h})

For *x* = *m*, *w*, *H*,

Additionally,

and

In some examples (e.g., in Fig. 1), we used a reference oscillator with *f*_{O} = 1 Hz, using the following parameter set adapted from (Gutierrez et al. 2013): *I*_{ext} = 0, *C* = 1 nF; (in mV): *E*_{leak} = −40, *E*_{Ca} = 100, *E*_{K} = −80, *E*_{h} = −20; (in μS): *g*_{leak} = 0.0001, *g*_{Ca} = 0.0025, *g*_{K} = 0.042, *g*_{h} = 0.025; and (in mV): *v*_{m} = 0, *h*_{m} = −10, *v*_{w} = 0, *k*_{w} = −7.5, *v*_{H} = −78, *k*_{H} = 10.5, *v*_{tw} = 0, *k*_{tw} = 30, *v*_{tH} = −42, *k*_{tH} = −87. Note that the choice of this model was arbitrary, and as we show in the results, the particular parameters of the model do not affect any of our findings. We also explored other model oscillator types and found that the choice of oscillator produces no qualitative change in the results. For the sake of clarity, we have not included these data.

The term *I*_{elec} = *g*_{elec}(*V*_{O} − *V*_{R}) denotes the electrical coupling current, and *g*_{elec} is the (bidirectional) gap junction conductance used to couple electrically O and R.

The resonator R was modeled as a linear neuron (Richardson et al. 2003; Rotstein and Nadim 2014). For most of the analysis (unless otherwise mentioned) and the dynamic clamp experiments (see below), a two-dimensional (2D) model was used (2)

Here, *V*_{R} = *V*_{R0} − *V**, and *W*_{R} = *w*_{R0} − *w**, indicating that the neural model is linearized around a steady state (resting value) of voltage and the recovery variable: (*V**, *w**) (Rotstein 2014). The capacitance *C*_{R} was kept fixed at 1 nF. In some examples (e.g., in Fig. 1), we used a reference resonator with *f*_{res} = 1 Hz, using the following parameters: *g*_{L} = 5.582 nS, *g*_{1} = 6.918 nS, *τ*_{1} = 236 ms.

Note that for a linear system, the shifting of the resting voltage (*V**) has no effect on the impedance profile. Clearly, the resting voltage *V** of R can be adjusted to set the frequency of the coupled R–O network. For instance, a depolarized value of *V** (compared with the oscillation voltages of O) would speed up the oscillation through the electrical coupling of R and O. This effect is well known and is not of interest for our study. Instead, we were interested in knowing if shifting the *f*_{res} (while keeping all other impedance attributes constant) has an effect on the oscillation frequency. Therefore, we always chose a value of *V** to minimize its impact on the oscillations. In the R–O model network, we chose *V** to be equal to the mean value of the oscillation voltage of O (*V*_{O}) in isolation (Fig. 1). A similar adjustment was made in the dynamic clamp experiments, as described below.

We chose a linear resonator for our simulations, because for such a resonator, described in *Eq. 2*, the *f*_{res} and the impedance amplitude *Q*_{Z} = *Z*_{max} − *Z*_{0} [where *Z*_{max} = *Z*(*f*_{res}) and *Z*_{0} = *Z*(0)] can be varied independently and calculated directly by changing the parameters *g*_{L}, *g*_{1}, and *τ*_{1} (Fig. 2) (Rotstein 2014). Note, however, that changing *f*_{res} or *Q*_{Z} does affect other attributes associated with the impedance profile (e.g., the half-width of the profile). Although these effects are unavoidable, we chose the parameters so as to minimize such additional changes in the profile.

In results, where we required multiple linear resonators with distinct parameter values, we used a 3D linear resonator. Multiple combinations of parameters in a 3D model can be used to generate very similar impedance profiles. This cannot be achieved in a 2D model because there are fewer degrees of freedom. The 3D linear resonator is given by (3)

The parameters *g*_{L}, *g*_{1}, *g*_{2}, *τ*_{1}, and *τ*_{2} were allowed to vary to adjust the impedance profile, whereas as in the 2D case, the capacitance *C*_{R} was kept fixed at 1 nF. When we used different parameter sets in the 3D resonator to produce similar impedance profiles, the impedance (*Z*) values were required to match at five points along the impedance profile. These points were chosen to be the two extreme points, 0 and 5 Hz; the *f*_{res}; and the two equidistant points between *f*_{res} and the extreme points [*f*_{res}/2 and (*f*_{res} + 5)/2].

For a linear resonator, it is not necessary to use a simulation protocol, injecting sinusoidal or chirp currents, to measure the impedance profile *Z*(*f*) and the associated attributes (such as *f*_{res} and *Z*_{max}). These attributes can be calculated directly and explicitly from the model parameters, as described in the Appendix of Rotstein and Nadim (2014). The calculations for 3D linear systems are shown in Richardson et al. (2003).

#### Experimental protocols.

Adult male crabs (*Cancer borealis*) were purchased from local seafood markets and kept in aquaria filled with chilled (12°C), artificial sea water until use. Before dissection, crabs were anesthetized by placing on ice for 30 min. The dissection was performed using standard protocols, as described previously (Blitz et al. 2004; Tohidi and Nadim 2009). After dissection, the stomatogastric nervous system, including the commissural ganglia, esophageal ganglion, and stomatogastric ganglion—the nerves connecting these ganglia and motor nerves—were pinned down in a 100-mm petri dish coated with clear silicon elastomer (Sylgard 184; Dow Corning, Auburn, MI). The stomatogastric ganglion was then desheathed to expose the neurons for impalement. During the experiment, the stomatogastric nervous system was superfused with normal *C. borealis* saline (11 mM KCl, 440 mM NaCl, 13 mM CaCl_{2}·^{2}H_{2}O, 26 mM MgCl_{2}·^{6}H_{2}O, 11.2 mM Trizma base, and 5.1 mM maleic acid, pH 7.4–7.5), kept at 10–13°C. The pyloric dilator (PD) and lateral pyloric (LP) neurons were identified by matching their intracellular activity with the extracellular action potentials on the corresponding motor nerves.

Intracellular recordings were done using Axoclamp 900A amplifiers (Molecular Devices, Sunnyvale, CA). Intracellular electrodes were prepared by using a P97 Flaming/Brown micropipette puller (Sutter Instrument, Novato, CA) and filled with 0.6 M K_{2}SO_{4} and 20 mM KCl solution. The resistance of the electrode was kept at 10–25 MΩ. Extracellular recordings from the motor nerves were done using a Differential AC Amplifier Model 1700 (A-M Systems, Carlsborg, WA), obtained with 0.005 in. stainless-steel wire electrodes inserted inside and outside of a small petroleum jelly well built on a section of the nerve.

The two PD neurons are anatomically and functionally indistinguishable; they exhibit similar intrinsic properties and make and receive similar synaptic connections (Eisen and Marder 1984; Rabbah and Nadim 2007). After both PD neurons and the LP neuron were identified, one PD neuron was impaled by two electrodes: one used for current injection and the other for recording the membrane potential. The other PD neuron was impaled by one electrode to inject the same current simultaneously, which allowed for the control of the two electrically identical PD neurons with only three electrodes (Rabbah and Nadim 2005, 2007). When necessary, a small direct current was injected into both PD neurons to bring the baseline *f*_{net} to 1 Hz. In addition, the LP neuron was impaled by one electrode and injected with a hyperpolarizing current (−10 nA) to remove the LP to PD synapse functionally (Nadim et al. 2011).

#### Linear resonator parameters.

A 2D linear resonator (*Eq. 2*) was coupled to the PD neurons through dynamic clamp electrical coupling. The linear resonator, as used in the dynamic clamp software, was defined by
(4)

where *C* is the capacitance; *v*_{R} and *w*_{R} are the voltage and recovery variables of the resonator neuron, respectively; *g*_{L} is the effective leak conductance; *v*_{rest} is the resting potential; *g*_{1} is the effective resonant conductance; and *τ* is the time constant of the recovery variable. The variable *v*_{PD} is the measured biological membrane potential of the PD neuron, and *g*_{elec} is the (bidirectional) gap junction conductance used to couple the resonator to PD neurons.

In all experiments, the input resistance *Z*_{0} of the linear resonator was fixed at 5 MΩ by choosing appropriate values of *g*_{L} and *g*_{1} (see Tables 1 and 2), *v*_{rest} was set to the mean voltage of the uncoupled PD neuron before each experiment (see *Model neurons* above).

The values of *Z*_{max} and *f*_{res} of the linear resonator were varied independently by changing the parameters *g*_{L}, *g*_{1}, and *τ*_{1}. The different combinations of the parameters of the resonator and their corresponding *Z*_{max} and *f*_{res} used in the experiment are listed in Tables 1 and 2.

#### Dynamic clamp.

The dynamic clamp technique (Goaillard and Marder 2006; Sharp et al. 1993) was used to couple the linear resonator with both PD neurons by introducing an electrical synaptic current into the network. The current injected through the dynamic clamp was calculated based on the membrane potentials of the biological PD neuron and the linear resonator model neuron (*Eq. 4*) in real time

where *v*_{PD} is the membrane potential recorded in the PD neuron, and *v*_{R} is the membrane potential of the linear resonator. The parameter *g*_{elec} was set to 0.2 μS, unless otherwise specified. The electrical coupling was implemented in a reciprocal manner. With each set of distinct linear resonator parameters, the linear resonator was coupled to the PD neurons for 40 s. The PD neuron activity was recorded for at least 10 s, immediately before coupling and through the dynamic clamp coupling.

The dynamic clamp was implemented using NetClamp software (Gotham Scientific, Hasbrouck Heights, NJ; http://gothamsci.com/NetClamp) on a 64-bit Windows 7 personal computer using an NI PCI-6070-E board (National Instruments, Austin, TX). Data were acquired and stored using a Digidata 1440A Digitizer (Molecular Devices) at a 5-kHz sampling rate.

#### Data analysis and statistics.

The PD waveform was analyzed using custom-written Matlab (MathWorks, Natick, MA) scripts. The uncoupled baseline frequency was calculated as the mean cycle frequency of the PD neuron in the 10-s interval before dynamic clamp injection. The *f*_{net} after coupling was calculated based on the mean cycle frequency for the last 10 s of the dynamic clamp injection. Graphs were prepared using either Matlab or SigmaPlot (Ver. 13; Systat Software, San Jose, CA). All statistical analysis was performed using SigmaPlot.

## RESULTS

#### f_{net} is influenced by f_{res} and Q_{Z}.

To explore how the attributes of the impedance profile of R affect this relationship, we independently changed both *f*_{res} and resonance amplitude (*Q*_{Z}; see Fig. 2) and measured the change in *f*_{net} (Fig. 3). Note that *Q*_{Z} was changed by changing *Z*_{max} and keeping *Z*_{0} at a fixed value.

We found that for all values of *Q*_{Z}, increasing *f*_{res} always resulted in a corresponding increase in *f*_{net} (Fig. 3*B*). Similarly, the decrease of *f*_{res} resulted in a decrease in *f*_{net}. Although the overall effect was modest (approximately ±15% change in *f*_{net} as a result of ±60% change in *f*_{res} for the *Z*_{max} = 120 MΩ case shown in Fig. 3*B*), it indicated a clear, monotonic influence of *f*_{res} on the *f*_{net}.

Additionally, this effect was more prominent as *Q*_{Z} increased. As a consequence, for each fixed value of *f*_{res}, increasing *Q*_{Z} typically increasingly biased the value of *f*_{net} toward *f*_{res}. This could be seen clearly for the cases of *f*_{res} = 0.5 and 1.5 Hz (Fig. 3*C*). However, this was not an unequivocal or linear relationship. For example, the increase of *Q*_{Z}, in the case where *f*_{res} = 1 Hz (which was = *f*_{O}), also resulted in a small increase in *f*_{net} (Fig. 3*C* corresponding with 3*A*).

#### The effect of f_{res} on f_{net} is independent of the oscillator parameters.

To examine whether the monotonic effect of *f*_{res} on *f*_{net} was dependent on oscillator type, we generated 20 different model oscillators with a variety of parameter sets (Fig. 4*A*) and tuned the time constant of each oscillator so that its frequency *f*_{O} was 1 Hz (Fig. 4*B*). All models were given in *Eq. 1* but with different parameter values and included both type I and type II oscillators (Ermentrout and Terman 2010; Rinzel and Ermentrout 1998). The parameter values are provided in Table 3.

We then coupled each oscillator to the resonator R and examined the effect of shifting *f*_{res} on *f*_{net}. We found that although there was a small variability in the influence of *f*_{res} on *f*_{net}, depending on oscillator type, the overall effect remained unchanged (Fig. 4*C*). In fact, the coefficient of variation of Δ*f*_{net}, over the range of *f*_{res} examined, was only 0.0874 for all oscillator model neurons that were tested.

#### The effect of f_{res} on f_{net} is independent of the parameters of the resonator as long as the impedance profile is unchanged.

So far, we have established that changes in *f*_{res} influence *f*_{net} in an electrically coupled, two-cell network and that this effect is independent of the oscillator type. Because we change the value of *f*_{res} by adjusting the parameters of the linear resonator R, it is possible, however, that the effect observed is not due to resonance but a direct consequence of the parameter changes. To investigate this possibility, we explored the effect of different parameter sets in the resonator R that leads to the same *f*_{res} and the same overall impedance profile.

To do so, we used 3D linear resonators so that the larger number of parameters (5 instead of 3 for 2D linear resonators) would allow enough flexibility to fix the impedance profile at five distinct values, including *f*_{res} (see materials and methods). We subsequently generated a set of 20 linear resonator neurons, each with a distinct set of parameters value (Fig. 5*A*) but with matching impedance profiles (Fig. 5*B*). Note that all of these 3D resonators were of the type described in *Eq. 3* but with different choices of parameter values. These parameter values are provided in Table 4.

We coupled each resonator to the same oscillator neuron O and then shifted *f*_{res} to smaller or larger values, again choosing the resonator with the lower or higher *f*_{res} value, each from a set of 20 distinct model resonators (with matching impedance profiles). These results clearly show that the effect of *f*_{res} on *f*_{net} depended only on the impedance profile of the resonator and not on which parameters of R determined the impedance profile (Fig. 5*C*). Although Fig. 5*C* only shows three values of *f*_{res}, for additional values, the overall relationship between *f*_{net} and *f*_{res} is similar to those shown in Fig. 3*B*.

#### f_{res} can influence the frequency of a hybrid oscillatory network consisting of a biological oscillator electrically coupled to a model resonator.

To examine the predictions of our modeling results, we used the pacemaker neuron group of the crab pyloric network. The pacemaker group is composed of one anterior burster and two PD neurons that are strongly, electrically coupled and produce stable and synchronous bursting oscillations (Marder and Bucher 2007). We isolated the pacemaker group from the pyloric network by hyperpolarizing the only neuron that provides synaptic feedback onto it and electrically coupled the two PD neurons to our model resonator R using the dynamic clamp technique (see materials and methods).

We chose four versions of the resonator R, all with input resistance *Z*_{0} = 5 MΩ and *Z*_{max} = 25 MΩ but with four distinct *f*_{res} = 0.5, 1, 1.5, and 2 Hz (Fig. 6). The parameters describing the resonator are provided in Table 2. As with the model network discussed above, we chose the resting potential of R to be equal to the average value of the membrane voltage of the biological PD neuron before we introduced the dynamic clamp coupling (Fig. 6*A*). Figure 6*A* also shows voltage traces of the PD neuron when coupled to two different R model neurons with *f*_{res} = 0.5 and 2 Hz. We found that as predicted by the network model, when the value of *f*_{res} was increased, the burst frequency of the PD neuron also increased significantly [one-way repeated-measures (RM) ANOVA, *n* = 6, F = 27.511, *P* < 0.001; Fig. 6*B*]. To rule out the effect of natural variability of the pyloric frequency, we also compared the PD frequency immediately before coupling with R, which showed no significant change (one-way RM ANOVA, *n* = 6, F = 0.528, *P* = 0.638; Fig. 6*B*).

#### The effect of f_{res} on the frequency of the hybrid network of a biological oscillator electrically coupled to a linear model resonator is enhanced by increasing Q_{Z}.

To address the question of how the resonance amplitude *Q*_{Z} affects the influence of *f*_{res} on the frequency of the biological oscillator, we changed the value of *Z*_{max} in the linear resonator R to 10, 20, and 30 MΩ, while keeping *Z*_{0} constant at 5 MΩ, and explored the influence of *f*_{res} on the *f*_{net} (Fig. 7). These results confirmed the model prediction that the influence of *f*_{res} on *f*_{net} (measured as the slope value Δ*f*_{net}/Δ*f*_{res} in Fig. 7*A*; see materials and methods) increases as the value of *Q*_{Z} is increased (one-way RM ANOVA, *n* = 7, F = 33.606, *P* < 0.001; Fig. 7*A*). Additionally, when we examined the influence of increasing *Q*_{Z} on *f*_{net} for a fixed value of *f*_{res}, we found that for each value of *f*_{res}, increasing *Q*_{Z} shifted the value of *f*_{net} closer to that value of *f*_{res} (Fig. 7*B*), once again, confirming the prediction of the model.

#### Influence of g_{elec} on the f_{res}–f_{net} relationship.

As a final step, we explored how the strength of the electrical coupling affected the influence of *f*_{res} on *f*_{net}. As in Fig. 7, we measured this influence by coupling two different R model neurons, with *f*_{res} values of 1 and 1.6 Hz, to O (with an intrinsic frequency of 1 Hz) and quantifying the slope value (Δ*f*_{net}/Δ*f*_{res} in Fig. 7*A*) for different values of *g*_{elec}. The influence of different *g*_{elec} values on the network activity is seen in Fig. 8*A*, which shows that as *g*_{elec} was increased, the voltages of O and R became closer, and the rhythm frequency became faster. Dynamic clamp experiments showed that a similar effect would be obtained when *g*_{elec} was increased in the biological hybrid network (Fig. 8*B*). We found that overall, as *g*_{elec} was increased, the slope value changed from 0 monotonically to a maximum value of 0.187 (Fig. 8*C*). Although these results are intuitively clear, they indicate that even when the electrical coupling was very strong, the *f*_{net} did not equal *f*_{res} but only moved toward that value.

## DISCUSSION

The understanding of the mechanisms of generation of neuronal rhythmic activity requires the identification of the role played by the intrinsic biophysical and dynamic properties of the participating neurons and their interaction with the network connectivity. One such dynamic property is the ability of neurons to exhibit membrane potential resonance in response to oscillatory inputs (Hutcheon and Yarom 2000). For certain neuron types, the resonant frequency has been shown to be correlated with the oscillatory frequency of the networks in which they are embedded (Engel et al. 2008; Erchova et al. 2004; Heys et al. 2010; Hu et al. 2009; Leung and Yu 1998; Moca et al. 2014; Schreiber et al. 2004; Tchumatchenko and Clopath 2014; Tikidji-Hamburyan et al. 2015; Tohidi and Nadim 2009). Despite speculations that resonance determines the frequency of oscillatory networks, a causal effect of the *f*_{res} of individual neurons on *f*_{net} has not been established. Moreover, resonance in individual neurons does not necessarily communicate to the spiking and network activity, even when the resonant frequency of individual neurons and the *f*_{net} fall in the same band (Beatty et al. 2015; Stark et al. 2013).

We set out to demonstrate this causal effect, both computationally and experimentally, in minimal networks, consisting of a linear resonator R electrically coupled to an oscillator O. The ability of the network to oscillate is due to the presence of the oscillator that acts as a pacemaker, because linear resonators are not able to generate persistent oscillations on their own. The resonant frequency *f*_{res} of R is determined by the model parameters (*g*_{L}, *g*_{1}, and *τ*_{1}), which in turn, affect the *f*_{net}. One has direct access to these model parameters but not to *f*_{res}. To address rigorously the hypothesis that increasing *f*_{res} causes monotonically increasing changes in *f*_{net}, we simultaneously changed the resonator's parameter values in such a way that *f*_{res} changed, whereas the remaining attributes of the impedance profile were affected minimally or not at all. This required changing more than one biophysical parameter at a time, along the so-called level sets for *f*_{res} in parameter space (Rotstein and Nadim 2014), and assured that the observed effects are directly due to the value of *f*_{res} and not because of changes in the shape of the impedance profile (such as the *Z*_{max} or the resonance amplitude *Q*_{Z} = *Z*_{max} − *Z*_{0}). The use of linear models greatly facilitated our ability to control *f*_{res} and other resonance attributes independently.

Our results consistently showed a monotonic dependence between *f*_{res} and *f*_{net}. This effect was enhanced by increasing values of *Q*_{Z}, for constant values of *f*_{res}, and it was independent of both the model parameters, as long as the shape of the impedance profiles remained unchanged, and the parameter set values used for the oscillator. These statements are strictly only valid for the oscillators and parameter regimes that we used. However, because of the wide range of parameters considered, we hypothesize that the statement remains true for a more general class of oscillators and resonators, both linear and nonlinear. The testing of this hypothesis requires consideration of a much larger number of cases, both theoretically and experimentally [see, for example, Tseng et al. (2010)], which is beyond the scope of this paper.

To our knowledge, our results show, for the first time, a causal relationship between the resonance properties of individual neurons and the oscillatory properties of the network in which they are embedded. Moreover, the paradigm that we developed for our study—by testing the effects of controlled changes in the peak location of the impedance profile without changing the remaining attributes (impedance profile shape)—has not been used before for the causal investigation of the effects of the neuronal *f*_{res} on the *f*_{net}. This method paves the way for the investigation of similar effects in other network types and to determine whether resonance is an epiphenomenon or whether it plays important, functional roles for oscillatory network dynamics.

The results of our study are restricted to electrically coupled networks of relatively simple neurons and serve as a proof of principle of what, we believe, is a more general phenomenon. It is important to establish the extent to which the principles extracted from our study are valid for larger networks of electrically coupled networks, such as those observed in the inferior olive (Devor and Yarom 2002; Kitazawa and Wolpert 2005; Leznik and Llinas 2005; Llinas and Welsh 1993), cerebellar cortex Golgi neurons (Dugue et al. 2009), and neocortical networks (Mancilla et al. 2007; Tchumatchenko and Clopath 2014), and for networks of neurons connected via chemical synapses, either excitatory or inhibitory, where the network dynamics depend on the presence of the additional time scales provided by the synaptic connections (Tseng et al. 2014). Additionally, our results have direct implications for neurons exhibiting dendritic resonance, because the connections between different neural compartments, such as dendrites and the soma, are mathematically similar to electrical coupling among different neurons (Timofeeva et al. 2013; Zhuchkova et al. 2013).

A detailed mathematical analysis is required to understand the dynamic mechanisms underlying the results presented in this study, such as which factors determine the exact value of the *f*_{net} and how the oscillatory and resonant properties of the participating neurons interact to produce this frequency. It is also important to examine how oscillations in electrically coupled networks that involve resonant neurons may be influenced by chemical synapses (Lefler et al. 2014), even in a simple network.

In contrast to previous studies, we simultaneously changed a group of model parameters to control the changes in the *f*_{res} or resonance amplitude, without changing the remaining attributes of the impedance profile. Biologically, these changes may be achieved by simultaneous modulation of multiple parameters, potentially in different voltage-gated ionic currents, to alter the *f*_{net} (Nadim and Bucher 2014). Clearly, such balanced changes influence the effective time scales at which the resonant neuron operates. Yet, how these effective time scales interact with the time scales of the oscillator or other electrically coupled neurons to produce a change in the *f*_{net} remains an open question.

## GRANTS

Support for this work was provided by the National Institute of Mental Health (R01-MH060605; to F. Nadim), National Institute of Neurological Disorders and Stroke (R01-NS083319; to F. Nadim), and National Science Foundation (DMS1313861; to H. G. Rotstein).

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

## AUTHOR CONTRIBUTIONS

H.G.R. and F.N. conception and design of research; Y.C. and X.L. performed experiments; Y.C. and X.L. analyzed data; X.L. and F.N. interpreted results of experiments; Y.C., X.L., and F.N. prepared figures; F.N. drafted manuscript; H.G.R. and F.N. edited and revised manuscript; Y.C., X.L., H.G.R., and F.N. approved final version of manuscript.

## ACKNOWLEDGMENTS

The authors thank Dirk Bucher for his comments on the manuscript.

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