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Math Biosci. Author manuscript; available in PMC 2017 August 1.

Published in final edited form as:

PMCID: PMC4940261

NIHMSID: NIHMS791205

The publisher's final edited version of this article is available at Math Biosci

See other articles in PMC that cite the published article.

A simplified model of the crustacean gastric mill network is considered. Rhythmic activity in this network has largely been attributed to half center oscillations driven by mutual inhibition. We use mathematical modeling and dynamical systems theory to show that rhythmic oscillations in this network may also depend on, or even arise from, a voltage-dependent electrical coupling between one of the cells in the half-center network and a projection neuron that lies outside of the network. This finding uncovers a potentially new mechanism for the generation of oscillations in neuronal networks.

Networks of neurons display a variety of oscillatory behaviors. For example, oscillations in the levels of calcium concentrations, gene expressions and in the membrane voltage across cell membranes are all commonly found in neuronal systems. Often these oscillations are rhythmic in that they display a consistent pattern at a prescribed frequency [1]. Central pattern generating (CPG) networks provide several examples that exhibit rhythmic activity. CPGs refer to networks of neurons in the central nervous system that produce patterned (usually oscillatory) activity in the absence of patterned sensory input. These networks play a critical role in generating a diverse array of motor functions such as digestion, locomotion, respiration and regulation of heartbeat in invertebrates [2]. A central question in the study of neural oscillations is what are the mechanisms that underlie the generation of rhythmic activity and how that activity is regulated. This study will focus on this general question in the context of the gastric mill rhythm (GMR; frequency 0.1 *Hz*) that arises in the stomatogastric ganglion (STG) in the crustacean central nervous system. In particular, we will show the existence of a new mechanism based on voltage-dependent electrical coupling for generation of oscillations within a neuronal network.

The gastric mill network consists of a small number of neurons in the STG that control muscles that move teeth to provide grinding of food (chewing) within the gastric mill stomach of crustaceans [3]. In the Jonah crab, a pair of neurons, the lateral gastric (*LG*) and Interneuron 1 (*INT*1) form a half-center oscillator (HCO) and are primary contributors to the GMR. These neurons are connected by reciprocally inhibitory synapses and, during gastric mill activity, display anti-phase bursting oscillations. They also receive input from various parts of the stomatogastric nervous system (STNS). In particular, *INT*1 receives rhythmic inhibition from the pacemaker anterior burster neuron (*AB*) of the pyloric CPG. Because the pyloric rhythm (frequency 1 *Hz*) is much faster than the gastric mill, the *AB* to *INT*1 input produces pyloric timed patterns in the *INT*1 bursting activity. Both *LG* and *INT*1 receive excitatory input from the modulatory commissural neuron 1 (*MCN*1) with *INT*1 receiving fast excitation and *LG* receiving slow modulatory excitation. Additionally, the *MCN*1 axon terminals are electrically coupled to *LG* in a manner that is dependent on the voltage of *LG* [5]. It is the role of this electrical coupling that is of particular interest to us in this paper.

Neurons that lie within a HCO typically utilize reciprocal inhibition to generate oscillations [6]. In particular, in a two cell HCO, when one of the cells is active, its inhibitory synapse suppresses the other. At some later time, the silent cell escapes or is released from inhibition and the roles of the two cells switch [7]. In the gastric mill network, *LG* and *INT*1 can oscillate in this manner with the ability to escape inhibition and generate oscillations, but only in the presence of the excitatory input provided by *MCN*1 [5, 8].

Although a number of modeling studies have explored the generation of oscillations in the gastric mill network [8, 9, 10, 11, 13], the role of the strong electrical coupling between the *MCN*1 axon terminals and the *LG* neuron has not been previously explored. In this study, we will show that voltage-dependent electrical coupling can provide an alternative mechanism for the generation of oscillations when the inhibition based HCO mechanism is incapable of doing so. In particular the *LG – INT*1 HCO can be rendered ineffective if 1) the inhibitory synapse form *INT*1 to *LG* is inactivated, or 2) if the excitability property of *LG* is reduced. In order to fully understand how electrical coupling affects this network, we will first consider a simple model to see how how electrical coupling between *LG* and *MCN*1 axon terminals affects the ability of oscillations to be created through the standard HCO inhibition based mechanism. We will discuss how the electrical coupling modulates the rhythmic properties of this oscillation. We will then remove the *INT*1 to *LG* synapse and show that rhythmic oscillations can still arise through the electrical coupling between *LG* and *MCN*1 axon terminals, but only if this coupling is voltage dependent, as has been reported experimentally [5]. We will then demonstrate the same in a biophysical model based on the Morris-Lecar equations [15]. For both models, we derive conditions on parameters showing why the electrical coupling must be voltage dependent to produce oscillations.

The modeling and analysis in this paper is based on the use of geometric singular perturbation theory. Exploiting inherent differences in timescales, we will derive sets of fast and slow equations that can be studied in the relevant phase space. For the simple model, this can be done on a two-dimensional phase plane and is the focus of Sections 3.1–3.4. The analysis in those sections follows the tradition of using relaxation oscillators with the individual neurons modeled as passive elements. The relaxation oscillations in this case arise due to the method of model reduction that incorporates a slow synaptic variable. In Section 3.5, the fast-slow analysis allows us to project the relevant dynamics onto two different phase planes to facilitate understanding of the model.

We describe the simple network that we shall initially consider. A key assumption for this model is that *INT*1 and *LG* are modeled as passive cells with no active currents or excitable properties. Thus if oscillations are to be generated, they must arise as a direct result of network interactions. By identifying variables that evolve on different time scales and by making a few other assumptions, we can use geometric singular perturbation theory to focus on the analysis of a reduced two-dimensional system of equations. These variables correspond to the voltage of *LG* and to the synaptic input that *LG* receives from MCN1 and are shown in solid in Fig. 1. The electrical coupling is also shown in solid in Fig. 1 as it can be defined in terms of the reduced quantities including the voltage of *LG*. Shown with dotted lines/circles are the other variables that we will incorporate into the solid variables and thus will not need to explicitly track.

Let *V _{L}* and

$$\epsilon \frac{d{V}_{L}}{dt}=-{I}_{\mathit{rest},L}({V}_{L})-{I}_{\mathit{syn},I\to L}({V}_{I},{V}_{L})-{I}_{\mathit{syn},M\to L}({V}_{M},{V}_{L},s)-{I}_{\mathit{elec}}({V}_{L},{V}_{M})$$

(1)

$$\epsilon \frac{d{V}_{I}}{dt}=-{I}_{\mathit{rest},I}({V}_{I})-{I}_{\mathit{syn},L\to I}({V}_{I},{V}_{L})-{I}_{\mathit{syn},AB\to I}({V}_{I},{s}_{AB\to I})$$

(2)

The intrinsic current *I _{rest,x}*(

$${s}_{AB\to I}(t)=\mathit{Heav}(\mathit{sin}(\frac{2\pi (t)}{1000})-0.5)$$

(3)

$${s}_{L\to I}({V}_{L})={[1+\mathit{exp}(\frac{{v}_{1}-{V}_{L}}{{k}_{1}})]}^{-1}$$

(4)

$${s}_{I\to L}({V}_{I})={[1+\mathit{exp}(\frac{{v}_{2}-{V}_{I}}{{k}_{2}})]}^{-1}$$

(5)

The remaining synaptic variable *s* requires some explanation. In the biological system, *MCN*1 exerts a slow excitatory effect on *LG* that is modulated by pre-synaptic inhibition from *LG* onto the *MCN*1 to *LG* synapse. Thus when *LG* is active, this excitation is slowly removed; when *LG* is silent, the excitation slowly builds. This is modeled by the variable *s* that evolves on a slow time scale and is the only slow variable in our model. Equations governing this variable are:

$$\frac{ds}{dt}=\{\begin{array}{ll}(1-s)/{\tau}_{r}\hfill & {V}_{L}\le {V}_{T}\hfill \\ -s/{\tau}_{f}\hfill & {V}_{L}>{V}_{T}\hfill \end{array}$$

(6)

In equation (1), the synaptic current is then given by

$${I}_{\mathit{syn},M\to L}={g}_{M\to L}s[{V}_{L}-{E}_{\mathit{exc}}].$$

(7)

Figure 1 shows an electrical coupling between *LG* and the *MCN*1 axon terminals. The electrical current is given by

$${I}_{\mathit{elec}}({V}_{L},{V}_{M})={g}_{\mathit{elec}}({V}_{L})[{V}_{L}-{V}_{M}].$$

(8)

This coupling is dependent on the voltage of *LG* and *MCN*1 in two different ways. First, the strength is an increasing function of *V _{L}*. The dependency of the conductance

$${g}_{\mathit{elec}}({V}_{L})={\overline{g}}_{\mathit{elec}}{n}_{\infty}({V}_{L})$$

(9)

where

$${n}_{\infty}({V}_{L})=(1-{g}_{\mathit{min}}){(1+\mathit{exp}(\frac{{v}_{el}-{V}_{L}}{{k}_{el}}))}^{-1}+{g}_{\mathit{min}}.$$

(10)

where *v _{el}* is the half activation value at which

While, equations (1)–(8) govern the flow of the gastric mill circuit, the dynamics can be simplified by exploiting the small parameter *ε* that demarcates the fast and slow time scales, as was first done by Kintos et. al. [11]. Set *ε* = 0 in (1)–(2). The latter of these equations can be rewritten in terms of *V _{L}* and of the independently controlled quantity

$$0=-{g}_{\mathit{rest},L}[{V}_{L}-{E}_{\mathit{rest},L}]-{g}_{I\to L}{s}_{I\to L}({h}_{1}({V}_{L},{s}_{AB\to I})))[{V}_{L}-{E}_{\mathit{inh}}]-{g}_{M\to L}s[{V}_{L}-{E}_{\mathit{exc}}]-{g}_{\mathit{elec}}({V}_{L},{V}_{M})[{V}_{L}-{V}_{M}]$$

(11)

$$\frac{ds}{dt}=\{\begin{array}{ll}(1-s)/{\tau}_{r}\hfill & {V}_{L}\le {V}_{T}\hfill \\ -s/{\tau}_{f}\hfill & {V}_{L}>{V}_{T}.\hfill \end{array}$$

(12)

Denote the right-hand side of (11) by *F* (*V _{L}, s*). The first equation constrains the flow to lie on

$$\frac{d{V}_{L}}{d\tau}=F({V}_{L},s)$$

(13)

$$\frac{ds}{d\tau}=0.$$

(14)

Equations (13–14) govern the fast jumps that a trajectory in the phase plane makes between different possible (stable) branches of the *V _{L}*-nullcline. For

The *V _{L}* nullcline is the set of points {(

$$s=\frac{-{g}_{\mathit{rest},L}[{V}_{L}-{E}_{\mathit{rest},L}]-{g}_{I\to L}{s}_{I\to L}({h}_{1}({V}_{L},{s}_{AB\to I})))[{V}_{L}-{E}_{\mathit{inh}}]-{g}_{\mathit{elec}}({V}_{L})[{V}_{L}-{V}_{M}]}{{g}_{M\to L}[{V}_{L}-{E}_{\mathit{exc}}]}.$$

(15)

The *s*-nullcline is simply the Heaviside function given by *s* = 1 when *V _{L}* <

The shape of the *V _{L}*-nullcline is dependent on our choice of parameters. It is known from prior modeling work of this system [11, 12], and of many others in different contexts, that when one of the nullclines is cubic shaped and the other is linear or sigmoidal that oscillations may occur if the nullclines intersect on the middle branch of the cubic. In the results section below we will show how various parameters related to both the synaptic and electrical coupling affect the shape of the

In Section 3.5, we will use the Morris-Lecar equations to model both *LG* and *INT*1. As a result of the added dimensionality of the model, we will not be able to reduce the analysis to a two-dimensional phase plane. However, similar to our analysis with the simple model, we will be able to show that the projection of the *LG* trajectory onto two distinct two-dimensional phase planes will be crucial to understanding the role of voltage-dependent electrical coupling. When parameters are chosen in the Morris-Lecar equations to reduce the excitability of *LG*, the inhibition based HCO becomes ineffective. In that case, as in the case of the simple model, electrical coupling will be able to produce oscillations but only when it is voltage-dependent. Details of the model will be provided in Section 3.5 and the Appendix.

For completeness and for ease in explaining the role of the voltage dependent electrical coupling, we begin by reviewing the case when * _{elec}* = 0 as described in [10]. Oscillations in this case arise as a direct consequence of the mutually inhibitory pair

First set *g _{AB}*

When the *AB* to *INT*1 inhibtion is present (*g _{AB}*

We next investigate the effect of adding electrical coupling to the network. First we consider the case when the electrical coupling is not voltage dependent. To do so, set *v _{el}* = −100. Since

Next, observe that electrical coupling and the MCN1 synapse have similar effects on the *V _{L}*-nullcline. Namely, increases in either

We can get a better understanding of the range of conductance values for which oscillations exist. Figure 3C shows a bifurcation diagram in *g _{M}*

The left boundary corresponds to the set of saddle-node values along the local maximum of the *V _{L}*-nullcline at

$$s=\frac{f({V}_{L})-{\overline{g}}_{\mathit{elec}}[{V}_{L}-{V}_{M}]}{{g}_{M\to L}[{V}_{L}-{E}_{\mathit{exc}}]},$$

(16)

where *f*(*V _{L}*) refers to the first two terms in the numerator on the left hand side of (15). A saddle-node point occurs when

$${\overline{g}}_{\mathit{elec}}=-\frac{{V}_{L}-{E}_{\mathit{exc}}}{{V}_{L}-{V}_{M}}{g}_{M\to L}+\frac{f({V}_{L})}{{V}_{L}-{V}_{M}}.$$

(17)

Next observe that

$$\frac{ds}{d{V}_{L}}=\frac{[df/d{V}_{L}-{\overline{g}}_{\mathit{elec}}][{V}_{L}-{E}_{\mathit{exc}}]-[f({V}_{L})-{\overline{g}}_{\mathit{elec}}[{V}_{L}-{V}_{M}]]}{{g}_{M\to L}{[{V}_{L}-{E}_{\mathit{exc}}]}^{2}}.$$

(18)

The condition *ds/dV _{L}* = 0 implies that the numerator of the above fraction equals zero which reduces to the relationship,

$$\frac{df}{d{V}_{L}}[{V}_{L}-{E}_{\mathit{exc}}]-f({V}_{L})-{\overline{g}}_{\mathit{elec}}[{V}_{M}-{E}_{\mathit{exc}}]=0.$$

(19)

Let
${V}_{L}^{\ast}({\overline{g}}_{\mathit{elec}})$ denote the solution of (19) and note it that does not depend on *g _{M}*

The top boundary of the oscillation region corresponds to the set of saddle-node points when the minimum of the cubic nullcline is tangent to *s* = 0. This curve is given by *F* (*V _{L},* 0) = 0 and

The region *R*1 is unbounded on the right. This is precisely because the local minimum of *s* at *s* = 0 is independent of *g _{M}*

To explore the role of voltage dependence on the electrical coupling in the *INT*1-*LG* generated rhythm, we let *v _{el}* =

Figure 4C shows the regions of oscillations for these cases. For this set of parameters, there are two primary differences between the voltage-dependent (*R*2) and independent (*R*1) cases. First, the left boundary is more steeply sloped and the top boundary sits at a higher * _{elec}* value compared to the voltage-independent case. Both are easily explained. In the voltage-dependent case, equation (17) becomes

$${\overline{g}}_{\mathit{elec}}=-\frac{{V}_{L}-{E}_{\mathit{exc}}}{{n}_{\infty}({V}_{L})[{V}_{L}-{V}_{M}]}{g}_{M\to L}+\frac{f({V}_{L})}{{V}_{L}-{V}_{M}}.$$

(20)

The condition *ds/dV _{L}* = 0 yields a solutions
${V}_{L}^{\ast}({\overline{g}}_{\mathit{elec}})$ which is again independent of

The intersection of the *V _{L}* nullcline with

$${\overline{g}}_{\mathit{elec}}{n}_{\infty}({V}_{L}^{\ast \ast})=\frac{{g}_{\mathit{rest},L}[{V}_{L}^{\ast \ast}-{E}_{\mathit{rest},L}]+{g}_{I\to L}{s}_{I\to L}({h}_{1}({V}_{L}^{\ast \ast},{s}_{AB\to I})))[{V}_{L}^{\ast \ast}-{E}_{\mathit{inh}}]}{{V}_{M}-{V}_{L}^{\ast \ast}}.$$

(21)

The value
${V}_{L}^{\ast \ast}$ increases with voltage dependence (specifically with *v _{el}* from (10)). As a result, the right-hand side of (21) increases since the numerator increases while the denominator decreases. In the voltage independent case,

The effect of voltage dependence can be amplified by making the *n*_{∞}(*V _{L}*) curve less steeply sloped. For instance, if

To this point, we have simply shown how electrical coupling affects the existing oscillations that arise through the *INT*1-*LG* HCO. A more important observation that we now make is that oscillations can arise in the absence of this HCO provided that the electrical coupling is voltage-dependent.

Consider equations (11)–(14) with *g _{I}*

$$s=\frac{-{g}_{\mathit{rest},L}[{V}_{L}-{E}_{\mathit{rest},L}]-{\overline{g}}_{\mathit{elec}}{n}_{\infty}({V}_{L})[{V}_{L}-{V}_{M}]}{{g}_{M\to L}[{V}_{L}-{E}_{\mathit{exc}}]}.$$

(22)

In this case, to see why voltage dependence is necessary for oscillations, first take the case where the electrical coupling is non-voltage dependent. Then *ds/dV _{L}* = [

Now take the case when the electrical coupling is voltage dependent. Then after some algebraic manipulation, the condition *ds/dV _{L}* = 0 yields

$${g}_{\mathit{rest},L}[{E}_{\mathit{exc}}-{E}_{\mathit{rest},L}]={\overline{g}}_{\mathit{elec}}\phantom{\rule{0.16667em}{0ex}}\left[\frac{d{n}_{\infty}}{d{V}_{L}}{[{v}_{L}-{E}_{\mathit{exc}}]}^{2}+[{V}_{L}+{n}_{\infty}({V}_{L})-{E}_{\mathit{exc}}][{V}_{M}-{E}_{\mathit{exc}}]\right].$$

For simplicity, take *E _{exc}* =

$${g}_{\mathit{rest},L}[{V}_{M}-{E}_{\mathit{rest},L}]={\overline{g}}_{\mathit{elec}}\frac{d{n}_{\infty}}{d{V}_{L}}{[{V}_{L}-{V}_{M}]}^{2}.$$

(23)

The left hand side is independent of * _{elec}*, while the right hand side increases with it. Further the right hand side has a zero at

Using equation (10), we can derive an estimate on how large * _{elec}* needs to be to obtain oscillations. The right hand side of (23) has a local maximum at

$${\overline{g}}_{\mathit{elec}}\ge \frac{4{g}_{\mathit{rest},L}{k}_{el}[{V}_{M}-{E}_{\mathit{rest},L}]}{{[{v}_{el}-{V}_{M}]}^{2}}.$$

(24)

This condition is fairly straightforward to interpret. Namely, the stronger the passive properties of *LG*, either through larger leak conductance *g _{rest,L}* or smaller leak reversal

We next explore the role of *INT*1 on the *MCN*1 *– LG* generated oscillation. We emphasize that, although the inhibition from *INT*1 to *LG* is restored, the parameters remain in range where the inhibition based HCO-based mechanism is not capable of producing oscillations. *INT*1 inhibition to *LG* raises the LB of *V _{L}*-nullcline as shown in Fig 5B. Now the trajectory (black) must increase to higher values of

When *AB* to *INT*1 inhibition is included, the trajectory is allowed to leave the left branch prematurely at one of the moments in time when *INT*1 is inhibited by *AB*. This results in a shorter interburst and burst duration very similar to what was described in Section 3.1. Note that the period is very similar to that obtained when *INT*1 to *LG* inhibition is completely absent (*g _{I}*

We now demonstrate that our main findings regarding the role of voltage dependent electrical coupling hold in a model in which *LG* and *INT*1 are modeled using biophysical equations. We model each of these cells using the two-dimensional Morris-Lecar equations, which are a commonly used set of equations that are derived in the Hodgkin-Huxley formalism. The voltage equation includes ionic currents for calcium, potassium and a leak current. There is a recovery variable associated with the activation of the potassium current. The equations for each cell are

$$\epsilon \frac{d{V}_{L}}{dt}=-{g}_{\mathit{leak},L}[{V}_{L}-{E}_{\mathit{leak},L}]-{g}_{Ca,L}{m}_{\infty}({V}_{L})[{V}_{L}-{E}_{Ca}]-{g}_{K}{w}_{L}[{V}_{L}-{E}_{K}]-{I}_{\mathit{syn},I\to L}({V}_{I},{V}_{L})-{I}_{\mathit{syn},M\to L}({V}_{M},{V}_{L},s)-{I}_{\mathit{elec}}({V}_{L},{V}_{M})+{I}_{\mathit{app},L}$$

(25)

$$\frac{d{W}_{L}}{dt}={\varphi}_{L}[{w}_{\infty ,L}({V}_{L})-{W}_{L}]/{\tau}_{\infty}({V}_{L})$$

(26)

$$\epsilon \frac{d{V}_{I}}{dt}=-{g}_{\mathit{leak},I}[{V}_{I}-{E}_{\mathit{leak},I}]-{g}_{Ca,I}{m}_{\infty}({V}_{I})[{V}_{I}-{E}_{Ca}]-{g}_{K}{w}_{I}[{V}_{I}-{E}_{K}]-{I}_{\mathit{syn},L\to I}({V}_{I},{V}_{L})-{I}_{\mathit{syn},AB\to I}({V}_{I},{s}_{AB\to I})+{I}_{\mathit{app},I}$$

(27)

$$\frac{d{W}_{I}}{dt}={\varphi}_{I}[{w}_{\infty ,I}({V}_{I})-{W}_{I}]/{\tau}_{\infty}({V}_{I}).$$

(28)

On the right-hand side of equations (25) and (27), the first three terms are specific to the Morris-Lecar equations, while the remaining terms have the same form as defined in Section 2.1. The specific details of the model and parameter values are provided in the Appendix. Of interest to us here is the shape of the nullclines of the two cells. For *INT*1, parameters are chosen such that in the absence of input (*g _{L}*

For *LG*, we consider two different parameter choices. In one case, in the absence of input, we choose *g _{Ca,L}* = 4.0 which is large enough so that the

The more interesting situation arises in the second case when we choose *g _{Ca,L}* = 0.5 so that the

In contrast, consider Fig. 6B1. Shown is the *V _{L}* nullcline when the electrical coupling is voltage-dependent for two different values of

In Fig. 6C, we restore the *INT*1 to *LG* synapse *g _{I}*

Just as in Section 3.4, we can determine conditions under which voltage-dependence allows the electrical coupling to produce oscillations. Consider the case *g _{I}*

$${W}_{L}=\frac{f({V}_{L})+h({V}_{L})}{{g}_{K}[{V}_{L}-{E}_{K}]}$$

(29)

The slope of this nullcline is given by

$$\frac{d{W}_{L}}{d{V}_{L}}=\frac{[{f}^{\prime}({V}_{L})+{h}^{\prime}({V}_{L})][{V}_{L}-{E}_{K}]-[f({V}_{L})+h({V}_{L})]}{{g}_{K}{[{V}_{L}-{E}_{K}]}^{2}}$$

(30)

To show that the *V _{L}* nullcline can be cubic shaped, we need to find conditions under which the derivative (29) changes sign. Observe that

$${f}^{\prime}({V}_{L})=-{g}_{\mathit{rest},L}-{g}_{Ca,L}[{m}_{\infty}^{\prime}({V}_{L})[{V}_{L}-{E}_{Ca}]+{m}_{\infty}({V}_{L})]<0$$

(31)

if *g _{Ca,L}* is sufficiently small. The derivative

$${h}^{\prime}({V}_{L})=-{\overline{g}}_{\mathit{elec}}{n}_{\infty}^{\prime}({V}_{L})[{V}_{L}-{V}_{M}]-{\overline{g}}_{\mathit{elec}}{n}_{\infty}({V}_{L})-{g}_{s}{s}^{\prime}[{V}_{L}-{E}_{\mathit{exc}}]-{g}_{s}s.$$

(32)

The first term in (32) is non-negative, while the remaining three are all negative (note that *s*′(*V _{L}*) < 0). Thus the sign of

Neuronal circuits involved in the generation of rhythmic behavior often involve half center oscillators that are composed of sets of reciprocally inhibitory neurons. There is an extensive and ongoing effort to understand the dynamics of half center oscillators in the context of central pattern generation [10, 11, 14, 16, 17]. In many cases, it has been noted that a careful coordination between network elements is necessary to generate and set the frequency of the network [18, 19, 20]. The role of electrical coupling in rhythmic networks has also been studied [21, 22] where the neurons were modeled as intrinsic oscillators. Electrical coupling was not needed to generate oscillations, but rather used to modulate the characteristics of the oscillation.

As part of a larger work on the role of feedback to projection neurons, Kintos and colleagues [10, 12] had shown how to employ phase plane analysis to understand the effect of *MCN*1 synaptic input on the *GMR*. In particular, they showed how to analyze *MCN*1 synaptic input and *AB* inhibition of *INT*1 to determine the frequency of the *GMR*. In this paper, we have extended this analysis to show how to incorporate the effect of *MCN*1 *– LG* voltage-dependent electrical coupling to determine the conditions under which electrical coupling in the absence of the *LG – INT*1 HCO can generate oscillations.

In the presence of an intact *LG – INT*1 HCO, we first considered the effect of non-voltage dependent electrical coupling. We showed that the non-voltage dependent electrical coupling acts to increase the *LG* burst duration while shortening its interburst duration. This occurs because the voltage of *LG* is driven towards the fixed, large voltage of *MCN*1. If the strength of the electrical coupling is too large, however, *LG* gets stuck in its burst phase. One advantage of the non-voltage dependent electrical coupling is that it can be used in conjunction with the *MCN*1 chemical synapse allowing for the generation of the *GMR* for a smaller amount of the chemical excitation. This is a ”cheaper” way to generate oscillations as it requires less synaptic resources. The bifurcation diagram in Fig. 3C shows the precise relationship between electrical and synaptic coupling needed to create oscillations. We showed that boundaries of this diagram are all roughly linear. In the case of voltage dependent electrical coupling, the right branch of the *LG* nullcline is affected much more significantly than the left branch. This allows for an increase in the *LG* burst duration and a larger range of values of * _{elec}* for the generation of network oscillations.

A significant finding of our study is that network oscillations can also be generated in the absence of coupling between *LG* and *INT*1 simply through the voltage dependent electrical coupling between *MCN*1 and *LG* and the slow excitation from *MCN*1, together with its removal due to the pre-synaptic inhibition of this excitation. We derived a condition on the minimum value of * _{elec}* in order for the

Our findings are not limited to the simple model in which *LG* and *INT*1 are modeled as passive cells that we first considered. We showed that voltage-dependent electrical coupling played the same role in a model in which these cells were described using the biophysically based Morris-Lecar equations. In order for voltage-dependent electrical coupling to create the mechanism for oscillations, we showed that *LG* must not be modeled as being excitable. This fact is consistent with the underlying biological properties of the *LG* neuron, which, in the absence of *MCN*1 or other modulatory input, shows no active properties (e.g. post-inhibitory rebound, voltage sags or plateaus) that are associated with slow bursting oscillations [4].

There are several natural extensions of this work. In previous work [23], based on experiments of Wood et. al. [24], we showed that *AB* inhibition to*MCN*1 provides an alternate mechanism to regulate the gastric mill frequency. In the current work, we did not include the inhibition from *AB* to *MCN*1. If the *AB* inhibition to *MCN*1 were included, the *LG* burst would end when *AB* inhibits *MCN*1. It would be necessary for *MCN*1 to be gated when *LG* is in its active state in order to maintain robust oscillations. Indeed, in the VCN-activated version of the gastric mill rhythm, the *AB* to *MCN*1 synapse is gated out during *LG* active phase [25]. It would be of interest to extend our current model to test whether this gating is truly necessary to maintain oscillations.

Another area that remains to be explored is the role of electrical coupling in the *MCN*1*/CP N*2 generated gastric mill rhythm. Kintos and Nadim [10] showed that the *LG – INT*1 HCO could be replaced by a tri-synaptic pathway that included the projection neuron *CP N*2. Of interest would be to see whether voltage dependence can replace one or more of those synaptic pathways.

Although the networks under consideration in this, and related papers, are relatively simple and only involve a small number of neurons, it is evident that the dynamics exhibited by them can be quite complicated. Moreover, the neural mechanisms that underlie the existence of oscillations are often hard to separate from those that simply modulate the rhythmic properties of these networks. Minimal modeling and mathematical analysis of small networks plays a critical role in allowing us to discern which inputs generate oscillations versus those that modulate oscillations by providing valuable insights into how these important central pattern generating networks operate.

- A simple model is used to understand how electrical coupling affects the ability of oscillations to be created through a standard half-center oscillator mechanism is incapable of doing so.
- Voltage-dependent electrical coupling is shown to provide an alternate mechanism for the generation of oscillations when an inhibition based half-center oscillator mechanism is incapable of doing so.
- This same result is then demonstrated in a biophysical model based on the Morris-Lecar equations.
- Conditions on the parameters in both models are derived that show why electrical coupling must be voltage dependent to produce oscillations.

This work was supported in part by the National Science Foundation, DMS 1122291 (AB, FN) and the National Institutes of Health, NIH R01-MH060605 (FN).

- CPG
- central pattern generating
- GMR
- gastric mill rhythm
- STG
- stomatogastric ganglion
- HCO
- half-center oscillator
- LG
- lateral gastric
- INT1
- interneuron 1
- STNS
- stomatogastric nervous system
- AB
- anterior burster
- MCN1
- modulatory commissural neuron 1

Numerical simulations were performed using XPPAUT [26]. For the simple model of passive cells used to produce Figs. 2–5 the following set of equations was used.

$$\frac{d{V}_{L}}{dt}=-{g}_{\mathit{rest},L}[{V}_{L}-{E}_{\mathit{rest},L}]-{g}_{M\to L}s[{V}_{L}-{E}_{\mathit{exc}}]-{g}_{\mathit{elec}}({V}_{L})[{V}_{L}-{V}_{M}]-{g}_{I\to L}{s}_{I\to L}({h}_{1}({V}_{L},{s}_{AB\to I}))[{V}_{L}-{E}_{\mathit{inh}}]$$

(33)

$$\frac{ds}{dt}=\frac{1-s}{{\tau}_{r}}\mathit{Heav}({V}_{T}-{V}_{L})-\frac{s}{{\tau}_{f}}\mathit{Heav}({V}_{L}-{V}_{T})$$

(34)

The term *V _{I}* =

$${h}_{1}({V}_{L},{s}_{AB\to I})=\frac{{E}_{\mathit{rest},I}+{\scriptstyle \frac{{g}_{L\to I}}{{g}_{\mathit{rest},I}}}{s}_{L\to I}({V}_{L}){E}_{\mathit{inh}}+{\scriptstyle \frac{{g}_{AB\to I}}{{g}_{\mathit{rest},I}}}{s}_{AB\to I}(t)P({V}_{L}){E}_{\mathit{inh}}}{1+{\scriptstyle \frac{{g}_{AB\to I}}{{g}_{\mathit{rest},I}}}{s}_{AB\to I}(t)P({V}_{L})+{\scriptstyle \frac{{g}_{L\to I}}{{g}_{\mathit{rest},I}}}{s}_{L\to I}({V}_{L})}.$$

Note here the presence of the function
$P({V}_{L})={(1+\mathit{exp}({\scriptstyle \frac{{V}_{L}-{v}_{3}}{{k}_{3}}}))}^{-1}$. This term is used to gate out the effect of *AB* input to *INT*1, and its subsequent effect on the *V _{L}* nullcline when

Table 1 shows parameter values that were common to all simulations of the simple model. Below that we show specific values used for * _{elec}* and

For Figures 2 to to4,4, we chose *g _{I}*

For Figure 5A: *g _{M}*

For the simulations shown in Fig. 6, the following set of equations was used:

$$\begin{array}{lll}\hfill C\frac{d{V}_{L}}{dt}& =\hfill & -{g}_{\mathit{leak},L}[{V}_{L}-{E}_{\mathit{leak}}]-{g}_{K}{w}_{L}[{V}_{L}-{E}_{K}]-{g}_{Ca,I}{m}_{\infty}({V}_{L})[{V}_{L}-{E}_{Ca}]\hfill \\ \phantom{\rule{0.16667em}{0ex}}\hfill & \phantom{\rule{0.16667em}{0ex}}\hfill & -{g}_{I\to L}{s}_{I\to L}I[{V}_{L}-{E}_{\mathit{inh}}]-{g}_{s}s[{V}_{L}-{E}_{\mathit{exc}}]-{g}_{\mathit{elec}}({V}_{L})[{V}_{L}-{V}_{M}]+{I}_{\mathit{app},L}\hfill \\ \hfill \frac{d{W}_{L}}{dt}& =\hfill & {\varphi}_{L}\frac{{w}_{\infty ,L}({V}_{L})-{W}_{L}}{{\tau}_{\infty}({V}_{L})}\hfill \\ \hfill C\frac{d{V}_{I}}{dt}& =\hfill & -{g}_{\mathit{leak},I}[{V}_{I}-{E}_{\mathit{leak}}]-{g}_{K}{w}_{I}[{V}_{I}-{E}_{K}]-{g}_{Ca,I}{m}_{\infty}({V}_{I})[{V}_{I}-{E}_{Ca}]\hfill \\ \phantom{\rule{0.16667em}{0ex}}\hfill & \phantom{\rule{0.16667em}{0ex}}\hfill & -{g}_{L\to I}{s}_{L\to I}[{V}_{I}-{E}_{\mathit{inh}}]-{g}_{AB\to I}{s}_{AB}(t)[{V}_{I}-{E}_{\mathit{syn}}]+{I}_{\mathit{app},I}\hfill \\ \hfill \frac{d{W}_{I}}{dt}& =\hfill & {\varphi}_{I}\frac{{w}_{\infty ,I}({V}_{I})-{W}_{I}}{{\tau}_{\infty}({V}_{I})}\hfill \end{array}$$

The synaptic variables are governed by

$$\begin{array}{lll}\hfill \frac{d{s}_{I\to L}}{dt}& =\hfill & \gamma (\frac{1}{1+\mathit{exp}({\scriptstyle \frac{{v}_{\mathit{ith}}-{V}_{I}}{{t}_{\alpha}}})}-{s}_{I\to L})\hfill \\ \hfill \frac{d{s}_{L\to I}}{dt}& =\hfill & \alpha (1-{s}_{L\to I})\mathit{Heav}({V}_{L}-{V}_{T})-\beta {s}_{L\to I}\mathit{Heav}({V}_{T}-{V}_{L})\hfill \\ \hfill \frac{ds}{dt}& =\hfill & \frac{1-s}{{\tau}_{r}}\mathit{Heav}({V}_{T}-{V}_{L})-\frac{s}{{\tau}_{f}}\mathit{Heav}({V}_{L}-{V}_{T}).\hfill \end{array}$$

The remaining terms are given by
${m}_{\infty}({V}_{x})=(1+\mathit{tanh}({\scriptstyle \frac{{V}_{x}-cv1}{cv2}}))/2,{\tau}_{\infty}({V}_{x})=\mathit{cosh}({\scriptstyle \frac{{V}_{x}-cv3}{2\mathit{cvv}}}),{w}_{\infty ,x}({V}_{x})=(1+\mathit{tanh}({\scriptstyle \frac{{V}_{x}-w{f}_{x}}{cv4}}))/2$, where the subscript *x* refers to either *L* or *I*. In addition, we used equations (3), (9) and (10). For Fig. 6A–B,*g _{I}*

**Conflict of Interest**

The authors declare that they have no conflict of interest.

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