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1Toronto Western Research Institute, University Health Network, 2Department of Physiology, 3Engineering Science Program, 4Department of Medicine (Neurology), 5Institute of Biomaterials and Biomedical Engineering, University of Toronto, Toronto, Ontario, Canada
Submitted 24 June 2005; accepted in final form 1 December 2005
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ABSTRACT |
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INTRODUCTION |
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; Connors and Long 2004; Hormuzdi et al. 2004). This direct coupling is mediated by gap junctions or clusters of channels, each of which consists of 12 connexin proteins. Several studies on gap junctions indicate that they are not simply static pores but are affected by age, pH, phosphorylation, and other factors (Connors and Long 2004). Indeed, they can be viewed as a dynamical signaling system that significantly contributes to neuronal network dynamics (Hormuzdi et al. 2004). Inhibitory networks play critical roles in generating population network rhythms (e.g., see Klausberger et al. 2003) and are known to be connected with gap junctions that can be located several hundred micrometers from their somata (Galarreta and Hestrin 2001; Hestrin and Galarreta 2005).
Interneurons, or inhibitory GABAergic neurons, in hippocampus are diverse in their cellular characteristics and have been shown and suggested to have active dendrites (Maccaferri et al. 2004; Martina et al. 2000; McBain and Fisahn 2001; Traub and Miles 1995). Furthermore, parvalbumin-positive GABAergic neurons, also known as basket cells, are known to contact one another via gap junctions on their distal dendrites (Fukuda and Kosaka 2000, 2003; Katsumaru et al. 1988; Kosaka and Hama 1985). Because gap junctions represent a form of direct, bidirectional electrical communication, it is reasonable to think of their primary role as that of mediating synchrony. However, theoretical and modeling studies show that although gap junctions can act to synchronize network output in firing (oscillating) neurons, they can also give rise to many other dynamic patterns including antiphase, bursting and other phase-locked states (Alvarez et al. 2002; Chow and Kopell 2000; Lewis and Rinzel 2003, 2004; Pfeuty et al. 2003; Saraga and Skinner 2004; Sherman and Rinzel 1992; Skinner et al. 1999). The particular network patterns that arise depend on cellular, intrinsic properties, which dictate intrinsic firing frequencies, as well as the strength and location of the gap junctions (Alvarez et al. 2002; Chow and Kopell 2000; Lewis and Rinzel 2003, 2004; Saraga and Skinner 2004; Sherman and Rinzel 1992; Skinner et al. 1999). To date, there are no modeling studies that examine how active, intrinsic properties of neuronal dendrites coupled with distal electrical coupling might affect network dynamics.
In this paper, we use two-cell network models of basket cells to explore how distal gap junction connections affect network output when their dendrites are passive or active by varying amounts. Our modeling work suggests that with distal dendritic gap junctions the level of active dendrites can control the sensitivity of network dynamics to gap junction modulation. Moreover, the presence of active dendrites might be required to obtain stable, phase-locked, near-synchronous states in such networks.
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METHODS |
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A 372-compartment model (1 somatic compartment and 371 dendritic compartments) was built using NEURON (Hines and Carnevale 1997). It is based on a hippocampal basket cell morphology taken from Gulyas et al. (1999) and http://www.koki.hu/
gulyas/ca1cells, as schematized in Fig. 1A and has a surface area of 18,069 µm2.
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 | (1) |
is the coupling conductance between different connected compartments, Iionic is the ionic current, and k denotes the particular compartment. The axial resistivity, rL, is 200 
cm. Note that i,j = a/(2 rL l2) if compartments i and j have the same length (l) and radius (a) (Dayan and Abbott 2001). The somatic compartment contains the traditional Hodgkin Huxley (HH) sodium (INa) and delayed-rectifier potassium (IK) currents, a leak current (IL), as well as an injected external current, Iext, in some cases. For some simulations, the dendrites were considered passive (i.e., only contain IL), whereas other simulations included active dendrites (i.e., contain INa, IK, and IL). Details for the kinetics and conductances of the ionic currents can be found in the APPENDIX. With passive dendrites, the basket cell model spontaneously fires at 12.7 Hz with a spike amplitude of 110 mV and an afterhyperpolarization (AHP) of 18 mV as measured from spike threshold (see Fig. 1B). It has an input resistance Rin = 245 M and a membrane time constant, 
m = 30 ms.
When considering active dendrites, we used a percentage of the sodium and potassium maximal conductances at the same density throughout the dendritic tree while the leak current and other passive properties were kept the same. Therefore passive dendrites are referred to as 0% active and fully active dendrites are referred to as 100% active. This choice of exploring active dendrites was made to allow us to focus on the level of active dendrites rather than particular dendritic ion channel types as done in other studies (Crook et al. 1998; Pfeuty et al. 2003). It is important to explore different ion channel amounts and distributions. However, although evidence exists that basket cells have active dendrites (Maccaferri et al. 2004), specifics are not yet available. Given the well-known diversity of hippocampal interneurons, we chose to "define" active dendrites as in the preceding text at this time. Also, any imbalance of inward and outward currents will affect the ability of the model to produce oscillations (action potentials) in the first place.
Electrotonic distances, L, were calculated in NEURON where electrotonic distance is defined as the natural logarithm of the voltage attenuation that is taken as the ratio of the voltage at the reference point (1 mV change) to the voltage at the endpoint (from a –65 mV holding potential). Note that this definition of electrotonic distance was developed to account for noninfinite cables and multiple branches with direction-dependent signal transfer (Carnevale and Johnston 1982; Carnevale et al. 1997). The electrotonic distance to, Vin, and from, Vout, the soma is shown in Fig. 1, C and , respectively, for the model with passive dendrites. In particular, Vin and Vout values for the dendrite shown in red in the schematic of Fig. 1A are shown in red in Fig. 1, C and D. At the end of this dendrite, Vin = 1.6 and Vout = 0.15.
Two-cell networks of model cells coupled by nonrectifying, dendritic gap junctions were constructed. The gap junction current, Igap,k between the kth dendritic compartments of cells 1 and 2 obeys the following equation(s)
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Distal (400 µm from soma) gap junctions mostly were examined here, but middle (200 µm from soma) and proximal (100 µm from soma) locations were also considered. These sites are used in several figures to illustrate the variation of voltage along the dendrite. The distal site was used to generate phase response curves. Our simulations minimally encompass a range of electrical connections designed to incorporate gap junctions with physiologically plausible values. These values are 10–3,000 pS based on a unitary gap junction conductance of 10–300 pS and 1–10 gap junction channels per electrical connection (Galarreta and Hestrin 2001; Srinivas et al. 1999), although these estimates may be conservative (Amitai et al. 2002). However, values up to 240 nS were explored in the simulations.
For the case of passive dendrites, different intrinsic frequencies were obtained by changing the level of injected current, Iext, into the soma (as in Fig. 5). This is not an unreasonable thing to do because it can be thought of as an estimate of the effect of summated synaptic input to the dendritic tree of the cell. However, with active dendrites, this is not easily justified since spikes can be dendritically initiated depending on the size and location of synaptic input (e.g., see Saraga et al. 2003). Therefore we did not try to adjust the intrinsic frequency of the cell using Iext when it had active dendrites. Instead, intrinsic frequency changes were brought about by changing the level of active dendrites as described in the preceding text. When two-cell heterogeneous networks were considered, the different intrinsic frequencies were due to different Iext values for the passive dendrite situation and different percentage active for the active dendrite situation.
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cm; 5-compartment models, rL = 160 Mcm; 10-compartment models, rL = 29 Mcm. Because two-compartment models required axial resistivities in excess of a million cm and that was beyond the limitation of the NEURON software, we used three compartments to represent our smallest (physical) neuron when considering the effects of geometry and when applying weak-coupling theoretical analyses.
Weak coupling theory has been used to analyze networks of neurons (e.g., Crook et al. 1998; Lewis and Rinzel 2003). This is a powerful theory as it allows one to say something about the coupled oscillator system from the behavior of the uncoupled system. It involves reduction to phase models and computing interaction functions from these reduced phase models. These interaction functions can then be used to predict stable phase-locked states. The theoretical predictions are valid for weak coupling but this often encompasses physiologically relevant regimes. The theory and method application is described in detail in Crook et al. (1998), Lewis and Rinzel (2003), and Ermentrout (2002). In this work, we have used XPPAUT (Ermentrout 2002) to compute interaction functions for our three-compartment models and so make predictions of network dynamics. These predictions are checked by performing the actual network simulations using XPPAUT or NEURON.
Simulation details
Simulations were performed with NEURON using a 25 µs time step. Initial membrane voltage was set to –65 mV for all cases. For simulations in which the two cells were homogeneous, a small perturbing current (100 pA for 5 ms) was injected into one of the cells within 100 ms from the start of the simulations so that the two cells would not fire in synchrony from the beginning. This was not necessary for the heterogeneous networks. Stable solutions were independent of the timing of this perturbing current.
To determine whether phase-locked behavior was present, the simulations were run for up to 100 s, and if the time lag between the action potential peaks was unchanged (or less than the step size resolution) for 3 s, phase-locking was taken to exist with a phase lag as measured. If not, then the behavior was characterized as not being phase-locked.
Note the following. To clarify our terminology, we use the term synchronous if the phase-lag is far from antiphase (approximately <20%), and we use the term pure synchrony for the case where the phase lag is zero. Using the term phase-locked encompasses behaviors in which lags could be zero (pure synchrony) up to 180° (antiphase).
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RESULTS |
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Active dendrites, intrinsic frequencies, and different dynamic regions
Whereas some active dendrite details of only one type of hippocampal interneuron are published at present (Martina et al. 2000), additional interneuron types also express active dendritic properties (Maccaferri et al. 2004). We expect the presence of active dendrites to affect the dynamics of networks coupled with distal gap junctions in various ways. In the absence of particular details, we explore a range of active dendrite levels considered as a percent of the maximal sodium and potassium conductances in the soma. This balanced exploration allows us to focus more generally on the level of active dendrites rather than the contribution of a particular dendritic ion channel type. How does changing the level of active dendrites affect the intrinsic firing frequency of the cell? One might expect that as the dendrites become more active, their firing frequency (as measured in the soma) would increase because the active dendrites would make it easier and faster to generate spikes. Figure 2A shows how the intrinsic frequency of the cell changes as the dendrites are made more active. Note that there is a highly nonmonotonic relationship. Specifically, there is first an increase in frequency with increasing active dendrite levels (0–0.5% active) and then from somewhere close to 0.8%, the frequency decreases till close to 5% active, after which it starts increasing again with further percentage-active increases.
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). Here our distal phase response curves are generated by injecting depolarizing inputs of a given amplitude and duration into the distal site of the cell (where the electrical coupling site would be) and measuring the phase shift that occurs. Note that this distal site is the same as shown in Fig. 2B, left. Stimuli of various amplitudes and durations were used (with similar results), but only two different (depolarizing) ones are shown in Fig. 2B, right. It is clear from these phase response curves that only certain levels of active dendrites allow both advances and delays to occur for perturbation phases near the somatic peak (i.e., <10% perturbation phase where 0% perturbation phase refers to the time of the somatic peak). The reason that delays for small perturbation phases could occur is due to the presence of refractoriness. If we think of the distal dendritic voltage response in Fig. 2B, left, as a "perturbation, " then we can see that only at certain levels of active dendrites is it possible to "delay" the somatic spiking (if appropriately timed and of sufficient amplitude in refractory period). This in turn implies that the intrinsic period can be extended so that the intrinsic frequency can actually decrease with increasing levels of active dendrites. In other words, the timing of the "distal perturbation" that is dictated by the level of active dendrites critically determines whether intrinsic frequencies would increase or decrease. However, for high enough levels of active dendrites the distal perturbation is large and closer to the somatic peak (relative to lower percentage-active values) as can be seen in Fig. 2B, left, for 10% active, and delays are not seen in the distal phase response curves (Fig. 2B, right). Thus intrinsic frequencies now increase with increases in percentage active. These observations suggest that there might be well-defined percentage-active regions that give rise to different classes of network dynamics. To get a better understanding of this nonmonotonic behavior, we made a reduced version of the full multi-compartment mode that consisted of only three compartments but such that the distal location is similar electrotonically, and the percentage active is done in the same way (see METHODS). The behavior of this three-compartment model is shown in Fig. 3. It is similar to the full multicompartment model. We obtain a nonmonotonic relationship between intrinsic frequency and percentage active (Fig. 3A) and phase response curves with negative phase shifts (Fig. 3B, right). The negative phase shifts in the distal phase response curves occur for intermediate values of percentage-active values (similar to the full model) during which there is a decrease in intrinsic frequency with increasing percentage active. If we consider that the beginning (with respect to increasing percentage-active values) of this intermediate region corresponds to when negative phase shifts occur, this intermediate region reflects the situation when the voltage amplitude (polarization) at the distal site equalizes and exceeds polarization amplitudes in sites more proximal to it. This can be seen for 1 and 1.5% active examples in Fig. 3B, left and right. These observations also hold for the full model described in the preceding text (as can be seen in Fig. 2).
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We want to understand how network dynamics change for variations in gap junction coupling strengths. Our explorations encompass physiological regimes (see METHODS), and we first examine passive dendrites where previous work has been done before moving on to describing network dynamics in which the dendrites are active.
Homogeneous networks with passive dendrites
Previous theoretical and modeling studies focused on passive dendrites and intrinsic frequency effects (brought about by changing the level of injected current). We know from these earlier works (Chow and Kopell 2000; Lewis and Rinzel 2003, 2004; Saraga and Skinner 2004) that pure synchrony can be achieved in two-cell homogeneous networks for lower intrinsic frequencies and more proximal gap junction locations. This is observed when single- and two-compartment representations are used for the individual cell. For these simpler individual cell representations, bistability at certain frequencies between antiphase and pure synchrony is also present for lower intrinsic frequencies.
Using our basket cell multicompartment model, we previously found that only pure synchrony and synchronous states occurred for gap junctions located more proximally (<about 200 µm from soma), but with more distal locations (400 µm from soma) synchronous and antiphase dynamical states were obtained (Saraga and Skinner 2004). Our simulations here show that with distal electrical connections the interneuronal networks achieve a pure synchrony state with increasing gap junction conductances if the intrinsic frequency is low enough (see Fig. 5). Therefore in accordance with previous knowledge, distal gap junction coupling can be seen to act as a "low-pass filter. " With intrinsic frequencies greater than 15 Hz, pure synchrony is not possible in our model networks for any size of gap junction conductance. With increases in gap junction conductances, the networks move away from the synchronous or pure synchrony state (depending on the intrinsic frequency) and move toward an antiphase phase-locked state. In Fig. 5, we show six different intrinsic frequencies on a semilog plot of percentage phase lag versus gap junction conductance, where percentage phase lag is calculated relative to the network period. A U-shaped relationship is present in which the U becomes more pronounced as the intrinsic frequency increases. At high enough intrinsic frequencies (60 Hz), the U becomes a straight line at 50% phase lag (i.e., antiphase). For these model networks with passive dendrites, the network frequency does not differ from the intrinsic frequency by any significant amount. There is a maximum of 1% difference between intrinsic and network frequencies, and this occurs at the highest gap junction conductances explored. Thus if percentage phase lags were calculated using intrinsic (rather than network) periods in Fig. 5, the plots would be essentially the same. Therefore the U shape in the percentage phase lag plot of Fig. 5 is not due to network frequency changes. Overall, we observe that there is a greatly expanded repertoire of phase-locked states present in the electrically coupled networks when the individual neuron has an extended geometry relative to single- or two-compartment model cells. This is partially due to the multicompartment nature of the models because only pure synchrony and antiphase patterns are obtained in two-cell networks using single-compartment models for the individual cells in which the spike shape characteristics do not change (Lewis and Rinzel 2003). Phase-locked patterns that are neither pure synchrony nor antiphase have been observed in two-compartment models with dendritic coupling (Lewis and Rinzel 2004) as well as in single-compartment models in which spike shape characteristics change dramatically (Chow and Kopell 2000).
This U-shaped relationship means that networks can exhibit the same phase lag at two different (high and low) electrical connection strengths. However, the existence of the high gap junction conductance may not occur given the physiologically plausible values for gap junctions (see METHODS). Why this U-shaped relationship occurs can be understood by appreciating that the electrical coupling affects the electrotonic characteristics of the cells as calculated in the outward direction from the soma, i.e., Vout (see Fig. 1D). This can be directly seen by observing the voltages along the dendritic tree. For example, consider networks in which the cells have intrinsic frequencies of 12.9 Hz so that there is a range of gap junction conductance values where pure synchrony is present (see Fig. 5). In Fig. 6, we show the membrane voltage in the soma and along the dendrite for four different gap junction conductances for these networks. For each case, Vout is also shown as measured at the distal site in one of the cells (see METHODS). The two cells are seen to go from a synchronous state to a state of pure synchrony and then back to a synchronous state with increasing gap junction conductances at this distal site. There is an increasing outward electrotonic distance with increasing gap junction conductance (see Fig. 6), indicating that there is more signal attenuation in the outward direction. Note that even when there is a 40-fold increase in gap junction conductance (from the 2nd to 3rd plot in Fig. 6), the distal voltage amplitude clearly decreases even though the network is still in a state of pure synchrony. The subsequent doubling of gap junction strength from the third to the fourth plot also results in a decrease in distal voltage amplitude so that the pure synchrony state can no longer be sustained. Although pure synchrony is considered lost because the somatic spikes are not aligned, the distal voltages are in a state of pure synchrony.
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As shown in the preceding text, the changing intrinsic frequency of the model cells with changing levels of active dendrites is nonmonotonic, and we used weak coupling theory to define LOW, MEDIUM, and HIGH percentage-active regions in terms of their network dynamics. In Fig. 7A, we show a three-dimensional (3D) plot of percentage phase lag versus percentage-active and gap junction conductance. The predicted two-cell network dynamics (for weak coupling) are apparent along the percentage-active axis. That is, for the LOW percentage-active region, there is an increase in the percentage phase lag, for the MEDIUM percentage active region, there is a decrease in the percentage phase lag, and for the HIGH percentage-active region, there is pure synchrony. Interestingly, these dynamic regions are also observed for higher gap junction conductance values where weak coupling theory may not strictly apply. The U-shaped dynamics described in the preceding text for passive dendrites can be seen along the ggap axis, being maintained for percentage-active values beyond passive. Thus the same phase lag can occur for both high and low gap junction conductances. However, unlike the passive dendrite situation, there are changes in the network period relative to the intrinsic period and with changes in the electrical connection strengths. This is shown in Fig. 7B for four different percentage-active values, 0.5, 1, 1.5, and 10% active, which correspond to LOW, MEDIUM, MEDIUM, and HIGH percentage-active regions, respectively, as defined in the preceding text. It is interesting to note that for the LOW and HIGH regions there is not a large difference between intrinsic and network periods until quite large gap junction conductances. However, for the MEDIUM regions, the network period changes significantly with gap junction conductance changes. Also note that for LOW and MEDIUM regions, the network period is always smaller than the intrinsic period. Thus frequency changes partially contribute to the U-shaped relationship seen in Fig. 7A along the ggap axis.
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14.5 to 50% (antiphase) for the gap junction conductances explored. In other words, at 1% active, the network dynamics are most sensitive to changes in gap junction conductance. As described in the preceding text, the MEDIUM percentage-active region refers to when percentage phase lags are decreasing with increasing percentage active, and phase response curves with negative phase shifts occur in this region as well as distal voltage amplitudes that are larger than more proximal ones. It may be possible to predict percentage-active values that give rise to network dynamics that are sensitive to distal gap junction conductance changes. Let us generate PPRCs from the two-cell networks with active dendrites that are distinct from the PRCs shown in Fig. 2B, right, which were obtained by injecting stimuli into distal sites. Unlike the situation when PRCs are generated with precisely controlled and timed stimuli, our stimuli are the distal voltages or distal electrical postsynaptic potentials (elPSPs) generated due to the cells being electrically coupled at the distal site. As such, we do not have direct control over the actual amplitude, duration and timing of the stimulus since the "stimulus" is simply whatever the elPSP response is at that distal site by virtue of the electrical coupling of various strengths. Thus points obtained on any particular PPRC are simply those in which an elPSP (i.e., the "stimulus") could be unambiguously seen in the two-cell network simulations for a particular percentage-active value for the range of gap junction conductances explored. Also one does not necessarily span the entire perturbation phase range with these elPSP stimuli because where they occur is not being directly controlled. In Fig. 8A, we show an example of phase-locked network oscillations in a two-cell network with the red cell lagging the blue cell. This particular example is for model cells with 1% active dendrites and with distal gap junctions of conductance 1,500 pS. The membrane voltage along the dendrite, with decreasing line thickness representing more distal sites, is shown in Fig. 8A. A distal "perturbation" or stimuli from the other cell can be seen for each cell. The time between this stimulus and the somatic spike is defined as b, the resulting phase shift (relative to the intrinsic period) is defined as c, and the intrinsic period is illustrated as a in Fig. 8A. With these definitions, we generate PPRCs for each percentage-active value where possible (i.e., where elPSP can be clearly seen). Note that we use a similar framework to PRC theory so that percentage phase shift and perturbation percentage phase are calculated relative to the intrinsic period. The PPRC for 1% active is shown in Fig. 8B where the point corresponding to the specific case (1,500 pS) illustrated in Fig. 8A is circled. The points on the PPRC plot of Fig. 8B reflect increasing ggap values starting from the smallest percentage phase shift values and outlining the "mushroom" shape of the PPRC. As observed in the preceding text, for LOW and MEDIUM percentage-active regions, the network period is always less than the intrinsic period. This means that percentage phase shifts will be positive for these regions. Due to our definitions and because the red cell lags the blue cell, the perturbation phase is <50% for the blue cell, whereas for the red cell, the perturbation phase is >50%.
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The effect of cell morphology or geometry on network dynamics is complex. As we are examining distal electrical coupling here, the actual physical location of gap junctions is a consideration that in turn depends on dendritic branch lengths and patterns. Together with the intrinsic characteristics of the cell, electrotonic lengths can be determined. To get some insight into the effect of geometry on our model results, we built several reduced versions of our full model (see METHODS). In each version, the outward electrotonic length was the same so that the distal location of the gap junctions was equivalent. However, because different numbers of (equal-sized) compartments were used, the physical length was different for each version. In Fig. 10, we show the percentage phase lag obtained in two-cell networks electrically coupled distally where the individual cell has 3, 5, and 10 compartments. For this illustration, 0.5% active dendrites are used. Note that as the dendrite length increases, a larger gap junction conductance is required for pure synchrony to be achieved. Also, longer (single) dendrites permit phase-locked states that are closer to antiphase for weak coupling. This is certainly not the whole story as branching patterns and different levels of active dendrites will also affect the dynamics. Perhaps quantitative morphological differences between different types of basket cells in different brain regions will become apparent as well as different levels of active dendrites and types of dendritic ion channels. We need full multicompartment reconstructions from which reduced versions can be derived and compared. However, these results do give us some sense of the effect of geometry on the network dynamics. In agreement with our previous work (Saraga and Skinner 2004), we see that gap junctions located at more distal physical locations require larger gap junction conductances to synchronize.
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Heterogeneous networks
It is known that heterogeneity (in terms of different intrinsic properties and afferent drives) in networks coupled by mutual inhibition can (mathematically) bring about different modes and mechanisms of attaining synchronous activity (e.g., see Skinner et al. 2005b; White et al. 1998). However, consideration of homogeneous mutual inhibitory networks help uncover interesting and nonintuitive dynamics in the first place (i.e., synchronous activity in purely inhibitory networks) (see Wang and Rinzel 1992). Consideration of heterogeneity is a step closer to more realistic physiological situations since biological networks do not consist of cells with identical properties. Therefore given our results above (i.e., U-shaped relationships, high sensitivity for certain percentage-active etc.) involving homogeneous networks coupled with distal gap junctions, we wondered whether our results would fall apart with the introduction of heterogeneity.
We introduce heterogeneity as a 6–9% difference in intrinsic frequencies brought about by differences in percentage- active in the two cells. First of all, with heterogeneous networks, one would not expect to get pure synchrony for any set of parameters. Second, not surprisingly we find that there is a reduced range of gap junction conductance values for which phase-locking does occur. Instead, we see a "meandering" or oscillating phase lag pattern so that there are no distinct lags to define a phase-locked relationship. In other words, the cells cannot entrain each other for the particular set of parameters used. In general, the more active the dendrites are (i.e., larger percentage-active), the smaller the gap junction conductance needs to be to obtain phase-locking. However, for the parameter values for which phase-locking does occur, we do still see the U-shaped relationship previously seen in the homogeneous networks. This can be seen in Fig. 11 in which percentage phase lag versus gap junction conductance is plotted for several different percentage-active levels in the heterogeneous networks. In the homogeneous networks, we observed that a larger range of phase lags occurred for ranges of gap junction conductance values when percentage-active values were in the MEDIUM percentage-active region. In the heterogeneous networks here, we see that this is also the case. The top curves in Fig. 11 are for percentage-active values that are in the MEDIUM region, and these curves are more U-shaped (i.e., express a larger range of phase lags) than the lower curves which are for percentage-active values in the HIGH region.
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13.5 to 21% for the range of gap junction conductances explored. Therefore the observations (of U-shaped relationships and sensitivity to gap junction conductance changes) obtained in the homogenous networks hold when heterogeneity is introduced. Finally, if the dendrites are made passive, phase-locked states do not occur in the electrically coupled networks at these distal sites with similar levels of heterogeneity. In summary, we can say that the modeling work indicates that it is advantageous to have gap junctions distally located if one wants to have a wide range of possible phase lags in phase-locked dynamic states as gap junction conductances are varied. There is a level of active dendrites in the MEDIUM percentage-active region that allows the network dynamics to respond most sensitively to modulations in the distal electrical connection strength. Furthermore, based on the observations with the heterogeneous networks, it may be the case that one requires active dendrites if phase-locking is to be obtained in networks formed with distal gap junctions.
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DISCUSSION |
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That PV-positive interneurons in hippocampus and neocortex are interconnected with gap junctions that can be distally located (Fukuda and Kosaka 2000 2003) has motivated this work. Freund (2003) has hypothesized that PV-containing basket cells represent the nonplastic clockwork without which no cortical operations are possible. We used a simplified model representation of a basket cell with a minimal set of voltage-gated channels in our modeling studies here. Although this model is far from complete in its representation of a basket cell, we think that our general results will hold as more detailed models of basket cells are developed when further experimental data becomes available. That is, additional details such as the addition of transient potassium, calcium, and other channels into the dendrites of multicompartment models will certainly affect characteristics of their dendritic spikes and other specifics, but the sensitivity of the network dynamics to gap junction conductance modulation is expected to remain if the distal phase response curve expresses delay characteristics (e.g., see Fig. 2B). Furthermore, some nonzero level of active dendrites is probably required for such networks to produce stable synchronous output (see Fig. 11). Exactly how distal is distal will depend on details of the model including morphology, but one could easily generate several phase response curves at different dendritic locations to determine what this might be. One could also use measures such as electrotonic distances available in NEURON to help gauge where such sites might be as has been done in another study in which spikelet amplitudes and delays were matched from experimental data (Saraga et al. 2005). With highly active dendrites, one could also use conduction speed to help predict where distal gap junction sites might be (Saraga et al. 2003) if sensitivity to gap junction modulation is to occur.
Do we expect our results to hold in larger networks? Although this can only be definitely answered by actually performing such simulations, several studies make us think that our two-cell model results should be manifest in larger networks. Using the theory of weakly coupled oscillators, Lewis and Rinzel (2003) and Pfeuty et al. (2003) examined pairs and large networks of phase-reduced models when electrically coupled. Although their neuronal models did not include extended geometries, they showed that one can predict the dynamic state of large networks of neurons coupled with electrical synapses by considering phase response functions which depend on the neuron's intrinsic properties (Pfeuty et al. 2003). Therefore we expect that the different phase-locked dynamic states (especially those approaching antiphase) and the sensitivity to gap junction conductance could give rise to population rhythms (i.e., synchronous activities) of different frequencies and durations. Chow and Kopell's (2000) two-cell analyses also have predicted correspondences in larger systems of all-to-all coupled neurons. Furthermore, we previously used simulations and bifurcation analyses to show that it is possible to link two-cell dynamics to larger network dynamics in which the networks are due to inhibitory coupling and the neurons have biophysical characteristics that are not identical (Skinner et al. 2005a). Thus it is not unreasonable to think that some linkage might also be possible with electrical coupling.
Population rhythms in hippocampus and neocortex arise due to some sort of coherent relationship between neurons in the underlying inhibitory neuronal networks (Buzsáki and Chrobak 1995). What are the important elements to consider in understanding how synchronous behaviors are produced in inhibitory networks? Modeling and experimental works have already suggested and shown how mutually inhibitory networks help the occurrence of this coherence (e.g., Freund 2003; Freund and Buzsáki 1996; Traub et al. 1996; Wang and Buzsáki 1996; White et al. 1998). Theoretical and modeling studies of networks with both electrical and inhibitory coupling have also been performed showing that electrical coupling does not always promote synchronous activity (Lewis and Rinzel 2003; Pfeuty et al. 2005). As usual, it depends on the characteristics of the underlying neurons. However, one also requires a large-scale focus and context as background activities (noise and heterogeneities) also need to be considered. With an inhibitory population rhythm context such as spontaneous rhythms that depend on electrical coupling as produced in an intact hippocampus preparation (Wu et al. 2002), it should be possible to determine background activities from the experimental data (Rudolph et al. 2004). In turn, this information can be used to directly link appropriate models to the experimental data and ultimately predict and understand the contribution of gap junctions to population rhythms. This will require shaping the model from experimental data and using intuition to give rise to key clues of the critical elements and deconstructing model details where possible to be able to perform mathematical analyses and thus gain insight into the underlying mechanisms (Kopell 2005).
The role of gap junctions is clearly not just about synchronization (Marder 1998), and we should consider expanding and refining this view to particular contexts involving locations and densities. For example, Migliore et al. (2005) have used multi-compartment models to examine gap junctions in the fine tuft dendrites of mitral cells and shown that they could play a major role in synchronizing action potential output despite their distal dendritic location. However, gap junctions may not play a role in synchronizing the output of subthreshold oscillations in inferior olivary neurons (Long et al. 2002). Furthermore, axonal electrical coupling may be needed for gamma oscillations to exist in vitro (Traub et al. 2003), and gap junctions may be needed to produce bursting oscillations in inhibitory networks (Skinner et al. 1999). LTP can be induced on interneurons (Perez et al. 2001), and it was recently shown that LTP changes the active properties of dendrites (Frick et al. 2004). In light of our present work showing that particular levels of active dendrites and distal gap junctions might "work together" to produce a wide range of phase lags in phase-locked states for changes in gap junction conductances, the suggested connection between active dendrites and memory capacity (Poirazi and Mel 2001) is intriguing.
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APPENDIX |
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 | (A1) |
 | (A2) |
 | (A3) |
 | (A4) |
 | (A5) |
 | (A6) |
 | (A7) |
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m = –0.1(V + 35)/{exp[–0.1(V + 35)] –1}, 
m = 4exp[–(V + 60)/18]. h is the sodium channel inactivation variable and h = 0.07exp[– (V + 58)/20], h = 1/{exp[–0.1(V + 28)] + 1}. n is the potassium channel activation variable and n = –0.01(V + 34)/{exp[–0.1(V + 34)]–1}, n = 0.125exp[–(V + 44)/80].
Kinetic models for these currents are based on Wang and Buzsáki (1996). The conductance values used are modified from Martina and Jonas (1997) and Martina et al. (1998), so that electrophysiological responses, such as current frequency relationship, spike amplitude, afterhyperpolarization, input resistance, and membrane time constant were approximately matched to those of hippocampal basket cells (Morin et al. 1996; van Hooft et al. 2000) when dendrites were kept passive.
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GRANTS |
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Address for reprint requests and other correspondence: F. Skinner, Div. of Cell and Molecular Biology, Toronto Western Research Institute, Toronto Western Hospital, 399 Bathurst St., MP13-317, Toronto, Ontario M5T 2S8, Canada (E-mail: fskinner{at}uhnres.utoronto.ca)
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ABSTRACT
INTRODUCTION
), 15.1 Hz (
), 22 Hz (
), 30 Hz (
), 40 Hz (
), and 60 Hz (
), the injected current to each model cell of the 2-cell network is 0.35, 3.45, 15, 31, 55.5, and 125 pA, respectively. Note the U-shaped relationship that exists at these different frequencies until the higher 60 Hz frequency, which produces antiphase spiking dynamics. Schematic of 2-cell network with a distal (
); 28 and 18% active (8.3% heterogeneity, 
); 10 and 5% active (7.2% heterogeneity,