We combined the Hodgkin–Huxley equations and a 36-state model of gap junction channel gating to simulate electrical signal transfer through electrical synapses. Differently from most previous studies, our model can account for dynamic modulation of junctional conductance during the spread of electrical signal between coupled neurons. The model of electrical synapse is based on electrical properties of the gap junction channel encompassing two fast and two slow gates triggered by the transjunctional voltage. We quantified the influence of a difference in input resistances of electrically coupled neurons and instantaneous conductance–voltage rectification of gap junctions on an asymmetry of cell-to-cell signaling. We demonstrated that such asymmetry strongly depends on junctional conductance and can lead to the unidirectional transfer of action potentials. The simulation results also revealed that voltage spikes, which develop between neighboring cells during the spread of action potentials, can induce a rapid decay of junctional conductance, thus demonstrating spiking activity-dependent short-term plasticity of electrical synapses. This conclusion was supported by experimental data obtained in HeLa cells transfected with connexin45, which is among connexin isoforms expressed in neurons. Moreover, the model allowed us to replicate the kinetics of junctional conductance under different levels of intracellular concentration of free magnesium ([Mg2+]i), which was experimentally recorded in cells expressing connexin36, a major neuronal connexin. We demonstrated that such [Mg2+]i-dependent long-term plasticity of the electrical synapse can be adequately reproduced through the changes of slow gate parameters of the 36-state model. This suggests that some types of chemical modulation of gap junctions can be executed through the underlying mechanisms of voltage gating. Overall, the developed model accounts for direction-dependent asymmetry, as well as for short- and long-term plasticity of electrical synapses. Our modeling results demonstrate that such complex behavior of the electrical synapse is important in shaping the response of coupled neurons.
In most models of neuronal networks, it is assumed that electrical synapses exhibit constant conductance, and that electric synaptic transmission is bidirectional and symmetric. However, experimental studies show that these assumptions are not always satisfied. For instance, some synapses formed of gap junction channels exhibit an instantaneous conductance–voltage rectification, which promotes a direction-dependent asymmetry of electrical signaling [1–4]. In addition, all members of the connexin (Cx) family forming gap junction channels exhibit a sensitivity of junctional conductance to the transjunctional voltage [5]. Moreover, voltage sensitivity of gap junctions can be strongly affected by chemical factors, e. g. by intracellular concentrations of H+, Ca2+ or Mg2+ [6–8]. Thus, electrical synapses are not just passive pores, but can exhibit dynamic changes of junctional conductance. Presumably, these changes in electrical synaptic strength could affect the transfer of an electrical signal. The purpose of our study was to develop a computational model for evaluation of such an interaction between electrical synapses and signal transmission between coupled neurons. The first quantitative models describing equilibrium [9,10] and kinetic [11] properties of junctional conductance dependence on transjunctional voltage were based on the assumption that the channel can be in two states, open and closed. Later, single channel studies have shown that transjunctional voltage causes channels to close to a subconductance (residual) state [12,13] with fast gating transitions, and to a fully closed state with slow gating transitions [14,15]. Thereafter, it was proposed that gap junction channels comprise two types of gating mechanisms, fast and slow, each exhibiting rectification of their unitary conductances depending on the voltage across them. These properties were described in a stochastic 16-state model (16SM) of gap junction channel gating [16] in which fast and slow gates operate between open (o) and closed (c) states. However, experimental data from our and other groups [17,18] allowed us to suggest that the slow gate operates between open (o), initial-closed (c1) and deep-closed (c2) states. Such a suggestion was implemented in a 36-state model (36SM) of voltage gating [19]. The 36SM allowed us to reproduce experimentally observed gating behavior of gap junction channels more adequately than 16SM, especially regarding the kinetics of conductance recovery, or a low fraction of functional channels clustered in junctional plaques. Earlier [20], we combined a 16SM of gap junction channel gating and rectification with the Hodgkin–Huxley (HH) equations [21]. The developed model (HH-16SM) allowed us to evaluate the kinetics of junctional conductance during the spread of excitation in neuronal networks. In this study, we replaced 16SM with 36SM for a better evaluation of junctional conductance kinetics. We applied the combined model (HH-36SM) to investigate the signal transfer between electrically coupled neurons in response to different types of presynaptic inputs, such as electrotonic signals or action potentials (APs). In this study, we analyse three main aspects of the 36SM with respect to the functional behavior of electrical synapses. Firstly, transjunctional voltage distribution across each channel gate can result to almost instantaneous asymmetric conductance-voltage rectification of gap junction channel. We showed that such rectification of gap junctions can affect the asymmetry of the electrical cell-to-cell signaling, especially the spread of a single AP. Secondly, in the 36SM, the gap junction channel can transit between open and closed states, and probabilities of these transitions depend on voltage across each channel gate. We demonstrated that closing of gap junction channels could be induced by transjunctional voltage spikes, which develop during the spread of excitation. More precisely, our modeling results showed that voltage spikes induced by the trains of APs can cause an accumulation of gap junction conductance decay. As a result, the junctional conductance can significantly decrease in just a few seconds, and substantially modulate electrical signaling between neurons. This short-term plasticity of electrical synapses was supported by our electrophysiological experiments in HeLa cells expressing connexin45. Thirdly, we suggested that some types of chemical modulation of electrical synapses could be explained by an assumption that the values of 36SM parameters depend on chemical factors. Under such a hypothesis, the chemical modulator would influence the junctional conductance by modifying voltage sensitivity properties of a gap junction channel. In this case, the chemically-induced variation of junctional conductance would be explained by the changed equilibrium of open and closed voltage sensitive channels, and not by a separate chemical gate. To illustrate the feasibility of this idea, we fitted the 36SM to explain the kinetics of connexin36 gap junctional conductance under different concentrations of free magnesium ions ([Mg2+]i). We demonstrated that a long-term (a few minutes) plasticity, which is induced by variation in [Mg2+]i, can be adequately reproduced through the changes of 36SM parameters. Thus, the presented model accounts for the complex behavior of electrical synapses under a wide variety of voltage and temporal conditions. Moreover, all these phenomena can be explained by the underlying mechanisms of gap junction channel voltage gating. Such a modeling approach allows one to evaluate the response of neuronal networks, which would be very difficult to measure experimentally.
Junctional conductance of electrical synapses was evaluated using a Markov chain 36-state model of voltage gating, which is detailed in [19]. The model describes the probabilistic behavior of gap junction channels in response to the transjunctional voltage. In the 36SM, the gap junction channel consists of two hemichannels, each enclosing one fast and one slow gates (Fig 1). Thus, the channel comprises four gates (fast left, slow left, slow right and fast right), all arranged in series (Fig 1B). The fast and the slow gates operate according to a linear kinetic schemes, o↔c and o↔c1↔c2, respectively (Fig 1A). Thus, gap junction channel can be in 36 (2∙3∙3∙2) different states, and overall junctional conductance is estimated as an averaged value of each state conductance weighted to their probabilities. Transition probabilities between system states depend on transjunctional voltage distribution across gates, which must be evaluated first. In general, the voltage distribution can be nonlinear due to rectification of unitary conductances of channel gates. The developed HH-36SM combines Hodgkin–Huxley equations that describe excitability of neurons and a 36-state model (36SM) of gap junction channel gating that evaluates conductance of the electrical synapse. More precisely, membrane voltages of the neurons are estimated using the Hodgkin-Huxley model. The resulting transjunctional voltage can affect the junctional conductance, which is evaluated using the 36SM. Thus, the HH-36SM allowed us to simulate electrical signal transfer between neurons connected through modulatable gap junctions.
Asymmetry of electrical synaptic transmission has been observed in numerous studies [41–44]. Such asymmetry might arise due to differences in input resistances (Rins) of coupled neurons [29], even when gap junctions themselves are symmetric. Rin depends on the conductivity of the plasma membrane and its surface area, as well as on the number of neighboring neurons connected through electrical synapses. Another source of electrical synaptic transmission asymmetry is related to instantaneous conductance–voltage rectification of gap junction channel, which results from the inhomogeneous distribution of charged amino acids lining the pore [45]. Such rectification of gap junction channels typically arises in heterotypic junctions under normal conditions [4,17,46], but it can also develop in homotypic gap junctions under an asymmetry of intracellular milieu, e. g. gradients of [Mg2+]i [47]. In addition, electrical signaling asymmetry across heterotypic channels can arise with repeated stimulation due to voltage gating (see Fig 7). This type of asymmetry in electric synaptic transmission is not instantaneous and depends on past history. The other factors that contribute to asymmetry of signaling do not have this property. Our data show that asymmetry in electrotonic cell-to-cell communication is more affected by the difference in Rins of coupled cells (see Figs 3D and 5D), while gap junctional rectification primarily influences an asymmetry of AP transfer between neurons (Fig 5A–5C). This can be explained by the conductance–voltage curves in Fig 2, which show that conductance changes are small at low voltages (±10 mV), which typically arise during measurements of coupling coefficients. Significant changes of junctional conductance can only be expressed at high voltages (±100 mV), which develop during the spread of excitation. We believe that these observations might have practical applications in electrophysiological experiments when studying the strength and rectification properties of electrical synapses. The aforementioned sources of functional asymmetry are independent by nature, e. g. Rin of a neuron directly depends on plasma membrane area, while synaptic rectification is determined by properties of gap junction channels [48]. Thus, they can act antagonistically promoting bidirectionality of electrical synapses, as was demonstrated in the teleost auditory system [4]. Alternatively, if rectification of gap junctions and differences in Rins acted synergistically, it could facilitate unidirectional AP transfer. Thus, unidirectionality, which is a genuine property of chemical synapses, could be executed through electrical synapses alone. Because electrical synaptic transmission is faster than chemical, unidirectional spread of AP through gap junctions might be useful in rapid response warranting behavior such as escape reflex [49,50]. Asymmetry of electrical synaptic transmission plays an important role in spike-timing regulation, as was demonstrated in neurons of the thalamic reticular nucleus [44]. In larger networks, even a small asymmetry would add up during the spread of excitation and could significantly affect the latency of AP transfer along neural pathways. This process could be crucial in temporal coding activities, such as coincidence detection, in which gap junctions are reported to play an important role [50]. Presumably, the effect of asymmetry of electrical signaling would be difficult to measure and observe experimentally in highly complex neuronal networks, and a simulation-based approach could provide valuable insights on the role of rectification in network dynamics [51]. It is well established that gap junctional conductance depends on voltage [10]. Our previous [20] and current modeling studies show that decay of junctional conductance can be induced by voltage gating of gap junction channels during bursting activity of neurons. To our knowledge, at least one study reported such spiking activity-dependent reduction of electrical synaptic strength in brain slices [52]. Our data showed that even in gap junctions formed of low-voltage-sensitive Cx36, this decay exceeds 10% while in more voltage-sensitive Cx isoforms it could reach ~50% over several seconds (Figs 6 and 7). The magnitude of junctional conductance decrease and duration of its recovery depends not only on Cx properties but also on the firing rates of neurons (Fig 6). Because the transfer of electrical signal and its asymmetry depends on junctional conductance [53], an activity-induced inhibition of electrical synapses can significantly diminish (Fig 6A-b) or even abolish AP transfer between neurons (Fig 8C-c). Such a role of electrical synaptic plasticity was acknowledged in [54] and was demonstrated by an activity-dependent decrease of junctional conductance together with enhanced asymmetry of electrical synaptic transmission in TRN slices [52]. Heterotypic gap junctions exhibit structure-determined voltage-gating asymmetry, which could result in even more diverse functional behavior with respect to plasticity and directionality than homotypic gap junction. As we showed in Fig 8, changes in junctional conductance and the response rate of neurons depends on the direction of AP spread with respect to the orientation of heterotypic gap junctions. Thus, heterotypic synapses could promote direction-dependent asymmetry of electrical signal transfer not only by its rectification properties but by asymmetric voltage gating as well. We presume that such processes might have an important functional role in sensory systems where heterotypic electrical synapses are detected [55,56]. Regulation of the strength of electrical synapses by a variety of chemical reagents is well established. Others and our data showed that junctional conductance decay caused by chemical uncouplers can be reversed by voltage, while some chemical factors can change voltage sensitivity of Cxs [8,38,39,57]. These observations, as well as the fact that all known chemical uncouplers close gap junction channels fully but not to residual conductance, suggest that some chemical factors act through the slow gate. We implemented this idea by simulating Mg2+-mediated changes in junctional conductance of Cx36 gap junctions using the 36SM. The obtained data revealed that an effect of [Mg2+]i can be relatively well reproduced (Fig 9) assuming variation in values of 36SM parameters, mainly V0 and probabilities of c1↔c2 transitions of slow gates. Moreover, because the voltage sensitivity of the gap junction channels is defined by the same parameters (see Fig 4 in [20] and Fig 6 in this paper), chemically modulated gating would also affect spiking activity-dependent short-term plasticity of the electrical synapse. Our modeling results showed that even a moderate change (±20%) in [Mg2+]i could result in very significant differences in the spread of APs between two neurons (see Fig 10). Thus, chemically modulated gating of Cx36 can expand the time window of electrical synaptic plasticity for as long as chemical factors are present, which could last for minutes or even hours. Therefore, even Cx36, which exhibits relatively low voltage sensitivity, could act as a highly modulatable constituent of neuronal networks due to chemically mediated gating. Our modeling results show that a persistent spiking activity or chemical factors could keep a significant proportion of gap junction channels in a closed state. We assume that this process could offer at least a partial explanation to a well-documented ‘low functionality’ of gap junctions, especially those expressed in excitable cells, such as neurons or cardiomyocytes. Low functionality refers to a small fraction of channels residing in the open (or high conductance) state. This applies to all connexins, such as Cx36 [58], Cx43 [26], Cx45 [57] and Cx57 [59], examined on this issue, and likely applies to other Cx isoforms. The strength of electrical synapses directly affects the level of synchronization in neuronal networks, which can underlie various physiological processes and pathological brain conditions. For example, increased cortical synchronization correlates with reduced information processing capability in the primary auditory cortex [60]. The rise in junctional conductance can lead to over-synchronization, which is associated with episodes of epileptic seizures. Interestingly, an activity-induced decrease in the coupling of electrical synapses through an intracellular Ca2+ mechanism was observed in the thalamic reticular nucleus of epileptic rats and was proposed to act as a compensatory mechanism to reduce excessive synchronization [61]. Thus, both voltage- and chemically induced gating of gap junction channels can play an important role in shaping activity of neuronal networks through modulation of neuronal synchrony. In addition, short-term plasticity induced through voltage gating of electrical synapses could contribute to lateral inhibition and resulting center-surround effect, which is important in sensory systems of the CNS. This hypothesis is supported by studies showing that more voltage-sensitive Cx isoforms are expressed in the structures associated with sensory functions. For example, one of the most voltage-sensitive Cxs, mouse Cx57 and its human homolog Cx62 are expressed in horizontal cells of the retina [62], while Cx45, which is significantly more voltage-sensitive than Cx36, predominates in the olfactory bulb [34]. The chemically mediated gating could play an important role in regulating longer term changes, especially in less-voltage-sensitive Cx36. For example, it was reported that Cx36 plays an important role in shifting between sleep and wake states [63]. We believe that the unique sensitivity of Cx36 to Mg2+ could contribute to this process. This view is supported by accompanying changes in ATP levels, which effectively influence [Mg2+]i. It was reported that ATP levels increase during the initial hours of sleep in wake-active regions of rat brain [64]. This should decrease [Mg2+]i and, consequently, increase conductance of Cx36 gap junctions. As a result, an increased synchronization could suppress activities in brain regions associated with the waking state, thus maintaining sleep. In this study, we used the Hodgkin–Huxley equations to describe excitability of neurons. The developed model can be adapted to various brain regions and circuits by choosing an appropriate set of ionic currents. For example, the inclusion of Ca2+ currents, which underlie bursting trains of APs in thalamic relay neurons [65], might be relevant for short-term plasticity as well as for chemical modulation of electrical synapses. Furthermore, major principles used to develop an HH-36SM can be applied in cardiac tissue modeling, provided that the Hodgkin–Huxley equations are replaced by those specific for cardiomyocytes [66,67]. Cardiomyocytes are predominantly connected through Cx43, Cx40 and Cx45, which are more voltage sensitive than Cx36; therefore, it might exhibit more expressed activity-dependent conductance decrease, especially during tachyarrhythmias. Furthermore, chemically mediated gating of cardiac gap junction channels, e. g. by acidification [68], could be important in describing enhanced arrhythmogenicity of the ischemic myocardium [69]. Obviously, the 36SM of gap junction channel voltage gating is a simplification of complex processes underlying changes of electrical synaptic strength. However, we believe that rectification and voltage gating properties of gap junction channel can be reasonably well reproduced using the 36SM. On the other hand, an inclusion of chemical modulation into 36SM is far less explored. So far we made only the first steps in this direction to explain Cx36 mediation by [Mg2+]i, and presented modeling results (Fig 10) are obtained from just a few data points. Moreover, cytosolic conditions are rarely defined by a single chemical factor, and various different reagents might affect electrical synapses synergistically or antagonistically. For example, our preliminary data suggest that [Mg2+]i effect on Cx36 gap junctions might depend on the pH level. In addition, modulation of electrical synapses by other chemical reagents, such as Ca2+ ions, might be more relevant for the spread of excitation than that of [Mg2+]i. In this study, we simulated electrical synaptic transmission between two cells connected through a soma-somatic gap junction. For a more realistic neuronal network simulation, it would be beneficial to include dendro-dendritic connections, which are far more prevalent in mammalian brain. Another important extension of our model would be an inclusion of chemical synapses. Presumably, this would allow one to study an interaction between chemical and electrical synapses, which was observed in numerous experimental studies [70]. However, all physiologically relevant extensions, and especially an increased number of cells and synapses, might require a large amount of computational recourses. To our knowledge, Hodgkin-Huxley type models are rarely applied for large neuronal network simulation due to computation time constraints. This problem would be enhanced by our modeling approach, because evaluation of junctional conductance using the 36SM consumes ~95 percent of overall computation time. We presume that simulation time could be decreased by two different approaches: 1) Creation of a more simplistic model of gap junction voltage gating, which would roughly describe relative changes of junctional conductance in response to a single AP. Somewhat similar approach is applied in mathematical models of chemical synapses [71]. This would allow one to combine a model of electrical synapse with integrate-and-fire type models, which are often used for simulation of large neuronal networks. 2) Application of advanced computation techniques, such as an extensive parallelization together with graphic processing unit computation.
In most computational models of neuronal networks, it is assumed that electrical synapses have a constant and ohmic conductance. However, numerous experimental studies demonstrate that connexin-based channels expressed in neuronal gap junctions can change their conductance in response to a transjunctional voltage or various chemical reagents. In addition, electrical synapses may exhibit direction-dependent asymmetry of signal transfer. To account for all these phenomena, we combined a 36-state model of gap junction channel gating with Hodgkin–Huxley equations, which describes neuronal excitability. The combined model (HH-36SM) allowed us to evaluate the kinetics of junctional conductance during the spread of electrical signal or in response to chemical factors. Our modeling results, which were based on experimental data, demonstrated that electrical synapses exhibit a complex behavior that can strongly affect the response of coupled neurons. We suggest that the proposed modeling approach is also applicable to describe the behavior of cardiac or other excitable cell networks interconnected through gap junction channels.
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Functional asymmetry and plasticity of electrical synapses interconnecting neurons through a 36-state model of gap junction channel gating