Application of stochastic automata networks for creation of continuous time Markov chain models of voltage gating of gap junction channels

Properties of cardiac conduction in a cell-based computational model

The conduction of electrical signals through cardiac tissue is essential for maintaining the function of the heart, and conduction abnormalities are known to potentially lead to life-threatening arrhythmias. The properties of cardiac conduction have therefore been the topic of intense study for decades, but a number of questions related to the mechanisms of conduction still remain unresolved. In this paper, we demonstrate how the so-called EMI model may be used to study some of these open questions. In the EMI model, the extracellular space, the cell membrane, the intracellular space and the cell connections are all represented as separate parts of the computational domain, and the model therefore allows for study of local properties that are hard to represent in the classical homogenized bidomain or monodomain models commonly used to study cardiac conduction. We conclude that a non-uniform sodium channel distribution increases the conduction velocity and decreases the time delays over gap junctions of reduced coupling in the EMI model simulations. We also present a theoretical optimal cell length with respect to conduction velocity and consider the possibility of ephaptic coupling (i.e. cell-to-cell coupling through the extracellular potential) acting as an alternative or supporting mechanism to gap junction coupling. We conclude that for a non-uniform distribution of sodium channels and a sufficiently small intercellular distance, ephaptic coupling can influence the dynamics of the sodium channels and potentially provide cell-to-cell coupling when the gap junction connection is absent.

The electrochemical wave traversing the heart during every beat is essential for cardiac pumping function and supply of blood to the body. Understanding the stability of this wave is crucial to understanding how lethal arrhythmias are generated. Despite this importance, our knowledge of the physical determinants of wave propagation are still evolving. One particular challenge has been the lack of accurate mathematical models of conduction at the cellular level. Because cardiac muscle is an electrical syncytium, in which direct charge transfer between cells drives wave propagation, classical bidomain and monodomain tissue models employ a homogenized approximation of this process. This approximation is not valid at the length scale of single cells, and prevents any analysis of how cellular structures impact cardiac conduction. Instead, so-called microdomain models must be used for these questions. Here we utilize a recently developed modelling framework that is well suited to represent small collections of cells. By applying this framework, we show that concentration of sodium channels at the longitudinal borders of myocytes accelerates cardiac conduction. We also demonstrate that when juxtaposed cells are sufficiently close, this non-uniform distribution induces large ephaptic currents, which contribute to intercellular coupling.

Ephaptic coupling rescues conduction failure in weakly coupled cardiac tissue with voltage-gated gap junctions

Electrical conduction in cardiac tissue is usually considered to be primarily facilitated by gap junctions, providing a pathway between the intracellular spaces of neighboring cells. However, recent studies have highlighted the role of coupling via extracellular electric fields, also known as ephaptic coupling, particularly in the setting of reduced gap junction expression. Further, in the setting of reduced gap junctional coupling, voltage-dependent gating of gap junctions, an oft-neglected biophysical property in computational studies, produces a positive feedback that promotes conduction failure. We hypothesized that ephaptic coupling can break the positive feedback loop and rescue conduction failure in weakly coupled cardiac tissue. In a computational tissue model incorporating voltage-gated gap junctions and ephaptic coupling, we demonstrate that ephaptic coupling can rescue conduction failure in weakly coupled tissue. Further, ephaptic coupling increased conduction velocity in weakly coupled tissue, and importantly, reduced the minimum gap junctional coupling necessary for conduction, most prominently at fast pacing rates. Finally, we find that, although neglecting gap junction voltage-gating results in negligible differences in well coupled tissue, more significant differences occur in weakly coupled tissue, greatly underestimating the minimal gap junctional coupling that can maintain conduction. Our study suggests that ephaptic coupling plays a conduction-preserving role, particularly at rapid heart rates.

Stochastic Model of Gap Junctions Exhibiting Rectification and Multiple Closed States of Slow Gates

Gap-junction (GJ) channels formed from connexin (Cx) proteins provide direct pathways for electrical and metabolic cell-cell communication. Earlier, we developed a stochastic 16-state model (S16SM) of voltage gating of the GJ channel containing two pairs of fast and slow gates, each operating between open (o) and closed (c) states. However, experimental data suggest that gates may in fact contain two or more closed states. We developed a model in which the slow gate operates according to a linear reaction scheme, o↔c1↔c2, where c1 and c2 are initial-closed and deep-closed states that both close the channel fully, whereas the fast gate operates between the open state and the closed state and exhibits a residual conductance. Thus, we developed a stochastic 36-state model (S36SM) of GJ channel gating that is sensitive to transjunctional voltage (Vj). To accelerate simulation and eliminate noise in simulated junctional conductance (gj) records, we transformed an S36SM into a Markov chain 36-state model (MC36SM) of GJ channel gating. This model provides an explanation for well-established experimental data, such as delayed gj recovery after Vj gating, hysteresis of gj-Vj dependence, and the low ratio of functional channels to the total number of GJ channels clustered in junctional plaques, and it has the potential to describe chemically mediated gating, which cannot be reflected using an S16SM. The MC36SM, when combined with global optimization algorithms, can be used for automated estimation of gating parameters including probabilities of c1↔c2 transitions from experimental gj-time and gj-Vj dependencies.

Reverberation of excitation in neuronal networks interconnected through voltage-gated gap junction channels

Voltage-dependent gap junction gating contributes to reverberation in neuronal circuits.

We combined Hodgkin–Huxley equations and gating models of gap junction (GJ) channels to simulate the spread of excitation in two-dimensional networks composed of neurons interconnected by voltage-gated GJs. Each GJ channel contains two fast and slow gates, each exhibiting current–voltage (I-V) rectification and gating properties that depend on transjunctional voltage (V_j). The data obtained show how junctional conductance (g_j), which is necessary for synchronization of the neuronal network, depends on its size and the intrinsic firing rate of neurons. A phase shift between action potentials (APs) of neighboring neurons creates bipolar, short-lasting V_j spikes of approximately ±100 mV that induce V_j gating, leading to a small decay of g_j, which can accumulate into larger decays during bursting activity of neurons. We show that I-V rectification of GJs in local regions of the two-dimensional network of neurons can lead to unidirectional AP transfer and consequently to reverberation of excitation. This reverberation can be initiated by a single electrical pulse and terminated by a low-amplitude pulse applied in a specific window of reverberation cycle. Thus, the model accounts for the influence of dynamically modulatable electrical synapses in shaping the function of a neuronal network and the formation of reverberation, which, as proposed earlier, may be important for the development of short-term memory and its consolidation into long-term memory.