Gap Junctions, Dendrites and Resonances: A Recipe for Tuning Network Dynamics
© Y. Timofeeva et al.; licensee Springer 2013
Received: 20 December 2012
Accepted: 12 April 2013
Published: 14 August 2013
Gap junctions, also referred to as electrical synapses, are expressed along the entire central nervous system and are important in mediating various brain rhythms in both normal and pathological states. These connections can form between the dendritic trees of individual cells. Many dendrites express membrane channels that confer on them a form of sub-threshold resonant dynamics. To obtain insight into the modulatory role of gap junctions in tuning networks of resonant dendritic trees, we generalise the “sum-over-trips” formalism for calculating the response function of a single branching dendrite to a gap junctionally coupled network. Each cell in the network is modelled by a soma connected to an arbitrary structure of dendrites with resonant membrane. The network is treated as a single extended tree structure with dendro-dendritic gap junction coupling. We present the generalised “sum-over-trips” rules for constructing the network response function in terms of a set of coefficients defined at special branching, somatic and gap-junctional nodes. Applying this framework to a two-cell network, we construct compact closed form solutions for the network response function in the Laplace (frequency) domain and study how a preferred frequency in each soma depends on the location and strength of the gap junction.
It has been known since the end of the nineteenth century and mainly from the work of Ramón y Cajal  that neuronal cells have a distinctive structure, which is different to that of any other cell type. The most extended parts of many neurons are dendrites. Their complex branching formations receive and integrate thousands of inputs from other cells in a network, via both chemical and electrical synapses. The voltage-dependent properties of dendrites can be uncovered with the use of sharp micropipette electrodes and it has long been recognised that modelling is essential for the interpretation of intracellular recordings. In the late 1950s, the theoretical work of Wilfrid Rall on cable theory provided a significant insight into the role of dendrites in processing synaptic inputs (see the book of Segev et al.  for a historical perspective on Rall’s work). Recent experimental and theoretical studies at a single cell level reinforce the fact that dendritic morphology and membrane properties play an important role in dendritic integration and firing patterns [3–5]. Coupling neuronal cells in a network adds an extra level of complexity to the generation of dynamic patterns. Electrical synapses, also known as gap junctions, are known to be important in mediating various brain rhythms in both normal [6, 7] and pathological [8–10] states. They are mechanical and electrically conductive links between adjacent nerve cells that are formed at fine gaps between the pre- and post-synaptic cells and permit direct electrical connections between them. Each gap junction contains numerous connexon hemi-channels, which cross the membranes of both cells. With a lumen diameter of about 1.2 to 2.0 nm, the pore of a gap junction channel is wide enough to allow ions and even medium-sized signalling molecules to flow from one cell to the next thereby connecting the two cells’ cytoplasm. Being first discovered at the giant motor synapses of the crayfish in the late 1950s, gap junctions are now known to be expressed in the majority of cell types in the brain . Without the need for receptors to recognise chemical messengers, gap junctions are much faster than chemical synapses at relaying signals.
Earlier theoretical studies demonstrate that although neuronal gap junctions are able to synchronise network dynamics, they can also contribute toward the generation of many other dynamic patterns including anti-phase, phase-locked and bistable rhythms . However, such studies often ignore dendritic morphology and focus only on somato-somatic gap junctions. In the case of dendro-dendritic coupling, simulations of multi-compartmental models reveal that network dynamics can be tuned by the location of the gap junction on the dendritic tree [13, 14]. Here, we develop a more mathematical approach using the continuum cable description of a dendritic tree (either passive or resonant) that can compactly represent the response of an entire dendro-dendritic gap junction coupled neural network to injected current using a response function. This response function, often referred as a Green’s function, describes the voltage dynamics along a network structure in response to a delta-Dirac pulse applied at a given discrete location. Our work is based on the method for constructing the Green’s function of a single branched passive dendritic tree as originally proposed by Abbott et al. [15, 16] and generalised by Coombes et al.  to treat resonant membrane (whereby subthreshold oscillatory behaviour is amplified for inputs at preferential frequencies determined by ionic currents such as ). This “sum-over-trips” method is built on the path integral formulation and calculates the Green’s function on an arbitrary dendritic geometry as a convergent infinite series solution.
In Sect. 2, we introduce the network model for gap junction coupled neurons. Each neuron in the network comprises of a soma and a dendritic tree. Cellular membrane dynamics are modelled by an ‘LRC’ (resonant) circuit. In Sect. 3, we focus on an example of two unbranched dendritic cells, with no distinguished somatic node, with identical and heterogeneous sets of parameters and give the closed form solution for network response with a single gap junction. The complete “sum-over-trips” rules for the more general case of an arbitrary network geometry are also presented. In Sect. 4, we apply the formalism to a more realistic case of two coupled neurons, each with a soma and a branching structure. We introduce a method of ‘words’ to construct compact solutions for the Green’s function of this network and study how a preferred frequency in each soma depends on the location and strength of the gap junction. Finally, in Sect. 5, we consider possible extensions of the work in this paper.
2 The Model
where is the conductance of the gap junction and and ( and ) are two segments of branch m (branch n) connected at the gap junction (see Fig. 1a). The expressions in (10) reflect continuity of the potential across individual branches m and n, and Eqs. (11)–(12) enforce conservation of current.
where describes the initial conditions on branch k and the sum is over all branches of the tree. Multiple external stimuli can be tackled by simply adding new terms with additional inputs to Eq. (13).
3 The Green’s Function on a Network
Earlier work of Coombes et al.  demonstrated that the Green’s function for a single cell with resonant membrane can be constructed by generalising the “sum-over-trips” framework of Abbott et al. [15, 16] for passive dendrites. Here, we demonstrate how this framework can be extended to a network level starting with the simple case of two identical cells.
3.1 Two Simplified Identical Cells
is the Laplace transform of the Green’s function for an infinite resonant cable. is the length of a path that starts at point x on one of the segments and ends at point y on segment . The trip coefficients which ensure that the boundary conditions at the gap junction hold are chosen according to the following rules (see Fig. 3):
if the trip reflects along on the gap junction back onto the same dendrite.
if the trip passes through the gap junction along the same dendrite.
if the trip passes through the gap junction from one cell to another cell.
3.2 Two Simplified Non-identical Cells
3.3 An Arbitrary Network Geometry
where and is the length of a path along the network structure that starts at the point on branch i and ends at the point on branch j. Note that the length of each branch of the network needs to be scaled by before is calculated for (29). It is also worth mentioning here that if all branches of a network have the same biophysical parameters, i.e. , the function defined by (19). The trip coefficients in (29) are chosen according to the following set of rules:
For any branching node at which the trip passes from branch i to a different branch k, is multiplied by a factor .
For any branching node at which the trip approaches a node and reflects off this node back along the same branch k, is multiplied by a factor .
where the sum is over all branches connected to the node.
For every terminal which always reflects any trip, is multiplied by +1 for the closed-end boundary condition or by −1 for the open-end boundary condition.
For the somatic node at which the trip passes through the soma from branch i to a different branch k, is multiplied by a factor .
For the somatic node at which the trip approaches the soma and reflects off the soma back along the same branch k, is multiplied by a factor .
where the sum is over all branches connected to the soma.
For the GJ node at which the trip passes through the gap junction from branch n to branch m, is multiplied by a factor . For the GJ node at which the trip passes through the gap junction from branch m to branch n, is multiplied by a factor .
For the GJ node at which the trip approaches the gap junction, passes it and then continues along the same branch m, is multiplied by a factor . For the GJ node at which the trip approaches the gap junction, passes it and then continues along the same branch n, is multiplied by a factor .
For the GJ node at which the trip approaches the gap junction and reflects off the gap junction back along the same branch m, is multiplied by a factor . For the GJ node at which the trip approaches the gap junction and reflects off the gap junction back along the same branch n, is multiplied by a factor .
We refer the reader to Coombes et al.  for a proof of rules for branching and somatic nodes. In Appendix B, we prove that the rules for generating the trip coefficients at the GJ node satisfy the gap-junctional boundary conditions.
4 Application: Two-Cell Network
4.1 Method of Words for Compact Solutions
Here, we introduce a method which allows us to construct compact solution forms for the Green’s functions of this two-cell network. We describe this method in detail by constructing the Green’s function for Cell 2 when is placed between the soma and the gap-junction as shown in Fig. 12. Introducing points from 1 to 4 on this network, we associate letters with different directions as follows:
From or from : letter A.
From or from : letter B.
From : letter W.
From : letter Y.
From : letter Z.
4.2 Network Dynamics
Resonant properties of each cell are analysed by studying a preferred frequency for each cell. This is defined as the frequency at which the corresponding power function, for Cell 1 and for Cell 2, reaches its maximum. This means that for each soma is simply a solution of one of the corresponding equations, and .
In this paper, we have generalised the “sum-over-trips” formalism for single dendritic trees to cover networks of gap-junction coupled resonant neurons. With the use of ideas from combinatorics, we have also introduced a so-called method of ‘words’ that allows for a compact representation of the Green’s function network response formulas. This has allowed us to determine that the position of a dendro-dendritic gap junction can be used to tune the preferred frequency at the cell body. Moreover we have been able to generate mathematical formula for this dependence without recourse to direct numerical simulations of the physical model. One clear prediction is that the preferred frequency increases with distance of the gap junction from the soma in a model with passive soma and resonant dendrites. In contrast for a system with a resonant soma and passive or resonant dendrite, the preferred frequency decreases as the gap junction is placed further away from the cell body.
There are a number of natural extensions of the work in this paper. One is an application to more realistic network geometries or more than just two neurons, as may be found in retinal networks. Here, it would also be interesting to exploit any network symmetries (either arising from the identical nature of the cells, their shapes, or the topology of their coupling) to allow for the compact representation of network response (and further utilising the method of ‘words’). Another is to incorporate a model of an active soma whilst preserving some measure of analytical tractability. Schwemmer and Lewis  have recently achieved this for a single unbranched cable model by coupling it to an integrate-and-fire soma model. The merger of our approach with theirs may pave the way for understanding spiking networks of gap junction coupled dendritic trees. Moreover, by using the techniques developed by them in  (using weakly coupled oscillator theory) we may further shed light on the role of dendro-dendritic coupling in contributing to the robustness of phase-locking in oscillatory networks.
Appendix A: Two Simplified Identical Cells with Passive Membrane
These solutions generalise earlier results of Harris and Timofeeva  applicable to a neural network, but with gap-junctional coupling at tip-to-tip contacts of two branches.
Appendix B: Proof of the “Sum-over-Trips” Rules at the Gap Junction
We prove here that the rules for generating the trip coefficients are consistent with these boundary conditions.
Trips that start out from X and move away from the GJ node are identical to trips that start out from the GJ node itself along segment . The only difference is that the trips in the first case are shorter by the length X. We denote the sum of such shortened trips by . The argument −X means that a distance X has to be subtracted from the length of each trip summed to compute (and not that the trips start at the point −X).
Trips that start out from X by moving toward the GJ node and then reflecting back along segment are also identical to trips that start out from the GJ node along segment except that these are longer by the length X. In addition, because of the reflection from the GJ node these trips pick up a factor according to the “sum-over-trips” rules. Therefore, the contribution to the solution from those trips is . Trips that start out from X by moving toward the GJ node and then continue moving along branch m, i.e. on segment , pick up a factor and the sum of such trips is given by . Finally, trips that start from X, move toward the GJ node and then leave the GJ node by moving out along segment or pick up a factor and contribute to the solution by the terms or .
which satisfies the boundary condition (64).
Substituting (73) and (74) together with (68) and (69) in Eq. (65) gives us the right equality. Similarly, we can prove the boundary condition (66).
YT would like to acknowledge the support provided by the BBSRC (BB/H011900) and the RCUK. DM would like to acknowledge the Complexity Science Doctoral Training Centre at the University of Warwick along with the funding provided by the EPSRC (EP/E501311).
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