# Response functions for electrically coupled neuronal network: a method of local point matching and its applications

## Abstract

Neuronal networks connected by electrical synapses, also referred to as gap junctions, are present throughout the entire central nervous system. Many instances of gap-junctional coupling are formed between dendritic arbours of individual cells, and these dendro-dendritic gap junctions are known to play an important role in mediating various brain rhythms in both normal and pathological states. The dynamics of such neuronal networks modelled by passive or quasi-active (resonant) membranes can be described by the Green’s function which provides the fundamental input-output relationships of the entire network. One of the methods for calculating this response function is the so-called ‘sum-over-trips’ framework which enables the construction of the Green’s function for an arbitrary network as a convergent infinite series solution. Here we propose an alternative and computationally efficient approach for constructing the Green’s functions on dendro-dendritic gap junction-coupled neuronal networks which avoids any infinite terms in the solutions. Instead, the Green’s function is constructed from the solution of a system of linear algebraic equations. We apply this new method to a number of systems including a simple single cell model and two-cell neuronal networks. We also demonstrate that the application of this novel approach allows one to reduce a model with complex dendritic formations to an equivalent model with a much simpler morphological structure.

### Keywords

Dendrites Gap junctions Network dynamics Sum-over-trips## 1 Introduction

Neuronal cells have a distinctive structure which differentiates them from any other cell types. The most extended parts of many neurons are dendrites, and their morphological complexity has fascinated scientists since the exemplary work of Ramón y Cajal [3]. Organised in a network, neurons receive and integrate thousands of neuronal inputs via both chemical and electrical synapses located primarily on dendrites. With the development of sharp micropipette electrodes, dynamic properties of dendritic membranes started to be revealed through intracellular recordings, and in the late 1950s experimental work was complemented with the pioneering theoretical work of Wilfrid Rall on the application of cable theory to dendritic modelling. Rall’s significant contribution to the topic of dendritic function is nicely summarised in the book of Segev et al. [15]. Recent experimental and theoretical/computational studies at a single cell level reinforce the fact that dendritic morphology combined with membrane properties plays an important role in dendritic integration (two books, edited by Stuart et al. [16] and Cuntz et al. [7], give informative overviews from both an experimental and a theoretical perspective). An additional level of complexity associated with synaptic connectivity needs to be taken into consideration when dynamics of neuronal networks, rather than single cell dynamics, are investigated.

The dendritic membrane of various types of neurons is known to be equipped with voltage-gated ion channels, nonuniformly distributed throughout dendritic arbours and often demonstrating nonlinear dynamics. Many models of neuronal cells with retention of complex dendritic formations are built by combining the linear (passive) properties of dendrites together with nonlinear (active) dynamics of ion channels. At the level of a single cell or at the network level, such models are restricted to being solved only by numerical methods, based on a compartmental approach [14]. Although the nonlinear properties of voltage-gated ion channels contribute considerably to neuronal input-output relations, it is important to recognise that the purely passive or resonant (quasi-active) properties of dendritic membranes provide the fundamental core for signal filtration and integration. Resonant dynamics of dendritic membrane are usually associated with the hyperpolarisation-activated \(I_h\) current and, from a mathematical perspective, can be described by linearising channel kinetics [9, 10, 11].

Here we focus on a network of neuronal cells with purely passive or resonant membrane dynamics coupled by dendro-dendritic electrical synapses, also known as gap junctions. Gap junctions are mechanical and electrically conductive links between adjacent neuronal cells that permit direct electrical connections between them. Having been 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 [8, 13]. Using the cable theory approach for modelling dendritic arbours, the response of an entire dendro-dendritic gap junction-coupled neuronal network to any injected current can be represented by 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 Dirac delta pulse applied at a given discrete location. One of the methods for constructing the Green’s function, the so-called ‘sum-over-trips’ approach, is built on a path integral formulation and was originally proposed by Abbott et al. [1, 2] for passive dendrites of a single cell and then generalised by Coombes et al. [6] for resonant membranes and Timofeeva et al. [19] for a neuronal network. This method calculates the response function as a convergent infinite series solution consisting of terms with various trips (paths) on a given branching structure and the associated coefficients obtained by the sum-over-trips rules. It has been shown at the single cell level that although convergence of this method is fast for simplified dendritic structures, the number of trips to guarantee a small convergence error for real morphologies might be large and have a strong effect on computational efficiency [4]. Here we propose an alternative method for calculating the Green’s function on a neuronal network coupled by dendro-dendritic gap junctions. This new method, named as a method of local point matching, is inspired by the sum-over-trips approach and utilises the trip coefficients of that method, but avoids the construction of any trips. Instead, the new method is based on the construction of a linear system of algebraic equations and therefore leads to compact solutions without an infinite number of terms.

In Sect. 2, we introduce the network model for gap junction-coupled neurons. Each neuron in the network consists of a soma and a dendritic arbour. Cellular membrane dynamics are modelled by a resonant electrical circuit. In Sect. 3, we develop a new method of local point matching from the generalised form of the sum-over-trips approach [19] for constructing the Green’s function for an arbitrary network. Applications of this new method are demonstrated in Sect. 4. We start with a simple single cell model consisting of a soma and dendrite and then move to a two-cell simplified network and, finally, to a more complex tufted network. Not only do we apply the local point matching method for constructing the Green’s functions in each case, but also use it to reduce the full two-cell tufted network model to an equivalent and much simpler model. The last two aforementioned sections include the key results and skip some mathematical details on the derivation of analytical results. We refer the interested reader to “Appendix” for detailed mathematical derivations. Finally, in Sect. 5, we provide a discussion of our results, as well as possible extensions of this work.

## 2 The model

*i*of each cell is governed by the following set of equations:

*i*. The term \(I_{\mathrm{inj},i}(x,t)\) models an external current applied to this branch. The dendritic structure of each cell is attached to an equipotential soma of the diameter \(a_\mathrm{S}\) modelled by the ‘LRC’ circuit with the parameters \(C_\mathrm{S}=C_\mathrm{soma} \pi a^2_\mathrm{S}\), \(R_\mathrm{S}=R_\mathrm{soma}/(\pi a^2_\mathrm{S})\), \(L_\mathrm{S}=L_\mathrm{soma}/(\pi a^2_\mathrm{S})\) and \(r_\mathrm{S}= r_\mathrm{soma}/(\pi a^2_\mathrm{S})\). Moreover, individual branches of different cells can be connected by gap junctions described by a conductance parameter \(g_\mathrm{GJ}\).

*j*. If a branch terminates at \(X={\widehat{\mathcal {L}}}_i\), we have either a no-flux (a closed-end) boundary condition

*m*(branch

*n*) on the left and right from a gap junction (see Fig. 1), respectively. The expressions in (10) reflect continuity of the potential across individual branches

*m*and

*n*, and Eqs. (11) and (12) enforce conservation of current.

The whole network in Fig. 1 can be viewed as a graph structure (which can be cyclic) with different types of nodes: a terminal, a regular branching node, a somatic node or the GJ node. The voltage dynamics along the network structure are described by linear equations, and therefore, the model’s behaviour can be studied by constructing the network response function known as the Green’s function, \({\widehat{G}}_{ij}(X,Y;t)\). This function describes the voltage response at the location *X* on branch *i* in response to a Dirac delta pulse applied to the location *Y* on branch *j* at time \(t = 0\).

## 3 Method of local point matching for finding the Green’s functions

*i*:

*k*of the network by its own characteristic function \(\gamma _k(\omega )\) as \({\mathcal {L}}_{k}=\gamma _{k}(\omega ){\widehat{\mathcal {L}}}_k\), it is possible to derive (see [6, 19]) that the Green’s function on a scaled network (\(x=\gamma _{i}(\omega )X\), \(y=\gamma _{j}(\omega )Y\)) takes the form of an infinite series expansion

*i*and ends at the point \(y=\gamma _j(\omega )Y\) on branch

*j*. The trips coefficients \(A_{\mathrm{trip}}(\omega )\) in (15) are chosen according to the following set of rules:

Initiate \(A_\mathrm{trip}(\omega )=1\).

*Branching node*: \(A_\mathrm{trip}(\omega )\) is multiplied by a factor \(2p_k(\omega )\) or \(2p_k(\omega )-1\) (see Fig. 2a), where \(p_k(\omega )\) is a branch factor defined by$$\begin{aligned} p_k(\omega )=\frac{z_k(\omega )}{\sum _n z_n(\omega )},\qquad z_k(\omega )=\frac{\gamma _k(\omega )}{r_{a,k}}. \end{aligned}$$(16)*Somatic node*: \(A_\mathrm{trip}(\omega )\) is multiplied by a factor \(2p_{\mathrm{S},k}(\omega )\) or \(2p_{\mathrm{S},k}(\omega )-1\) (see Fig. 2b), where$$\begin{aligned} p_{\mathrm{S},k}(\omega )= & {} \frac{z_k(\omega )}{\sum _n z_n(\omega )+z_\mathrm{S}(\omega )}, \end{aligned}$$(17)$$\begin{aligned} z_\mathrm{S}(\omega )= & {} C_\mathrm{S}\omega +R_\mathrm{S}^{-1}+(r_\mathrm{S}+L_\mathrm{S}\omega )^{-1}. \end{aligned}$$(18)*GJ node*: \(A_\mathrm{trip}(\omega )\) is multiplied by a factor \(p_{\mathrm{GJ},n}(\omega )\), \(1-p_{\mathrm{GJ},n}(\omega )\) or \(-p_{\mathrm{GJ},n}(\omega )\) (see Fig. 2c), whereand \(R_\mathrm{GJ}=1/g_\mathrm{GJ}\).$$\begin{aligned} p_{\mathrm{GJ},n}(\omega )=\frac{z_n(\omega )}{z_m(\omega )+z_n(\omega )+2R_\mathrm{GJ}z_m(\omega )z_n(\omega )} \end{aligned}$$(19)*Terminal*: \(A_\mathrm{trip}(\omega )\) is multiplied by \(+1\) for the closed-end boundary or by \(-1\) for the open-end boundary condition.

*y*can be divided into two classes separated by the direction of the last part of the trip. Placing two points \(v_j\) and \(w_j\) on segment

*j*as shown in Fig. 3, we consider one class which includes the trips with \(L_\mathrm{trip}(x,v_j^{\rightarrow y})\) approaching

*y*from the left (named as \(J_{v_j}\)) and the other class which includes the trips with \(L_\mathrm{trip}(x,w_j^{\rightarrow y})\) approaching

*y*from the right (named as \(J_{w_j}\)). Without constructing the actual trips, it is possible to show that all trips ending at

*y*, named as \(J_y\) and from (15) having the form

*k*and introducing two classes of trips \(J_{v_k}\) and \(J_{w_k}\) (see Fig. 4). Each unknown function \(J_{v_k}\) can then be written as a linear combination of the nearest unknown functions \(J_{w_n}\), \(J_{w_{n+1}}\) and \(J_{w_k}\) which are heading towards point \(v_k\). Similarly, the unknown function \(J_{w_k}\) can be written as a linear combination of the nearest unknown functions \(J_{v_k}\) and \(J_{w_{n-1}}\) heading towards point \(w_k\). This leads to a linear system of 2

*N*algebraic equations for all unknown functions \(J_{v_k}\) and \(J_{w_k}\) defined on each segment \(k=1,\dots ,N\), where

*N*is the number of dendritic segments in the network. Solving this linear system for \(J_{v_k}\) and \(J_{w_k}\), we can then find the unknown function \(J_y\) and, as a result, the Green’s function \(G_{ij}(x,y;\omega )\).

*Summary of method*

- 1.
The physical length \({\widehat{{\mathcal {L}}}}_{k}\) of each branch

*k*is scaled by its own characteristic function \(\gamma _{k}(\omega )\) given by Eq. (14). - 2.
Place a pair of points \((v_k,w_k)\) on each segment

*k*(see Fig. 4). Assume that \(v_k\) and \(w_k\) are placed infinitesimally close to both ends of the branch. Trips from \(v_k\) and \(w_k\) can move only towards each other (see red vectors in Fig. 4). Construct a system of linear algebraic equations for all \(J_{v_k}\) and \(J_{w_k}\). For example, the function \(J_{v_k}\) in Fig. 4 depends on a linear combination of the terms with \(J_{w_n}\), \(J_{w_k}\) and \(J_{w_{n+1}}\) (if the branch*i*with point*x*is absent; otherwise, an additional term \(a_{ik}f(x)\) must be included in the linear combination, where \(a_{ik}\) is a coefficient for a trip passing from segment*i*to segment*k*). The function \(J_{w_k}\) in Fig. 4 depends on a linear combination of the terms with \(J_{v_k}\) and \(J_{w_{n-1}}\). The constructed linear combinations for the unknown functions \(J_{v_k}\) and \(J_{w_k}\) include trip coefficients \(a_{nk}\) for trips passing from segment*n*to segment*k*and trip coefficients \(a_{kk}\) for trips reflecting at the end points of segment*k*. These coefficients are obtained from the sum-over-trips rules summarised in Fig. 2. - 3.
Solve the constructed linear system of algebraic equations and therefore find \(J_{v_j}\) and \(J_{w_j}\) for a pair of points \((v_j,w_j)\) placed on segment

*j*which includes point*y*, (\(0<y<{\mathcal {L}}_j\)), see Fig. 3. - 4.
Find the function \(J_y\) as \(J_y=f(y)J_{v_j}+f({\mathcal {L}}_j-y)J_{w_j}\) or, if

*x*is located on branch*j*, using \(J_y=f(y)J_{v_j}+f({\mathcal {L}}_j-y)J_{w_j}+f(x-y)\). - 5.
Find \(G_{ij}(x,y)\) as \(G_{ij}(x,y)=J_y/(2D_j\gamma _j)\).

- 6.
Rescale the coordinates \(X=x/\gamma _i(\omega )\) and \(Y=y/\gamma _j(\omega )\) and take the inverse Laplace transform (InvLT) of \(G_{ij}(X,Y;\omega )\) to obtain the Green’s function \({\widehat{G}}_{ij}(X,Y;t)\).

*y*is located at a node (i.e. \(y=0\) or \(y={\mathcal {L}}_j\)), due to the continuity of the potential at the boundaries the method can be easily applied by initially, assuming that

*y*is placed on segment

*j*slightly away from this node and, after the Green’s function is constructed, considering that \(y=0\) or \(y={\mathcal {L}}_j\). A similar approach can be used if point

*x*is also located at one of the nodes.

Note that spatially extended neurons coupled by gap junctions into an arbitrary neuronal network might develop a graph structure with cycles, and our method of local point matching (as well as the original sum-over-trips method) can support such structures.

## 4 Applications

### 4.1 A soma and dendrite model

*v*,

*w*) takes the following form

*x*and

*y*are placed at the soma. This dependence is obtained as a solution of the implicit equation

### 4.2 A two-cell simplified network

Here we demonstrate how our method can be applied to a two-cell network of either identical or nonidentical cells coupled by a dendro-dendritic gap junction. In each case, we obtain the compact solutions for the Green’s functions, Eqs. (40)–(43) for the two-cell identical network and Eqs. (49)–(56) for the two-cell nonidentical network, which can inform us about the roles of individual parameters on the network dynamics.

*N*attached semi-infinite dendrites as shown in Fig. 8. We assume that the biophysical properties of all dendritic segments are the same and that the physical lengths are scaled by the characteristic function \(\gamma (\omega )\) given by (14). The gap junction is located at some distance \({\mathcal {L}}_\mathrm{GJ}\) away from the cell bodies. We assume that this network can receive stimuli in four different locations mimicking distal (\(y_{1}\) and \(y_{2}\)) and proximal (\(y_{3}\) and \(y_{4}\)) inputs. Points of output \(x_{1}\) (for Cell 1) and \(x_{2}\) (for Cell 2) are located between either soma and the gap junction.

*k*, \({\mathcal {L}}_k\) is the distance between the gap junction and the soma of Cell

*k*, and \(p_{\mathrm{GJ},k}\) is given by Eq. (19).

### 4.3 A two-cell tufted network

*N*dendritic branches, one of which is the primary dendrite with the tuft spanning from its end. Two cells are coupled in their tufts by dendro-dendritic gap junctions (see Fig. 12a). As in the previous model, we assume that the biophysical properties of all dendritic segments are the same and that the physical lengths are scaled by the characteristic function \(\gamma (\omega )\). We consider that each cell has \(n_\mathrm{T}\) segments in its tuft, and \(n_\mathrm{GJ}\) of them possess identical single gap-junctional points located \(l_{0}\) away from the end of the primary dendrite. The primary dendrite of each cell has the length \({\mathcal {L}}\), while the other branches are semi-infinite. For simplicity, we consider that the membrane of both cells is purely passive (i.e. \(\gamma ^2(\omega )=(\tau ^{-1}+\omega )/D\)); however, it is straightforward to generalise it for the resonant case.

*i*and

*j*in the equations for the full model change from 1 to \(n_\mathrm{GJ}\), and \(p_\mathrm{T}\) is a branch factor of any tuft dendrite defined as in (16).

\(\theta =2n_\mathrm{GJ}p_\mathrm{T}f({\mathcal {L}}), \eta =2/(1-\zeta \delta )\) and \(\mu =(\zeta \theta \eta p_\mathrm{D}+2n_\mathrm{GJ}p_\mathrm{T}-1)p_\mathrm{GJ}f(2l_0)\).

*k*can receive a Dirac delta pulse at the location \(y_k\) away from the branch point with the primary dendrite. This tuft dendrite can be either with or without a gap junction. In the equivalent reduced model shown in Fig. 14, we consider two possible inputs corresponding to the location of \(y_k\), namely the input \(y_1\) applied to the branch without a gap junction and \(y_2\) applied to the branch with a gap junction.

*y*away from the primary dendrite of each cell, we obtain in this special case

## 5 Discussion

In this paper we have presented a novel method for calculating the Green’s functions for arbitrary neuronal networks with a passive or resonant cell membrane coupled by dendro-dendritic gap junctions. This method provides an alternative and complementary approach to the generalised sum-over-trip method [19]. Importantly, our new approach avoids the construction of an infinite number of trips and, being based on the construction of a linear system of algebraic equations, provides exact expressions for the network response function in the Laplace (frequency) domain without any issues of computational convergence. We have applied this new method of local point matching to a simple single cell model and two-cell neuronal networks (simplified and with tuft dendrites). Its application to the tufted network has also allowed us to reduce it to an equivalent network, but with a much simpler morphological structure. We have also illustrated that knowledge of the exact compact expressions for the Green’s function can provide important insights into the role of individual variations in cell parameters on the model’s dynamics.

There are a number of natural extensions of the work in this paper. One is an application to more realistic network geometries with more than just two cells, given that a computational implementation of the method of local point matching can provide a fast realisation of the Green’s function for the whole network. Having a complex network of multiple cells with a graph structure consisting of *N* dendritic segments, we need to construct and solve a linear system of 2*N* equations only once to find all unknown \(J_{v_{k}}\) and \(J_{w_{k}}\) functions. We can then simply construct the functions \(J_{y}\) for each dendritic segment to obtain \(G_{ij}(X,Y;\omega )\). Note that the point *X* can be placed on each dendritic segment before constructing a system of linear equations for \(J_{v_{k}}\) and \(J_{w_{k}}\). Switching off all *X* points except one on branch *i* in the solution for \(J_{y}\) allows one to find the Green’s function for the entire network straight away. The numerical inverse Laplace transform to obtain \({\widehat{G}}_{ij}(X,Y;t)\) is the only procedure in which a computational approximation appears. As has been previously pointed in Sect. 4.2, knowledge of a map from the preferred frequencies to the system’s parameters for a reconstructed neuronal network combined with subthreshold electrophysiological data might provide some estimates for important network’s parameters and additional work is required in this direction. Another possible extension is to incorporate active properties in dendrites and somas of cells in a network and analyse the propagation of dendritic action potentials as well as somatic spiking dynamics. The spike-diffuse-spike (SDS)-type model [5, 17] can be utilised for that, as although the voltage-gated channels in the SDS framework are modelled by piecewise linear instead of nonlinear dynamics, it has been shown that the speed of wave propagation in the SDS model is in excellent agreement with a more biophysically realistic nonlinear model [20]. Both these extensions will be reported on elsewhere.

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