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 gapjunctional coupling are formed between dendritic arbours of individual cells, and these dendrodendritic 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 quasiactive (resonant) membranes can be described by the Green’s function which provides the fundamental inputoutput relationships of the entire network. One of the methods for calculating this response function is the socalled ‘sumovertrips’ 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 dendrodendritic gap junctioncoupled 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 twocell 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 Sumovertrips1 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 voltagegated 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 voltagegated ion channels contribute considerably to neuronal inputoutput relations, it is important to recognise that the purely passive or resonant (quasiactive) properties of dendritic membranes provide the fundamental core for signal filtration and integration. Resonant dynamics of dendritic membrane are usually associated with the hyperpolarisationactivated \(I_h\) 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 dendrodendritic 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 dendrodendritic gap junctioncoupled 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 socalled ‘sumovertrips’ 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 sumovertrips 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 dendrodendritic gap junctions. This new method, named as a method of local point matching, is inspired by the sumovertrips 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 junctioncoupled 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 sumovertrips 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 twocell 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 twocell 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
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

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 \(p_{\mathrm{GJ},n}(\omega )\), \(1p_{\mathrm{GJ},n}(\omega )\) 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 closedend boundary or by \(1\) for the openend boundary condition.
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 \(J_{w_{n1}}\). 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 sumovertrips 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 \(J_y=f(y)J_{v_j}+f({\mathcal {L}}_jy)J_{w_j}\) or, if x is located on branch j, using \(J_y=f(y)J_{v_j}+f({\mathcal {L}}_jy)J_{w_j}+f(xy)\).
 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 \({\widehat{G}}_{ij}(X,Y;t)\).
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 sumovertrips method) can support such structures.
4 Applications
4.1 A soma and dendrite model
4.2 A twocell simplified network
Here we demonstrate how our method can be applied to a twocell network of either identical or nonidentical cells coupled by a dendrodendritic gap junction. In each case, we obtain the compact solutions for the Green’s functions, Eqs. (40)–(43) for the twocell identical network and Eqs. (49)–(56) for the twocell nonidentical network, which can inform us about the roles of individual parameters on the network dynamics.
4.3 A twocell tufted network
\(\theta =2n_\mathrm{GJ}p_\mathrm{T}f({\mathcal {L}}), \eta =2/(1\zeta \delta )\) and \(\mu =(\zeta \theta \eta p_\mathrm{D}+2n_\mathrm{GJ}p_\mathrm{T}1)p_\mathrm{GJ}f(2l_0)\).
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 dendrodendritic gap junctions. This method provides an alternative and complementary approach to the generalised sumovertrip 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 twocell 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 2N 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 spikediffusespike (SDS)type model [5, 17] can be utilised for that, as although the voltagegated 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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