Computational Convergence of the Path Integral for Real Dendritic Morphologies
© Q. Caudron et al.; licensee Springer 2012
Received: 19 June 2012
Accepted: 11 September 2012
Published: 22 November 2012
Neurons are characterised by a morphological structure unique amongst biological cells, the core of which is the dendritic tree. The vast number of dendritic geometries, combined with heterogeneous properties of the cell membrane, continue to challenge scientists in predicting neuronal input-output relationships, even in the case of sub-threshold dendritic currents. The Green’s function obtained for a given dendritic geometry provides this functional relationship for passive or quasi-active dendrites and can be constructed by a sum-over-trips approach based on a path integral formalism. In this paper, we introduce a number of efficient algorithms for realisation of the sum-over-trips framework and investigate the convergence of these algorithms on different dendritic geometries. We demonstrate that the convergence of the trip sampling methods strongly depends on dendritic morphology as well as the biophysical properties of the cell membrane. For real morphologies, the number of trips to guarantee a small convergence error might become very large and strongly affect computational efficiency. As an alternative, we introduce a highly-efficient matrix method which can be applied to arbitrary branching structures.
Discovered more than a century ago by Santiago Ramón y Cajal , dendrites form the vast majority of the surface area of a neuron, with the dendritic trees of some motoneurons representing up to 97% of total neuronal surface area and 75% of the total neuronal volume . These complex branching structures are responsible for transferring electrical activity between synapses and the soma. As technology evolved, interest in dendrites began to gather momentum, with the invention of sharp micropipette electrodes in the early 1950s allowing intracellular recordings to be made. It was the breakthrough work of Wilfrid Rall  on the application of cable theory to dendritic modelling that provided significant insight into the role of dendrites in processing synaptic inputs, the historical perspective of which is summarised in a book by Segev, Rinzel and Shepherd . Recent experimental and theoretical studies reinforce the fact that dendritic morphology and membrane properties play an important role in dendritic integration [5, 6]. We refer the reader to the book Dendrites , devoted exclusively to these formations and revealing their biological complexity at different scales.
It has also been known for some time that nonlinear voltage-gated ion channels are present in the dendrites of various types of neurons , and many recent dendritic models are constructed by combining the linear (passive) properties of dendrites together with nonlinear (active) dynamics of membrane channels. Although the nonlinear properties of ion channels contribute considerably to neuronal input-output relations, it is important to recognise that the passive properties of dendritic membranes provide the fundamental core for signal filtration and integration, and thus remain an essential component in understanding electrical signalling in dendrites .
When branched dendritic fibres are modelled by passive cable equations, the voltage response across the branching structure for any form of applied current can be calculated via a convolution operation, as long as the Green’s function for the given dendritic tree is found. This approach provides an alternative to the compartmental method, based on the discrete spatial approximation of the potential [10, 11]. It is not always trivial to construct such a Green’s function for realistic dendritic geometries. Arbitrarily-branching systems are inherently difficult to solve, a fact recognised early by Rall, who proposed a method of mapping the branching structure onto an equivalent cylinder provided that certain geometrical restrictions were satisfied . The work of Koch and Poggio , based on the graphical calculus of Butz and Cowan , focused on the calculation of the response function for complete dendritic trees in the Laplace (frequency) domain. Later, Rall’s method of equivalent cylinders was extended by releasing the constraints on diameters of individual branches and by constructing the Green’s function, again in the Laplace domain [15, 16]. An alternative method for constructing the Green’s function for a branching structure with a shunted soma was proposed by Evans and coauthors [17–19]. In this series of papers, the response function was found in the form of an eigenfunction expansion, which converges particularly rapidly for large times. For smaller times, a Laplace-domain series solution provides better accuracy, agreeing well with an earlier “sum-over-trips” method for constructing the Green’s function directly in the time domain, proposed by Abbott et al. . This sum-over-trips framework is built on a path integral formulation and enables the calculation of the Green’s function on an arbitrary dendritic geometry as a convergent infinite series solution. Cao and Abbott  presented an algorithm for a computational realisation of the sum-over-trips approach, based on the division of trips into four classes. They applied this algorithm to a number of sample dendritic trees, the largest of which had 22 branches, in contrast to real dendritic geometries, which might have more than 400 terminals alone , with a large variation in branch length. This complexity in neuronal morphologies across different types of neurons is expected to affect the convergence of computational implementations of the sum-over-trips framework.
In this paper, we introduce and investigate a number of efficient algorithms for calculating the Green’s function on dendritic trees using the sum-over-trips formalism. In Sect. 2, we review the theoretical framework and the four-classes algorithm of Cao and Abbott , and introduce alternative algorithms for the sum-over-trips method in Sect. 3. We begin with a modification on the four-classes algorithm aimed at improving its time complexity by developing a formal grammar to derive the trips. Then, a length-priority ordering of the trips using Eppstein’s algorithm  for finding the k shortest trips on a graph is proposed. We also derive a stochastic approach for sampling trips on the tree based on a Monte-Carlo approach. Finally, a highly-efficient deterministic method for discretised tree structures is described. We assess the convergence of the introduced algorithms on different dendritic geometries in Sect. 4, where we also compare the delay and attenuation of voltage spread on four reconstructed dendritic morphologies. Finally, in Sect. 5, we provide a discussion of our results, as well as possible extensions of this work.
2 The Sum-over-trips Framework
where is a terminal on branch k.
where is the length of a trip along the tree that starts at point on branch i and ends at point on branch j. Note that the length of each branch needs to be scaled by its own electrotonic space constant before is calculated for Eq. (6). A constructed trip is allowed to reflect on or pass through any node on the tree an arbitrary number of times. The coefficients depend on the constructed trip and are determined according to the following rules :
From any starting point, .
For every node at which the trip passes from branch m to branch k where , is multiplied by a factor .
For every node at which the trip reflects along on a node back onto the same branch n, is multiplied by a factor .
For every terminal, is multiplied by +1 for the closed-end boundary condition or by −1 for the open-end boundary condition.
However, note that the sum-over-trips method for constructing the Green’s function in the time domain only works for uniform characteristic time constant τ across the entirety of the dendritic tree. The generalisation of this framework to support a quasi-active membrane, instead of a passive membrane, releases this restriction and different cell membrane properties can be chosen on each branch . However, this means that the construction of the Green’s function as an infinite series solution can only be performed in the Laplace domain.
where is a location of a stimulus on branch j.
The Green’s function calculated by Eq. (5) for any branching structure with finite length branches includes an infinite number of terms. It is possible to show that this infinite series solution converges faster than , for sufficiently-high k, the number of nodes visited by the trip. We demonstrate this in the Appendix for an arbitrary tree with nodes of degree or less. This generalises Abbott’s convergence analysis , where it was shown that, for an infinite binary tree, the sum of coefficients is for trips visiting any number of nodes.
2.1 Four-Classes Algorithm
Cao and Abbott  introduced an algorithm for constructing the Green’s function using the sum-over-trips method. Their algorithm is based on finding the shortest trips between any two points of measurement x and current injection y on a tree. Starting from the most direct, shortest trip from x to y, passing through the minimum number of nodes, four classes of trips are defined by allowing a trip to leave the point x in either direction and approach y from either direction along their respective branches. These initial trips, therefore, form the first and shortest trips in their respective classes; longer trips are generated incrementally from these. New additional trips can pass the points x and y any number of times and are allowed to change direction at any node. We will refer to this method as the four-classes algorithm.
From these main trips, the four-classes algorithm generates all trips by inserting what are described as “excursions” into the trips. If A and B are adjacent nodes in a tree, then an excursion could be added to the trip to generate the trip , representing a reflection on node B towards A, reflecting at the terminal A back towards B, passing through this node and finally onto point y. This process can be iterated indefinitely, generating a trip with two more nodes each time. If this process is applied to every node on every trip with n and nodes, then every trip with and nodes will be generated. Thus, from the four shortest trips on the tree, it is possible to construct all trips up to some threshold number of nodes in length explicitly. The lengths and coefficients of these trips can then be calculated from their full trip descriptions, allowing the Green’s function given by Eq. (5) to be approximated.
3 Algorithmic Realisations
Here, we suggest possible modifications to the four-classes algorithm of Cao and Abbott  as well as introduce novel alternative algorithms for the sum-over-trips formalism.
3.1 Formal Language Theory Approach
This problem can be avoided by assigning each branch a direction. If the branch AB is given the direction , then the excursion is disallowed. The choice of direction for each branch is unambiguous on acyclic structures: apart from the branch on which x is found, each branch must be directed away from x. The branch upon which x resides is directed away from y. This ensures that each node has a sequence of excursions that allow the algorithm to generate trips including it. The allocation of direction to each branch can be performed before the process of generating trips and may coincide with finding the four main classes of trips. These modifications require that the graph be acyclic, since “away from a point” is not generally definable on a graph with cycles. There do exist cyclic graphs for which an unambiguous grammar can generate the language of trips, but these are not relevant to the study of single dendritic trees.
The two presented modifications of the four-classes algorithm are sufficient to prevent the generation of any duplicate trips, without any trips being missed. Together, they provide an unambiguous context-free grammar generating the language of trips.
3.2 Length-Priority Method
Since the coefficients decay at most with (although the number of trips increases with ), the dominating term in the Green’s function (5) is the exponential decay in . The four-classes algorithm  does not generate trips in monotonic order in length, since trips are constructed by adding the same excursion to all four classes of trips. If, for example, a Class 2 trip is significantly longer than its Class 1 counterpart, due to x being along a long edge but close to a node, then a longer Class 2 trip will be generated before a potentially shorter Class 1 trip having an additional excursion on a shorter branch. In general, trips are likely to be disordered in length if the branches upon which x or y reside are substantially longer than at least one other branch on the tree, or if x or y are much closer to one of their adjacent nodes than to the other.
Here, we propose to realise the sum-over-trips framework by a length-priority method. In this implementation, trips are generated and the corresponding terms are added to the infinite series solution (5) in monotonic order in length . This is achieved by incorporating Eppstein’s algorithm  for finding the k shortest trips on a graph in time, with n being the number of nodes and m the number of edges on the branching structure.
Both the four-classes algorithm and the improvements described in the language-theoretic approach rely on storing trips explicitly as sequences of nodes. This consumes space and time for k trips with n nodes but allows on-the-fly calculation of coefficients . This is contrary to Eppstein’s algorithm , which stores trips using an implicit representation and allows us to find the k shortest trips implicitly using only space and time for each trip. The current implementation, based on Eppstein’s algorithm, requires time to calculate coefficients despite the savings on space due to the implicit trip representation. However, Eppstein provides a method for computing any property that can be described by a monoid in time per trip. Such a description of coefficient calculation exists, and its use would supplement the current to decrease in space requirements with an analogous decrease in time complexity. The savings in space already allow the length-priority method to scale better than the four-classes algorithm.
3.3 Monte-Carlo Method
The path integral formulation of the solution to the cable equation introduced by Abbott et al.  is derived via consideration of a Feynman–Kac representation of the solution in terms of random walkers on the dendritic geometry. Hence, it is natural to consider Monte-Carlo approaches to evaluating this path integral. Instead of a length-ordered series solution as provided by the length-priority approach, the Green’s function (5) can be constructed using a stochastic algorithm. The aim of this approach is to sample from trips in such a way that the probabilistically more likely samples coincide with the trips that contribute most to the series solution (5).
where is the probability of the realisation ω, and denotes the expectation operator. Therefore, the Monte-Carlo strategy is to sample, sequentially or in parallel, the random function in order to construct this expectation.
3.4 Matrix Method
where the sum over l is over all possible trip lengths . On a dendritic tree, discretised as in compartmental models  or in a manner similar to the discretisation of the tree into segments in NEURON , grouping trips according to their lengths allows us to count the number of trips of a given length l without having to explicitly construct them.
This method uses a modified directed edge adjacency matrix of the discretised tree in order to compute the sum of coefficients of trips of a given length. It requires all compartments to have the same fixed length Δx, although this restriction can be relaxed in a generalisation presented at the end of this section. The extremities of compartments define the position of nodes; there is a directed edge in both directions between adjacent nodes.
We begin by defining as the set of nodes and ℰ as the set of directed edges in the discretised tree. Edges are ordered pairs of nodes: is a directed edge from u to v, with . For any edge , we denote the reverse edge by . Trips are taken to begin from a point x along a starting edge and end at a point y along a goal edge for . We say that or if x resides along edge . Based on the locations of and , the orientations of s and g are defined such that the shortest trip satisfies . Therefore, the shortest trip always starts on edge s, that is, in the direction, and approaches y along the edge g, in the direction. This is equivalent to a Class 1 trip; Class 2 trips leave x along the edge, arrive at ; Class 3 trips go from to ; Class 4 trips, finally, go from to . The locations of the points and along their respective edges are given as a fraction of the branch length such that denotes the distance from x to and is the distance between node and point y. We distinguish between k, the number of edges travelled in a particular trip, from the length of the trip . Because x and y reside along their respective edges, the total length of a trip that travels along k edges is less than if the full distance along k edges had been travelled. That is, for any combination of x, y and for all k.
Because the set of edges ℰ is both ordered and finite, then can be thought of as a vector, where for . The i th element of the vector corresponds to the sum of coefficients for all trips originating at x on s and ending along the i th edge , having travelled over k edges. The vector consists mostly of zeros, with a one only in the entry corresponding to the edge s, as the coefficient of moving in this direction remains 1, while all other moves are invalid by travelling over only one edge, and hence have coefficient 0.
Note that the above rules apply to the transpose of .
By selecting a small Δx, branches may be approximated by a discretisation using an integer number of edges of length Δx. As in compartmental models, this allows the full morphology of the dendritic tree to be approximated, in a trade-off between high speed (large Δx) and accuracy (small Δx). As , however, this approach tends to the computational complexity of naively integrating the cable equation using numerical methods. As in numerical simulations, where reducing Δx in order to increase accuracy brings about a necessary and associated change in Δt, the same is true of the matrix method: selecting a small Δx and hence increasing , implies that must be increased.
This algorithm can be generalised to accept several discrete edge lengths , at an exponential cost in the number of different lengths n, allowing “caricature” neurons to be constructed from a small number of different edge lengths. Our description of this method is focused on the case where x and y are located on different branches. For computations where x and y are required to exist on the same edge, the edges can be discretised such that x and y appear on different segments. In all cases with bounded node degree, Q is a sparse matrix with only a few entries per row, and entries altogether, making the complexity for the calculation of all coefficients by using highly-efficient sparse linear algebra algorithms.
3.4.1 Example calculation
4 Convergence of Methods
The computational convergence of any algorithm impacts both the accuracy of its results and the speed at which these results are obtained. While the series solution to the Green’s function (5) is proven to converge for a sufficiently-high number of terms (see the Appendix), this assumes optimal ordering of terms in the solution. Because the coefficients are impossible to compute until a trip is constructed, generating terms for the series solution in descending order of magnitude is inherently difficult. The four-classes, length-priority and matrix methods generate trips in order of increasing , with the aim of ordering trips by their terms, which decreases monotonically in the length of the trip. The Monte-Carlo method uses a stochastic method to order trips by their probabilities, with more likely trips contributing more to (5). However, none of these approaches order the trips optimally, and hence their accuracy relies not on the theoretical convergence of the mathematical method, but on the computational convergence of the algorithm that implements it.
where T is the final simulation time, is NEURON’s numerical solution to very high accuracy and is the integral of the accurate NEURON solution. This convergence measure is therefore relative to the amplitude of the “real” solution, and thus errors ε are comparable between different neuronal types.
Parameter sets of the binary tree in Fig. 4A
Parameter set A
Parameter set B
Parameter set C
Branch length L
Branch diameter a
Diffusion coefficient D
Membrane time constant τ
Membrane capacitance C
Length-priority method on a binary tree: number of trips required for a given accuracy
Relative error threshold ε
Parameter set A
>5 × 107
Parameter set B
>5 × 107
Parameter set C
Figures 5D–G show ε for the structures in Figs. 4B–E respectively. They demonstrate that convergence is non-trivial on complex branching structures. Figure 5D shows that the length-priority method makes consistently less error on the amacrine cell geometry, in contrast to the convergence of the Purkinje cell, shown in Fig. 5E, where the four-classes method generates less error for all numbers of trips. Both of these show strongly irregular convergence and high-amplitude oscillation in the errors ε in the amacrine cell. For both methods, the Purkinje cell shows a plateau in error for Green’s functions with few trips, indicating that either these trips are of small magnitude or that their voltage traces alternate between undershooting or overshooting the correct solution between subsequent trips. This indicates that neither the length-priority or the four-classes methods are good heuristics for ordering terms in the Green’s function. This is further hinted at by the oscillating property of the error, which implies that there are regions where trips that increase the error are more frequent than trips that reduce it.
The pyramidal cell’s convergence shows very discontinuous behaviour (Fig. 5F), particularly in the length-priority method. The large jump in error when approximately 350 trips are included in the Green’s function was found to be caused by the first and shortest Class 2 trip included thus far, with all prior trips belonging to Class 1. This behaviour is likely to arise if there exist very short branches along the shortest and most direct trip, and thus many Class 1 trips are generated first, being shorter than the first Class 2 trip. Whilst one of the motivating reasons for considering a length-priority approach was to generate trips fully by length order, this heuristic makes no attempt to include the coefficient in its ordering. This is an example of a pathologically large change in the coefficients value for a Class 2 trip which contributes a very significant amount to the Green’s function. The four-classes approach, which enforces generation of trips of all four classes at every added excursion, does not show such a drastic drop in error. However, the error plot is still very discontinuous, and this may be a characteristic of situations as we have just described, where points x and y are placed on branches having a very different length to those on the most direct trip, or when these points are placed very close to a node. Whether injection and measurement points are located on branches that are significantly longer or shorter than those along the shortest trip, both the four-classes and the length-priority methods will generate trips in an “unnatural” order, subsampling the trips where current will spread the most, but oversampling in areas of the tree with very short branches. This pathological feature may not be inherently present in the real neuronal morphology, but may have been created during digital reconstruction from slice image data if, for example, a change of radius were found along the branch. Therefore, this pathology may not be representative of the neuronal geometry, but becomes a function of the reconstruction.
The tangential cell’s convergence, shown in Fig. 5G, shows almost identical errors for both the four-classes and the length-priority methods, indicating that trips are generated in a similar order regardless of method. Contrary to the example with the pyramidal cell, this behaviour is likely to occur when x and y are placed on branches that are significantly shorter than those that arise on the shortest path, such that the length-priority method returns trips of Class 1, 2, 3 and 4 in sequential order, as these increases in length are shorter than adding an excursion along the direct trip.
Our results clearly indicate that the convergence of the realisation of the sum-over-trips framework by either the four-classes or the length-priority method strongly depends on a dendritic geometry. For real morphologies, the number of trips required quickly becomes very large to the point where guaranteeing convergence to within some small error threshold may become computationally expensive.
4.1 Structural-Electrotonic Properties
It acts as a measure of the amount a transient signal’s amplitude diminishes as it travels between two points. is also additive for a point z between x and y, that is, .
In this paper, we introduced a number of efficient algorithms for the computational realisation of the sum-over-trips framework and assessed their convergence. We started with some modifications of the four-classes algorithm of Cao and Abbott  to avoid constructing duplicate trips. An unambiguous context-free grammar was derived, which is able to generate all trips uniquely and in monotonic order of length. We then developed the length-priority method, in which trips are constructed purely in length order rather than in classes. Both methods were found to demonstrate very nonuniform convergence which was highly-dependent on the dendritic morphology, as well as the biophysical properties of the cell membrane. Oscillations of the convergence error make it difficult to predict the number of trips required for constructing the Green’s function on a particular geometry. Dendritic structures with longer branches of uniform diameters will converge faster, that is, for a smaller number of trips in the series solution. Instead of sampling the trips in some well-defined order, we also derived a stochastic method of sampling the trips based on a Monte-Carlo approach. Finally, we proposed an extremely efficient matrix method which computes the trip coefficients, , for trees where all branches are integer-multiples in length to some base length, Δx.
Although we considered dendrites to be passive in this study, the proposed algorithms can be easily generalised to support quasi-active (resonant) dendrites with a calculation of the Green’s function in the Laplace domain . Moreover, it is straightforward to include an isopotential soma in the Laplace-domain series solution. A soma can be considered as a special node with the factors on the branches connected to the soma defined as , where and are the capacitance and the resistance of the somatic membrane and ω is the Laplace transform’s frequency variable. Similar factors can also be found for when the leaky-end boundary condition, referred to as natural termination by Tuckwell , is imposed at the terminals. A knowledge of the Green’s function for a given dendritic structure allows one to efficiently find the sub-threshold voltage response along the entire tree for any number of various inputs, either analytically or via a computation of the convolution integral. This obviates the need for the brute-force numerical simulations of an underlining set of PDEs. Such simulations may be computationally expensive, particularly since they have to be re-initiated each time a new stimulus is introduced. In the case of supra-threshold inputs, which can activate voltage-gated channels known to be present in dendrites of many neurons, the Spike-Diffuse-Spike (SDS) type model [34, 35] can be utilised for analysing the propagations of dendritic action potentials. 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 a wave propagation in the SDS model is in excellent agreement with a more biophysically realistic nonlinear model . However, an analytically tractable SDS model combined with a fast algorithm for constructing the Green’s function on real geometries provides a computationally efficient framework for studying wave scattering in dendrites.
Although networks of spatially-extended neural cells can be numerically simulated, there are currently few mathematical studies of such networks. A natural extension might be to consider a network of branched neurons coupled by gap-junctions. The sum-over-trips formalism can then be generalised to support a presence of new boundary conditions. Recent results of Harris and Timofeeva  can be applied to the case of tip-to-tip coupling of the dendritic branches. The proposed algorithms can then be modified by including additional sum-over-trips rules. It is worth mentioning that the computational schemes presented in this paper are able to handle cyclic graphs, which may form as a result of gap-junction coupling across several neurons. While the matrix method is expected to be the most efficient for a network of symmetric or regular structures, realistic reconstructions of discretised trees remain within computational reach. For example, given a sparse random matrix, of correct density and of size , equivalent to a dendritic tree with edges, the calculation of for all j up to only takes fifteen seconds on a desktop computer. For comparison, the Purkinje cell reconstruction in Fig. 4D has just under branches. For the case of very large, complex irregular structures it might be possible to employ a recently developed technique of reducing the complexity of large dendrites  before applying the sum-over-trips methodology.
Appendix: Mathematical Convergence of the Sum-over-trips Series Solution
This makes B the coefficient of the lower bound on trip length in terms of the number of nodes in a trip. Intuitively, Bk equals the minimum distance between any two nodes where, for this purpose, we count x and y as nodes.
With , the second sum in (26) converges absolutely for all constants . Therefore, the series in (26) is absolutely convergent for sufficiently-high k.
and the path integral converges faster than in the worst case, with the number of nodes k visited by the trips.
QC and SPCB 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). SRD acknowledges funding from the EPSRC and the MRC through the Doctoral Training Centre in Neuroinformatics at the University of Edinburgh. YT would like to acknowledge the support provided by the BBSRC (BB/H011900) and the RCUK.
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