Iterations of Diagrammatic Solutions of the Cable Equation Using Arbitrary Dendritic Geometry


Literature Review, 2015

14 Pages


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Abstract

Cable theory, an intricate neuronal postulate, elucidates neuritic characteristics, that of which, membrane voltage along a dendrite is most salient. This subjective review focuses on the progression of diagrammatic, i.e., analytical and graphical, solutions of the linear cable equation. Geometrical calculus, the mathematical underpinnings of neural cable theory, provided the proper tools to advance dendritic computational paradigms. The first implemented computational model was rudimentary, which obtained respective variables in the Laplace, or frequency, domain. The following work, somewhat heuristic, reformulated the linear cable equation in attempt to increase computational efficacy and produce results in the temporal domain. The derivation of such a complex dendritic neuronal model has ranging implications. However, in many respects, such a computational model is a methodological asset, which can be used implicitly in other models providing a stringently detailed paradigm, both biophysically and phenomenologically. In effect, neural system dynamics can be simulated and once adequately described, can be used to discern neuronal perturbations that, potentially, yield ill results psychologically, psychiatrically, and neurologically.

Introduction

The brain, the encephalon, is a remarkable organ in complexity and convolution. Like any biological substrate, the scientific strivings are to understand this organ microscopically, i.e., cellularly and molecularly. As the realm of inquiry reached such scientific levels, the idea of neural computation arose. In other words, as we began to delve into neuritic connections and transmitter-induced propagation, teased apart neuronal morphology and topography, and attempted to understand connectivity and its relation to functionality, one began to wonder: how does inputted information result in such stringent calculation and yield a result? Although this paper does not encompass such broad doctrines, entirely, the scope of this review is restricted to the dendritic processes that protrude from the neuronal soma, or body, and how it computes information with respect to linear cable theory. Furthermore, this review reports the progression of diagrammatic solutions of the linear cable equation from the derivations of its mathematical underpinnings to the implementation of a computational paradigm in the temporal domain.

Cable theory, initially developed in aide of the transatlantic telegraph system [1], describes electrical characteristics (i.e., currents, voltages, etc.) of passive neurites, particularly dendrites, by modeling neuronal morphology as compartmentalized segments [2,3,4]. In explanation of such a difficult theory, a generalized overview of the basic cable equation ( eq. 1 and eq. 2) is needed: where V is the membrane voltage, x is the spatial position along a dendrite,rm is the segmental membrane resistance,ri is the segmental intramembranous resistance,cm is the segmental membrane capacitance, λ is the length constant, which is indicative of electrotonic properties, and τ is a time constant, which indicates the temporal change of membrane potential in response to input currents. Because of the complicated nature of the aforementioned cable equation, solutions have been produced in a myriad of ways - numerically, analytically, and graphically [5-8]. This review is concerned primarily with diagrammatic solutions to the linear cable equation. However, to reach such solutions, the derivation of advanced mathematics was first required.

Butz and Cowan [9] were integral in the development of diagrammatic solutions of the linear cable equation. They formulated geometrical calculus, or as referenced in their 1974 publication, simple graphical calculus, by replicating passive neurites as cylindrical protrusions. The most salient aspect of this study was the discernment of the dendritic branch transfer function, which is, in a lack of a more mathematical definition, the solution to the cable equation. This particular solution allowed for the computation of voltage responses imputed by arbitrary dendritic current inputs [9]. Thus, Butz and Cowan [9] devised a computational framework in the Laplace, or frequency, domain with the ability to solve the cable equation diagrammatically, which lead to scientific investigation and expansion of such formulations.

Koch, C., & Poggio, T. (1985)

Koch and Poggio [5] derived a simplistic algorithm that solved the linear cable equation, diagrammatically, for neurites, i.e., for dendrites, of arbitrary geometry. The methodological novelty of this work is in the extension and reformulation of the diagrammatic paradigm, composed of geometrical calculus, posed by Butz and Cowan in 1974 [9]. The specific augmentation involved the inclusion of linear membrane impedances [5]; this contrasts the framework developed by Butz and Cowan [9], which only utilized characteristic impedances. The derivations in this study yielded a model of increased computational efficiency, as well as equivalency and simplicity. The scheme devised by Koch and Poggio [5] computes by recurrent application of four elementary equations, whilst attaining a membrane potential with a voltage equation, which included the transfer function.

The aforementioned elementary equations, or rules, which construct the algorithm, initially calculate an impedance of a terminal dendritic branch (Table 1.1). The model proceeds to rule 2, which integrates the impedances of each dendritic branch extending from a node (Table 1.2) [5]. The previously mentioned rules were then repetitively applied until the dendritic geometry satisfied rule 3 (Table 1.3). This rule solved for a voltage, which is related in part to the input current. Lastly, rule 4 was implemented, which solved for the voltage at an arbitrary point along the dendrite (Table 1.4), which could only proceed if the voltage relative to rule 3 was known. Rule 4 also encompassed the unfolding of the dendritic structure, yielding a transfer function. This transfer function, using the voltage expression (eq. 3), could be used to calculate the voltage at position, y, with a current input at position, x [5]. In the previous equation (eq. 3), Vj( ω) is the voltage at position, j, Kij (ω) is the transfer function between dendritic branches interconnecting position, i, and position, j, andIi(ω) is the inputted current at position, i, all of which are in the frequency domain. Figure 1 illustrates the diagrammatic process of this paradigm.

Abbott, L.F., Farhi, E., & Gutmann, S. (1991)

Abbott, Farhi, and Gutmann [6] stated very explicitly the importance of the Green's function, which is equivalent to the previously discussed transfer function, in computing a solution to the linear cable equation. Thus, this study ascertained a path integral by deploying an alternative, but complementary, mathematical approach, which involved numerical formulation, and yielded the Green's function for a dendritic tree of arbitrary geometry. With such an approach, the Green's function, G( x, y, t), was an integration of a measure of all paths connecting position, x, to position, y, in time, t. With this explanation, it’s easy to cerebrate the ideation that this methodological approach, in contrast to Koch and Poggio [5], is in the temporal domain [6].

Therefore, in defining the previously mentioned integral, the path must be described, which was done so by utilizing the Brownian motion measure in attainment of the Feynman path integral. The path integral method provided simplistic diagrammatic rules, which produced a rapid and efficient technique for solving complex cable problems [6].

The main intention of the diagrammatic rules set by Abbott et al. [6] was to express the Green's function for a dendritic tree of arbitrary geometry as a sum of terms in which the Green's function for an infinitely long cable, G0, was included. The Green's function ( eq. 4), Gij(x, y, t ), was computed by summating trip-equating terms from all possible trips from position, x, on segment, i, of the neurite to position, y, on segment, j. The parameters in eq. 4: G is the Green's function as a function of position, x, position, y, and time, t, Atrip is a weighted function, the factor of which changes with the passing of or reflecting from a node or terminal, and G0, a Gaussian-based distributed Green's function, which is a function of Ltrip, the length of a trip in electrotonic length constants and time, t, for an arbitrary dendritic tree, i.e., in its morphology [6]. The diagrammatic aspect of this solution is in the essence of a trip, which starts at a known position, x, traveling in any direction, only changing at a node or a terminal, where the trip could pass and remain in its direction or reflect and, thus, change its directionality, and ends at position, y [6]. Although the derivation of the path integral method was successful, the computational framework was yet to be developed.

Abbott, L.F. (1992)

Abbott [7], after presenting the path integral method a year earlier [3], which yielded an accurate Green's function in the temporal domain, further developed methodology pertinent to diagrammatic solutions of the linear cable equation. After showing the applicability of path integral mathematics to the cable problem, Abbott [7] extended the diagrammatic rules of the 1991 publication [6] to a computational framework of the aforementioned diagrammatic paradigm, which could potentially compute the voltage on a cable of arbitrary geometry as a function of time by solving the linear cable equation (eq. 5) - where Ii( x,t) is the injected current on segment, i, at position, x, at time, t, and v is the neuronal resting potential on segment, i, at position, x, at time, t. This would eliminate the need to deploy an inverse Laplace, or Fourier, transform, while concurrently mitigating transform errors.

Abbott [7] derived the Green's function for a single segment of infinite length (eq. 6) where the solution of the linear cable equation, i.e., the Green's function, on a dendritic tree of any size and geometry could be obtained by summating said equation (i.e., eq. 6). The diagrammatic rules in this methodological proof publication were termed sum over trips (eq. 4) and encompassed the same rules as Abbott et al. [6], which is pragmatic considering the predication of this paper was the extension of the paradigm to include a computational framework [7]. With the successful construction of said framework, the logical successive step was to, simply, implement it.

Cao, B.J., & Abbott, L.F. (1993)

Cao and Abbott [8] developed a novel, computational model that measured the short-time behavior of the membrane potential of an arbitrary dendritic tree, which was an extension of Abbott's previous work [6,7]. The innovation was in the development of a paradigm in the temporal domain [8], which contrasted previous implementations in the Laplace domain [5]. Preceding models, for example the Koch and Poggio [5] paradigm, was forced to utilize the inverse Fourier transform to describe dendritic voltage characteristics in the temporal domain, which resulted in measurement errors. Moreover, this paper computationally implements

Abbott's sum over trips method [6,7], where each trip represents a value in a series of all possible trips and when integrated, provides the exact Green's function [6-8]. This particular function, inputted into Cao and Abbott's derived voltage equation (eq. 7), was used to calculate the membrane voltage [8]. The diagrammatic rules of this computational paradigm are presented hereafter.

As an extension of the path integral [6] and the computational paradigm thereof [7], the diagrammatic rules governing the calculation of the Green's function is analogous to that of the previous two studies [6-8]. Model simulations of all possible trips within length confines resulted in the ascertainment of the computationally procured Green's function, G, voltage at position, y, V(y, 0), and input current at position, y, I( y, s). Resultantly, the de novo voltage equation (eq. 7) was subsequently solved using said parameters. As an extension of the computational framework, the production of the membrane potential was yielded by the solution of the linear cable equation ( eq. 5) [8].

Discussion

As the mathematical underpinnings needed to diagrammatically solve the linear cable equation were being derived by Butz and Cowan [9], scientists were diligently exploring the depths of our cerebral tissue, investigating the possible causation and, effectively, the result of neuronal signal propagation. Subsequently, Koch and Poggio [5] implemented, reformulated, and expanded Butz and Cowan’s [9] geometrical calculus to include membrane impedances, yielding a more detailed and stringent computational paradigm, which allowed for a more realistic calculation of the biophysical mechanisms encompassed by neurites. Their model was novel in its computation of membrane voltages in the Laplace, or frequency, domain [5]. In search of neuronal computations in the temporal domain, Abbott et al. [6] devised a path integral applicable to neuronal cable theory and constructed the beginnings of a novel computational framework.

Abbott, in an individually published paper [7], revised and honed the diagrammatic rules set forth in the study of the preceding year [6], which yielded the sum over trips method.

Successively, Cao and Abbott [8] furthered the aforementioned framework by deriving a voltage equation encompassing, most saliently, the Green’s function from the sum over trips method. They implemented this computational model, which resulted in an efficacious paradigm capable of computing membrane voltages, in rigorous detail, of dendrites.

Because this review is methodological in nature, it is not explicitly obvious the implications of these diagrammatic solutions. Computational work, as can be intuited, allows scientists to make and test predictions based on empirical measures. As neuronal models, at a basic level, become adequate in their description of neuronal dynamics, the complexity thereof can be increased as more empirical data is collected, resulting in a biophysical model descriptive of cellular and molecular processes that which describe the phenomenological dynamics of the respective neural substrate. For example, if a model is accurate in describing the summation, integration, and computation of information, as well as the neuronal output of the primary visual cortex on a simplistic level, one can increase its complexity as empirical data is procured in attempt to retain dynamical preciseness. As such a model becomes more complex, in congruence with novel experimental findings, one can begin to perturb the system in attempt to explicate psychological, psychiatric, and neurological ailments (e.g., simple hallucinations, which are of interest to me). As Abbott and colleagues [6] reason, to adequately understand the behavior of neural networks it is integral to first understand the input-output relationship of a single neuron (e.g., understanding what set of synaptic inputs would result in adequate integration and, thus, a neuronal response, or output). In order to understand the neuronal dynamics of one neuron, each aspect of its functionality must be made intelligible, which dendritic processes are an implicit constituent thereof.

References

[1] Ma Y, Huguenard JR: Reemerging role of cable properties in action potential initiation . Proceedings of the National Academy of Sciences 2013, 110:3715-3716.

[2] Rall W: Branching dendritic trees and motoneuron membrane resistivity . Exp. Neurol. 1959, 1:491-527.

[3] Rall W: Theoretical significance of dendritic trees for neuronal input-output relations . In Neuronal Theory and Modeling. Edited by RF Reiss. Stanford University Press; 1964.

[4] Rall W:Core conductor theory and cable properties of neurons. In Handbook of Physiology. Edited by E Kandel and S Geiger. American Psychological Society; 1977.

[5] Koch C, Poggio T: A simple algorithm for solving the cable equation in dendritic trees of arbitrary geometry . Journal of Neuroscience Methods 1985, 12:303-315.

[6] Abbott LF, Farhi E, Gutmann S:The path integral for dendritic trees. Biol. Cybern. 1991, 66:49-60.

[7] Abbott LF: Simple diagrammatic rules for solving dendritic cable problems . Physica, 185:343-356.

[8] Cao BJ, Abbott LF:A new computational method for cable theory problems. Biophysical Journal 1993, 64:303-313.

[9] Butz EG, Cowan JD: Transient potentials in dendritic systems of arbitrary geometry . Biophysical Journal 1974, 14:661-689.

[10] Scott JG, Kelly RC, Smith MA, Zhou P, Kass, RE: False discovery rate regression: An application to neural synchrony detection in primary visual cortex . Journal of the American Statistical Association 2014.

[11] Phillips RT, Chakravarthy VS: An activity-dependent computational model of development of the retinotopic map along the dorsoventral axis in the primary ventral cortex . BMC Neuroscience 2014, 15:189.

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Details

Title
Iterations of Diagrammatic Solutions of the Cable Equation Using Arbitrary Dendritic Geometry
College
West Virginia University
Author
Year
2015
Pages
14
Catalog Number
V315980
File size
869 KB
Language
English
Tags
Cable Theory, Computational Neuroscience, Neural Engineering
Quote paper
Ryan Ellison (Author), 2015, Iterations of Diagrammatic Solutions of the Cable Equation Using Arbitrary Dendritic Geometry, Munich, GRIN Verlag, https://www.grin.com/document/315980

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