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(The FASEB Journal. 2007;21:866-871.)
© 2007 FASEB

Role of transport performance for neuron cell morphology

E. Louis*, C. Degli Esposti Boschi{dagger}, G. J. Ortega{ddagger} and E. Fernández§,1

* Departamento de Física Aplicada, Instituto Universitario de Materiales (IUMA) and Unidad Asociada of the Consejo Superior de Investigaciones Científicas, Universidad de Alicante, Alicante, Spain;

{dagger} CNR-INFM, Unità di Ricerca CNISM di Bologna, Bologna, Italy;

{ddagger} Departamento de Física, F.C.E.N. Universidad de Buenos Aires and CONICET, Pabellón I, Ciudad Universitaria, Bueno Aires, Argentina; and

§ Instituto de Bioingeniería, Universidad Miguel Hernández, Campus de San Juan, Alicante, Spain

1Correspondence: Universidad Miguel Hernández, Department of Histology and Institute of Bioengineering, Fac. Medicina, San Juan 03550, Alicante, Spain. E-mail: e.fernandez{at}umh.es


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 
The compartmental model is a basic tool for studying signal propagation in neurons; if the model parameters are adequately redefined, it can also be helpful in the study of electrical or fluid transport in other biological systems. Here we show that the input resistance in different networks that simulate the morphology of neurons is the result of the interplay between the relevant conductances, neuron morphology, and neuron size. The results suggest that neurons may grow in such a way that facilitates the current flow to the synapses, concurrently minimizing power consumption.—Louis, E., Boschi C. D. E., Ortega, G. J., Fernández, E. Role of transport performance for neuron cell morphology.


Key Words: compartmental model • passive properties


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 
ALMOST A CENTURY AGO, W. D’ARCY THOMPSON, in his now classic essay (1) , set down a basis for the study of morphology of living systems by means of physical and mathematical methods. By the same time, J. B. S. Haldane broadened the scope of this new field, introducing the relationship between "size" and "form" (2) . Since then, much research has been done (3 4 5 6 7 8 9 10 11 12 13) , including the latest interest in the nonequilibrium growth phenomena in several biological systems, which has attracted much attention in the last 20 years (14 15 16 17 18 19 20 21 22) .

The nervous system is a good example of a complex structure that contains many different types of neurons with tree-like morphologies (Fig. 1 ), whose function is largely dependent on their structure. Consequently, to understand how a neuron integrates its myriad synaptic inputs to generate an appropriate response, a thorough understanding of the cell’s morphology and geometry is required (5 , 10 11 12) . Furthermore, recent work suggests that the structure and function of neuronal networks are governed by basic principles such as "wiring economy" (3 , 4) and "constraint minimization" or "resource allocation" (5 6 7) . These ideas were anticipated nearly 100 years ago by the great neuroanatomist Santiago Ramón y Cajal, "We realized that all of the various conformations of the neuron and its various components are simply morphological adaptations governed by laws of conservation for time, space and material" (8) . Thus, interest in the shape, size, and geometry of dendritic branching patterns of these cells arises from a variety of reasons (9) . Anatomists are interested in the morphological characterization and differences among neuronal classes, as well as in morphological variations within these classes. Neurophysiologists are interested in how dendritic morphology is involved in synaptic connectivity and in the integration and processing of neural information within neuronal networks. Developmental neuroscientists seek to discover the rules of development and the mechanisms by which neurons attain their final appearance. Computer scientists are interested in algorithms able to describe and predict the dynamic behavior of single neurons and assemblies of neurons. The enormous amount of structural and functional variation of nervous cells is a major challenge providing strong motivation to search for "fundamental rules" that unify design features across different neuron types. Despite these efforts, a full understanding is still far from being achieved.


Figure 1
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  		Figure 1. Examples of neurons, simulated networks, and the compartmental model used to describe the passive electrical behavior of neurons. A) Golgi impregnation of a human retinal ganglion cell. B) Neurobiotin labeling of horizontal cells in turtle retina. C) Typical diffusion-limited aggregation (DLA) fractal. D) Compact network generated by the Eden model. E) Detail of panel C. F) High magnification of the insert in panel E. G) Compartmental electrical representation of the DLA segment showed in panel F. Several membrane patches are connected in series via axial resistances. The extracellular space is assumed to be isopotential.

The actual morphology of neurons depends mainly on their own intrinsic properties and their connectivity. Their cell membranes are lipid bilayers with proteins floating around them, some of which are ion channels. The cytoplasm is composed mainly of water, electrolytes, charged proteins, cytoskeleton, endoplasmic reticulum, and other organelles. Therefore, electrical signals propagate through the neuron cytoplasm, which has an axial conductance gc. Current can also flow through the membrane, with a conductance, gm (11) . Such a system can be described adequately by the so-called "cable with faults model," a cable through which current flows that has losses due to defective insulation (10 , 23) . In the absence of voltage-dependent currents, two parameters characterize this model: a resistance that describes the cable itself and a resistance associated with the faults. When applied to neurons, these parameters are the axial resistance and membrane resistance mentioned above. In addition, a conductance associated with the terminal segments, output conductance gout, must be included (10 11 12) . Note that the values of those conductances may cover many orders of magnitude, depending on the system and the environmental conditions (13) .

A variant of the cable model widely used to describe the electrical behavior of neurons is the compartmental model (11) . This model, which is simply the discretized version of the cable with a faults model, has long been a basic tool to 1) study the passive electrical properties of neurons; 2) reconstruct signals that originated in one part of a passive neuronal tree but have been recorded at another point, usually because of technical constrains; 3) producing an electrical skeleton onto which active conductances can be grafted to build up realistic computational models of different neurons. The compartmental model incorporates in many cases a feature that may play a significant role—namely, the thinning of neurons branches (increasing in resistance) as the neuron grows. However, although several versions of the compartmental model have been used to calculate the passive electrical properties of a great variety of neurons (10 , 11 , 23 24 25 26 27 28 29 30) , a detailed comparison between the electrical performance of different cell morphologies is lacking. Such a study could throw light on the mechanisms that force neurons to branch.

The effects of electrical propagation in cells and networks of cells have been widely studied (10–13, 25, 28–31) using the following experimental approach. A current is injected into a given point or cell, and the subsequent voltage deflections at that site and sites at increasing distances from it are measured. These data give the input conductance g0 and, combined with a measure of the distance, the length constant {lambda}. From these two values, one can, in principle, derive the membrane and axial resistances. This procedure can be complicated greatly by the complex effects of tissue morphology and the presence of fluctuations in the conductances (13 , 26 , 27) , those probably being the reasons for the scarce data in the literature about the relevant parameters.

The present study investigates how the electrical power dissipated by current flow in different networks with largely different morphologies depends on the parameters of the compartmental model. We examined whether patterns mimicking the morphological appearance of neurons grow in such a way that transport efficiency is maximized or, equivalently, the resistance to electrical transport is minimized (4 , 7 , 10 11 12 , 17 , 18) . Our results show that the input resistance and current flow toward the boundaries, which quantify the passive response of a neuron to a current feed, determine to some extent its growth and main morphology.


   MATERIALS AND METHODS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS
 DISCUSSION
 REFERENCES
 
Since neurons tend to ramify, creating large and complicated trees (see Fig. 1A, B ), systems with fractal geometry can be a useful model for their morphological properties (14) . Fractal theory (21 , 22) has already found widespread applications in biology (14 , 19 , 20) , particularly in the field of neuroscience (14 , 15 , 19 , 20) . In our computer simulations we used diffusion-limited aggregation (DLA) networks with fractal dimension D = 1.71 (Fig. 1C ), a well-established physical model that can provide a good description of neurite outgrowth (19 , 20 21 22) in which the factors controlling growth are those inherent in a diffusion equation (i.e., chemical gradients of neurotrophic factors, electrical fields). We also used a deterministic fractal having D = 1.47 (21 , 22) . As it has been claimed that some types of neurons are space-filling objects (15) , we also used the Eden model (Fig. 1D ), which takes the main physics underlying the formation of many living systems (21 , 22) and represents a rapidly saturating system with compact geometry and a complex boundary having a fractal character (22) . All networks were grown on the square lattice (having coordination number p=4). To have another compact network for comparison, we also consider simple clusters of the square lattice. Subsequently, we built up on each network the arrangement of electrical resistances, a large number of sites (compartments) with membrane conductance gm, coupled through junctions of contact conductance gc (the cytoplasm conductance) (Fig. 1E-G ). An additional conductance associated only with the terminal compartments, output conductance gout, was added. Within the basic approximation of considering only passive elements, we should take into account that the terminal compartments (ending in tips) may represent electrical synapses (gap junctions) or mimic chemical synapses.

For the results presented here, both the extracellular space and the terminal ends were assumed to be at the same potential V = 0, since choosing different potentials will only add a new parameter without an increase in insight. Furthermore, to elucidate the genuine role played by the different morphologies, we limited ourselves to simple ohmic conductors and did not take into account the thinning of neuron branches as the neuron grows. Thus, if a current I is injected at site i with membrane conductance gm, which we assume to be located at the center of the network, Kirchoff’s current law applied to the model outlined above leads to a set of coupled linear equations for the potential at all sites Vj. The equation for the site i at which the current I is injected reads (23) ,

Formula 1(1)
where the first sum in the right-hand side of the above equation runs over the ni nearest neighbors of site i belonging to the network, and the second term accounts for its contact with the extracellular space and covers the remaining nearest neighbors up to the actual coordination p of the substrate (Fig. 1G ). The equation associated with a general element j, where no current is injected, is exactly as in Eq. 1 , with I = 0. In solving this system of linear equations for Vj, with j = 1,..,N, we used the fast sparse matrix techniques now available (32) . As a result, the input resistance of the aggregate R0 = Vi/I, is obtained. Note that due to linearity (Ohm’s law), the resulting potentials Vi will be proportional to I and R0 will be independent of the actual value chosen for the injected current. Hence, at fixed I, the problem of minimizing the input resistance is equivalent to that of selecting the network’s geometry that exhibits the smaller (electrical) power consumption W = Vi I = R0I2. Hereafter, we will take gc as the unit of conductance.

As noted above, the compartmental model investigated here is nothing more than the discretized version of the cable with a faults model. In principle, if the whole neuron is divided into sufficiently small compartments, the compartmental model converges to the continuous cable model. However, as compartments get smaller, the membrane and contact (or axial) conductance should be changed, as neurons are usually characterized by a specific membrane resistance that is proportional to the area of the compartment (given in {Omega} cm2) and an axial resistance that is proportional to its length (given in {Omega} cm) (10) . In the ensuing discussion, we investigate the performance of a set of compartments arranged on a physical network containing N sites (note that the word network is not used to denote an arrangement of cells but instead an arrangement of compartments used to describe a single neuron). In our analysis we assume that increasing N does not mean doing a finer division of the neuron, but rather increasing its global size, keeping the compartment size, and thus the characteristic conductances, constant.

After redefining the model parameters, the present approach can be applied to other cases—for instance, to arrangements of cells connected through gap junctions (24 25 26 27) such as pancreatic ß-cells in Langerhans islets (24 , 31) . In such a case, contact conductance characterizes the junction whereas the membrane conductance has a meaning identical to that in the case of a single neuron. On the other hand, the output conductance describes the connection of the cell arrangement, or tissue, to the outer world (other tissues, organs, etc.). The latter conductance, however, is in most cases quite small. Note that in this case we are dealing with a network of cells. We also mention here the flow of liquids on a given network. In such a case, voltage should be replaced by pressure, electric current by fluid flow, and electrical resistances by capillary (or tube) resistances. Otherwise the equations are formally equivalent. Heat transport could also be considered within the same framework. With this analogy in mind, we analyzed values of relative conductances covering many orders of magnitude even outside the range of reasonable values observed in real neurons (10 11 12 13) .


   RESULTS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
DISCUSSION
 REFERENCES
 
Let us first discuss the dependence of the input resistance on the relative conductances by holding the number of compartments fixed. Results of the numerical calculations illustrated in Fig. 2 correspond to aggregates having N = 1252 sites. An extremely small value like gm = e–28 gc has been included to show the effect on the low membrane conductance end. In the limit of large gout, no current flows through the membrane and the input conductance depends only on the number of compartments the current should flow through before reaching the neuron’s terminals. When the output conductance decreases, the plots indicate that networks can be classified into two groups: compact and ramified. At small values of gout, ramified networks perform better (smaller input resistance) for the lower value of the membrane conductance (Fig. 2A ) whereas the opposite occurs for gm = e–13 gc (Fig. 2B ). In the latter case, the input resistance becomes constant at low values of the output conductance, as current flows mainly through the cell membrane (for the square network, this occurs even at the smaller value of the membrane conductance, as it has less terminal connections; see Fig. 2A ).


Figure 2
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  		Figure 2. Input resistance R0 of networks with n = 1252 sites (compartments) vs. the conductance that couples the network to the extracellular space gout (A, B) and membrane conductance gm (C, D). A, B) Membrane conductance was fixed at gm = e–28 gc and gm = e–13 gc, respectively; C, D) output conductance was kept constant at gout = e–17 gc and gout = e–5 gc. Symbols correspond to DLA aggregates (thick chain line), a deterministic fractal (thin chain line), a perfect square (thin continuous line), and the Eden model (thick continuous line). The gray and gray/white shaded areas mark the typical range of gm/gc for neurons and ß-cells, respectively (see Discussion).

Figure 2C, D shows the results of the input resistance R0 vs. membrane conductance for DLA and Eden aggregates of the same size as above and output conductances of gm = e–17 gc (Fig. 2C ) and gm = e–5 gc (Fig. 2D ). A crossing of the results for the two types of aggregate is again observed for the lower value of the output conductance (Fig. 2C ). However, when the latter is increased (Fig. 2D ), no crossing occurs and the compact aggregate shows a lower input resistance in the whole range of membrane conductances shown in the figure. For very high membrane conductances, the whole current flows through the axial resistance, and all simulations show an input resistance, which decreases as gm increases.

We have also investigated how these results depend on the system size. In Fig. 3 we show results of the input resistance R0 vs. the number of compartments in the network N for systems with a very small membrane conductance gm = e–17gc and an output conductance either 20-fold smaller (Fig. 3A ) or equal to the contact conductance (Fig. 3B ). In the first case, R0 decreases approximately as 1/N no matter which morphology system is considered. This is readily understood by noting that in such a case the system behaves as a network of N resistances gm–1 in parallel, as all compartments are at approximately the same voltage. When gout = gc, no current flows through the cell membrane and all is carried to the extracellular space through neuron’s terminals. As a consequence, compact systems show an input resistance that increases logarithmically with N, a scaling behavior typical of 2-dimensional systems, whereas in ramified networks R0 becomes constant beyond a value of N, which depends on the actual ratio gm/gout. This is further illustrated by the results for intermediate values of gout. As shown in Fig. 3C, D , input resistance undergoes a crossover from a power law to either a constant R0 in ramified systems or, as shown in Fig. 3B , a logarithmic behavior in the case of compact systems. Note that while the power law, followed by ramified systems at sufficiently small N, always close to 1/ N (a sign of good scaling down to small sizes), the exponent steadily decreases with gout in compact aggregates. The crossover occurs at a characteristic size that depends on the actual values of the conductance and on the network type (whether compact or ramified).


Figure 3
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  		Figure 3. Input resistance R0 of networks vs. the number of sites N for several values of the characteristic conductances. DLA (diamonds), deterministic fractal (crosses), a perfect square (squares), and the Eden model (circles). A) gm = e–17 gc and gout = e–20 gc. All results collapse (only some are shown) on the curve R0 = 2.4 x 107 N–0.98 gc–1. B) gm = e–17 gc and gout = gc. Curves fitted to the numerical results are R0 = (0.078lnN + 0.17) gc–1 (perfect square) and R0 = (0.1lnN –0.05) gc–1 (Eden model). For ramified systems (both DLA and deterministic), R0 readily becomes constant (0.32 and 0.34, respectively). C) gm = e–17 gc and gout = e–14 gc. The square and the Eden model follow a power law in the whole range with fittings R0 = 1.8 x 106 N–0.79 gc–1 and R0 = 3.0 x 105 N–0.67 gc–1, respectively, whereas R0 for ramified systems (DLA or deterministic fractal) is constant beyond N {cong} 104 and follows the power law R0 = 4.9 x 105 N–0.97 gc–1 for smaller systems. D) gm = e–17 gc and gout = e–8 gc. Results for the ramified systems (DLA and deterministic) follow the power law R0 = 1.34 x 103 N–0.92 gc–1 for N < 103, and remain constant for larger systems.

A possible way to summarize the essentials of our simulations is to arrange a phase diagram in the plane (gm/gc, gout/gc) indicating the regions where the representative fractal morphology—namely, the DLA one—is favored with respect to the compact one (Eden in our study) in the sense that it yields a smaller input resistance or, equivalently, power consumption. The resulting phase diagram is plotted in Fig. 4 for two values of N. It is seen that the phase boundaries, especially the ones in the region of small output conductance, depend on the system size. Note that the small gout region over which DLA performs better shrinks when the network size increases, in line with the results discussed above. An additional important issue is to specify regions of the parameter space within which a large portion of the injected current I is delivered to the output contacts. Figure 5 shows a diagram obtained under the requirement that half or more of the total current goes through output contacts, both for ramified (DLA clusters, region defined by diamonds) or compact (Eden clusters, region defined by circles) networks, and over the same parameter ranges considered in Fig. 4 . Regions where DLA is efficient do indeed greatly overlap with those where a majority of the current goes through the terminal contacts. Compact clusters are more efficient from the two vantage points in a smaller region bounded approximately by gm/gc < e–8 and e–6 < gout/gc < e–3.


Figure 4
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  		Figure 4. Regions of the parameter space over which either ramified (DLA clusters) or compact (Eden clusters) networks show a better performance (evaluated by the input resistance or power consumption). Results for networks of 2500 (small circles) and 15625 compartments (large circles) are shown. Filled circles correspond to the case gm = 0. Gray and gray/white shaded areas mark the typical range of gm/gc for neurons and ß-cells, respectively (see Discussion).


Figure 5
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  		Figure 5. Regions of the parameter space over which half or more of the total current goes through terminal junctions in ramified (DLA clusters, region defined by diamonds) or compact (Eden clusters, region defined by circles) networks. Results for networks of 2500 (small symbols) and 15625 compartments (large symbols) are shown. Gray and gray/white shaded areas mark the typical range of gm/gc for neurons and ß-cells, respectively (see Discussion).


   DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
REFERENCES
 
In this paper we have addressed the question of which is the optimal neuron structure as far as the power consumption is concerned, namely, which is the network geometry that yields the smaller input resistance R0, and, at the same time, produces an optimal voltage spread at the boundaries. To this end we have adopted the compartmental model implemented both in ramified (fractal) and compact networks of resistors mimicking the morphological appearance of a wide variety of neurons, and varied the system size and the relevant conductance over a rather wide range. The mechanism of this optimization may resemble those involved in the formation of arterial structures or nonliving tree structures such as river junctions and electric discharge patterns (17) .

Our results show crossings between the curves R0 vs. gout/gc (Fig. 2A, B ) and R0 vs. gm/gc (Fig. 2C, D ) for the two groups of ramified or compact aggregates, indicating that the optimal structure may vary depending on the relative values of the system conductances and size (Fig. 3) . At this stage, note that most neurons, which usually have a low membrane conductance, show ramified structures (10 11 12 13) , whereas other networks of gap junction-connected cells such as the pancreatic ß-cells, which have a much larger membrane conductance (24 , 31) , show compact structures. To quantify this point, we extracted from the literature several experimental data. Regarding neurons, we obtained –17 < ln(gm/gc) < –8.5. These bounds were derived from refs. 10 11 12 13 , 33 , 34 by taking compartments of equal length and diameter (~1–10 µm; because the range over which gm/gc varies is wide, even an order of magnitude difference between those two magnitudes would not have had much effect). For a neuronal area of ~10–3 cm2 (34) , the number of compartments ranges from 103 to 105. Figures 4 and 5 tell us that for low values of gout, ramified structures perform better. For pancreatic ß-cells, the data reported in ref. 24 , 31 ) give ~–1 < ln(gm/gc) < 1 (data for pancreatic ß-cells in their active and passive phases are included in this range). Then, assuming a very small output conductance, we conclude from Fig. 4 that compact networks do a better job (in this case, voltage spread may be a less relevant parameter).

These results suggest that input resistance and current flow to the boundaries, which quantify the passive response of a neuron to a current feed, determine to some extent its growth morphology.

Finally, we believe that although the model investigated here is certainly an oversimplification of both the complex morphology and the electrical nature of neurons, it may help us to understand how the dendritic morphology influences the input-output function of a neuron and shed light on a problem to which little attention has been drawn: the dependence of transport performance on neuron morphology.


   ACKNOWLEDGMENTS
 
This work was supported by the Spanish Comisión Interministerial de Ciencia y Tecnología through grants PB96 0085, 1FD97 1358, NAN2004–09306-C05–04, SAF2005–08370-C02–01 and by the European Commission through the projects "TMR Network-Fractals c.n. FMRXCT980183," "FET Open project COSIN IST-2001–33555," and "NEUROPROBES IST-027017." C.D.E.B. acknowledges a postdoctoral position at the Universidad de Alicante, during which this work was started.

Received for publication May 16, 2006. Accepted for publication September 29, 2006.


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 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 

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