A Biophysical Model of Neuronal Dendrites' Integrative Properties: Relations to Morphological Data

A BIOPHYSICAL MODEL OF NEURONALDENDRITES’ INTEGRATIVE PROPERTIES:RELATIONS TO MORPHOLOGICAL DATA

Patrick Mouchet1 and Jerôme Yelnik2

1Laboratoire de Physiologie Section Neurophysiologie, Université Joseph Fourier,Grenoble, Pavillon de Neurologie, CHU de Grenoble, BP 217, 38043 Grenoble Cedex,France. Email: [email protected] U289, Pavillon Claude Bernard, Hopital de la Salpêtrière, 47, Boulevard del’Hopital, 75013 Paris, France.

ABSTRACTWe used a biophysical model to probe the basic integrative properties of primate pallidal

neurons in order to obtain a better understanding of Basal Ganglia physiology. The first resultswe present here deal mainly with the way dendritic morphology influences these properties.Neuronal morphology has been quantitatatively analyzed in 3D. Single fast excitatory synapticinputs resulting in AMPA receptors activations have been simulated, without regenerativevoltage dependent conductances. Dendrites of both pallidal segments (GPi and GPe) showed astrong dependence of the synaptic efficacy upon distance from soma, but even the most distaldendritic synaptic sites were able to substantially depolarize the cell body. The mean synapticefficacy was the same in both populations, but the attenuation of propagated post-synapticpotentials was higher in GPi neurons. All these features were very dependent on the dendriticdiameters which appear to constitute a key parameter in these neuronal populations both withrespect to the integration of afferent information and to the differences between cells inperforming this task.

1. INTRODUCTIONThe work presented here concerns the basic problem of structure-functions

relations. In the present instance, the question is: does the particular morphologyexhibited by a neuronal type influence the functions performed by these cells? Itespecially concerns the neuronal dendritic arbors because they constitute the maininformation processing part of the neuron since they bear nearly 90% of the synapsesby which neural messages are passed to the receiving cell. Dendrites display a verylarge topological as well as metrical diversity. Indeed, they range from rather simpletrees with a few branches to extremely complex arborizations, and their length rangeextends across more than one millimeter. Though this has been recognized as soon asGolgi staining methods had become available in the last part of the XIXth century, alink between these astonishing geometries and the information processing abilities ofthe corresponding cells has been denied by many authors. According to this tenet,dendritic extensions are conceived solely as a mean of increasing the receiving surfaceof the cell, with the possible additional power to allow for the segregation of inputsfrom different anatomical origins. So, in the absence of some learning and/orbiochemical processes, all synaptic inputs would have equal importance in the

c©2004 Kluwer Academic Publishers. Printed in the Netherlands.Acta Biotheoretica 52: 313–322, 2004.

314 MOUCHET AND YELNIK

response of the recipient cell which may thus be reduced to the isopotential unit that iscommon in neural network approaches. The opposite view rose at the same early time,but received a strong theoretical foundation much later with Rall’s work (Segev et al.,1995). Surprisingly, in spite of great advances in the experimental tools allowing anever finer study of dendritic physiology the debate is still continuing today and wewish to contribute to it. Indeed, we are currently modelling neurons whosemorphology has been analysed in a very detailed and quantitative way by one of us(Yelnik et al., 1984), providing data very convenient to deal with such questions.These cells are the main, if not exclusive, neuronal component of the primate GlobusPallidus. This is a diencephalic anatomical structure which constitutes an importantpart of a larger brain circuit called the Basal Ganglia (BG) in which we were primarilyinterested because of its prominent role in some neurological disorders, the bestknown example of which being Parkinson’s disease. Since BG physiology andphysiopathology remain poorly understood in spite of a wealth of data, we haveundertaken a modelling study in order to propose hypotheses about informationprocessing in this brain circuitry.

The Globus Pallidus is made up of two parts with very similar neuronal elementsbut markedly different functions, as may be deduced from their anatomicalconnections (Figure 1). One part is the external segment (GPe), which processesinformation mainly inside the BG circuitry. On the contrary, the other part, the internalsegment (GPi), integrates the information processed upstream (a task it does in parallelwith the Substantia Nigra which is another BG station) and sends the resultingmessages to the target centres located outside the BG. The data we have are veryaccurate when using biophysical models to obtain insights into the way the pallidalspecific dendritic geometry could shape the integrative behaviour of these cells. In afirst step we have addressed the question of the location dependence, in pallidaldendritic trees, of single synaptic inputs.

Figure 1. Schematic repre-sentation of a part of the BGcircuitry. The BG processinformation coming mainlyfrom the cerebral cortex andthe thalamus (dotted arrows).The main input station is thestriatum (caudate nucleus andputamen), though projectionsalso exist from the cerebralcortex to the subthalamicnucleus (STN). GPe isnarrowly connected to all BGcomponents and occupiesthus a key position inside thecircuit, while GPi transmitsthe processed informationoutside (dotted arrow).

A BIOPHYSICAL MODEL OF NEURONAL DENDRITES’ INTEGRATIVE PROPERTIES 315

2. MATERIAL AND METHODS

Source and main features of the morphological dataOur morphological data are the result of the 3D quantitative analysis of primate

and human pallidal neurons. In previous studies (Yelnik et al., 1984), pallidal neuronsfrom both species have been shown to be identical. Here we used seven human GPeGolgi stained neurons and 14 macaque GPi neurons stained either with the Golgimethod (7 neurons) or biocytin filling (7 neurons). Previous studies (unpublished)have found no morphological difference between Golgi and biocytin labelled pallidalneurons. We found the same results with the samples used here. Since our simulationscould find no evidence of significant differences between both samples, all GPi wereconsidered as constituting the same cell group. Dendrites were exhaustivelyreconstructed from 250 µm serial sections (François et al., 1984) and their lengths anddiameters measured at regularly spaced intervals. The somatic membrane area wasalso computed from serial sections passing through the cell body.

Figure 2. Perspective view of a computer reconstructed typical GPe neuron (central drawing).Two dimensional planar representations of the dendritic arborizations including that in the mainplane (I13DA5PL14), upper part: “PLAN PRINCIPAL”, and in the two perpendicular planes(RX90 and RY90) are shown around (from Yelnik et al., 1984, with permission).


Pallidal neurons have very long dendrites (up to 1200 µm) which are poorly tomoderately branched. The mean number of dendritic trees per cell is around four.Interestingly, principal component analysis shows that for each cell, dendritic trees liein a narrow cylindrical section of about 1500*1000*250 dimension (in µm) (Figure 2).The main planes of these disks are roughly parallel to the pallidal boundary, so thedendrites are in position to intercept the wealth of incoming fibres which runperpendicular to the disks.

Model and simulationsThe evolution of the transmembrane potential V(x,t) is modelled according to

the classical one dimension cable equation:

λ τ22

2 0∂∂

−∂∂

− =Vx

Vt

V x tm ( , )

where x is the distance along the dendrite (in cm) and t the time (in ms), τ m RmCm=

is the membrane time constant, λ =RmRi

D4

is the electrotonic space constant with

Rm: resistance of 1 cm2 of membrane (in ohm.cm2), Cm: specific capacitance of themembrane (in µ F/cm2), Ri : axial cytoplasmic resistivity (ohm.cm), D: dendriticdiameter (in µm). Intracellulary injected and regenerative membrane currents are notconsidered here. When synaptic inputs have to be modelled, a term representing thecorresponding current is added. In the absence of published detailed kinetic parametersconcerning the synapses present on pallidal dendrites, we modelled these synapticcurrents by an alpha function:

I Gt

e V Vsyn

p

t

psyn= −

−

max ( ),τ

τ1

where Gmax is the maximal conductance (in nS) of the considered synapse, τ p the

current time to peak and Vsyn the equilibrium potential of the involved ions. Dendriticbranching and the junctions with the cell body are modelled by stating the continuityof currents at the junction points.

Choice of parameters

Since experimental estimations of the electrical parameters present in the cableequation are not available for primate pallidal neurons we used a parameter set chosenaccording to two criteria: the plausibility with respect to experimental estimationsmade with other neuronal types and the ability to allow for a good reproduction of invitro intracellular recordings made with rodent pallidal neurons (Nakanishi et al.,1990). The present results have been obtained with Rm : 40.000 ohm.cm2, Ri :200 ohm.cm and Cm : 1 µF/cm2, with a resting potential of -57 mV.

We modelled the main synaptic inputs known to physiologically drive the firing ofpallidal neurons, which are the glutamatergic synapses made by subthalamic axonalendings on the whole extent of the pallidal dendrites. More precisely, we modelled the


activation of AMPA receptors according to Bernander et al. (1994), with Gmax : 1.5 nS,

τ p : 0.5 ms and Vsyn : 0 mV.

Simulations

The dendritic arborizations issued from the 3D reconstruction were parcelled into5 µm length segments where diameters were interpolated from the measured ones (inpreliminary simulations we have found this segment length to maximize the resultsaccuracy). The cable equation was numerically solved according to the methoddescribed by Hines (1984).

3. RESULTSOur simulations were mainly concerned with the variations of the synaptic efficacy

along the pallidal dendritic arbors. For any dendritic site, the synaptic efficacy isdefined as the peak amplitude of the post-synaptic potential (PSP) elicited at the somafrom that site. The way dendritic architecture sets the synaptic efficacy is estimated bythe attenuation ratio, which is the ratio of the input site peak to the somatic PSP peak.

Figure 3. Times decays of postsynaptic potentials (lower part) evoked from different distancesfrom soma on a dendrite (upper part, arrow) of a GPi neuron (planar representation). Distances(in micrometers) at which simulated PSPs have been induced are indicated on the correspondingcurves for the somatic PSPs (Soma) and for the most distal and proximal sites for the input sites(Dendrite).


For all modelled dendrites, synaptic efficacy was strongly dependent upon thedistance from soma. An example is shown in Figure 3. However, as shown with thisexample, though pallidal dendrites have often very long dendrites, distal synapticefficacy does not reach very low levels. This is clearly apparent when the distalefficacy is expressed as a fraction of the efficacy of the most proximal site of the samedendrite. Indeed, distal efficacy very seldom represents less than 10% of the proximalefficacy.

Figure 4. Efficacy (A), attenuation ratio (B) and mean dendritic diameters (C) profiles for GPi(solid lines) and GPe (dotted lines) neurons. For all dendritic paths of all neurons in eachpopulation, PSPs have been evoked at distances from soma increasing by 25 µm steps. For eachdistance, all somatic peaks have been averaged, as well as their ratios to the corresponding inputsite peaks (attenuation ratio). Mean dendritic diameters have been computed in the same way.

The dependence of the synaptic efficacy upon the distance from soma as well asthe absolute values of the somatic PSPs peaks evoked from dendrites were verysimilar in GPi and GPe neurons. This is represented in Figure 4A which shows nosignificant differences between the mean efficacy profiles of both populations.However, when plotting the attenuation ratio against the distance from soma, GPi andGPe dendrites markedly differ (Figure 4B). Indeed, the attenuation is significantlyhigher in GPi neurons. This means that PSPs elicited at the input site are higher in GPineurons. Since dendritic length is of the same magnitude in both populations, thisdifference is due to the main quantitative difference between them, namely the smallermean dendritic diameters of GPi neurons (Figure 4C). There is indeed a slight butsignificant difference, as evidenced with a Student’s t test (p < 0.001) between GPi(mean: 1.205 µm, SEM: 0.056 µ m) and GPe (mean: 1.74 µm, SEM: 0.078 µm)diameters.


Figure 5. Efficacy and diameter profiles of the dendrite paths of the GPi dendritic tree shown inthe upper part (planar representation). Numbers refer to the branching code used to labeldendritic branches.

A strong sensitivity of the dendritic responsiveness to the diameter is furtherevidenced by other simulation results. For instance, the markedly different efficacyprofiles observed on the second dendritic tree of the GPi neuron displayed in Figure 5are clearly paralleled by the profiles of the diameters measured in this tree. Theseresults are in good accordance with a previous theoretical study (Holmes, 1989)suggesting an important role of dendritic diameters with respect to synaptic efficacy.In this study it has been shown that for a given dendritic length and a given dendriticlocation a diameter distribution exists (on the whole neuron) which maximizes theefficacy at that location. The optimal diameter varies with the synaptic input site. Wesearched here for whether the measured diameters maximize the synaptic efficacy ofsome parts of the pallidal dendrites and we found that this was true for most GPineurons. Interestingly, the maximization concerned the distal part of the dendrites inmany cases (more than 65% for all dendrites). As a typical example, Figure 6 showsthe somatic PSPs evoked from the tip of the long dendrite of the GPi neuron shown in


Figure 3 for the measured diameters and for diameters modified according to factorsbetween 2 and 0.5. Moreover, this dendrite is also an example of a very often observedsituation in that when diameters are modified in a way slight enough to keep the distalefficacy at the same maximal level, the modified diameters increase the efficacy ofproximal sites. In GPe, this property was found for about 30% of the dendrites.

Figure 6. Effect of modifying dendritic diameters upon the synaptic efficacy of the distal part ofthe dendrite shown on Figure 3. On each graph the somatic PSP evoked from the dendritic tipwith the measured diameters is indicated with the solid line. In A and B the somatic PSPobtained after changing the diameters in the whole neuron are represented with the dotted lines,with the modification factors used. In C, the dotted lines correspond to diameters divided by 1.1.With this factor, somatic PSPs (left) are very close but their ratios to the depolarisation evokedfrom the proximal part of the same dendrite (right) is higher for the measured diameters.

4. DISCUSSIONThe model used in this work is a very usual one which comes from Rall’s seminal

work (Segev et al., 1995). It is applied here to very detailed quantitativemorphological data whose electrical counterpart is lacking. Indeed, the preciseexperimental data needed to estimate the electrotonical parameters of the modelledneurons are not available, and they seem unlikely to be obtained since their derivationwould necessitate in vitro slice recordings, a probably precluded method when dealingwith primate brains. However, we used an electrical parameter set very compatiblywith recordings done on pallidal rodent neurons, and also well in the range of knownbiological values reported in a variety of species (Spruston et al., 2001). At this stepwe did not try to add models of regenerative voltage dependent conductances. Thoughthey are also available in an incomplete way from rodent experiments, a purelyelectrotonic characterization is first wished for, for two main reasons. Firstly, the


dendritic electrotonic structure constitutes the reference framework in which otherelectrical features are expressed. Secondly, this electrotonic behaviour is dependentupon the morphological data we dispose of in a very straightforward way. Moreover,we modelled the response of pallidal neurons to very fast excitatory synaptic inputs,the effects of which are the most sensitive in cell morphology.

The basic simulations we performed accordingly already suggest some features ofbiological relevance. First, for all modelled pallidal neurons the synaptic efficacy isstrongly dependent upon the distance from soma of the corresponding synaptic site.This was expected due to the great length of pallidal dendrites but had to beestablished since it is not the case for other Central Nervous System neuronal typesinvestigated in the same way (Jaffe and Carnevale, 1999). Indeed, in some neurons theefficacies of distal and proximal synapses are very close to each other, a phenomenoncalled passive normalisation. However, and perhaps more importantly from aphysiological point of view, the somatic depolarisations resulting from pallidal distalactivations are not negligible with respect to the ones evoked from proximal dendriticsites. Indeed, the ratio of the corresponding peaks are very rarely below 10%. So,excitatory inputs received on the whole extent of the pallidal dendritic arborizationsare able to efficiently influence the somatic membrane potential, without the help ofregenerative voltage dependent conductances. This is in contrast, for example, withobservations made on cortical pyramidal neurons which have also very long dendrites,but for which distal synaptic input effects on the cell body necessitate a regenerativepropagation (Williams and Stuart, 2002). Secondly, though very similar from thispoint of view, since their efficacy profiles are not significantly different, GPi and GPeneurons differ in a subtle way because dendritic PSPs are more attenuated in theformer population. Accordingly, PSPs peaks at the synaptic sites located at equaldistances from soma are higher in GPi cells. As a physiological consequence, we mayhypothetize that at a given distance from the cell body, while the mean single synapticefficacy is similar in both populations, GPe neurons integrate incoming excitatoryinformation more linearly than GPi neurons do. Thirdly, these conclusions dependstrongly upon a key morphological parameter which is the dendritic diameter. Indeed,the differences we report between pallidal neurons, and even between dendrites of thesame neuron, are mainly due to slight but significant diameter differences. Moreover,the main features of pallidal neurons responses to fast single synaptic are greatlylinked to this parameter, as evidenced for instance by the effect of changing it upon thesimulation results. The underlying mechanism is the increase of both input resistanceat synaptic sites and propagation attenuation from there when diameters are decreased,and our results are in line with the previous theoretical study made by Holmes (1989).This critical role of dendritic diameters, which is a global property of the dendritictrees, is all the more evident here as the other morphological parameters, andespecially the dendritic lengths are very similar in the studied population. Whiledendritic lengths and branching differences between neurons are often very obvious,diameter variations are hardly noticeable. Moreover, their careful measure is difficultand time consuming. However, the present results confirm (Holmes, 1989) that theyconstitute a very important parameter by which dendritic architecture shapes theneuronal integrative properties. So, as soon as we are interested in the functionalaspects of dendrite morphology, by contrast for instance with purely growth questionsconcerns, their painstaking estimation is needed.


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A Biophysical Model of Neuronal Dendrites' Integrative Properties: Relations to Morphological Data

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