Electrodiffusion of Synaptic Receptors:A Mechanism to Modify Synaptic Efficacy?

LEONID P. SAVTCHENKO,1 SERGEY M. KOROGOD,1 AND DMITRI A. RUSAKOV2*1Laboratory of Biophysics and Bioelectronics, Dnepropetrovsk State University and International Center for Molecular

Physiology, National Academy of Sciences of Ukraine, Dnepropetrovsk, Ukraine2Division of Neurophysiology, National Institute for Medical Research, London, UK

KEY WORDS synapse; glutamate; long-term potentiation; AMPAR; lateral electro-diffusion; Brownian motion; receptor anchoring

ABSTRACT We analysed physical forces that act on synaptic receptor-channelsfollowing the release of neurotransmitter. These forces are: 1) electrostatic interactionbetween receptors, 2) stochastic Brownian diffusion in the membrane, 3) transientelectric field force generated by currents through open channels, 4) viscous drag forceelicited by the flowing molecules and 5) strong in-membrane friction. By consideringa-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) type receptors, we showthat, depending on the size and electrophoretic charge of the extracellular receptordomain, release of an excitatory neurotransmitter (glutamate) can induce receptorclustering towards the release site on a fast time scale (8–100 ms). This clusteringprogresses whenever repetitive synaptic activation exceeds a critical frequency (20–100s-1, depending on the currents through individual channels). As a result, a higherproportion of the receptors is exposed to higher glutamate levels. This should increase by50–100% the peak synaptic current induced by the same amount of released neurotrans-mitter. In order for this mechanism to contribute to long-term changes of synapticefficacy, we consider the possibility that the in-membrane motility of the AMPA receptorsis transiently increased during synaptic activity, e.g., through the breakage of receptoranchors in the postsynaptic membrane due to activation of N-methyl-d-aspartic acidreceptors. Synapse 35:26–38, 2000. r 2000 Wiley-Liss, Inc.

INTRODUCTION

Function-dependent changes in synaptic circuitryhave long been considered as the basis of memoryformation in the brain (Hebb, 1949). The cellularmachinery of such changes has been extensively stud-ied, mostly by exploring the paradigm of hippocampallong-term potentiation (LTP), an increase in the synap-tic response observed hours after a short burst ofhigh-frequency stimulation (Bliss and Lømo, 1973; forreview see Bliss and Collingridge, 1993). Althoughmuch is known about mechanisms that are essential forthe induction of LTP, the cellular basis of its expressionhas prompted considerable debate in the literature(McNaughton, 1993; Kullmann and Siegelbaum, 1995;Nicoll and Malenka, 1995; Kullmann et al., 1996). Mostrecent findings seem to favour the hypothesis that anincreased number (or sensitivity) of postsynaptic gluta-mate receptors, in particular a-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid receptors (AMPARs),underlies the increased synaptic response (Malinow,1998). However, identifying cellular mechanisms thatwould link changes in AMPARs to the observed rapid

increase in synaptic responses following induction ofLTP remains a challenge, and a number of candidatecascades have been implicated (Bortolotto et al., 1994;Petralia et al., 1999).

The phenomenon of lateral electrophoresis of recep-tor molecules in cell membranes has long been known(Poo and Robinson, 1977; Jaffe, 1977). In the neuro-muscular junction, Poo and colleagues demonstratedthat the response of acetylcholine (ACh) receptors canbe altered through receptor aggregation in the postsyn-aptic membrane following the external application ofelectric field pulses (Young and Poo, 1983a). In fluores-cent labelling experiments, it was shown that suchaggregation is due to lateral electrodiffusion of receptormolecules (Young and Poo, 1983b). Further studiessuggested electrodiffusion (Poo and Young, 1990) andelectroosmosis (McLaughlin and Poo, 1986) as major

Contract grant sponsors: MRC (UK), BBSRC (UK), INTAS-93–246, and HFSP.

*Correspondence to: Dr. Dmitri Rusakov, Division of Neurophysiology, Na-tional Institute for Medical Research, The Ridgeway, Mill Hill, London NW7 1AA,UK. E-mail: drusako@nimr.mrc.ac.uk.

Received 18 August 1998; accepted 30 March 1999

SYNAPSE 35:26–38 (2000)

r 2000 WILEY-LISS, INC.

candidate mechanisms by which synaptic receptorscould change their positions in the membrane. A theo-retical model based on these data was described whichrelates the cluster patterns of the ACh receptors to theeffect of ACh leaking from the growth cones duringsynapse formation. Stollberg (1995) also studied theo-retical aspects of diffusion-driven receptor aggregationin cell membranes. The estimated aggregation timewas tens of minutes to one hour, which agreed well withexperimental data (Poo and Young, 1990). These impor-tant findings prompt a logical question: Can suchreceptor aggregation be induced by synaptic release ofneurotransmitter? In other words, can electrodiffusionand electro-osmosis induce clustering of postsynapticreceptors on a time scale compatible with synapticactivity itself? And, if so, what are the consequencesregarding synaptic efficacy? In hippocampal synapses,comparison of short- and long-term potentiation ofAMPA- and NMDA-mediated components of synaptictransmission allowed Xie et al. (1997) to put forward ahypothesis that LTP is associated with the alignment ofAMPARs with the sites of neurotransmitter release.However, no experimental technique at present candirectly address this issue, and we believe it is impor-tant to analyse the plausibility of such phenomena froma biophysical standpoint.

Here we consider physical forces acting within agroup of receptor-channels, as exemplified by AMPARs,following the release of glutamate into the synapticcleft. Our physical model develops from work by From-herz and Zeiler (1994), who described what is to datethe most complete nonlinear case of interaction be-tween charged molecules. We show that electric fieldsarising in the microenvironment of the ‘‘typical’’ centralsynapse during synaptic activation can be sufficient toinduce lateral rearrangement of the AMPA type recep-tors. Furthermore, neurotransmitter release repeatedwith a frequency that exceeds a certain critical valuecan induce a progressive clustering under the preferredrelease site. This increases the average probability ofchannel opening, and therefore increases the synapticcurrent in response to the same amount of neurotrans-mitter. However, electrodiffusion mechanisms couldexplain long-term changes in synaptic efficacy only ifthe lateral receptor mobility is reversibly increasedduring intense synaptic activation. To account for this,a candidate cellular mechanism is considered wherebysynaptic activation of N-methyl-d-aspartic acid (NMDA)receptors releases molecular anchors that hold thesynaptic receptors at their ‘‘strategic’’ sites.

METHODS: THEORETICAL CONSIDERATIONS

In our model, synaptic efficacy is considered in termsof the total synaptic current, Isyn, through N postsynap-tic receptor-channels (AMPARs) following the release ofa fixed amount of neurotransmitter (glutamate) into

the synaptic cleft. We postulate that neurotransmitteris released within a small presynaptic membrane area(synaptic vesicle pore) giving a unimodal concentrationprofile in the lateral direction within the synaptic cleft(Wahl et al., 1996; Uteshev and Pennefather, 1996;Kleinle et al., 1996). The total synaptic current isdetermined by this concentration profile and by thekinetics of AMPAR channel opening.

Component forces acting on receptors

Consider area S of the synaptic membrane (withdielectric constant em) containing N receptor-channelsadjacent to the synaptic cleft of thickness d (withdielectric constant ec .. em), as illustrated in Figure1a. The electrophoretic charge q normally associatedwith the extracellular receptor domain, generates alocal electric field potential Vsi. Another electric fieldpotential, Vdi, is generated by the ion current througheach ith open channel, I 5 2gA (Vm 2 Vr) where Vm isthe membrane potential, gA is the open-state channelconductance, and Vr is the reversal membrane poten-tial. Therefore, each channel in the postsynaptic mem-brane is exposed to the accumulated electric fielddefined by the potential oi51

N (Vdi 1 Vsi). This field elicitsforce Fe that acts upon each ith channel and can berepresented by the two corresponding terms, the station-ary one, Fs, and the dynamic one, Fd:

Fei 5 Fdi 1 Fsi 5 2qe 1grad oj51, jÞi

N

Vdj 1 grad oj51, jÞi

N

Vsj2 (1)

where e is the elementary charge, 1.6·10-19 C.The second component force that act on each receptor

is due to Brownian, or stochastic thermal, motion and isdenoted here as Fbrown. The third component force,denoted Fdrag, is due to the net movement of extracellu-lar charged particles (induced by electric forces), whichdrag membrane receptor molecules alongside. Thisdrag force obeys the Stokes law Fdrag 5 6 phb(v2vi)where b is the size parameter of the extracellularreceptor domain, v is the extracellular particles veloc-ity, and vi is the velocity of the ith receptor. Therefore,the accumulated force FS that acts on each ith channel,is given by expression:

FS 5 Fei 1 Fbrown 1 Fosm (2a)

where b is the size parameter of the extracellularreceptor domain, v is the extracellular particles veloc-ity, and vi is the velocity of the ith receptor. In turn, thevelocity of lateral displacement, vi, in a bilayer mem-brane is commonly derived from Langevin’s equationwhich, in the limiting case of strong friction, gives(Arnold, 1974):

fsvi 5 FS (2b)

27SYNAPTIC RECEPTOR CLUSTERING

where fs 5 KbTD-1 (Einstein’s formula) is the frictionfactor, Kb 5 1.38·10-23 J·K-1 is the Boltzmann constant,T is the temperature, and D is the diffusion coefficient.Equation (2), therefore, relates the lateral velocity ofindividual receptors vi to the parameters of synapticenvironment and other physiological variables. Thediagram in Figure 1b illustrates the main participantsin this system, and the explicit form of each term in Eq.(2) is derived in Appendix A.

Release and diffusion of neurotransmitter

The concentration transient of neurotransmitter (glu-tamate) in the synaptic cleft, f(r, t), was approximatedaccording to Kleinle et al. (1996):

f(r, t) 5 u g(r2rc) t1/2 exp (2t/t) (3)

where g(r-rc) 5 exp(-(r-rc)2/d*) is the 2D (in the X-Yplane) Gaussian distribution centred at rc (centre ofarea S), d* is the Gaussian variance in lateral direc-tions, u.0 is a normalising factor, and t is the timecourse parameter. In this simplified approximation, d*(shape factor) does not depend on time as one wouldexpect (Rusakov and Kullmann, 1998a) but we consid-ered Eq. (3) sufficiently accurate for the purposes of ourmodel. Parameters of the release were set so that thepeak concentration could open ,80% of all presentAMPARs in accordance with their kinetics (see below).This corresponded to the peak glutamate level ,40mM, which is lower than its level within synapticvesicles (60–200 mM; Clements et al., 1992). We alsonote that release of neurotransmitter in our model

occurs at a given frequency, which only approximatelyreflects the likely situation where the release occursstochastically.

Stochastic opening of receptor channels

Modelling AMPAR currents normally involves a com-plex kinetic scheme (Jonas et al., 1993; Rusakov andKullmann, 1998a). However, desensitisation of AMPARhas a negligible effect when its onset rate is signifi-cantly slower than the decay rate of the response to abrief application of glutamate, ,5 ms (Hestrin, 1992).In this case, a simplified scheme including three reac-tion steps, is plausible (Heckmann et al., 1996):

A 1 R

k1≤=;≤k21

AR 1 A

k2≤=;≤k22

A2R

k3≤=;≤k23

A2O (4)

where notations are: R, an AMPAR in the closed state,with no bound glutamate molecules; A, glutamate; ARand A2R, the closed-state receptor with one or twoglutamate molecules bound, respectively; and A2O, theopen-state receptor with two glutamate moleculesbound. In the present study, kinetic parameters pro-posed by Kleinle at al. (1996) were used (Table 1) tomeet the condition that the maximum probability ofreceptor opening does not exceed 80%. For simulationpurposes, we employed the stochastic model by DeFe-lice and Isaac (1993) as follows. In the absence ofglutamate, all receptors are in the state R; after release

Fig. 1. Physical interactions between postsynaptic receptor chan-nels. A. Geometry of the system. Presynaptic terminal and postsynap-tic membrane are separated by the synaptic cleft of width d; Npostsynaptic receptor channels (shaded circles) are scattered over amembrane area S; neurotransmitter is released within area S8,S inthe presynaptic membrane where synaptic vesicle exocytosis is mostlikely to occur. B. Major forces acting on receptor channels that bear a(positive) charge in their extracellular domain. Binding of a neurotrans-

mitter molecule (NM) to the receptor induces its stochastic openingand therefore rapid influx of Na1 (and/or Ca21) ions; this influx elicitsa transient dynamic electric field force (Fd ) and a viscous drug force(Fdrag) that attract the neighbouring receptor; at the same time, thereceptors are repulsed through a static Colombian force (Fs ), and areaffected by the stochastic Brownian force (Fbrown) and the in-membranefriction force (Ffric).

28 L.P. SAVTCHENKO ET AL.

of mediator, they move to the state AR with theprobability proportional to k1 or remain in the R statewith the probability proportional to exp(2k1ADt). Withina short time interval Dt after glutamate is released, thereceptor could be in one of the four states: R, AR, A2R orA2O, with the probabilities proportional to exp(2k1ADt),exp2(k-11Ak2) Dt, exp2(k31k-2) Dt, and exp(2k-3Dt),respectively. After that, the receptor can move, with thecorresponding probability, in the new state. For in-stance: the R-state receptor can move to the AR statewith chances proportional to 1-exp(2k1ADt), and theAR-state receptor can move either to the R, or to theA2R state, with the probabilities proportional tok21 (k21 1 Ak2) or Ak2 (k21 1 Ak2) respectively. A MonteCarlo procedure was used in order to simulate thesetransitions with the corresponding probabilities (DeFe-lice and Isaac, 1993). The total synaptic current, Isyn,was computed as Isyn 5 W(Dt)I, where I is the currentthough an individual channel (see above) and W(Dt) isthe number of open-state AMPARs that occur duringtime interval Dt.

Initial and boundary conditions

The initial conditions were set by simulating a uni-form distribution of N receptor-channels within a circu-lar area of radius Rinitial in the postsynaptic membrane.The sealed boundary conditions were applied, withRmax.Rinitial being the maximum radius permitted.Parameters employed in our model are listed in Table 1and will be further discussed below. The Langevin’s Eq.(2b) was solved numerically using the algorithm pro-posed by Ermak (1975). A detailed theoretical analysisof the model’s behaviour is presented in Appendix B.

Geometric indicator of clustering

For the sake of simplicity, we considered the averagedistance from all receptors to the centre of the postsyn-

aptic membrane area as a clustering parameter L:

L 51

N oj51

N

rj,

where rj 5 Î(xc 2 xj) 1 (yc 2 yj)2 is the distance be-tween the jth receptor (with coordinates xj and yj) andthe centre (with co-ordinates xc and yc). Parameter Lwas sufficient for our purposes because our simulationexperiments always observed only one cluster of recep-tors.

RESULTS

We investigated complex behaviour of AMPARs inthe postsynaptic membrane by decomposing it intoseveral simpler, ‘‘component’’ behaviours. First, electro-diffusion of the open-state AMPARs that interact intheir own electric field was assessed. Second, a uni-formly distributed neurotransmitter was introducedinto the system to induce stochastic opening of thereceptors. Finally, a unimodal (Gaussian) distributionof neurotransmitter in the synaptic cleft was consid-ered, and the effect of receptor rearrangements on thesynaptic response was examined.

Clustering of open AMPARsin their own electric field

In these simulation experiments, all AMPARs wereset open and uniformly distributed in the postsynapticmembrane at t 5 0. Immediately after that, they wereallowed to move in the membrane plane while obeyingthe forces represented in our model. Figure 2A showsthe time course of clustering parameter L for severalvalues of the peak single channel current, I (the choiceof I as a convenient varied parameter is addressed inDiscussion). These results show that, within a rela-

TABLE I. Main symbols and parameters of the model

Parameter Symbol Value Units

Effective synaptic cleft width D 10 nmDielectric constant, extracellular solution ec 80 –Temperature T 310 KRadius of individual AMPARa Rc 5 nmElectrophoretic charge in the extracellular AMPAR domain q 4 el. chargeMaximum current through individual AMPAR I 0.5–10 pACoefficient of lateral diffusion D 1–100 nm2 · s21

Total number of AMPARs N 200 –Maximum radius of the post-synaptic area occupied by AMPARs Rmax 300 nmTime constant of glutamate release t* 0.7 msElectrochemical j-potential j 240 . . . 100 mVGaussian variance of the glutamate profile within the synaptic

cleft dp 4900 nm2

AMPAR kinetic rate constants (see Eq. 4)k1 50.5 mM21 · ms21

k21 119.3 ms21

k2 24.1 mM21 · ms21

k22 6.8 ms21

k3 14.84 mM21 · ms21

k23 1.9 ms21

aPhysical radius; the effective radius is larger by the Debay’s length.

29SYNAPTIC RECEPTOR CLUSTERING

tively short period of time (7–50 ms, depending on I),the receptors are aggregated in one cluster of themaximum allowed density near the centre of area S, asillustrated in Figure 2B. Within the cluster, they ap-pear to be packed in a crystalline-like order (Fig. 2B,right panel), as was predicted theoretically by From-herz and Zeiler (1994). Our simulations also demon-strate that the receptors did not cluster at I , 0.5 pA(data not shown), and this critical value of I could alsobe derived analytically from Eq. B3. Although thesesimulations explore an unlikely situation where allreceptors are in the open state, the results demonstratethe parameter domain and the time scale at which thereceptor clustering is plausible.

Clustering of AMPARs following a spatiallyuniform neurotransmitter release

In the next series of simulations, the time course ofglutamate release was compatible with synaptic re-lease but the spatial distribution of the neurotransmit-ter within the cleft was uniform (t 5 0.7 ms andg(r-rc) 5 1 in Eq. 3) producing stochastic opening ofAMPAR channels. Our task was to compare, given theunchanged coefficient of lateral diffusion, the time scaleof AMPAR clustering and that of declustering. Figure3A,B shows the time course of the total synaptic currentIsyn (A) and of the clustering parameter L (B) during fiveneurotransmitter release pulses applied every 10 ms,and typical lateral patterns of AMPARs in this situationare illustrated in Figure 3C. The characteristic timecourse of declustering following the closure of AMPARsis also shown in Figure 3B (dotted line). This timecourse indicates that, given the peak channel current of10 pA, a single neurotransmitter release results in an

increase of the lateral AMPAR density lasting for 40–50ms. In other words, repetitive glutamate release withfrequency that exceeds 20–25 s-1 should induce progres-sive clustering of AMPARs in similar conditions. Thiscritical frequency increases with lower values of peakcurrent I (see Discussion). Since the glutamate levelwithin the cleft was spatially uniform in this series ofexperiments, stochastic opening of individual receptorswas also spatially uniform. Therefore, any rearrange-ment of AMPARs in these conditions did not affect Isyn.A more realistic case is considered below, where thespatial profile of glutamate within the synaptic cleft isunimodal.

Clustering of AMPARs following a spatiallyunimodal neurotransmitter release

In this series of simulation experiments, the unimo-dal (Gaussian) profile of neurotransmitter was formedwithin the synaptic cleft (Eq. 3) following exocytosis,according to Kleinle et al. (1996). Figure 4 shows thecorresponding simulation results for five values of I.The data demonstrate that Isyn increases with repeti-tive releases of equal glutamate amounts. This reflectsthe increased probability of receptor opening as a resultof AMPAR clustering towards the centre of glutamaterelease, as shown by the time course of L in Figure 4Band further illustrated in Figure 4C. The data alsoshow that Isyn reaches its maximum when L 5 Ïd*(where d* is the lateral Gaussian variance of theneurotransmitter level, see Eq. 3), or, in other words,when most of the AMPARs are accumulated within thecharacteristic area of d* (dotted circle in Fig. 4C).

Fig. 2. Clustering of open receptor channels (AMPA-type) in theirown electric field. A. Time course of the characteristic inter-receptordistance, L (see text), for five values of the single channel current, I. B.Example of the initial (t 5 0, left panel), and the resulting (t 5 50 ms,right panel) lateral distribution of open receptor channels in thepost-synaptic membrane (at I 5 4 pA). Dotted circle indicates the area(radius Rinitial) occupied by the receptors at t 5 0. Total current through

all receptors, Isyn, remains unchanged. Other model parameters were:number of receptors N 5 200, receptor radius Rc 5 5 nm, Rinitial 5 200nm, boundary Rmax 5 300 nm, coefficient of lateral diffusion D 5 100nm2/ms, extracellular electrophoretic charge q 5 4e, zeta-potentialj5-100 mV, G2 1 5 200 Ohm · cm, cleft width d 5 20 nm, temperatureT 5 310K.

30 L.P. SAVTCHENKO ET AL.

Longer stimulation trains andthe role of diffusion coefficient

Figure 5 shows the simulated time course of AMPARclustering under repetitive glutamate release (A) andAMPAR dispersion in the absence of glutamate release(B), for five different coefficients of lateral AMPARdiffusion D. These data indicate that longer trains ofstimuli result in a higher degree of clustering (e.g.,compare Figs. 5A and 4B) ultimately reaching themaximum allowed density, and therefore a higherdegree of potentiation (data not shown). The resultsalso demonstrate that the kinetics of the receptorrearrangements depends on the values of D: lowerdiffusion coefficient will proportionately slow down thesystem behaviour that otherwise remains qualitativelysimilar. Although the results presented here regardonly one frequency of synaptic activation, 100 s-1, theeffect of lower frequencies on receptor clustering can besimply extrapolated from these data since the channelcurrents induced by individual synaptic releases areeffectively independent, as long as they are .8 msapart (see Figs. 3A–4A).

DISCUSSIONCellular mechanisms involved

in receptor rearrangements

Unless synaptic receptors, including AMPARs, areanchored to their ‘‘strategic’’ sites, they will diffuseaway from the synapse within minutes or hours (oreven seconds), depending on the lateral diffusion coeffi-cient, as illustrated in Figure 5. However, immunospe-cific labelling at the electron microscope level indicatesa high degree of co-localisation between AMPAR iso-forms and the postsynaptic densities (PSDs) at centralsynapses (Bernard et al., 1997; Nusser et al., 1998). Thecandidate molecular anchor to hold AMPARs at PSDshas recently been identified as a glutamate receptor-interacting protein, GRIP (Dong et al., 1997), which isalso associated with an AMPA receptor binding protein(ABP) related to the PSD (Srivastava et al., 1998).Another protein (PICK1) that interacts with C terminiof AMPARs has also been directly implicated in themodulation of synaptic transmission through synaptictargeting of the receptors (Xia et al., 1999).

Fig. 3. Lateral clustering of neurotrans-mitter activated receptor channels (AMPAtype) under the even distribution of gluta-mate within the synaptic cleft. A. Time courseof total synaptic current, Isyn, for five values ofsingle channel current I (shown) in responseto five brief (t 5 0.7 ms) pulses of glutamaterelease separated by 10 ms intervals. Giventhe same single channel current I, peak val-ues of Isyn remain effectively unchangedthroughout each train. B. Time course of thecharacteristic inter-receptor distance, L, forfive values of single channel current I (shown).Dotted line shows the experiment (at I 5 10pA) where no subsequent pulses were deliv-ered after the first glutamate release indicat-ing the difference between the clustering time(tc 5 8 ms, shown by arrow) and the de-clustering time (td 5 42 ms). C. Lateraldistribution of receptors at t 5 50 ms, at I 5 2pA (left panel) and I 5 10 (right panel).Dotted circles indicate the area of initialdistribution (radius Rinitial). Glutamate re-lease parameters were, according to equation3: g(r 2 rc) 5 1 (uniform distribution) and t 50.7 ms. Other model parameters were thesame as in Figure 2.

31SYNAPTIC RECEPTOR CLUSTERING

Clearly, if AMPARs remain restrained by the molecu-lar anchors during the time of synaptic activation, theiractivity-dependent lateral electrodiffusion is unlikely.For significant electrodiffusion to follow neurotransmit-ter release, the anchoring link should be transientlybroken in a use-dependent manner. A potential molecu-lar mechanism of such breakage has been suggested:The state of the postsynaptic cytoskeleton can bemodulated through activation of NMDAreceptors, whichin turn involves MAP2 (Bigot et al., 1991; Quinlan andHalpain, 1996), a key linking protein in the meshworkof microtubules and neurofilaments. In an in vitromodel system (cultured hippocampal neurons), it hasbeen shown that incubation with NMDA for severalminutes results in a loss of dendritic spines, which isaccompanied by a selective loss of actin filaments atsynapses (Halpain et al., 1998). This may have directimplications for the receptor anchor breakage mecha-nisms, since the actin meshwork has been associated

with the anchoring of both NMDA and AMPA receptors(Wyszynski et al., 1997; Allison et al., 1998). Whetherthis sequence of events occurs at individual synapsesduring intense activation remains to be establishedexperimentally, but whenever it is the case, mecha-nisms described in a model could contribute to use-dependent synaptic strengthening. On the contrary,LTP induced by pairing postsynaptic cell depolarisationwith low frequency stimulation (for review, see Blissand Collingridge, 1993) cannot be explained by ourmodel.

Synaptic depression/depotentiation

The example shown above (Fig. 4) illustrates how thepresent model explains synaptic potentiation. Can itexplain use-dependent synaptic depression? In ourparadigm, this would imply repulsion of receptors inlateral directions following a certain pattern of synaptic

Fig 4. Lateral clustering of neurotransmit-ter activated receptor channels (AMPA type)under the Gaussian distribution of glutamatewithin the synaptic cleft (with its center atthe release cite). A. Time course of the totalcurrent, Isyn, for five values of single channelcurrent I (shown). Given the fixed value of I,the peak Isyn increases with each subsequentglutamate release pulse. B. Time course ofthe characteristic inter-receptor distance, L,for five values of the single channel current, I(shown). C. Lateral distribution of the recep-tors (at I 5 6 pA) at t 5 0 ms (left panel) andt 5 60 ms (right panel). Dashed circle indi-cates the area of initial distribution (radiusRinitial). Dotted circle indicates the area of thehighest glutamate concentration, Îd*,70 nm(the charcterstic length of the radial Gauss-ian decay of the glutamate level, according toequation 3). Other model parameters werethe same as in Figures 2 and 3.

32 L.P. SAVTCHENKO ET AL.

activity; for instance, the pattern associated with thephenomenon of long-term depression (Bear and Abra-ham, 1996). In the conditions of the model, repulsionwill occur whenever the AMPAR membrane anchors arebroken but no significant inward current flows throughthe receptors themselves. Experimental findings thatrelate activation of NMDA receptors to disruption of thecytoskeleton in dendritic spines (Halpain et al., 1998)lend support to this hypothesis. The unbinding kineticsof high-affinity NMDA receptors is known to be muchslower than that of AMPARs, which have a much loweraffinity. Therefore, single or low-frequency activation ofNMDAreceptors could induce relatively prolonged (tensof milliseconds) deanchoring of AMPARs while remain-ing below the critical frequency required for clustering(see Fig. 3). As a result, AMPARs will repulse and thesynaptic strength will be reduced. Reported stimula-tion protocols for induction of long-term depression, inparticular low-frequency trains, are in line with thisexplanation (Bear and Abraham, 1996). Another plau-sible, but less direct, explanation for depotentiation inour model is to recruit an independent second messen-ger system. The related discussion, however, is outsidethe scope of the present study.

Long-term changes

To maintain the induced change in synaptic strength,the lateral receptor pattern formed during synapticactivation should hold together after synaptic activityhas ceased. In other words, movements of the receptors

after the cessation of synaptic activity should be re-stricted, otherwise they will repulse due to electrostaticinteraction and Brownian forces. Figure 6 illustrateshow the repulsion of AMPARs affects the synapticresponse 2 sec after repetitive activity (five pulses) hasended. Apparently, long-term synaptic changes willrequire restoration of the anchoring links followingreceptor rearrangement. Although it is plausible tosuggest that the submembrane actin mesh, which isassociated with the AMPAR anchoring (Wyszynski etal., 1997; Allison et al., 1998) and can be disrupted byNMDA receptor activation (Halpain et al., 1998), isrestored after the receptor activation has ceased, thishypothesis remains to be ascertained experimentally.

Critical parameter relationshipsrevealed by the model

In our simulation examples, we used the value ofsingle channel current I as a varied parameter. Theconductance of an individual receptor varied from 5 pSto 100 pS, whereas experiments suggest 10–40 pS.Although this choice may not appear the most sensiblefrom a physiological viewpoint, parameter I is in factinvolved in all model equations so that its changesrepresent a convenient tool to explore all importantvariables. Therefore, the extended range of this param-eter has allowed us to keep other parameters of themodel unchanged. In particular, the critical condition ofclustering depicted in Eq. (B3) in our model can be

Fig. 5. Clustering of AMPA-type receptor channels during repeti-tive glutamate release pulses (A) and their de-clustering in theabsence of glutamate release (B) for five different coefficients of lateraldiffusion (shown, in nm2/ms). Notations: L, the characteristic inter-receptor distance. In A, data for diffusion coefficient 0.1 nm2/ms are

not shown: in these conditions the receptors show predominance ofrepulsion over clustering. Dotted line in A, the lowest possible value ofL. The single channel current is I 5 6 pA, and the glutamate profilefollows Gaussian distribution as in Figure 4. Other parameters are thesame as in Figure 4.

33SYNAPTIC RECEPTOR CLUSTERING

rewritten with respect to a dimensionless parameter inthe form:

g 52p

S

RT

F

Gd

1q 2 1 2zF

RT2 rstI

(5)

Equation 5 postulates that if g , 1, AMPAR tend tocluster, and if g $ 1, AMPARs tend to repulse in thepostsynaptic membrane. Therefore, varying parameterI in our examples can be thought of as varying param-eters S, z, or q, according to Eq. (5).

In our model, synaptic strengthening critically de-pends on the frequency of presynaptic stimulationbecause the latter defines the rate of progressive cluster-ing. Progressive clustering will occur only when param-eter L decreases between subsequent release events.Our calculations show that, given the time of decluster-ing to the initial state of ,40 ms, the critical frequencyß that lowers L is ,22 s-1 (Fig. 3B). Because both therates of clustering and that of de-clustering dependsimilarly on the diffusion coefficient D (see Fig. 6), thevalue of D will have no significant effect on ß. However,another parameter, the value of current I, affects onlythe rate of clustering (Figs. 3,4). Therefore, lowervalues of I or related parameters (see Eq. 5) willcorrespond to higher values of critical frequency ß in aroughly proportionate manner. A more detailed analy-sis that relates the critical parameter values to themodel behaviour is given in Appendix B.

Plausible parameter domainand limitations of the model

In our study, AMPARs occupy a 400 nm wide area ofthe postsynaptic membrane, a characteristic size ofPSDs in central synapses. Significant electrodiffusionof receptor molecules within such a small area isplausible only due to the high field density inside anarrow (10–20 nm wide) synaptic cleft. For example, ina 10 nm cleft containing the medium of 200 Vcmspecific resistance, the current through 100 open recep-tor channels with conductivity 50 pS each generates theelectric field density E,100 V/cm (see Eq. A11). This istwo orders of magnitude higher than the value (0.2–2V/cm) estimated by Poo and Young (1990). Such apowerful electric field could induce dense clustering ofreceptors with negligible electrostatic repulsion. In thiscase, the receptors will be driven by electrophoresiswith the velocity of v 5 qD(F/RT)E. Although thelateral diffusion coefficient for AMPARs is unknown, itwas measured to be within the range of 70 6 30 nm2/msfor NMDARs (Benke et al., 1993). In this case, givenD 5 100 nm2/ms, the characteristic time of clusteringwill be in the range of 100 ms. The effect will be evenstronger whenever the medium specific resistance ishigher than the likely lower limit of 200 Vcm consid-ered in our model. Note also that similar phenomenaare unlikely to be induced on the inner surface ofplasmalemma because characteristic dimensions of thelumen are at least an order of magnitude higher thandimensions of the synaptic cleft and, therefore, field

Fig. 6. Longer-term activity-dependent changes in synapticstrength. Left panel illustrates the synaptic response to a brief‘tetanus’ of five glutamate release pulses (with the Gaussian profile ofglutamate in the cleft) applied in conditions of I 5 6 pA, D 5 100nm2/ms. Right panel shows the synaptic response to a single gluta-

mate pulse two seconds after the tetanus, in conditions where thelateral diffusion coefficient after the tetanus was kept at one of the fourvalues (shown, in nm2/ms). Other parameters were the same as inFigures 4 and 5.

34 L.P. SAVTCHENKO ET AL.

perturbations caused by channel currents will be negli-gible.

Apparently, the inward current through any otherreceptors, e.g., NMDARs, could induce clustering ofAMPARs in our model. However, we do not include theeffect of voltage-dependent conductances on the recep-tor movements in the model. This could induce aregenerative process of membrane depolarisation which,in the limiting case, should reverse the membranepotential and thus invert the sign of the receptorcurrent. Another uncertainty is that the accurate valueof the electrophoretic charge of the AMPAR extracellu-lar domain is unknown, although it is thought that theextracellular parts of AMPARs do not possess nega-tively charged sialic acid residues (Hollman and Hine-mann, 1994). A potential error could also arise from thefact that the simulated glutamate profile (Eq. 3) adoptsthe diffusion coefficient of glutamate in free medium.Indeed, diffusion could be significantly retarded in thesynaptic cleft due to interactions with extracellularmatrix macromolecules (Rusakov and Kullmann,1998b). In essence, our study should be considered as ageneral analysis of forces acting on postsynaptic recep-tors following synaptic activation, with an example ofone possible scenario occurring as a result of theseforces.

ACKNOWLEDGMENTS

The authors thank Drs. Dimitri Kullmann and AlanFine for reading the manuscript and for valuablecomments. This study was inspired by a discussionwith Prof. T.W. Berger at a St. Brieaux HFSP meeting,1996.

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APPENDIX A. COMPONENT FORCES ACTINGON THE POSTSYNAPTIC RECEPTOR

Electrostatic potential

Since charge q is exposed to the extracellular electro-lyte solution and Vsi is small compared to KbT/e ? 25mV (e 5 1.6 10-19 C), the Debay-Huckel’s expression forVsi takes the form (Debay and Huckel, 1923):

Vsi (r) 5q ? e ? exp (2k0r0)

2pe0ec 0r0

ri

0r0(A1)

where the characteristic decay distance, k-1, representsthe Debay’s length, and e0 5 8.85·10-12 C2·J-1·m-1 is thepermittivity of vacuum. Potential Vsi given by expres-sion (A1) is twice the value predicted by the Debay-Huckel’s theory because charge q is excluded from themembrane phase. The Debay’s length is defined asfollows (Debay and Huckel, 1923):

k21 5Î e0ec KbT

2e2 on

zn2 Cn

(A2)

where zn and Cn are the valence and the concentrationof the nth ion species.

It follows from expressions (A1-A2) that

grad Vsi 5 2Vsi 1k 11

0r02ri

0r0(A3)

Therefore, the coordinate components of the electricfield gradient produced by all N channels at the point ofthe ith channel are (considering the plane of membraneas the X-Y plane):

­Vsi

­ x5 2

q ? e

2pee0o

j51, jÞi

N exp (2k0rj0)

0rj0 1k 11

0rj2

3(xi 2 xj)

0rj0

(A4a)

­Vsi

­ y52

q ? e

2pee0o

j51, jÞi

N exp (2k0rj0

0rj0 1k 11

0rj02(yi 2 yj)

0rj0(A4b)

where 0rj0 5 Î0xi 2 xj02 1 0yi 2 yj0

2 22Rc is the minimumdistance between the ith and the jth AMPARs, and Rc isthe characteristic radius of AMPARs.

Transient potential generatedby channel currents

In order to assess the spatial profile of the nonstation-ary electric field potential in the synaptic cleft (deter-mined as Vd 5 SVdi at each time point), we consider theplausible situation where the intracellular medium ismuch larger than the synaptic cleft. In this case,according to Qian and Sejnowski (1989), Vm 5 –Vd 5–SVdi and Vd is equal to the change of the total chargein the synaptic cleft divided by the membrane capacity:

­Vd

­ t5

Fd

Cm

­ on

Cn zn2

­ t(A5)

where F 5 9.65·104 C·mol-1 is Faraday’s constant, zn

represents the valence of the nth ion species, Cn is theconcentration of nth ion species, and Cm is the mem-brane capacity. The movement of ions in the synapticcleft is described by the Nernst-Plank electrodiffusionequation as described below.

To determine potential Vd, we shall first determinethe electrical field potential produced by the currentthrough each ith open AMPAR, Vdi. Given the negligibleeffects of the electric field density perpendicular to themembrane (dVdi/dz 5 0), the polar coordinates withthe origin placed at the ith receptor centre represent aconvenient coordinate system. According to physiologi-cal data,AMPARs conduct a voltage-independent, alpha-function type current (Brown and Johnston, 1983)which implies that the conductivity of each open recep-tor does not depend on Vd. This allows the linear casewhere potential Vd is the sum of local potentials Vdi

produced by individual channels (see Fromherz andZeiler, 1990, for the nonlinear case).

Now we shall consider how the transition ‘‘closed-open-closed’’ in the ith receptor-channel alters the ionconcentration. Given the charge Q 5 IDt transferredduring the open state period Dt, the concentration

36 L.P. SAVTCHENKO ET AL.

profile C for each nth ion species is described by theNernst-Plank electrodiffusion equation:

­Cn

­ t5 Dn

1

r

­

d rr 1­Cn

­ t1

F znCn

RT

­Vd

­r 21

Q

2pzFRcdd (ri 2 Rc)d(Dt)

(A6)

where d represents delta-function.Substitution of (A6) into (A5), and subsequent summa-

tion over each species of ions in the synaptic cleft gives:

­Vdi

­ t5

d

Cm

1

r

­

d rr 1G ­Vdi

­ r1

RT

F

­ G

­ r 21

Q

2p CmRcd (ri 2 Rc)d(Dt)

(A7)

where G 5 F2 ⁄ RTon Dnzn2Cn is the conductivity of the

synaptic cleft.Assuming that G is constant, we obtain

­Vdi

­ t5

d G

Cm

1

r

­

­r 1r­Vdi

­ r 2 1Q

2pCmRcd (ri 2 Rc)d(Dt) (A8)

The solution to this equation is

Vdi 5Q

2pCmQtexp 12 ri

2 1 Rc2

4Qt 2 I0 1riRc

2Qt2 (A9)

where I0 is Bessel’s function, and Q 5 Gd/Cm is thepropagation coefficient, which is measured in the sameunits as the coefficient of diffusion. Because QDt .. S(where S is the postsynaptic membrane area), thesteady-state approximation dVdi/dt ? 0 can be em-ployed, which reduces Eq. (A8) to

Gd

r

­

d r 1r­Vdi

­ r 2 1I

2p Rcd (ri 2 Rc) 5 0 (A10)

Therefore, the radial gradient of the electrical fieldpotential induced by the ith open receptor in thesynaptic cleft is given by the expression:

dVdi

dr5 2

I

2p d G 0r0

ri

0r0(A11)

It follows from expression (A11) that, in the case of linearsummation, the electric field potential gradient producedby all N receptors at the point of the ith receptor takesthe form (the plane of membrane is the X-Y plane):

­Vd

­ x5 2

I

2pdG oj51, jÞi

N 1

0rj0

(xi 2 xj)

0rj0(A12a)

­Vd

­ y5

I

2p dG oj51, jÞi

N 1

0rj0

(yi 2 yj)

0rj0 (A12b)

Brownian force

A stochastic force Fbrown guides random thermalmotion of AMPAR-type molecules in the postsynapticmembrane. The magnitude of this force is known tofollow the Gaussian distribution, and its autocorrela-tion function is a d-function with a ‘‘strength’’ factor B 52fs·Kb·T. The average of the Brownian force is zero, andB defines the intensity of random pushing with respectto the surrounding lipids (independent of electrical fieldforces). For Brownian motion, the expected squaredradius of displacement, ,r2., is determined by expres-sion ,r2. 5 4Dt where D is the diffusion coefficient,and t is time.

Osmotic drag force

The electric field induces movement of the peri-membrane extracellular layers of charged particleswith velocity v, so that dv ⁄ dx 5 v ⁄ K where x is thedistance to the membrane, and K is the Debay’s length(,1 nm for cell membranes). The velocity v of the fluxjust outside the peri-membrane double layer in field Eis given by v 5 2ere0bE ⁄ h where e0 is the permittivityof vacuum, er is the dielectric constant of the aqueousphase, and h is the viscosity of the extracellular me-dium. As mentioned in the main text, the drag forceacting upon the ith receptor in the aqueous phase isFdrag 5 6phb(v-vi). Assuming that the membrane viscos-ity is much greater than that of the extracellularmedium, the velocity of the receptor will be given, aftera few elementary transformations and taking intoaccount Eq. (A11), by the formula:

vi 5 2De

KbT 1q 2 zF

RT2I

2pdG oi

ri

0r00r0(A13)

where z is the z-potential of the postsynaptic membrane(normally between 240 and 2120 mV).

APPENDIX B. ANALYSIS OF THE MODELGeneral solution

To analyse conditions of AMPAR clustering in thepostsynaptic membrane, a deterministic set of equa-tions was obtained from the stochastic Eqs. (2), (A11),and (A13) using standard techniques which also ex-cluded the electrostatic interaction between receptors:

­rs

­ t5 D 5Drs 2

F

RT 1q 2 zF

RT2 (div(rs grad Vd))6 (B1a)

Cm

­Vd

­ t5 GdDVd 1 rs pVd (B1b)

where D is the Laplacian operator, rs is the surfacedensity of AMPAR channels, R ? 8.31m2·kg·s-2·K-1·mol-1

is the gas constant. Equation (B1) takes into accountthat each AMPAR has only one conformation whichcorresponds to the open channel state.

37SYNAPTIC RECEPTOR CLUSTERING

The model described by Eq. (B1) represents a nonlin-ear, nonstationary system, which could feature differ-ent behaviours depending on the adopted parameters.We shall begin, therefore, by establishing major con-straints, which would provide a realistic, physiologi-cally plausible prediction of the AMPAR distribution inthe postsynaptic membrane.

Limiting cases

To define conditions that are essential for AMPARclustering, we followed the approach described previ-ously (Fromherz, 1988; Korogod and Savtchenko, 1997).The condition of instability, which is a necessary condi-tion of clustering, with respect to the spatially inhomo-geneous perturbation in an infinite domain is:

(q 2 RTz/F 2 1) I . 0 (B2)

It is believed that the electrophoretic charge q relatedto molecules like AMPARs is positive at physiologicallevels of pH (Ryan et al., 1988). As long as the extracel-lular domain is not negatively charged, the necessarycondition (B2) is commonly met because AMPARs con-duct the inward Na1 current and the value of z rangesform 240 to 2100 mV (Hille, 1992). Therefore, suchreceptors should cluster due to electroosmotic forces.

However, clustering or other electrodiffusion phenom-ena become negligible when the receptor density doesnot provide conditions for their significant interactionswithin the cleft. In other words, perturbations due tochannel opening are not always sufficient to overrideforces that hold the initial receptor pattern. Equation(B1) allows us to establish the critical condition thatwould enable receptors to cluster. This condition, whichwas obtained using the small perturbation analysisdescribed earlier (Turing, 1953; Murray, 1989), takesthe form:

S . 2 1pRT

F 22 Gd

1q 2RTz

F2 12 rstI

(B3)

where rst 5 N ⁄ S is the average lateral density of AM-PARs. Expression (B3) allows the estimation of theparameter range necessary for the channel clusteringto occur. For instance, assuming G-1 5 200 V·cm, q 5 7,rst 5 1,000 µm-2, I 5 10 pA, d 5 20 nm, z 5 2100 mV, weobtain S ,0.0225 µm2. This implies that AMPARpatterns will depend on glutamate release only if themembrane area occupied by the receptors is greaterthan 0.0225 µm2 (which gives N 5 23).

The opposite limiting case of electrostatic interac-tions between equally charged AMPARs occurs when

the average distance between molecules approachesDebay’s radius: in this case, repulsion overrides allother interactive forces. For instance, if ec 5 80 and theelectrolyte concentration in the cleft is ,200 mM,Debay’s radius, k-1, approaches ,1 nm. Sutcliffe et al.(1996) recently performed stochiometric reconstructionof several molecular subunits which are likely to consti-tute AMPARs. Their analyses suggest the sizes of theextracellular AMPARs domains to be in the range of10–12 nm. This value, added to Debay’s length, deter-mines, therefore, the highest plausible density of thereceptor channels.

Critical conditions of clustering

To establish the critical role of electrostatic interac-tion between AMPARs for receptor rearrangements, wefirst excluded the Brownian force. Second, in order tointroduce the parameter conditions of clustering, wedefined two characteristic distances: 1) the averagedistance between two AMPARs in the form ,r8. 5

(S ⁄ N)1/2, and 2) distance ,r88. satisfying the followingequation:

1

N oi51

N

1grad oj51, jÞi

N

Vdj 1 grad oj51, jÞi

N

Vsj2 5 0. (B4)

Equation (B4) represents the condition where the result-ing electric field force equals zero.

If FS 5 Fe 1 Fbrown 5 2q(grad oj51,jÞiN Vdj1grad oj51,jÞi

N

Vsj) 1 Fbrown and all N receptors are open, their cluster-ing is likely to occur when r8 . r88 (or S . r88 N), and,similarly, declustering is likely to follow when r8 , r88.For instance, setting G-1 5 200 V·cm, q 5 7, z 5 2100mV, N 5 23, I 5 10 pA, and d 5 20 nm, we obtain fromEq. (B4) the condition S ? 38 nm (r88 ? 7.9 nm). Thisimplies that electrostatic interactions are likely toinduce declustering if the initial channel density.15,000 µm-2. Because this value is apparently higherthan the physiologically plausible channel density, elec-trostatic interactions between real open AMPARs arelikely to assist, rather than prevent, the process ofAMPARs clustering (given the parameters listed above).Note, however, that different phenomena would occur ifthe electrophoretic charge of the receptor was locatedwithin the cell membrane, which dielectric constant ismuch lower (em ? 2) than that of the extracellularsolution. To get analytical insights into the Brownianterm was a less straightforward task; therefore, to solvethis equation, we employed a Monte-Carlo simulationapproach.

38 L.P. SAVTCHENKO ET AL.