Proc. Nati. Acad. Sci. USAVol. 91, pp. 3941-3945, April 1994Neurobiology

A model of dendritic spine Ca2+ concentration exploring possiblebases for a sliding synaptic modification threshold

(N-methyl-D-aspartate receptor/synaptic plasticity/long-term potentiation)

JOSHUA I. GOLD* AND MARK F. BEARtDepartment of Neuroscience and Institute for Brain and Neural Systems, Brown University Box 1953, Providence, RI 02912

Communicated by Leon N Cooper, September 27, 1993

ABSTRACT We used a biophysical model of an isolateddendritic spine to assess quantitatively the impact of changes inspine geometry, Ca2+ buffer concentration, and channel ki-netics on Ca2+ dynamics following high-frequency activation ofN-methyl-D-aspartate receptors. We found that varying thebuffer concentration in the postsynaptic density from 50 to 500,uM can result in an 8-fold difference in the peak Ca2+concentration following three pulses -at 100 Hz. Similarly,varying the spine neck diameter from 0.1 to 0.55 jAm can resultin a 15-fold difference in the peak Ca2+ concentration. Theamplification of peak Ca2+ concentration also depended ontemporal summation of N-methyl-D-aspartate-mediated excit-atory postsynaptic currents. Variation of the current durationon the order of 100 msec can significanly affect summation ata given stimulation frequency, resulting in a 10-fold differencein the peak Ca2+ concentration at 100 Hz. It is suggested thatactivity-dependent modifications of these parameters may beimportant for the regulation of synaptic plasticity in the brain.

"Hebbian" synaptic modification, in which an excitatoryconnection is strengthened whenever pre- and postsynapticelements are synchronously active, has been shown bycomputational models to be useful to account for aspects oflearning, memory, and neural development (for reviews, seerefs. 1-3). Success of these models usually requires, inaddition, mechanisms to constrain synaptic weights so thatthe network of modifiable synapses reaches a stable equilib-rium. Thus, two questions of neurobiological interest emergefrom this work: (i) is there a neurobiological basis forHebbian synaptic modification and (ii) how are Hebbiansynaptic modifications regulated?The answer to the first question is clearly yes. Long-term

potentiation (LTP), a form of synaptic plasticity first de-scribed in the hippocampus (4, 5), appears to have thecharacteristics expected of a Hebbian modification (for re-views, see refs. 6 and 7). Induction of LTP in area CA1 ofhippocampus (8-10), in dentate gyrus (11), and in someregions of neocortex (12-14) occurs following simultaneouspresynaptic input activity and strong postsynaptic depolar-ization. There are a number oflines ofevidence that this formof LTP in hippocampus is triggered by a brief surge inintracellular Ca2+ during a tetanus (15-19). In addition,activation of N-methyl-D-aspartate (NMDA) receptors is arequisite step for the induction of this form of plasticity (20).Thus, the discovery that a substantial, voltage-dependentCa2+ conductance is activated by agonists acting at theNMDA channel paired with significant postsynaptic activity(21, 22) appeared to account for all the properties of HebbianLTP (23).

Recently, however, this simple model has been broughtinto question. Two separate theoretical analyses of the Ca2+

dynamics that might result from NMDA receptor activationon dendritic spines have led to the same conclusion; namely,the "nonlinear" properties of the NMDA receptor-mediatedCa2+ flux are not sufficient to account for the "nonlinear"stimulation requirements for induction of LTP (24, 25). Bothgroups suggested that an additional nonlinear factor, such asa Ca2+ diffusion barrier at the spine neck or the existence ofa saturable Ca2+ buffer in the spine head, could account forthe properties of LTP (making the critical assumption, ofcourse, that the relation between the Ca2+ concentration andthe amount or probability of LTP is monotonic).We were struck by the possibility that regulation of these

same parameters might offer an answer to the second ques-tion posed above, concerning how LTP is regulated. Thetheoretical underpinnings for this supposition are providedby the work of Cooper and colleagues (26-30). This work hasprovided an analysis of a theoretical form of synaptic plas-ticity in which the occurrence of Hebbian modification iscontingent upon the activation of the postsynaptic neuronbeyond a critical value termed the modification threshold.The key property that imparts stability in these models is theproposition that the value of the modification threshold (setfor the entire cell) is some function of recent postsynapticactivity. Based on the models of Ca2+ dynamics in spinesreviewed above, the proposal was made that regulation ofexpression of Ca2+-binding proteins as a function of cellactivity might be a physical basis for such a "sliding"modification threshold (31).The aim of the present work was to use a computational

model of Ca2+ dynamics in dendritic spines to examine thefeasibility of such a proposal. We elaborate on previousmodels to explore in more detail how parameters such asbuffers, spine geometry, and the time course of NMDAreceptor-mediated excitatory postsynaptic currents (EPSCs)affect the rise in Ca2+ concentration that might accompany atetanus. The results show that as little as a 2-fold change inspine neck diameter, buffer concentration, and/or NMDA-mediated EPSC decay time will substantially alter the peakrise in Ca2+ and may therefore significantly affect the abilityto induce LTP.

MATERIALS AND METHODSThe simulations performed were based on the idealizedgedankenexperiment presented in Fig. 1A and concerned thedynamics of Ca2+ following tetanic stimulation of a singleaxospinous synapse. The experiment is highly idealized:stimulation is at a single synapse which contains a fullcomplement of NMDA receptors; NMDA receptor activa-

Abbreviations: EPSC, excitatory postsynaptic current; LTP, long-term potentiation; NMDA, N-methyl-D-aspartate; Vm, membranepotential.*Present address: Department ofNeurobiology, Stanford UniversityMedical Center, Stanford University, Stanford, CA 94305-5401.tTo whom reprint requests should be addressed.

3941

The publication costs of this article were defrayed in part by page chargepayment. This article must therefore be hereby marked "advertisement"in accordance with 18 U.S.C. §1734 solely to indicate this fact.

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3942 Neurobiology: Gold and Bear

AAction poter

Bitials Dendrtc

Buffers spine

O Ca2+ O Diffusion0 Influx0 via NMDA

receptors Pumps

Ideal voltage clamp

123 4 5T6 7

0.55 PM

FIG. 1. (A) Gedankenexperiment to investigate the Ca2+ dynamics resulting from high-frequency activation ofNMDA receptors on the headof a dendritic spine. (B) Eight-compartment model of the dendritic spine. The spine was considered to be cylindrical, with cross-sectional sizes(length x width) through the center as 0.55 atm x 0.55 ,am for the spine head and 0.73 ,am x 0.1 ,um for the neck. Neck diameter was variedfor some simulations.

tion alone causes Ca2+ influx; a perfect voltage clamp keepsthe membrane potential (Vm) constant, which allows theelectrotonic contribution of coactivation of nearby synapsesand the colocalized AMPA receptors-i.e., depolarization ofthe cell-to be effectively ignored; and measurements ofCa2+ concentration are done within the single spine.The experiment was modeled by means of a hybrid of the

computational approaches used by Holmes (32), Holmes andLevy (24), and Zador et al. (25) to describe Ca2+ influxfollowing NMDA receptor activation, diffusion into the den-drite, sequestration by Ca2+-binding substances, and effiux

through pumps. An eight-compartment model of a singledendritic spine was used (Fig. 1B).

Synaptic activation was modeled as an increase in Ca2+ incompartment 1 of the model as derived from electrophysio-logical and modeling studies of NMDA receptor propertiesand subsequent Ca2+ increases. Although there is some

debate as to the relative contributions of influx throughvoltage-gated channels, influx through NMDA receptor-mediated channels, and release from intracellular stores (33,34), there is evidence that for the range of depolarizationlevels examined, changes in Ca2+ concentration are reason-

ably approximated as a fixed percentage (2%) of totalNMDA-mediated current (35). The NMDA-mediated con-

ductance was modeled as a function of time, transmitterbinding, and voltage (adapted from ref. 25):

{aet/li + (1- a)e-/T2}1- et/Tn

1 + iMg2 + ]e-YV[1]

in which T = 250 msec, T2 = 50 msec, r3 = 7.0 msec, =

0.33/mM, y = 0.06/mV, and gn = 0.2 nS (36-38). In Eq. 1,the numerator describes the kinetics of the receptor, whichhas a high affinity for glutamate and a slow rate of unbinding.The double-exponential decay function was used so that therelative contributions of the slow and fast terms could bemanipulated (39) by varying a between 0 and 1. The denom-inator reflects the level of depolarization needed to relievethe Mg2+ block. Because Vm was "clamped" in every case,the NMDA current was directly proportional to the conduc-tance from Ohm's law and was calculated by assuming areversal potential of 0 mV (22). Brief episodes of repetitivestimulation were modeled as the sum of the individualwaveforms.The dynamics of Ca2+ concentration following NMDA

receptor activation were modeled by using equations fromHolmes and Levy (24). The effects of spine head and necksize were considered in relation to the diffusion of Ca2+ outof the spine head. For compartment i, the change in Ca2+concentration due to diffusion to adjacent compartments isgiven by the one-dimensional diffusion equation

d[Ca],

+ ([Ca]i- [Ca]i+i) .

i,i+lJ

[2]

in which D is the diffusion constant for Ca2+ (related to theaverage distance a Ca2+ ion will travel in a given amount oftime) and equals 0.6 ,um2/msec, Vi is the volume of the ithcompartment, and A/8 is the coupling coefficient betweencompartments i and j, computed as 2(al, + ajlj)/(li + l4)2,where Ai and Aj are the cross-sectional areas and 1i and 14 arethe lengths of compartments i and j. Head and neck sizeranges were taken from electron microscopy studies (40).Two types ofATP-driven pumps are thought to exist on the

postsynaptic element. Both seem to operate at similar ve-locities and were simulated by a single term:

d[Ca],

= -i

_kp.[Cfa]i -[Ca]r)[dt

in which kpi is the rate constant computed from the velocityof the ATP-driven pump (1.4 x 10-4 cm/sec) (41) multipliedby Ai/Vi, where Ai is the membrane area of compartment iand Vi is its volume.

All Ca2+-binding proteins, including calmodulin, calbindin,and calcineurin, were lumped into a single buffering term.Calmodulin is normally the most highly concentrated ofthesehigh-affinity buffers, generally found in the cytosol at 30-50jM. Because calmodulin has four Ca2+ binding sites, buff-ering capacity was varied in our simulations around 120-200AM (42). Studies have revealed a significant amount ofcalmodulin in the postsynaptic density (the first 100 nm fromthe tip of the spine head); therefore, buffer concentrationswere taken to be higher in compartments 1 and 2 thanelsewhere. The kinetics of the binding and unbinding of Ca2+from these buffers are described by the following two equa-tions; the first describes the change in Ca2+ concentrationdue to sequestration by the buffers, and the second describesthe change in the amount of available buffer after applicationof Ca2+:

d[Ca]idt

=- kbf[Ca]IB]i + kbb([BJi - [B]1)

dt - kbf[Ca],tB]i + kbbjBdi- [B],),

[4]

[5]

in which B is the available buffer, Bt is total buffer, and kbf

and kbb are the forward and backward binding constants,

8

Proc. Natl. Acad Sci. USA 91 (1994)

QCali [Cali-,)

[31

9NMDA(t) = g,

-D A

vi 8ij-1

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respectively; kbN was set at 0.5 msec-1, and kbf was set at 0.5AM-1-msec-1. Although experiments have still not revealedtheir true values, the values chosen have been used in anumber of modeling studies (24, 43).

RESULTSWe began by replicating the findings of Holmes and Levy (24)and Zador et al. (25), showing that two mechanisms-diffusion into the dendrite and sequestration by buffers-most profoundly modulated the steep rise in Ca2+ concen-tration in the spine head. These mechanisms operated on timescales of <1 msec and significantly influenced the nonlinearCa2+ rise required for LTP induction. Pumps, on the otherhand, reduced the Ca2+ concentration much more slowly(>100 msec to return Ca2+ to resting levels) and thereforewere less able to affect transient changes in Ca2+ concentra-tion significantly (data not shown). These results are sup-ported by experimental evidence which suggests that follow-ing rapid influx, Ca2+ ions will immediately diffuse away fromthe membrane, then very quickly (within 10 msec) bind tobuffer proteins and become sequestered in the smooth en-doplasmic reticulum, and finally over the next 0.1-1.0 sec betransported into the extracellular fluid via pumps (44).We next focused on buffers, diffusion, and the time course

of NMDA receptor-mediated currents as means to regulatepeak Ca2+ transients in dendritic spines. All of these Ca2+-modulating mechanisms were incorporated in the subsequentsimulations. The concentrations of both Ca2+ and Ca2+buffers were monitored in all eight compartments; simula-tions were compared via peak Ca2+ concentrations attainedin the distal tip of the spine head following the tetanus.

In the first set of simulations, changes in Ca2+ bufferconcentrations in the postsynaptic density were studiedsystematically while the other parameters were held con-stant, but in a range considered to be plausible physiologi-cally. The results (Fig. 2 B and C) demonstrate that varyingthe concentration offree buffer in the spine from 50 to 500 ,uMcan result in an 8-fold decrease in peak Ca2+ and an 11-folddecrease in the accumulated Ca2+ concentrations followingstimulation (three pulses at 100 Hz; Vm = -30 mV). Thesimulations also demonstrate that a steep rise in Ca2+ comesas a direct result of saturation of the Ca2+ buffers.

In a second set of simulations, the buffer concentration wasfixed (325 ,uM) and changes in spine neck diameter werestudied systematically (Fig. 2 D and E). From these exper-iments it is clear that rapid diffusion of Ca2+ ions out of thespine and into the dendrite is extremely sensitive to thediameter of the spine neck. In fact, a thin neck of 0.1 ,um,corresponding to 20% of the spine head diameter, allowedsignificant accumulation of Ca2+ in the head, to peak levelsof 29.8 ,uM, whereas a wide neck, equal in diameter to thespine head (0.55 ,um), allowed very rapid diffusion so thatpeak levels reached only 1.9,uM.

Next, the effects of varying the decay time of the NMDA-mediated current were simulated (Fig. 3). The decay of thecurrent was modeled as a double-exponential function with afast and slow component. When the slow component domi-nated (90% of the total current), peak Ca2+ concentrationrose to 29.7,M. When the fast component made the largercontribution (80% of the total current), peak Ca2+ reachedonly 3.1 ,uM. As has been described in detail (24), therelationship between stimulus frequency and the time courseof the NMDA receptor-mediated EPSCs following individualstimuli determines the amount of waveform summation-andtherefore the amount of Ca2+ influx-arising from tetanicstimulation.

Fig. 4 shows the results of simulations in which both bufferand spine neck were varied. These simulations clearly showthat Ca2+ transients in dendritic spines are very sensitive tosmall changes in these parameters.

AX _a

C) C

C) _

)o -

)o:

100 200 30

B60-

2 . 40-~

0*0

C, 60-0

420-

20-C 0m0.

D- 50-

401' 30-

A 20-010101

E0t),c!5C

0.

50-- 40-

_=30-C 20-U 10-

Time (msec)

100 200Time (msec)

0T

300

100 200 300 400 500PSD [buffer] (AM)

100 200Time (msec)

0 0.2 0.4Spine neck width (gm)

300

0.6

FIG. 2. Effects of varying the concentration of free Ca2+ bufferand spine neck diameter on Ca2+ concentration following tetanicstimulation. (A) Ca2+ flux resulting from NMDA receptor activationthat was simulated as a tetanic burst of activity with three pulses at100 Hz, with the cell depolarized to -30 mV. NMDA conductanceswere modeled with relatively long decay times (from Eq. 1, 72accounted for 80o of the total decay current). (B) Instantaneousconcentration of Ca2+ in compartment 1 of the model changingthrough time, shown for a variety of buffer concentrations (from 501M for the top trace to 500 1LM for the bottom, varied in 50-pMincrements). The spine neck width was fixed at 0.1 g&m, and pumpswere included. (C) Peak instantaneous level of Ca2+ plotted as afunction of initial free buffer concentrations. (D) Instantaneousconcentration of Ca2+ in compartment 1 of the model changingthrough time, shown for a variety of neck widths (from 0.05 pm forthe top trace to 0.6 pm for the bottom, varied in 0.05-pum increments).Initial buffer concentration was set to 325 p.M in the postsynapticdensity, and pumps were included. (E) Peak instantaneous level ofCa2+ plotted as a function of spine neck width.

To ensure that these conclusions were robust and notsensitive to slight variations in the assumptions made aboutother features of the model, we ran additional simulations asa number of key parameters were adjusted. In particular, weperformed simulations while varying the number of compart-ments (range, 3-8), spine head width (0.2-1.0pum), intracel-

i4l. l-

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3944 Neurobiology: Gold and Bear

A300

1=~2007E Ec100

0

0 100 200Time (msec)

B_ 40-

E 30-1-

i+ 20-

d d 10-

0-

C

rAn0

4),

c

._

I ,

100 200Time (msec)

300

0.25 0.5 0.75Fraction ofEPSC decay described by

fast component (ar)

FIG. 3. Effects of varying the decay time ofthe NMDA receptor-mediated EPSC on Ca2+ concentration following tetanic stimulation;these simulations include pumps and initial buffer concentration asin Fig. 2D and a spine neck width of 0.1 pm. (A) Ca2+ influx intocompartment 1 viaNMDA receptor channels for slow and fast decaytimes (a = 1 and a = 0, respectively, from Eq. 1). (B) Instantaneousconcentration of Ca2+ in compartment 1 of the model changingthrough time, shown for a variety of EPSC decay times (the fractionof total decay described by the slow time constant from Eq. 1 wasvaried from 1.0 in the top trace down to 0.0 in the bottom trace inincrements of 0.1). (C) Peak instantaneous level of Ca2+ plotted asa function of the EPSC decay time.

lular Ca2+ at rest (0-0.5 juM), Ca2+ diffusion constant (0.1-1.0 mm2/msec), pump rate constant (1.4-1.8 psec'1), bufferforward binding constant (0.1-1.0 msec-1), and buffer back-ward binding constant (0.1-1.0 msec'). Fortunately, thesensitivity of the model to most of these constants was verylow. On the other hand, spine head width (and, correspond-ingly, volume) did significantly affect Ca2+ levels; however,this effect can be considered as similar to changing theamount of free buffer, which was studied systematically.

DISCUSSIONIn contrast to the earliest models of spine function, whichexplored their possible role in regulating synaptic currentflow into the dendrite (e.g., refs. 45-47), the latest generationofmodels has been focused on the role of spines in regulatingcalcium dynamics (e.g., refs. 7, 32, and 48). This work hassuggested three specific functions for spines in this regard: (i)to amplify synaptic Ca2+ signals, due to the small volume ofthe spine and restricted diffusion out the spine neck; (ii) tocompartmentalize Ca2+ signals, restricting steep Ca2+ in-creases to the spine; and (iii) to isolate changes in spine Ca2+concentration from Ca2+ entering elsewhere, such as throughvoltage-gated channels on the dendritic shaft. Given theassumption that a steep rise in Ca2+ concentration in thespine head is the trigger for LTP, recent models (24, 25) havedemonstrated that these spine functions could provide therequired nonlinearity of LTP induction.

tic'

FIG. 4. Intracellular Ca2+ concentration vs. time for all compart-ments for two different values ofbuffer concentration and spine neckdiameter. (A) Buffer at 150 pM and neck width of0.2 pm. (B) Bufferat 350 ,IM and neck width of 0.4 .m.

Not explicitly considered in previous modeling efforts wasthe extent to which physiological regulation ofspine propertiesmight alter the ability to induce Ca2+-dependent synapticplasticity. Our aim here was to address this issue by investi-gating quantitatively the sensitivity of spine Ca2+ transients tovariations in buffer concentration, spine geometry, and thetime course of NMDA receptor-mediated EPSCs over alimited range of values. In limiting the scope of our investi-gation to only these mechanisms, we made a number ofsimplifying assumptions to avoid issues not directly related toour goals. For example, coactivation of a-amino-3-hydroxy-5-methyl4isoxazolepropionate receptors, which plays an im-portant role in the generation of NMDA-mediated currentsphysiologically (by providing the depolarization needed torelieve the Mg2+ block), is ignored in our model. However,this issue has been examined in considerable detail elsewhere(24, 25) and would complicate our model without significantlyaffecting the phenomena with which we were directly con-cerned-in particular, the peak Ca2+ concentration arisingfrom tetanic stimulation (25). Similarly, the relatively minoreffects ofCa2+ accumulation in the dendrites are ignored hereby treating compartment 8 of the model, representing thedendritic shaft, as a Ca2+ sink of infinite capacity; again, thisissue is treated in more detail in other models (24, 25).The results of our model suggest that relatively small

changes in buffer concentration, spine geometry, and thetime course ofNMDA receptor-mediated EPSCs can have asignificant effect on the peak Ca2+ concentration in a den-dritic spine after synaptic activation of NMDA receptors.There is ample experimental evidence that these parametersare variable within the central nervous system: a wide varietyofspine geometries (40), Ca2+ buffer concentrations (44), andNMDA-mediated EPSC decay times (39, 49) are found. Ourmodel supports the hypothesis that these variations are offunctional significance.Recent evidence suggests that the variations in these

parameters may reflect their ongoing activity-dependent reg-ulation. For example, high-frequency synaptic stimulationthat is sufficient to activateNMDA receptors and induce LTPin CA1 neurons also results in both transient and lastingchanges in spine morphology (50, 51). Of particular interestis the reported increase in "sessile" spines, interpreted to

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reflect a shortening and widening of the spine neck followingstimulation. A consequence of this change in spine shapewould be a marked increase in the diffusion of Ca2+ from thesubsynaptic region that, according to our model, could resultin up to a 15-fold decrease in the peak instantaneous Ca2+concentration following stimulation. Such a modification ofspine shape, if it were to occur within seconds of stimulation,could explain why LTP is often more difficult to induce aftertransient activation of NMDA receptors (52) and possiblycould account for why LTP appears to saturate. According tothis view, saturation does not reflect a physical limit in theparameter that is modified to increase synaptic effectiveness;rather, saturation reflects a change in spine geometry thatdoes not permit the sharp rise in Ca2+ that is required totrigger further LTP.Ca2+ buffers such as the Ca2+-binding protein calbindin

D28k also appear to be regulated by activity. For example,immunoreactive calbindin decreases markedly in visual cor-tical neurons that are deprived ofpatterned visual stimulation(53). Conversely, strong electrical stimulation of the per-forant path has been shown to cause an almost 2-fold increasein the calbindin mRNA in dentate gyrus granule cells (54).Our model suggests that activity-dependent regulation ofCa2+ buffer within this range can have a substantial impact onthe peak instantaneous Ca2+ concentration that follows te-tanic stimulation. This may explain why the type of stimu-lation used to raise calbindin mRNA also can inhibit subse-quent induction of LTP (55).NMDA-receptor mediated EPSCs are regulated by activity

during early postnatal development (39, 49). In visual corticalneurons these EPSCs last significantly longer in neonatesthan in adults. In newborns, the slower component (x 250msec) contributes >90% of the total current, whereas inadults the contribution of the slow component is <20%o of thetotal, and the current is dominated by the fast component (r

50 msec). This developmental decline in EPSC durationevidently depends on activity: no change was observed whenthe animals were reared in the dark or when the cortex wastreated with tetrodotoxin during early postnatal life. Ourresults suggest that this change in the time course of synapticcurrents during normal postnatal development will have asubstantial impact on the change in postsynaptic Ca2+ con-centration that follows tetanic stimulation. This may providea partial explanation for the reported developmental declinein LTP probability and magnitude in visual cortex (56).The preceding discussion suggests that increased postsyn-

aptic activity affects spine shape, buffer concentration, andthe time course ofNMDA receptor-mediated EPSCs, and themodel indicates that the resulting changes in these parame-ters have a similar consequence: to markedly decrease thetransient increases in Ca2+ in the subsynaptic region thatfollow synaptic activation. Assuming that LTP is triggered bya threshold rise in Ca2+, this regulation could provide a typeof homeostasis of synaptic strength such that increasedpostsynaptic activity, which naturally follows induction ofLTP, also makes the induction of further LTP more difficult.Our work therefore supports the hypothesis that regulation ofspine shape, Ca2+ buffer concentration, and time course ofNMDA receptor-mediated EPSCs by the history of postsyn-aptic activity are plausible neurobiological bases for the"sliding modification threshold" of Bienenstock et al. (27).The challenge now is to show experimentally that activity-dependent changes in one or more of these parametersactually regulates synaptic plasticity in a manner consistentwith the properties of the sliding threshold.We wish to thank L. N. Cooper and N. Intrator for their support

and encouragement of this project. This work was supported by theU.S. Office of Naval Research.

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