Geometry of Dendritic Spines Affects Calcium Dynamics in Hippocampal Neurons: Theory and Experiments

N. Volfovsky,1 H. Parnas,2 M. Segal,3 and E. Korkotian3

 1Department of Computer Science and  2Department of Neurobiology, The Hebrew University, Jerusalem; and  3Department of Neurobiology, The Weizmann Institute, Rehovot, Israel


    ABSTRACT
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

Volfovsky, N., H. Parnas, M. Segal, and E. Korkotian. Geometry of Dendritic Spines Affects Calcium Dynamics in Hippocampal Neurons: Theory and Experiments. J. Neurophysiol. 82: 450-462, 1999. The role of dendritic spine morphology in the regulation of the spatiotemporal distribution of free intracellular calcium concentration ([Ca2+]i) was examined in a unique axial-symmetrical model that focuses on spine-dendrite interactions, and the simulations of the model were compared with the behavior of real dendritic spines in cultured hippocampal neurons. A set of nonlinear differential equations describes the behavior of a spherical dendritic spine head, linked to a dendrite via a cylindrical spine neck. Mechanisms for handling of calcium (including internal stores, buffers, and efflux pathways) are placed in both the dendrites and spines. In response to a calcium surge, the magnitude and time course of the response in both the spine and the parent dendrite vary as a function of the length of the spine neck such that a short neck increases the magnitude of the response in the dendrite and speeds up the recovery in the spine head. The generality of the model, originally constructed for a case of release of calcium from stores, was tested in simulations of fast calcium influx through membrane channels and verified the impact of spine neck on calcium dynamics. Spatiotemporal distributions of [Ca2+]i, measured in individual dendritic spines of cultured hippocampal neurons injected with Calcium Green-1, were monitored with a confocal laser scanning microscope. Line scans of spines and dendrites at a <1-ms time resolution reveal simultaneous transient rises in [Ca2+]i in spines and their parent dendrites after application of caffeine or during spontaneous calcium transients associated with synaptic or action potential discharges. The magnitude of responses in the individual compartments, spine-dendrite disparity, and the temporal distribution of [Ca2+]i were different for spines with short and long necks, with the latter being more independent of the dendrite, in agreement with prediction of the model.


    INTRODUCTION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

The ability of neurons to maintain a low intracellular calcium concentration is a combined function of calcium binding proteins, pumps, exchangers, and intracellular calcium stores (Sharp et al. 1993; Simpson et al. 1995). Activation of calcium channels causes a large and transient rise in [Ca2+]i. Calcium influx has been shown to constitute a necessary step in long-term synaptic plasticity in central neurons (Bliss and Collingridge 1993; Lynch et al. 1983). Being the primary site of excitatory synaptic interactions, dendritic spines are likely to play a major role in neuronal plasticity, and their ability to regulate calcium is of prime interest for the understanding of mechanisms underlying plasticity.

The spine head is a closed cellular compartment separated from its parent dendrite by a thin neck (Harris and Kater 1994). The spine shape, length, and neck diameters were proposed to be critically important for processes of synaptic transduction (Segev and Rall 1988). Indeed some morphological changes were detected in size and density of dendritic spines in correlation with neuronal activity, learning, and memory (Lowndes and Stewart 1994; Moser et al. 1994; Segal 1995b). Recent experiments with high spatial and temporal resolution microscopy suggest that dendritic spines are unique calcium compartments, regulating local [Ca2 +]i changes independently of parent dendrites (Denk et al. 1995; Guthrie et al. 1991; Jaffe et al. 1994; Korkotian and Segal 1998; Muller and Connor 1991; Segal 1995a; Yuste and Denk 1995). Earlier models addressed the possibility that the spines are independent calcium compartments (Gamble and Koch 1987; Holmes and Levy 1990; Koch and Zador 1993; Schiegg et al. 1995; Wickens 1988). In fact, direct measurements of coupling between spines and parent dendrites were conducted with photobleaching of fluorescent compounds in the spine head (Svoboda et al. 1996). This coupling may, however, be different for calcium ions than for a fluorescent compound. Immunocytochemical studies revealed the existence of ryanodine receptors in dendrites and in spine heads (Sharp et al. 1993; Korkotian and Segal 1998). Release of calcium from ryanodine-sensitive stores can play an important role during synaptic activation accompanying calcium influx. We first investigated the role of calcium stores in the temporal distribution of Ca2+ in spine and its associated dendritic shaft. To discern the specific role of the stores we initially conducted experiments and computer simulations where the sole source of Ca2+ is that released from the stores. We further examined theoretically and experimentally the effect of spine neck length on the distribution of [Ca2+]i in the two compartments. Finally, we extended the model to present simulation and experimental results that demonstrate that the morphology of spine retains its decisive role in regulating Ca2+ dynamics even when influx of Ca2+ through membrane channels in the spine head is considered.


    METHODS
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

Three-dimensional spine shape

To describe biological spine shapes (Fig. 1, A1-A3) as faithfully as possible and yet to simplify the complex calculations involved in a three-dimensional (3D) model, we reduced it to an axiosymmetric model where the spine head was described as a sphere, the neck was described as a cylinder, and the dendrite was treated as a fragment of a disk set at 90° to the spine. This fragment of the dendrite bearing a single spine is represented in Fig. 2A and is denoted as "spine domain."



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Fig. 1. Dendritic spines in cultured hippocampal neurons respond to caffeine. A: 3 dendritic spines (1-3) taken from cells loaded with Calcium Green-1 (CG-1) and reconstructed from serial confocal sections of dendritic segments having long (1), middle (2), and short (3) spine necks. B: spine-dendrite segment seen in A1, line scanned at 0.8 ms/line, before and after puff application (bar) of 10 mM caffeine. The entire frame, top to bottom, comprises 512 lines. Segments a-c correspond to the region of interest for the analysis illustrated in C, with a being the dendrite, b the spine neck, and c the spine head. The regions of interest in B were quantified and plotted as a function of time since the application of caffeine (C). D and E: 2 spines with long (D) and short (E) necks, imaged at a fast frame rate (200 ms/frame), in control condition before application of caffeine (1), soon after (2), and 1 frame later (3). The frames 1-3 in both D and E are presented as DF/F relative to predrug fluorescence.



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Fig. 2. Three-dimensional (3D) dendritic spine model. A: 3D representation of a spine domain. B: schematic description of all processes that are considered in the model. Numerals correspond to the equation numbers in the text. C: model spine-dendrite collapsed into a 2D one. Axial symmetric presentation of the spine domain.

Mathematical model

The model that relates to the experiments described in BIOLOGICAL EXPERIMENTS includes the following processes: 1) processes associated with calcium stores, 2) diffusion, 3) Ca2+ buffering, and 4) Ca2+ extrusion.

Calcium stores were located in the spine head and in the adjacent dendrite (Fig. 2B). No stores were included in the neck because line scan recording did not show a significant indication for a short latency calcium responses in the neck (Fig. 1, B and C). Following Bezprozvanny et al. (1991) and DeYoung and Keizer (1992), we distinguish between two processes, uptake to the stores from the cytoplasm and efflux from the stores. The uptake is ATP-dependent and described by
<IT>J</IT><SUB>uptake</SUB><IT>‖</IT><SUB><IT>&Ggr;</IT><SUB>store</SUB></SUB> = − <FR><NU><IT>X</IT><SUB>store</SUB><IT>·C</IT><SUP><IT>2</IT></SUP></NU><DE><IT>C</IT><SUP>2</SUP>+<IT>K</IT><SUB>store</SUB></DE></FR> (1)
where Gamma store is the intracellular store boundaries, C (µM) is concentration of Ca2+. Xstore (µM/s) and Kstore (µM) are the maximal rate of uptake and the half-saturation constants, respectively. The second process is Ca2+-induced Ca2+ release (CICR) from the store to the cytoplasm
<IT>J</IT><SUB>store</SUB>‖<SUB>&Ggr;<SUB>store</SUB></SUB> = <IT>O</IT>(<IT>C</IT>) · &ngr;<SUB>max</SUB><IT>·</IT>(<IT>C</IT><SUB>store</SUB><IT>−</IT><IT>C</IT>)<IT>+</IT><IT>J</IT><SUB>leak</SUB> (2)
where nu max (s-1) is the maximal rate of Ca2+ release, Cstore is concentration of Ca2+ in the store, Jleak is a small constant current out of the stores, and O(C) describes the number of open channels as a function of C. Cstore is determined by
<FR><NU>d<IT>C</IT><SUB>store</SUB></NU><DE><IT>dt</IT></DE></FR><IT>=</IT>−(<IT>J</IT><SUB>store</SUB><IT>‖</IT><SUB><IT>&Ggr;</IT><SUB>store</SUB></SUB><IT>+</IT><IT>J</IT><SUB>uptake</SUB><IT>‖</IT><SUB><IT>&Ggr;</IT><SUB>store</SUB></SUB>)<IT>·&egr;</IT><SUB>store</SUB> (3)
where epsilon store is the ratio of the area of the store boundary Gamma store to the volume of store (the volume of the store is equal in the spine and the dendrite and assumed to be 10% of the total volume of spine head, but variations of this condition are described subsequently).

The diffusion process in the spine domain (Fig. 2A) is described by the classical diffusion equation for cylindrical coordinates
<FENCE><FR><NU>d<IT>C</IT></NU><DE>d<IT>t</IT></DE></FR></FENCE><SUB><IT>d</IT></SUB><IT>=</IT><FR><NU><IT>1</IT></NU><DE><IT>r</IT></DE></FR> <FENCE><FR><NU><IT>d</IT></NU><DE><IT>d</IT><IT>r</IT></DE></FR> <FENCE><IT>rD</IT> <FR><NU><IT>d</IT><IT>C</IT></NU><DE>d<IT>t</IT></DE></FR></FENCE> + <FR><NU>d</NU><DE>d&thgr;</DE></FR> <FENCE><FR><NU><IT>D</IT></NU><DE><IT>r</IT></DE></FR> <FENCE><FR><NU><IT>d</IT><IT>C</IT></NU><DE><IT>d&thgr;</IT></DE></FR></FENCE></FENCE><IT>+</IT><FR><NU><IT>d</IT></NU><DE><IT>d</IT><IT>z</IT></DE></FR> <FENCE><IT>rD</IT> <FR><NU><IT>d</IT><IT>C</IT></NU><DE><IT>d</IT><IT>z</IT></DE></FR></FENCE></FENCE> (4)
where D (µm2/s) is the diffusion coefficient of Ca2+. The index d denotes changes in Ca2+ concentration because of diffusion.

The mechanism of Ca2+ buffering is given by
<IT>nC</IT><IT>+</IT><IT>B </IT><LIM><OP><IT>⇔</IT></OP><LL><IT>k</IT><SUB>−</SUB></LL><UL><IT>k</IT><SUB>+</SUB></UL></LIM><IT>BC</IT><SUB>n</SUB> (5)
where B is the free buffer, BCn is the Ca2+-bound buffer, and n is the number of calcium binding sites. We use n = 1 for calcineurine and n = 4 for calmodulin. k+ µM-1·s-1 and k- (s-1) denote the binding and dissociation rate constants, respectively. For any buffer B (µM), the differential equations of the kinetic scheme (Eq. 5) are
<FENCE><FR><NU>d<IT>C</IT></NU><DE><IT>d</IT><IT>t</IT></DE></FR></FENCE><SUB><IT>r</IT></SUB><IT>=</IT><FR><NU><IT>1</IT></NU><DE><IT>n</IT></DE></FR> (<IT>−</IT><IT>k</IT><SUB>+</SUB><IT>·</IT><IT>C</IT><SUP><IT>n</IT></SUP><IT>·</IT><IT>B</IT><IT>+</IT><IT>k</IT><SUB><IT>−</IT></SUB><IT>·</IT><IT>BC</IT><SUB><IT>n</IT></SUB>) (6)

<FR><NU>d<IT>B</IT></NU><DE><IT>d</IT><IT>t</IT></DE></FR><IT> = </IT>−<IT>k</IT><SUB>+</SUB><IT>·</IT><IT>C</IT><SUP><IT>n</IT></SUP><IT>·</IT><IT>B + k</IT><SUB>−</SUB><IT>·</IT><IT>BC</IT><SUB><IT>n</IT></SUB> (7)

<FR><NU>d<IT>BC</IT><SUB><IT>n</IT></SUB></NU><DE><IT>d</IT><IT>t</IT></DE></FR><IT>=</IT><IT>k</IT><SUB>+</SUB><IT>·</IT><IT>C</IT><SUP><IT>n</IT></SUP><IT>·</IT><IT>B − k</IT><SUB>−</SUB><IT>·</IT><IT>BC</IT><SUB><IT>n</IT></SUB> (8)
The index r denotes the change in [Ca2+]i because of buffering by B. The diffusion of the free or Ca2+-bound diffusible buffer is described by an equation such as Eq. 4.

Ca2+ extrusion is conducted by pumps located throughout the entire spine and dendrite membrane and given by
<IT>J</IT><SUB>pump</SUB><IT>‖</IT><SUB><IT>&Ggr;</IT><SUB>pump</SUB></SUB><IT>=</IT>− <FR><NU><IT>X</IT><SUB>pump</SUB><IT>·</IT><IT>C</IT></NU><DE><IT>C + K</IT><SUB>pump</SUB></DE></FR> (9)
where Gamma pump is the membrane surface and Xpump (µM/s) and Kpump (µM) are the maximal rate and the half-saturation constants of the pump, respectively.

Ca2+ dynamics is given by the sum of Eqs. 4 and 6
<FR><NU>d<IT>C</IT></NU><DE><IT>d</IT><IT>t</IT></DE></FR><IT>=</IT><FENCE><FR><NU><IT>d</IT><IT>C</IT></NU><DE><IT>d</IT><IT>t</IT></DE></FR></FENCE><SUB><IT>d</IT></SUB><IT>+</IT><FENCE><FR><NU><IT>d</IT><IT>C</IT></NU><DE><IT>d</IT><IT>t</IT></DE></FR></FENCE><SUB><IT>r</IT></SUB> (10)
with Eq. 9 describing processes taking place only in the spine domain boundary (see Fig. 2B).

To account for the results described in BIOLOGICAL EXPERIMENTS, influx of Ca2+ through spine head was added to the previous model. On the basis of previous studies by Miyakawa et al. (1992) and Jaffe and Brown (1997) we assumed tentatively that the main Ca2+ influx is through voltage-gated channels.

Ca2+ flux through voltage-dependent channels situated in the membrane of the spine head and dendrite is described by
<IT>J</IT><SUB>in</SUB><IT>‖</IT><SUB><IT>&Ggr;</IT><SUB>in</SUB></SUB><IT>=</IT><IT>g</IT>(<IT>t</IT>) · <FENCE><IT>V − </IT><FR><NU><IT>RT</IT></NU><DE><IT>ZF</IT></DE></FR><IT> ln </IT><FENCE><FR><NU><IT>C</IT><SUB>out</SUB></NU><DE><IT>C</IT>(<IT>t</IT>)</DE></FR></FENCE></FENCE> (11)
where Gamma in are the segments of membrane on the spine's head and on the dendrite, which include voltage-dependent channels, V is the membrane potential, Cout (µM) is the extracellular concentration of Ca2+, g(t) is the time-dependent conductivity of the voltage-dependent channels, R is the gas constant, T is the absolute temperature (in degrees Kelvin), Z is the valence of Ca2+, F is the Faraday constant, and C(t) is the concentration of intracellular Ca2+ beneath the head membrane at time t. Values of the parameters used in the simulation are presented in Table 1.


                              
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Table 1. Parameters of the model

Solution of the model equations

The specific geometry of a real spine (e.g., Figs. 1A and 2A) allows us to present the model as possessing axial symmetry around the z-axis. There is also no swirling motion in the considered problem. This permits the use of an axial-symmetrical approximation that reduces the 3D diffusion equation (Eq. 4) to a 2D axial-symmetrical one (Eq. 4a and Fig. 2C)
<FR><NU>d<IT>C</IT></NU><DE><IT>d</IT><IT>t</IT></DE></FR><IT>=</IT><IT>D</IT><FENCE><FR><NU><IT>d</IT></NU><DE><IT>d</IT><IT>r</IT></DE></FR> <FENCE><IT>r</IT> <FR><NU><IT>d</IT><IT>C</IT></NU><DE><IT>d</IT><IT>r</IT></DE></FR></FENCE><IT>+</IT><FR><NU><IT>d<SUP>2</SUP></IT><IT>C</IT></NU><DE><IT>d</IT><IT>z</IT><SUP><IT>2</IT></SUP></DE></FR></FENCE> (4a)
Although this is a good approximation to the real spine, it neglects possible effects of asymmetry and inhomogeneity in the spine head, but these should not cause significant disparities from the real spine.

To solve the problem numerically the spine domain was subdivided into a number of simply shaped, small regions, finite elements (Fig. 2C). Computations were carried out with FIDAP software (version 7.6) (Engelman 1996) modified in our laboratory (for more details see Aharon and Bercovier 1993; Aharon et al. 1996).

As seen in Fig. 2, stores in the dendrite are presented by a single-store compartment situated at the base of the spine and in the boundaries of the dendrite fragment. As in reality, the stores are continuously distributed along the entire dendrite.

As mentioned previously the "spine domain" consists of the spine itself and a fragment of the entire dendrite. The question is what length of dendrite fragment to consider in the simulations such as the solution in the spine domain be minimally dependent on the diameter of the fragment. Toward this end, Eqs. 1-10 were solved in spine domain with a dendrite fragment diameter of 2-µm length. Then the fragment diameter was increased to 4 µm, and the equations were solved again. We found that Ca2+ concentration at the peak was ~2% higher for the 2 µm. The rate of decay, however, was very similar. We therefore used, for the simulations, the 2-µm diameter.

Biological experiments

Three-week-old E19 hippocampal cultures grown on glass coverslip were prepared as described previously (Papa et al. 1995). A glass coverslip was transferred into the recording chamber, placed in a confocal laser scanning microscope (Leica; Heidelberg, Germany), and perfused continuously at room temperature with medium containing (in mM) 129 NaCl, 4 KCl, 1 MgCl 2, 2 CaCl, 4.2 glucose, 10 HEPES, and 0.5 µM tetrodotoxin (TTX). pH was adjusted to 7.4 with NaOH, and osmolarity was adjusted to 330 mosmol with sucrose. Caffeine (Sigma, 5-10 mM) was prepared in the recording medium. It was loaded in a pressure pipette with a tip diameter of 1-2 µm, which was placed ~20 µm from the spine-dendrite segment studied, at right angle to the spine dendrite axis, equidistant from both compartments. A micropipette containing 10 mM Calcium-Green-1 (CG-1) impaled somata of individual cells and the dye was iontophoresed for 1-2 min with negative currents of 1-2 nA. The fluorescence of the cell was continuously monitored during dye filling. Images of 256 × 256 pixels were taken with a ×63 water immersion objective. The intensity of the argon-ion laser was reduced to 2-3% of nominal value. Experiments were conducted >= 1 h after loading of a cell with the dye and withdrawal of the micropipette to allow its equilibration into thin secondary branches of the dendritic tree and recovery from the initial penetration injury. Dendrites were 3D reconstructed at the beginning and end of the experiment with 5-10 successive optical sections with 0.1-µm steps. To reveal fast changes in fluorescence after caffeine application, single lines were scanned between the head of the spine and its parent dendrite at a rate of ~0.9 ms/line. Given the low-intensity laser light, lines of spine-dendrite segments could be scanned repeatedly without significant photodynamic damage or bleaching of the dye (Korkotian and Segal 1998). Fluorescence intensity was quantified with Leica analysis software. Changes in Calcium Green-1 (CG-1) fluorescence were standardized by dividing the net fluorescence by the pretreatment fluorescence with background subtracted, separately for the spine and its parent dendrite. Autofluorescence, measured in bis-(o-aminophenoxy)-N,N,N',N'-tetraacetic acid-loaded cells under identical conditions, was insignificant. In some experiments, TTX was omitted from the recording medium, and spontaneous changes in fluorescence resulting from back-propagating action potentials and excitatory postsynaptic potentials (EPSPs) were recorded. Further details of the methodology are presented elsewhere (Korkotian and Segal 1998).


    RESULTS
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

Model simulations: increase in [Ca2+]i is obtained via Ca2+ release from internal stores

To simulate the spatiotemporal distribution of Ca2+ under the experimental conditions where no influx of Ca2+ occurs but Ca2 + is released from the internal stores, model equations (Eqs. 1-10 and Table 1) were solved. As in the experiments to be described, Ca2+ stores were open in response to caffeine. Accordingly, Eq. 2 was modified such that O(C) is constant and independent of C.

Thus
<IT>J</IT><SUB>store</SUB><IT>‖</IT><SUB><IT>&Ggr;</IT><SUB>store</SUB></SUB><IT>=</IT><IT>O</IT><IT>·&ngr;</IT><SUB>max</SUB><IT>·</IT>(<IT>C</IT><SUB>store</SUB><IT>−</IT><IT>C</IT>)<IT>+</IT><IT>J</IT><SUB>leak</SUB> (2a)
Another experimental aspect that needs attention is the inclusion of a diffusible buffer, CG-1, in these experiments. We include this buffer in the model employing naturally the concentration used experimentally together with stationary buffers as described previously.

Three cases were examined: a long spine (1.5-µm length of neck, Fig. 3A), a medium spine (1 µm, Fig. 3B), and a short spine (0.1 µm, Fig. 3C). Spine head diameter was in all cases 0.6 µm, and spine neck diameter was 0.1 µm. In the spine head the average [Ca2+]i is presented (Fig. 3, line 1). To best represent the relevant [Ca2+]i in the dendrite we selected a volume, at the spine base, equal to the volume of the spine head. Line 2 depicts average [Ca2+]i in this volume.



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Fig. 3. Spatiotemporal distribution of [Ca2+]i calculated by FIDAP. Stores are activated by caffeine. A-C, solid line: with 10 µM diffusible buffer CG-1; dotted lines: stationary buffer. D-F: without CG-1. A and D: spines with long neck (1.5 µm); B and E: spines with medium neck (1 µm); C and F: spines with short neck (0.15 µm). 1: averaged [Ca2+]i in the spine; 2: average [Ca2+]i in the dendrite in the same volume as in the spine head. The following spine parameters (from Gamble and Koch 1987; Zador et al. 1990) were used: head diameter, 0.6 µm; neck diameter, 0.1 µm; dendrite diameter, 0.6 µm. Initial intracellular calcium concentration in the spine and in the dendrite is 0.06 µM. At the beginning of simulations stores are full (initial concentration 500 µM). Diffusion coefficient, D = 400 µm2/s (Eqs. 4 and 10). Two stationary buffers, calcineurin and calmodulin, were considered. Concentration of calcineurin, B1 =10 µM; concentration of calmodulin, B2 = 25 µM. The forward binding rate constants and the dissociation rate constants for calcineurin, k1+ = 50 µM-1·s-1 and k1- = 25 s-1, respectively. For calmodulin, forward binding and dissociation rate constants are k2+ = 50 µM-1·s-1 and k1- = 500 s-1 (Eqs. 5-8). The following parameters for extrusion (Eq. 9) were applied: maximal rate of Ca2+ extrusion, Xpump =15 µM/s; half-saturation constant, Kpump = 0.9 µM. Uptake to the stores (Eq. 9), the maximal uptake Xstore = 0.9 µM/s, and half-saturation constant is Kstore =0.3 µM. Maximum rate of Ca2+ release, vmax = 6 s-1 (Eq. 2) (from DeYoung and Keizer 1992).

Figure 3, A-C, demonstrates that the length of the spine affects the interaction between [Ca2+]i in the spine and the dendrite; the difference in peak [Ca2+]i between spine and dendrite is maximal in the longer spines (Fig. 3, A and B). Most importantly, the peak [Ca2+]i in the dendrite is affected by the spine length in that it is highest in the dendrite with the shortest spine. This is interpreted to indicate that the short spine contributes to the calcium surge in the parent dendrite more than the long spine. Not surprisingly, the peak of [Ca2+]i in the spine is not much affected by the length of the neck. The decay of [Ca2+]i in both spines and dendrite is faster in the short spine in comparison with either medium or long spines. Finally, the decay of [Ca2+]i in the spines is faster than in the dendrites in all spine-dendrite configurations.

As the diffusible buffer CG-1 is not present under normal physiological conditions, we repeated the simulations of Fig. 3, A-C, but without inclusion of CG-1. The results of the simulations are seen in Fig. 3, D-F.

Basically, the results with and without a diffusible buffer are rather similar. In particular, the same difference among spines of different length is preserved in both cases. Also, the time to peak of Ca2+ concentration and the ratio between the level of Ca2+ in the spine head and in the dendritic shaft are very similar for various spines. Also, the time constant of Ca2+ decay is ~15% slower in the presence of 10 µM of CG-1. Naturally, for the same spine length the overall concentrations of Ca2+ both in the spine and in the dendrite are lower when the diffusible buffer is added (Fig. 3, A-C) than in its absence (Fig. 3, D-F). The lower Ca2+ concentration should, however, be attributed mainly to the overall higher concentration of buffer in the simulations of Fig. 3, A-C, and less to the fact that one of the buffers is diffusible. (Compare dotted to solid lines in Fig. 3, A-C). The dotted lines correspond to the higher (10 µM more) concentration of buffer but all of it stationary. The solid lines corresponded to the same higher concentration of the buffer, but the 10 µM additional buffer is diffusible. It can be seen that the main reduction in [Ca2+]i was caused by the addition of 10 µM buffer (compare Fig. 3, A-C, with D-F).

We continue our study with the more physiological case where no diffusible buffer is added to the system. To evaluate quantitatively the effects of neck length on each of the compartments we present in Fig. 4 the results of Fig. 3 but where each compartment is shown alone (Fig. 4, spine head, A, C, and E; dendrite, B, D, and F). To demonstrate the similarity in behavior when CG-1 is included or being absent we show in Fig. 4, A and B, simulations with CG-1. The quantitative analysis is carried on, however, only for the case when CG-1 is not included.



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Fig. 4. Calcium kinetics in the different compartments. Spatiotemporal distribution of [Ca2+]i in spine head (A and C) and in dendrite (B and D). A and B with 10 µM of CG-1 and C and D without; a-c correspond to the spines in the inserts, and d corresponds to the dendrite alone. E: semilog plots of the decay in spines taken from C, normalized to the starting point of 480 ms. F: semilog plots of the decay in dendrites taken from D. The model predicts different behaviors of spines and dendrites in response to a calcium load, depending on the length of their dendritic spines.

To evaluate the time constant of Ca2+ decay, the semilog plots of lines a-c from Fig. 4, C and D, were drawn from 480 ms to the end of the curve. The initial time point of 480 ms was selected because it occurs after the [Ca2+]i peak and therefore reflects the decay phase better than the peak itself does. Also, calcium in the stores is at steady-state at this time (not shown). The use of semilog plots to evaluate the time constant of decay provides an approximation. It assumes that only one process governs the decay, and this process is of an exponential nature. In fact, several processes govern the decay (see MATHEMATICAL MODEL), and not all of them are of exponential nature. Indeed, the semilog plots in Fig. 4, E and F, do not exhibit an absolute straight line. Nevertheless, because we are not interested in discerning the contribution of each of the processes alone to the net decline in Ca2+ concentration, we use the "lumped time constant" extracted from Fig. 4, E and F, as an indicator to the differences in the rate of decline of [Ca2+]i in the various compartments.

Concerning the spine head, Fig. 4E shows that the shortest time constant of decay, 900 ms, is for the short spine, and it increases as the neck elongates (1,050 ms for medium spine and 2,010 ms for long spine).

The effect of neck length on the dendritic calcium is basically similar to that of the spine head (Fig. 4D). Specifically, shortest time constant of decay is for the dendrite with a short spine (970 ms) and obviously for a dendrite alone (960 ms). For the dendrite with longer spines, the time constants for decay were 1,330 and 2,100 ms for the medium and long spine, respectively (Fig. 4F). Thus the long and medium spines have greater influence on duration of the response of the parent dendrite compared with the short spine whose calcium decay at the later stage is of approximately the same speed as the dendrite alone.

Biological experiments

We now examine whether the model conclusions, that the length of the spine neck affects Ca2+ dynamics in both spine and dendrite, can be validated experimentally.

As seen from Delta F/F processed (Fig. 1, D1 and E1) and 3D reconstructed images (Fig. 1, A1-A3) the dye distributed evenly along the dendrites and spines. Sizes and diameters of spine heads and necks varied significantly in the same optical fields. Spines and their parent dendrites responded transiently to brief (100-200 ms) application of caffeine (5 mM), demonstrating calcium release from ryanodine-sensitive stores (Korkotian and Segal 1998). Of 44 spine-dendrite pairs, 35 (80%) responded to caffeine by a transient increase in fluorescence. In the majority of cases the fluorescence response was initiated simultaneously in the two compartments (Fig. 1, B and C) with the same latency (~40 ms) from the onset of drug application, indicating that it is unlikely that a calcium wave is generated in one compartment and travels into the other one, as suggested previously (Jaffe and Brown 1994, 1997). The response peaked within ~300 ms and recovered to half-maximum within 760 ± 40 (SE) ms.

To test predictions of the model, we analyzed the data separately for short (0.1-0.4 µm) and long (>0.7 µm) spine necks. A subset of spines, where the line was scanned through the neck, was analyzed. Other cases, where necks were not scanned, were included in a larger sample. The rise time and decay of the grouped responses with long (Fig. 5A, 5 spines) and short (Fig. 5B, 8 spines) necks were similar to those predicted by the model (Figs. 3 and 4). Different dynamics of [Ca2+]i surge in individual spines with long (top line) and short (bottom line) length of spine necks are illustrated in Fig. 1, D and E. The entire population of spines was divided into long (>0.7 µm, n = 9, group 1), middle (0.3-0.7 µm, n = 11, group 2), and short (0.1-0.3 µm, n = 15, group 3) spine necks as in Fig. 1, A1-A3. The time between the peak and one-half recovery (Fig. 5C) in response to caffeine was measured for each spine-dendrite pair. In the spine heads, the duration of response, between the peak and one-half recovery, was longer in the longest neck group compared with the shortest spines (t = 3.5, P < 0.01). Likewise, in the dendrites, those with the longest spine necks were slower to recover compared with either the medium group (t = 2.65, P < 0.01) or the short group (t = 4.47, P < 0.01). These results confirm the predictions of the model.



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Fig. 5. Effect of neck length on a transient calcium surge in spiny neurons: A: spines with a long neck (>0.7 µm, n = 5) have a larger disparity from their parent dendrites, which raise [Ca2+]i to lower levels. B: spines with short necks (<0.4 µm, n = 8) respond to caffeine with a transient rise in [Ca2+]I in a similar time course in both spines and parent dendrites. C: summary of time between peak and one-half recovery in all the spine-dendrites sampled, for long (left, 1), middle (2), and short (3) spine necks. The first group is distinctly different from the other two groups in duration of response as well as a difference between the spines and the dendrites (hatched bars).

Robustness of the model

As shown previously, the model predictions agree with experimental results. The question that still arises is how robust is the model and how sensitive are its conclusions to changes in parameters values. Figure 3 already shows that adding a diffusible buffer affects the overall concentration of Ca2+ but does not alter the key conclusions. In Fig. 6 we examined the effect of additional parameters on Ca2+ dynamics. For clarity, Fig. 6A displays the "control" case. In the rest of Fig. 6 the concentration of the stationary buffer (Fig. 6B), the diffusion coefficient for Ca 2+ (Fig. 6C), the dissociation constant of the stationary buffer (Fig. 6D), and the concentration of the diffusible buffer (Fig. 6E) were modified. It can be seen that the overall concentration of Ca2 was naturally affected by changing the values of the various parameters. However, our main conclusion regarding the effect of spine length on Ca2+ dynamics was not altered. This holds also for the case where all the stationary buffer was replaced by a diffusible buffer (not shown). Changing the values of the parameters associated with extrusion has similarly no effect on the basic results.



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Fig. 6. Sensitivity of model behavior to changes in parameter values. A: "control," the same as Fig. 4, C and D. B: endogenous buffer concentration was increased and decreased by 50% over control. C: Ca2+ diffusion constant was increased and reduced by 50%. D: dissociation constants of endogenous buffers were changed by 50% (because of changes in forward buffering rate constants). Blue lines in B-D indicate decrease over control conditions, and red lines indicate increase for each of 3 different spine lengths. E: addition of 20 µM (red line) and 50 µM of CG-1 (blue line).

As Ca2+ stores play an important role in this study we tried to find how changes in size and differential location of the stores will affect Ca2+ dynamics. Morphological studies (Spacek and Harris 1997) indicate that spines may have a smaller volume of endoplasmic reticulum than the parent dendrites. Also, spines may vary in their size and relative density of their calcium stores. Experimentally, the averaged response to caffeine in spines and dendrites do not vary by much, indicating that our initial choice of equal stores in spines and dendrite is not unrealistic. Nonetheless, we tested the effect of a differential volume of calcium stores in spines and dendrites. We used two arbitrary conditions; in the first (Fig. 7), the volume of the calcium stores in the spines was reduced to one-half that of the parent dendrite (thus reducing the surface of the stores to 2/3 of the initial value). In this case, calcium changes in the spines were smaller than under "control" conditions, but the differences among them remained the same (compare Fig. 3, right panel, and Fig. 7). The effect of changes in the shorter spine on the parent dendrite was smaller, and the parent dendrites of the longer spines were not affected.



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Fig. 7. Reducing spine calcium store affects spine-dendrite interaction. The volume of the store in the spine was reduced by 50% compared with the dendritic store. A-C: spines with long, medium, and short necks are depicted along with their corresponding dendritic segments, respectively.

The other condition tested involved the lack of calcium stores either in the spines or in the dendrites. In the case of no stores in the spine (Fig. 8A), the calcium change seen in dendrite caused a significant change in the spine with short neck and a marginal effect in the spine with a long neck. The dendrites had the same responses irrespective of the spines. In the opposite case, where stores were not present in the dendrites, there was a sizable change in the dendrite adjacent to the short spine and none in the dendrite next to the long spine (Fig. 8B). It is interesting to note that the magnitude of the calcium response in the spine with the long neck is largest compared with the other two cases.



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Fig. 8. Differential elimination of stores in spines or dendrites affect spine-dendrite interaction. A-C: no stores in the spines. Top to bottom: long, medium, and short spine necks. D-F: no stores in the dendrites, same spine-dendrite configuration. Right panel is control plots as in Fig. 3, D-F. The short spine affects calcium changes in dendrites, whereas the long spine has the largest calcium surge. : presence of calcium stores in the particular compartments.

Our model demonstrates the significance of Ca2+ stores and spine morphology to Ca2+ dynamics under conditions where Ca2+ is released from internal stores. The question is whether Ca2+ stores and spine morphology play similarly important roles under physiological conditions where Ca2+ enters via voltage-dependent Ca2+ channels. These investigations are discussed in the following section.

Model simulations: Ca2+ influx and CICR release from stores

To simulate fast synaptic activation of voltage-gated calcium channels and study their interactions with calcium released from the stores, Eq. 11 was invoked, and the term O(C) in Eq. 2 was described as seen in Eq. 15. As in previous studies (Gamble and Koch 1987; Zador et al. 1990), g(t) in Eq. 11 is presented by an alpha -function. Thus
<IT>g</IT>(<IT>t</IT>)<IT>=</IT><IT>g</IT><SUB><IT>p</IT></SUB> <FR><NU><IT>e</IT></NU><DE><IT>t</IT><SUB><IT>p</IT></SUB></DE></FR> <IT>t </IT><IT>exp</IT><FENCE><FR><NU>−<IT>t</IT></NU><DE><IT>t</IT><SUB><IT>p</IT></SUB></DE></FR></FENCE> (12)
where gp (nS) is the peak conductivity, e is the base of natural logarithm, and tp (ms) is the channel mean open time. We assumed the voltage-dependent channels to be distributed uniformly on the spine head and on the dendritic shaft.

To calculate CICR (Eq. 2), description of O(C) is needed. We derived O(C) from the following kinetic scheme
<IT>R+C </IT><LIM><OP><IT>⇔</IT></OP><LL><IT>k</IT><SUB>−<IT>1</IT></SUB></LL><UL><IT>k</IT><SUB>+<IT>1</IT></SUB></UL></LIM> <IT>RC </IT><LIM><OP><IT>⇔</IT></OP><LL><IT>k</IT><SUB><IT>c</IT></SUB></LL><UL><IT>k</IT><SUB><IT>o</IT></SUB></UL></LIM><IT> O+C </IT><LIM><OP><IT>⇔</IT></OP><LL><IT>k</IT><SUB>−<IT>i</IT></SUB></LL><UL><IT>k</IT><SUB><IT>i</IT></SUB></UL></LIM><IT> I</IT> (13)
where R denotes free receptor; RC, O, and I represent bound receptor, open, and blocked (inactive) channels, respectively, and k are the corresponding rate constants. C, as before, stands for intracellular Ca2+ concentration. Here, as assumed previously by Bezprozvanny et al. (1991), Ca2+ binding to R causes opening of the channels, and at higher concentrations (above 100 µM) Ca2+ blocks the channels. The differential equations derived from the scheme are as follows
<FR><NU>d<IT>RC</IT></NU><DE><IT>d</IT><IT>t</IT></DE></FR><IT>=</IT><IT>k</IT><SUB> + <IT>1</IT></SUB><IT>·</IT><IT>R</IT><IT>·</IT><IT>C − k</IT><SUB>−<IT>1</IT></SUB><IT>·</IT><IT>RC − k</IT><SUB><IT>o</IT></SUB><IT>·</IT><IT>RC + k</IT><SUB><IT>c</IT></SUB><IT>·</IT><IT>O</IT><IT>·</IT><IT>C</IT>

<FR><NU>d<IT>O</IT></NU><DE><IT>d</IT><IT>t</IT></DE></FR><IT>=</IT><IT>k</IT><SUB><IT>o</IT></SUB><IT>·</IT><IT>RC + k</IT><SUB>−<IT>i</IT></SUB><IT>·</IT><IT>I − k</IT><SUB><IT>i</IT></SUB><IT>·</IT><IT>O</IT><IT>·</IT><IT>C − k</IT><SUB><IT>c</IT></SUB><IT>·</IT><IT>O</IT>

<FR><NU>d<IT>I</IT></NU><DE><IT>d</IT><IT>t</IT></DE></FR><IT>=</IT><IT>k</IT><SUB><IT>i</IT></SUB><IT>·</IT><IT>O</IT><IT>·</IT><IT>C − k</IT><SUB><IT>−i</IT></SUB><IT>·</IT><IT>I</IT> (14)
In addition, according to the law of conservation
<IT>RC + O + I + R = R</IT><SUB><IT>T</IT></SUB>
where RT denotes the total receptor concentration. From the steady-state solution of the previous system we could compute O, which is the number of open channels in steady state for any concentration of Ca2+, C. In doing so we assume quasi-steady state, that is, change in C is lower in comparison with the changes in O.

Thus,
<IT>O</IT>(<IT>C</IT>)<IT>=</IT><FR><NU><IT>K</IT><SUB><IT>2</IT></SUB><IT>·</IT><IT>K</IT><SUB><IT>3</IT></SUB><IT>·</IT><IT>R</IT><SUB><IT>T</IT></SUB><IT>·</IT><IT>C</IT></NU><DE><IT>C</IT><SUP><IT>2</IT></SUP><IT>+</IT>(<IT>K</IT><SUB><IT>2</IT></SUB><IT>·</IT><IT>K</IT><SUB><IT>3</IT></SUB><IT>+</IT><IT>K</IT><SUB><IT>3</IT></SUB><IT>−</IT><IT>K</IT><SUB><IT>2</IT></SUB>)<IT>·</IT><IT>C + </IT><FENCE><IT>K</IT><SUB><IT>3</IT></SUB> <FR><NU><IT>k</IT><SUB><IT>c</IT></SUB></NU><DE><IT>k</IT><SUB><IT>1</IT></SUB></DE></FR><IT>+</IT><IT>K</IT><SUB><IT>1</IT></SUB><IT>·</IT><IT>K</IT><SUB><IT>2</IT></SUB><IT>·</IT><IT>K</IT><SUB><IT>3</IT></SUB></FENCE></DE></FR>
where K1, K2, and K3 are correspondingly
<IT>K</IT><SUB><IT>1</IT></SUB><IT>=</IT><FR><NU><IT>k</IT><SUB><IT>1</IT></SUB></NU><DE><IT>k</IT><SUB><IT>−1</IT></SUB></DE></FR><IT>, </IT><IT>K</IT><SUB><IT>2</IT></SUB><IT>=</IT><FR><NU><IT>k</IT><SUB><IT>c</IT></SUB></NU><DE><IT>k</IT><SUB><IT>o</IT></SUB></DE></FR><IT>, </IT><IT>K</IT><SUB><IT>3</IT></SUB><IT>=</IT><FR><NU><IT>k</IT><SUB><IT>−i</IT></SUB></NU><DE><IT>k</IT><SUB><IT>i</IT></SUB></DE></FR>
Consequently we got the following equation for O(C)
<IT>O</IT>(<IT>C</IT>)<IT>=</IT><FR><NU><IT>A</IT><SUB><IT>1</IT></SUB><IT>·</IT><IT>C</IT></NU><DE><IT>C</IT><SUP><IT>2</IT></SUP><IT>+</IT><IT>A</IT><SUB><IT>2</IT></SUB><IT>·</IT><IT>C + A</IT><SUB><IT>3</IT></SUB></DE></FR> (15)
where Ai are the coefficients that are dependent on the rate constants of Eq. 14 and were evaluated from the experimental data (Bezprozvanny et al. 1991). It should be noted that Eq. 15, the outcome of the Eq. 14, exhibits a bell-shaped behavior as noticed experimentally by Bezprozvanny et al. (1991).

The simulations of influx combined with release of calcium from stores are illustrated in Fig. 9. It can be seen that, even with the faster time course than in the previous simulations, the morphology of the spine still has a marked effect on the magnitude and decay kinetics of calcium in spines and in their parent dendrites. Furthermore, the presence of stores has a pronounced effect on calcium dynamics, although the initial calcium surge is produced by an influx of calcium rather than by release from stores; the long spine has a slower [Ca2+]i decay than the short one after calcium influx into the spine head and its release from stores, when the parent dendrite is not depolarized (Fig. 9A) or partly depolarized (Fig. 9B, only 1/3 of the channels open). Removal of calcium stores from both compartments results in a strong decrease in calcium responses in both compartments. However, an increase in calcium influx compensates partly for the lack of difference between short and long spines in the absence of stores (Fig. 9C). Re-introduction of stores to the case of enhanced influx causes an amplification of the difference in calcium response (Fig. 9E). For the latter case the decay time constants were 138 and 95 ms for the longer and shorter spines, respectively, and 342 and 145 ms for their parent dendrites.



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Fig. 9. Spine morphology affects a transient rise of [Ca2+]i caused by influx through synaptically activated, voltage-gated calcium channels. In all simulations a: spine with a long neck, b: spine with a short neck, 1: spine head, 2: dendrite, arrow: influx, with thickness relative to the amount of influx channels. Presence of stores is indicated by a  inside the spine-dendrite. A: influx of calcium only in the spines, with stores present in both the spine and dendrite. Calcium rises only in the dendrite of the short spine, which by itself decays faster than the spine of a long neck. B: more realistic case, with calcium influx into both spine and dendrite (e.g., through voltage-gated channels) and active calcium stores in both compartments. A larger rise in both dendrites is seen. C: influx is activated in both spine and dendrite in the absence of stores. D: influx of calcium is numerically enhanced to reach values seen in B in the absence of stores. E: stores added to the case seen in D to illustrate the differential role of stores in short and long spines. Note different [Ca2+]i scales in E and different time scale in this figure compared with previous ones. The depolarizing pulse is of 12 ms. Voltage-gated channels are assumed to open instantaneously.

Biological experiments

The validity of the predictions of the model for the synaptic influx case was tested in spine-dendrite segments during spontaneous calcium surges found in the cultured cells when synaptic and spike activity are not blocked by TTX. In such conditions, cells tend to fire action potentials spontaneously and evoke EPSPs in follower cells. Both small, synaptically originated events and large, spike-associated events were recorded in both short and long spines. There was a clear difference between the two spine types in latency, magnitude, and similarity to the response in the parent dendrite in the case of the small, probably EPSP-associated calcium change, but the spike-associated, large calcium change was similar in the two compartments (Fig. 10). In all 11 spine-dendrite pairs tested the long spines were more independent of the parent dendrites than the short ones, as predicted by the model.



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Fig. 10. Spontaneous individual calcium surges are different in short and long spines. A: 2 images of a dendritic segment illustrating a line scan taken for a long spine (A1, bottom left) and a short one (A2, top right, see arrows). B: line scans taken for the long (B1) and the short (B2) spines during a small calcium surge, which is probably associated with a synaptic response. Note the disparity between the spine and dendritic response in the long spine (B3) and the more similar and faster response in the 2 compartments in the short spine (B4). Note also that in both cases the calcium surge has a shorter latency and larger response in the spine than in the parent dendrite. The increase of [Ca2+]i in the dendrite of the longer spine is much slower than in that of the shorter one. C: larger calcium surge, probably associated with a back-propagating action potential, illustrating a larger and shorter-latency response in the short (C2 and C4) than in the long (C1 and C3) spines, whereas responses in both dendrites are about the same. In both cases, the latency of calcium surge is shorter in the parent dendrite than in its associated spine.


    DISCUSSION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

This study combines a computational model of a dendritic spine and experimental results obtained by a transient release of calcium from stores in dendritic spines of cultured hippocampal neurons. The model takes into account the presence of calcium stores in both spines and dendrites and simulates the spatiotemporal distribution of [Ca2+]i in the spine and the dendrite in response to a transient calcium rise. This rise was produced in the initial simulations by calcium released from stores and in advanced simulations by activation of voltage-gated calcium channels, which in turn activate the stores. The model predicts that the length of the spine neck affects both spine and dendritic responses to a calcium surge so that short spines are more similar to their parent dendrites than long ones and contribute to an increase in the response magnitude of the parent dendrites. On the other hand, long spines express a larger calcium response than their parent dendrites. Also, the time course of calcium recovery is slower in spines with long necks than those with short necks. In both short and long spines, calcium stores appear to strongly affect calcium distribution during synaptic and CICR activation. The model is supported by experimental results on calcium released from stores and on influx of calcium during synaptic and spike activity in spines and dendrites of cultured hippocampal neurons.

The effects of spine distance on the magnitude and duration of calcium surge in the spine and dendrite are intuitively simple. The close distance between the spine and its parent dendrite in the case of the short spine makes the calcium surge faster, larger, but shorter in duration. This is due to the fast diffusion of calcium from the spine head into and along the dendritic shaft. With the medium-long spines, the larger response in the spine head than the response in the dendrite is due to the fact that, unlike the shaft, the spine head is a closed spherical compartment. The longer distance causes the same amount of free calcium to move slower out of the spine compartment into the dendrite, making the increase higher and the recovery in both compartments slower than the case of the short spines. Clearly, the extrusion through a longer neck also causes less calcium to reach the dendrite. These mechanisms act also when calcium influxes transiently through membrane channels.

Several models of calcium homeostasis in dendritic spines were proposed recently (Gold and Bear 1994; Holmes and Levy 1990; Koch and Zador 1993; Wickens 1988; Zador et al. 1990). They were initiated by the realization that the spine neck may not constitute a significant modulatory barrier for the transfer of synaptic charge from the spine head into the dendrite (Barrionuevo et al. 1986; Brown et al. 1988) and by the experimental observations that suggested that the spine may constitute a unique cellular calcium compartment (Guthrie et al. 1991; Muller and Connor 1991). Our model is different from previous ones in that it contains calcium stores as an essential compartment in the spine (for the possibility of calcium stores in spines see Jaffe and Brown 1997; Jaffe et al. 1994) and that in fact it links influx of calcium through voltage-gated channels in the spine-dendrite to release of calcium from stores. In a previous model, which focuses on influx pathways as the main source of loading the spine with calcium (Schiegg et al. 1995), it is predicted that the stores cause prolongation of the calcium surge, which is important for the formation of long-term potentiation of reactivity to afferent stimulation in central neurons (Brown et al. 1988). Experimental data also suggest that calcium stores are important for the formation of long-term potentiation (Bliss and Collingridge 1993). With this exception, other models do not ascribe a significant function to calcium stores in dendritic spines.

A major difference between our model and previous ones concerns the dynamics of calcium changes in the dendrites. Although it was obvious that the dendritic calcium should be affected by spine calcium (e.g., Jaffe and Brown 1994), in previous models [Ca2+]i in the dendrite was not affected much by changes in morphology of the spine (Gold and Bear 1994; Holmes and Levy 1990), and therefore the dendrite was ignored altogether (Gamble and Koch 1987). Experimentally, both examples, where [Ca2+]i rise spreads from the spine into the dendrite and where the calcium rise is restricted to the spine, were reported (Guthrie et al. 1991; Yuste and Denk 1995). We found that the temporal distribution of [Ca2+]i in the dendrite in response to a calcium surge produced within the dendrite itself is strongly affected by the morphology of its attached spine. In reality, when the volume of the dendrite and its stores are several orders of magnitude larger than the spine, the impact of the latter compartment on dendritic [Ca2+]i might be negligible. Such can be the case with spines growing on large proximal dendrites. This however is the exception, as in most cases the spines are attached to dendrites of similar diameters, containing about the same concentrations of calcium stores (Spacek and Harris 1997) (see Fig. 1).

The disparity between the calcium response of the long spine and its parent dendrite is strikingly different from the similarity seen in the case of the short spine and its parent dendrite. This result is seen in both the model and the experimental data. Concerning the experiments, there could be several biological explanations for this disparity; it is possible, for example, that dendrites with long spines accumulate more dye than the other dendrites, and therefore calcium changes are buffered more in these dendrites, leading to smaller and slower responses. However, this is not likely because the protocol for the dye loading allows ample time for an even distribution of the dye. It is also possible that dendrites with long spines are different in concentrations of calcium stores than other dendrites. This is also less likely because most spines are found to contain calcium stores in about the same density (Spacek and Harris 1997). Finally, the CICR mechanism (Verhatsky and Shmigol 1996) may contribute to the short spines more than to the long ones. Clearly, some dendrites are endowed with short and some with long spines, such that a systematic biological difference between them is quite possible. However, the model indicates that no such added biological mechanism is needed to explain the observed results, and a mere morphological consideration is sufficient to explain the effect of spines on dendritic responses to a calcium surge.

The current results demonstrate the importance of the morphology of dendritic spines in regulating calcium dynamics in the parent dendrites in that small variations in spine length may have a significant effect on the way the dendrite handles calcium. Such small variations have been shown to take place in living dendritic spines (Segal 1995b) so that an ongoing control by spine of its parent dendrite is to be expected. Assuming that the duration of [Ca2+]i surge may be critically important for causing changes in calcium-dependent biochemical processes in the cell, in connection with long-term plasticity (Bliss and Collingridge 1993) it becomes extremely important to be able to control the amplitude and duration of this calcium surge, which is provided by the spines.

Although the cultured neuron is an ideal test system for examining the interaction between the single spine and its parent dendrite, the situation will be more complex when the real dendrite, endowed with higher density of spines, will be examined. One issue of importance for the interpretation of the current results is that dendritic spines are not present in isolation, and several spines are likely to be activated by the same application of caffeine. Therefore what if short spines reside near long ones? How effective will the individual spine be in its ability to regulate dendritic calcium under such conditions? Luckily, the cultured neuron has quite often very few spines, allowing the examination of the effect of individual spines on dendritic calcium. Also, in most cases, in vivo as well as in the cultured neuron, spines of similar lengths and properties are likely to cluster together on dendritic segments (Segal, unpublished observations). Nonetheless, the issue of interactions among spines with different physical properties remains an exciting subject for future modeling and experiments.


    ACKNOWLEDGMENTS

We thank V. Greenberger for preparation of the cultures.

This work was supported by grants from the Binational US-Israel Science Foundation and the Israel Academy of Science to M. Segal.


    FOOTNOTES

Address reprint requests to E. Korkotian.

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

Received 26 May 1998; accepted in final form 2 February 1999.


    REFERENCES
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

0022-3077/99 $5.00 Copyright © 1999 The American Physiological Society