338
Brain Research, 519 (1990) 338-342
Elsevier BRES 24120
Is the function of dendritic spines to concentrate calcium? William R. Holmes Mathematical Research Branch, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, MD 20892 (U.S.A.)
(Accepted 20 February 1990) Key words: Dendritic spine; Calcium; Plasticity; Long-term potentiation; Synapse; Computer simulation
Although dendritic spines are thought to play an important role in synaptic transmission and plasticity, their function remains unknown. Theoretical investigations of spine function have focused on the large electrical resistance provided by the narrow constriction of the spine neck. However, this narrow constriction is also thought to provide a large diffusional resistance. The importance of this diffusional resistance was investigated theoretically with models. When calcium currents were activated on dendritic spines, peak spine head Ca2+ concentration was an order of magnitude larger in 'long-thin' spines than in 'mushroom-shaped' or 'stubby' spines. The same currents activated on dendrites produced even smaller local Ca2÷ concentration changes. Although the diffusional resistance of the spine neck was important for producing these differences in [Ca2÷], the amplitude and duration of the Ca2÷ current relative to the number of Ca2÷ binding sites determined whether Ca2÷ would be concentrated near synapses. Given the importance of Ca2÷ for long-term potentiation, the ability of spines to concentrate Ca2+ may play a key role in processes leading to learning and memory storage.
The function of dendritic spines has been the subject of much speculation ever since spines were discovered a century ago. Beginning with the theoretical investigations of Rail and Rinzel ~4-t7, there have been a number of studies exploring the role of spines in synaptic transmission and plasticity (see refs. 1, 2 for reviews). Usually, these studies have focused on the electrical resistance of the thin spine neck. However, besides providing an electrical resistance to current flow, the thin spine neck provides a diffusional resistance to the flow of ions and molecules. Some have suggested that by restricting the flow of materials, Ca 2÷ in particular, into or out of the spine head, the thin spine neck might effectively isolate the spine head and thus provide a localized environment in which reactions specific to a particular synapse could Occur 1"2"5"21'26. The Gamble and Koch model 4 showed how large, transient increases in spine head [Ca 2÷] might be attained. However, the effects of different spine shapes on spine head [Ca 2÷] increases and the consequences of placing the synaptic Ca 2+ current on the dendrite rather than on the spine have not been explored. These issues are the focus of the present theoretical study. If the degree to which Ca 2÷ can become concentrated in the spine head were controlled by the diffusional resistance of the spine neck, then spine shape would determine the rates and types of Cae+-dependent reactions that are activated at a synapse on a particular spine.
Because Ca2+-dependent reactions are thought to lead to the induction of long-term potentiation (LTP) (i.e. ref. 8), regulation of [Ca 2÷] by the diffusional resistance of the spine neck could be exceedingly important. To explore the importance of the diffusional resistance of the spine neck, [Ca 2÷] changes were modeled in the spine heads of 3 differently shaped spines. The dimensions of the 3 spines were taken to be the average dimensions for long-thin, mushroom-shaped, and stubby spines as measured and categorized on hippocampal dentate granule cells 3. With a standard value for intracellular resistivity (70 t2cm), the spine neck resistance was less than 70 Mr2 for each of the 3 spine shapes. A compartmental model similar to the Gamble and Koch model 4 was used to predict [Ca 2÷] in a dendritic spine for a given Ca 2+ current. The model had 20 compartments, 8 for the dendritic spine and 12 for the dendrite. Calcium was assumed to enter the outermost spine head compartment, after which it could diffuse into a neighboring compartment, become bound to buffer, or be pumped out of the cell. Equations for free calcium concentration and free buffer concentration in each compartment were solved using a fourth-order Runga-Kutta method. A technical description of this model is given elsewhere 6. The synaptic Ca 2+ current used in the model is illustrated in Fig. 1. Values for the free parameters in the model are given in Table I; the basis for these choices is the same
Correspondence: W.R. Holmes, Mathematical Research Branch, Bldg. 31 Rm. 4B-54, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, MD 20892, U.S.A.
0006-8993/90/$03.50 (~) 1990 Elsevier Science Publishers B.V. (Biomedical Division)
339 A
0.20 ~" 0.16
30.
~= 20. :zL
e~
0.12 -O-10
o 0.08 0
0.04
0
40
80 120 T i m e (ms)
160
200
160
200
0.00 0
40
80 120 Time (ms)
160
200
Fig. 1. Hypothetical synaptic calcium current. Illustrated is the predicted Ca z÷ component of the multi-ion ionic current flow through NMDA receptor channels at a synapse on a long-thin spine obtained in a hippocampal dentate granule cell model6 when 96 synapses on the same dendrite were co-activated at 200 Hz. Calcium currents nearly identical to this current were used for the mushroom-shaped and stubby spines; the differences that occurred were slight because the spine neck resistance was less than 70 MI2 in all 3 cases and the conductance at an individual synapse (NMDA + non-NMDA) was never larger than 0.3 nS. This calcium current and those used to obtain the results illustrated in Fig. 3 are quite small, but they represent only a fraction of the total ionic current that one is likely to see at a single activated synapse.
as given in the previous description of the m o d e l 6. T h e results pictured in Fig. 2 show that spine shape plays an i m p o r t a n t role in controlling the magnitude of localized increases in [Ca 2÷] in the spine head. The synaptic Ca 2÷ current p r o d u c e d a large, steep increase in calcium concentration in the spine h e a d of the long-thin spine (Fig. 2 A ) . T h e r e were only small differences in [Ca 2÷] a m o n g the spine h e a d c o m p a r t m e n t s , but there was a steep calcium concentration gradient down the spine neck. T h e p e a k [Ca 2+] in the dendritic c o m p a r t m e n t s was less than 1% of the p e a k [Ca 2+] seen in the spine head. F o r a m u s h r o o m - s h a p e d spine, the Ca 2÷ current p r o d u c e d a p e a k [Ca z+] in the spine h e a d that was less than o n e - t e n t h that f o u n d with the long-thin spine (Fig. 2B). The calcium concentration gradient d o w n the spine neck was much smaller in this case, and slightly larger concentrations of Ca 2+ were found in the
TABLE I Values of free parameters used in the model Parameter
Value
Buffer concentration (compartments 1-2) Buffer concentration (compartments 3-20) Buffer forward rate constant, kbf Buffer backward rate constant, kb b Buffer affinity Ka (kbb/kbf) Pump rate constant Ca2+ diffusion coefficient
200/~M 100#M 0.5/tM-l.ms -1 0.5.ms -1 1.0/*M 1.4 x 10-4 cm/s 0.6pmZ/ms
1
0
7
40
80 120 T i m e (ms)
14
r
~-
0 0
40 ;
80 120 T i m e (ms)
160
200
Fig. 2. Free Ca2+ concentration in long-thin, mushroom-shaped,
and stubby spines for a given C a 2+ current. Note the different ordinate scales. The synaptic calcium current was that illustrated in Fig. 1. The numbers refer to compartments in the model with the locations as shown. The dimensions (taken from ref. 3) for the spine head and neck were 0.55 x 0.55 and 0.1 x 0.73 pm (long-thin), 0.77 x 0.77 and 0.2 x 0.43pm (mushroom), and 0.76 x 0.99pm (stubby, head and neck together) and are shown to scale. dendritic c o m p a r t m e n t s . T h e p e a k [Ca 2+] in the spine head of the stubby spine was n e a r l y the s a m e as that for the m u s h r o o m - s h a p e d spine. H o w e v e r , spine h e a d [Ca 2+] fell m o r e rapidly in the stubby spine because there was no spine neck constriction to restrict diffusion (Fig. 2C). W h e n the same Ca 2+ current was a p p l i e d to a small dendritic c o m p a r t m e n t ( c o m p a r a b l e to c o m p a r t m e n t 1 in Fig. 2 A ) instead of to the tip of the spine h e a d , p e a k [Ca 2+] in the c o m p a r t m e n t having the Ca 2+ influx again was less than o n e - t e n t h of that seen with the long-thin spine. The results were robust with respect to changes in the free p a r a m e t e r s of the m o d e l in all but two cases. A s shown in Table II, p e a k spine h e a d [Ca 2÷] r e m a i n e d over 10 times higher in the long-thin spine than in the stubby
340 A
TABLE II
30. a
Effects of varying the free parameters in the model 20-
Freeparameter change
Standard model Double current Half current Double buffer concentration Half buffer concentration Double current, double buffer concentration Half current, half buffer concentration Double pump rate constant Half pump rate constant Buffer binding Ka = 0.5 Buffer binding Ka = 2.0 Buffer binding Ka = 5.0
Peak [Ca2+] in compartment I
Ratio peak [Ca2+]. Long-thin Mushroom- Stubby tong-tl~m spine shaped spine spine to stubby 28.4 101.6 2.85
1.59 8.67 0.63
1.57 4.60 0.64
*~ 10-
18.1 22.1 4.4
b
0 0
40
80
120
160
200
160
200
Time (ms)
3.70
0.90
0.94
3.9
50.5
5.06
2.52
20.0
56.7
2.43
2.54
22.3
B
30
a
=20
/
7--,
% 14.4
1.16
1.00
14.5
26.1
1.59
1.57
16.6
29.7
1.60
1.58
18.9
28.3
1.22
1.27
22.4
28.9
2.29
1.99
14.5
30.9
4.09
2.69
11.5
spine with two-fold changes in the free parameters of the model except when the buffer concentration was doubled or when the Ca 2÷ current amplitude was halved. In these two cases, the difference in peak spine head [Ca 2÷] between long-thin and stubby spines was about 4-fold. However, when both buffer concentration and Ca 2+ current amplitude were doubled or when both Ca 2÷ current amplitude and buffer concentration were halved, the results were again qualitatively similar to those of Fig. 2. Thus, the qualitative results were sensitive only to particular combinations of buffer concentration values and Ca 2÷ current amplitudes. The results did not depend on the particular Ca e÷ current waveform used in the model. Whether the Ca 2÷ current was a half-rectified sine wave (100 Hz, with equal on and off times) or a constant current pulse lasting 100 ms, peak spine head [Ca 2÷] was over 10 times larger in the spine head of the long-thin spine than in the stubby spine (compare curves a and e in Fig. 3A,B). In all of these results, spine head [Ca 2÷] was much more highly concentrated in the long-thin spine than in the mushroom-shaped or stubby spines because of differences in spine shape. However, these results did not distinguish between differences caused by the diffusionai resistance of the spine neck and those caused by different spine head volumes. As shown by curves b and c in Fig. 3A and B, the diffusional resistance of the spine neck
O10
bc
0,
1 40
80
120
Time (ms)
Fig. 3. The [Ca 2+] in the spine heads of long-thin, mushroomshaped, and stubby spines for two hypothetical Ca 2÷ currents. The [Ca 2+] was observed at the tip of the spine head (compartment 1). In (A), the Ca 2+ current was a constant current pulse of 0.06 pA lasting 100 ms. In (B), the current was a half-rectified sine wave of 100 Hz with a peak of 0.16 pA (on for 5 ms, off for 5 ms). Key to the labeled curves: (a) long-thin spine, (b) long-thin spine with neck diameter changed to 0.2/~m, (c) long-thin spine with head diameter changed to 0.77/~m, (d) mushroom-shaped spine, (e) stubby spine.
began to play an important role in concentrating C a 2+ only after the Ca e÷ current duration was long enough to saturate the Ca 2+ binding sites in the spine. In Fig. 3A, spine head [Ca 2+] for the long-thin spine reached a steady-state oscillation after about 100 ms (curve a). When the spine neck diameter was doubled to equal that of the mushroom-shaped spine, spine head [Ca 2÷] again reached a steady-state oscillation, but the peak was reduced by one-third (curve b in Fig. 3A). A similar reduction was found when the current was a constant current pulse (Fig. 3B). These reductions in peak [Ca 2+] occurred because of the smaller diffusional resistance of the spine neck. However, at 35-40 ms the peak spine head [Ca 2÷] was almost the same for the two spine neck diameters. A current duration of 35-40 ms would have been too short to produce differences in peak spine head [Ca 2+] because of the different spine neck diameters. On the other hand, when the diameter of the head of the long-thin spine was increased to equal that of the mushroom-shaped spine (curve c in Fig. 3A), it took much longer for a steady-state oscillation in [Ca 2÷] to develop. For a given buffer concentration, increasing spine head volume increased the number of Ca 2+ binding
341 sites. Increasing the number of binding sites had the effect of delaying the rise to a steady-state oscillation; a similar (but not identical) effect has been noted when buffer concentration was increased in a constant volume 23. At 100 ms in Fig. 3A and at 60 ms in Fig. 3B, [Ca 2÷] was more than 10-fold higher in the long-thin spine than in the long-thin spine with increased head diameter (compare curves a and c in Figs. 3A,B). If current duration were 100 ms in Fig. 3A and 60 ms in Fig. 3B, one might conclude that spine head volume is much more important than spine neck diameter for peak spine head [Ca2+], but this effect would be due to the increased number of Ca 2÷ binding sites. For current durations of 200 ms or longer (or alternatively, lower buffer concentrations), the binding sites would be occupied and the differences in peak spine head [Ca z÷] due to differences in spine head volume would be much reduced. If the function of spines is to provide a locus for large, highly localized [Ca 2÷] increases, then (1) Ca 2÷ channels must exist on dendritic spines; (2) Ca 2÷ currents of sufficient strength and duration must be generated at synapses on dendritic spines; and (3) given such Ca 2÷ currents, the processes limiting short-term [Ca 2÷] changes in the spine head must not be fast enough or have sufficient capacity to prevent large short-term changes in spine head [Ca 2÷] from occurring. These 3 points need to be verified experimentally. The amplitude of the current could be estimated if the synaptic conductance were known. The current duration might depend on the frequency of afferent activation, but it could be limited by channel inactivation lz, channel desensitization 9, or changes in the driving force due to accumulation of ions 13. If long-duration Ca 2÷ currents exist, then long-term buffering mechanisms not included in this model (i.e. by organelles) might be important for limiting increases in spine head [Ca/÷]. At the present time, data for spine head buffer capacity, buffer rate constants, pump capacity, and pump rate are not available. Nevertheless, it would seem that Ca 2÷ currents with sufficient strength and duration to overwhelm Ca 2+ buffer and pump capacity might occur with high frequency co-activation of large numbers of synapses, i.e. conditions that are thought to lead to the induction of long-term potentiation. The C a 2+ current given in Fig. 1 represents an attempt to quantitatively estimate the magnitude and time course of such a current. The fact that Ca 2÷ influx is thought to be necessary for the induction of LTP7 argues that the required currents should exist. Furthermore, concentrations of intracellular Ca 2÷ in the micromolar range have been measured by optical imaging methods in cerebellar Purkinje cells and CA1 hippocampal pyramidal cells following afferent stimulation ~8'19'24'25. If this influx occurred at spines, then
it would seem likely that synaptic Ca z+ currents could have the strength and duration to overwhelm the shortterm Ca 2+ buffer and pump capacities, whatever these capacities might be. The present results show how spines can concentrate calcium near synapses. Once synaptic currents have the strength and duration to overwhelm the Ca z÷ binding sites in the spine, the diffusional resistance of the spine neck can significantly amplify spine head [CaZ+]. The long-thin spine in the present simulations had both a larger spine neck diffusional resistance and a smaller spine head (and hence fewer Ca a+ binding sites) than the mushroom-shaped or stubby spines. Both of these factors enabled the long-thin spine to concentrate Ca a+ to much higher levels than the mushroom-shaped or stubby spines for the current durations used in the model. Calcium concentrations in the spine head of long-thin spines could reach levels that would be toxic if they occurred in other parts of the cell. However, localized to spine heads, these calcium levels could perform important functions. It is known that many reactions in a cell, particularly those thought to be related to plasticity, depend on free ionized calcium 8. Because spine head [Ca z÷] levels following a given synaptic Ca 2÷ current are very sensitive to spine shape, the rates and types of Caa+-dependent reactions that are activated by transient [Ca a÷] increases may be significantly different in different spines. Other theoretical studies have shown the importance of a large electrical spine neck resistance for synaptic transmission and information processing (e.g. refs. 10, 11, 20, 22). However, the electrical resistances of the spines modeled here were less than 70 MK2, which is far smaller than usually assumed in theoretical studies. If the spine neck resistance truly is very small, then it will play a minor role in information transfer. Nevertheless, it is possible that the intraceUular resistivity is much higher in spines than elsewhere in the neuron because of the presence of the spine apparatus and other organelles. The importance of the electrical resistance of the spine neck for synaptic transmission and information processing will not be known until an accurate measurement of spine neck resistance can be made. The present theoretical investigation lends strong support to the idea that spines exist to concentrate calcium near synapses and suggests that spine shape plays an extremely important role in determining the rates and types of CaE+-depen dent reactions that can be activated in a particular spine. The key to the function of spines almost certainly lies with the spine neck resistance, whether it be electrical or diffusionai; however, if the electrical resistance is small, the diffusional resistance of the spine neck, by allowing Ca 2+ to become concentrated in the spine head, may be the critical factor explaining the function of spines.
342 I thank W. Rail, J. Rinzel, and W.B. Levy for their comments and suggestions on preliminary versions of this manuscript. I also wish
to thank the Frederick Cancer Research Facility for use of their computing resources.
1 Brown, T.H., Chang, V.C., Ganong, A.H., Keenan, C.L. and Kelso, S.R., Biophysical properties of dendrites and spines that may control the induction and expression of long-term synaptic potentiation. In P.W. Landfield and S.A. Deadwyler (Eds.), Long-term Potentiation: From Biophysics to Behavior, Alan R. Liss, New York, 1988, pp. 201-264. 2 Coss, R.G. and Perkel, D.H., The function of dendritic spines: a review of theoretical issues, Behav. Neural Biol., 44 (1985) 151-185. 3 Desmond, N.L. and Levy, W.B., Granule cell dendritic spine density in the rat hippocampus varies with spine shape and location, Neurosci. Lett., 54 (1985) 219-224. 4 Gamble, E. and Koch, C., The dynamics of free calcium in dendritic spines in response to repetitive synaptic input, Science, 236 (1987) 1311-1315. 5 Harris, K.M. and Stevens, J.K., Dendritic spines of rat cerebellar Purkinje cells: serial electron microscopy with reference to their biophysical characteristics, J. Neurosci., 8 (1988) 4455-4469. 6 Holmes, W.R. and Levy, W.B., Insights into associative longterm potentiation from computational models of NMDA receptor-mediated calcium influx and intracellular calcium concentration changes, J. Neurophysiol., 63 (1990). 7 Lynch, G., Larson, J., Kelso, S., Barrionuevo, G. and Schottler, E, Intracellular injections of EGTA block induction of hippocampal long-term potentiation, Nature (Lond.), 305 (1983) 719-721. 8 Malenka, R.C., Kauer, J.A., Perkel, D.J., Mauk, M.D., Kelly, P.T., Nicoll, R.A. and Waxham, M.N., An essential role for postsynaptic calmodulin and protein kinase activity in long-term potentiation, Nature (Lond.), 340 (1989) 554-557. 9 Mayer, M.L. and Westbrook, G., The action of N-methylD-aspartic acid on mouse spinal neurones in culture, J. Physiol. (Lond.), 261 (1985) 65-90. 10 Miller, J.P., Rail, W. and Rinzel, J., Synaptic amplification by active membrane in dendritic spines, Brain Research, 325 (1985) 325-330. 11 Perkel, D.H. and Perkel, D.J., Dendritic spines: role of active membrane in modulating synaptic efficacy, Brain Research, 325 (1985) 331-335. 12 Pitier, T.A. and Landfield, P.W., Probable Ca2÷-mediated inactivation of Ca 2÷ currents in mammalian brain neurons, Brain Research, 410 (1987) 147-153. 13 Qian, N. and Sejnowski, T., Electrodiffusion model of electrical
conduction in neuronal processes. In C.D. Woody, D.L. Alkon and J.L. McGaugh (Eds.), Cellular Mechanisms of Conditioning and Behavioral Plasticity, Plenum, New York, 1988, pp. 237244. 14 Rail, W., Dendritic spines, synaptic potency and neuronal plasticity. In C.D. Woody, K.A. Brown, T.J. Crow and J.D. Knispel (Eds.), Cellular Mechanisms Subserving Changes in Neuronal Activity, University of California Brain Information Service, Los Angeles, 1974, pp. 13-24. 15 Rail, W., Dendritic spines and synaptic potency. In R. Porter (Ed.), Studies of Neurophysiology, Cambridge University, London, 1978, pp. 203-209. 16 Rail, W. and Rinzel, J., Dendritic spines and synaptic potency explored theoretically, Proc. Int. Congr. Physiol. Sci. (XXV Intl. Congr.), 9 (1971) 466. 17 Rail, W. and Rinzel, J., Dendritic spine function and synaptic attenuation calculations, Soc. Neurosci. Abstr., 1 (1971) 64. 18 Regehr, W.G. and Tank, D.W., Calcium in hippocampal pyramidal cells during tetanic stimulation: implications for LTP, Soc. Neurosci. Abstr., 15 (1989) 398. 19 Regehr, W.G., Connor, J.A. and Tank, D.W., Optical imaging of calcium accumulation in hippocampal pyramidal cells during synaptic activation, Nature (Lond.), 341 (1989) 533-536. 20 Segev, I. and Rail, W., Computational study of an excitable dendritic spine, J. Neurophysiol., 60 (1988) 499-523. 21 Shepherd, G.M., The Synaptic Organization of the Brain, Oxford University, New York, 1979. 22 Shepherd, G.M. and Brayton, R.K., Logic operations are properties of computer-simulated interactions between excitable dendritic spines, Neuroscience, 21 (1987) 151-165. 23 Simon, S.M. and Llinas, R.R., Compartmentalization of the submembrane calcium activity during calcium influx and its significance in transmitter release, Biophys. J., 48 (1985) 485-498. 24 Tank, D.W., Sugimori, M., Connor, J.A. and Llinas, R.R., Spatially resolved calcium dynamics of mammalian Purkinje cells in cerebellar slice, Science, 242 (1988) 773. 25 Tank, D.W. and Regehr, W.G., Optical imaging of ion concentration dynamics in hippocampal brain slice, Soc. Neurosci. Abstr., 15 (1989) 398. 26 Wickens, J., Electrically coupled but chemically isolated synapses: dendritic spines and calcium in a rule for synaptic modification, Prog. Neurobiol., 31 (1988) 507-528.
Brain Research, 519 (1990) 338-342
Elsevier BRES 24120
Is the function of dendritic spines to concentrate calcium? William R. Holmes Mathematical Research Branch, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, MD 20892 (U.S.A.)
(Accepted 20 February 1990) Key words: Dendritic spine; Calcium; Plasticity; Long-term potentiation; Synapse; Computer simulation
Although dendritic spines are thought to play an important role in synaptic transmission and plasticity, their function remains unknown. Theoretical investigations of spine function have focused on the large electrical resistance provided by the narrow constriction of the spine neck. However, this narrow constriction is also thought to provide a large diffusional resistance. The importance of this diffusional resistance was investigated theoretically with models. When calcium currents were activated on dendritic spines, peak spine head Ca2+ concentration was an order of magnitude larger in 'long-thin' spines than in 'mushroom-shaped' or 'stubby' spines. The same currents activated on dendrites produced even smaller local Ca2÷ concentration changes. Although the diffusional resistance of the spine neck was important for producing these differences in [Ca2÷], the amplitude and duration of the Ca2÷ current relative to the number of Ca2÷ binding sites determined whether Ca2÷ would be concentrated near synapses. Given the importance of Ca2÷ for long-term potentiation, the ability of spines to concentrate Ca2+ may play a key role in processes leading to learning and memory storage.
The function of dendritic spines has been the subject of much speculation ever since spines were discovered a century ago. Beginning with the theoretical investigations of Rail and Rinzel ~4-t7, there have been a number of studies exploring the role of spines in synaptic transmission and plasticity (see refs. 1, 2 for reviews). Usually, these studies have focused on the electrical resistance of the thin spine neck. However, besides providing an electrical resistance to current flow, the thin spine neck provides a diffusional resistance to the flow of ions and molecules. Some have suggested that by restricting the flow of materials, Ca 2÷ in particular, into or out of the spine head, the thin spine neck might effectively isolate the spine head and thus provide a localized environment in which reactions specific to a particular synapse could Occur 1"2"5"21'26. The Gamble and Koch model 4 showed how large, transient increases in spine head [Ca 2÷] might be attained. However, the effects of different spine shapes on spine head [Ca 2÷] increases and the consequences of placing the synaptic Ca 2+ current on the dendrite rather than on the spine have not been explored. These issues are the focus of the present theoretical study. If the degree to which Ca 2÷ can become concentrated in the spine head were controlled by the diffusional resistance of the spine neck, then spine shape would determine the rates and types of Cae+-dependent reactions that are activated at a synapse on a particular spine.
Because Ca2+-dependent reactions are thought to lead to the induction of long-term potentiation (LTP) (i.e. ref. 8), regulation of [Ca 2÷] by the diffusional resistance of the spine neck could be exceedingly important. To explore the importance of the diffusional resistance of the spine neck, [Ca 2÷] changes were modeled in the spine heads of 3 differently shaped spines. The dimensions of the 3 spines were taken to be the average dimensions for long-thin, mushroom-shaped, and stubby spines as measured and categorized on hippocampal dentate granule cells 3. With a standard value for intracellular resistivity (70 t2cm), the spine neck resistance was less than 70 Mr2 for each of the 3 spine shapes. A compartmental model similar to the Gamble and Koch model 4 was used to predict [Ca 2÷] in a dendritic spine for a given Ca 2+ current. The model had 20 compartments, 8 for the dendritic spine and 12 for the dendrite. Calcium was assumed to enter the outermost spine head compartment, after which it could diffuse into a neighboring compartment, become bound to buffer, or be pumped out of the cell. Equations for free calcium concentration and free buffer concentration in each compartment were solved using a fourth-order Runga-Kutta method. A technical description of this model is given elsewhere 6. The synaptic Ca 2+ current used in the model is illustrated in Fig. 1. Values for the free parameters in the model are given in Table I; the basis for these choices is the same
Correspondence: W.R. Holmes, Mathematical Research Branch, Bldg. 31 Rm. 4B-54, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, MD 20892, U.S.A.
0006-8993/90/$03.50 (~) 1990 Elsevier Science Publishers B.V. (Biomedical Division)
339 A
0.20 ~" 0.16
30.
~= 20. :zL
e~
0.12 -O-10
o 0.08 0
0.04
0
40
80 120 T i m e (ms)
160
200
160
200
0.00 0
40
80 120 Time (ms)
160
200
Fig. 1. Hypothetical synaptic calcium current. Illustrated is the predicted Ca z÷ component of the multi-ion ionic current flow through NMDA receptor channels at a synapse on a long-thin spine obtained in a hippocampal dentate granule cell model6 when 96 synapses on the same dendrite were co-activated at 200 Hz. Calcium currents nearly identical to this current were used for the mushroom-shaped and stubby spines; the differences that occurred were slight because the spine neck resistance was less than 70 MI2 in all 3 cases and the conductance at an individual synapse (NMDA + non-NMDA) was never larger than 0.3 nS. This calcium current and those used to obtain the results illustrated in Fig. 3 are quite small, but they represent only a fraction of the total ionic current that one is likely to see at a single activated synapse.
as given in the previous description of the m o d e l 6. T h e results pictured in Fig. 2 show that spine shape plays an i m p o r t a n t role in controlling the magnitude of localized increases in [Ca 2÷] in the spine head. The synaptic Ca 2÷ current p r o d u c e d a large, steep increase in calcium concentration in the spine h e a d of the long-thin spine (Fig. 2 A ) . T h e r e were only small differences in [Ca 2÷] a m o n g the spine h e a d c o m p a r t m e n t s , but there was a steep calcium concentration gradient down the spine neck. T h e p e a k [Ca 2+] in the dendritic c o m p a r t m e n t s was less than 1% of the p e a k [Ca 2+] seen in the spine head. F o r a m u s h r o o m - s h a p e d spine, the Ca 2÷ current p r o d u c e d a p e a k [Ca z+] in the spine h e a d that was less than o n e - t e n t h that f o u n d with the long-thin spine (Fig. 2B). The calcium concentration gradient d o w n the spine neck was much smaller in this case, and slightly larger concentrations of Ca 2+ were found in the
TABLE I Values of free parameters used in the model Parameter
Value
Buffer concentration (compartments 1-2) Buffer concentration (compartments 3-20) Buffer forward rate constant, kbf Buffer backward rate constant, kb b Buffer affinity Ka (kbb/kbf) Pump rate constant Ca2+ diffusion coefficient
200/~M 100#M 0.5/tM-l.ms -1 0.5.ms -1 1.0/*M 1.4 x 10-4 cm/s 0.6pmZ/ms
1
0
7
40
80 120 T i m e (ms)
14
r
~-
0 0
40 ;
80 120 T i m e (ms)
160
200
Fig. 2. Free Ca2+ concentration in long-thin, mushroom-shaped,
and stubby spines for a given C a 2+ current. Note the different ordinate scales. The synaptic calcium current was that illustrated in Fig. 1. The numbers refer to compartments in the model with the locations as shown. The dimensions (taken from ref. 3) for the spine head and neck were 0.55 x 0.55 and 0.1 x 0.73 pm (long-thin), 0.77 x 0.77 and 0.2 x 0.43pm (mushroom), and 0.76 x 0.99pm (stubby, head and neck together) and are shown to scale. dendritic c o m p a r t m e n t s . T h e p e a k [Ca 2+] in the spine head of the stubby spine was n e a r l y the s a m e as that for the m u s h r o o m - s h a p e d spine. H o w e v e r , spine h e a d [Ca 2+] fell m o r e rapidly in the stubby spine because there was no spine neck constriction to restrict diffusion (Fig. 2C). W h e n the same Ca 2+ current was a p p l i e d to a small dendritic c o m p a r t m e n t ( c o m p a r a b l e to c o m p a r t m e n t 1 in Fig. 2 A ) instead of to the tip of the spine h e a d , p e a k [Ca 2+] in the c o m p a r t m e n t having the Ca 2+ influx again was less than o n e - t e n t h of that seen with the long-thin spine. The results were robust with respect to changes in the free p a r a m e t e r s of the m o d e l in all but two cases. A s shown in Table II, p e a k spine h e a d [Ca 2÷] r e m a i n e d over 10 times higher in the long-thin spine than in the stubby
340 A
TABLE II
30. a
Effects of varying the free parameters in the model 20-
Freeparameter change
Standard model Double current Half current Double buffer concentration Half buffer concentration Double current, double buffer concentration Half current, half buffer concentration Double pump rate constant Half pump rate constant Buffer binding Ka = 0.5 Buffer binding Ka = 2.0 Buffer binding Ka = 5.0
Peak [Ca2+] in compartment I
Ratio peak [Ca2+]. Long-thin Mushroom- Stubby tong-tl~m spine shaped spine spine to stubby 28.4 101.6 2.85
1.59 8.67 0.63
1.57 4.60 0.64
*~ 10-
18.1 22.1 4.4
b
0 0
40
80
120
160
200
160
200
Time (ms)
3.70
0.90
0.94
3.9
50.5
5.06
2.52
20.0
56.7
2.43
2.54
22.3
B
30
a
=20
/
7--,
% 14.4
1.16
1.00
14.5
26.1
1.59
1.57
16.6
29.7
1.60
1.58
18.9
28.3
1.22
1.27
22.4
28.9
2.29
1.99
14.5
30.9
4.09
2.69
11.5
spine with two-fold changes in the free parameters of the model except when the buffer concentration was doubled or when the Ca 2÷ current amplitude was halved. In these two cases, the difference in peak spine head [Ca 2÷] between long-thin and stubby spines was about 4-fold. However, when both buffer concentration and Ca 2+ current amplitude were doubled or when both Ca 2÷ current amplitude and buffer concentration were halved, the results were again qualitatively similar to those of Fig. 2. Thus, the qualitative results were sensitive only to particular combinations of buffer concentration values and Ca 2÷ current amplitudes. The results did not depend on the particular Ca e÷ current waveform used in the model. Whether the Ca 2÷ current was a half-rectified sine wave (100 Hz, with equal on and off times) or a constant current pulse lasting 100 ms, peak spine head [Ca 2÷] was over 10 times larger in the spine head of the long-thin spine than in the stubby spine (compare curves a and e in Fig. 3A,B). In all of these results, spine head [Ca 2÷] was much more highly concentrated in the long-thin spine than in the mushroom-shaped or stubby spines because of differences in spine shape. However, these results did not distinguish between differences caused by the diffusionai resistance of the spine neck and those caused by different spine head volumes. As shown by curves b and c in Fig. 3A and B, the diffusional resistance of the spine neck
O10
bc
0,
1 40
80
120
Time (ms)
Fig. 3. The [Ca 2+] in the spine heads of long-thin, mushroomshaped, and stubby spines for two hypothetical Ca 2÷ currents. The [Ca 2+] was observed at the tip of the spine head (compartment 1). In (A), the Ca 2+ current was a constant current pulse of 0.06 pA lasting 100 ms. In (B), the current was a half-rectified sine wave of 100 Hz with a peak of 0.16 pA (on for 5 ms, off for 5 ms). Key to the labeled curves: (a) long-thin spine, (b) long-thin spine with neck diameter changed to 0.2/~m, (c) long-thin spine with head diameter changed to 0.77/~m, (d) mushroom-shaped spine, (e) stubby spine.
began to play an important role in concentrating C a 2+ only after the Ca e÷ current duration was long enough to saturate the Ca 2+ binding sites in the spine. In Fig. 3A, spine head [Ca 2+] for the long-thin spine reached a steady-state oscillation after about 100 ms (curve a). When the spine neck diameter was doubled to equal that of the mushroom-shaped spine, spine head [Ca 2÷] again reached a steady-state oscillation, but the peak was reduced by one-third (curve b in Fig. 3A). A similar reduction was found when the current was a constant current pulse (Fig. 3B). These reductions in peak [Ca 2+] occurred because of the smaller diffusional resistance of the spine neck. However, at 35-40 ms the peak spine head [Ca 2÷] was almost the same for the two spine neck diameters. A current duration of 35-40 ms would have been too short to produce differences in peak spine head [Ca 2+] because of the different spine neck diameters. On the other hand, when the diameter of the head of the long-thin spine was increased to equal that of the mushroom-shaped spine (curve c in Fig. 3A), it took much longer for a steady-state oscillation in [Ca 2÷] to develop. For a given buffer concentration, increasing spine head volume increased the number of Ca 2+ binding
341 sites. Increasing the number of binding sites had the effect of delaying the rise to a steady-state oscillation; a similar (but not identical) effect has been noted when buffer concentration was increased in a constant volume 23. At 100 ms in Fig. 3A and at 60 ms in Fig. 3B, [Ca 2÷] was more than 10-fold higher in the long-thin spine than in the long-thin spine with increased head diameter (compare curves a and c in Figs. 3A,B). If current duration were 100 ms in Fig. 3A and 60 ms in Fig. 3B, one might conclude that spine head volume is much more important than spine neck diameter for peak spine head [Ca2+], but this effect would be due to the increased number of Ca 2÷ binding sites. For current durations of 200 ms or longer (or alternatively, lower buffer concentrations), the binding sites would be occupied and the differences in peak spine head [Ca z÷] due to differences in spine head volume would be much reduced. If the function of spines is to provide a locus for large, highly localized [Ca 2÷] increases, then (1) Ca 2÷ channels must exist on dendritic spines; (2) Ca 2÷ currents of sufficient strength and duration must be generated at synapses on dendritic spines; and (3) given such Ca 2÷ currents, the processes limiting short-term [Ca 2÷] changes in the spine head must not be fast enough or have sufficient capacity to prevent large short-term changes in spine head [Ca 2÷] from occurring. These 3 points need to be verified experimentally. The amplitude of the current could be estimated if the synaptic conductance were known. The current duration might depend on the frequency of afferent activation, but it could be limited by channel inactivation lz, channel desensitization 9, or changes in the driving force due to accumulation of ions 13. If long-duration Ca 2÷ currents exist, then long-term buffering mechanisms not included in this model (i.e. by organelles) might be important for limiting increases in spine head [Ca/÷]. At the present time, data for spine head buffer capacity, buffer rate constants, pump capacity, and pump rate are not available. Nevertheless, it would seem that Ca 2÷ currents with sufficient strength and duration to overwhelm Ca 2+ buffer and pump capacity might occur with high frequency co-activation of large numbers of synapses, i.e. conditions that are thought to lead to the induction of long-term potentiation. The C a 2+ current given in Fig. 1 represents an attempt to quantitatively estimate the magnitude and time course of such a current. The fact that Ca 2÷ influx is thought to be necessary for the induction of LTP7 argues that the required currents should exist. Furthermore, concentrations of intracellular Ca 2÷ in the micromolar range have been measured by optical imaging methods in cerebellar Purkinje cells and CA1 hippocampal pyramidal cells following afferent stimulation ~8'19'24'25. If this influx occurred at spines, then
it would seem likely that synaptic Ca z+ currents could have the strength and duration to overwhelm the shortterm Ca 2+ buffer and pump capacities, whatever these capacities might be. The present results show how spines can concentrate calcium near synapses. Once synaptic currents have the strength and duration to overwhelm the Ca z÷ binding sites in the spine, the diffusional resistance of the spine neck can significantly amplify spine head [CaZ+]. The long-thin spine in the present simulations had both a larger spine neck diffusional resistance and a smaller spine head (and hence fewer Ca a+ binding sites) than the mushroom-shaped or stubby spines. Both of these factors enabled the long-thin spine to concentrate Ca a+ to much higher levels than the mushroom-shaped or stubby spines for the current durations used in the model. Calcium concentrations in the spine head of long-thin spines could reach levels that would be toxic if they occurred in other parts of the cell. However, localized to spine heads, these calcium levels could perform important functions. It is known that many reactions in a cell, particularly those thought to be related to plasticity, depend on free ionized calcium 8. Because spine head [Ca z÷] levels following a given synaptic Ca 2÷ current are very sensitive to spine shape, the rates and types of Caa+-dependent reactions that are activated by transient [Ca a÷] increases may be significantly different in different spines. Other theoretical studies have shown the importance of a large electrical spine neck resistance for synaptic transmission and information processing (e.g. refs. 10, 11, 20, 22). However, the electrical resistances of the spines modeled here were less than 70 MK2, which is far smaller than usually assumed in theoretical studies. If the spine neck resistance truly is very small, then it will play a minor role in information transfer. Nevertheless, it is possible that the intraceUular resistivity is much higher in spines than elsewhere in the neuron because of the presence of the spine apparatus and other organelles. The importance of the electrical resistance of the spine neck for synaptic transmission and information processing will not be known until an accurate measurement of spine neck resistance can be made. The present theoretical investigation lends strong support to the idea that spines exist to concentrate calcium near synapses and suggests that spine shape plays an extremely important role in determining the rates and types of CaE+-depen dent reactions that can be activated in a particular spine. The key to the function of spines almost certainly lies with the spine neck resistance, whether it be electrical or diffusionai; however, if the electrical resistance is small, the diffusional resistance of the spine neck, by allowing Ca 2+ to become concentrated in the spine head, may be the critical factor explaining the function of spines.
342 I thank W. Rail, J. Rinzel, and W.B. Levy for their comments and suggestions on preliminary versions of this manuscript. I also wish
to thank the Frederick Cancer Research Facility for use of their computing resources.
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