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Impact of Time-Dependent Changes in Spine Density and Spine Shape on the Input-Output Properties of a Dendritic Branch: A Computational Study

¹Department of Mathematics, San Diego State University–Imperial Valley Campus, Calexico, California; ²Department of Pathobiology and Diagnostic Investigation, Michigan State University, East Lansing, Michigan; and ³Department of Mathematics and Statistics, Arizona State University, Tempe, Arizona

Submitted 12 April 2004; accepted in final form 24 November 2004

	ABSTRACT

TOP ABSTRACT INTRODUCTION METHODS RESULTS Calcium-mediated spine... DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
Populations of dendritic spines can change in number and shape quite rapidly as a result of synaptic activity. Here, we explore the consequences of such changes on the input–output properties of a dendritic branch. We consider two models: one for activity-dependent spine densities and the other for calcium-mediated spine-stem restructuring. In the activity-dependent density model we find that for repetitive synaptic input to passive spines, changes in spine density remain local to the input site. For excitable spines, the spine density increases both inside and outside the input region. When the spine stem resistances are relatively high, the transition to higher dendritic output is abrupt; when low, the rate of increase is gradual and resembles long-term potentiation. In the second model, spine density is held constant, but the stem dimensions are allowed to change as a result of stimulation-induced calcium influxes. The model is formulated so that a moderate amount of synaptic activation results in spine stem elongation, whereas high levels of activation result in stem shortening. Under these conditions, passive spines receiving modest stimulation progressively increase their spine stem resistance and head potentials, but little change occurs in the dendritic output. For excitable spines, modest stimulation frequencies cause a lengthening of both stimulated and neighboring spines and the stimulus eventually propagates. High-frequency stimulation that causes spines to shorten in the stimulated region decreases the amplitude of the dendritic output slightly or drastically, depending on initial spine densities and stem resistances.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS Calcium-mediated spine... DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
The innate properties of the dendritic trees of neurons within the CNS are highly variable and allow a wide range of functions. Some are very spiny and some are not; some conduct action potentials and some rely on electrotonic spread of input to the soma and the axon. Some, like inhibitory interneurons, are specialized to receive input quickly and to conduct action potentials at extraordinarily high frequencies with sustained depolarization, from 200 to 500 Hz (Jonas et al. 2004). Dendritic action potentials can be generated by voltage-gated calcium channels (Helmchen et al. 1999; Magee and Johnston 1995) and/or by voltage-gated sodium channels (Golding and Spruston 1998; Huguenard et al. 1989; Jung et al. 1997), and even in systems in which dendritic action potentials do not occur, voltage-gated channels modify the shapes of subthreshold synaptic potentials (Gonzalez-Burgos and Barrionuevo 2001). The density of voltage-gated channels in dendritic membranes is not necessarily uniform. Sodium channels and A-type potassium channels can be activated and distributed more densely near the postsynaptic specializations (Alonso and Widmer 1997; Frick et al. 2004; Hanson et al. 2004), which enhances the likelihood of spiking. This diversity and precise distribution of channel types allows for the different capabilities needed in the nervous system.

For many years investigators have engaged the problem of identifying the features of neuronal networks that are capable of changing in response to input from the periphery, and that could underlie learning. This search has resulted in many activity-dependent mechanisms that are invoked to varying degrees in different networks. Changes both in molecular properties of dendritic membranes and in overall structure and number of dendritic spines have been seen.

Increases in spine density can be observed in conjunction with increased synaptic activation associated with early development, increased electrical activity arising from block of inhibition, increased sensory input, providing more complex environments, or experimental induction of long-term potentiation (LTP) (Annis et al. 1994; Engert and Bonhoeffer 1999; Knott et al. 2002; Maletic-Savatic et al. 1999; Toni et al. 1999) and others reviewed by Nimchinsky et al. (2002). It has been suggested that spines may not increase in number by a splitting mechanism but rather by the emergence of dendritic protrusions (Harris et al. 2003). Decreases in spine density are observed in association with decreases in activity with block of sodium channels, deafferentation, developmental "pruning," or sensory deprivation, but may also occur after high levels of activity—excitotoxicity such as seizures, stimulation resulting in excessive glutamate release, or application of kainic acid (Müller et al. 2000; Nimchinsky et al. 2002; Oliva et al. 2002).

Direct observation of living spines with fluorescent probes has allowed us to see that their shapes can change with remarkable rapidity, within seconds (Fischer et al. 1998; Kaech et al. 2001; Krucker et al. 2000). Growth and movement of filopodia or spines can occur within minutes, either as a developmental phenomenon (Dailey and Smith 1996) or as a result of stimulation (Engert and Bonhoeffer 1999; Maletic-Savatic et al. 1999). Several studies suggest that modest activation of glutamate receptors, which gives rise to a small influx of calcium, and possibly release of calcium from internal stores, favors lengthening of spines, but excessive stimulation and concomitant large increases in calcium cause retraction or collapse of spines (Halpain et al. 1998; Korkotian and Segal 1998, 1999; Segal et al. 2000; see review in Nimchinsky et al. 2002).

Changes in spine shape will certainly affect the spine's cytoplasmic environment as a result of increased or decreased degrees of compartmentalization. However, changes in the electrical signals of both affected and neighboring spines will also occur. Simulations show that it may be advantageous, at least if action potential conduction is the goal, for the dendrite to cluster and isolate some of the voltage-gated channels to the spine head (Baer and Rinzel 1991; Segev and Rall 1988; Tsay and Yuste 2002), so changes in the degree of synaptic isolation stemming from shape changes will have significant impact.

In this study we have focused on simulating the output of a generalized dendritic segment if morphological properties of the spines are presumed to change in some of the ways that have been observed in various systems. We considered the impact of time-dependent changes in spine density and spine shape on the properties of a dendritic branch under conditions in which the entire dendrite was assumed to have passive membrane properties, and under conditions in which voltage-gated sodium channels were clustered in the spine head. Both modest and excitotoxic levels of stimulation were explored.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS Calcium-mediated spine... DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
We formulate two mathematical models: the first is a model for activity-dependent spine densities and the second for calcium-mediated restructuring of individual spines. Both models are built on a continuum formulation for the interaction of many spines (Baer and Rinzel 1991).

Consider a passive dendritic cable of length l (µm), with both ends sealed, studded with a population of dendritic spines. The spine density is defined as the number of spines per unit physical length. Over a short segment x, the spines deliver current xI_ss to the dendrite, where I_ss represents the current flowing through an individual spine stem. The stem current (I_ss) is expressed as an I · R voltage drop across the spine stem resistance R_ss (M), given by

(1)

where V_sh and V_d (mV) are the membrane potential in the head and dendritic base, respectively. The spine stem is modeled as in previous studies (Baer and Rinzel 1991; Segev and Rall 1988) as a lumped ohmic resistor, neglecting the stem's membrane and cable properties. If the potential in the spine head is larger than the potential in the dendritic shaft (V_sh > V_d), then I_ss > 0 and the current is flowing from spine head to spine base. Conversely, if the potential in the base is larger than the potential in the head (V_d > V_sh), then I_ss < 0 and the current flow is from base to head. If I_ss = 0, then no current is passing through the spine stem.

The electrical potential V_d(x, t) in a passive dendrite studded with spines per unit physical length satisfies the cable equation

(2)

Here R_i ( · cm) is the specific cytoplasmic resistivity; R_m ( · cm²) is the resistance across a unit area of passive membrane; C_m (µF/cm²) is the specific membrane capacitance; and d (µm) is the diameter of the dendrite. The dendrite is thought of as a distal branch. Parameter values for the cable are identified in Table 1.

(3)

Here, represents the number of spines over length (or number of spines per unit electrotonic length; denoted here by spines/e.l.). It is assumed throughout this paper that both ends of the dendrite are sealed, with a uniform resting potential of zero in the cable and the spine heads.

We modeled the spine head as an isopotential compartment with surface area A_sh (µm²) and specific membrane capacitance C_m (µF/cm²); individual spines have a capacitance of C_sh = A_shC_m (µF). An equation for the membrane potential in a single spine is obtained from a current balance relation for the capacitive, ionic, spine stem, and synaptic currents given by

(4)

The term I_ion represents ionic currents passing through the head membrane and I_syn represents synaptic current. If the membrane is passive, then I_ion = V_sh/R_sh, where R_sh is the membrane resistance of the head. When modeling excitable membrane in the spine heads, we used Hodgkin–Huxley kinetics (Hodgkin and Huxley 1952) for voltage-dependent ion channel currents

(5)

Here, _i and V_i are maximal conductances and reversal potentials, respectively, for sodium, potassium, and leakage currents. We followed Baer and Rinzel (1991) and used, except where noted, increased channel densities ( = 2.5) and a temperature of 22°C; this corresponds to approximately 328 sodium channels per spine head. For the dendritic geometry defined here, this corresponds to 1.6 channels/µm if the channels were moved from the spines to the dendritic shaft. For example, a density of 54 spines/e.l. corresponds to 86 sodium channels/µm² and a density of 100 corresponds to 160 sodium channels/µm², and so forth.

In the continuum description, we can prescribe different distributions of spines and different synaptic input patterns. However, the spine density can vary significantly with X. We simulate the activation of a cluster of synapses by applying to all spines in the activation region, X₀ X X₀ + X

(6)

where V_syn is the synaptic reversal potential and g_syn is a brief synaptic conductance generated by the -function

(7)

which reaches a peak synaptic conductance g_p, t_p ms after activation. Equations 6 and 7 model a typical synaptic current observed in experiments and are similar to the equations for synaptic current used in other models. Kinetic and physical parameter values for spine heads and ionic and synaptic currents may be found in Table 1.

The synaptic input I_syn(X, t), given by Eqs. 6 and 7, is applied in all simulations to the spine heads over the region 0 X 0.2 periodically with g_syn peaking at t_p = 0.2 ms into each activation period, allowing the system to return to rest between activations. A graph of I_syn during 6 ms of an initial activation cycle for passive spines is compared with spine head potential response over the same time frame in Fig. 1. We define activation cycle as the total length of time from the beginning of a simulated synaptic input up to the beginning of the next. It is comparable to the stimulus interval. Note that the synaptic current I_syn peaks at about t_p = 0.2 ms, and returns to rest before t = 2 ms. The head potential (V_sh) for spines under synaptic activation, although slower, has fallen to 0.23 mV at t = 6 ms, which is close to rest. We apply no shorter than a 10-ms activation cycle to avoid affects of summation of the input. In the sections considering calcium-mediated restructuring, while g_p and t_p remain unchanged, we apply different activation frequencies; the duration of activation cycles in those sections may be found in the figures.

View larger version (10K): [in this window] [in a new window]   FIG. 1. Both synaptic current Isyn and spine head potential Vsh return to rest within one activation period. For passive spines the time course for Isyn (see Eqs. 6 and 7) is compared with the time course of the postsynaptic Vsh. Synaptic current returns to rest after 2 ms, whereas the head potential approaches rest after 6 ms. The figure illustrates that a 10 ms period between synaptic inputs is sufficient for the system to return to rest.

It is convenient to use the spine stem current (I_ss) as a measure, over long periods of time (minutes to hours), of the electrical activity between the spine head and dendritic base (Kuske and Baer 2002; Wu and Baer 1998). Herein we explored the effects of having this electrical interaction control the local recruitment of new spines and synapses (see Fig. 2, A and B), and the loss of existing ones over time.

View larger version (26K): [in this window] [in a new window]   FIG. 2. Simulated morphological changes. A: starting point for each of the simulations is a number of evenly spaced spines with similar spine stem shapes. B: a subset of these spines at a particular location will receive synaptic input. In the first series of simulations, we examine the situation in which the dendrite receiving input responds to that activity by growing new spines in response to the current flows that are generated; a local increase in spine density is thereby created. C: in the second series of simulations, synaptic activity causes a proportionate increase in cytoplasmic calcium. If the increase in calcium ions is small, the spine stems elongate and become thinner, thus increasing spine stem resistance. D: if the influx of calcium is large, greater than a "threshold" amount, the spine stems collapse into the dendrite. We assume that the presynaptic axonal processes are plastic, and that the synaptic connection is not terminated by change in spine stem shape.

A continuum model for activity-dependent spine densities is

(8)

(9)

(10)

The initial distribution of spine density, _d, the maximum and minimum spine density parameters, _max and _min, which bound , are given in Table 1. Note that if I_ss = 0, is constant in time, but could still be spatially nonuniform.

The system of Eqs. 8–10 is nonlinear because Eq. 10 is nonlinear. This is true even if the membrane properties of the spine heads and cable are assumed to be passive. However, changes slowly because is at least 3 orders of magnitude larger than the membrane time constants for the spine heads and dendrite (e.g., _m = 2.5 ms and = 1,250 ms). For the passive membrane case, Eqs. 8 and 9 constitute a linear subsystem that acts on a fast time scale (milliseconds), and Eq. 10 for is a single equation acting on a slow time scale (seconds). Changes in spine density depend on changes in I_ss, which in turn depend on the integrative properties of the surrounding membrane and synaptic activity. This formulation does not require to be continuous in space (X).

In most of our simulations for this model, the synaptic input will be repeated every 10 ms, a period short enough to capture the dynamics of V_sh and V_d but long enough to allow those potentials to return to resting values. The spine density appears constant on a 10-ms time scale. The spine density's dynamics are resolved on the time scale of (on the order of seconds in our simulations). Increasing slows down the change in spine density. In an earlier variation of the model (Verzi 2000) the effect of varying = 1/ was explored. It was found that there exists * such that for < * the dynamical properties of the system remain invariant. In this paper, a sufficiently large value of was chosen that was biologically plausible and computationally efficient. To integrate the system we used a semi-implicit Crank–Nicholson (for Eq. 8) and Adams–Bashforth (for time integration) finite-difference method. This method is sufficiently fast without sacrificing the accuracy of computations.

Model for calcium-mediated restructuring

Although activity affects—and is affected by—the distribution and density of spines along the dendrite, so also are the structures of individual spines. Recent experiments implicate the intraspine calcium level as a mediator for changes in dendritic spine structure (reviewed in Nimchinsky et al. 2002). Spines of cultured hippocampal neurons have been monitored over several hours (Korkotian and Segal 1999). Release of calcium from internal stores, in response to pulse applications of caffeine, induced a small transient rise in Ca²⁺ (200–400 nM), and an increase in the length of spine stems in <5 min. Conversely, Halpain et al. (1998) induced a rapid collapse of dendritic spine stems (also within 5 min) by stimulating cultured neurons with glutamate. This caused maximal calcium influx, raising intraspine calcium to much higher levels.

A diagrammatic model has been proposed for spine restructuring based on the above experiments (Harris 1999) and is illustrated in Fig. 2, C and D. A moderate amount of synaptic activation may result in spine stem elongation. However, a high level of activity may cause too much calcium influx and induce spine stem shortening or loss, perhaps as a result of actin depolymerization. A continuum model for a uniform distribution of spines (with spine density _c constant in time) consistent with the above hypothesis is

(11)

(12)

(13)

(14)

In Eq. 13, the change in intraspine calcium C_a is dependent on activity, as measured by |I_ss|. In the absence of activity (I_ss = 0) calcium decays slowly to a minimal value, denoted by C_min. In Eq. 14, the spine stem resistance R_ss is a dependent variable that reflects changes in spine stem structure. The stem resistance approaches steady state in the absence of activity because C_a approaches C_min when I_ss = 0. An earlier mathematical formulation made the counterintuitive assumption that spines grow to their maximum length (i.e., R_ss approaches R_max) in the absence of activity (Verzi and Baer 2004). When synaptic activity is present the stem resistance approaches steady state if R_ss approaches R_max or R_min, or if I_ss drives C_a to C_crit.

This model builds on the simplified Wu–Baer model (Wu and Baer 1998) for a single spine with an activity-dependent stem conductance. Only here, activity-dependent calcium is viewed as a second messenger that regulates changes in spine stem resistance (reciprocal of conductance). We identify a critical intraspine calcium level, C_crit, that is threshold or critical to whether local spines become long and thin or short and stubby. The stem resistance increases for C_a < C_crit (subcritical), modeling spine stem elongation, and decreases for C_a > C_crit (supercritical), modeling spine stem shortening, as described by the Harris diagrammatic model.

The system of Eqs. 11–14 is nonlinear because Eq. 14 is nonlinear; Eqs. 13 and 14 constitute a slow subsystem for variables R_ss and C_a. Parameter values for this model are found in Table 1 and in the figure legends. Kinetic and physical parameters for the cable and spine heads are identical to those used in the model for activity-dependent spine densities. We also assume that the dendritic cable has sealed end-boundary conditions and that the dendritic shaft and spines have zero rest potentials. The system was integrated using the semi-implicit method described earlier.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS Calcium-mediated spine... DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
Activity-dependent density

We use Eqs. 8–10 to explore the dynamics of a dendritic cable with activity-dependent spine density. Here, density is constrained to increase slowly in response to current flow toward the dendrite (I_ss > 0), as would occur with excitatory synaptic input, but decreases when current flows from dendrite to spine (I_ss < 0), as would occur in response to local potential changes or to inhibitory input. The effects of changes in spine numbers in both the presence and the absence of voltage-gated channels, and for short, low-resistance spines versus long high-resistance spines will be compared (Table 2).

Figure 3A shows the initial spine density over the cable (left), and the amplitude and time course for spine head potential at various points (V_sh, center) and along the dendrite (V_d, right) during the first 6 ms of the initial activation cycle. The location X = 0.1 is at the center of the synaptically activated region; X = 0.4 is immediately downstream from the activated region; and X = 2.0 is much farther downstream, where it could be considered as the "output" of the region in question.

View larger version (35K): [in this window] [in a new window]   FIG. 3. Repetitive synaptic input to passive spines with activity-dependent spine densities: changes in spine density remain local to the input site. A cable of dimensionless length 3 ( = 179.3 µm) and diameter 0.36 µm, with both ends sealed, has R = 1233 M. (Rm = 2,500 · cm2, Ri = 70 · cm). The dendrite has a uniform distribution of 54 passive spines; spine density (number per ) is initially = 18. Spine resistance is Rss = 1,240 M. Spines near X = 0 (0 X 0.2) are periodically activated every 10 ms with peak conductance 0.074 nS (region indicated by the bar), with Isyn given by Eqs. 6 and 7. A: initially (1st input) the spine density is uniformly distributed (left panel). Potentials in the head (middle panel) and dendrite (right panel) are shown for 3 spatial locations: at X = 0.1 in the middle of the stimulated cluster, and X = 0.4, and X = 2.0, which correspond to unstimulated regions. B: after 600 ms (60 inputs), and with d = 1 pA and = 1.25 sec, the spine density at the input site increases from 18 (3–4 spines) to 20.5 (4 spines), but remains relatively constant outside the stimulus region. C: after 1,200 ms (120 inputs) the spine density increases to 23.2 (gaining only one more spine). Increase in density of the stimulated spines causes only a 1.26-mV increase in head potentials in the spines adjacent to the stimulated cluster, and the output of the dendrite is increased by only 0.13 mV.

Figure 4 graphs the evolution with time of peak head and dendritic membrane potentials, and spine density, at the same 3 spatial locations as in Fig. 3, over 500 activation cycles. The increase in the synaptic potentials in the spine heads and dendritic shaft is roughly linear, especially at distances far from the input site. At X = 2.0 there is little or no response to the stimuli. The increase in spine density is also roughly linear at X = 0.1, with only a negligible change in density downstream for spines with passive membrane. The increase in spines in the stimulated region (from 3.6 to 9 spines over 500 activations), however, is sufficient to have an impact on the amplitude of the passive spread of depolarization to the neighboring spines (compare X = 0.1 and X = 0.4 in Fig. 4, A and B). On the other hand, the local increase in density by itself does not have much effect on the output of the dendrite because the resulting spread of potential down the dendrite, from this increase in synaptically activated spines, leads only to a small 0.07 mV rise in dendritic potential at X = 2.0.

View larger version (23K): [in this window] [in a new window]   FIG. 4. Dynamic stimulus–response curve for slowly changing spine densities (passive spines). For passive spines with activity-dependent densities the peak head and dendritic potentials increase slowly and roughly linearly as the spine density increases. Same parameter values as Fig. 3. Time evolution of the peak head and dendritic membrane potential and spine density is plotted for 3 spatial locations: the middle of the input site X = 0.1 (solid), adjacent to the input site X = 0.4 (dotted), and distant from the site X = 2.0 (dashed). A: rise in peak head potential is nearly linear at all 3 spatial points along the dendrite. However, the rate of increase decreases with distance from the input site. B: peak dendritic potential rises linearly at approximately the same rate as the peak head potentials at the same spatial location, but with smaller peak values. C: after 500 inputs the spine density increases, but only in the vicinity of the input site. Number of spines at the input site increases from about 4 (at t = 0) to 9 (at t = 5,000 ms), but remain at about 18 spines/electrotonic length (e.l.) outside the input region.

ACTIVITY-DEPENDENT DENSITIES: EXCITABLE SPINES. The next series of figures considers spines with voltage-gated channels. The active membrane in the spine heads is modeled with Hodgkin–Huxley kinetics, with I_ion given by Eq. 5. We have explored both the implications of channel density and of the spine stem length on the model's predictions.

In Fig. 5 the spine stem resistance, R_ss, is 1,240 M. At the initial spine density of 18 excitable spines/e.l., no action potential is generated in the stimulated spines after a single stimulus, but the peak amplitudes in the stimulated and adjacent spines are greater as a result of the addition of voltage-gated channels (compare Fig. 3A to Fig. 5A). However, the voltage output of the dendrite is still small.

View larger version (42K): [in this window] [in a new window]   FIG. 5. Repetitive synaptic input to excitable spines with activity-dependent spine densities: spine density increases both inside and outside the input region, forging a path for impulse propagation. A passive cable, with the same geometric and electrical parameters as in Fig. 3, has an initial uniform distribution of 54 excitable spines. However, the excitable spines have Hodgkin–Huxley (HH) membrane in their spine heads; ion channel densities are = 2.5 HH values (HH kinetics for squid at 22°C). As in Fig. 3, the stem resistance is Rss = 1,240 M and spines are activated every 10 ms with a peak conductance of 0.074 nS over the input region 0 X 0.2; indicated by the bars on the time axis. A: initially (1st input) the spine density is uniformly distributed (left). Postsynaptic response is subthreshold in the head (middle) and dendrite (right) as shown for 3 spatial locations. B: after 30 inputs, the spine density in the input region increases from 18 to just over 20, but drops slightly outside the region. This local increase in spine density is sufficient to generate a local action potential at X = 0.1. Impulse propagates to X = 0.4 but then fails. C: after 60 inputs the spine density increases both inside and outside the input region. Propagation is successful to X = 2.0, resulting in a substantial increase in the output of the dendrite at that point, from 1.81 to 23.74 mV.

After 60 stimuli (Fig. 5C), action potential generation in the spine heads extends to at least X = 2.0 because of density increases that reach as far as X = 2.0. The spines downstream are now generating their own membrane current (I_ion) that dominates, on average for each stimulus, the current source from the dendrite. The spines located at X > 2.2 have average stem currents directed the opposite way because these spines have not yet generated their own membrane current, and thus the temporary drop in density there.

The effect of simply assuming the presence of voltage-gated channels in the spine heads may be seen by comparing Fig. 5C with Fig. 3C. Spines are added to the unstimulated region as well as to the input region. The evolving density profile increases maximum spine head potential downstream at X = 0.4 by 46.85 mV, and the maximum dendritic potential farther downstream at X = 2.0 by 23.23 mV, compared with initial activation. The increase in peak head and dendritic potentials is nonlinear at all 3 spatial points (Fig. 6). The curves increase rapidly as the spine density, at the input site, crosses a threshold for the generation of an action potential. Peak head potentials at X = 0.4 (dotted) have greater magnitude than spine heads at X = 0.1 because of an increase in current flowing downstream from the cluster of spines accumulating within the input region.

View larger version (24K): [in this window] [in a new window]   FIG. 6. Dynamic stimulus–response curve for slowly changing spine densities (excitable spines and large Rss). Same parameter values as Fig. 5. Time evolution of the peak head and dendritic potentials, and spine density are plotted for X = 0.1 (solid), X = 0.4 (dotted), and X = 2.0 (dashed). A: rise in peak spine head potential is nonlinear at all 3 spatial points. Rapid increase in potential after a small number of stimuli is attributed to local action potential generation. Voltage peaks in spine heads increase to about 70 mV as the spine densities increase with time. B: peak dendritic potentials. After about 1,000 ms, the peak dendritic potentials at all 3 locations reach values quite close to each other, differing by <10 mV. At 5,000 ms, the output in the dendrite is near 40 mV, a significant increase considering there was little or no output to start with. C: spine density at X = 0.1 increases at a slower rate than in the passive case (compare with Fig. 4C). However if stimulation continues, the spine densities increase outside the input region, whereas in the passive case the densities remain nearly constant.

When very long stimulus times are considered the spine densities at X = 0.1 and X = 2.0 are seen to asymptote to _max (See Fig. 7A). However, at X = 0.4 the spine density grows at a slower rate because of its proximity to the input region. At the input region the spines generate, in unison, a large flux of current that drives current outward through adjacent spine stems. This outward current causes the adjacent spine heads to generate their own action potentials, which act as a counterbalance. The net effect is that the magnitude of the spine stem current (I_ss) of adjacent spines is reduced, slowing down the growth of (see Eq. 10).

View larger version (27K): [in this window] [in a new window]   FIG. 7. Relationships between spine densities and voltage-gated channel densities. Left panels: long-time evolution of the spine density for X = 0.1 (solid), X = 0.4 (dotted), and X = 2.0 (dashed). Right panels: spatial profile of the spine density at 400,000 ms. A: long-time evolution of the spine density for the same parameters as in Fig. 5 with continuous stimulation at 10 ms intervals. Downstream from the input site the spine density approaches max = 100. B: channel densities are reduced by changing in Eq. 5 to = 0.6. This lowers the sodium conductance to 720 pS/µm2 and the potassium conductance to 216 pS/µm2. At the output site (left panel, dashed) the spin density decays to min = 0. The spatial profile shows that after 40,000 input cycles the spine density has decayed to min for all points to the right of X = 1.5, a consequence of repetitive propagation failure near X = 1.5. C: if the maximum spine density is increased to max = 180 the effectiveness of the active channels in propagating action potentials is recovered as well as the downstream growth of spines. As in A the spines downstream approach their maximum value max.

There will be an interaction between the assumed density of voltage-gated channels and the spine density because jointly they determine the total number of channels. Increasing _max allows more spines to be recruited, effectively increasing the density of sodium channels. In Fig. 7C the spine density is allowed to increase to _max = 180. At X = 2.0 (dashed) initially the spine density decays, but it eventually reverses and grows to _max. The reason for this initial decay is that successful action potential propagation to X = 2.0 required over 20,000 activation cycles (simulation not shown) compared with <60 cycles for the higher sodium conductance case (see Fig. 6). Note that even though it takes about 100 times longer to forge a pathway for propagation for the low sodium conductance case, the shape of spine density profiles in Fig. 7, A and C are very similar at t = 400,000 ms.

In model dendrites, when the spines are postulated to have voltage-gated channels, it has been found that propagation of an action potential from spine to spine is precluded if spine stem resistance (R_ss) is either too large or too small. The likelihood of propagation is also affected by the density of spines along the dendrite (Baer and Rinzel 1991). The value we used for R_ss in Fig. 5 did not allow an action potential to propagate for the initial spine density, but promoted propagation at higher densities.

In the next series of simulations, the effect of assuming that spines have lower spine stem resistances is considered. For the channel densities giving a conductance value of = 2.5, if spine stem resistance is 100 M, no active response occurs initially either in the stimulated spine heads or in adjacent ones for these initial conditions. The spine heads at X = 0.1 have a peak potential of 8.65 mV, and peak potential in the dendrite at X = 2.0 is negligible for the first stimulus (Fig. 8A).

View larger version (39K): [in this window] [in a new window]   FIG. 8. Repetitive synaptic input to excitable spines with lower stem resistance. A passive cable, but with spines having excitable membranes as in Fig. 5, has an initial uniform distribution of 54 excitable spines, only now the spine stem resistances are uniformly reduced to Rss = 100 M. For this lower stem resistance the membrane potentials in the head and dendritic shaft are nearly identical. Spines are periodically activated as in Fig. 5. A: initially (1st input), the spine density is uniformly distributed (left panel). Postsynaptic response is subthreshold in the head (middle) and dendrite (right). In the dendrite, the amplitude of the potential at X = 2.0 is indicated (arrow). B: after 60 inputs, the spine density in the input region increases from 18 to just over 20, but this local increase in spine density is insufficient to generate a local action potential at X = 0.1 (cf. Fig. 5B). C: after 360 inputs, the spine density reaches 37 and an action potential is generated at X = 0.1 and propagates to X = 0.4, but fails before reaching X = 2.0. D: after 480 inputs, propagation reaches the target X = 2.0, with spine density increasing adjacent to and to the right of the input region, but decaying farther down the dendrite. Note that the peak of the action potential in the spine heads is greatest at the input site for lower values of Rss. Compare with Fig. 5C, where it is the adjacent spines at X = 0.4 that have the highest peak potentials.

Figure 9 plots peak head and dendritic membrane potentials, and spine density for R_ss = 100 M. Spine density at the input site is higher after 3 s than for the passive case, but the rate of change in density decreases once spines reach threshold for an action potential (cf. Fig. 4). The most significant effect is that the increase in dendritic output takes longer to begin and climbs more slowly than when R_ss = 1,240 M.

View larger version (24K): [in this window] [in a new window]   FIG. 9. Dynamic stimulus–response curve for slowly changing spine densities (excitable spines and lower Rss). Same parameter values as Fig. 8. Time evolution of the peak head and dendritic potential and spine density is plotted for X = 0.1 (solid), X = 0.4 (dotted), and X = 2.0 (dashed). A: rise in peak head potential is nonlinear at all 3 spatial points but the transition to larger values occurs much later than for the large stem resistance case (cf. Fig. 6). B: response curve for the peak dendritic potential is nearly the same as in the head because of the low stem resistance. Although the potential in the dendrite does not increase as dramatically as in the case with higher spine stem resistances, there is a region of the curve between 3,000 and 5,000 ms that the output voltage rises quite steadily with each successive stimulus. C: spine density at the input site increases similarly to the passive case until the density is sufficiently large to generate an action potential. Spine density begins to increase outside the input region after propagation is initiated (between 4,000 and 5,000 ms).

	Calcium-mediated spine restructuring

TOP ABSTRACT INTRODUCTION METHODS RESULTS Calcium-mediated spine... DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
In the model for spine restructuring, Eqs. 11–14, there are 2 slow variables, R_ss and C_a, and the spine density is assumed constant (in time and space). We consider 75 spines uniformly distributed along a cable of electrotonic length 3 (_c = 25).

RESTRUCTURING WITH PASSIVE SPINES.    For the passive spine case, I_ion = V_sh/R_sh in Eq. 12, and as in previous sections we simulate the periodic activation of synapses on the spine heads over the region 0 X 0.2. The frequency of activation, with the concurrent influx of Ca²⁺, is the primary control parameter in contrast to the previous series. Figure 10 illustrates the dynamics of the model at X = 0.1 in response to 3 different frequencies applied to the input region. Initially, there is a uniform distribution of spines and all spines are uniform in structure (i.e., R_ss is constant for all X along the dendritic shaft of electrotonic length 3). At a low frequency of 5 Hz (inputs repeat every 200 ms) there is a small increase in calcium (Fig. 10A). This causes an increase in the stem resistance from initially 750 M to just above 1,000 M (Fig. 10B). Also note that at t = 3,500 ms, when the synaptic stimulation ceases, the stem resistance approaches steady state as calcium decreases to its minimum value.

View larger version (19K): [in this window] [in a new window]   FIG. 10. Repetitive synaptic input to passive spines modifies intraspine Ca2+ levels, which mediate changes in spine stem resistance. A passive cable of electrotonic length 3 (with diameter and R, Rm, Ri the same as in Fig. 3) has a uniform distribution of 75 passive spines (c = 25). Unlike previous simulations, the spine density is kept constant. Initially, the spine stem resistance is Rss = 750 M for all spines. Simulations for 3 input frequencies are shown. Between 500 and 3,500 ms, and between 4,800 and 5,000 ms (indicated by the solid bars on the time axis) spines are periodically activated at 5 Hz (every 200 ms), 50 Hz (every 20 ms), and 125 Hz (every 8 ms). Peak synaptic conductance and Isyn are the same as in Fig. 3. A: calcium response in the center of the input site (X = 0.1) for the 3 different input frequencies. High-frequency input (125 Hz) drives Ca above the critical value of 300 nM (dashed). Calcium remains below criticality for the lower input frequencies (5 and 50 Hz). At 3,500 ms, the input is turned off and calcium returns to its resting value of 5 nM. B: stem resistance response to calcium concentration at the input site. Effect of the 125 Hz input is to initially increase Rss (because Ca < 300), it then peaks at 1,000 ms, which coincides with the critical calcium value, then the resistance decreases rapidly as calcium rises above criticality and the spines collapse into the dendrite. Lower-frequency input (5 and 50 Hz) monotonically increases Rss. Stem resistance rises more rapidly in response to the 50 Hz input. With cessation of activity at 3,500 ms the stem resistance continues to rise, but quickly levels out as Ca approaches its rest value.

For passive spines, the mathematical model predicted that calcium-mediated restructuring would remain largely local. This is illustrated in Fig. 11 for the subcritical input frequency of 50 Hz. Because the dendrite and spines have passive membrane properties, the membrane potential decreases exponentially away from the input site. This is seen in Fig. 11 by comparing peak spine head and dendritic membrane potentials at X = 0.1 and X = 2.0. For example, for each input cycle displayed, the peak potential in the heads or dendrite at X = 2.0 does not exceed 0.6 mV, whereas at X = 0.1 (middle of input region) the peak potential is about 10 mV in the dendrite and >10 mV in the heads. Also note that the time courses for the head and shaft potentials are nearly identical at X = 0.4 and X = 2.0, for each input cycle. Thus the spine stem current I_ss, our measure of electrical activity, is negligible away from the input region [recall I_ss = (V_sh – V_d)/R_ss]. Thus I_ss is near 0 in Eq. 13, forcing C_a to approach C_min outside the input region. The right side of Eq. 14 approaches zero, which explains why R_ss does not change outside the input region in Fig. 11. Thus our simulations for passive spines indicate that, although a spine may restructure as a result of synaptic activation, the restructuring remains in or near the input region and with little change to the electrical response of the system. The spread of potential to neighboring spines is actually decreased by increased spine stem resistance.

View larger version (38K): [in this window] [in a new window]   FIG. 11. Repetitive synaptic input to passive spines with calcium-mediated stem restructuring: changes in spine stem resistance remain local to input site. Passive spines on a passive cable are periodically activated at a frequency of 50 Hz (every 20 ms); all other parameters are the same as in Fig. 10. A: 1st input occurs at 500 ms. Stem resistance is initially 750 M for all 75 spines, which are uniformly distributed over 0 < X < 3 (left). Potentials in the spine head (middle column) and dendrite (right column) are shown for the middle of the stimulated region (solid: X = 0.1), just outside the input region (dotted: X = 0.4) and distal to the region (dashed: X = 2.0). B: after 2,000 ms (75 inputs) the stem resistance in the input region increases from 750 M to just under 1,500 M, but there is little change outside the region. In the stimulated region the peak head potential increases by almost 3 mV, and just outside the region there is a small decrease of <1 mV. There is a negligible change in dendritic potentials. C: after 3,400 ms (145 inputs) the rise in stem resistance remains local and the peak head potential in the stimulated region increases by another 2 mV. However, outside the input region the change in the peak head and dendritic potentials are small. There is actually a decrease in amplitude after the spine stem resistance increases in the stimulated spines.

View larger version (37K): [in this window] [in a new window]   FIG. 12. Repetitive synaptic input to excitable spines with calcium-mediated stem restructuring: stem resistance increases both inside and outside the input region. Excitable spines on a passive cable are periodically activated at a frequency of 20 Hz (every 50 ms). Excitable spines have HH kinetics, as described in Fig. 5. Spines were set to have an initial stem resistance of 750 M. A: initially (1st input at 500 ms) the stem resistance is uniform for all spines (left). Postsynaptic response is subthreshold in the head (middle) and dendrite (right) as shown for 3 spatial locations. B: after 1,400 ms (18 inputs) the stem resistance in the input region increases about 67%, and the output is increased almost 6-fold. Unlike the passive case (cf. Fig. 11) the stem resistance increases also significantly outside the input region. This increase in resistance is sufficient to generate a local enhanced potential at X = 0.1 that propagates to X = 0.4. C: after 3,400 ms (58 inputs), the stem resistance grows to large values inside and outside the input region. Propagation in the heads is successful to X = 2.0, resulting in a significant increase in output of the dendrite at that point, from 4.25 to 31.76 mV.

View larger version (28K): [in this window] [in a new window]   FIG. 13. Dynamic stimulus–response curve for calcium-mediated stem restructuring (excitable spines and Rss initially large). Same parameters and input frequency (20 Hz) as in Fig. 12. All spines have an initial stem resistance of 750 M. Time evolution of the peak dendritic potential, calcium concentration, and stem resistance is plotted for the center of the input region X = 0.1 and distally at X = 2.0. A: between 1,200 and 1,800 ms, the peak dendritic potentials increase rapidly in the input region and distally. At approximately t = 1,500 ms there is a sharp steplike transition for X = 2.0, which indicates that potentials generated in the input region are beginning to affect voltage-gated channels in spine heads in the region near X = 2.0. B: impulses propagating to X = 2.0 (and other distal locations) cause an increase in calcium concentration there. Although calcium concentration increases locally and distally, at this frequency of input the calcium concentration remains below its critical value (300 nM, dashed). C: because calcium remains below criticality, the stem resistance increases for all spines along the cable, and thereby facilitates the generation of spine head action potentials and impulse propagation.

View larger version (22K): [in this window] [in a new window]   FIG. 14. Dynamic stimulus–response curve for calcium-mediated stem restructuring (excitable spines and initially lower Rss). Same parameters as Fig. 13, except initially Rss = 100 M. Time evolution of the peak dendritic potential, calcium concentration, and stem resistance is plotted for X = 0.1, X = 0.4, and X = 2,0. A: transition to larger peak potentials occurs much later than for the case of initially large stem resistances (cf. Fig. 13). B: this delay arises from the slower increase in calcium concentration and stem resistance (C).

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