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Calcium-mediated spine stem restructuring

2005, Mathematical and Computer Modelling

PERGAMON Mathematical and Computer Modelling 0 (2005) 1–0 www.elsevier.com/locate/mcm Calcium-Mediated Spine Stem Restructuring D. W. Verzi∗ Department of Mathematics and Statistics San Diego State University, Imperial Valley Campus 720 Heber Avenue, Calexico, CA 92231, U.S.A. S. M. Baer Department of Mathematics and Statistics Arizona State University Tempe, AZ 85287, U.S.A. (Received and accepted February 2004) Abstract—A spine is a protrusion from the dendritic (or somatic) surface of a neuron. In recent experiments, caffeine-induced calcium released from internal stores was shown to cause elongation of dendritic spine stems in slice cultures. Still another experiment indicates that glutamate-induced increases in calcium may cause spine stem shortening. Harris draws a schematic model to explain these seemingly conflicting results, indicating that a small amount of activity may increase free calcium within the spines and cause spine stem elongation, but an excessive amount of activity may increase intraspine calcium beyond a critical level and cause spine stem shortening (see [1, Figure 2]). This paper develops a mathematical model for a fixed population of spines along the dendrite, each with a dynamic structure and calcium level. The system is integrated over time and space to observe an interdependent relationship between calcium, morphology and chemical/electrical activity. Results of simulation qualitatively capture phenomena observed in recent experiments and exhibit periodic oscillations in potential when the spines have excitable membrane properties by allowing spine structure to transition through threshold geometries for generation of action potentials in a bidirectional manner. As in recent experiments, a variety of chemical and structural profiles emerge, depending on membrane properties, patterns of synaptic input, and initial conditions considered. c 2005 Elsevier Science Ltd. All rights reserved. ° Keywords—Synapse restructuring, Intraspine calcium, Dendritic spines. 1. INTRODUCTION Experimental evidence is mounting that the physical structure of dendritic spines is modifiable in response to chemical and/or electrical activity [2–4]. In response to activity, spine head volume (and membrane surface area) can increase or decrease; spine necks can change shape from long and slender to short and stubby, and a single synapse on a single spine head can transform into *Author to whom all correspondence should be addressed. DWV was supported by NSF DBI-9602226: Research Training Grant—Nonlinear Dynamics in Biology, awarded to the University of California, Davis; and Faculty Grant-in-Aid 242122 from San Diego State University. SMB was supported by NSF DMS-9320597. The authors gratefully acknowledge M.B. Rheuben of Michigan State University for helpful discussions, and Armando Solorzano of El Centro, CA for graphics. c 2005 Elsevier Science Ltd. All rights reserved. 0895-7177/05/$ - see front matter ° PII:00 Typeset by AMS-TEX 2 D. W. Verzi and S. M. Baer a spine with multiple heads and multiple synapses [5,6]. Consequently, populations of spines on dendritic trees are subject to activity-dependent processes. The mechanisms underlying structural changes in spines may be due to one or more cellular influences. One possible mechanism may be the activation of calcium-dependent enzymes as a result of increased calcium levels at the sites of excitatory synapses [7]. Recent experiments implicate intraspine calcium levels as a mediator for changes in dendritic spine structure [8]. Korkotian and Segal monitored and 3-D reconstructed spines in cultured hippocampal neurons over several hours on a confocal laser scanning microscope. Release of calcium from internal stores in response to pulse applications of caffeine induced a small, transient rise in Ca2+ (200−400nM), but caused a significant increase in the length of spine stems in less than 5 minutes [2]. An opposite response was observed in a similar experiment performed by Halpain, Hippolito, and Saffer [8]. They induced a rapid collapse of dendritic spines (also within 5 minutes) by stimulating cultured neurons with glutamate. This caused calcium influx, raising intraspine calcium to much higher levels. Harris proposes a model for spine stem restructuring based on the above experiments [1]. A primary goal of this paper is to formulate a mathematical description of Harris’ model, and to investigate possible interactions between electrical/biochemical activity and structural changes in dendritic spines. The system builds on Wu and Baer’s model [9] for a single spine with an activity-dependent stem conductance, and on the cable model [10] to explore how calciummediated changes in the structure of a population of spines along a dendrite may influence patterns of electrical activity, and how electrical activity due to synaptic events and excitable membrane dynamics may, over time, influence spine structure and calcium dynamics. The model qualitatively recovers chemical and morphological phenomena observed in recent experiments and exhibits periodic oscillations in potential when the spines have excitable membrane properties. As in recent experiments, a variety of chemical and structural profiles emerge, depending on membrane properties, patterns of synaptic input, and initial conditions considered. The equations are formulated in Section 2 and tested in simulation in Section 3, followed by Section 4, Discussion. A table of parameters, summary of the model, and stability analysis may be found in the Appendix. 2. METHODS Confocal videos from slice culture experiments confirm that activity-dependent processes regulate the structure of the dendrite in hippocampal neurons [1]. A small amount of electrical/chemical activity increases a spine’s level of free calcium within the cytosol, and causes the spine stem to elongate. However, a higher level of activity may cause calcium influx and induce spine stem shortening or loss, perhaps due to actin depolymerization [1]. A mathematical description of these observations may be built upon cable theory, a system of partial differential equations which permits the density of spines, membrane potentials, and stem resistance to vary continuously in space and time [10]. Since the length and shape of the spine stem is correlated to its resistance and ultimately to stem current flow, it is of interest to explore how this model for a continuum of spines may be adapted to simulate Harris’ model (see [1, Figure 2]). 2.1. The Cable Model The electrical potential Vd (X, t) in a passive dendrite of electrotonic (dimensionless) length L = l/λ, studded with n̄ spines per unit length satisfies the cable equation, τm ∂ 2 Vd ∂Vd = − Vd + R∞ n̄Iss , ∂t ∂X 2 (1) see [10], where Iss is the spine stem current, τm is the membrane time constant, and R∞ is the cable input resistance. The fixed spine density (n̄) represents the average number of spines per Spine Stem Restructuring 3 Figure 1. Schematic diagram of a single dendritic spine. A spine is a protrusion from the dendritic (or somatic) surface of a neuron. Dimensions vary, with typical ranges that include stem lengths of order 1.0µm, diameters of order 0.1µm and head surface areas of order 1.0µm2 . Isyn denotes input current into the spine head. Spines may be modelled with passive or excitable membrane properties by varying the formulation for Iion . In this paper, the magnitude of the stem current (|Iss | = |Vsh − Vd |/Rss ) measures local activity (regardless of direction), and stem resistance (Rss ) measures spine structure, decreasing (to model stem shortening) when calcium is above a critical value, and increasing (to model stem elongation) when calcium is below that value. unit length at X = x/λ. For simulations in this paper, λ = 180µm. (For a detailed derivation of the cable equation, see [10].) An equation for the membrane potential in each spine head is obtained from a current balance relation for the capacitive, ionic, synaptic and spine stem currents given by Csh ∂Vsh = −Iion − Isyn − Iss . ∂t (2) See [11]. (See Figure 1 for the schematic of a single spine.) The spine stem current is computed as an Ohm’s law voltage drop over the stem resistance, Iss = Vsh − Vd . Rss (3) See [11]. The term Iion represents the ionic currents passing through the head membrane and Isyn the synaptically applied current. In a simulation involving passive spines Iion = Vsh /Rsh , but if the spines have excitable membrane properties, Hodgkin-Huxley kinetics (set to 22o C) simulate voltage-dependent ion channel currents, ¡ ¢ Iion (Vsh , X, t) = γAsh (Vsh − VN a ) ḡN a m3 h + (Vsh − VK ) ḡK n4 + (Vsh − VL ) gL . (4) See [12]. A table of parameters may be found in the Appendix. Synapses over an interval X0 ≤ X ≤ (X0 + ∆X) are activated periodically by applying to the spines in that interval, Isyn (Vsh , X, t) = gsyn (X, t) (Vsh − Vsyn ) , (5) 4 D. W. Verzi and S. M. Baer where Vsyn is the synaptic reversal potential and gsyn is a brief synaptic conductance generated by the α-function t gsyn (X, t) = gp e(1−(t/tp )) . (6) tp See [11]. In this paper, spines in the interval 0.0 ≤ X ≤ 2.0 are activated every T = 10ms. 2.2. DYNAMIC MORPHOLOGY The model now breaks from cable theory [10] to view the spine stem current (Iss ) as an important measure, over time (minutes to hours), of the electrical activity between the spine head and dendritic base. The following subsystem, appended to the cable model, explores the possibility that this electrical interaction controls slow, local changes in spine structure and calcium level, ∂Ca = ǫ1 (η |Iss | − ρ) (Ca − Cmin ) , ∂t ∂Rss = −ǫ2 (Ca − Ccrit ) (Rss − Rmin ) (1 − Rss /Rmax ) . ∂t (7) (8) Free intraspine calcium (nM) changes proportional to activity (regardless of direction), relative to a minimal level, as measured by (η|Iss | − ρ), increasing when |Iss | > ρ/η and leaking away slowly when |Iss | < ρ/η. The ratio of spine stem resistance to input resistance (Rss /R∞ ) controls how electrically connected the spine is to the dendrite [4,10,11]. In the model, R∞ is fixed, while Rss varies over time and space, so that the difference between spine head and base potentials (Vsh − Vd ) becomes negligible as Rss → Rmin , imposing a kinetic upper bound on calcium (i.e., a |Iss | = |Vsh − Vd |/Rss becomes small enough so that ∂C ∂t is negative). The factor (Ca − Cmin ) prevents calcium from becoming negative in areas receiving no activity over a sustained period of time. The stem resistance Rss is generally computed as the ratio of the specific cytoplasmic resistance to the cross-sectional area, integrated over the length of the stem [11]. Stem resistance has previously been used as a measure for stem structure [9,13,14], since spines with long, narrow stems have a higher resistance to current flow than spines with short, broad stems. Equation (8) models slow, bounded changes in spine structure as measured by stem resistance. Activity-dependent calcium regulates changes in stem resistance, and a critical intraspine calcium level (Ccrit ) controls the direction of change, decreasing for Ca > Ccrit , modelling spine stem shortening, and increasing for Ca < Ccrit , modelling spine stem elongation, as in Harris’ model [1]. A low level of synaptic input into spines, over an extended period of time, may increase calcium levels, and the resulting spine stem lengthening may lead synaptic input to depolarize a spine with excitable membrane properties. Threshold conditions for action potential (AP) generation in the spine heads is sensitive to conductance loading provided by the spine stem [11]. Thus, in the model, a fixed density of spines (n̄) with little free calcium and below threshold for AP may elongate from a small amount of periodic synaptic input over time and reach threshold for generating an AP (See Figures 2a and 2b). The increased stem current from repetitive spiking in each cycle of activity will ramp-up the level of internal free calcium until it passes Ccrit , and the spine stems begin to shorten over time, as in Figures 2c and 2d. When the spine stem no longer provides the necessary conductance loading, APs will cease, and activity will fall below a minimum level (η|Iss | < ρ). If periodic synaptic input continues over time at a low level, the calcium will again fall below Ccrit , and the spine stems will elongate, repeating the cycle of morphological adaptation. Thus, for spines with excitable membrane, the model can exhibit burst oscillations between periods of active and silent phases, allowing calcium to modulate signaling within the dendritic tree. A summary of the fast and slow system, including initial and boundary conditions, along with a table of parameters may be found in the Appendix. Spine Stem Restructuring 5 Figure 2. Morphological adaptation allows the spines to respond to, and regulate levels of local activity. Spine stem structure is related to the amount of free calcium present in the spines, and changes in calcium levels are proportional to the amount of local activity. (a) A small amount of synaptic activity may raise calcium levels slightly, causing spine stems to elongate, and stem resistance (Rss ) to increase. (b) If synaptic activity persists over time, calcium may continue to increase beyond a critical amount. (c) If the spines have excitable membrane properties, increased stem length (resistance) may depolarize the spine head beyond threshold for generation of an AP. (d) In any case, increased levels of synaptic activity, or AP responses increase calcium beyond a critical level for the spines, and stems begin to shorten, lowering stem resistance. (e) Stems may be partially or completely retracted into the dendrite, so that synaptic input is no longer directed at an isolated compartment, reducing local activity and calcium levels. When stem resistance falls below threshold for depolarizing excitable spines, AP generation ceases, lowering the level of activity and decreasing the amount of calcium present. If low-level activity continues, over time, calcium will increase and stems will elongate, to enter another cycle of morphological adaptation. Figure modified with permission from Proc. Natl. Acad. Sci. (1999) 96-22 p. 12214. Copyright 1999 National Academy of Sciences, U.S.A. 3. RESULTS Of initial interest is how the above system may affect local activity and structure for spines with passive membrane properties. Figure 3 compares the results for two separate simulations of 63 spines, located uniformly along a dendrite of electrotonic length L = 3 (n̄ = 21 spines/e.l.), for two different levels of synaptic activity. The spines at 0.0 ≤ X ≤ 0.2 are activated every 10ms with Isyn (Equation (5),(6)), with peak synaptic input (gp from equation (6)) occurring at tp = 0.2ms in each activation cycle. Since the spine head membrane is considered to be passive, Iion = Vsh /Rsh . For the first simulation, calcium and stem resistance are initially uniform at Ca = 800nM (above Ccrit = 300nM) and Rss = 1600MΩ, respectively. Peak synaptic activation reaches gp = 0.074nS in each cycle, modelling a weaker synapse. Figure 3 (left), shows a time course for head potential, 6 D. W. Verzi and S. M. Baer 60 Vsh (mV) Vsh (mV) 60 5 0 t (ms) 500 5 0 500 t (ms) (a) Ca (nM) 1000 Ca (nM) 1000 Ccrit 0 0 t (ms) 500 0 Ccrit 0 t (ms) 500 (b) 2000 Rss (MΩ) Rss (MΩ) 2000 400 0 t (ms) 400 500 0 t (ms) 500 (c) Figure 3. Intraspine calcium levels depend on activity, and mediate structural change. Sixty-three spines with passive membrane properties are located uniformly along a dendrite of electrotonic length L = 3. Spines over 0.0 ≤ X ≤ 0.2 are synaptically activated every 10ms with Equation (5),(6), for two separate simulations. Time courses are shown for head potential, free intraspine calcium and stem resistance over 50 cycles of activation at X = 0.0. Graphs on the left show results for a simulation with peak synaptic activation (gp ) reaching 0.074nS in each cycle, and initial values for calcium, and stem resistance uniformly set to Ca = 800nM (above Ccrit ) and Rss = 1600MΩ, respectively. Graphs on the right are for a simulation with gp reaching 0.37nS in each cycle, and initial values for calcium and stem resistance uniformly set to Ca = 200nM (below Ccrit ) and Rss = 1000MΩ, respectively. (a) Maximum values for Vsh in each cycle are proportional to gp , and also to Rss , since current flows out of stimulated spines more rapidly when resistance is low. (b) Calcium decreases toward a lower bound (Cmin = 0.0) when η|Iss | − ρ < 0.0 (left) and increases when η|Iss | − ρ > 0.0 (right). (c) Stem resistance decreases when Ca > Ccrit , and increases when calcium is subcritical. intraspine calcium and stem resistance for the spines under synaptic activation at X = 0.0 over 50 activation cycles. Calcium decreases throughout the simulation (approaching Cmin = 0.0), since the stem current is less than ρ/η. When Ca > Ccrit , during the first 15 cycles, Rss decreases to a minimum of 950MΩ, modelling spine stem shortening, and then increases during the remaining Spine Stem Restructuring 7 35 cycles (approaching Rmax = 1800MΩ) when Ca < Ccrit , modelling spine stem elongation (Figures 3b and 3c). In Figure 3a, the maximum amplitude for Vsh in each cycle of activation is proportional to Rss , since current flows more rapidly from spines when stem resistance is low. For the second simulation, calcium and stem resistance are initially uniform at Ca = 200nM (below Ccrit = 300nM) and Rss = 1000MΩ, respectively, and gp = 0.37nS, modelling a higher level of synaptic activity. Figure 3 (right) again shows a time course for head potential, intraspine calcium and stem resistance for spines under synaptic activation over 50 applications of Isyn . Now, calcium increases throughout the simulation, reaching a critical level after 25 activations. Now, stem resistance increases until Ca crosses Ccrit from below, and then decreases for the remainder of the simulation. 3.1. Spines with Excitable Membrane Properties Since 90% of excitatory synapses terminate on dendritic spines [4], this section explores the effects that calcium-mediated dynamic morphology may have upon the generation and propagation of APs when the spines have excitable membrane properties. In the next simulation, average spine density is fixed uniformly at n̄ = 21 excitable spines/e.l., with initial uniform values of Rss = 800MΩ and Ca = 100nM. Spines are synaptically activated every 10ms with a weaker input as in the second passive simulation (gp = 0.074nS), but now the spines can generate and propagate an AP when properly stimulated, using Hodgkin-Huxley kinetics [12] to model ionic current in the spine heads (equation (4)). Figure 4 graphs a time course for 150 activation cycles for head potential, calcium and stem resistance under synaptic input. For a fixed density of spines, an excitable response depends on the stem resistance [11], now a dynamic variable. The initial value for Rss is below threshold to generate an AP for 21 spines/e.l., so that peak head potential is less than 20mV for the first 10 cycles of activation (Figure 4a). Calcium is initially less than Ccrit and decreases during this “silent” phase (absence of APs), so that resistance increases (stems elongate) at X = 0.0 (Figures 4b and 4c). When stem resistance reaches a threshold value of approximately 1000MΩ, the spines begin firing APs, causing calcium to increase. Stem resistance and peak values for head potential and stem resistance continue to rise, until Ca = Ccrit . The “active” phase of repetitive spiking in head potential continues until t = 800ms, as resistance slowly decreases (stems shorten), for calcium above a critical level. Calcium reaches a peak value of 820nM just before activity begins a second silent phase. Then, calcium and resistance decrease until Ca < Ccrit again at t = 1000ms. The results from Figure 4 raise several questions. Will the bursting pattern for active and silent phases continue over time, and what effect does this pattern of impulses have upon the spines “downstream” from the stimulated region? Figure 5 compares results under synaptic activation and downstream for 500 cycles of activation for the same simulation shown in Figure 4. Figures 5a and 5b graph head and dendritic potentials over time at X = 0.0 (left) and X = 1.0 (right). Figure 5c shows a state space diagram for the slow variables of calcium and resistance over the same period of time at these same locations. Let us first consider what is happening in the stimulated region (X = 0.0 left). In Figure 5a, the initial resistance is just below threshold for generation of APs, so that η|Iss | < ρ and calcium decreases while Rss increases since Ca < Ccrit (Quadrant 1, Figure 5c) until spines in the stimulated region begin firing APs. Since η|Iss | > ρ (on average over each activation cycle) in the presence of APs, both measured quantities continue to rise (Quadrant 2) until Ca = Ccrit and Rss is approximately 1770MΩ. Spiking continues while Ca increases and Rss decreases (Quadrant 3) until stem resistance once again falls below threshold for AP generation (approximately 1000MΩ) and Ca reaches its peak value of approximately 820nM after 80 cycles of activation (c.p., Figure 5c to Figure 4b). A silent phase then begins in the absence of AP generation, as both measured quantities decrease (Quadrant 4) until Ca = Ccrit and Rss is approximately 570MΩ. Once Ca < Ccrit , resistance begins rising toward threshold, and a period of high level activity begins 8 D. W. Verzi and S. M. Baer Vsh (mV) 80 10 0 1500 t (ms) (a). Potential is proportional to stem resistance, initially low for 10 synaptic activations (c.p., to Figure 3). An active phase of 80 APs begins when stem resistance for spines in the stimulated region crosses threshold for AP generation (in (c)). A silent phase of 40 cycles (400ms) begins when Rss falls below threshold. Ca (nM) 1000 Ccrit 0 0 t (ms) 1500 (b). Calcium decreases during silent phases in head potential, and increases to a maximum of 820nM after 90 cycles of synaptic input when spiking terminates. Calcium returns to a minimum of 50nM over the following 400ms, before increasing again when spiking returns to the spines under synaptic activation. Rss (MΩ) 2000 400 threshold 0 t (ms) 1500 (c). Stem resistance increases for 52 cycles of synaptic input when Ca < Ccrit , asymptotically approaching Rmax = 1800M Ω. Stem resistance decreases for the next 60 cycles when Ca > Ccrit , reaching a minimum value of 580MΩ after 112 cycles, before beginning to rise again. Figure 4. Calcium mediates bidirectional changes in structure and activity when the spines have excitable membrane properties. Sixty-three excitable spines are uniformly located along a dendrite of electrotonic length L = 3. Ionic current in the spine heads is modelled with equation (4)). Initial values are uniformly Ca = 100nM and Rss = 800MΩ (below threshold for n̄ = 21 spines per unit length). Spines are synaptically activated with a low level of input as in Figure 3 (right) (gp = 0.074nS), and time courses are shown for head potential, free calcium and stem resistance over 150 activation cycles at X = 0.0. Spine Stem Restructuring X=0.0 9 X=1.0 80 Vsh (mV) Vsh (mV) 80 10 0 t (ms) 5000 10 0 t (ms) 5000 0 t (ms) 5000 (a) 80 10 Vd (mV) Vd (mV) 80 0 t (ms) 5000 10 (b) Ccrit Rss (MΩ) 2 threshold * i.c. 0 i.c. * 4 1 400 2000 3 Rss (MΩ) 2000 Ca (nM) 1000 400 0 Ca (nM) 100 (c) Figure 5. A cycle for structural change induces periodic spiking in spines with excitable membrane properties. Time series for head and dendritic potential are shown over 500 activation cycles (in (a) and (b), respectively), along with a state space diagram for calcium and stem resistance (in (c)) at two different locations, for the same simulation in Figure 4 (gp = 0.074nS). Results are shown for X = 0.0, under synaptic activation (left), and for X = 1.0, downstream (right). (a) Alternating periods of active and silent phases in AP generation and propagation occur over time as Rss increases and decreases through threshold values. The magnitude of the silent phases at X = 1.0 indicates that synaptic input for 0 ≤ X ≤ 0.2 is severely attenuated by rising stem resistance for X > 0.2. (b) Periods of high and low potential at the base of the spines are graphed over the same time evolution. The magnitude of the silent phases at X = 1.0 demonstrates that synaptic input (in the absence of APs) is severely attenuated by the (fixed) cable input resistance (R∞ ). (c) State diagrams show the relationship between calcium and stem resistance over the same time period. Behavior is cyclic under synaptic activation (left) as Rss crosses threshold and Ca crosses Ccrit . Downstream (right), morphology tends to (Cmin , Rmax ) (long spines with minimal free calcium), due to attenuated activity during silent phases and a persistent rise in stem resistance. 10 D. W. Verzi and S. M. Baer again after 120 applications of Isyn (Figure 5a). The potential in the dendrite at the base of the spines (Vd ) is graphed in Figure 5b to compare differences from X = 0.0 to 1.0. Downstream from synaptic activation, at X = 1.0, there are several noticeable differences in the graphs. The shapes of the active and silent phases are different, the magnitude for potential during silent phases is much smaller in both the spines and the dendrite, and the phase plane indicates that (Ca , Rss ) is tending to (0.0, Rmax ). While the spines downstream are brought to threshold by APs firing in the stimulated region, the edges of the active phases are sharper (c.p. left), an indication of the “all or nothing response” that occurs in the absence of synaptic input, i.e., spines at X = 0.0 have a few cycles where they are generating APs just below threshold for propagation on the leading and trailing edges of each active phase. Potentials in both the spines and the dendrite are greatly reduced during silent phases (c.p., left) since current flowing down the dendrite is attenuated by increasing values for Rss . At the end of the initial silent phase, resistance at X = 1.0 is approximately 1200MΩ, but approaches Rmax by the end of the second silent phase (Figure 5c right). Calcium is initially less than Ccrit , and decreases throughout the simulation, causing Rss to increase toward its upper bound, further isolating the spines downstream and locally reducing |Iss |. There are brief periods of time in each AP response when Ca increases slightly (note the “noise” in Figure 5c), but on average, calcium decreases and resistance increases over each 10ms-cycle, so that (Ca , Rss ) → (0.0, Rmax ) , at X = 1.0. A stability analysis for critical points in the slow subsystem is included in the Appendix. 4. DISCUSSION This paper considers one way to model the interdependence of activity and morphology in the dendrite by theoretically studying specifically observed phenomena. Intraspine calcium levels respond to local changes in activity and mediate spine structure, as measured by stem resistance. The model qualitatively captures chemical and morphological phenomena observed in recent experiments. Intraspine calcium levels decrease in response to weaker synaptic activation (gp = 0.074nS) and increase in response to stronger synaptic inputs (gp = 0.37nS) or AP generation in the spine heads. Stem resistance increases (modelling stem elongation) when calcium is subcritical, and decreases (modelling stem shortening) when calcium is supercritical. The model predicts a higher percentage of spines more electrically connected to the dendrite (shorter spine stems) in areas receiving a sustained high level of activity. Results also predict that areas of the dendrite experiencing a sustained lack of electrical activity (whether synaptic activation or current flowing along the dendrite) will have a higher percentage of spines with long, narrow stems. The result that stem resistance increases monotonically outside the stimulated region (see Figure 5c) is consistent with experimental observations that dendrites are more spiny when synapses are inactivated, possibly to compensate for lost activity [15]. The length of time for chemical and structural transition depends on the magnitudes of ǫ1 and ǫ2 , respectively, selected here for computational efficiency. To achieve the changes shown in this paper over five to ten minutes, as observed in recent experiments [2,8], one would need to set the rates of change in calcium and resistance to 10−5 , and 10−6 , respectively (three orders of magnitude lower than values used for this paper.) By allowing the morphology of the dendrite to transition through identified threshold geometries for generating an AP, new pathways are created for wave propagation when the spines are modelled with excitable membrane properties. Results shown here are for a dendrite of physical length 3λ, where λ is approximately 180µm. This means that the signal propagation to X = 1.0 shown in Figure 5 represents a wave of APs extending to a physical length of x = 0.18mm, Spine Stem Restructuring 11 resulting from synaptic activation (Isyn ) of five synapses located between x = 0.0 and 0.036mm, along with the evolving structural profile from previous synaptic events. In this paper, η and ρ were selected so η|Iss | − ρ was negative for passive spines receiving low levels of synaptic input, and positive for passive spines receiving higher levels of activation (using gp as a measure of magnitude). Using the same values for η and ρ and the lower value for gp , the model can create and then dissolve a path for impulse propagation when the spines have excitable membrane properties, by allowing stem resistance for activated spines to transition through identified threshold values for generating an AP. For spines with excitable membrane properties, η|Iss |−ρ is positive in the presence of low-level synaptic input and membrane response APs, and negative in the absence of either of these two quantities. The interdependent changes in activity and structure presented here kinetically limit the time period for sustained higher levels of activity, and prevent calcium from reaching a toxic level for the neuron. Shortening the stems causes spines to become more electrically connected to the dendrite, thereby reducing activity and calcium levels in an isolated compartment of the cell. Restructuring neural circuitry modulates signaling, and may also serve as a mechanism to prevent necrosis. The simple proportional relationship between calcium and activity, presented here as a first step in the modelling process, cannot capture all of the dynamic processes involved in a system with activity-dependent morphology. Nevertheless, the strength of this model is that, like the experimental studies, it helps to dissect the complex phenomenon of these processes. Using a continuum model allows us to exhibit a variety of morphologies and membrane properties with just a few differential equations. 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Rall, Computational study of an excitable dendritic spine, J. Neurophysiol. 60, 499–523, (1988). 12. A. Hodgkin and A. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. (Lond.) 117, 500–544, (1952). 13. R. Kuske and S.M. Baer, Asymptotic analysis of noise sensitivity in a neuronal burster, B. Math. Biol. 64, 447–481, (2002). 14. D.W. Verzi, Modelling activity-dependent synapse restructuring, B. Math. Biol., (October, 2003). 15. S.A. Kirov and K.M. Harris, Dendrites are more spiny on mature hippocampal neurons when synapses are inactivated, Nat. Neur. 2 (10), 878–883, (2000). 12 D. W. Verzi and S. M. Baer 16. P. Waltman, A Second Course in Elementary Differential Equations, Academic Press, Inc., Orlando, FL, (1986). 5. APPENDIX 5.1. Table of Parameters Ri specific cytoplasmic resistivity Rm passive membrane resistance Cm specific membrane capacitance R∞ input resistance λ length constant τm membrane time constant d 70Ω·cm 2500Ω·cm2 1µF cm2 Rm /(λπd) p Rm d/4Ri Rm C m diameter of the dendrite 0.36µm Ash surface area of each spine head 1.31µm2 Csh capacitance of each spine head Ash Cm γ channel density 2.5 VN a sodium reversal potential VK potassium reversal potential 115mV VL leakage reversal potential 10.5989mV Vsyn synaptic reversal potential 100mV gN a maximal sodium conductance 120mS/cm2 gK maximal potassium conductance 36mS/cm2 gL maximal leakage conductance 0.3mS/cm2 gp peak synaptic conductance tp time to peak in each activation cycle T time between synaptic activations -12mV (see figure legends) 0.2ms 10ms R∞ input resistance of the dendrite 1233MΩ Rsh resistance of each spine head 1.02x1011 Ω Rmax stem resistance upper bound 1800MΩ Rmin stem resistance lower bound 500MΩ ǫ1 rate of change in calcium 0.01 ǫ2 rate of change in stem resistance 0.001 Ccrit critical intraspine calcium level 300nM ρ measure of minimal local activity 1.0 η scaling parameter for stem current 1.0x109 5.2. Summary of Equations ∂Vd ∂t ∂Vsh Csh ∂t ∂Ca ∂t ∂Rss ∂t τm = ∂ 2 Vd − Vd + R∞ n̄Iss , ∂X 2 (9) = −Iion − Isyn − Iss , (10) = ǫ1 Ca (η|Iss | − ρ) , (11) = −ǫ2 (Ca − Ccrit )(Rss − Rmin )(1 − Rss /Rmax ). (12) Calcium is bounded below by Cmin = 0.0, with a critical intraspine calcium level, Ccrit = 300nM. Spine stem structure is modelled by Rss , increasing for Ca < Ccrit and decreasing otherwise. Stem resistance is bounded between Rmin = 500MΩ and Rmax = 1800MΩ. The density of Spine Stem Restructuring 13 spines per unit length (n̄) is fixed over time, but may vary in X. Both ends of the dendrite are sealed, so that the voltage gradient is always zero at the boundaries, with a resting potential of zero in the dendritic shaft and the spines, i.e., ∂Vd (0, t) = 0.0, ∂t ∂Vd (L, t) = 0.0, ∂t Vsh (X, 0) = 0.0, Vd (X, 0) = 0.0. (13) (14) (15) (16) If the spines are passive, then Iion = Vsh /Rsh , but if the spines have excitable membrane properties, the ionic current is modelled using Hodgkin-Huxley kinetics [12] ¡ ¢ Iion (Vsh , X, t) = γAsh (Vsh − VN a ) ḡN a m3 h + (Vsh − VK ) ḡK n4 + (Vsh − VL ) gL (17) with increased channel densities [10]. Synaptic input is simulated by applying Isyn (Vsh , X, t) = gp t (1−(t/tp )) e (Vsh − Vsyn ) tp (18) every 10ms. The system comes to full rest between activations, since synaptic input reaches its maximum value at tp = 0.2ms. Equations are integrated using a semi-implicit Crank-Nicholson/Adams-Bashforth finite differencing method, verified against a fully explicit method. For stability, ∆X = .04 and ∆t = .005. Since a significant change in spine structure has been observed over minutes to hours and individual APs are on a time scale of milliseconds (ms), the computation time for a simulation could be on the order of hours. To identify values for ǫi ≪ 1 (i = 1, 2) that reduce computation time, but preserve the basic dynamics of the system as ǫi → 0.0, results were compared during corresponding cycles for different simulations, varying ǫi . For example, results for the time course of a propagating wave of APs over 60 cycles of synaptic events using ǫi are superimposed over results of every other cycle from a second simulation, using ǫi /2. As ǫi is successively reduced, the animations converge, and ǫi is chosen to be computationally efficient, yet sufficiently small to preserve the qualitative structure of results for longer time periods as ǫi → 0.0. 5.3. Stability Analysis for the Slow Variables The equations for change in calcium and resistance constitute a slow subsystem. Fixing X at one point along the dendrite, insight may be obtained into the stability of stationary points at that location by considering the average contribution of the fast variables to the measure for change in the slow system over each cycle of length T , avg |Vsh − Vd | = 1 T Z Ti |Vsh − V d| dt. (19) Ti−1 Numerical simulations indicate that |Vsh − Vd |avg is approximately piecewise constant for a fixed location X, with c1 representing the value for active phases of bursting, and c2 the value during silent phases. Further, the response of the system over the synaptically activated region behaves as though it were space-clamped, i.e., the state variables are independent of X in that region. Thus one may average over the fast variables (activity) since the slow variables (Ca and Rss ) are relatively constant within activation cycles. In general, c1 and c2 are different, but insights into 14 D. W. Verzi and S. M. Baer the dynamics of the slow system may be obtained by setting c1 = c2 and averaging over the fast system to obtain the following ordinary differential equations for dynamic morphology, ¶ µ avg |Vsh − Vd | dCa = ǫ1 η − ρ (Ca − Cmin ) dT Rss dRss = −ǫ2 (Ca − Ccrit ) (Rss − Rmin ) (1 − Rss /Rmax ) . dT (20) (21) This system has critical points at ∗ )= (Ca∗ , Rss ¶ µ η|Vsh − Vd |avg , Ccrit , ρ (Cmin , Rmax ), and (Cmin , Rmin ). To first order, the stability of these points in the non-linear system is related to the eigenvalues of the operator matrix (Jacobian) for the corresponding linearized system, evaluated at the fixed points [waltman]. The generalized Jacobian for the above system is ¶ µ   ǫ1 (Ca∗ − Cmin )|Vsh − Vd |avg η|Vsh − Vd |avg −ρ − ǫ1 ∗ ∗   Rss Rss (22)   ǫ ǫ2 2 ∗ ∗ ∗ ∗ (Rss − Rmin ) (Rmax − Rss ) − (Ca − Ccrit )(Rmax + Rmin − 2Rss ) − Rmax Rmax Evaluating the Jacobian at the point ∗ (Ca∗ , Rss ) ¶ µ η|Vsh − Vd |avg , = Ccrit , ρ yields    0 − ǫ2 ∗ (R∗ − Rmin ) (Rmax − Rss ) Rmax ss −  ǫ1 ρ2 (Ccrit − Cmin ) η 2 |Vsh − Vd |avg   0 (23) The trace of the matrix is zero and the determinant is negative, so that the eigenvalues are real and of opposite sign. Since one eigenvalue is positive, independent of the location along the dendrite, this critical point is unstable in the linear system, and, therefore, unstable at each location X in the nonlinear system [16]. For the second fixed point, (Cmin , Rmax ), the Jacobian is ¶   µ η|Vsh − Vd |avg − ρ 0 ǫ   1 Rmax (24)   ǫ2 0 − (Cmin − Ccrit )(Rmin − Rmax ) Rmax with eigenvalues along the diagonal. The sign of (a22 ) is always negative since Cmin < Ccrit and Rmin < Rmax , but the sign of the (a11 ) depends on the level of local activity when Rss = Rmax . Excitable spines at 0.0 ≤ X ≤ 0.2 are spiking in response to synaptic activation when Rss = Rmax , so that (a11 ) > 0 in the presence of local synaptic activation and AP generation. Therefore, the fixed point is unstable for synaptically activated spines. However, spines downstream are spiking in response to activity flowing along the dendrite so that the difference in local head and dendritic potentials is, on average, smaller in the absence of synaptic activation. For spines downstream, η|Vsh − Vd |avg < ρ/Rmax , for parameter values in this paper. Therefore, (a11 ) < 0.0 and the fixed point is stable. A similar analysis for the remaining critical point indicates that (Cmin , Rmin ) also has eigenvalues along the diagonal with (a11 ) as above, but (a22 ) = −ǫ2 (Cmin − Ccrit )(Rmax − Rmin )/Rmax > Spine Stem Restructuring 15 X=0.0 2000 Rss (MΩ) Init. Cond (Ca,Rss) (800,800) (100,800) (400,900) 400 0 Ca (nM) 1200 Figure 6. Closed-curve behavior is persistent for the morphological system under synaptic activation. Additional trajectories are superimposed for the same simulation as in Figure 5c (gp = 0.074nS) at X = 0.0, varying the initial conditions for calcium and resistance to observe the behavior of the system under synaptic activation. The dark, solid trajectory repeats the simulation from Figure 5 over 500 activations. The dotted outer trajectory also runs for 500 cycles. The dashed inner trajectory begins near the center of the region and runs for 1200 synaptic activations. 0.0, since Rmax > Rmin . The Jacobian has at least one positive eigenvalue, independent of location along the dendrite, and this point is always unstable. In summary, with the above assumptions at each location X under synaptic activation, the system has no stable fixed points in a bounded region of the plane, since Rmin ≤ Rss ≤ Rmax , Cmin ≤ Ca , and calcium is kinetically bounded above by the periodic behavior of activity, for ∗ ) inside this region permits the chosen parameter values. Removing the unstable point (Ca∗ , Rss application of the Poincare-Bendixson theorem [16] to conclude that trajectories in this region would be either periodic, or have an omega limit set that is periodic. In Figure 6, three results are plotted for the same simulation as in Figure 5 (smoothing the trajectories for comparison) with different initial conditions for calcium and resistance. The outer two trajectories were run for 500 activation cycles, and the inner trajectory (dashed) was run for 1200 cycles. Since, in general, c1 6= c2 in equation (7),(8), the trajectories are not exactly periodic over time. Fixing X at a point not under synaptic activation, the above system has only one stable fixed point. Unless the system is started exactly at one of the other two critical points, trajectories will tend to (Cmin , Rmax ) over time (see Figure 5c, right).