Postsynaptic calcium dynamics in cytoplasm and in stores are expressed by Kusters et al. (2005):

Here, I_NMDAR is the calcium current through NMDARs, J_IP3R denotes calcium flux from the ER to the cytoplasmic space through IP₃Rs, J_SERCA is the calcium flux pumped from the cytoplasmic space into the ER, J_IKER is the leak of calcium ions from the ER to cytoplasmic space, Z_Ca is calcium ion valence, F is Faraday's constant, Vol_Cyt is the volume of synapses, Vol_ER is the volume of stores, A_ER is the surface area of the store exposed in synapses, [Ca2+]ER and [Ca2+]Cyt are calcium concentrations of ER and cytoplasmic region, respectively, [Ca2+]basal is the basal concentration of calcium in cytoplasmic region and the decay constant is τ_Ca. The values of the different parameters in equations (1, 2) using in this paper are furnished in the Table 1.

I_NMDAR is given by Shouval et al. (2002)

where V is the postsynaptic potential and glutamate is released at a time t = t_glut. Here P₀ is the fraction of NMDARs that change to the open state after glutamate binding and G_NMDAR is the conductance of the open NMDARs from calcium ions. θ₁ is a step function: θ₁(t) = 1 when t ≥ t_glut, and θ₁(t) = 0 when t < t_glut. Glutamate unbinding is characterized by two decay constants τglutf and τgluts, and the fast and slow unbinding components Iglutf and Igluts. The term H(V) describes Mg²⁺ unblock due to changes in the postsynaptic potential and is modeled by

where E_Ca is the reversal potential of calcium, V is the postsynaptic membrane potential, and [Mg²⁺] is the external magnesium ion concentration. All parameter values used for simulation of I_NMDAR are listed in Table 2.

The expressions for each calcium fluxes from the store are

with

And w is the fraction of activated IP₃Rs; its dynamics is modeled by

with

IP₃ is used to mimic activation of mGluRs, we describe the change in IP₃ by production and degradation as follows (De Young and Keizer, 1992):

Where [IP₃] is concentration of IP₃ in cytoplasm, I_r is the rate constant for degradation of IP₃, I_p is the maximal production generated by each activation of mGluRs, and f(t) controls the timing of activation of mGluRs and is described by a step function: If t_glut + 2 ms > t > t_glut, f(t) = 1; otherwise f(t) = 0. Details of the parameters are listed in Table 3.

Postsynaptic potential change has been modeled by (Shouval et al., 2002)

where V_rest is the resting membrane potential, V_rest = −65 mV. EPSP is the net depolarized voltage generated by binding glutamate to receptors in post synapse. The EPSP is:

where s is the maximal potential of depolarization, s = 50 mV, τ₁ and τ₂ are membrane potential decay constants, τ₁ = 50 ms and τ₂ = 5 ms, and t_glut is the time that glutamate released.

In the present modeling, the CaMKII/PP2A reaction network exactly follows the work of Pi and Lisman (2008). The cytoplasmic calcium dynamics in the post synaptic site affects the rates of autocatalytic reactions that convert CaMKII and PP2A from their non-phosphorylated to phosphorylated forms. For CaMKII the concentration of the phosphorylated form, which is the active form, is denoted by K^*, while for PP2A the non-phosphorylated form is active, and its concentration is denoted by P^*. All equations describing the evolution of K^* and P^* were taken directly from Pi and Lisman (2008):

where k₁, k₂, k₃, k₄, k₁₁, k₁₂, k₁₃, and k₁₄ are rate constants for each reaction. K_m1, K_m2, K_m, K_m11, and K_m12 are equilibrium constants for kinase and phosphatase. K₀ and P₀ indicate the basal concentration of active kinase and phosphatase. K_tot and P_tot indicate the total amount of kinase and phosphatase.

Dependence of AMPARs on K^* and P^* is described as:

where A and A_tot indicate the AMPARs on the synaptic membrane and the total amount. c₁ and c₂ are scaling factors, c₃ and c₄ are rate constants for processes independent of kinase and phosphatase activities. All parameters of CaMKII/PP2A reaction network are listed in Table 4.

All the calculations were implemented in the C programming language, with the differential equations expressed as different equations, and solved by using the fourth order Runge–Kutta method in steps of 0.01 ms. According to the related experimental procedures, theta-burst stimulation (TBS) was used as triggers for elevation of calcium concentration in cytoplasm. One unit of TBS is consisted of 3 pulses (100 Hz) delivered at 200 ms intervals. For application of each specific units of TBS, the variation of cytoplasmic calcium concentration in the post-synaptic site, the values of concentration of active kinase and phosphatase, and synaptic efficacy were calculated as functions of time, respectively.

To better understand the transition of synaptic strength, it is useful to consider how the original Pi and Lisman model switches in simpler situations. We begin with a study on the responses of the original CaMKII/PP2A network to a simple calcium pulse (P_Ca) with different amplitude and the same duration (100 ms). Simulation results are shown in Figure 2. The results for P_Ca = 4.5 μM are consistent with those reported by Pi and Lisman (2008), changes of phosphatase and kinase activities are shown in Figure 2A. Both enzymes were nearly inactive in the basal state. During elevation of P_Ca, the kinase and phosphatase become strongly activated. At the end of the induction period, P_Ca returns to the basal level, phosphatase activation returns back to its basal level subsequently but the kinase activation remains at a high level. In this way a LTP is induced.

However, further transient elevations of calcium concentration failed to induce a LTP state. The simulation result of changes in phosphatase and kinase activities to a transient increase of P_Ca to 6.5 μM is shown in Figure 2B. Similarly, two enzymes are inactive in the basal state. During P_Ca elevation, the kinase and phosphatase also become strongly activated. At the end of the induction period, P_Ca returns to its basal level and both phosphatase and kinase return to their basal states subsequently. So there is no sustained change in synaptic strength.

To further characterize the transitions between different states, we systematically varied amplitude of P_Ca with the duration fixed at 100 ms. For P_Ca between 0.1 and 1.3 μM, the system is stabled at basal state. For P_Ca in a range between 1.4 and 2.4 μM, the system is switched from the basal state to LTD states. For P_Ca between 3.0 μM and 6.2 μM, it is switched to LTP states. However, in conditions with P_Ca > 6.2 μM, neither LTP nor LTD is inducible (Figure 2C), the system returns to its basal state, suggesting that significant elevation of calcium concentration may depotentiate synaptic strength to basal levels.

Depotentiation from a LTP state was also tested by inputting a large P_Ca at a LTP level. As shown in Figure 2D, after a LTP is induced from the basal state by a proper P_Ca (4.5 μM for 100 ms), a large P_Ca (6.5 μM for 100 ms) reverses the potentiated state to its basal level, this mechanism of depotentiation is different from the ones induced by small elevations of P_Ca (Pi and Lisman, 2008), which induce LTD from a basal synaptic strength.

In order to directly simulate the experimental induction of depotentiation from a LTP state, we applied the induction procedure used in the experiments to the modified tristable system (including calcium dynamics) in simulation. A postsynaptic cytoplasmic calcium concentration dynamics produced by NMDARs activation and calcium release from internal calcium stores was considered and coupled in the original system. We then observed very different responses induced by application of different amount of TBS.

We first consider the responses to a brief TBS. A train of corresponding elevations of cytoplasmic calcium concentration is induced, and the amplitudes of these elevations are moderate (Figure 3A1). Details of the result are presented in Figure 3A2. Before application of the stimulation, no transmitters are released, thus NMDARs are closed, while the basal concentration of IP₃ is very low. The IP₃Rs are also closed, therefore, there are neither Ca²⁺ influx through NMDARs nor Ca²⁺ release through IP₃Rs. During TBS delivery, NMDARs are activated to mediate a series of Ca²⁺ influx through NMDARs (Figure 3A2 top), while IP₃Rs is still closed (Figure 3A2 bottom) as the concentration of IP₃ is not high enough, although it shows growing pulsatile elevations (Figure 3A2 middle). The results show that, a comparatively high elevation in cytoplasmic calcium concentration is induced by brief TBS, which is mainly mediated by activation of NMDARs.

As shown in Figure 3A3, before elevation of cytoplasmic calcium concentration, both enzymes are nearly inactive in the basal state. During the elevation of cytoplasmic calcium concentration, kinase and phosphates become strongly activated. Then cytoplasmic calcium concentration decline to the basal level, phosphates returns back to the basal level subsequently, while the kinase still stays at the activated state. Therefore, synaptic efficacy is switched to a LTP state.

Then, we turn to consider the responses to a prolonged TBS. Correspondingly, a series of elevations of cytoplasmic calcium concentration is induced by this prolonged TBS, and the amplitudes of the elevations gradually increase to significantly high levels (Figure 3B1). We then analyzed in details the sources of Ca²⁺. Before the stimulation, both NMDARs and IP₃Rs are closed and there is no Ca²⁺ influx. As this prolonged TBS is delivered, NMDARs are activated to mediate stronger Ca²⁺ influx (Figure 3B2 top), meanwhile, IP₃Rs are gradually activated (Figure 3B2 bottom) as the concentration of IP₃ is accumulated to a higher lever (Figure 3B2 middle). Therefore, a more significant elevation in cytoplasmic calcium concentration is eventually formed by activation of both NMDARs and IP₃Rs.

Changes in responses of phosphates and kinase caused by this significant elevation in cytoplasmic calcium concentration are shown in Figure 3B3. Both enzymes are nearly inactive in the basal state. During this significant elevation in cytoplasmic calcium concentration, the kinase and phosphates become strongly activated. However, both of them return back to their basal states subsequently as cytoplasmic calcium concentration declines to basal levels. Therefore, synaptic efficacy remains at the basal state.

Previous behavioral experiments have also shown that brief novelty acquisition was insufficient to reverse LTP, but a prolonged novelty exposure did reverse an established LTP (Qi et al., 2013). Similar phenomena were also observed in hippocampus slices by using electrical stimulations. In the CA1 region of hippocampus slices from young adult rats, TBS, which mimics a physiologically relevant frequency of neuronal activity exhibited in the hippocampus of behaving animals (Kandel and Spencer, 1961; Ranck, 1973; Bikbaev et al., 2008), induces long-lasting reversal of previously induced LTP (Barr et al., 1995). To test for whether a prolonged TBS can reverse LTP while a brief TBS cannot, we first simulated the effect of a brief TBS on established LTP. As shown in Figure 4A, activation of kinase was initially set at an activated state and phosphates at a basal state, corresponding to a LTP state of synaptic strength. When an elevation of cytoplasmic calcium concentration is induced by a brief TBS, kinase becomes slightly inactivated and phosphatase strongly activated. After cytoplasmic calcium concentration declines to the basal state, kinase returns to its initial activated state and phosphatase returns to its basal state. In other words, before and after brief TBS, activation levels of both kinase and phosphatase are unchanged. Hence, a brief TBS is not able to induce a transition from the established LTP state to the basal state.

Then, we simulated the effect of a prolonged TBS on an established LTP. The results are presented in Figure 4B. Before application of TBS, activation of kinase was still set at an activated state and phosphatase at a basal state. During a significant elevation by a prolonged TBS, kinase becomes strongly inactivated but phosphatase becomes strongly activated. When cytoplasmic calcium concentration declines to its basal level, activated forms of both kinase and phosphatase are switched to their basal states. The results clearly show that a prolonged TBS can reverse an established LTP by switching kinase from its activated state to its basal state, which is induced by a significant elevation of calcium concentration at the post synaptic site.

In order to show the dynamics of depotentiation graphically, the nullclines of CaMKII/PP2A network at different Ca²⁺ concentrations is given in Figure 5. Firstly, we studied the case with unchanged synaptic efficacy after action of Ca²⁺ pulses. In this case, the system stays at the basal state before application of Ca²⁺ pulses. At the basal Ca²⁺ level, nullclines (Figure 5A, left panel) intersects at five points; three are stable points (filled circles at top left, bottom left, and bottom right) and two are unstable (open circles). In the basal state, the system is stable at the bottom left point. During a high Ca²⁺ elevation, the nullclines are deformed to create only one stable state (Figure 5A, middle panel, light blue and light red lines, and gray dot), and the system finally stays at the new unique stable state (Figure 5A, middle panel, gray trace). After action of Ca²⁺ pulses, the three stable states restore, but the system stays at the top left stable point (LTP) (Figure 5A, top right panel). However, during a significant elevation of Ca²⁺ concentration, nullclines are deformed to create another unique stable state (Figure 5A, middle panel, blue and red lines, and black dot), and the system moves from this stable state (Figure 5A, middle panel, black trace) to and stays at the bottom left stable point (basal state), when the original three stable points restores after action of Ca²⁺ pulses (Figure 5A, bottom right panel).

Then, we studied reversal of the established LTP by application of Ca²⁺ pulses. At the LTP level, the system was stable at the top left stable point (Figure 5B, left panel). During action of high Ca²⁺ pulses, high Ca²⁺ elevations deform the nullclines to create one stable state (Figure 5B, middle panel, light blue and light red lines, and gray dot), and the system moves to this new stable state (Figure 5B, middle panel, gray trace). After action of Ca²⁺ pulses, nullclines restore the three stable states and the system returns to the top left stable point (LTP) (Figure 5B, top right panel). During action of significant Ca²⁺ pulses, large Ca²⁺ elevations deform nullclines to create the unique stable state (Figure 5B, middle panel, blue and red lines, and black dot), and the system moves to this new stable state (Figure 5B, middle panel, black trace). After action of Ca²⁺ pulses, nullclines restore the three stable states, but the system moves to and stays at the bottom left stable point (basal state) (Figure 5B, bottom right panel).

Attractive basins of each stable state were depicted in the K^* − P^* plane when calcium concentration was set at the basal level (Figure 6). We have known that nullclines form three stable states (filled circles at top left, bottom left, and bottom right of left panels in Figures 5A,B) at basal Ca²⁺ level. The K^* − P^* plane is then divided into three regions; points located within the red region will evolve to the stable point at top left (named LTP basin), those in the green region to the stable point at bottom left (named basal state basin), and those in the blue region to the one point at bottom right (named LTD basin), respectively.

Then we turned to observe the distribution of the stable points created by Ca²⁺ elevations in the K^* − P^* plane. As the amplitude of Ca²⁺ elevation increases from 3 to 10 μM gradually, the stable points induced by Ca²⁺ elevation moves from bottom left to top right, from the LTP basin to the basal state basin. The stable point induced by 4.5 μM Ca²⁺ elevation is in the LTP basin (white dot in left), while the stable point induced by 6.5 μM Ca²⁺ elevation is in the basal state basin (white dot in right, graph inserted in Figure 6). After the action of Ca²⁺ elevation, nullclines restore the three stable states, and the system will evolve from its new initial position (the position determined by Ca²⁺ elevation) to one of the three stable states, according to the location of this initial position in the K^* − P^* plane. The new initial position determined by high Ca²⁺ elevation is in the LTP basin and the system evolves to LTP. The new initial position determined by significant Ca²⁺ elevation is in the basal state basin, therefore, the system evolves to the basal state and can manifest in this way the depotentiation from a LTP state. Just as in the original CaMKII/PP2A network model, Ca²⁺ elevation is still the primary controller, which controls the synaptic strength through activation of kinase and phosphatase. In experimentations synaptic strength is not directly manipulated by Ca²⁺ elevation but by presynaptic simulation, including TBS. The above analysis provides a link between experiments and the generic dynamics revealed by the tristable system (Pi and Lisman, 2008).

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.