Fig 1.
This figure shows the critical flux terms utilized in the formulation of the model. In addition, the relevant fluxes affected by Aβ are highlighted.
Table 1.
Fig 2.
Calcium dynamics for constant IP3 levels.
Properties of the Ca2+ model when IP3 concentrations are fixed and no Aβ is present (i.e., a = 0). A and B show the numerical solution with p = 5 and p = 10, respectively. For these parameter values, the frequency of oscillations increases as p increases. C illustrates the partial bifurcation diagram for the model with p as the bifurcation parameter. The middle curve crossing the diagram correspond to the steady-state values (solid for stable, dashed for unstable). Also shown are the maximum and minimum amplitudes of the periodic orbits of the model (solid curves above and below the steady-state curve). Key bifurcation points are labeled HBp1 and HBp2 (Hopf), and PD (period-doubling). The shaded area corresponds to the regions where MMOs are present. D shows MMOs for p = 18.5.
Fig 3.
The effects of Aβ on steady-state levels.
This figure shows the effects of Aβ on the steady-state Ca2+ levels in the absence of IP3. A shows a bifurcation diagram with a as the bifurcation parameter. Two Hopf bifurcations, labeled HBa1 and HBa2, give rise to a Hopf bubble with relatively large oscillation amplitude. The steady-state level quickly becomes unphysical as the amount of Aβ increases towards a = 1.3. B shows stable Ca2+ oscillations for a = 1.15. C shows the solution when a = 1.276.
Fig 4.
A shows a two parameter bifurcation diagram when p and a are varied. The solid curves in the diagram correspond to the Hopf bifurcation manifolds and are labeled HB1m and HB2m. The dashed lines correspond to manifolds of period doubling points. The shaded regions (between the dashed lines and near the bottom of HB2 m) correspond to regions where MMOs occur. B shows a bifurcation diagram for a = 0.45. Notice that for this value of a, four Hopf bifurcations exist and three of them have been labeled with HB1 while the last is labeled HB2. This figure also illustrates the intermediate region of MMOs between the labels PD1 and PD2.
Fig 5.
Calcium oscillations in the presence of Aβ.
This figure shows various solutions when a = 0.45. A shows small amplitude oscillations when p = 5 while B shows a stable-steady state solution when p = 20. C and D both show MMOs with multiple sub-threshold oscillations when p = 26 and p = 45.5, respectively. E shows aberrant Ca2+ signals when p = 45.8. F shows sustained Ca2+ oscillations when p = 50.
Fig 6.
Effects of a and p on Ca2+ oscillation amplitudes.
This figures shows two examples of sustained Ca2+ oscillations for two sets of parameter selections of a and p. A shows Ca2+ oscillations with peak amplitudes around 2 corresponding to a = 1 and p = 30. B shows similar oscillations with peak amplitude closer to 3 corresponding to a = 1.2 and p = 20.
Fig 7.
The effects of kα on Ca2+ signals in the presence of Aβ.
This figure shows a bifurcation diagram and four Ca2+ traces. A shows a partial bifurcation diagram for the parameter kα. In this figure, a number of period doubling points have been labeled as PD. The shaded region corresponds to MMOs. The single Hopf bifurcation point has been labeled with HB. B-E show four solutions for the parameter values kα = 0.5, 0.9, 1, and 1.25, respectively. D shows aberrant Ca2+ signals.
Fig 8.
Changing the maximal reaction rate of the RyR.
This figures shows the effects of changing the parameter k2 on model solutions. A shows a two parameter bifurcation diagram with kα and k2 as the bifurcation parameters. The shaded region corresponds to the region of MMOs. The solid curve corresponds to the manifold of Hopf bifurcation points. The two dots on the left side represents the locations of the parameter values of k2 used to generate the Ca2+ traces in Fig 7C and Fig 8B. The two dots on the right side represents the locations of the parameter values of k2 used to generate the Ca2+ traces in Fig 7D and Fig 8C. B and C show the stable periodic oscillations that occur when k2 is increased to k2 = 0.5 and k2 = 0.65, respectively.
Table 2.
Parameter values of the IP3 model (23).
Fig 9.
A shows Ca2+ and B shows IP3 as a function of time in the absence of Aβ. A spike of Ca2+ occurs once enough IP3 has accumulated but does not lead to sustained oscillations. The amount of IP3 settles to a steady-state level in B.
Fig 10.
Calcium signals in the presence of Aβ with dynamic IP3.
This figure shows a bifurcation diagram and four Ca2+ traces for the model with dynamic IP3. A shows a partial bifurcation diagram where two oscillatory regions are separated by a single steady-state region. The shaded regions near each of the four Hopf bifurcations correspond to MMOs. Numerous period doubling bifurcations occur around the shaded regions. Two important limit points have been labeled LP1 and LP1. These points are important in the description of solutions. B-D show model solutions for a = 0.75, 0.875, 1, and 1.256, respectively. The various patterns in these figures are predicted by the bifurcation diagram in A.
Table 3.
Solution behavior for model with dynamic IP3.
Fig 11.
Ca2+ flux due to changes in membrane potential.
This figures shows the Ca2+ current Ica in response to changes in membrane potential. A-C show changes in the membrane potential of the astrocytic model when a current of 300 nA is applied for a duration of 0.1 second (A), 1 second (B), and 10 seconds (C). The current was applied at t = 2 seconds and is represented by the black bar at the bottom of each figure. D-F show the corresponding VGCC current Ica as defined by (15).
Fig 12.
The influence of IP3 on Ca2+ signals with membrane potential.
This figures shows the impact of membrane potential on Ca2+ signaling. In each figure, membrane potentials are included as a response to a sustained applied current of 300 nA lasting from t = 100 to t = 150. Figs A, B, and C show intracellular Ca2+ signals when p = 5, p = 10, and p = 12 with a = 0, respectively. Figs D, E, and F show the response when p = 13, p = 13.5, and p = 14, respectively. Note that the inclusion of the membrane potential filters MMOs and can help establish transient single-mode oscillations upon the termination of the applied current (D,E).
Fig 13.
The influence of Aβ on Ca2+ signals with membrane potential.
This figures shows the impact of membrane potential on Ca2+ signaling. In each figure, changes in membrane potential are included as a response to a sustained applied current of 300 lasting from t = 100 to t = 150. Figs A, B, and C show intracellular Ca2+ signals when p = 10 and when a = 0.2, a = 0.25, and a = 0.28, respectively. Figs D, E, and F show the response when p = 15 and when a = 0.2, a = 0.25, and a = 0.28, respectively.
Fig 14.
The impact of scaling Ica on model solutions.
This figures shows various model solutions when the scaling parameter ps is altered. In each figure, p = 10, and the amount of Aβ is fixed at a = 0.258. A-D show the response of (18) and (19) when ps = 0.5, ps = 0.75, ps = 1, and ps = 1.5, respectively. Notice that B shows that Ca2+ can enter aberrant oscillatory patterns when the membrane is stimulated by a constant applied current. C shows that Ca2+ signals can enter MMOs with altered amplitudes when the applied current stimulus is turned off.