Fig 1.
Schematic representation of the considered pathways.
The scheme depicts how Adenosine 2A Receptor (A2aR) activation by extracellular Adenosine (ADO) increases cAMP levels via activation of adenylate cyclases (AC), which in turn increases the opening probability of HCN and M channels via direct binding and PKA activation, respectively. The experiments in [4, 5] target this pathway through specific drugs to understand how both M and HCN channels participate in the modulation of neuronal electrical activity. The chemical compounds employed are the A2aR agonist CGS21680 (CGS), the HCN channel inhibitor ZD7288 (ZD), the AC activator Forskolin (FSK), and the M channel antagonists XE991 (XE) and Linopirdine (Lin).
Fig 2.
The figure presents simulated voltage traces under control conditions (first row, black curves) and when cAMP is elevated as in the experiments with CGS application (second row, red curves) at four different values of IApp.
Fig 3.
Ionic currents during APs and MMOs.
Voltage (upper row; black and red curves) and principal currents (middle and bottom rows; see legend for interpretation of colors) in simulations with IApp = 250 μA/cm2, in control and CGS-stimulated conditions, are shown entirely in the left panels, whereas the right panels show zooms on the dynamics near the onset of APs and the SAOs involved in MMOs.
Fig 4.
Blocking HCN and M channels affects electrical patterns at basal and elevated cAMP levels.
The figure presents the model simulations using IApp equal to 115 and 300 μA/cm2, in the presence of either the HCN blocker ZD7288 (ZD; left panels) or the M current blocker XE991 (XE; right panels). The first row shows simulations without activation of the cAMP-dependent pathway, whereas those presented in the second row are for CGS application.
Fig 5.
M-channel inhibition changes the signature of MMOs in stimulated conditions.
Simulations of the model with raised cAMP in the absence (left, blue) or presence of partial M channel inhibition, as in the experiments with Forskolin (FSK) application in the absence or presence of Linopirdine (Lin) [5], are presented. Note how the grey trace presents more full APs (LAOs) and fewer subthreshold oscillations (SAOs).
Fig 6.
The FF of the 5D model under various pharmacological stimulations at different levels of applied current. The black curve shows FFs for the unstimulated (control) condition. Red, magenta and turquoise curves are associated with GCS21680 (GCS; increased cAMP), XE991 (XE; M channel block), and simultaneous XE991+CGS21680 (XE/CGS) application, respectively.
Fig 7.
Simulations of the 3D model under control (left, black) and CGS-stimulated (right, red) conditions, compare with Fig 2.
Fig 8.
Slow-fast 3D subsystem bifurcation diagrams.
A: 1P-BD of the slow-fast subsystem for IApp = 300 μA/cm2 in the control condition with as parameter and
constrained to the dashed, black line in panel B. The red (respectively, black) curve indicates stable (unstable) equilibria, and the green (blue) curves are minima and maxima of stable (unstable) periodic solutions. The inset on the right shows a zoom on the region where the equilibrium changes stability. The inset on the left provides a zoom around the period-doubling cascade. HB: Hopf bifurcation, SNPO: saddle-node bifurcation of periodic orbits, PD: period doubling bifurcation. B:
plane with projections of 5D model simulations for IApp = 300 μA/cm2 in control (full, black curve) and CGS-stimulated (red) conditions. The dashed lines indicate the linear approximations
. The background shows the 2-parameter bifurcation diagram (2P-BD) for the slow-fast subsystem with
as parameters, obtained by following the most relevant bifurcations found in the 1P-BD (panel A). C: BD as in panel A with 5D simulation projected onto the
plane for the control (upper) and CGS-stimulated (lower) scenarios. The orange curve is the
nullcline. D and E: As panels B and C, but for IApp = 250 μA/cm2.
Fig 9.
Model dynamics depends on the degree of cAMP-activation of M channels.
Two-parameter BD with bifurcation parameters (IApp, ΔgM) for gM = 50 mS/cm2, EHCN = −50 mV, gHCN = 23 mS/cm2, and ΔgHCN = 12 mS/cm2 (the default values used throughout the paper, see Table 1). Black and red curves indicate respectively HBs in the control conditions (cAMP = 0) and with raised cAMP (cAMP = 1). The curves in blue and magenta represent, similarly, the PDs (PD1 in Fig 8) in the control and stimulated cases. The combination of IApp and ΔgM where MMOs occur is highlighted with shaded blue (control) and magenta (elevated cAMP) areas.
Fig 10.
This figure presents the 3D-model critical manifold with IApp = 250 μA/cm2 in the CGS-stimulated case. The red curve is the corresponding trajectory from Fig 7. Attracting (
) and repelling (
) submanifolds are presented with red and grey shaded surfaces. The black dashed curves correspond to the fold lines
. The folded node and the unstable fixed point are presented using a blue circle and an orange square, respectively. The right panel shows a zoom on the region near the trajectory.
Fig 11.
Critical manifold and folded singularities of the 3D model.
The presented phase-plane plots show the trajectory of the 3D model with parameters as in Fig 7 projected onto the (h, V) plane for the control case (black curve, upper panels) and in the presence of CGS (red, lower), observed globally (left) and locally with a zoom on the region where SAOs appear for the CGS-stimulated case (right). The dashed black curve represents the fold of the critical manifold. The red and gray shaded regions correspond to the attracting,
, and repelling,
, submanifolds, respectively. The unstable equilibrium point is given by the orange square, while the blue dot indicates the FN. The blue curve is the strong canard that together with the fold line delimits the funnel region indicated by the blue horizontal shading lines. The arrows indicate the direction of the flow.
Fig 12.
Canards and attracting and repelling manifolds for the 3D Model.
The upper panels show the local reconstruction of (shaded red) and
(shaded black) for IApp = 250 μA/cm2 (left) and IApp = 300 μA/cm2 (right). In each panel, the big blue dot indicates the FN. The vertical black dashed line identifies a plane ΣFS near the FN, and the inset presents the corresponding intersections between ΣFS and, respectively,
(red) and
(black). The colored lines ξ4 − ξ7 and ξ2 − ξ5, obtained for IApp equal to 250 and 300 μA/cm2, respectively, represent the canard orbits identified via the intersection points between
and
in the plane ΣFS. In the lower panels, the canard orbits ξ0 − ξ4, the associated rotational sectors R1 − R4 and the local dynamics of the 3D model are presented for the considered values of IApp. The voltage traces (insets) and the corresponding phase-plane trajectories are decomposed into color-coded segments beginning in small circles and ending in crosses to explain how the dynamics and signatures of the MMOs are related to the identified geometrical structures.
Table 1.
T is the temperature of the experiment, T(ref,1) is the reference temperature used to derive the activation/inactivation properties of the slow-K+ and fast-Na+ channels, while T(ref,2) is the reference temperature used to characterize the electrophysiological properties of the HCN channels [4].
Table 2.
Translation of drug combinations applied in the various experiments into parameter sets used to simulate the model.