Table 1.
Parameter values of the voltage-dependent ionic currents included in Eq (1).
Whenever two values are provided for a single parameter, the first corresponds to pre-runup (t = 0 min after patch clamping), while the second (between parentheses) corresponds to post-runup (t = 25 min after patch clamping).
Table 2.
Parameter values of the full system (1) that do not change during runup while exhibiting tonic firing.
Fig 1.
Time courses of the membrane voltage (V) during A) pre- and B) post-runup when the full system (1) is tonically firing. C) The AP-cycles of the same time courses in A and B plotted in the -plane. Black: pre-runup; blue: post-runup.
Fig 2.
Review of the main four features of the full system (1): A) Type I excitability with a SNIC bifurcation during pre- (faded lines) and post-runup (dark lines). Solid (dashed) black/gray lines correspond to branches of stable/unstable equilibria, respectively; solid green lines correspond to envelopes of stable limit cycles. B) Non-monotonic first-spike latency with respect to the holding potential (Vhold) during pre- (gray) and post-runup (back). C) The boundary between responsive (to the right) and non-responsive (to the left) regimes when a pair of inhibitory and excitatory pre-synaptic inputs of various magnitudes are applied during pre- (C1) and post-runup (C2). D) color-maps of the no-spiking (NS), single spiking (SS) and tonically spiking (TS) regimes during pre- (D1) and post-runup (D2), color-coded based on the duration of the first-spike latency calibrated according to the color-bar to the right. The system generates a single spike for Itest < ISNIC when the stable manifolds of the saddle and saddle-node do not coincide and the initial condition lies between them. Time series simulations of these properties are available in [13, 16, 17].
Fig 3.
Two types of bursting recorded in CSCs.
A) Square-wave bursting recorded in two CSCs bathed in 2 mM 4-AP, exhibiting short (top) and long (bottom) active phases. B) Pseudo-plateau bursting recorded in two CSCs bathed in 2 mM 4-AP and 300 μM Cd2+, both exhibiting similar profiles.
Fig 4.
Two time courses of the membrane voltage (V) produced by the full system (1) during post-runup.
Both exhibit bursting when A) (gHVA, gA) = (0.26, 6.0) μS.cm−2, and B) (gHVA, gA) = (0.235, 12.2) μS.cm−2.
Fig 5.
The effects of gHVA on the dynamics of the full system (1).
A) Bifurcation diagram of the full system (1) with respect to gHVA using the L2-norm of the state variables. The branch of equilibria (black) undergoes two Hopf bifurcations, labeled HB1 and HB2. The equilibria are unstable (dotted) between HB1 and HB2 and stable (solid) otherwise. The envelope of unstable periodic orbits (dotted dark green) emanating from HB1 undergoes a saddle-node bifurcation of periodic orbits (SNP) and terminates at a homoclinic bifurcation, denoted HC. The envelope of periodic orbits (dark olive) generated from HB2 is unstable, but becomes stable at a saddle-node bifurcation of periodic orbits (SNP) and terminates at a homoclinic bifurcation very close to the right of HB1 (not shown). The light olive line is the continuation of the dark olive line obtained by numerical integration of the full system (1). Both correspond to the envelope of pseudo-plateau BPOs (labeled PB). The ⊃-shaped curve (light green) is an isola of POs corresponding to tonic firing (labeled T); it consists of two branches separated by a saddle-node bifurcation of periodic orbits (SNP∞). The upper branch is stable and becomes unstable at a period-doubling bifurcation (PD). The set of (colorful) curves to the right of PD within the box shows a family of more isolas of square-wave BPOs (labeled SW) (for better visualization, see the magnification of this box in Fig 6A). B) Time course of a pseudo-plateau BPO for gHVA = 0.4 μS.cm−2.
Fig 6.
Square-wave bursting in the CSC model defined by the full system (1) when gHVA is varied.
A). Magnification of the set of isolas of square-wave BPOs shown within the box in Fig 5A. Each isola is an envelope of stable periodic orbits (solid) on the top that becomes unstable (dotted) either at a saddle-node bifurcation of periodic orbits (SNPi, i = 3, …) or period-doubling bifurcation (PDj, j = 1, …) to the right and left of each isola in a non-orchestrated manner (i.e., in a non-consistent pattern). The number of spikes of active phases of BPOs associated with these isolas is indicated with a number on top. B) Time courses of two square-wave BPOs when the number of spikes per burst is 14 (top) and 10 (bottom) obtained at gHVA = 0.232 μS.cm−2 and gHVA = 0.253 μS.cm−2, respectively.
Fig 7.
The effects of gK(Ca) on the dynamics of the full system (1).
Bifurcation diagram of the full system (1) with respect to gK(Ca) using the L2-norm of the state variables. Two subcritical Hopf bifurcations HB1 and HB2 are present along the branch of equilibria (black). The equilibria are unstable (dotted) between HB1 and HB2 and stable otherwise (solid). The envelope of periodic orbits emanating from HB1 (olive) is unstable and terminates at a homoclinic bifurcation, denoted by HC. The family of periodic orbits emanating from HB2 is unstable. After a drastic change in the L2-norm of the state variables, the envelope becomes stable at a period-doubling bifurcation (PD). This envelope becomes unstable at a saddle-node bifurcation of periodic orbits (SNP1) before undergoing the spike-adding process, and then becomes stable again at another saddle-node bifurcation of periodic orbits (SNP2). The two instances of drastic changes in the L2-norm of the state variables inside the right black box are further magnified in the inset. The set of (colored) curves inside the left gray box shows isolas of square-wave BPOs with different number of spikes. These isolas are further magnified in Fig 8A.
Fig 8.
Square-wave bursting in the CSC model defined by the full system (1) when gK(Ca) is varied.
A) Magnification of the set of isolas of square-wave BPOs in the left black boxed region of Fig 7. The top segment of each isola is an envelope of stable periodic orbits (solid) that becomes unstable (dotted) at a saddle-node bifurcation of periodic orbits (SNPi, i = 3, …) from the right and period-doubling bifurcation (PDj, j = 1, …) from the left. The number of spikes for each isola is indicated by the number on top. B) Time courses of two square-wave BPOs when the number of spikes per burst is 12 (top) and 3 (bottom) obtained at gK(Ca) = 0.9 μS.cm−2 and gHVA = 6 μS.cm−2, respectively.
Fig 9.
Color-map of the various regimes of behavior associated with the full system (1) in the (gK(Ca), gA)-plane, when gHVA = 0.249 μS.cm−2.
The square-wave bursting regime, labeled SW, is color-coded based on the number of spikes in its corresponding BPOs, calibrated according to the color-bar to the right. For further clarity, the number of spikes in various regions of SW are displayed. The number of spikes increases as gK(Ca) decreases such that for small values of gK(Ca), the system transitions to tonic firing (labeled T). The yellow and red curves are two-parameter continuation of the period-doubling and saddle-node bifurcations of periodic orbits detected in Fig 8A, respectively.
Fig 10.
Bifurcation diagram of the fast subsystem, represented by the membrane voltage V, when hA is varied as a parameter and Ca is kept fixed at a random but physiologically-relevant value of 0.2 μM.
The Z-shaped curve (black) is the branch of equilibria of the fast subsystem consisting of three branches separated by two saddle-node bifurcations, denoted SN1 and SN2. The solid and dotted black lines indicate stable and unstable branches of equilibria, respectively. The lower and middle branches are of stable and saddle type, respectively. The upper branch is stable for lower values of hA and becomes unstable (of saddle type) at a subcritical Hopf bifurcation, denoted HB, where an envelope of unstable periodic orbits emanates and terminates at a homoclinic bifurcation (HC1). An isolated envelope of periodic orbits comprised of an upper and a lower ⊂-shaped loop exists near hA-values of SN1. The isolated envelope has an inner envelope of unstable periodic orbits that terminates at a homoclinic bifurcation (HC2) to the right, but undergoes a saddle-node bifurcation of periodic orbits (SNP) to the left, forming a stable outer envelope. As shown in the inset, this envelope becomes unstable at a period doubling bifurcation (PD) and terminates to the right at another homoclinic bifurcation, denoted HC3.
Fig 11.
A three-dimensional representation of the bifurcation diagram shown in Fig 10 for a whole range of [Ca2+]i (Ca).
The gray surface is the critical manifold with three sheets of equilibria separated by two folds, whereas the green surface is a family of isolated envelopes of periodic orbits (as suggested by Fig 10). The lower/middle sheets of the critical manifold are attracting/repelling, respectively, while the upper sheet is attracting to the left of a curve of Hopf bifurcations (not shown), and repelling to the right. The sheets of unstable periodic orbits emanating from the curve of Hopf bifurcations are not shown. A square-wave bursting trajectory (blue) is superimposed on the bifurcation diagram. The arrows indicate the direction of flow and dark and light shades of blue discern between active and silent phases of the BPO, respectively. The inset shows the time courses of V, hA and Ca corresponding to the square-wave BPO shown in the main panel.
Fig 12.
Two-parameter continuation of the bifurcation points in Fig 10.
A) Two-parameter continuation of the bifurcation points SN1, SN2, HB, SNP, HC1, HC2 and HC3 of the fast subsystem detected in Fig 10 plotted in the hA, Ca-plane. B) The burst cycle (blue) is superimposed on an enlargement of the figure in A; the arrows indicate the direction of flow and dark and light shades of blue distinguish between active and silent phases of the square-wave BPO, respectively. The curves produced by the bifurcation points are color-coded according to the legend in each panel.
Fig 13.
Pseudo-plateau bursting in the CSC model defined by the full system (1).
A) Time course of the membrane voltage V when gK = 12 μS.cm−2, gK(Ca) = 7 μS.cm−2 and gHVA = 0.22 μS.cm−2. B) Bifurcation diagram of the fast subsystem, represented by the variable V, with respect to hA using the same values of maximum conductances listed in A. As in Fig 10, the stable (unstable) branches of equilibria are plotted as solid (dotted) lines, while envelopes of unstable periodic orbits are plotted as green dotted lines. Notice the absence of the isola near SN1 seen in Fig 10. C) The BPO shown in panel A superimposed on the critical manifold of system (1) shown as a gray surface in the (hA, Ca, V)-space. The family of envelopes of unstable periodic orbits emanating from the curve of Hopf bifurcations lying on the upper sheet of the critical manifold is shown as a green surface. D) Two-parameter continuation of the bifurcation points SN1, SN2, HB and HC detected in panel B plotted in the (hA, Ca)-space along with one pseudo-plateau bursting cycle (blue); the flow direction is indicated with the arrows and active and silent phases are shaded dark and light blue, respectively. The bifurcation curves are color-coded according to the legend.
Fig 14.
Chaotic bursting in the CSC model defined by the full system (1).
A) Time course of the membrane voltage V when exhibiting chaotic bursting obtained by setting gK(Ca) = 3 μS.cm−2 and gHVA = 0.215 μS.cm−2. B) The return map of the interspike interval (ISI) for the time course of V in A.
Fig 15.
The effects of varying gHVA on the dynamics of the full system (1).
A) Color-map of the various regimes of behavior in the (gK(Ca), gHVA)-plane when gA = 10.2 μS.cm−2. The square-wave bursting regime, labeled SW, is color-coded based on the number of spikes in their corresponding bursting orbits calibrated according to the color-bar to the right. Additional regimes are also present, including a tonic firing regime (labeled T), a pseudo-plateau bursting regime (labeled PB), a quiescent regime with elevated steady state (labeled ESS in black) and a chaotic bursting regime (labeled CB in gray). Since the full system (1) fires regularly in regimes T and PB, these regimes are also color-coded according to the color-bar to the right. For further clarity, the number of spikes in various regions of SW are displayed. B) Time course of the membrane voltage V showing pseudo-plateau bursting obtained by setting gHVA = 0.77 μS.cm−2, while keeping the other maximum current conductances at their default values listed in Table 2. C) Electrical recording of one CSC showing pseudo-plateau bursting activity induced by the application of 20 mM 4-AP in the absence of a hyperpolarizing step current.