CROSS-CURRENTS BETWEEN BIOLOGY AND MATHEMATICS: THE CODIMENSION OF PSEUDO-PLATEAU BURSTING

Abstract

A great deal of work has gone into classifying bursting oscillations, periodic alternations of spiking and quiescence modeled by fast-slow systems. In such systems, one or more slow variables carry the fast variables through a sequence of bifurcations that mediate transitions between oscillations and steady states. A rigorous classification approach is to characterize the bifurcations found in the neighborhood of a singularity; a measure of the complexity of the bursting oscillation is then given by the smallest codimension of the singularities near which it occurs. Fold/homoclinic bursting, along with most other burst types of interest, has been shown to occur near a singularity of codimension three by examining bifurcations of a cubic Liénard system; hence, these types of bursting have at most codimension three. Modeling and biological considerations suggest that fold/homoclinic bursting should be found near fold/subHopf bursting, a more recently identified burst type whose codimension has not been determined yet. One would expect that fold/subHopf bursting has the same codimension as fold/homoclinic bursting, because models of these two burst types have very similar underlying bifurcation diagrams. However, no codimension-three singularity is known that supports fold/subHopf bursting, which indicates that it may have codimension four. We identify a three-dimensional slice in a partial unfolding of a doubly-degenerate Bodganov-Takens point, and show that this codimension-four singularity gives rise to almost all known types of bursting.

Figure 1

Time series (a) and bifurcation diagram (b) for square-wave (fold/homoclinic) bursting in a conductance-based model. Time series (c) and bifurcation diagram (d) for pseudo-plateau (fold/subHopf) bursting. Parameters as in [, Fig. 1], except that here the square-wave bursting trajectory is calculated for f_c = 0.0052.

Figure 2

Time series (a) and bifurcation diagram (b) for the square-wave (fold/homoclinic) burster defined in Golubitsky et al. [7] with μ₂ = 1/3 and b = 3 in (2). For the slow path we set ν = −1.4+10 μ₁ and identified μ₁ with the slow variable z as in (3) with ${\stackrel{‒}{\mu }}_1$ = −0.071, A = 0.006 and ∈ = 0.01. Time series (c) and bifurcation diagram (d) for a ‘clean’ fold/homoclinic burster with no large surrounding periodic orbit. This case is obtained from (4) with μ₂ = ⅓ and b = 3, which is (2) with time reversed as well as the orientations of μ₁, ν and x. Here, we used ν = —0.7 + 10 μ₁ and z as in panels (a) and (b), with ${\stackrel{‒}{\mu }}_1$ = −0.062, A = 0.015 and ∈ = 0.01.

Figure 3

Pseudo-plateau bursting exists in the (partial) unfolding (4) of a doubly-degenerate Bogdanov–Takens singularity. Panel (a) shows the bifurcation diagram on the unit sphere in (μ₁, μ₂, ν)-space with b = 0.75. Color in on-line version: red, homoclinic (HC); blue, Hopf bifurcation (H; solid: supercritical, dashed: subcritical); black, saddle-node (SN); orange, Bogdanov–Takens point (BT); green, saddle node of periodics (SNP); degenerate Hopf and cusp are labeled DH and C, respectively. Panels (b) and (c) show time series and underlying bifurcation diagram of pseudo-plateau bursting corresponding to the path indicated in panel (a).

Figure 4

Bifurcation diagram of (4) with b = 0.75 and ν = −0.09; the bifurcation diagram on the unit sphere with b = 0.75 from Figure 3(a) is shown for reference in panel (a) with the (μ₁, μ₂)-plane shown in panel (b) and an enlargement in panel (c). Note that there are two Bogdanov–Takens points, BT_r and ${\mathsf{BT}}_r^{\mathrm{far}}$, on the same side SN_r of the cusp point C. Two burst paths are indicated, which are the fold/subHopf and fold/homoclinic bursters illustrated in Figure 5. Colors and symbols as in Figure 3.

Figure 5

Time series and one-parameter bifurcation diagrams illustrating the two path segments indicated in Figure 4; we used (4) with b = 0.75 and ν = −0.09, and μ₂ = 0.5 for pseudo-plateau bursting in panels (a) and (b), and μ₂ = 0.24 for square-wave bursting in panels (c) and (d), respectively. The slow variable z = μ₁ is defined in (3); we used ${\stackrel{‒}{\mu }}_1$ = −0.01, A = 0.15 and ε = 0.1 for μ₂ = 0.5, and ${\stackrel{‒}{\mu }}_1$ = −0.0395, A = 0.0066 and ε = 0.01 for μ₂ = 0.24.

Figure 6

Bifurcation diagram of (4) with b = 0.75 and μ₂ = 0.0675; the bifurcation diagram on the unit sphere with b = 0.75 from Figure 3 is shown for reference in panel (a) with the (μ₁, ν)-plane shown in panel (b) and an enlargement in panel (c); the Bogdanov–Takens point BT_r is the same point as the one labeled BT_r in the ν-slice shown in Figure 4. Colors and symbols as in Figure 3.

Figure A1

Bifurcation diagram of (4) in the (μ₁, ν)-plane with b = 0.75 and μ₂ = 0.0675 fixed; see also Figure 6. The horizontal paths indicated in the figure correspond to the bursters shown in Figures A2–A10 and have been labeled accordingly.

Figure A2

Fold/fold bursting in which there is no Hopf bifurcation of the fast subsystem but spikes are generated by transients of the fast subsystem. The path satisfies ν = −0.15, A = 0.008, ${\stackrel{‒}{\mu }}_1$ = 0 and ε = 0.025. The spikes disappear as ε ↔ 0.

Figure A3

Tapered (fold/fold) bursting with ν = −0.13, A = 0.008, ${\stackrel{‒}{\mu }}_1$ = 0.001 and ε = 0.002.

Figure A4

Square-wave (fold/homoclinic) bursting with ν = −0.105, A = 0.0013, ${\stackrel{‒}{\mu }}_1$ = −0.006 and ε = 0.005.

Figure A5

Parabolic (circle/circle) bursting with ν = −0.08, A = 0.008, ${\stackrel{‒}{\mu }}_1$ = −0.0025 and ε = 0.0025.

Figure A6

Fold/homoclinic bursting with no plateau because the limit cycles surround all three steady states. The path satisfies ν = −0.03, A = 0.002, ${\stackrel{‒}{\mu }}_1$ = −0.006 and ε = 0.002.

Figure A7

Fold/fold-cycle bursting, historically called “Type 4″, with ν = 0.2, A = 0.03, ${\stackrel{‒}{\mu }}_1$ = 0.005 and ε = 0.02.

Figure A8

A form of subHopf/fold-cycle bursting with folds in the steady-state curve so that some of the fast-subsystem limit cycles surround three steady states. The path satisfies ν = 0.3, A = 0.05, ${\stackrel{‒}{\mu }}_1$ = 0.02 and ε = 0.01.

Figure A9

“Type III” (subHopf/fold-cycle) bursting with ν = 0.35, A = 0.043, ${\stackrel{‒}{\mu }}_1$ = 0.05 and ε = 0.001.

Figure A10

“Parabolic amplitude” bursting, a phenomenological variant of fold/homoclinic bursting, with μ₂ = 0.2, ν = −0.132, A = 0.006, ${\stackrel{‒}{\mu }}_1$ = 0.023 and ε = 0.05. This case does not correspond to a path in Figure A1; μ₂ was changed to move both Hopf bifurcations to the right of the left knee (small periodic branch corresponding to the right Hopf bifurcation not displayed).