The dynamics underlying pseudo-plateau bursting in a pituitary cell model
- Wondimu Teka^{1},
- Joël Tabak^{2},
- Theodore Vo^{3},
- Martin Wechselberger^{3} and
- Richard Bertram^{4}Email author
DOI: 10.1186/2190-8567-1-12
© Teka et al; licensee Springer. 2011
Received: 27 June 2011
Accepted: 8 November 2011
Published: 8 November 2011
Abstract
Pituitary cells of the anterior pituitary gland secrete hormones in response to patterns of electrical activity. Several types of pituitary cells produce short bursts of electrical activity which are more effective than single spikes in evoking hormone release. These bursts, called pseudo-plateau bursts, are unlike bursts studied mathematically in neurons (plateau bursting) and the standard fast-slow analysis used for plateau bursting is of limited use. Using an alternative fast-slow analysis, with one fast and two slow variables, we show that pseudo-plateau bursting is a canard-induced mixed mode oscillation. Using this technique, it is possible to determine the region of parameter space where bursting occurs as well as salient properties of the burst such as the number of spikes in the burst. The information gained from this one-fast/two-slow decomposition complements the information obtained from a two-fast/one-slow decomposition.
Keywords
Bursting Mixed mode oscillations Folded node singularity Canards Mathematical model1 Introduction
Bursting is a common pattern of electrical activity in excitable cells such as neurons and many endocrine cells. Bursting oscillations are characterized by the alternation between periods of fast spiking (the active phase) and quiescent periods (the silent phase), and accompanied by slow variations in one or more slowly changing variables, such as the intracellular calcium concentration. Bursts are often more efficient than periodic spiking in evoking the release of neurotransmitter or hormone [1–3].
where V is the membrane potential, n is the fraction of activated delayed rectifier K^{+} channels, and c is the cytosolic free Ca^{2+} concentration. The velocity functions are nonlinear, and ε_{1} and ε_{2} are parameters that may be small.
The variables V, n, and c vary on different time scales (for details, see Section 2). By taking advantage of time-scale separation, the system can be divided into fast and slow subsystems. In the standard fast/slow analysis one considers ε_{2} ≈ 0, so that V and n form the fast subsystem and c represents the slow subsystem. One then studies the dynamics of the fast subsystem with the slow variable treated as a slowly varying parameter [12, 15–18]. This approach has been very successful for understanding plateau bursting, such as occurs in pancreatic islets [19], pre-Bötzinger neurons of the brain stem [20], trigeminal motoneurons [21] or neonatal CA3 hippocampal principal neurons [14], Figure 1(b). It has also been useful in understanding aspects of pseudo-plateau bursting such as resetting properties [11], how fast subsystem manifolds affect burst termination [17], and how parameter changes convert the system from plateau to pseudo-plateau bursting [12].
An alternate approach, which we use here, is to consider ε_{1} ≈ 0, so that V is the sole fast variable and n and c form the slow subsystem. With this approach, we show that the active phase of spiking arises naturally through a canard mechanism, due to the existence of a folded node singularity [22–25]. Also, the transition from continuous spiking to bursting is easily explained, as is the change in the number of spikes per burst with variation of conductance parameters. Thus, the one-fast/two-slow variable analysis provides information that is not available from the standard two-fast/one-slow variable analysis in the case of pseudo-plateau bursting.
2 The mathematical model
Parameter values for the lactotroph model.
Parameter | Value | Description |
---|---|---|
C _{ m } | 5 pF | Membrane capacitance of the cell |
g _{ Ca } | 2 nS | Maximum conductance of Ca^{2+} channels |
V _{ Ca } | 50 mV | Reversal potential for Ca^{2+} |
ν _{ m } | -20 mV | Voltage value at midpoint of m_{∞} |
s _{ m } | 12 mV | Slope parameter of m_{∞} |
g _{ K } | 4 nS | Maximum conductance of K^{+} channels |
V _{ K } | -75 mV | Reversal potential for K^{+} |
ν _{ n } | -5 mV | Voltage value at midpoint of n_{∞} |
s _{ n } | 10 mV | Slope parameter of n_{∞} |
τ _{ n } | 43 ms | Time constant of n |
g _{K(Ca)} | 1.7 nS | Maximum conductance of K(Ca) channels |
K _{ d } | 0.5 μ M | c at midpoint of s_{∞} |
g _{ BK } | 0.4 nS | Maximum conductance of BK-type K^{+} channels |
ν _{ b } | -20 mV | Voltage value at midpoint of f_{∞} |
s _{ b } | 5.6 mV | Slope parameter of f_{∞} |
f _{ c } | 0.01 | Fraction of free Ca^{2+} ions in cytoplasm |
α | 0.0015 μ M fC^{-1} | Conversion from charge to concentration |
k _{ c } | 0.16 ms^{-1} | Rate of Ca^{2+} extrusion |
The variables V, n and c vary on different time scales. The time constant of V is given by τ_{ V } = C_{ m } /g_{ Total } , where g_{ Total }= g_{ K }n + g_{ BK }b_{∞}(V) + g_{ Ca }m_{∞}(V) + g_{K(Ca)}s_{∞}(c). During a bursting oscillation, the minimum of g_{ Total } is 0.483 pS and the maximum is 3 pS. Hence, $\frac{{C}_{m}}{max{g}_{Total}}\le {\tau}_{V}\le \frac{{C}_{m}}{min{g}_{Total}}$, or 1.7 ms ≤ τ_{ V } ≤ 10.4 ms, for C_{ m } = 5 pF, a typical capacitance value for lactotrophs. The time constant for n is τ_{ n } = 43 ms. For the variable c, the time constant is $\frac{1}{{f}_{c}{k}_{c}}=\frac{1}{\left(0.01\right)\phantom{\rule{2.77695pt}{0ex}}\left(0.16\right)}$ ms = 625 ms. Thus, n and c change more slowly than V. This time scale separation between V and (c, n) can be accentuated when C_{ m } is made smaller than the default 5 pF, i.e., when C_{ m } → 0, τ_{ V } gets smaller and V varies much faster. Thus, we can view the capacitance C_{ m } as a representative of the dimensionless singular perturbation parameter ε_{1} in this model (Eq. 1.1).
All numerical simulations and bifurcation diagrams (both one- and two-parameter) are constructed using the XPPAUT software package [26], using the Runge-Kutta integration method, and computer codes can be downloaded from the following website: http://www.math.fsu.edu/~bertram/software/pituitary. The surface in Figure 9 was constructed using the AUTO software package [27]. All graphics were produced with the software package MATLAB.
3 Geometric Singular Perturbation Theory
The reduced system
This yields two constant V values and two equations for n in the form of n = n(c). Thus, the fold curves (L^{±}) are (V^{±}, c, n^{±} (c)) where V^{-} and V^{+} are constant V values. The curve L^{+} is projected vertically (along the fast variable V) onto the lower sheet to obtain the projection curve P(L^{+}), and similarly for the (L^{-}) projection onto the upper sheet. Figure 2 shows the critical manifold, the fold curves and the projections of the fold curves.
The desingularized system describes the flow on the critical manifold. Because of the time rescaling, the flow on the middle sheet, where $\frac{\partial f}{\partial V}>0$, must be reversed to obtain the equivalent reduced flow.
FoLded singuLarities and canards
A folded singularity is classified as a folded node if the eigenvalues are real and have the same sign, a folded saddle if the eigenvalues are real and have opposite signs, or a folded focus if the eigenvalues are complex [22, 23, 25, 29]. For parameter values used in Figure 2, the system has a folded node (with negative eigenvalues) on L^{+} (FN, blue point, in Figure 2), and a folded focus on L^{-}(not shown).
Singular periodic orbits, relaxation oscillations, and mixed mode oscillations
A singular periodic orbit (Figure 2, black curve with arrows) can be constructed by solving the desingularized system for the flow on the top and bottom sheets of the critical manifold, and then projecting the trajectory from one sheet to the other along fast fibers when the trajectory reaches a fold curve. The singular periodic orbit is the closed curve constructed in this way. This process was discussed in detail in [22, 28, 30]. Briefly, the trajectory moves along the bottom sheet until L^{-} is reached. At this point the reduced flow is singular ($\frac{\partial f}{\partial V}=0$). The quasi-steady state assumption f(V, c, n) = 0 is no longer valid and there is a rapid motion away from the fold curve L^{-}. This rapid motion is seen as vertical movement to the top sheet (the dynamics are governed by the layer problem, see [22, 28]). The trajectory moves to a point on P(L^{-}) and from there is once again governed by the desingularized equations, moving along the top sheet until L^{+} is reached. The fast vertical downward motion along fast fibers returns the trajectory to a point on P(L^{+}) on the bottom sheet, completing the cycle.
When the singular periodic orbit reaches L^{-} it jumps up to a point on P(L^{-}). If this point on P(L^{-}) is in the singular funnel, then the orbit will move through the FN. Otherwise it will not. Let δ denote the distance measured along P(L^{-}) from the phase point on P(L^{-}) of the singular periodic orbit to the strong canard (SC in Figure 3). When the phase point is on the strong canard, δ = 0. Let δ > 0 when the phase point is in the singular funnel and δ < 0 when the phase point is outside the singular funnel. Singular canards are produced when δ > 0.
In Figure 3(a) the singular periodic orbit jumps to a point on P(L^{-}) outside of the singular funnel (δ < 0), so it does not enter the FN. This orbit is a relaxation oscillation [31]. In Figure 3(b)δ > 0, so the orbit is a singular canard. Away from the singular limit, this singular canard perturbs to an actual canard that is characterized by small oscillations about L^{+}[22]. The combination of these small oscillations with the large oscillations that occur due to jumps between upper and lower sheets yields mixed mode oscillations [24, 32]. The small oscillations have zero amplitude in the singular case, which grows as $\sqrt{{C}_{m}}$ for C_{ m } sufficiently small [23]. A discriminating condition between relaxation and mixed mode oscillations is δ = 0, where the singular periodic orbit jumps to P(L^{-}) on the SC curve.
4 Analysis of the desingularized system and folded nodes
For values g_{ K } < 0.5131 nS, there is a stable node on CN1 (A_{1}), which would be on the top sheet of the critical manifold. There are also two folded saddles on L^{+} (B_{1} and C_{1}) and two folded foci on L^{-} (D_{1} and E_{1}). When g_{ K } is increased to 0.5131 nS the stable node A_{1} moves down and to the left and the folded saddle B_{1} moves to the left. These two equilibria coalesce at a transcritical bifurcation (TR1). This transcritical bifurcation corresponds to a bifurcation of folded singularities called a type II folded saddle-node [22, 30, 33]. Following this bifurcation, the folded singularity is a folded node. For g_{ K } = 4 nS, the equilibria on L^{+} are the folded node (B_{3}) and the folded saddle (C_{3}). The equilibrium on CN1 (A_{3}) is now a saddle point. There is no qualitative change of equilibria on L^{-}.
When g_{ K } is increased to 7.588 nS the equilibria B_{3} and C_{3} coalesce at a saddle-node bifurcation point (SN1). This is a standard saddle-node bifurcation of folded singularities and is called a type I folded saddle-node [22, 30, 33]. As g_{ K } is increased to 43.1 nS, the folded focus D_{5} moves to the left and changes to a folded node at D_{6}. The saddle points on CN1 move downward and to the left as g_{ K } is increased. For g_{ K } = 129.2 nS, the saddle point A_{6} coalesces with the fold node D_{6} at a second transcritical bifurcation (TR2); again a type II folded saddle-node. Beyond this, the ordinary singularity (A_{8}, A_{9}) is stable and the folded singularity becomes a folded saddle. Moreover the folded focus E_{6} has become a folded node (E_{7}). As g_{ K } is increased further to 137.2 nS, there is a second type I saddle-node bifurcation (SN2) at which the folded node and the folded saddle coalesce and disappear. For the values g_{ K } > 137.2 nS, the only equilibrium is on CN1 and is an ordinary stable node (A_{9}). This is on the bottom sheet of the critical manifold.
The folded node (B) and the saddle point (A) coalesce at a transcritical bifurcation (type II folded saddle-node) when g_{ BK } = 3.96 nS (TR1 in Figure 6(d)). Beyond this, the ordinary singularity (A) is a stable node that lies on the top sheet of the critical manifold. When g_{ BK } = 20 nS the folded singularities are either saddles or foci, Figure 6(e). For g_{ BK } ≈ 32.12 nS the two folded foci on L^{-}change to folded nodes. Finally, when g_{ BK } is increased to 32.1224 nS, the fold curves L^{+} and L^{-} merge. As a result, the folded saddles coalesce with the folded nodes at type I folded saddle-node bifurcations (SN3 and SN4 in Figure 6(f)). Beyond this, there is only a stable node (A in Figure 6(g)). The disappearance of the L^{+} and L^{-} curves correspond to the disappearance of the fold in the critical manifold.
For g_{ K } and g_{ BK } values in regions A, D and E there is only a stable node and the full system is in a depolarized (A) or hyperpolarized (D or E) steady state. In the left portion of region C there is a folded focus which becomes a folded node in the right portion of C. This family of folded singularities is on L^{-}. In region D there is a folded node on L^{-} for negative values of c. Region B consists of the folded nodes on L^{+}, and it is the key region for the existence of mixed mode oscillations, since δ > 0 for much of this region (shown below).
5 Twisted slow manifolds and secondary canards
The folded nodes discussed above are important since they yield small oscillations (for C_{ m } > 0) in all trajectories entering the singular funnel. In this section we explain the genesis of those oscillations (for more details, see [22, 23, 28, 32]).
The primary weak canard corresponds to the weak eigendirection of the folded node. It is at the intersection of the invariant manifolds ${\mathsf{\text{S}}}_{a,{C}_{m}}^{+}$ and ${\mathsf{\text{S}}}_{r,{C}_{m}}$ and serves as their axis of rotation. All other canards twist about the primary weak canard; they follow ${\mathsf{\text{S}}}_{a,{C}_{m}}^{+}$ as it twists and then follow ${\mathsf{\text{S}}}_{r,{C}_{m}}$ for a distance as it twists. The primary strong canard, which corresponds to the strong eigendirection of the folded node, moves along ${\mathsf{\text{S}}}_{a,{C}_{m}}^{+}$ to ${\mathsf{\text{S}}}_{r,{C}_{m}}$ without any rotation (SC, green curve in Figure 9). Other, secondary, canards rotate a number of times, depending on how close they are to the primary strong canard. A secondary canard that makes k small rotations in the vicinity of the folded node is called the k ^{ th } secondary canard. Figure 9 shows the first (ξ_{1}, gray), second (ξ_{2}, purple) and third (ξ_{3}, olive) secondary canards that make one, two and three rotations, respectively. For C_{ m } > 0, but small, there are S_{ max } - 1 secondary canards which divide the funnel region between the primary canards into S_{ max } subsectors [24]. The first subsector is bounded by the strong canard SC and the first secondary canard ξ_{1} and trajectories entering here have one rotation. The second subsector is bounded by ξ_{1} and ξ_{2} and trajectories entering here have two rotations. The last subsector is bounded by the last secondary canard and the primary weak canard. The maximal rotation number is achieved in the last subsector; trajectories entering here have S_{ max } rotations [23, 28, 32].
Figure 9 also shows a portion of the pseudo-plateau burst trajectory (PPB, black curve) for C_{ m } = 2 pF. It enters the funnel region in the rotational subsector bounded by ξ_{1} and ξ_{2}, and hence, makes two full rotations and then leaves the repelling sheet as it moves towards the lower attracting manifold ${\mathsf{\text{S}}}_{a,{C}_{m}}^{-}$. These rotations are the small oscillations or "spikes" during the burst active phase.
6 The boundaries of mixed mode oscillations
For a periodic mixed mode oscillation (i.e., pseudo-plateau bursting) solution to exist, there must be a folded node singularity and the periodic orbit must enter the funnel. In this section we construct curves in the two-parameter g_{ K } -g_{ BK } plane that form boundaries for the existence of mixed mode oscillations.
In the bursting region the active phase duration and the number of spike per burst vary with respect to the values of g_{ K } and g_{ BK } . The size of each circle represents the active phase duration, and the color of the circle (from cyan to dark red) represents the number of spikes in a burst. A burst with larger number of spikes has longer active phase duration, and in an actual cell this determines the amount of Ca^{2+} influx and hormone released. The bursts that have the shorter active phase duration and the smaller number of spikes occur near the right branch of the PD curve. These bursts are represented by smaller cyan circles in Figure 12. For example, when g_{ BK } = 1 nS and g_{ K } = 6 nS the system produces bursting oscillations with three spikes per burst (as in Figure 4(f)). When one moves away from the right to the left branch of the PD curve by increasing g_{ BK } or decreasing g_{ K } the burst duration becomes longer and the number of spikes in a burst becomes larger. The longest active phase duration is about 8.4 sec and the largest number of spikes per burst is about 36, represented by the largest dark red circle. These values will change when C_{ m } is changed.
The region between the HB and the left branch of the PD curves is bistable between bursting and continuous spiking. Orange circles with small black circles at the centers represent bistable solutions that are simulated by varying the initial conditions. This shows that the borders of the bursting region are the HB and the right branch of the PD curves. The dark blue circles represent bursting oscillations without small oscillations since the amplitudes of the spikes are almost zero, i.e., the small oscillations are too small to see.
The results that are shown in Figure 12 are very consistent with the analysis of the mixed mode oscillations in Figure 11. The HB and TR1 curves overlap, demonstrating that for small C_{ m } the HB of the full system corresponds to a type II saddle-node bifurcation of the desingularized system. Also, the HB curve and the left branch of the PD curve are almost indistinguishable for small C_{ m } . For these C_{ m } values (C_{ m } < 0.001 pF), the right branch of the PD curve converges to the δ = 0 curve of the desingularized system. Hence, the left and right borders of the MMOs in the singular limit C_{ m } → 0 pF correspond to the left and right borders of the bursting region of the full system for C_{ m } > 0, with the exception that the bursting region is smaller for larger values of C_{ m } . Also, the bistable region between the PD and HB curves only exists as the left PD moves away from the HB, which occurs as C_{ m } is increased.
In Figure 11 the MMOs region delimited by the TR1 and δ = 0 curves can be divided into subregions that have different numbers of small oscillations. For parameter values in the subregion near the curve δ = 0 the periodic orbit enters the funnel region near the strong (primary) canard. This subregion corresponds to the first subsector of the funnel region, and for C_{ m } > 0 only one small oscillation occurs in a burst. This corresponds to the jump from the lower attracting sheet to the upper attracting sheet and is not due to the folded node. When one moves leftward by decreasing g_{ K } , δ increases and the periodic orbit enters the funnel region through other subsectors. As a result, the number of small oscillations in a burst increases. When one moves to the subregion near or on the TR1 curve by decreasing g_{ K } further, the periodic orbit enters the funnel region through the last subsector. The number of small oscillations is closer to S_{ max } , the maximum number of spikes in a burst as determined by the eigenvalues of the folded node. Moreover, increasing g_{ BK } has the same effect as decreasing g_{ K } . These trends in the number of small oscillations obtained from an analysis of the desingularized system [28] are expressed far from the singular limit as shown in Figure 12 where C_{ m } = 5 pF. Here the longest bursts occur near the HB curves, as predicted.
7 A comparison with a two-fast/one-slow variable analysis
Using a one-fast/two-slow variable analysis we have shown the genesis of the spikes in a burst and how the number of spikes in a burst varies in the g_{ K } -g_{ BK } parameter space. The regions for steady states, pseudo-plateau bursting (mixed mode oscillations) and spiking are clearly identified in this parameter space (Figure 11). This has been done by investigating the qualitative changes of the desingularized system when parameters g_{ K } (Figure 5) and g_{ BK } (Figure 6) are varied, which are summarized in Figure 7.
In Figure 13(a), with g_{ K } = 0.1 nS, there is an intersection of the c-nullcline on the upper stable branch at location A_{1}. This is a stable equilibrium of the full 3-dimensional system, and corresponds to A_{1} in the analysis shown in Figure 5. Thus, both types of analysis indicate that the system will come to rest at a depolarized steady state when g_{ K } = 0.1 nS.
When g_{ K } is increased there is a subcritical Hopf bifurcation on the upper branch with emergent unstable periodic solutions of the fast-subsystem. This is shown in Figure 13(b) for the case g_{ K } = 4 nS. Pseudo-plateau bursting occurs for this and nearby values of g_{ K } . The full system unstable equilibrium (A_{3}) corresponds to A_{3} in Figure 5.
The superimposed burst trajectory in Figure 13(b) only weakly follows the fast-subsystem bifurcation diagram. Most notably, there are no stable periodic solutions of the fast subsystem, only bistability between two steady states. Also, the trajectory never follows the lower branch of stationary solutions and greatly overshoots the lower knee.
The subcritical Hopf bifurcation migrates leftward when g_{ K } is increased to 5.1 nS. The unstable branch of periodics goes through a saddle-node bifurcation, yielding a branch of stable periodic solutions of the fast subsystem (Figure 13(c)). There is bistability between upper and lower branches of the z-curve which is typically a necessary condition for bursting with this type of analysis. However, bursting is not produced for this value of g_{ K } . Instead, the system spikes continuously.
This example illustrates that features well described by the one-fast/two-slow variable analysis are not at all well described by a standard two-fast/one-slow variable analysis. Most notably, the transition from bursting to spiking is well characterized in the one-fast/two-slow variable analysis as the point at which δ = 0. Note that this is not a bifurcation point of the desingularized system, but reflects the jump point from the lower sheet of the slow manifold to the the upper sheet. In contrast, the bursting to spiking transition is not predicted from the two-fast/one-slow analysis, and indeed the periodic spiking trajectory of the full system occurs over a range of the fast-subsystem bifurcation diagram that contains only stable equilibria. The one-fast/two-slow approximation is good even at higher values of C_{ m } , for example, when C_{ m } = 5 pF (Figure 12). Similar remarks apply for smaller values of C_{ m } , where the one-fast/two-slow approximation becomes more accurate while the two-fast/one-slow approximation does not. The two-fast/one-slow approximation becomes more accurate when c is much slower than both V and n, but in this case only a stable steady solution or a relaxation oscillation is produced.
8 Discussion
The canard mechanism has been used to understand mixed mode oscillations in several neuronal models [30, 37, 40–44]. In these examples, the small oscillations correspond to subthreshold oscillations that occur between the electrical impulses. We have previously analyzed pseudo-plateau bursting in a pituitary lactotroph model using canard theory [28]. However, the model used was a simplification in which the cytosolic free Ca^{2+} concentration was treated as a fixed parameter and the second slow variable (in addition to the variable n used here) was an inactivation variable for an A-type K^{+} current. In the current paper, we again focused on pseudo-plateau bursting in a pituitary lactotroph model, but now with emphasis on a BK-type K^{+} current. In this analysis, we have examined the effects of changing the parameters C_{ m } , g_{ K } and g_{ BK } . The parameter g_{ BK } is important for producing bursting oscillations in actual pituitary cells in which bursting is converted to spiking when BK-type K^{+} channels are blocked [45].
Here, using C_{ m } to control the separation in time scales, we identified two slow variables (n, c) and one fast variable (V). Using the one-fast/two-slow variable analysis we showed that pseudo-plateau bursting is a canard-induced mixed mode oscillation. There are two main requirements for the existence of these bursting oscillations [22–24, 32]. One is that the desingularized system must have a folded node singularity, i.e., the eigenvalue ratio (μ) has to be positive. The second requirement is that the singular periodic orbit should enter the singular funnel and pass through the folded node, i.e., δ should be positive. In short, canard-induced mixed mode oscillations exist if both μ and δ are positive.
Using this technique we can understand several features of the burst and several trends that occur as parameters are varied. When both μ and δ are positive, small oscillations are produced during the active phase of a burst and their amplitude is proportional to $\sqrt{{C}_{m}}$ for C_{ m } sufficiently small [23]. We obtained the bursting borders in the (g_{ K } , g_{ BK } )-plane (Figure 11 and 12), and predicted how the active phase duration and the number of spikes per burst vary with changes in parameters.
The singular perturbation analysis performed here is technically more effective and informative in the singular limit (i.e., for sufficiently small values of C_{ m } ) [22, 23]. However, it provides useful information even far from this limit, as we showed in Figures 11 and 12. Eventually, as the singular parameter (C_{ m } ) is increased sufficiently, new dynamics will be introduced, and the insights from the singular analysis are no longer valid.
The one-fast/two-slow decomposition used here contrasts with the two-fast/one-slow variable analysis used previously for pseudo-plateau bursting [10–13]. Our analysis explains the origin of the small-amplitude spikes that occur during the active phase of pseudo-plateau bursting, the transition between spiking and bursting, and information about how the number of spikes per burst varies with parameters. While the two-fast/one-slow variable analysis provides little information on these things, it does provide valuable information about how one can make a transition between plateau and pseudo-plateau bursting as one or more parameters are changed [12]. It also provides information about complex phase resetting properties [11] and the termination of spikes in a burst [17]. Both fast/slow decompositions are approximations, however, to a system that evolves on three time scales. Some studies [13, 17, 18] focus on the dynamics of the full system, and illustrate the complexity of the seemingly simple set of equations. The advantage of obtaining useful information of the full system by a two-fast/one-slow or one-fast/two-slow decomposition points to the fact that system (2.1)-(2.3) actually evolves on three time scales: V fast, n intermediate and c slow. This can also be seen by the magnitude of μ which is bounded from above by μ_{ max } ≈ 0.07 (Figure 8(a)). Hence, we are close to folded saddle-node regimes (type I and type II) [33, 38] and a more detailed bifurcation analysis may explain the relation between the two-fast/one-slow and one-fast/two-slow splitting. This is left for future work.
Declarations
Acknowledgements
This work was supported by NSF grant DMS 0917664 to RB and NIH grant DK 043200 to RB and JT.
Authors’ Affiliations
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