Canard-Induced Mixed Mode OscillationsIn Pituitary Lactotrophs
Theodore Vo1 Martin Wechselberger1
Wondimu Teka2 Richard Bertram2 Joël Tabak3
1School of Mathematics & StatisticsUniversity of Sydney
2Department of MathematicsFlorida State University
3Department of Biological ScienceFlorida State University
SIAM LIFE SCIENCES CONFERENCEAUGUST 10, 2012
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
Motivation
Theo Vo MMOs In A 3-Timescale System
(a) http://www.empowher.com/media/reference/apoplexy
(b) http://www.scholarpedia.org/article/Models_of_hypothalamus
(b) Structure
(a) Location
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
Motivation
1 6.7−70
−60
−50
−40
−30
−20
−10
V (mV)
1 sec
(a)
40 mV
5 sec
(b)
Figure 1: (a) Pseudo-plateau bursting in a GH4 pituitary cell line. (b) Plateau bursting ina neonatal CA3 hippocampal principal neuron. Reprinted with permission from [14].
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 ϵ2 ≈ 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-Botzinger neurons of the
brain stem [20], trigeminal motoneurons [21] or neonatal CA3 hippocampal principal
neurons [14], Fig. 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 ϵ1 ≈ 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
3
Journal of Mathematical Neuroscience (2012) 1:12
Conversely, LH secretion from gonadotrophs is underpositive hypothalamic control by Ca2C-mobilizing gonado-tropin-releasing hormone (GnRH) receptors [8], and onlyin neonatal gonadotrophs is GnRH-stimulated gonado-tropin secretion inhibited by the pineal hormone mela-tonin [9]. Here, we review biophysical findings that helpto clarify what endows lactotrophs and somatotrophs, butnot gonadotrophs, with the ability to secrete in theabsence of ligand stimulation.
Excitability of pituitary cells and basal secretionRecent results obtained with perfused anterior pituitarycells indicate that spontaneous GH and PRL secretion ismuch higher than basal LH secretion [10]. Figure 1acompares the basal secretion of these three hormones inperfused pituitary cells. Most basal GH and PRL secretionis extracellular Ca2C dependent. Application of tetrodo-toxin (TTX), a specific voltage-gated NaC channel blocker,does not alter the pattern of spontaneous GH, PRL or LHsecretion. By contrast, application of the Ca2C channelblockers, nifedipine and Cd2C, inhibits basal GH and PRLsecretion without affecting basal LH secretion [10]. Theseresults indicate that the main fraction of spontaneous GHand PRL in vitro release reflects regulated, Ca2C influx-dependent exocytosis.
The extracellular Ca2C dependence of basal PRL andGH release, but not LH release, is consistent with findingsthat cultured somatotrophs [11], lactotrophs [12] andimmortalized pituitary cells [13,14], in addition to in situsomatotrophs [15], spontaneously fire action potentials(APs), whereas most unstimulated gonadotrophs frommale rats are quiescent [16]. However, basal gonadotropinsecretion is also low in cells from female animals, eventhough they fire APs spontaneously [10]. A comparison ofspontaneous electrical membrane activity in all threehormone-secreting cell types under identical recordingconditions using the perforated patch, whole-cell configur-ation and pituitary cells from randomly cycling femalerats is shown in Figure 1b. Most somatotrophs andlactotrophs fire Ca2C-dependent bursts of APs with afrequency of w0.3 Hz, and w50% of the gonadotrophsexhibit spontaneous single AP firing with a frequency of0.7 Hz [10].
In general, excitable cells secrete in a regulatedmannerthrough AP-driven Ca2C influx and Ca2C-dependentexocytosis. There are some differences in molecularmechanisms underlying exocytosis and recapture ofsynaptic and secretory vesicles [17]. In addition, atsynapses a single AP and a train of single APs aresufficient to trigger secretion, whereas in neuroendocrineand endocrine cells prolonged depolarization is required toactivate the exocytotic pathway [18]. The rise in intra-cellular Ca2C concentration ([Ca2C]i) is needed for twosteps in exocytosis: priming the vesicles to the site ofrelease and fusion of vesicles with the plasma membrane[19]. Thus, AP-driven secretion depends on the [Ca2C]i inthe vicinity of the secretory vesicles. Consistent with this,it appears that the pattern of basal anterior pituitaryhormone secretion is determined by differences in theability of APs to increase [Ca2C]i in these three cell types.As shown in Figure 1c, somatotrophs and lactotrophs
generate slow resting membrane potential oscillations,with superimposed bursts of APs, with an averagedduration of seconds, accompanied by high-amplitude[Ca2C]i signals that range from 0.3 mM to 1.2 mM. Bycontrast, gonadotrophs fire high-amplitude, single spikeswith a duration of milliseconds, which generate low-amplitude [Ca2C]i signals ranging from 20 nM to 70 nM.Immortalized pituitary cells can exhibit both single spikeand plateau bursting patterns of firing [20].
** *
15 s 15 s 15 s
1 s 1 s 0.1 s
0
-40
-80
0
-40
-80
0
1
0
1
(a)
(b)
(c)
Somatotrophs Lactotrophs Gonadotrophs
0
-40
-80
0
1
1 min
30 min
300
150
0
Secr
etio
n (n
g/m
l)
-Ca2+ -Ca2+ -Ca2+
-Ca2+ -Ca2+ -Ca2+
V m (m
V)V m
(mV)
[Ca2
+ ] (µ
M)
[Ca2
+ ] (µ
M)
[Ca2
+ ] (µ
M)
V m (m
V)
Figure 1. Characterization of electrical activity, calcium signaling and secretion inresting pituitary cells. (a) Effects of removal of extracellular Ca2C on basal GH (leftpanel), PRL (central panel) and LH (right panel) release. Cells were transientlyperfused with Ca2C-deficient medium containing 100 mM EGTA (-Ca2C). Secretionwas normalized to account for differences in the sizes of somatotroph, lactotrophand gonadotroph populations in anterior pituitary cell preparations. (b,c) Simul-taneous measurements of membrane potential (Vm) and [Ca2C]i in singlesomatotrophs (left panels), lactotrophs (central panels), and gonadotrophs (rightpanels). (b) Extracellular Ca2C dependence of electrical activity and Ca2C signaling.(c) Plateau bursting versus single AP firing. Asterisks in the upper panel illustrateselected APs and corresponding Ca2C transients, which are shown in the bottomtraces on an extended timescale. (Notice the difference in the timescale forgonadotrophs.) Derived from [10].
Review TRENDS in Endocrinology and Metabolism Vol.16 No.4 May/June 2005 153
www.sciencedirect.com
Conversely, LH secretion from gonadotrophs is underpositive hypothalamic control by Ca2C-mobilizing gonado-tropin-releasing hormone (GnRH) receptors [8], and onlyin neonatal gonadotrophs is GnRH-stimulated gonado-tropin secretion inhibited by the pineal hormone mela-tonin [9]. Here, we review biophysical findings that helpto clarify what endows lactotrophs and somatotrophs, butnot gonadotrophs, with the ability to secrete in theabsence of ligand stimulation.
Excitability of pituitary cells and basal secretionRecent results obtained with perfused anterior pituitarycells indicate that spontaneous GH and PRL secretion ismuch higher than basal LH secretion [10]. Figure 1acompares the basal secretion of these three hormones inperfused pituitary cells. Most basal GH and PRL secretionis extracellular Ca2C dependent. Application of tetrodo-toxin (TTX), a specific voltage-gated NaC channel blocker,does not alter the pattern of spontaneous GH, PRL or LHsecretion. By contrast, application of the Ca2C channelblockers, nifedipine and Cd2C, inhibits basal GH and PRLsecretion without affecting basal LH secretion [10]. Theseresults indicate that the main fraction of spontaneous GHand PRL in vitro release reflects regulated, Ca2C influx-dependent exocytosis.
The extracellular Ca2C dependence of basal PRL andGH release, but not LH release, is consistent with findingsthat cultured somatotrophs [11], lactotrophs [12] andimmortalized pituitary cells [13,14], in addition to in situsomatotrophs [15], spontaneously fire action potentials(APs), whereas most unstimulated gonadotrophs frommale rats are quiescent [16]. However, basal gonadotropinsecretion is also low in cells from female animals, eventhough they fire APs spontaneously [10]. A comparison ofspontaneous electrical membrane activity in all threehormone-secreting cell types under identical recordingconditions using the perforated patch, whole-cell configur-ation and pituitary cells from randomly cycling femalerats is shown in Figure 1b. Most somatotrophs andlactotrophs fire Ca2C-dependent bursts of APs with afrequency of w0.3 Hz, and w50% of the gonadotrophsexhibit spontaneous single AP firing with a frequency of0.7 Hz [10].
In general, excitable cells secrete in a regulatedmannerthrough AP-driven Ca2C influx and Ca2C-dependentexocytosis. There are some differences in molecularmechanisms underlying exocytosis and recapture ofsynaptic and secretory vesicles [17]. In addition, atsynapses a single AP and a train of single APs aresufficient to trigger secretion, whereas in neuroendocrineand endocrine cells prolonged depolarization is required toactivate the exocytotic pathway [18]. The rise in intra-cellular Ca2C concentration ([Ca2C]i) is needed for twosteps in exocytosis: priming the vesicles to the site ofrelease and fusion of vesicles with the plasma membrane[19]. Thus, AP-driven secretion depends on the [Ca2C]i inthe vicinity of the secretory vesicles. Consistent with this,it appears that the pattern of basal anterior pituitaryhormone secretion is determined by differences in theability of APs to increase [Ca2C]i in these three cell types.As shown in Figure 1c, somatotrophs and lactotrophs
generate slow resting membrane potential oscillations,with superimposed bursts of APs, with an averagedduration of seconds, accompanied by high-amplitude[Ca2C]i signals that range from 0.3 mM to 1.2 mM. Bycontrast, gonadotrophs fire high-amplitude, single spikeswith a duration of milliseconds, which generate low-amplitude [Ca2C]i signals ranging from 20 nM to 70 nM.Immortalized pituitary cells can exhibit both single spikeand plateau bursting patterns of firing [20].
** *
15 s 15 s 15 s
1 s 1 s 0.1 s
0
-40
-80
0
-40
-80
0
1
0
1
(a)
(b)
(c)
Somatotrophs Lactotrophs Gonadotrophs
0
-40
-80
0
1
1 min
30 min
300
150
0
Secr
etio
n (n
g/m
l)
-Ca2+ -Ca2+ -Ca2+
-Ca2+ -Ca2+ -Ca2+
V m (m
V)V m
(mV)
[Ca2
+ ] (µ
M)
[Ca2
+ ] (µ
M)
[Ca2
+ ] (µ
M)
V m (m
V)
Figure 1. Characterization of electrical activity, calcium signaling and secretion inresting pituitary cells. (a) Effects of removal of extracellular Ca2C on basal GH (leftpanel), PRL (central panel) and LH (right panel) release. Cells were transientlyperfused with Ca2C-deficient medium containing 100 mM EGTA (-Ca2C). Secretionwas normalized to account for differences in the sizes of somatotroph, lactotrophand gonadotroph populations in anterior pituitary cell preparations. (b,c) Simul-taneous measurements of membrane potential (Vm) and [Ca2C]i in singlesomatotrophs (left panels), lactotrophs (central panels), and gonadotrophs (rightpanels). (b) Extracellular Ca2C dependence of electrical activity and Ca2C signaling.(c) Plateau bursting versus single AP firing. Asterisks in the upper panel illustrateselected APs and corresponding Ca2C transients, which are shown in the bottomtraces on an extended timescale. (Notice the difference in the timescale forgonadotrophs.) Derived from [10].
Review TRENDS in Endocrinology and Metabolism Vol.16 No.4 May/June 2005 153
www.sciencedirect.com
Conversely, LH secretion from gonadotrophs is underpositive hypothalamic control by Ca2C-mobilizing gonado-tropin-releasing hormone (GnRH) receptors [8], and onlyin neonatal gonadotrophs is GnRH-stimulated gonado-tropin secretion inhibited by the pineal hormone mela-tonin [9]. Here, we review biophysical findings that helpto clarify what endows lactotrophs and somatotrophs, butnot gonadotrophs, with the ability to secrete in theabsence of ligand stimulation.
Excitability of pituitary cells and basal secretionRecent results obtained with perfused anterior pituitarycells indicate that spontaneous GH and PRL secretion ismuch higher than basal LH secretion [10]. Figure 1acompares the basal secretion of these three hormones inperfused pituitary cells. Most basal GH and PRL secretionis extracellular Ca2C dependent. Application of tetrodo-toxin (TTX), a specific voltage-gated NaC channel blocker,does not alter the pattern of spontaneous GH, PRL or LHsecretion. By contrast, application of the Ca2C channelblockers, nifedipine and Cd2C, inhibits basal GH and PRLsecretion without affecting basal LH secretion [10]. Theseresults indicate that the main fraction of spontaneous GHand PRL in vitro release reflects regulated, Ca2C influx-dependent exocytosis.
The extracellular Ca2C dependence of basal PRL andGH release, but not LH release, is consistent with findingsthat cultured somatotrophs [11], lactotrophs [12] andimmortalized pituitary cells [13,14], in addition to in situsomatotrophs [15], spontaneously fire action potentials(APs), whereas most unstimulated gonadotrophs frommale rats are quiescent [16]. However, basal gonadotropinsecretion is also low in cells from female animals, eventhough they fire APs spontaneously [10]. A comparison ofspontaneous electrical membrane activity in all threehormone-secreting cell types under identical recordingconditions using the perforated patch, whole-cell configur-ation and pituitary cells from randomly cycling femalerats is shown in Figure 1b. Most somatotrophs andlactotrophs fire Ca2C-dependent bursts of APs with afrequency of w0.3 Hz, and w50% of the gonadotrophsexhibit spontaneous single AP firing with a frequency of0.7 Hz [10].
In general, excitable cells secrete in a regulatedmannerthrough AP-driven Ca2C influx and Ca2C-dependentexocytosis. There are some differences in molecularmechanisms underlying exocytosis and recapture ofsynaptic and secretory vesicles [17]. In addition, atsynapses a single AP and a train of single APs aresufficient to trigger secretion, whereas in neuroendocrineand endocrine cells prolonged depolarization is required toactivate the exocytotic pathway [18]. The rise in intra-cellular Ca2C concentration ([Ca2C]i) is needed for twosteps in exocytosis: priming the vesicles to the site ofrelease and fusion of vesicles with the plasma membrane[19]. Thus, AP-driven secretion depends on the [Ca2C]i inthe vicinity of the secretory vesicles. Consistent with this,it appears that the pattern of basal anterior pituitaryhormone secretion is determined by differences in theability of APs to increase [Ca2C]i in these three cell types.As shown in Figure 1c, somatotrophs and lactotrophs
generate slow resting membrane potential oscillations,with superimposed bursts of APs, with an averagedduration of seconds, accompanied by high-amplitude[Ca2C]i signals that range from 0.3 mM to 1.2 mM. Bycontrast, gonadotrophs fire high-amplitude, single spikeswith a duration of milliseconds, which generate low-amplitude [Ca2C]i signals ranging from 20 nM to 70 nM.Immortalized pituitary cells can exhibit both single spikeand plateau bursting patterns of firing [20].
** *
15 s 15 s 15 s
1 s 1 s 0.1 s
0
-40
-80
0
-40
-80
0
1
0
1
(a)
(b)
(c)
Somatotrophs Lactotrophs Gonadotrophs
0
-40
-80
0
1
1 min
30 min
300
150
0Se
cret
ion
(ng/
ml)
-Ca2+ -Ca2+ -Ca2+
-Ca2+ -Ca2+ -Ca2+
V m (m
V)V m
(mV)
[Ca2
+ ] (µ
M)
[Ca2
+ ] (µ
M)
[Ca2
+ ] (µ
M)
V m (m
V)
Figure 1. Characterization of electrical activity, calcium signaling and secretion inresting pituitary cells. (a) Effects of removal of extracellular Ca2C on basal GH (leftpanel), PRL (central panel) and LH (right panel) release. Cells were transientlyperfused with Ca2C-deficient medium containing 100 mM EGTA (-Ca2C). Secretionwas normalized to account for differences in the sizes of somatotroph, lactotrophand gonadotroph populations in anterior pituitary cell preparations. (b,c) Simul-taneous measurements of membrane potential (Vm) and [Ca2C]i in singlesomatotrophs (left panels), lactotrophs (central panels), and gonadotrophs (rightpanels). (b) Extracellular Ca2C dependence of electrical activity and Ca2C signaling.(c) Plateau bursting versus single AP firing. Asterisks in the upper panel illustrateselected APs and corresponding Ca2C transients, which are shown in the bottomtraces on an extended timescale. (Notice the difference in the timescale forgonadotrophs.) Derived from [10].
Review TRENDS in Endocrinology and Metabolism Vol.16 No.4 May/June 2005 153
www.sciencedirect.com
Conversely, LH secretion from gonadotrophs is underpositive hypothalamic control by Ca2C-mobilizing gonado-tropin-releasing hormone (GnRH) receptors [8], and onlyin neonatal gonadotrophs is GnRH-stimulated gonado-tropin secretion inhibited by the pineal hormone mela-tonin [9]. Here, we review biophysical findings that helpto clarify what endows lactotrophs and somatotrophs, butnot gonadotrophs, with the ability to secrete in theabsence of ligand stimulation.
Excitability of pituitary cells and basal secretionRecent results obtained with perfused anterior pituitarycells indicate that spontaneous GH and PRL secretion ismuch higher than basal LH secretion [10]. Figure 1acompares the basal secretion of these three hormones inperfused pituitary cells. Most basal GH and PRL secretionis extracellular Ca2C dependent. Application of tetrodo-toxin (TTX), a specific voltage-gated NaC channel blocker,does not alter the pattern of spontaneous GH, PRL or LHsecretion. By contrast, application of the Ca2C channelblockers, nifedipine and Cd2C, inhibits basal GH and PRLsecretion without affecting basal LH secretion [10]. Theseresults indicate that the main fraction of spontaneous GHand PRL in vitro release reflects regulated, Ca2C influx-dependent exocytosis.
The extracellular Ca2C dependence of basal PRL andGH release, but not LH release, is consistent with findingsthat cultured somatotrophs [11], lactotrophs [12] andimmortalized pituitary cells [13,14], in addition to in situsomatotrophs [15], spontaneously fire action potentials(APs), whereas most unstimulated gonadotrophs frommale rats are quiescent [16]. However, basal gonadotropinsecretion is also low in cells from female animals, eventhough they fire APs spontaneously [10]. A comparison ofspontaneous electrical membrane activity in all threehormone-secreting cell types under identical recordingconditions using the perforated patch, whole-cell configur-ation and pituitary cells from randomly cycling femalerats is shown in Figure 1b. Most somatotrophs andlactotrophs fire Ca2C-dependent bursts of APs with afrequency of w0.3 Hz, and w50% of the gonadotrophsexhibit spontaneous single AP firing with a frequency of0.7 Hz [10].
In general, excitable cells secrete in a regulatedmannerthrough AP-driven Ca2C influx and Ca2C-dependentexocytosis. There are some differences in molecularmechanisms underlying exocytosis and recapture ofsynaptic and secretory vesicles [17]. In addition, atsynapses a single AP and a train of single APs aresufficient to trigger secretion, whereas in neuroendocrineand endocrine cells prolonged depolarization is required toactivate the exocytotic pathway [18]. The rise in intra-cellular Ca2C concentration ([Ca2C]i) is needed for twosteps in exocytosis: priming the vesicles to the site ofrelease and fusion of vesicles with the plasma membrane[19]. Thus, AP-driven secretion depends on the [Ca2C]i inthe vicinity of the secretory vesicles. Consistent with this,it appears that the pattern of basal anterior pituitaryhormone secretion is determined by differences in theability of APs to increase [Ca2C]i in these three cell types.As shown in Figure 1c, somatotrophs and lactotrophs
generate slow resting membrane potential oscillations,with superimposed bursts of APs, with an averagedduration of seconds, accompanied by high-amplitude[Ca2C]i signals that range from 0.3 mM to 1.2 mM. Bycontrast, gonadotrophs fire high-amplitude, single spikeswith a duration of milliseconds, which generate low-amplitude [Ca2C]i signals ranging from 20 nM to 70 nM.Immortalized pituitary cells can exhibit both single spikeand plateau bursting patterns of firing [20].
** *
15 s 15 s 15 s
1 s 1 s 0.1 s
0
-40
-80
0
-40
-80
0
1
0
1
(a)
(b)
(c)
Somatotrophs Lactotrophs Gonadotrophs
0
-40
-80
0
1
1 min
30 min
300
150
0
Secr
etio
n (n
g/m
l)
-Ca2+ -Ca2+ -Ca2+
-Ca2+ -Ca2+ -Ca2+
V m (m
V)V m
(mV)
[Ca2
+ ] (µ
M)
[Ca2
+ ] (µ
M)
[Ca2
+ ] (µ
M)
V m (m
V)
Figure 1. Characterization of electrical activity, calcium signaling and secretion inresting pituitary cells. (a) Effects of removal of extracellular Ca2C on basal GH (leftpanel), PRL (central panel) and LH (right panel) release. Cells were transientlyperfused with Ca2C-deficient medium containing 100 mM EGTA (-Ca2C). Secretionwas normalized to account for differences in the sizes of somatotroph, lactotrophand gonadotroph populations in anterior pituitary cell preparations. (b,c) Simul-taneous measurements of membrane potential (Vm) and [Ca2C]i in singlesomatotrophs (left panels), lactotrophs (central panels), and gonadotrophs (rightpanels). (b) Extracellular Ca2C dependence of electrical activity and Ca2C signaling.(c) Plateau bursting versus single AP firing. Asterisks in the upper panel illustrateselected APs and corresponding Ca2C transients, which are shown in the bottomtraces on an extended timescale. (Notice the difference in the timescale forgonadotrophs.) Derived from [10].
Review TRENDS in Endocrinology and Metabolism Vol.16 No.4 May/June 2005 153
www.sciencedirect.com
Trends in Endocrinology and Metabolism (2005) 16:152-159
Theo Vo MMOs In A 3-Timescale System
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
Pituitary Lactotroph Model
Theo Vo MMOs In A 3-Timescale System
444 J Comput Neurosci (2010) 28:443–458
(a) (b)
Fig. 1 (a) Perforated patch electrical recording of bursting in a GH4 pituitary cell. (b) Bursting produced by the model with the defaultparameters shown in Table 1, with C = 6 pF, gK = 4.4 nS, and gA = 18 nS
intracellular Ca2+ concentration is fixed or eliminated(Toporikova et al. 2008). This is unusual, since theslow variation of the intracellular Ca2+ concentrationis typically responsible for clustering impulses intoperiodic episodes of activity. Previously, we analyzedsome of the properties of this unusual form of burst-ing (Toporikova et al. 2008). In the current articlewe extend this in two ways. First, we show how thebursting is situated in parameter space relative to othertypes of behaviors, and demonstrate how the numberof spikes per burst varies in parameter space. Second,we take advantage of the different time scales withinthis model to analyze the mechanism for the bursting.We show that the bursting is actually a canard-inducedmixed mode oscillation (MMO) where a MMO patterncorresponds to a switching between small-amplitudeoscillations and large relaxation oscillations. Using geo-metric singular perturbation analysis (Fenichel 1979;Jones 1995; Wechselberger 2005; Brons et al. 2006),we demonstrate the origin of the MMO, identify theregion in parameter space where the MMO exists, andshow how the number of small amplitude oscillations(or spikes) varies in parameter space. In so doing, weidentify the mechanism for this type of pseudo-plateaubursting, and also perform analyses at the singular limitto explain behaviors seen away from this limit. Mixedmode oscillations have been described previously forneural models and data (Erchova and McGonigle 2008;Wechselberger 2005; Rubin and Wechselberger 2007;Rotstein et al. 2008; Ermentrout and Wechselberger2009; Drover et al. 2005; Brons et al. 2008; Krupa et al.2008; Guckenheimer et al. 1997), but this is the firstexample where the MMOs form bursting oscillations.
2 The mathematical model
The model is a minimal description of the electricalactivity and Ca2+ dynamics in a pituitary lactotroph(Toporikova et al. 2008). This is based on anothermodel (Tabak et al. 2007), which produces burstingover a range of parameter values. For much of thisrange the bursting is driven by slow activity-dependentvariation in the Ca2+ concentration. However, fora subset of this range the Ca2+ concentration canbe clamped and bursting persists. In the model byToporikova et al. this latter form of bursting was exam-ined, and the Ca2+ variable removed, since variation inCa2+ was not necessary to produce the bursting. This isthe model we use here.
The model includes variables for the membrane po-tential (V) of the cell, the fraction of activated K+ chan-nels of the delayed rectifier type (n), and the fraction ofA-type K+ channels that are not inactivated (e). Thedifferential equations are
CdVdt
= −(ICa + IK + IA + IL) (2.1)
τededt
= e∞(V) − e (2.2)
τndndt
= n∞(V) − n (2.3)
where ICa is an inward Ca2+ current and all othercurrents are outward K+ currents. IK is a delayed rec-tifier current, IA is an A-type current that inactivateswhen V is elevated, and IL is a constant-conductance
MMOsBursting
dd
ICa = gCam∞(V )(V − VCa)
dd
IK = gK n(V − VK )
dd
ISK = gSK s∞(c)(V − VK )
dd
IA = gAa∞(V )e (V − VK )
CmdVdt
= − (ICa + IK + ISK + IA)
dndt
=n∞(V )− n
τn
dedt
=e∞(V )− e
τe
dcdt
= −fc (αICa + kc c)
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
Dynamic and Calcium-Clamped MMOs
-80
-60
-40
-20
Time
V (
mV
)
0.2
0.3
0.4
Time
c (Μ
M)
-80
-60
-40
-20
Time
V (
mV
)0.2
0.3
0.4
Timec
(ΜM
)
Theo Vo MMOs In A 3-Timescale System
Calcium-clamped MMODynamic MMO
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
Bifurcation Structure
0 2 4 6 8 100
5
10
15
20
25
gK (nS)
g A (
nS)
Dep�9
Bursting Spiking
Hop
fBif
urca
tion
Perio
d-D
oubl
ing
Time HmsL
V (
mV
)
Time HmsL
V (
mV
)
Time HmsL
V (
mV
)
Theo Vo MMOs In A 3-Timescale System
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
A 3-Timescale Problem
FAST SYSTEM
εdVdtI
= f (V , n, e, c)
dndtI
= g1(V , n)
dedtI
= g2(V , e)
dcdtI
= δ h(V , c)
FAST SYSTEM
ddτV =
Cm
g< 1 ms
ddτn ≈ 43 ms
ddτe ≈ 20 ms
ddτc =
1fckc
≈ 625 ms
0 < ε = τVτe
� 1, 0 < δ = τeτc
� 1
0 400 800 1200-80
-60
-40
-20
Time HmsL
V (
mV
)
Theo Vo MMOs In A 3-Timescale System
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
Rationale
3-Timescale Bursting Model
1 Fast3 Slow
3 Fast1 Slow
1 Fast2 Intermediate
1 Slow
Origin & Properites of
Bursting
ε→ 0
δ 6= 0
δ → 0
ε 6= 0
ε = 0
δ → 0
δ = 0
ε→ 0
Bifurcation Theory
G.S.P.T
Bifurcation Theory
G.S.P.T
Bif
urca
tion
The
ory
G.S
.P.T
Theo Vo MMOs In A 3-Timescale System
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
Geometric Singular Perturbation Theory
‘SLOW’ SYSTEM
εdVdtI
= f (V , n, e, c)
dndtI
= g1(V , n)
dedtI
= g2(V , e)
dcdtI
= δh(V , c)
FAST SYSTEM
dVdtF
= f (V , n, e, c)
dndtF
= εg1(V , n)
dedtF
= εg2(V , e)
dcdtF
= εδ h(V , c)
REDUCED PROBLEM
11
0 = f (V , n, e, c)
dndtI
= g1(V , n)
dedtI
= g2(V , e)
dcdtI
= δh(V , c)
LAYER PROBLEM
dVdtF
= f (V , n, e, c)
dndtF
= 0
dedtF
= 0
dcdtF
= 0
Theo Vo MMOs In A 3-Timescale System
tF = ε tI−−−−−−−−→
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
Geometric Singular Perturbation Theory
‘SLOW’ SYSTEM
εdVdtI
= f (V , n, e, c)
dndtI
= g1(V , n)
dedtI
= g2(V , e)
dcdtI
= δh(V , c)
FAST SYSTEM
dVdtF
= f (V , n, e, c)
dndtF
= εg1(V , n)
dedtF
= εg2(V , e)
dcdtF
= εδ h(V , c)
3D REDUCED PROBLEM
11
0 = f (V , n, e, c)
dndtI
= g1(V , n)
dedtI
= g2(V , e)
dcdtI
= δh(V , c)
1D LAYER PROBLEM
dVdtF
= f (V , n, e, c)
dndtF
= 0
dedtF
= 0
dcdtF
= 0
Theo Vo MMOs In A 3-Timescale System
ε↓0
ε↓0
tF = ε tI−−−−−−−−→
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
The Layer Problem
dVdtF
= f (V , n, e, c)
dndtF
= 0
dedtF
= 0
dcdtF
= 0
S = Sa ∪ L ∪ Sr ={(V ,n,e, c) ∈ R4 : f (V ,n,e, c) = 0
}Attracting branch, Sa := {(V , n, e, c) ∈ S : fV < 0}
Repelling branch, Sr := {(V , n, e, c) ∈ S : fV > 0}
Fold surface, L := {(V , n, e, c) ∈ S : fV = 0}
CAUTION: THESE PLOTS SHOW 3D SLICES OF A 4D PHASE SPACE
Theo Vo MMOs In A 3-Timescale System
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
The Reduced System
REDUCED SYSTEM
11
0 = f (V , n, e, c)
dndtI
= g1(V , n)
dedtI
= g2(V , e)
dcdtI
= δ h(V , c)
PROJECTION
−fVdVdtI
= Fδ(V , n, e, c)
11
0 = f (V , n, e, c)
dedtI
= g2(V , e)
dcdtI
= δ h(V , c)
Fδ(V , n, e, c) := fng1 + feg2 + δfch
Theo Vo MMOs In A 3-Timescale System
ddtI
f = 0−−−−−−→
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
The Reduced System
REDUCED SYSTEM
−fVdVdtI
= Fδ(V , n, e, c)
dedtI
= g2(V , e)
dcdtI
= δ h(V , c)
DESINGULARIZED
dVdt∗I
= Fδ(V , n, e, c)
dedt∗I
= −fV g2(V , e)
dcdt∗I
= −δ fV h(V , c)
I Ordinary singularities
E := {(V , n, e, c) ∈ S : g1 = g2 = h = 0}
I Folded singularities
Mδ := {(V , n, e, c) ∈ S : fV = Fδ = 0}
Theo Vo MMOs In A 3-Timescale System
tI = −fV t∗I−−−−−−−→
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
Singular Orbit Construction
3D Reduced
3D Reduced
1D Layer 1D Layer---
666
CAUTION: We are plotting 3D projections of a 4D system!
Theo Vo MMOs In A 3-Timescale System
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
Perturbations – Mixed Mode Oscillations
Theo Vo MMOs In A 3-Timescale System
3600 3800 4000 4200 4400 4600 4800−80
−60
−40
−20
0
Time (ms)
V (
mV
)
ε = 0.005
3600 3800 4000 4200 4400 4600 4800−80
−60
−40
−20
0
Time (ms)
V (
mV
)
ε = 0.002
3600 3800 4000 4200 4400 4600 4800−80
−60
−40
−20
0
Time (ms)
V (
mV
)
ε = 0.0001
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
Perturbations – Slow Manifolds
Theo Vo MMOs In A 3-Timescale System
ε = 0Sa
Sr
Γ
0.225 0.23 0.235
−18
−15
−12
n
V (
mV
)
Sr ∩ Σ
Sa ∩ Σ
Γ
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
Perturbations – Slow Manifolds
Theo Vo MMOs In A 3-Timescale System
ε = 0.0005Sε
a
Sεr
Γ
0.225 0.23 0.235
−18
−15
−12
n
V (
mV
)
Sεr ∩ Σ
Sεa ∩ Σ
Γ
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
Perturbations – Slow Manifolds
Theo Vo MMOs In A 3-Timescale System
ε = 0.001Sε
a
Sεr
Γ
0.225 0.23 0.235
−18
−15
−12
n
V (
mV
)
Sεr ∩ Σ
Sεa ∩ Σ
Γ
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
Perturbations – Slow Manifolds
Theo Vo MMOs In A 3-Timescale System
ε = 0.002Sε
a
Sεr
Γ
0.225 0.23 0.235
−18
−15
−12
n
V (
mV
)
Sεr ∩ Σ
Sεa ∩ Σ
Γ
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
Perturbations – Canards
Theo Vo MMOs In A 3-Timescale System
ε = 0.002Sεa
Sεr
Γ
γ0 γ1
γ2
γ3
0.225 0.23 0.235
−18
−15
−12
n
V (
mV
)
Sεr ∩ Σ
Sεa ∩ Σ
Γ
γ0
γ1
γ2
γ3
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
Rationale
3-Timescale Bursting Model
1 Fast3 Slow
3 Fast1 Slow
1 Fast2 Intermediate
1 Slow
Origin & Properites of
Bursting
ε→ 0
δ 6= 0
δ → 0
ε 6= 0
ε = 0
δ → 0
δ = 0
ε→ 0
Bifurcation Theory
G.S.P.T
Bifurcation Theory
G.S.P.T
Bif
urca
tion
The
ory
G.S
.P.T
Theo Vo MMOs In A 3-Timescale System
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
Geometric Singular Perturbation Analysis
‘FAST’ SYSTEM
εdVdtI
= f (V , n, e, c)
dndtI
= g1(V , n)
dedtI
= g2(V , e)
dcdtI
= δh(V , c)
SLOW SYSTEM
εδdVdtS
= f (V , n, e, c)
δdndtS
= g1(V , n)
δdedtS
= g2(V , e)
dcdtS
= h(V , c)
3D LAYER PROBLEM
εdVdtI
= f (V , n, e, c)
dndtI
= g1(V , n)
dedtI
= g2(V , e)
dcdtI
= 0
1D REDUCED PROBLEM
11
0 = f (V , n, e, c)
11
0 = g1(V , n)
11
0 = g2(V , e)
dcdtS
= h(V , c)
Theo Vo MMOs In A 3-Timescale System
δ↓0
δ↓0
tS = δ tI−−−−−−−−→
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
Layer and Reduced Flows
εdVdtI
= f (V , n, e, c)
dndtI
= g1(V , n)
dedtI
= g2(V , e)
dcdtI
= 0
ææææææææææ
ææææææææææ
ìììììììììì
SSa
SSr
SSa
LL+
LL−
SSH
SS = {(V ,n,e, c) ∈ S : g1(V ,n) = g2(V ,e) = 0}
Fold points, LL := {(V , n, e, c) ∈ SS : det Df = 0}
Hopf Bifurcation, SSH := {(V , n, e, c) ∈ SS : fV = O(ε)}
Theo Vo MMOs In A 3-Timescale System
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
Singular Orbits
0.27 0.34 0.41-80
-60
-40
-20
c HΜML
V (
mV
)
SS
1D Reduced
1D Reduced
3DL
ayer
3DL
ayer
Γ(ε,δ)
•SSH
Theo Vo MMOs In A 3-Timescale System
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
Dynamic Hopf Bifurcation
ìììììììììì
0.27 0.34 0.41 0.48-80
-60
-40
-20
0
c HΜML
V (
mV
)
SS
SSH
Γ(ε,δ)
ìììììììììì
0.27 0.34 0.41-80
-60
-40
-20
0
c HΜML
V (
mV
)
ìììììììììì
SS SSH
Γ(ε,δ)
-80
-60
-40
-20
0
Time
V (
mV
)
-80
-60
-40
-20
0
Time
V (
mV
)
Theo Vo MMOs In A 3-Timescale System
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
Averaging
ìììììììììì
0.261 0.268 0.275 0.282-80
-60
-40
-20
c HΜML
V (
mV
)
SS
LL−
SSH
Γ(ε,δ)
Theo Vo MMOs In A 3-Timescale System
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
Calcium-Clamped MMOs
dcdtI
=1
T (c)
Z T (c)
0h(V (s, c), c) ds ≡ h(c)
0.265 0.272 0.279
-0.02
0.00
0.02
c HΜML
h
0.25 0.275 0.3-80
-60
-40
-20
0
c HΜML
V (
mV
)
SS •SSH Averaged�
Γ(ε,δ)
h = 0
10
121314
0 0.2 0.4 0.6 0.8-80
-60
-40
-20
e
V (
mV
)
Averaged���
Γ(ε,δ)
Theo Vo MMOs In A 3-Timescale System
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
Rationale
3-Timescale Bursting Model
1 Fast3 Slow
3 Fast1 Slow
1 Fast2 Intermediate
1 Slow
Origin & Properites of
Bursting
ε→ 0
δ 6= 0
δ → 0
ε 6= 0
ε = 0
δ → 0
δ = 0
ε→ 0
Bifurcation Theory
G.S.P.T
Bifurcation Theory
G.S.P.T
Bif
urca
tion
The
ory
G.S
.P.T
Theo Vo MMOs In A 3-Timescale System
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
The Double Limit
ε = 0, δ 6= 0
1-FAST/3-SLOW LAYER
dVdtF
= f (V , n, e, c)
dndtF
= 0
dedtF
= 0
dcdtF
= 0
ε 6= 0, δ = 0
3-FAST/1-SLOW LAYER
εdVdtI
= f (V , n, e, c)
dndtI
= g1(V , n)
dedtI
= g2(V , e)
dcdtI
= 0
1-FAST/3-SLOW REDUCED
11
0 = f (V , n, e, c)
dndtI
= g1(V , n)
dedtI
= g2(V , e)
dcdtI
= δ h(V , c)
3-FAST/1-SLOW REDUCED
11
0 = f (V , n, e, c)
11
0 = g1(V , n)
11
0 = g2(V , e)
dcdtS
= h(V , c)
Theo Vo MMOs In A 3-Timescale System
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
The Double Limit
ε = 0, δ = 0
1D FAST SUBSYSTEM
dVdtF
= f (V , n, e, c)
dndtF
= 0
dedtF
= 0
dcdtF
= 0
ε = 0, δ = 0
2D INTERMEDIATE SUBSYSTEM
11
0 = f (V , n, e, c)
dndtI
= g1(V , n)
dedtI
= g2(V , e)
dcdtI
= 0
2D INTERMEDIATE SUBSYSTEM
11
0 = f (V , n, e, c)
dndtI
= g1(V , n)
dedtI
= g2(V , e)
dcdtI
= 0
1D SLOW SUBSYSTEM
11
0 = f (V , n, e, c)
11
0 = g1(V , n)
11
0 = g2(V , e)
dcdtS
= h(V , c)
Theo Vo MMOs In A 3-Timescale System
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
Geometric Structures Persist
2D INTERMEDIATE
11
0 = f (V , n, e, c)
dndtI
= g1(V , n)
dedtI
= g2(V , e)
dcdtI
= 0
DESINGULARIZED
11
0 = f (V , n, e, c)
dVdt∗I
= F0(V , n, e, c)
dedt∗I
= −fV g2(V , e)
dcdt∗I
= 0
S (3D phase space)
ææææææææææ
ææææææææææ
ìììììììììì
SSa
SSr
SSa
LL+
LL−
SSH
SS (1D phase space) M0 (canard solutions)
Theo Vo MMOs In A 3-Timescale System
Proj & Desing−−−−−−−→
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
Dynamic & Calcium-Clamped MMOs
Dynamic MMOs
0.27 0.34 0.41-80
-60
-40
-20
c HΜML
V (
mV
)
SS
M0LL−•
M II0•
Γ(ε,δ)
ΓF(0,0)
ΓF(0,0)
ΓI(0,0)
ΓI(0,0)Xz
ΓI(0,0)
ΓS(0,0)
ΓS(0,0)
h = 0
0 0.2 0.4 0.6 0.8-80
-60
-40
-20
e
V (
mV
)
M II0•
Γ(ε,δ)ΓF(0,0)
ΓF(0,0)
ΓI(0,0)
ΓI(0,0)
ΓI(0,0)
ΓS(0,0)
Calcium-clamped MMOs
0.25 0.275 0.3-80
-60
-40
-20
c HΜML
V (
mV
)
SSM0
LL−•
M II0
Γ(ε,δ)ΓF(0,0)
ΓI(0,0)
h = 0
0 0.2 0.4 0.6 0.8-80
-60
-40
-20
e
V (
mV
)
M II0
•
Γ(ε,δ)
ΓF(0,0)
ΓI(0,0)
Theo Vo MMOs In A 3-Timescale System
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
Selling Point
.0 2 4 6 8 10
0
5
10
15
20
25
gK (nS)
g A (
nS)
Dep��9
DynamicMMO
Bis
tabi
lity
Cal
cium
-cla
mpe
d
SpikingESH
B
0 2 4 6 8 100
5
10
15
20
25
gK (nS)
g A (
nS)
Dep�9
Bursting Spiking
Hop
fBif
urca
tion
Perio
d-D
oubl
ing
Time HmsL
V (
mV
)
Time HmsL
V (
mV
)
Time HmsL
V (
mV
)
Theo Vo MMOs In A 3-Timescale System
(ε, δ) 6= (0,0)(ε, δ) = (0,0)
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
Recapitulation
Time
V (
mV
)
Time
c (Μ
M)
Time
V (
mV
)
Time
c (Μ
M)
I Motivation: explain dynamics
I 1-Fast/3-Slow: canard theory
I 3-Fast/1-Slow: delay phenomena
I 3-Timescale: best of both worlds
It never hurts to look at your problem from multiple points of view!
Theo Vo MMOs In A 3-Timescale System
MMOs In A3-Timescale
System
Theo Vo
MotivationMMO Model
Bifurcations
1-Fast/3-SlowLayer Flow
Reduced Flow
Canards
3-Fast/1-SlowG.S.P.T
Dynamic MMOs
Averaging
3-TimescaleDouble Limit
Inheritance
Summary
Thank You!
Theo Vo MMOs In A 3-Timescale System