Canard-Induced Mixed Mode Oscillations In Pituitary ......2012/08/10 · MMO Model Bifurcations 1-Fast/3-Slow Layer Flow Reduced Flow Canards 3-Fast/1-Slow G.S.P.T Dynamic MMOs Averaging

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


System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


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






** *

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)


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)









** *

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)


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)









** *

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)


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)




Trends in Endocrinology and Metabolism (2005) 16:152-159



System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


Inheritance

Summary

Pituitary Lactotroph Model


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)


System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


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

)


Calcium-clamped MMODynamic MMO


System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


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

)



System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


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

)



System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


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



System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


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


tF = ε tI−−−−−−−−→


System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


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


ε↓0

ε↓0

tF = ε tI−−−−−−−−→


System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


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



System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


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


ddtI

f = 0−−−−−−→


System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


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}


tI = −fV t∗I−−−−−−−→


System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


Inheritance

Summary

Singular Orbit Construction

3D Reduced

3D Reduced

1D Layer 1D Layer---

666

CAUTION: We are plotting 3D projections of a 4D system!



System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


Inheritance

Summary

Perturbations – Mixed Mode Oscillations


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


System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


Inheritance

Summary

Perturbations – Slow Manifolds


ε = 0Sa

Sr

Γ

0.225 0.23 0.235

−18

−15

−12

n

V (

mV

)

Sr ∩ Σ

Sa ∩ Σ

Γ


System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


Inheritance

Summary



ε = 0.0005Sε

a

Sεr

Γ

0.225 0.23 0.235

−18

−15

−12

n

V (

mV

)

Sεr ∩ Σ

Sεa ∩ Σ

Γ


System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


Inheritance

Summary



ε = 0.001Sε

a

Sεr

Γ

0.225 0.23 0.235

−18

−15

−12

n

V (

mV

)

Sεr ∩ Σ

Sεa ∩ Σ

Γ


System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


Inheritance

Summary



ε = 0.002Sε

a

Sεr

Γ

0.225 0.23 0.235

−18

−15

−12

n

V (

mV

)

Sεr ∩ Σ

Sεa ∩ Σ

Γ


System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


Inheritance

Summary

Perturbations – Canards


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


System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


Inheritance

Summary

Rationale


1 Fast3 Slow

3 Fast1 Slow


1 Slow


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



System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


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)


δ↓0

δ↓0

tS = δ tI−−−−−−−−→


System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


Inheritance

Summary

Layer and Reduced Flows

εdVdtI

= f (V , n, e, c)

dndtI

= g1(V , n)

dedtI

= g2(V , e)

dcdtI

= 0

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)}



System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


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



System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


Inheritance

Summary

Dynamic Hopf Bifurcation


0.27 0.34 0.41 0.48-80

-60

-40

-20

0

c HΜML

V (

mV

)

SS

SSH

Γ(ε,δ)


0.27 0.34 0.41-80

-60

-40

-20

0

c HΜML

V (

mV

)


SS SSH

Γ(ε,δ)

-80

-60

-40

-20

0

Time

V (

mV

)

-80

-60

-40

-20

0

Time

V (

mV

)



System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


Inheritance

Summary

Averaging


0.261 0.268 0.275 0.282-80

-60

-40

-20

c HΜML

V (

mV

)

SS

LL−

SSH

Γ(ε,δ)



System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


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)



System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


Inheritance

Summary

Rationale


1 Fast3 Slow

3 Fast1 Slow


1 Slow


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



System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


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)



System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


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)



System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


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)




SSa

SSr

SSa

LL+

LL−

SSH

SS (1D phase space) M0 (canard solutions)


Proj & Desing−−−−−−−→


System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


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)



System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


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

)


(ε, δ) 6= (0,0)(ε, δ) = (0,0)


System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


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!



System

Theo Vo

MotivationMMO Model

Bifurcations


Reduced Flow

Canards


Dynamic MMOs

Averaging


Inheritance

Summary

Thank You!


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