Control of stability of intracellular
Ca-oscillations and electrical
activity in a network of coupled
cells.
Stan Gielen
Dept. of Biophysics
Martijn Kusters
Wilbert van Meerwijk
Dick Ypey
Lex Theuvenet
May 27, 2005
Overview
• Summary of Ca-dynamics in NRK cell
• Dynamics of Ca-oscillations and action potentials
• coupling between Ca-oscillations and action
potentials
• Stability of Ca-dynamics in the cell
• Alternative model for cells with IP3-oscillations
• Coupling between two oscillators
• Propagation of electrical activity in network of
layers
– oscillators as pacemakers which initiate propagation ?
– instability due to coupling ?
May 27, 2005
Model for Normal Rat Kidney Cell
• NRK-cell = fibroblast
• similar to Cells of Cajal
• NRK cells form a network coupled by gapjunctions
May 27, 2005
Model for Membrane NRK cell
May 27, 2005
Components of the model
CaER
(1000 μM)
Gleak
Kcyt
GKir
(0.1 μM)
Cacyt
GCl(Ca)
Clex
May 27, 2005
BCacyt
B
ATP
GCaL
Caex
PMCA
(1000 μM)
This model focusses on the dynamics of the cell membrane,
including the L-type Ca-channel and other ion channels with
the following components:
• PMCA pump : pump Ca out of cytosol into extracellular
space
• Ca2+ L-type channel: Vca-L = +55 mV
• Cl(Ca) channel
: VCl = -20 mV
CaER
• Leak channel
• Kir channel : VK = -75 mV
Gleak
• Ca-buffer in the cytosol
Kcyt
GKir
Cacyt
BCacyt
B
GCl(Ca)
Clex
GCaL
Caex
May 27, 2005
PMCA
Components of the model for the NRK Membrane
I leak  Gleak (V  Eleak )
I K  GK
K 
KO  K
(V  E K )
5.4  K   K
0 .1
1  exp(0.06{V  E K  50})
 K  3 exp(0.0002 (V  E K  100 ))  exp
KO
RT
EK 
1000 ln(
)
F
120
Cacyt
J PMCA  C PMCA
Cacyt  K PMCA
May 27, 2005
• Leak current
• Potassium
channel
0.0002 (V  E K  10)
1  exp( 0.06(V  E K  50))

• PMCA-pomp
Components of the model for the NRK Membrane
I Ca ( L )  GCa ( L ) m  h  wCa (V  ECa ( L ) )
1
1  exp( (V  15) / 5.24)
m 0.01(1  exp( (V  10) / 5.9))
m 
0.035(V  10)
1
h 
1  exp((V  37) / 5.24)
0.01
h 
0.02  0.0197 exp( {0.0337(V  10)}2 )
1
wCa 
1  K wCa Cacyt
m 
I Cl ( Ca )  GCl ( Ca )
May 27, 2005
Cacyt
Cacyt  K Cl
(V  ECl ( Ca ) )
• Ca2+ L type
channel
• Cl(Ca) kanaal
Current clamp Ipulse=6 pA
• When we current clamp, the activation gate of the Ca L
type opens, giving rise to an inflow of Ca through the Ca L
type channel.
• As a consequence, an action potential will be generated
CaER
Gleak
K
GKir
GCl(Ca)
Cacyt
B
PMCA
GCaL
Caex
May 27, 2005
BCacyt
Current clamp Ipulse=6 pA
Cacyt
Action potential
Buffered
Ca
PMCA
current
I Ca ( L )  GCa ( L ) m  h  wCa (V  55mV )
ICl
IK
ILeak
May 27, 2005
I Cl (Ca )  GCl (Ca )
I leak
Cacyt
Cacyt  K Cl
 Gleak (V  Eleak )
I K  GK
(V  20mV )
KO  K
(V  EK )
5.4  K   K
J PMCA  C PMCA
Cacyt
Cacyt  K PMCA
Current clamp Ipulse=6 pA
Cacyt
Buffered
Ca
PMCA
current
ICl
IK
ILeak
May 27, 2005
Action potential
Inflow of
Ca through
L-type Ca
channel
Plateau due to
Nernst
potential of
Ca-dependent
Cl-channel
Current clamp Ipulse=6 pA
Cacyt
Action potential
Buffered
Ca
Important !
PMCA
current
I Ca ( L )  GCa ( L ) m  h  wCa (V  55mV )
ICl
IK
ILeak
May 27, 2005
I Cl (Ca )  GCl (Ca )
I leak
Cacyt
Cacyt  K Cl
 Gleak (V  Eleak )
I K  GK
(V  20mV )
KO  K
(V  EK )
5.4  K   K
J PMCA  C PMCA
Cacyt
Cacyt  K PMCA
Adding a Ca2+ buffer eliminates the plateau
May 27, 2005
De Roos et al. 1998
The effect of a Ca-buffer
With Ca buffer
Shorter
plateauphase
May 27, 2005
Without Ca buffer
Model for intracellular Ca2+oscillations
May 27, 2005
Model for Ca-oscillations from ER
CaER
(1000 μM)
Glek
•
•
•
•
•
•
ATP
Glek
IP3
Glk
SERCA
receptor
Cacyt
B
(0.1 μM)
SERCA pump
IP3-receptor
leakage of Ca from the ER into the cytosol
PMCA pump
leakage of Ca from extracellular space into the cytosol
Ca-buffer in the cytosol
May 27, 2005
BCacyt
ATP
PMCA
This model focusses on the dynamics of Ca in the ER and
cytosol by transport through the IP3 receptor.
The model has the following components:
•
•
•
•
•
SERCA pump
IP3-receptor
leakage of Ca from the ER into the cytosol
PMCA pump
leakage of Ca from extracellular space into the
cytosol
• Ca-buffer in the cytosol
May 27, 2005
Components of the model for the IP3-oscillator
• IP3-receptor
f 
Cacyt
Cacyt  K fIP3
IP3
IP3  K wIP 31
IP3
IP3  K wIP 32
w 
IP3
Kw
 Cacyt
IP3  K wIP 32
Kw
20
w 
Kw
IP3
 0.1Cacyt
IP3  K wIP 3
J IP3  CIP3 f 3 w3 (CaER  Cacyt )
• Leakage from ER
• SERCA-pomp
J leak , ER  Cleak , ER (CaER  Cacyt )
A conversion factor of 0.1 transforms an increase/decrease of
Ca27,ER2005
into a decrease/increase of Cacyt.
May
Intracellular Ca-oscillations
May 27, 2005
Harks et al., 2004
Stability analysis of IP3 receptor
f 
Cacyt
Cacyt  K fIP3
IP3
IP3  K wIP 31
IP3
IP3  K wIP 32
w 
IP3
Kw
 Cacyt
IP3  K wIP 32
Kw
20
w 
May 27, 2005
Kw
IP3
 0.1Cacyt
IP3  K wIP 3
Ca-oscillations as a function of IP3
May 27, 2005
Cacyt
CaER
JSERCA, JIP3
May 27, 2005
CaER
Buffered Ca
Cacyt
Buffered Ca
IP3-mediated calcium oscillations
JPMCA,JSOC,JLeak
IP3-mediated calcium oscillations
Concentration IP3 low
Cacyt Buffered Ca
JSERCA, JIP3
May 27, 2005
CaER
JPMCA,JSOC,JLeak
high
Cacyt Buffered Ca
JSERCA, JIP3
CaER
JPMCA,JSOC,JLeak
Overview
• Summary of Ca-dynamics in NRK cell
• Dynamics of Ca-oscillations and action potentials
• coupling between Ca-oscillations and action
potentials
• Stability of Ca-dynamics in the cell
• Alternative model for cells with IP3-oscillations
• Coupling between two oscillators
• Propagation of electrical activity in network of
layers
– oscillators as pacemakers which initiate propagation ?
– instability due to coupling ?
May 27, 2005
Stability of Ca-dynamics in the cell
Whole cell model
Action potentials
Ca-oscillations
May 27, 2005
Complete Model
Caer
JCalker
SERCA
IP3R
Glk
IP3
Kcyt
GKir
Cacyt
BCacyt
B
GCl(Ca)
Clex
May 27, 2005
GCalk
GCaL
Caex
PMCA
steady-state behavior
Without IP3, the steady-state is easily found by
solving JSERCA=Jleak,ER and JPMCA=Jleak,membrane
ER/cytosol:
Gleak (CaER  Cacyt )  J
membrane/cytosol:
I
max
PMCA
max
SERCA
Cacyt
Cacyt  K PMCA
Cacyt
Cacyt  K SERCA
 Glk (1000  Cacyt )
This gives a single, stable solution for Cacyt and CaER :
Cacytosol = 0.1 μM; CaER= 1300 μM
May 27, 2005
Stability of
2+
Ca concentrations
Action potential triggers Ca oscillation Ca oscillation triggers action potential
Caer
Caer
SERCA
JCalker
Cacyt
Cacyt
GCalk
PMCA
Caex (1000 μM)
May 27, 2005
SERCA
JCalker
GCalk
PMCA
Caex (1000 μM)
Additional channel to stabilize Ca-dynamics
Caer
JCalker
SERCA
IP3R
GSOC
Glk
IP3
BCacyt
Kcyt
GKir
Cacyt
B
GCl(Ca)
Clex
May 27, 2005
GCalk
GCaL
Ca
PMCA
Whole cell model with SOC/CRAC
channel
Action potentials
Ca-oscillations
May 27, 2005
Components of the model
Stable attractor
dV/dt = 0
dCacyt/dt=0
May 27, 2005
Membrane potential
IP3 = 0
Cacytosol (μMol)
Components of the model
No stable attractor
dV/dt = 0
dCacyt/dt=0
May 27, 2005
Membrane potential
IP3 receptor oscillates
Cacytosol (μM)
Components of the model
ip3
3
60
IP3 high
Membrane potential
40
Blue for V
Stable attractor at – 20 mV
dV/dt = 0
dCacyt/dt=0
20
0
20
40
60
80
0
May 27, 2005
0.5
1
Red for ca
1.5
Cacytosol (μMol)
2
Stability analysis of IP3 receptor
f 
Cacyt
Cacyt  K fIP3
IP3
IP3  K wIP 31
IP3
IP3  K wIP 32
w 
IP3
Kw
 Cacyt
IP3  K wIP 32
Kw
20
w 
May 27, 2005
Kw
IP3
 0.1Cacyt
IP3  K wIP 3
Summary
• Stability of Ca-dynamics for all possible
natural conditions requires a coupling
between Ca-concentration in ER and
extracellular Ca.
• Without IP3: stable condition corresponds
to V=-70 mV; Cacyt=0.1 μM
• Higher IP3 concentrations provide
oscillations or stable point at V= -20 mV
May 27, 2005
Overview
• Summary of Ca-dynamics in NRK cell
• Dynamics of Ca-oscillations and action potentials
• coupling between Ca-oscillations and action
potentials
• Stability of Ca-dynamics in the cell
• Alternative model for cells with IP3-oscillations
• Coupling between two oscillators
• Propagation of electrical activity in network of
layers
– oscillators as pacemakers which initiate propagation ?
– instability due to coupling ?
May 27, 2005
Alternative model for coupling
between IP3-oscillator
(Ca-oscillations) and membrane
oscillator (action potentials)
May 27, 2005
Problem
Many cell types do not oscillate in isolation, but do so in a
synchronized manner only when electrically coupled in a
network (e.g. β-pancreatic cells in islets of Langerhans and
aortic smooth muscle cells).
– Cells in isolation are quiet or oscillate at lower
frequencies.
Paradox: If identical cells oscillate in phase, there are no
currents ! How then can electrical coupling be crucial for
the synchronous oscillations ? Moreover: if there are phase
differences, they will be eliminated by the electrical
coupling !
May 27, 2005
Basic mechanism
dCacyt
 J (Cacyt , CaER )  KCacyt  U
dt
dCaER
  J (Cacyt , CaER )
dt
dJ (Cacyt , CaER )
0
dCacyt
J(Cacyt,CaER) = interaction term between Ca concentrations
with
dJ (Cacyt , CaER )
dCacyt
0
reflecting Ca-induced Ca-release
KCacyt = efflux of Ca from cell
U = constant, Ca-mediated electrical current
May 27, 2005
Loewenstein & Sompolinsky, PNAS, 2001
Calcium and Voltage oscillations
in non-excitable cell
Cytosolic Ca (μM)
Ca in stores (μM)
Rest-state is unstable fixed-point
Small perturbations in cytosolic Ca cause oscillations
May 27, 2005
Loewenstein et al., PNAS 98, 2001
Calcium and Voltage oscillations
Cytosolic Ca (μM) Ca in stores (μM)
Non-excitable cell
Excitable cell
with Voltagedependent Cacurrent en Kca
channel
Hyperpolarization
decreases by
electrical coupling
May 27, 2005
IK_Ca hyperpolarizes
membrane potential,
which de-activates Cainflux into cell
However, adding a
shunt conductance
i
I coupling
  gij (V i  V j )
j
destabilizes the fixed
point
Calcium and Voltage oscillations
Cytosolic Ca (μM) Ca in stores (μM)
Excitable cell
with Voltagedependent Cacurrent en Kca
channel
with Voltagedependent Cacurrent en Kca
channel but with
shunt
conductance
May 27, 2005
Addition of ashunt
conductance
1.
Reduces the effect
of Cacyt on
membranbe
potential
2.
Suppresses
efficacy of
negative feedback
by IK_Ca
3.
Enables
oscillations
Voltage and Ca oscillations in network of
two electrically coupled cells
Hyperpolarization due to
Ca-influx
Hyperpolarization
due to electrical
coupling
Ca oscillations out-of-phase; electrical oscillations inphase at double frequency
May 27, 2005
Multi-stability in
network with 6 coupled
cells.
In a large network different
realizations of out-of-phase calcium
oscillations are possible and
therefore the network possesses
many stable states. The stable state
in which the system will eventually
settle is determined by the initial
conditions.
Note the differences in membrane
potential !
May 27, 2005
Cell
1
2
3
4
5
6
1
2
3
4
5
6
Summary
• Cells are
– intrinsically stable (near –70 mV ; Loewenstein
et al. PNAS 2001) or
– intrinsically oscillating ?
• Electrical coupling
– enables oscillations and propagation of activity
to otherwise silent cells or
– disables oscillations and propagating activity in
a network of pacemaker cells ?
• Ca oscillations out of phase ! Why ?
May 27, 2005
Overview
• Summary of Ca-dynamics in NRK cell
• Dynamics of Ca-oscillations and action potentials
• coupling between Ca-oscillations and action
potentials
• Stability of Ca-dynamics in the cell
• Alternative model for cells with IP3-oscillations
• Coupling between two oscillators
• Propagation of electrical activity in network of
layers
– oscillators as pacemakers which initiate propagation ?
– instability due to coupling ?
May 27, 2005
Coupling between two oscillators
Inhibition and electrical coupling
May 27, 2005
Neuronal synchronization due to
external input
T
ΔT
Synaptic input
May 27, 2005
Δ(θ)= ΔT/T
Neuronal synchronization
T
Δ(θ)= ΔT/T
ΔT
Depolarizing
stimulus
Phase
advance
Hyperpolarizing
stimulus
May 27, 2005
Phase shift as a function of the
relative phase of the external
input.
Neuronal synchronization
T
ΔT
Δ(θ)= ΔT/T
Suppose:
• T = 95 ms
• external trigger: every 76 ms
• Synchronization when
ΔT/T=(95-76)/95=0.2
• external trigger at time 0.7x95
ms = 66.5 ms
May 27, 2005
Inhibitory coupling
for two identical leaky-integrate-and-fire neurons
Out-of-phase stable
In-phase stable
May 27, 2005
Lewis&Rinzel, J. Comp. Neurosci, 2003
Phase-shift function
for inhibitory coupling
dG( * )
0
d
for stable attractor
Increasing constant
input to the LIFneurons
I=1.2
I=1.4
I=1.6
May 27, 2005
Bifurcation diagram for two
identical LIF-neurons with inhibitory coupling
May 27, 2005
Bifurcation diagram for two
identical LIF-neurons with inhibitory coupling
Time
constant for
inhibitory
synaps
May 27, 2005
Electrical coupling for spiking neurons
by gap junctional coupling
Out-of-phase stable
May 27, 2005
In-phase stable
Phase-shift function
for electrical coupling
+40 mV
1.
0 mV
1.
-70 mV
May 27, 2005
2.
2.
effect of suprathreshold part of
spike tends to
synchronize activity
effect of subthreshold part of
spike tends to
desynchronize
activity
Phase-shift function
for electrical coupling
I=1.05
I=1.15
I=1.25
effect of suprathreshold part of
spike tends to
synchronize activity
effect of subthreshold part of
spike tends to
desynchronize
activity
effect of both
components
May 27, 2005
Bifurcation diagram for two
identical LIF-neurons with electrical coupling
May 27, 2005
Bifurcation diagram for two
identical LIF-neurons with electrical coupling
May 27, 2005
If natural frequencies do not match
Time courses of
hypathocyte x1 (solid
line) and of x2 (dashed
line) at P1=1.5 μM and
P2=2.5 μM.
(a) Harmonic locking of
1:3 (γCA=0.025 s-1);
(b) harmonic locking
of 1:2 (γCA=0.05 s-1);
(c) phase locking of 1:1
(γCA=0.09 s-1).
(d) Devil’s staircase, a
ratio N/M (where N is the
spike number of x1 and M
is the spike number
of x2) as a function of the
coupling strength γCA at
given IP3 level: P1=1.5
μM, P2=2.5 μM.
May 27, 2005
Wu et al., Biophys. Chem. 113, 2005
Coupling strength
Bifurcation diagram for two
identical LIF-neurons with inhibitory and electrical coupling
Inhibitory
coupling only
Electrical
coupling only
May 27, 2005
Electrical coupling in addition to
synaptic (inhibitory) interactions
anti-phase, weak
electrical coupling
in-phase , strong
electrical coupling
no electrical coupling
anti-phase , weak
electrical coupling
in-phase , strong
electrical coupling
May 27, 2005
Brem & Rinzel, J. Neurophysiol. 91, 2004
Electrical coupling in addition to
synaptic interactions
Anti-phase and inphase both stable
Stable antiphase
Stable inphase
The stronger is the synaptic inhibition, the larger is the
May 27, 2005
electrical
coupling required to stabilize in-phase behavior
Summary
• Gap-junctions between two cells tend to
synchronize the two oscillators
• synchronizing effect is stronger when there
is a plateau phase in the action potential
May 27, 2005
Overview
• Summary of Ca-dynamics in NRK cell
• Dynamics of Ca-oscillations and action potentials
• coupling between Ca-oscillations and action
potentials
• Stability of Ca-dynamics in the cell
• Alternative model for cells with IP3-oscillations
• Coupling between two oscillators
• Propagation of electrical activity in network of
layers
– oscillators as pacemakers which initiate propagation ?
– instability due to coupling ?
May 27, 2005
What happens for two pacemaker
cells with excitatory and gapjunctional coupling ?
May 27, 2005
Two pacemaker cells
May 27, 2005
Synchronization of two oscillators
No coupling
Small
conductance
gap junction
Small
conductance
gap junction
May 27, 2005
Simple result for excitatory and
electrical coupling
• Two pacemaker cells synchronize easily
May 27, 2005
Synchronization of activity in a
network of cells
May 27, 2005
Network of NRK-cells
May 27, 2005
One pacemaker, surrounded by 6 followers
May 27, 2005
Two pacemaker cells
Rgap
Ri
May 27, 2005
Rcell
V
Network of NRK-cells
Rgap
Rgap
Rgap
Rcell
Rcell
Ri
Rcell
Experimental observation: a single pacemaker cell
May 27, 2005
cannot initiate propagation of action potential firing
Resistance of gap-junction should
not be too high and not too low !
Rgap
Rgap
Rgap
Rcell
Rcell
Ri
May 27, 2005
In the heart: Rcell is high !
Rcell
Synchronization in a network of
different coupled oscillators
May 27, 2005
Spontaneous oscillations and
synchronization in NRK networks
NRK cell with intracellular (IP3)
oscillator and plasma membrane
Casyst
Caer
Membrane
potential
Network
with NRK cells
May 27, 2005
Oscillations and
synchronization
Standing problems
• Cells are intrinsically stable, but become unstable
due to coupling in a network ?
• Or: cells are unstable but synchronize in a network
to act as pacemakers for propagating activity ?
• What is the role of electrical/gap-junctional
coupling and Ca-diffusion through gap junctions
in propagation of action potential firing ?
• How to recognize pacemakers and followers ?
• Pace-makers seem to “move” in a network
May 27, 2005
May 27, 2005
Complete model
Caer
JCalker
SERCA
IP3R
GSOC
Glk
IP3
BCacyt
Kcyt
GKir
Cacyt
B
GCl(Ca)
Clex
May 27, 2005
GCalk
GCaL
Ca
PMCA
Further topics for study
• Compartimentalization:
– coupling of ER with cell membrane for store-operated
channels
– discrete sources and sinks (stores)
– discrete channels : distance between channel clusters is
larger than the diffusion length of free Ca2+
• stability of intracellular Ca2+ control
• relation between stochastic character of channel dynamics
and deterministic periodic behavior of Ca-oscillations
May 27, 2005
References
• Falcke (2004) Reading the patterns in living cells —the
physics of Ca2+ signaling. Advances in Physics, 53,
255–440
• Loewenstein, Yarom, Sompolinsky (2001) The generation
of oscillations in networks of electrically coupled cells.
PNAS 98, 8095-8100.
May 27, 2005
Components of the model for the NRK Membrane
I CRAC  1 /( convflux)  CCRAC
1
(V  ECa )
CaER  K CRAC
I Ca ( L )  GCa ( L ) m  h  wCa (V  ECa ( L ) )
1
1  exp( (V  15) / 5.24)
m 0.01(1  exp( (V  10) / 5.9))
m  
0.035(V  10)
1
h 
1  exp((V  37) / 5.24)
0.01
h 
0.02  0.0197 exp( {0.0337(V  10)}2 )
1
wCa 
Cacyt
1
1
Cacyt
IMay
GCl (Ca )
(V  ECl (Ca ) )
2005
Cl ( Ca27,
) 
Cacyt  K Cl
• CRAC kanaal
• Ca2+ L type channel
m 
• Cl(Ca) kanaal
Components of the model in the
cell membrane
• CRAC channel
J CRAC  CCRAC
• Leakage into cytosol
• PMCA-pomp
J
1
Nernst
(V  ECa
)
CaER  K CRAC
PMCA
May 27, 2005
J lk  Clk (1000  Cacyt )
 C PMCA
Cacyt
Cacyt  K PMCA
Overview of parameter values
for membrane
Nernst
Eleak
 0V
for ER
Nernst
ECa
( L )  0.05V
R  8.31m 2 kg s -2 K -1mol -1
T  293K
EClNernst
( Ca )  0.02V
F  96480C / mol
Gleak  0.05nS
GK  2.2nS
GCa ( L )  0.50nS
GCl (Ca )  10.0nS
K Cl  35M
Cm  20 pF
K O  35M
May 27, 2005
CSERCA  0.6 M / s
Clek  0.002s 1
K SERCA  0.2 M
TB  20 M
K on  0.032( M  s )
1
K off  0.06 s 1
C PMCA  1.27 Ms
C IP 3  10s 1
1
K PMCA  0.2 M
CCRAC  0.55Ms 1
K CRAC  10 M
convflux  2  0.00123 1000
K w  1s 1
K wIP 31  5M
K wIP 32  15M
K fIP3  0.5M
Dynamics of IP3 regulated Ca2+ release
May 27, 2005
Ca-oscillations as a function of IP3
May 27, 2005
Oscillations in a large network
May 27, 2005
Parameter fitting
-3
1
0.8
8
m(V)
x 10
 (V)
m
6
m (s)
0.6
4
0.4
2
0.2
0
-150
-100
-50
0
50
100
1.2
-50
0
8
h(V)
h (s)
0.8
0.6
0.4
May 27, 2005
-100
50
100
150
10
1
0.2
-150
0
-150
150
 (V)
h
6
4
2
-100
-50
0
50
Vclamp (mV)
100
150
0
-100
-50
0
Vclamp (mV)
50
100
Ca-action potentials
triggered by Ca-release from the ER
GCaL 20 mV
GCl(Ca) -20 mV
GKIR -70 mV
May 27, 2005
Phase diagram for closed-cell model
May 27, 2005
Sneyd
et al., PNAS, 2004
Ca2+ is involved in the control of
•
•
•
•
•
•
•
Muscle contraction
memory storage
egg fertilization
enzyme secretion by acinar cell in pancreas
coordination of cell behavior in the liver
cell apoptosis
second messenger : coding and transfer of information
from cell membrane to nucleus
• etc., etc., etc.
Yet, high cytosolic concentrations prohibit normal functioning
of the cell. How can this be made compatibel ?
May 27, 2005
See Martin Falcke, Advances in Physics, 53, 2004
Different forms of Ca2+ oscillations
sinusoidal
oscillations in a
parotid gland
hepatocyte stimulated
with norepinephrine
endothelial cell
stimulated with
histamine
May 27, 2005
Ca-dynamics
• Ca-oscillations in non-excitable cells
• Ca-inflow in excitable cells (actionpotential generation) without intracellular
Ca-oscillations.
• Ca-oscillations in cells with actionpotentials and with IP3-mediated Caoscillations.
May 27, 2005
Overview
• Summary of Hodgkin-Huxley model
• Dynamics of Ca-oscillations and action
potentials
• coupling between Ca-oscillations and action
potentials
• Stability of Ca-dynamics in the cell
• Propagation of electrical activity in network
of layers
– oscillators as pacemakers which initiate propagation ?
– instability due to coupling ?
May 27, 2005
Membrane voltage equation
0 mV
0 mV
IC
INa
K
V mV
V mV
-Cm dV/dt = gmax, Nam3h(V-Vna) + gmax, K n4 (V-VK ) + g leak(V-Vleak)
May 27, 2005
Gating kinetics
m
State:
Open
Probability:
m
m
dm  (1 m)   m
 m
m
dt
dh   (1 h)   h
h
h
dt
m
Closed
(1-m)
m 
m
 m  m
1

 m  m
Channel Open Probability: m.m.m.h=m3h
May 27, 2005
m
V (mV)
Actionpotential
May 27, 2005
Simplification of Hodgkin-Huxley
Fast variables
• membrane potential V
• activation rate for Na+
m
Slow variables
• activation rate for K+ n
• inactivation rate for
Na+ h
-C dV/dt = gNam3h(V-Ena)+gKn4(V-EK)+gL(V-EL) + I
dm/dt = αm(1-m)-βmm
dh/dt = αh(1-h)-βhh
dn/dt = αn(1-n)-βnn
May 27, 2005
Phase diagram for the Morris-Lecar model
May 27, 2005
Phase diagram
May 27, 2005
Phase diagram
of the MorrisLecar model
May 27, 2005
Buffer dynamics
Cacyt  B  CaB
with
Kon = 0.032 (μMol s)-1
Koff = 0.06 s-1
May 27, 2005
Phase-plane plot for membrane dynamics
(Morris-Lecar model)
May 27, 2005
Ca L type channel
activation (m∞) and inactivation (h ∞)
m
∞
h
May 27, 2005
∞
V (mV)
The effect of Kon on the action potential
Kon = 0.032 (μMol.s)-1
Kon = 3.2 (μMol.s)-1
Longer AP
All Ca buffered
More Ca
buffered
Shorter AP
Kon = 0.32 (μMol.s)-1
May 27, 2005