Reduced models: electrically excitable cells

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Reduced models electrically excitable cells

• Basic references
• 1) Keener and Sneyd, Mathematical Physiology
• 2) Dynamical systems in neuroscience the
  geometry of excitability and bursting by Eugene
  M. Izhikevich

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Huxley
Hodgkin
Alan Lloyd Hodgkin and Andrew Huxley described
the model in 1952 to explain the ionic mechanisms
underlying the initiation and propagation of
action potentials in the squid giant axon. They
received the 1963 Nobel Prize in Physiology or
Medicine for this work.
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Where does it go from here?

• Simplified models - FHN, Morris Lecar,
  Mitchell-Schaffer-Karma
• More detailed models - Noble, Beeler-Reuter,
  Luo-Rudy, .
• Forced oscillations of single cells - APD
  alternans, Wenckebach patterns.
• Other simplified models - Integrate and Fire,
  Poincare oscillator
• Networks and spatial coupling (neuroscience,
  cardiology, )

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FitzHugh-Nagumo model
Figure 1 Circuit diagram of the tunnel-diode
nerve model of Nagumo et al. (1962).
Fast variable
FitzHugh modified the Van der Pol equations for
the nonlinear relaxation oscillator. The result
had a stable resting state, from which it could
be excited by a sufficiently large electrical
stimulus to produce an impulse. A large enough
constant current stimulus produced a train of
impulses (FitzHugh 1961, 1969).
Slow (recovery) variable
http//www.scholarpedia.org/article/FitzHugh-Nagum
o_model
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Notation for FitzHugh-Nagumo model
dV/dt (F(V,W)Iapp)/t dW/dt G(V,W)/tW
The choice of F(V,W) and G(V,W) now determines
our model
To simplify notation, in the following slides, I
set t 1 and summarize the constants into one
FitzHugh-Nagumo model F(V,W) V - V3/3 -
W G(V,W) (a bV - W)c c is the ratio of t and
tW
dV/dt F(V,W)Iapp dW/dt G(V,W)
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How do we analyze this class of models?

• Phase plane Study the dynamics in the
  (V,w)-plane rather than V or w versus time

• Nullclines Determine the curves along which one
  ofthe time derivatives is 0

• Steady states At the intersections of the two
  nullclines both derivatives are 0, so the system
  is at rest

• Direction arrows The nullclines divide up the
  plane, and the direction of flow in each region
  can be determined

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Phase plane analysis
W
V
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Phase plane analysis Nullclines
dW/dt0
W
dV/dt0
V
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Phase plane analysis Nullclines
small Iapp 1 fixpoint
dW/dt0
W
dV/dt0
V
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Phase plane analysis
large Iapp (and blt1) Potentially 3 fixpoints
dW/dt0
W
dV/dt0
V
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Phase plane analysis
bgt1 1 fixpoint (no matter Iapp)
W
V
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Phase plane analysis
bgt1 1 fixpoint (no matter Iapp)
W
V
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Phase plane analysis
bgt1 1 fixpoint (no matter Iapp)
W
V
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Phase plane analysis Stability of fixed points
For a fixed point to be stable, the real part of
both eigenvalues of the matrix must be
negative, For the FitzHugh-Nagumo model, we
have
dV/dt F(V,W)Iapp dW/dt G(V,W)
FitzHugh-Nagumo model F(V,W) V - V3/3 -
W G(V,W) (a bV - W)c c is the ratio of t and
tW
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FitzHugh-Nagumo model
Figure 3 Absence of all-or-none spikes in the
FitzHugh-Nagumo model.
Figure 4 Excitation block in the FitzHugh-Nagumo
model.
Fast variable
Figure 2 Phase portrait and physiological state
diagram of FitzHugh-Nagumo model (modified from
FitzHugh 1961).
Slow (recovery) variable
http//www.scholarpedia.org/article/FitzHugh-Nagum
o_model
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FitzHugh-Nagumo model
Figure 6 Spike accommodation to slowly
increasing stimulus in the FitzHugh-Nagumo model.
Figure 5 Anodal break excitation
(post-inhibitory rebound spike) in the
FitzHugh-Nagumo model.
Fast variable
Slow (recovery) variable
http//www.scholarpedia.org/article/FitzHugh-Nagum
o_model
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Morris-Lecar model

• The Morris-Lecar model is a two-dimensional
  "reduced" excitation model applicable to systems
  having two non-inactivating voltage-sensitive
  conductances. The original form of the model
  employed an instantaneously responding
  voltage-sensitive Ca2 conductance for excitation
  and a delayed voltage-dependent K conductance
  for recovery. The equations of the model are

• This model was named after Cathy Morris and
  Harold Lecar, who derived it in 1981. Because it
  is two-dimensional, the Morris-Lecar model is one
  of the favourite conductance-based models in
  computational neuroscience.
• Morris, C. and Lecar, H. (1981) Voltage
  oscillations in the barnacle giant muscle fibre.
  Biophys. J. 35 193 - 213. 17

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Morris-Lecar Lab model files

• GMorris_Lecar_Init.m - This model was run using
  globals for the many parameters. The init file
  starts everything out with I150, Vinit-60 and
  Winit.01. The operator can change any variable
  in the command window.
• GMorris_Lecar.m - This program can be used to
  generate pictures as shown in Summary figure A, B
  and C. Though designed for one set of parameters,
  by opting not to erase, the figures will be
  superimposed. The program calls Gmlfun to
  simulate system, GMorris_Lecar_Jac to plot the
  phase plane and nullclines, which will be shown
  in an individual plot (Figure 3) with values for
  the intersections and eigenvalues, and
  vector_field to draw little two second arrows at
  start of simulation.
• GMorris_Lecar_Jac.m - Subroutine designed to plot
  nullclines of system and find intersection points
  and eigenvalues. Solution is done in symbolic
  language which can be slow. Calls Intersect
  to find intersection points.
• Intersect.m - Subroutine to find the intersection
  point of the nullclines. . Solution is done in
  symbolic language which at times has difficulty
  with very complicated equations such as these
  tend to be. Particularly has trouble between
  I270 and 290 on this example.
• GMorris_Lecar_Bif.m - Calculates bifurcation plot
  shown in Summary figure D by brute force. Calls
  Intersect to get the steady state solution if
  it were to exist. Calls Gmlfun to find out what
  maximum and minimum values are.
• Gmlfun.m - Called by programs listed above
  whenever the Matlab ODE simulation is to be
  performed.
• vector_field.m - plots initial 2 seconds for a
  simulation.
• GMorris_Lecar_fW.m - Computes graph that display
  the current. The operator can enter a separate
  fixed w value. The default is w.35 as
  demonstrated in the figure. Other parameters are
  set as above, e.g., set I150 in the command
  window to get the figure shown. Go back to
  GMorris_Lecar to get Figure 2.10B (by changing
  phi from .04 to .004).

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Morris-Lecar summary
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Morris-Lecar with Iapp0
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Morris-Lecar with Iapp100 pA
Increasing the applied current translates the
V-nullcline upward, so that the nullcline
intersection is on the middle branch of the
V-nullcline. The steady state is unstable. The
stable solution is a limit cycle.
Motion along the limit cycle is periodic.
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Hopf Bifurcation
This transition from a stable steady state to a
stable limit cycle through variation of a
parameter is called a Hopf bifurcation. It is one
way in which periodic motion can arise from a
previously stationary system, or vice versa.
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Morris-Lecar with Iapp250 pA
Increasing the applied current more puts the
intersection on the right branch of the
V-nullcline. Here the steady state is again
stable. The system has gone through a second Hopf
bifurcation.
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Bifurcation Diagrams
These different behaviors and how they change
with variation of a parameter can be summarized
with a Bifurcation Diagram.
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Stationary Branches
For each value of the bifurcation parameter Iapp
plot the V value of the steady state solution. If
stable make the curve solid if unstable make the
curve dashed. Stability changes at bifurcation
points.
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Periodic Branches
Next plot the maximum and minimum V values
of periodic solutions. Stability changes at
two Saddle Node of Periodics (SNP) bifurcations.
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Bistability
For some values of Iapp the system is bistable.
For a single value of the parameter, a stable
steady state and a stable limit cycle coexist.
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Bistability in the Phase Plane
The unstable limit cycle (dashed) separates
the basin of attraction of the steady state from
that of the stable limit cycle. It is the
separatrix.
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Time Courses of Bistable System
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Coupled Morris-Lecar b-cells
Time scale is of order of milliseconds
Time scale is of order of seconds
Time scale is of order of minutes
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Calcium Has Multiple Kinetic Components
Zhang et al, Biophys. J. 842852, 2003
Bertram et al, Biophys J. 87(5)3074-87, 2004.
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Tsaneva-Atanasova et al. (2006) Biophys J. 90
3434-3446