Overview

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Overview

• Reduction of the HH model to a 2-D model
• Principles of phase plane analysis geometric
  aspects of neuronal dynamics emergence of
  oscillations
• A mathematical description of bursting in
  excitable cells

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HH model
Threshold behaviour
V(0)5 mV
V(0)4 mV
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(No Transcript)
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The slow variables h, n
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Two qualitative observations

• 1) the time scale of the dynamics of the gating
  variable m is much faster than that of the
  variables n, h.
• 2) the time constants and
  are roughly the same, whatever the voltage V.
  Moreover, the asymptotic functions
• ,
  
• are similar.

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The idea
use quasi steady-state approximation

• 1)
• 2) lump together the slow variables

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The V-equation can be rewritten as
or
which must be coupled to the equation for the
recovery variable w,
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Two dimensional models of excitable media
fast variable
slow variable
Fitzhugh Nagumo
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2D models

• Phase plane, trajectories
• Vector fields, nullclines
• Fixed points, stability, linearization
• Periodic orbits, limit cycles
• Bifurcations

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Phase-plane
integration of dynamical equations
the infinitesimal displacement is in the
direction of the flow
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w- nullcline
V- nullcline
fixed point both the components vanish
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Stability of fixed points
Introduce the displacement from the F.P.
Linearize the system at the F.P.
Find the eigenvalues of the matrix
and eigenvector
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An example of bifurcation
saddle-node bifurcation
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Hopf bifurcation
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Homoclinic bifurcation
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The Morris-Lecar model
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A spike in the ML (type II) model
graded threshold
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Phase-portrait of the ML (type II) model
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ML (type II) bifurcation diagram
amplitude of oscillations
The rest state turns unstable here, via a
(subcritical) Hopf bifurcation
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ML (type II) response function narrow frequency
range
onset of oscillation with non-zero frequency
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Bistability in ML (type II) model
R
UPO Unstable periodic orbit SPO Stable
periodic orbit R Stable rest state
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A bistable cell model can switch from rest to
oscilation and then back to rest
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ML (type I)
U
T
R
unstable manifold
stable manifold
R stable rest state T saddle point
threshold U unstable node
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ML (type I) threshold behaviour
-20
-22.1
The model shows a distinct threshold
-22.2
-25
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When the stimulus increases, the stable and the
threshold points coalesce, and a zero-frequency
limit cycle is formed
narrow channel slow dynamics
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Low frequency repetitive firing
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ML (type I) bifurcation diagram
high-voltage stable state
unstable oscillations
unstable states
stable oscilllations
a stable rest state exists here
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ML (type I) response function
Oscillations arise with zero frequency
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Bursting cells

• Widespread among neurons and non-neuronal cells
• Involved in generation of rythmic pattern (e.g.
  breathing, pacemaking, hormone secretion .. )
• The membrane is switching periodically between an
  active state (rapid oscillations) and a rest
  state (lack of oscillations)

Active phase
Silent phase
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Bursting with bistability the hystheresis loop
Busting requires more than two dynamical variables
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• Ca2 is the fundamental second messenger
  release
• of hormones and neurotransmitters, cell motility
  or
• contractions, gene expression and development, .
• Ca2 ions flow into the cell via V-dependent
  channels
• Ca2 is highly buffered inside the cytoplasm,
• low intracellular concentration

phenomenological model
buffering
influx
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Ca2 controls the activation of ionic channels
adaptation
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Calcium dynamics provide a slow, negative
feedback on membrane excitability
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A dynamical interpretation
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Bursting without bistability

• Without the hystheresis loop, a negative feedback
  would only produce adaptation
• In order to push the system into the oscillatory
  regime, we need a positive feedback, e.g. a
  slowly activating Ca2 current

slow time scale
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Bursting without bistability