Neural Modeling

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Neural Modeling
Suparat Chuechote
Introduction
• Nervous system - the main means by which
humans and animals coordinate short-term
responses to stimuli.
• It consists of :
- receptors (e.g. eyes, receiving signals from
outside world)
- effectors (e.g. muscles, responding to these
signals by producing an effect)
- nerve cells or neurons (communicate
between cells)
Neurons
•
•
Source: http://en.wikipedia.org/wiki/Neurons
Neuron consist of a
cell body (the soma)
and cytoplasmic
extension ( the axon
and many
dendrites) through
which they connect
(via synapse) to a
network of other
neurons.
Synapsesspecialized
structures where
neurotransmitter
chemicals are
released in order to
communicate with
target neurons
Neurons
• Cells that have the ability to transmit action potentials
are called ‘excitable cells’.
• The action potentials are initiated by inputs from the
dendrites arriving at the axon hillock, where the axon
meets the soma.
• Then they travel down the axon to terminal branches
which have synapses to the next cells.
• Action potential is electrical, produced by flow of ion
into and out of the cell through ion channels in the
membrane.
• These channels are open and closed and open in
response to voltage changes and each is specific to a
particular ion.
Hodgkin-Huxley model
• They worked on a nerve cell with the largest axon
known the squid giant axon.
• They manipulated ionic concentrations outside the
axon and discovered that sodium and potassium
currents were controlled separately.
• They used a technique called a voltage clamp to
control the membrane potential and deduce how ion
conductances would change with time and fixed
voltages, and used a space clamp to remove the
spatial variation inherent in the travelling action
potential.
Hodgkin-Huxley model
dV
Cm
 gNa m3h(V VNa )  gK n 4 (V VK )  gL (V VL )
dt
H-H variables:
dm
 m (V )
 m (V )  m
dt
dh
 h (V )  h (V )  h
dt
dn
 n (V )  n (V )  n
dt
V-potential difference
m-sodium activation variable
h-sodium inactivation variable
n-potassium activation variable
Cm-membrane capacitance
3
gNa= gNa m h sodium conductance
gK=

gK n 4 potassium conductance
gL = leakage conductance
Suppose V is kept constant. Then m tends exponentially to m(V) with time constant m(V), and
similar interpretation holds for h and n. The
function m and n increase with V since they are
activation variable, while h decreases.
Hodgkin-Huxley model
• Running on matlab
Hodgkin-Huxley model
Experiments
showed that gNa
and gK varied
with time and V.
After stimulus,
Na responds
much more
rapidly than K .
Fitzhugh-Nagumo model
• Fitzhugh reduced the Hodgkin-Huxley models to two variables,
and Nagumo built an electrical circuit that mimics the behavior
of Fitzhugh’s model.
• It involves 2 variables, v and w.
• V - the excitation variable represents the fast variables and may
be thought of as potential difference.
• W - the recovery variable represents the slow variables and may
be thought of as potassium conductance.
• Generalized Fitzhugh-Nagumo equation:
dv
dw

 f (v,w),
 g(v,w)
dt
dt
Fitzhugh-Nagumo model
• The traditional form for g and f
- g is a straight line g(v,w) = v-c-bw
- f is a cubic f(v,w) = v(v-a)(1-v) -w, or
f is a piecewise linear function
f(v,w) =H(v-a)-v-w, where H is a heaviside
function
Consider the numerical solution when f is a cubic:
dv
  f (v,w)  v(v  a)(1 v)  w
dt
dw
 g(v,w)  v  bw
dt
Fitzhugh-Nagumo model
t
• Defining a short time scale by T  and defining

V(T) = v(t), W(T) = w(t), we obtain:
•
dV
 f (V,W )  V (V  a)(1 V )  W
dT

dW
 g(V,W )  (V  bW )
dT
• The two systems of ODE will be used in different
phases of the solution (phase 1 and 3 use short time
scale, phase 2 and 4 use long time scale).

Fitzhugh-Nagumo model
•
There are 4 phases of the solutions
-phase 1: upstroke phase - sodium channels open, triggered by partial
depolarization and positively charged Na+ flood into the cell and hence
leads to further increasing the depolarization (the excitation variable v is
changing very quickly to attain f = 0).
-phase 2: excited phase - on the slow time scale, potasium channel
open, and K+ flood out of the cell. However, Na+ still flood in and just
about keep pace, and the potential difference falls slowly (v,w are at the
highest range).
-phase 3: downstroke phase-outward potassium current overwhelms
the inward sodium current, making the cell more negatively charged.
The cell becomes hyperpolarized (v changes very rapidly as the solution
jumps from the right-hand to the left-hand branch of the nullcline f=0).
-phase 4: recovery phase-most of the Na+ channels are inactive and
need time to recover before they can open again (v,w recovers from
below zero to the initial v, w at 0).
Fitzhugh-Nagumo model
h1
h2
h3
Numerical solution for f(v,w) = v(v-a)(1-v) -w and g(v,w) = v-bw with
=0.01, a =0.1, b =0.5. The equations have a unique globally stable
steady state at the origin. If v is perturbed slightly from the stead state,
the system returns there immediately, but if it is perturbed beyond v =
h2(0) = 0.1, then there is a large excursion and return to the origin.
Fitzhugh-Nagumo model
• There are 3 solutions of f(v,w) = 0 for w*≤w≤w* given
by v =h1(w), v=h2(w) and v=h3(w) with h1(w)≤ h2(w)≤
h3(w).
• Time taken for excited phase:
– We have f(v,w) = 0 by continuity v=h3(w), and w satisfies w’ =
g(h3(w),w) = G3(w). Hence w increases until it reaches w*,
beyond which h3(w) ceases to exist. The time taken is
w*
1
t2  
dw
w0 G3 (w)
Fitzhugh-Nagumo model
Fitzhugh-Nagumo model
Fitzhugh-Nagumo model
• When g is shifted to the left:
• g(v,w) = v -c -bw
• The results have different behavior. In
recovery phase, w would drop until it reached
w*, and we would then have a jump to the
right-hand branch of f =0. This repeats
indefinitely and have a period of oscilation
equal to:
w*
1
1
tp   (

)dw
G1 (w)
w* G3 (w)
Fitzhugh-Nagumo model
The solution
have a unique
unstable steady
state at (0.1,0),
surrounded by
a stable
periodic
relaxation
oscillation.
A numerical solution of the oscillatory FitzHugh-Nagumo
with f(v,w) = v(v-a)(1-v) -w and g(v,w) = v-c-bw.
Fitzhugh-Nagumo model
Reference
• Britton N.F. Essential Mathematical
Biology, Springer U.S. (2003)

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