Neural Modeling

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
•
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
Source: http://en.wikipedia.org/wiki/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
C m dV
  g
Na m
3 h ( V

V
Na
)
 g
K n
4
( V

V
K
)
 g
L
( V

V
L
) dt
H-H variables:
 m
( V ) dm dt
 m

( V )
 m
V-potential difference m-sodium activation variable
 h
( V ) dh dt
 h

( V )
 h h-sodium inactivation variable n-potassium activation variable
C m
-membrane capacitance
 n
( V ) dn dt
 n

( V )
 n g
Na g m
3 h g
K g
K n
4 g
L
= leakage conductance

Suppose V is kept constant. Then m tends exponentially to m
 similar interpretation holds for h and n. The function m activation variable, while h
 decreases.

 and n

(V) with time constant
 m
(V), and increase with V since they are

Hodgkin-Huxley model
• Running on matlab

Hodgkin-Huxley model
Experiments showed that g
Na and g
K 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 dt
 f ( v , w ), dw dt
 g ( v , w )

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 dt
 f ( v , w )
 v ( v
 a )(1
 v )
 w dw
 g ( v , w )
 v
 bw dt


Fitzhugh-Nagumo model
• Defining a short time scale by T

 t
V(T) = v(t), W(T) = w(t), we obtain: and defining
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 =h
1
(w), v=h
2
(w) and v=h
3
(w) with h
1
(w)≤ h
2
(w)≤ h
3
(w).
• Time taken for excited phase:
– We have f(v,w) = 0 by continuity v=h
3
(w), and w satisfies w’ = g(h
3
(w),w) = G
3
(w). Hence w increases until it reaches w * , beyond which h
3
(w) ceases to exist. The time taken is t
2
 w *  1 dw
G
3
( w ) w 0


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: t p

* w

(
1
G
3
( w ) w
*

1
G
1
( w )
) dw


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