Post on 15-Dec-2015
transcript
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• 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.
• Synapses-specialized 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
€
Cm
dV
dt= −g Nam3h(V −VNa ) − g K n4 (V −VK ) − gL (V −VL )
€
τm (V )dm
dt= m∞(V ) −m
τ h (V )dh
dt= h∞(V ) − h
τ n (V )dn
dt= n∞(V ) − n
H-H variables:
V-potential difference
m-sodium activation variable
h-sodium inactivation variable
n-potassium activation variable
Cm-membrane capacitance
gNa= sodium conductance
gK= potassium conductance
gL = leakage conductance€
g Nam3h
€
g K n4
Suppose V is kept constant. Then m tends exponentially to m(V) with time constant τm(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:
€
ε 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:
€
ε dv
dt= f (v,w) = v(v − a)(1− v) −w
dw
dt= g(v,w) = v −bw
Fitzhugh-Nagumo model
• Defining a short time scale by and defining V(T) = v(t), W(T) = w(t), we obtain:
•
• 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).
€
T =t
ε
€
dV
dT= f (V ,W ) =V (V − a)(1−V ) −W
dW
dT= εg(V ,W ) = ε(V −bW )
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
Numerical solution for f(v,w) = v(v-a)(1-v) -w and g(v,w) = v-bw with ε=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.
h1 h2 h3
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
€
t2 =1
G3(w)w0
w*
∫ dw
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:
€
tp = (1
G3(w)−
1
G1(w)w*
w*
∫ )dw
Fitzhugh-Nagumo model
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.
The solution have a unique unstable steady state at (0.1,0), surrounded by a stable periodic relaxation oscillation.
Fitzhugh-Nagumo model
Reference
• Britton N.F. Essential Mathematical Biology, Springer U.S. (2003)