Model Neurons and Fast-Slow Systems
Click here to receive the DynaSys input file (updated 4/7/97)
Introduction
Neurons are the fundamental units of information processing in the human body and as such have been the subject of much study. A. L. Hodgkin and A. F. Huxley in the early 1950's found that a neuron processes information by controlling the flow of charged ions through its cell membrane, hence generating an electrical signal. Hodgkin and Huxley proposed that a neuron can be modeled via an equivalent electrical circuit and ultimately by a system of four differential equations. This work is the foundation of modern neuroscience and earned them the Nobel Prize. Since this pioneering work more and more detailed models of specific types of neurons have been developed through a combination of experiments and mathematics. The advent of the digital computer has allowed the numerical study of these models, providing new insights into the function of neurons.
A neuron transmits information by generating ``action
potentials.'' In general, neurons exhibit two modes of operation,
excitable and bursting. An excitable neuron is quiet (does not
generate action potentials) except in response to an outside
stimulus (for example from a second neuron through a synapse). A
bursting neuron generates periodic trains of action potentials.
This is often the case in neurons that help regulate rhythmic
body functions such as heart contractions. Many neurons exhibit
both of these behaviors depending on various factors. A common
experiment is to apply different levels of current through an
electrode to a neuron to determine whether and where a neuron
transitions from excitable to bursting.
The Model
In this lab we will study the Morris-Lecar equations, a system of two differential equations similar to the Hodgkin-Huxley equations, to explore how a neuron responds to external stimulus. The Morris-Lecar equations were originally formulated to describe electrical activity in barnacle muscle fiber and are sometimes used as a simple caricature of the envelope of bursting neurons and only explicitly model the flow of potassium (K+) and calcium (Ca2+) ions. The variable v in the given equations denotes the voltage of the neuron while the variable w is known as a recovery variable and describes the percentage of open channels selectively permeable to Ca2+.
w' = ελ(v)(w∞(v) - w)
The functions m∞(v), w∞(v), and λ(v) are given by
The parameter I represents injected current into the model neuron. Both I and ε will be treated a parameters in the exercises. The values of the other constants are given in the table below.
Constant |
|
|
|
|
|
|
|
|
|
|
Value |
-0.01 |
0.15 |
-0.12 |
0.30 |
0.22 |
2.00 |
-0.70 |
0.50 |
-0.50 |
1.10 |
Problem 1. Plot the functions
m∞(v) and
w∞(v).
What happens to the graphs of these functions as
v2 and
v4
approach zero respectively?
Problem 2. Compute 
and 
.
Problem 3. Speculate on the
purpose of these functions in these equations.
Analysis
Excitable Neurons.
Enter the Lecar-Morris equations into your ordinary
differential equations solver. For your convenience, a phase
plane is displayed below for the equations, if you prefer to use
it instead of your own solver.
Begin by setting I
= 0.25 and ε = 0.1. Your phase space window should be
-1 ≤ v ≤ 1, -0.5 ≤ w ≤ 1. Set your integrator to integrate for a total time
of t=40. You might want to use an integrator specifically
designed for ``stiff'' differential equations such as
Gear's method.
Problem 1. Draw both the v
and w nullclines and compute the stability of the
critical point.
Problem 2. Compute the
solutions to the Morris-Lecar equations with initial conditions
of (-0.5,0.1) and (-0.5,0.2). Plot these solutions both in the (v,w) phase plane and plot the graphs of v
vs t. For each initial condition describe the relationship
between the trajectory in phase space and the v vs t
plot. What are the differences and similarities between the two
solutions? What is the physical interpretation?
Problem 3. Keeping the graphs
of the nullclines on the screen draw trajectories from at least 6
initial conditions on a vertical line left of the critical point extending beyond the
minimum of the v-nullcline. Describe what you see. What do
you think the physical interpretation of this phenomenon is?
Problem 4. In the previous
exercise you noticed that some points on your line went directly
to the critical point while others made a long excursion around
the right branch of the v nullcline before approaching the
critical point. This phenomenon extends through much of the phase
plane. Try to determine a curve or region that partitions the
phase plane by this criterion. This may have to be done by a
combination of computing and using a printout of your nullclines
to record how each initial condition behaves. Are there any
points that don't fit into either category?
Problem 5. Repeat
exercise (3) for ε = 0.05 and ε = 0.02
(you may need to increase your total integration time
as ε decreases). How does the separation between the two
regions change? What do you think separates theses regions in the
limiting case ε → 0.
Advanced Analysis
You probably noticed that for small ε
trajectories consisted of fast, almost
horizontal ``jumps'' to the v-nullcline and slow
``drifts'' along the v-nullcline. This behavior is typical
of differential equations of the form
| x' | = | f(x,y) | |
| (1) | |||
| y' | = | ε g(x,y) |
for 0 < ε << 1.
Note that if ε ≡ 0
then y' ≡ 0
and hence y acts as a parameter.
Problem 1. What is the
interpretation of the v-nullcline when ε ≡ 0
in the Morris-Lecar equations? Describe the dynamics
for each value of w. What can you say about these dynamics
and the dynamics you observed numerically for small ε?
Problem 2. Let τ = εt
and use the chain rule on
equations (1) to show that
If ε ≡ 0 this becomes
If we can solve (2) for x
in terms of y then there exists a function 
such that 
. In other words, the x-coordinate of the flow
is given by 
. This means that the flow
is on the curve with velocity given by 
. The curve is known as the slow manifold and the flow on this
manifold is known as the slow flow. For the Morris-Lecar
equations the Implicit Function Theorem can be used to show that
there exists functions 
and 
that parameterize the left and right branches of the v-nullcline
in terms of w. You won't be able to compute these
functions, however the qualitative dynamics on each of these
branches can be determined by considering the geometry of both
nullclines. Do this to show that the slow flow on the left branch
is towards the critical point at the intersection of the two
nullclines and that the flow on the right branch is upward to the
local maxima of the v-nullcline.
Problem 3. The ``singular
dynamics'' are determined by combining the results of parts one
and two. Describe the singular orbit originating from the points
(-0.5,0.1) and (-0.5,0.2). How do these compare with the
numerically computed orbits originating from the same points for
small ε? Numerically compute the orbits from these points for
various values of ε. At what value
of ε do you think the singular approximation ceases to be
valid?
Mathematical modeling of Biological
Neural Networks
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