BME 6938
Neurodynamics
Instructor: Dr Sachin S Talathi
Recap
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One Dimensional Flow: Bifurcations, Normal form,
Stability
One dimensional neuron model:
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Inward activation
Inward Inactivation
Outward Activation
Outward Inactivation
XPPAUTO for neuron modeling
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Auto
Bifurcation diagram, bistability, hysterisis
Two Dimensional Dynamical Systems
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Also called planar systems
f and g describe the evolution of two-dimensional state
variables: x(t), y(t)
f and g are vector fields that describe how the trajectory
will evolve
Two dimensional vector fields:
Nullclines
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Nullclines are curves along which the vector fields are
completely horizontal or completely vertical
The set of points where the vector field changes its
horizontal direction defines the x-nullcline
x-nullcline partitions the phase space into two parts
where either x increases or x decreases
x-nullcline is defined through
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Similarly y-nullcline is defined through
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Example
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Consider the following two dimensional ODE
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Draw the nullclines for the dynamical system. Infer the
stability of the fixed point from the phase plot by looking
at the evolution of the vector field
XPPAUTO to see Nullclines/Direction
Fields of a Neuron Model
V-Nullcline
n-Nullcline
Setting up XppAuto for TwoD-Neuron
Model (Ex3.ode)
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The ODE file: Copy the ode file from course website
(Ex3.ode)
Neglect the comment in the ode file on type-1 and type-II
dynamics for now. We will look into it later in the class.
For time being select the parameters for type-1 dynamics
In the nUmerics menu select Dt=0.1; Total Time=100;
Bounds=1000
Go to main menu and press (W)(W) to set X-axis:[0
100] and y-axis [-80 120]
Click (H)(C) to open a new plot window. Click (v)(2d) to
set the axes: x-axis -80:100 y-axis:-.5:1
We can get oscillations in Two-D (Limit
cycles and )
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Limit cycles: Isolated periodic orbits in the phase space
Simple example
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r˙ = r(1- r 2 )
q˙ = 1
•Limit cycles are inherently a nonlinear
phenomenon
•Usually it is difficult to guess the existence
of limit cycles by looking at the set of
ODEs
•We will learn more about limit cycles and
their role in neuronal dynamics
Center: Continuum of periodic orbits in the phase space
• Can be observed in linear systems
• The amplitude of oscillations typically
depend on initial conditions.
Simple example:
x˙ = y
y˙ = -w 2 x
Relaxation oscillators: Intuitive way of thinking about
the origin of spikes by neurons
x˙ = f ( x, y)
y˙ = mg( x, y) m << 1
•Relaxation oscillators are special because of huge separation of time scales
•Neurons also exhibit fast and slow time scales
•Neurons are not relaxation oscillators, because mu is not small enough;
•There is finite rise time for membrane potential
•Time permitting we will consider relaxation oscillators in details when we study
bursting neuron models
Recap
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Two Dimensional Flow
Direction Field; Nullclines
Mentioned limit cycles in passing (closed trajectories in
phase space)
XPPAUTO to draw nullclines and direction fields.
Phase portrait for two dimensional
system: General Features
A
B
D
C
Fixed points A, B and C satisfy
D represents a closed trajectory; solution for which
Two Dimensional Phase Portrait
Additonal Properties
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Trajectories in phase portrait do not cross each other
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(Consequence of existence and uniqueness property of
solution for initial value problem in dynamical systems)
Fate of a trajectory enclosed in a close in a closed orbit
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Poincare-Bendixon Theorem: If a trajectory is enclosed by a
closed orbit and there is no fixed point inside the trajectory,
then the trajectory will eventually approach the closed orbit
Also tells us that a closed orbit encloses a fixed-point
(Another theorem, btw)
Two Dimensional Linear systems
With initial conditions
A is
constant matrix
General solution is
Simple example
Uncoupled two-d system
where
Solution:
Phase portrait-Bifurcation
Trajectory approaches
the stable fixed point in
direction tangent to the
slower axis
General Case: Some Linear Algebra
Diagonalize A:
where
=2x2 diagonal matrix
OR
What choice for P will diagonalize A?
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Select P to be the eigen-vector of A that satisfies the
following linear equation
Eigen value of A
Eigen vector of A
Find eigen vectors and eigen values by solving the characteristics equation
3 possibilities
OR
Choice of
We have:
Case I.
for three cases considered
Example
æ -2 1ö
A=ç
÷
è 0 2ø
Eigen values are: l1 = -2 l2 = 2
æ 1ö
æ 1ö
Eigen vectors are: e1 = ç ÷ e2 = ç ÷
è 0ø
è 4ø
æ 1 1ö
P=ç
÷
è 0 4ø
æ a bö
U
=
For a general invertible 2x2 matrix
ç
÷
è c dø
P -1 =
U -1 =
1 æ d -bö
ç
÷
ad - bc è -c a ø
1 æ 4 -1ö
ç
÷
4 è0 1ø
æ
-2t
tA
tL -1 ç e
Easy to show e = Pe P =
ç
è 0
(e
2t
)
- e -2t ö
÷
4
÷
e2t
ø
The solution to x˙ = Ax
x(t) = e tA x0
Complex conjugate eigen-values
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Case II
Claim: There exists P such that
Let z be a complex eigenvector of A such that
Let
Example
æ 2 1ö
A=ç
÷
è -2 0ø
Eigen values are:l1 = 1+ i l2 = 1- i
Eigen vectors corresponding to l1 = 1+ i is z = (1 -1+ i )
The solution to x˙ = Ax
1
1
æ 1 -1ö
æ
ö
tA
P=ç
P -1 = ç
÷
÷
x(t)
=
e
x0
è0 1ø
è 1 0ø
T
&
æ 1 -1ö
L= ç
÷
è1 1 ø
-1
Verify L = P AP and show
tA
tL
-1
t æ cos t + sin t
e = Pe P = e ç
è -2sin t
ö
÷
cos t - sin tø
sin t
Degenerate eigenvalues
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Case III.
Trivial case:
for any choice
of e1 and e2
Nontrivial case: There exists matrix P=[v1 v2] such that
Note:If we are willing to work with complex eigen vectors then it is possible to dia
gonalize any matrix A with distinct eigen values
Example
æ 5 -3ö
A=ç
÷
è3 1 ø
l1 = l2 = 2
Eigen values are:
Lets choose e2 = (1 -1)
T
Note:
(A - lI)e2 ¹ 0
e1 = (A - lI)e2
æ 2 1ö
L= ç
÷
è 0 2ø
e1 = (6 6)
T
æ6 1ö
P=ç
÷
è 6 -1ø
-1 æ -1 -1ö
& P =
ç
÷
12 è -6 6 ø
-1
2t
2t
æ
ö
e
1+
3t
-3te
(
)
tA
tL -1
-1
Verify L = P AP and show e = Pe P = ç
2t
e2t (1- 3t )÷ø
è 3te
Intuitive way to think of degenerate case
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The trajectories are almost trying to spiral into the fixed
point (if stable) but do not quite make it
It represents a border line case between a node and
spiral
Classification Scheme for fixed points
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We saw how to get explicit solution to any 2-dimensional
linear ODE.
However if we are interested in global properties of the
trajectories: explicit solutions are unnecessary
The eigen values of A give us all the information about the
stability properties of the fixed points of the system and
we can devise classification scheme based on the
eigenvalues for the system
Classification Scheme for fixed points
D = l1l2
t = l1 + l2
•Case I: D < 0
saddle node
•Case II: D > 0
•(a) t 2 - 4D > 0
either stable or unstable node
•(b) t 2 - 4D < 0
either stable or unstable spiral
•(c) t 2 - 4D = 0 border line between nodes and spirals
•Case III: D = 0
one of the eigenvalues is zero; no isolated fixed point
but a series of fixed points (centers)
Summary of the classification scheme
centers
Degenerate nodes
Dynamics around fixed point based on
previous analysis
Linear Stability Analysis for Nonlinear
Two-Dimensional System
Evaluated at the fixed point
(x*, y*)
Few comments
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The Linearized system does represent the dynamics of
the nonlinear system locally around the fixed point
correctly (Stable manifold theorem or the HartmanGrobman Theorem) when the real part of eigenvalues are
non-zero.
Fixed point in these case are referred to as the hyperbolic
fixed points. The contribution from nonlinear higher
order terms is negligible locally around hyperbolic fixed
points.
Nonhyperbolic fixed points are those for which the real
part of eigenvalue is zero. They are more sensitive to
higher order nonlinear terms eg: Center’s Bifurcation
points etc..
Invariant Manifolds
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The eigenvectors of A corresponding to the cases with
non-degenerate eigenvalues considered earlier represent
invariant manifolds for the dynamical system.
Eg. Lets say the phase space is 2 dimensional made up of
dynamical variables x and y. If initial condition is on x-axis
and the flow as the system evolves remains on x-axis then
x-axis is the invariant manifold of the dynamical system.
In other words, orbits that start on the manifold remain in
it.
Examples of Invariant Manifold
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Invariant manifolds of saddle (1-Dimensional manifolds)
Spirals (II-Dimensional manifolds)
Stable and Unstable Manifolds of
Saddle
Unstable manifold: v1
Stable manifold: v2
Stable manifold of saddle is also referred to as the seperatrix; since it separates
the phase plane into different regions of long term behavior
Revisit a simple example
We saw the phase space for this system earlier in our class. Lets Revisit and now
Identify the stable and unstable manifolds of the saddle node
(This time using XPPAUTO)
Important of stable manifold of saddle
in Neurodynamics
Note:
Threshold is
not a single
voltage value.
It is a curve in
Phase space;
defined by the
Stable manifold
Of saddle
Homoclinic and Heteroclinic
Trajectories
A trajectory is homoclinic if it originates from and terminates at the same equilibrium point
A trajectory is heteroclinic if it originates at one equilibrium and terminates at a
different equilibrium point