BME 6938Neurodynamics

Instructor: Dr Sachin S Talathi

Recap One Dimensional Flow: Bifurcations, Normal

form, Stability One dimensional neuron model:

Inward activation Inward Inactivation Outward Activation Outward Inactivation

XPPAUTO for neuron modeling Auto Bifurcation diagram, bistability, hysterisis

Two Dimensional Dynamical Systems 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 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

Similarly y-nullcline is defined through

Example Consider the following two dimensional ODE

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) 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 ) Limit cycles: Isolated periodic orbits in the

phase space

Center: Continuum of periodic orbits in the phase space

•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

• Can be observed in linear systems• The amplitude of oscillations typically depend on initial conditions.

Simple example

Simple example:

Relaxation oscillators: Intuitive way of thinking about the origin of spikes by neurons

•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 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

AB

CD

Fixed points A, B and C satisfy

D represents a closed trajectory; solution for which

Two Dimensional Phase PortraitAdditonal Properties Trajectories in phase portrait do not cross

each other (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 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

General solution is

A is constant matrix

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

OR

=2x2 diagonal matrix

What choice for P will diagonalize A?

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 for three cases considered

Case I.

We have:

Example

Eigen values are:

Eigen vectors are:

For a general invertible 2x2 matrix

Easy to show

The solution to

Complex conjugate eigen-values Case II Claim: There exists P such that

Let z be a complex eigenvector of A such that

Let

Example

Eigen values are:

Eigen vectors corresponding to is The solution to

&

Verify and show

Degenerate eigenvalues 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 diagonalize any matrix A with distinct eigen values

Example

Eigen values are:

&

Verify and show

Lets choose

Note:

Intuitive way to think of degenerate case 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 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

•Case I: saddle node•Case II:

•(a) either stable or unstable node•(b) either stable or unstable spiral•(c) border line between nodes and spirals

•Case III: 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 The Linearized system does represent the

dynamics of the nonlinear system locally around the fixed point correctly (Stable manifold theorem or the Hartman-Grobman 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 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 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 separatesthe 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 singlevoltage value.

It is a curve inPhase 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