BME 6938 Neurodynamics 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

Example of Nullcline and corresponding vector fields for a 2d-neuron model..More later Regions a and d Regions a and b

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 [ ]  Click (H)(C) to open a new plot window. Click (v)(2d) to set the axes: x-axis -80:100 y-axis:-.5:1

Goal

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 A B C D Fixed points A, B and C satisfy D represents a closed trajectory; solution for which

Two Dimensional Phase Portrait Additonal 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 e 1 and e 2  Nontrivial case: There exists matrix P=[v 1 v 2 ] 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 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: v 1 Stable manifold: v 2 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