LS30B Midterm

Lecture 2 increasing sigmoid

1/1+X^n

Lecture 10 Definition of the coordinates and the coordinate vector relative to a basis (p.2)

U V are vectors Xu and Xv are coordinates of the vector

Lecture 1.2 H/P/G model

-H/P/G axis -The gonads make a certain hormone G. -The pituitary gland makes a certain hormone, P. -The hypothalamus makes a certain hormone, H. -The control sequence is -More of H causes more of P to be made. -More if P causes more of G to be made. -More of G causes less of H to be made. -We will assume that -Rate of production of P is proportional to the level of H (with constant of proportionality 1) -The rate of production of G is proportional to the level of P (with constant of proportionality 1) -The rate of production of H decreases with the level of G. -We will assume that: -The decrease in each hormone level is proportional to the current level of that hormone (with constants k1, k2, k3). *In summary, you do not need to know the H/P/G or the Mackey-Glass model by heart, but, given the equations, you do need to be able to explain the physiological significance of each term in them. In particular, you do need to be able to explain how time delay and negative feedback are represented in the equations, and how you can adjust them (particularly in the Mackey-Glass model).term-26

Lecture 4 Linear Function *be able to check

A function f : R → R is linear if 1)f(X+Y)=f(X)+f(Y) 2) f(aX) = af(X)

Lecture 5 Know the definition of a basis for a vector space, and the example of standard basis for Rn

A set of vectors B1,...,Bk∈ is a basis for the vector space R^n if -every vector X∈R^n is a linear combination of B1,...,Bk for precisely one set of scalars a1,...,ak

Lecture 5 Linear Combination

A vector X∈ R^n is a linear combination of vectors V1,...,Vk∈R^n if there exist a1,...,ak∈R such that X = a1V1 + ... + akVk

Lecture 5 Be able to determine, by appealing to the definition, whether a given algebraic expression is a linear combination of the given variables

Again, a function f : R → R is linear if 1)f(X+Y)=f(X)+f(Y) 2) f(aX) = af(X)

Lecture 1.1 limit cycle attractor (LCA)

Closed loop that pulls other trajectories towards it. a state point X(t0) undergoes stable oscillation if the trajectory of X(t0) is a closed loop in Um THAT IS, A POUNT UNDERGOES STABLE OSCILLATION IF THE TRAJECTORY IS A CLOSED LOOP THA PULLS OTHER TRAJECTORIES TOWARDS IT. - Be able to explain why stable oscillation corresponds to a limit cycle attractor.

Lecture 3 oscillating spring with no friction

Do need to know that this model exhibits non-stable oscillation and the state space of this model has a neutral center equilibrium point.

"decreasing sigmoid function"

Lecture 1.2 You do need to know the general form of decreasing sigmoid functions, in particular that the negative feedback in the H/P/G is represented by a decreasing sigmoid function, and the steepness of negative feedback is controlled by the parameter n.

Lecture 1.1 stable oscillation

Represented by a limit cycle attractor in the state space (how it differs from non stable oscillation)

Lecture 2 Hopf bifurcation

The core idea is that systems like the H/P/G will have stable oscillation, hence LCA, when the negative feedback and time delay are both sufficiently large. The diagram on the last slide says it all.

Lecture 4 Know the definition of n-dimensional vector space over R, and be able to give examples; 123 that is, recognize that the line is R , the plane is R , and the 'ordinary' space is R , each equipped with the two algebraic operations: vector addition and scalar multiplication

The n-dimensional vector space over R is the n-dimensional Euclidean space R^n equipped space R^n equipped with the operations of vector addition and scalar multiplication.

Lecture 1.2 Explain how time delay is present in the H/P/G: the 'middleman' hormone P

Three variable system implies time delay (implicit) given the complexity of 3 variables

Lecture 2 Mackey-Glass equation (explain what the terms mean) -How is Negative feedback and time delay present?

X = The concentration of carbon dioxide in the blood X' = Body metabolism of CO2 - ventilation of CO2 -Assume that the body produces CO2 at a constant rate L Modeling the effect of ventilation: -Ventilation rate V = breaths/minute -V is controlled by the CO2 concentration in the blood -The more CO2 there is, the higher the rate of V. -X outside the parenthesis = CO2 level in the lungs -X inside the parenthesis = CO2 level "seen' by the brain THE BRAIN WILL ALWAYS BE A LITTLE "BEHIND" THE LUNGS. *Parameter n measures the steepness of negative feedback. *Parameter τ measures the time delay *For low values of n and tau ---> stable equilibrium. *For high value of n and tau (both)---> oscillation. *In summary, you do not need to know the H/P/G or the Mackey-Glass model by heart, but, given the equations, you do need to be able to explain the physiological significance of each term in them. In particular, you do need to be able to explain how time delay and negative feedback are represented in the equations, and how you can adjust them (particularly in the Mackey-Glass model).

Lecture 1.1 concept of an attractor in the state space of a model

a set A in the state space such that the trajectory of any point close enough to A will get closer and closer to A (but will never touch, cross trajectories because of determinism)

Lecture 1.1 Point attractor

a stable equilibrium point

Lecture 3 spring (reed) with "N-shaped" friction

limit cycle attractor X′(t) = V (t) V′(t)=−X(t)−(V(t)3 −V(t)) *Explain how this allows the possibility of holding a note with a clarinet.

Lecture 4 scalar multiplication

multiplication of a vector by a scalar

Lecture 3 Spring (reed) with linear function

stable spiral X′(t) = V (t) V ′(t) = −X(t) − V (t)

Lecture 4 vector addition

the combining of vector magnitudes and directions

Lecture 3 Spring (reed) with negative linear friction

unstable spiral X′(t) = V (t) V ′(t) = −X(t) − (−V (t))