APMA 0410 Final Exam
19 December 2013
Problem 1
Using the method of judicious guessing, find the general solution of:
! = 2 + ! cos
Problem 2
Do a complete analysis, including eigenvector-
eigenvalue study and pha

APMA 0410 MIDTERM WARMUP SOLUTIONS
October 7, 2013
Three Poisson neurons, A, B and C, fire independently of each other. Their rates
are, respectively, 3 spikes/s, 5 spikes/s, and 9 spikes/s.
Problem 1.
We denote the numbers of spikes of A, B and C during

APMA 0410 MIDTERM 2013
SOLUTIONS
Problem 1.
We record for 1 second from two populations of neurons, A and B. Each population
contains 100 neurons. All neurons fire independently of each other. During the 1
second of recording, each neuron fires at most on

HW6
Solutions
Problem 6.1.
"
$
$
#
$
$
%
dx
= F ( x, y ) = x x 3 y
dt
dy
= G ( x, y ) = y
dt
Vertical nullcline: y = x x3. Horizontal nullcline: y = 0.
There are three equilibria: (1, 0), (0, 0), (1, 0).
1 3x 2 1
The Jacobian is: J =
0
1
At (0, 0 ):
1

HW5
Problem 5.1
a = -2; b = -2; c = -1; d = -3; e = 4; f = 0;
Vertical nullcline:
y=x+2
Horizontal nullcline: y = x/3
(T, D) = (5, 4). The equilibrium is (3, 1) and it is a stable node.
Eigenvalues
Eigenvectors
Separatrices
1 (slow)
(2, 1)
y = x/2 + 1/2
4

Problem 4.1. Qualitative analysis of
dx/dt = x2 (9 x2) (1 + x).
From the diagram to the right we see
that there are four equilibria and their
stability is as follows:
3 is stable;
1 is unstable;
0 is stable from the left (from below)
and unstable from the

APMA 0410 HW3 SOLUTIONS
1.
We record the spiking activity of two Poisson Neurons, C and D, with
millisecond accuracy. The probability of firing of C in a one-millisecond
interval is pC = 0.05. The probability of firing of D in a one-milliseco

APMA 0410 Final Exam 2010
Problem 1. Do a qualitative analysis of the following system of first-order linear
differential equations:
dx/dt = 2x + 2y
dy/dt = 3x + 2y.
Problem 2. Do a qualitative analysis (including eigenvalue/eigenvector analysis and
drawi

APMA 0410 Final Exam 2010
Problem 1. Do a qualitative analysis of the following system of first-order linear
differential equations:
dx/dt = 2x + 2y
dy/dt = 3x + 2y.
Problem 2. Do a qualitative analysis (including eigenvalue/eigenvector analysis and
drawi

APMA 0410 Final Exam Solutions
19 December 2013
Problem 1
Using the method of judicious guessing, find the general solution of:
! = + !
The derivative of ! cos is ! cos ! sin , and when we differentiate

APMA 0410 MIDTERM WARMUP
PROBLEMS
October 7, 2013
Three Poisson neurons, A, B and C, fire independently of each other. Their rates are,
respectively, 3 spikes/s, 5 spikes/s, and 9 spikes/s.
Problem 1.
We denote the numbers of spikes of A, B and