CSYS5040 Criticality in Dynamical Systems Assignment 2

Due Date: This is assignment is due in TurnItIn by Friday, September the 17th. This
assignment is worth 25% of your final mark.

You must do all of your working in a Mathematica notebook that I can run (no pdfs of
Mathematica notebooks).

The * for some questions indicate the relative difficulty of the question. This is an individual
assessment; your answers must reflect your own work. Marks will be based on the
correctness of each answer, the effort put into exploring each question, and the originality
of the examples you choose to look at. You are strongly encouraged to read beyond the
class material to get a higher grade.

Question 1 (7.5%): The dynamics of a stochastic differential equation
a. Choose the constant valued parameters (i.e. , ! and ) of a linear stochastic
differential equation of the form: d = d + d where is the strength of the
stochastic (noise) term, ! is the initial starting point and then plot the time series of
the solution, making sure is not equal to zero (it can be positive or negative
though). Choose some numerical value (for ‘boundary’) that has the same sign as , run some simulations of your solution, does your solution ever cross the boundary and if so at what value of does your solution cross it? Without simulating your
solution, how can you know when to expect the solution to cross the boundary?
b. * Repeat part a. except that the stochastic differential equation is now non-linear,
i.e. d = ( + ) d + d or uses even higher order terms in , e.g. "or #such
as d = (" + + ) d + d you will score higher the more sophisticated your
model is in this part and consequently for part 1 c. below. Do not implement the
same equations here that are needed for Question 2.a.
c. Write a single paragraph on one application of the methods you used in parts a. and
b. For example you can look up: “drift diffusion” and “neural network”, “two
alternative forced choice task” (e.g. here: http://tinyurl.com/yygku3oa ), “Ornstein-
Uhlenbeck process”, or see here: http://en.wikipedia.org/wiki/Ornstein–
Uhlenbeck_process

Question 2 (7.5%): Plotting non-linear functions for a non-linear map
a. From the article we looked at in Week 4 I want you to implement one of the non-
linear stochastic neural models. Do not implement the same equations you used for
1.b. For this first step all you have to do is replicate the work I showed you in class
but using either 1 or 2 (depending on which model you chose to implement)
stochastic non-linear equations that you will you find in the articles listed below
where a decision is reached once a “decision variable” crosses a boundary threshold.
This question is not intended to be difficult to understand but it can still be tricky to
implement, just modify the code I’ve already given you in class to reflect the non-
linear model you’re implementing. For the different equations for each system see
pages 705 to 707 of the article here
http://sites.engineering.ucsb.edu/~moehlis/moehlis_papers/psych.pdf
Or look at the different equations listed on the Wikipedia page under “Other
Models” here: http://en.wikipedia.org/wiki/Two-alternative_forced_choice
b. * Find an article (using Google Scholar etc.) that has used the model you
implemented in part 2a. and illustrate some aspect of the results from that article
using the model you’ve just implemented.
c. Discuss the impact of the model you’ve used in the context of the article you’ve
found, e.g. you might discuss why this stochastic model was used rather than some
other model, or what the parameters mean in a practical setting, or what new
interpretation the model has provided in the area of study etc.

Question 3 (10%): Parameters in non-linear dynamical systems
a. **Based on your answer to Question 1 for the non-linear system, write a
Mathematica function that shows what happens when a parameter value, e.g. or changes and the system switches from one equilibrium state to another. See the
Mathematica notebook from Week 1, at the end of the notebook there is a
stochastic diffusion processes with more than one stationary state, we didn’t discuss
this model but I want you to base your answer on the ideas outlined there. Note that
the stationary state the system was attracted to (settled in) depended on the initial
starting point !. For this question, using a system with more than one stationary
state, I want you to let the system find its stationary state (i.e. it is in equilibrium)
and then change the parameter values so that the system switches from one
equilibrium point to another by going through a tipping point. The minimum you
need to do to pass this question is to plot the time series of the system passing
through this tipping point.