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DEPARTMENT OF COMPUTER SCIENCE Spring Semester 2018-2019 MODELLING AND SIMULATION OF NATURAL SYSTEMS 2 hours
Answer BOTH questions.
All questions carry equal weight. Figures in square brackets indicate the per- centage of available marks allocated to each part of a question.
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1. a)
General concepts for modelling natural systems.
(i) Describe what a natural system is and give two distinct examples. Give one example of a non-natural system that in your view could be modelled as a natural system.
[10%]
(ii) State and briefly explain four valid reasons for modelling a natural system.
[20%]
(iii) You are presented with two models fitting a set of data. One of these models has 15 free parameters and error 0. The other has 4 free parameters and 2% error. Give a definition of what a free parameter is. What are the disadvantages of a model with too many free parameters? Briefly discuss which of the two models you would use.
[30%]
Numerical methods for solving differential equations (Euler’s method).
(i) Explain Euler’s method. As part of your answer, provide the corresponding algorithm. How should one choose the discretisation time-step ∆t? Justify your answer.
[20%]
(ii) Use Euler’s method to solve the differential equation dx = ax, for initial con-
dt
ditions t0 = 0, x0 = 1 and a = 100. Find the values of x(t) for t=0.1 and t=0.2. (Hint: choose the discretisation time-step ∆t in a convenient way for this purpose.) How does the choice of ∆t affect the quality of the solution?
[10%]
(iii) A first order linear differential equation can be solved with Euler’s method, or analytically (via integration). How do the two solutions differ in terms of accuracy?
[10%]
b)
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2. a)
Figure 1 (last page of this document) shows the population of a species x at time tn+1 versus the population of the same species at time tn (index n refers to the time discretisation).
(i) Define mathematically when a system is in steady-state and explain what this means in the context of modelling populations. Explain what the steady-state points are.
[10%]
(ii) Use Figure 1 to find the steady-state points of the population. Which are stable and which are unstable (if any)? Interpret and justify your response. (Hint: you may want to copy the figure in your answer sheet and use it to justify your response.)
[15%]
(iii) Which differential equation describes the growth of the population x (shown in Figure 1)? Justify your response. The plot in Figure 1 can be described as a line y = βx. Express mathematically the parameter(s) of the differential equation as a function of β.
[25%]
Solve the differential equation αdx = x(1 − βx). Also, find the stable steady-state
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point(s), if any. Assume α > 0 and β > 0 (Hint: you don’t need to solve the equation
to do the stability analysis). [25%]
(i) What is a spiking neuron model? Briefly describe the integrate-and-fire model.
(ii) Describe the problem of neuronal coding. As part of your reply, give one example according to which information could be transmitted in brain systems.
[10%]
b)
c)
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[15%]
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Figure 1: Population size x(tn+1) vs x(tn) of observed species. END OF QUESTION PAPER
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Population size at t n
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Population size at t n+1