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
(ii) State and brieﬂy explain four valid reasons for modelling a natural system.
(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.
b) 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
(ii) Use Euler’s method to solve the diﬀerential equation dx
= ax, for initial con-
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?
(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
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
(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
(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
(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.
b) Solve the diﬀerential equation αdx
= x(1 − βx). Also, ﬁnd the stable steady-state
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%]
c) (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.
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