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nemenman>Ilya at 13:34, 17 November 2012

2012-11-17T13:34:27Z

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{{PHYS434-2012}}

#Consider the following Langevin differential equation that we discussed in class: <math>\frac{dx}{dt}=-kx+c\eta</math>, where <math>\eta</math> is a Wiener process, that is, it is a Gaussian variable with <math>\langle\eta\rangle=0</math>, <math>\langle\eta(t)\eta(t')\rangle=\delta(t-t')</math>.
#*Write simple program that would solve this equation using Euler stepping. That is, following our discussion in one of the previous homeworks, for a given <math>k</math>, <math>c</math> and the temporal step size <math>\Delta t</math>, we can define <math>x_n\equiv x(n\Delta t)</math>, and then <math>\frac{x_{n+1}-x_{n}}{\Delta t}= -kx_n+\frac{c}{\sqrt{\Delta t}}\nu</math>, where <math>\nu</math> is a Gaussian random variable with zero mean and unit variance. This then gives <math>x_{n+1}=x_{n} -kx_n\Delta t+c\sqrt{\Delta t}\nu</math>, which can be turned into a simple for-loop code for simulating a sequence of x's.
#*Simulate <math>N=1e5</math> steps of this dynamics of <math>x</math>. Plot <math>x</math>. Describe what you see.
#*Take a Fourier transform of this <math>x</math> using the Matlab build-in <math>fft</math> function.
#*Plot (in log-log scale) the power spectrum (that is <math>\left|x_\omega\right|^2\equiv x_\omega x_{-\omega}</math>.
#*Using the expressions we derived in class, show that the power spectrum for this <math>x</math> should look like <math>\langle x_\omega x_{-\omega}\rangle=\frac{c^2}{k^2+\omega^2}</math>. Compare this to your plot. Do they look similar? Think about why or why not. (Hint -- they will be similar for about half of your plot).
#Write down an expression for the mutual information through an enzymatic amplifier, derived in class. Suppose the input signal is band limited, so that its spectrum is <math>\langle \Delta E_{a,\omega}\Delta E_{a,-\omega}\rangle= \frac{\epsilon^2}{1+\omega^2\tau_E^2}</math>, where <math>\epsilon</math> is some constant depending on the system kinetic rates. We suppose that there's no input noise, and only intrinsic noise in the amplifier. Remembering that <math>g_0\frac{1}{\tau}\le {\rm const}</math> (recall what this constant is), can you find the best setting for an amplifier (that is, the choice of <math>\tau</math>), which would improve the mutual information between the input or the output of the amplifier? That is, can filtering improve the information? Now suppose there is input noise <math>\delta E_{a,\omega}</math>. Does it change the result? Explain your results.

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Physics 434, 2012: Homework 8 - Revision history

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nemenman>Ilya at 13:34, 17 November 2012