Physics 434, 2012: Homework 8

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Back to Physics 434, 2012: Information Processing in Biology.

1. Consider the following Langevin differential equation that we discussed in class: ${\frac {dx}{dt}}=-kx+c\eta$, where $\eta$ is a Wiener process, that is, it is a Gaussian variable with $\langle \eta \rangle =0$, $\langle \eta (t)\eta (t')\rangle =\delta (t-t')$.
   • 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 $k$, $c$ and the temporal step size $\Delta t$, we can define $x_{n}\equiv x(n\Delta t)$, and then ${\frac {x_{n+1}-x_{n}}{\Delta t}}=-kx_{n}+{\frac {c}{\sqrt {\Delta t}}}\nu$, where $\nu$ is a Gaussian random variable with zero mean and unit variance. This then gives Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{n+1}=x_{n} -kx_n\Delta t+c\sqrt{\Delta t}\nu} , which can be turned into a simple for-loop code for simulating a sequence of x's.
   • Simulate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N=1e5} steps of this dynamics of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} . Plot Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} . Describe what you see.
   • Take a Fourier transform of this Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} using the Matlab build-in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle fft} function.
   • Plot (in log-log scale) the power spectrum (that is $\left|x_{\omega }\right|^{2}\equiv x_{\omega }x_{-\omega }$.
   • Using the expressions we derived in class, show that the power spectrum for this Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} should look like Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle x_\omega x_{-\omega}\rangle=\frac{c^2}{k^2+\omega^2}} . 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).
2. 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \Delta E_{a,\omega}\Delta E_{a,-\omega}\rangle= \frac{\epsilon^2}{1+\omega^2\tau_E^2}} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon} 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_0\frac{1}{\tau}\le {\rm const}} (recall what this constant is), can you find the best setting for an amplifier (that is, the choice of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau} ), 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta E_{a,\omega}} . Does it change the result? Explain your results.