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ref: Gilletti-2006.09 tags: electrode micromotion histology GFAP variable reluctance date: 01-04-2013 02:28 gmt revision:2   [head]

PMID-16921202 Brain micromotion around implants in the rodent somatosensory cortex.

• Used a differential variable reluctance transducer (DVRT) in adult rats (n = 6) to monitor micromotion normal to the somatosensory cortex surface
• Reluctance e.g. AC inductance varied with a floating bobbin (or so -- they do not list the details of this COTS device).
• Pulsatile surface micromotion was observed to be in the order of 10-30 um due to pressure changes during respiration and 2-4 um due to vascular pulsatility.
• Large inward displacements of brain tissue between 10-60 um were observed in n = 3 animals immediately following the administration of anesthesia

____References____

  Gilletti A, Muthuswamy J, Brain micromotion around implants in the rodent somatosensory cortex.J Neural Eng 3:3, 189-95 (2006 Sep)

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ref: work-0 tags: gaussian random variables mutual information SNR date: 01-16-2012 03:54 gmt revision:26       [head]

I've recently tried to determine the bit-rate of conveyed by one gaussian random process about another in terms of the signal-to-noise ratio between the two. Assume $x$ is the known signal to be predicted, and $y$ is the prediction.

Let's define $SNR(y) = \frac{Var(x)}{Var(err)}$ where $err = x-y$ . Note this is a ratio of powers; for the conventional SNR, $SNR_{dB} = 10*log_{10 } \frac{Var(x)}{Var(err)}$ . $Var(err)$ is also known as the mean-squared-error (mse).

Now, $Var(err) = \sum{ (x - y - sstrch \bar{err})^2 estrch} = Var(x) + Var(y) - 2 Cov(x,y)$ ; assume x and y have unit variance (or scale them so that they do), then

$\frac{2 - SNR(y)^{-1}}{2 } = Cov(x,y)$

We need the covariance because the mutual information between two jointly Gaussian zero-mean variables can be defined in terms of their covariance matrix: (see http://www.springerlink.com/content/v026617150753x6q/ ). Here Q is the covariance matrix,

$Q = \left[ \array{Var(x) & Cov(x,y) \\ Cov(x,y) & Var(y)} \right]$

$MI = \frac{1 }{2 } log \frac{Var(x) Var(y)}{det(Q)}$

$Det(Q) = 1 - Cov(x,y)^2$

Then $MI = - \frac{1 }{2 } log_2 \left[ 1 - Cov(x,y)^2 \right]$

or $MI = - \frac{1 }{2 } log_2 \left[ SNR(y)^{-1} - \frac{1 }{4 } SNR(y)^{-2} \right]$

This agrees with intuition. If we have a SNR of 10db, or 10 (power ratio), then we would expect to be able to break a random variable into about 10 different categories or bins (recall stdev is the sqrt of the variance), with the probability of the variable being in the estimated bin to be 1/2. (This, at least in my mind, is where the 1/2 constant comes from - if there is gaussian noise, you won't be able to determine exactly which bin the random variable is in, hence log_2 is an overestimator.)

Here is a table with the respective values, including the amplitude (not power) ratio representations of SNR. "

 SNR Amp. ratio MI (bits) 10 3.1 1.6 20 10 3.3 30 31 5.0 40 100 6.6 90 31e3 15
Note that at 90dB, you get about 15 bits of resolution. This makes sense, as 16-bit DACs and ADCs have (typically) 96dB SNR. good.

Now, to get the bitrate, you take the SNR, calculate the mutual information, and multiply it by the bandwidth (not the sampling rate in a discrete time system) of the signals. In our particular application, I think the bandwidth is between 1 and 2 Hz, hence we're getting 1.6-3.2 bits/second/axis, hence 3.2-6.4 bits/second for our normal 2D tasks. If you read this blog regularly, you'll notice that others have achieved 4bits/sec with one neuron and 6.5 bits/sec with dozens {271}.