Footnotes

... problem.¹

Here our educators have to solve the problem of how to test our problem solving skills instead of our skill in memorizing. This is a topic for another paper.

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... reflex ^1.1

The siphon is used to facilitate the snail's breathing. When aplysia breathes, water is drawn across the gill from the front and exits through the siphon. The siphon is usually outside of the snail's shell or mantle. However, when gently touched, the snail will withdraw and protect its siphon for a short period of time. If this touch is preceded by an electric shock to the tail, the snail will withdraw its siphon for a longer period of time. The snail will continue to have this exagerated response for up to a day following the shock, and thus, is an example of short term memory. Multiple shocks given over multiple days cause this exagerated response to become even more exagerated and retained for much longer. Even after a week since the electric shocks, the snail will continue to exhibit the exagerated behavior, and this is an example of long term memory.

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... calcium ^1.2

We are assuming calcium, Ca, is the agent responsible for short term memory because it is probably critical for neurotransmitter release required to signal adjacent neurons. More intraceulluar Ca would trigger an increase in the release of pre-synaptic neurotransmitter which would then activate more post-synaptic receptors, giving a larger post synaptic response.

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... potential ^1.3

A brief primer in electrophysiology: Membrane Potential is the result of a difference in the relative concentrations of positively and negitively charged particles on opposite sides of a cell's plasma membrane. Cells that have the ability to maintain an a transmembrane gradient in charged ions (and thus generate a membrane potentials) and can rapidly change their membrane potential following a suprathreshold stimulation are called excitable cells.

An excitable cell is either in the rest state where the transmembrane potential is -50 to -80 mV or in the excited state where the transmembrane potential can become as large as +40 mV for a few milliseconds. The electrical response to suprathreshold stimulation is called an action potential (see figure 2.6.3) and is caused by the rapid influx of a + charge carrier (either Na or Ca). The restoration of the charge balance is accomplished by a slower efflux of a + charge carrier (K) from the intracellular fluid. However, the charge redistributuion alone is insufficient to keep the cell healthy.

The charge carriers must also be redistributed - which is a regulatory process that takes place in the background of cellular activity. Because charge flow during the action potential is down concentration gradients, it is physically impossible to restore the charge carriers without active transport up the concentration gradient by actively exchanging ions between the extracellular and intracellular fluids. The Na-K transporter is an example of an active transporter that exchanges extracellular K for intracellular Na.

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... discontinuous ^1.4

Our friend, Valentin Krinsky, was the first to articulate this

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... wave ^1.5

Spirals form from fragments because the ends of the fragment propagate more slowly than the interior segments of the wave. Why? because the ends must excite not only the cells in front of them but also the cells to the side - and, because the cell has a limited charge available to excite adjacent cells, more time is required to transfer this charge to the larger audience of adjacent cells

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... -^1.6

This site in the heart is composed of what are called pacemaker cells

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... excitability ^1.7

Four examples for such an asymmetry are: inexcitable obstacles that the wave collides with, cellular coupling, as described by Maddy Spach, dispersion of refractoriness, or a spatially inhomogeneous distribution of potassium channels.

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... switches ^1.8

Both excitable cells and DNA transcription involve switches. Switches are either on or off. The phase plane of any system with two stable states requires a third, intermediate state that is unstable and possibly oscillatory.

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... gradients ^2.1

In the presence of both an ionic concentration gradient and an electric field, two currents are possible, one derived from passive diffusion of charge carriers down the concentration gradient and one derived from the attraction of a charge carrier by the electric field.

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... potential ^2.2

The reversal potential is the transmembrane potential required to create a current based on charge attraction that exactly balances the diffusive flow of charge carriers down the concentration gradient. For example, consider a higher concentration of Na outside of the cell than inside. The reversal potential required to stop the diffusive current is described by

$\displaystyle V_{\textrm{Na}} = \frac{RT}{F} \ln\frac{[\textrm{Na}]_o}{[\textrm{Na}]_i},$

where

is the Rydburg constant,

is absolute temperature,

is the Faraday constant, $[\textrm{Na}]_o$ is the concentration of Na outside of the cell and $[\textrm{Na}]_i$ is the Na concentration inside. The equation is is derived by equating the diffusive current with the current created by an electric field. A full treatment of this equation can be found in Appendix D.1.

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... factor.^2.3

The specific mechanics of this solving an ordinary differential equation using an integration factor is fully described in Section 2.10.3.

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... it.^2.4

See Section 2.10.1 for a complete overview of the general method of phase plane analysis.

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... unstable ^2.5

See Section 2.10.1 for a full explanation of the terms stable and unstable.

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... value ^3.1

That is, a greater value, or lesser value, or both, depending on the model and the type of hypothesis you are testing. The details of this will be explained in the next few paragraphs.

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... situations.^3.2

A lot of the material in this section was plagiarized from the web page: http://www.stat.yale.edu/Courses/1997-98/101/chigf.htm, author unknown.

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... 1.^3.3

Most computer languages have standard routines that do this. For example, rand() in Perl and in C there is rand() and random(), which both return random numbers between 0 and RAND_MAX

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... independent ^3.4

Independent simply means that knowing the value of one specific data point does not tell you anything about the value of any of the other data points. For example, if our data consisted of the results of tossing a coin, knowing that

landed heads would not tell us a thing about whether

landed heads or tails.

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...X.^3.5

For example, if your data set was two heads when a coin is tossed twice, then the probability of the data is

, since the probability of getting heads on any one toss is 1/2.

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... involved.^3.6

There is a third alternative, called ``variational method'' which is interesting, but I don't quite fully understand well enough to write about at this time

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...3.6 ^3.7

For now we will discuss how the method works for functions of three variables, but it works fine on functions with two variables (as you'll see in Example 3.11.1.1) and trivially extends to functions with more.

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... results.^3.8

Complete derivations of these results can be found in Appendix D.3 and D.4.

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... line ^3.9

It is important to note, that just because the data does not all fall on a single line, doesn't mean that the model is not linear. There could have been errors in measurement, both human and mechanical, that cause the data to deviate from a line.

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... examples.^3.10

However, see Example 3.13.5.3 for the solution to this current conundrum!

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... coefficients ^3.11

For a function,

, to be considered linear with respect to its coefficients means that if the function were considered to be a function of the coefficients, $g({\bf c})$ , then $g(\alpha {\bf c}) = \alpha g({\bf c})$ . For example, the function,

, can be written in terms of c, $g({\bf c}) = c_0 + c_1x$ and $g(\alpha {\bf c}) = \alpha c_0 + \alpha c_1 x = \alpha (c_0 + c_1x) = \alpha f({\bf c})$ . Another example of a function that is linear with respect to its coefficients is $f(x) = c_0\sin(x) + c_1e^{x}$ , because $g(\alpha {\bf c}) = \alpha g({\bf c})$ . An example of a function that is not linear with respect to its coefficients is $f(x) = \sin(c_0x) + c_1x^2$ , since $g(\alpha {\bf c}) = \sin(\alpha c_0x) + \alpha c_1x^2 \ne \alpha(\sin(c_0x) + c_1x^2) = \alpha g({\bf c})$

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... measurements ^3.12

In Section 3.13.6, Linear Models with Multiple Dependent Variables we will generalize the test developed here for multiple dependent variables.

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... noise.^3.13

I like to use this concept instead of calling $\boldsymbol{\epsilon}$ ``error'' which it is not. It simply reflects the limits of our ability to capture the totality of what is going on. With perfect models, we'd be able to capture the thermal noise generated by molecular motion and have a perfect fit. So - errors - NO, noise - YES.

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... thus ^3.14

We can easily verify that this solution for $\hat{\boldsymbol{\beta}}$ is a minimum by taking the second derivative of Equation 3.13.4 with respect to $\boldsymbol{\beta}$ and observing that when $\bf{X}$ is not completely filled with zeros, the resulting quantity, $2{\bf X}'{\bf X}$ , will be positive.

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... distributed ^3.15

It is possible to use distributions other than the normal as long as each $\epsilon_i$ is an independent variable with mean 0 (zero) and variance $\sigma^2$ . These conditions are called Gauss-Markov Conditions. However, when you use a normal distribution, the least squares estimates are the same as the maximum likelihood estimates and are thus best unbiased estimators, which is a good thing.

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... equal ^3.16

This would amount to an ANOVA test. See Example 3.13.5.4 for a full treatment of this.

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... methods.^3.17

That our solution provides the maximum probability is easily verified in the manner demonstrated in Example 3.9.2.1

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... zero ^3.18

To see this, all that is needed is to multiply them out and some minor cancellation. See D.5 for a full derivation of this fact.

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... to ^3.19

See D.6 for a full derivation of this reduction.

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...^3.20

Don't worry too much about this, we'll derive the value for this constant before too long

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...-distribution ^3.21

-distribution is defined as the ratio of two independent chi-square variables, each divided by its degrees of freedom. That is,

$\displaystyle F_{m,n} = (u/m)/(v/n),$

where $u \sim\chi^2_m$ , $v \sim\chi^2_n$ and

and

are independent.

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... true.^3.22

Just as a gentle reminder, the value, $F_{m,n-p,\alpha}$ and the ratio in Equation 3.13.21 represent points on an

-axis. The value, $F_{m,n-p,\alpha}$ , represents a cut-off point, and anything larger, and thus further away from the mean, is determined to not come from the same distribution.

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... model ^3.23

This use of the word modern is perhaps wishful thanking as it is the author's opinion that this model should be considered thus. In practice, most people, for historical reasons, use alternative models for ANOVA. See Appendix E, for a full discussion on this topic.

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... singular ^3.24

This can be seen by first adding together the first five columns in X, which will give you a vector of 1s. Adding the last three columns of X together also gives you a vector of 1s. Thus, adding the first five columns and subtracting the last three columns will result in a vector of 0s, satisfying the definition of a singular matrix.

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... learn.^3.25

A large number of different interaction plots, as well as their potential interpretations is given in Appendix F.

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... are:^3.26

The data given here were stolen from David Dickey and Jimmy Joi's web page: http://www.stat.ncsu.edu/ $\sim$ st512_info/ dickey/crsnotes/notes_5.htm.

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... is ^3.27

See Appendix D.8 for the derivation

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... pdf ^4.1

xmgrace doesn't do the best job exporting PDF images so it is sometimes better to export an EPS image an use epstopdf to create the PDF version. You will just have to do it both ways and decide which looks better.

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... computer.^4.2

A lot of the material in this section was plagiarized from the web page: CVS-RCS-HOWTO.html, written by Alavoor Vasudevan.

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... rank ^D.1

The rank of a matrix is the number of linearly independent rows or columns. For any matrix, the number of linearly independent rows is equal to the number of linearly independent columns.

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