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In [[statistics]], '''Cochran's theorem''', devised by [[William G. Cochran]],<ref name="Cochran">{{cite journal|last=Cochran|first=W. G.|authorlink=William Gemmell Cochran|title=The distribution of quadratic forms in a normal system, with applications to the analysis of covariance|journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]]|date=April 1934|volume=30|issue=2|pages=178–191|doi=10.1017/S0305004100016595}}</ref> is a [[theorem]] used to justify results relating to the [[probability distribution]]s of statistics that are used in the [[analysis of variance]].<ref>{{cite book |author= Bapat, R. B.|title=Linear Algebra and Linear Models|edition=Second|publisher= Springer |year=2000|isbn=978-0-387-98871-9}}</ref>
{{about|the biological application of the Nernst equation|the general equation|Nernst equation}}


== Statement ==
In a [[biological membrane]], the '''reversal potential''' (also known as the '''Nernst potential''') of an [[ion]] is the [[membrane potential]] at which there is no net (overall) flow of that particular ion from one side of the membrane to the other. In the case of [[post-synaptic]] [[neuron]]s, the reversal potential is the membrane potential at which a given [[neurotransmitter]] causes no net current flow of ions through that neurotransmitter [[Receptor (biochemistry)|receptor's]] [[ion channel]].<ref name="Purves" >{{cite book | author = Dale Purves, et al. | title = Neuroscience, 4th Edition | publisher = Sinauer Associates | pages = 109–11 | year = 2008 | isbn = 978-0-87893-697-7}}</ref>
Suppose ''U''<sub>1</sub>, ..., ''U''<sub>''n''</sub> are [[statistical independence|independent]] standard [[normal distribution|normally distributed]] [[random variable]]s, and an identity of the form


:<math>
In a single-ion system, ''reversal potential'' is synonymous with '''equilibrium potential'''; their numerical values are identical. The two terms refer to different aspects of the difference in membrane potential. ''Equilibrium'' refers to the fact that the net ion flux at a particular voltage is zero. That is, the outward and inward rates of ion movement are the same; the ion flux is in equilibrium. ''Reversal'' refers to the fact that a change of membrane potential on either side of the equilibrium potential reverses the overall direction of ion flux.<ref name="Purves" />
\sum_{i=1}^n U_i^2=Q_1+\cdots + Q_k
</math>


can be written, where each ''Q''<sub>''i''</sub> is a sum of squares of linear combinations of the ''U''s. Further suppose that
The reversal potential is often called the "Nernst potential", as it can be calculated from the [[Nernst equation]]. Ion channels conduct most of the flow of simple ions in and out of [[cell (biology)|cells]]. When a channel type that is selective to one species of ion dominates within the membrane of a cell (because other ion channels are closed, for example) then the voltage inside the cell will equilibrate (i.e. become equal) to the reversal potential for that ion (assuming the outside of the cell is at 0 volts). For example, the [[resting potential]] of most cells is close to the [[potassium|K]]<sup>+</sup> (potassium ion) reversal potential. This is because at resting potential, potassium conductance dominates. During a typical [[action potential]], the small resting ion conductance mediated by potassium channels is overwhelmed by the opening of a large number of [[sodium|Na]]<sup>+</sup> (sodium ion) channels, which brings the membrane potential close to the reversal potential of sodium.


:<math>
The relationship between the terms "reversal potential" and "equilibrium potential" only holds true for single-ion systems. In multi-ion systems, there are areas of the cell membrane where the summed currents of the multiple ions will equal zero. While this is a reversal potential in the sense that membrane potential reverses direction, it is not an equilibrium potential because not all (and in some cases, none) of the ions are in equilibrium and thus have net fluxes across the membrane. When a cell has significant permeabilities to more than one ion, the cell potential can be calculated from the [[Goldman-Hodgkin-Katz equation]] rather than the Nernst equation.
r_1+\cdots +r_k=n
</math>


where ''r''<sub>''i''</sub> is the [[rank (linear algebra)|rank]] of ''Q''<sub>''i''</sub>. Cochran's theorem states that the ''Q''<sub>''i''</sub> are independent, and each ''Q''<sub>''i''</sub> has a  [[chi-squared distribution]] with ''r''<sub>''i''</sub> [[degrees of freedom (statistics)|degrees of freedom]].<ref name="Cochran"/> Here the rank of ''Q''<sub>''i''</sub> should be interpreted as meaning the rank of the matrix ''B''<sup>(''i'')</sup>, with elements ''B''<sub>''j,k''</sub><sup>(''i'')</sup>, in the representation of ''Q''<sub>''i''</sub> as a [[quadratic form]]:
==Mathematical models==
The term ''driving force'' is related to equilibrium potential, and is likewise useful in understanding the current in biological membranes. Driving force refers to the difference between the actual membrane potential and an ion's equilibrium potential. It is defined by the following equation:


:<math>Q_i=\sum_{j=1}^n\sum_{k=1}^n U_j B_{j,k}^{(i)} U_k .</math>
:<math>{I_{ion}} = {g_{ion}} ({V_m}-{E_{ion}})\,</math> where <math>{V_m}-{E_{ion}}</math> is the Driving Force.


Less formally, it is the number of linear combinations included in the sum of squares defining ''Q''<sub>''i''</sub>, provided that these linear combinations are linearly independent.
In words, this equation says that: the ionic current (I<sub>ion</sub>) is equal to that ion's conductance (g<sub>ion</sub>) multiplied by the driving force, which is represented by the difference between the membrane potential and the ion's equilibrium potential (i.e. V<sub>m</sub>-E<sub>ion</sub>). Note that the ionic current will be zero if the membrane is impermeable (g<sub>ion</sub> = 0) to the ion in question, regardless of the size of the driving force.


===Proof===
A related equation (which is derived from the more general equation above) determines the magnitude of an [[end plate current]] (EPC), at a given membrane potential, in the [[neuromuscular junction]]:


We first show that the matrices ''B''<sup>(''i'')</sup> can be [[Matrix_diagonalization#Simultaneous_diagonalization|simultaneously diagonalized]] and that their non-zero [[eigenvalue]]s are all equal to +1. We then use the [[Basis (linear algebra)|vector basis]] that diagonalize them to simplify their [[Characteristic function (probability theory)|characteristic function]] and show their independence and distribution.<ref>Craig A.T. (1938) On The Independence of Certain Estimates of Variances. Ann. Math. Statist. 9, pp. 48-55</ref>
:<math>EPC = {g_{ACh}} ({V_m}-{E_{rev}})\,</math>


Each of the matrices ''B''<sup>(''i'')</sup> has [[rank (linear algebra)|rank]] ''r''<sub>''i''</sub> and so has exactly ''r''<sub>''i''</sub> non-zero [[eigenvalue]]s. For each ''i'', the sum <math>C^{(i)} \equiv \sum_{j\ne i}B^{(j)}</math> has at most rank <math>\sum_{j\ne i}r_j = N-r_i</math>. Since <math>B^{(i)}+C^{(i)} = I_{NxN}</math>, it follows that ''C''<sup>(''i'')</sup> has exactly rank N-''r''<sub>''i''</sub>.
where EPC is the end plate current, g<sub>ACh</sub> is the ionic conductance activated by [[acetylcholine]], V<sub>m</sub> is the membrane potential, and E<sub>rev</sub> is the reversal potential. When the membrane potential is equal to the reversal potential, V<sub>m</sub>-E<sub>rev</sub> is equal to 0 and there is no driving force on the ions involved.<ref name="Purves" />


Therefore ''B''<sup>(''i'')</sup> and ''C''<sup>(''i'')</sup> can be [[Matrix_diagonalization#Simultaneous_diagonalization|simultaneously diagonalized]]. This can be shown by first diagonalizing ''B''<sup>(''i'')</sup>. In this basis, it is of the form:
==Use in research==
:<math>\begin{bmatrix}
When V<sub>m</sub> is at the reversal potential (V<sub>m</sub>-E<sub>rev</sub> is equal to 0), the identity of the ions that flow during an EPC can be deduced by comparing the reversal potential of the EPC to the equilibrium potential for various ions. For instance several excitatory [[ionotropic]] ligand-gated [[neurotransmitter]] [[Receptor (biochemistry)|receptors]] including [[glutamate receptor]]s ([[AMPA]], [[NMDA]], and [[kainate]]), [[nicotinic]] [[acetylcholine]] (nACh), and [[serotonin]] (5-HT<sub>3</sub>) receptors are nonselective cation channels that pass Na<sup>+</sup> and K<sup>+</sup> in nearly equal proportions, giving an equilibrium potential close to 0 mV. The inhibitory ionotropic ligand-gated neurotransmitter receptors that carry Cl<sup>–</sup>, such as [[GABA]]<sub>A</sub> and [[glycine]] receptors, have equilibrium potentials close to the resting potential (approximately –70 mV) in neurons.<ref name="Purves" />
\lambda_1      & 0          & ... & ...          & ... &0 \\
0              & \lambda_2  & 0  & ...          & ... & 0 \\
0              &  ...      & ... & ...          & ... & 0\\
0              &  ...      &0    & \lambda_{r_i} & 0 &... \\
0 & ...      &    & 0            & 0...&0\\
0 & ...      &    & 0            & ...&...\\
0 & ...      &    & 0            & 0...&0
\end{bmatrix}.</math>


Thus the lower <math>(N-r_i)</math> rows are zero. Since <math>C^{(i)} = I - B^{(i)}</math>, it follows these rows in ''C''<sup>(''i'')</sup> in this basis contain a right block which is a <math>(N-r_i)\times(N-r_i)</math> unit matrix, with zeros in the rest of these rows. But since ''C''<sup>(''i'')</sup> has rank N-''r''<sub>''i''</sub>, it must be zero elsewhere. Thus it is diagonal in this basis as well. Moreover, it follows that all the non-zero [[eigenvalue]]s of both ''B''<sup>(''i'')</sup> and ''C''<sup>(''i'')</sup> are +1.
This line of reasoning led to the development of experiments (by Akira Takeuchi and Noriko Takeuchi in 1960) that proved that acetylcholine-activated ion channels are approximately equally permeable to Na<sup>+</sup> and K<sup>+</sup> ions. The experiment was performed by lowering the external Na<sup>+</sup> concentration, which lowers (more negative) the Na<sup>+</sup> equilibrium potential and produces a negative shift in reversal potential. Conversely, increasing the external K<sup>+</sup> concentration raises (more positive) the K<sup>+</sup> equilibrium potential and produces a positive shift in reversal potential.<ref name="Purves" />


It follows that the non-zero eigenvalues of all the ''B''-s are equal to +1. Moreover, the above analysis can be repeated in the diagonal basis for <math>C^{(1)} = B^{(2)} + \sum_{j>2}B^{(j)}</math>. In this basis <math>C^{(1)}</math> is the identity of an <math>(N-r_i)\times(N-r_i)</math> vector space, so it follows that both ''B''<sup>(''2'')</sup> and <math>\sum_{j>2}B^{(j)}</math> are simultaneously diagonalizable in this vector space (and hence also together ''B''<sup>(''1'')</sup>). By repeating this over and over it follows that all the ''B''-s are simultaneously diagonalizable.
==See also==
 
*[[Electrochemical potential]]
Thus there exists an [[orthogonal matrix]] S such that for all i between 1 and ''k'': <math> S^\mathrm{T}Q_i S </math> is diagonal with the diagonal having 1-s at the places between <math>r_1 + ... r_{i-1} +1 </math> and <math>r_1 + ... r_i </math>.
*[[Cell potential]]
 
*[[Goldman equation]]
Let <math>U_i^\prime</math> be the independent variables <math>U_i</math> after transformation by S.
 
The characteristic function of ''Q''<sub>''i''</sub> is:
:<math>\begin{align}
\varphi_i(t) =& (2\pi)^{-N/2} \int dU_1 \int dU_2 ... \int dU_N e^{i t Q_i} \cdot e^{-\frac{U_1^2}{2}}\cdot e^{-\frac{U_2^2}{2}}\cdot ...e^{-\frac{U_N^2}{2}} = (2\pi)^{-N/2} \left(\prod_{j=1}^N \int dU_j\right) e^{i t Q_i} \cdot e^{-\sum_{j=1}^N \frac{U_j^2}{2}} \\
=& (2\pi)^{-N/2} \left(\prod_{j=1}^N \int dU_j^\prime\right) e^{i t\cdot \sum_{m = r_1+...r_{i-1}+1}^{r_1+...r_i} (U_m^\prime)^2} \cdot e^{-\sum_{j=1}^N \frac{{U_j^\prime}^2}{2}}  \\
=& (1 - 2 i t)^{-r_i/2}
\end{align}</math>
 
This is the [[Fourier transform]] of the [[chi-squared distribution]] with ''r''<sub>''i''</sub> degrees of freedom. Therefore this is the distribution of ''Q''<sub>''i''</sub>.
 
Moreover, the characteristic function of the joint distribution of all the ''Q''<sub>''i''</sub>-s is:
:<math>\begin{align}
\varphi(t_1, t_2... t_k) =& (2\pi)^{-N/2} \left(\prod_{j=1}^N \int dU_j\right) e^{i \sum_{i=1}^k t_i \cdot Q_i} \cdot e^{-\sum_{j=1}^N \frac{U_j^2}{2}} \\
=& (2\pi)^{-N/2} \left(\prod_{j=1}^N \int dU_j^\prime\right) e^{i \cdot \sum_{i=1}^k t_i \sum_{k = r_1+...r_{i-1}+1}^{r_1+...r_i}  (U_k^\prime)^2} \cdot e^{-\sum_{j=1}^N \frac{{U_j^\prime}^2}{2}}  \\
=& \prod_{i=1}^k (1 - 2 i t_i)^{-r_i/2} = \prod_{i=1}^k \varphi_i(t_i)
\end{align}</math>
 
From which it follows that all the ''Q''<sub>''i''</sub>-s are statistically independent.
 
<!--
Cochran's theorem is the converse of [[Fisher's theorem]]. -->
 
== Examples ==
 
=== Sample mean and sample variance ===
If ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> are independent normally distributed random variables with mean μ and standard deviation σ
then
 
:<math>U_i = \frac{X_i-\mu}{\sigma}</math>
 
is [[standard normal]] for each ''i''. It is possible to write
 
:<math>
\sum_{i=1}^n U_i^2=\sum_{i=1}^n\left(\frac{X_i-\overline{X}}{\sigma}\right)^2
+ n\left(\frac{\overline{X}-\mu}{\sigma}\right)^2
</math>
 
(here <math>\overline{X}</math> is the [[Arithmetic mean|sample mean]]). To see this identity, multiply throughout by <math>\sigma^2</math> and note that
 
:<math>
\sum(X_i-\mu)^2=
\sum(X_i-\overline{X}+\overline{X}-\mu)^2
</math>
 
and expand to give
 
:<math>
\sum(X_i-\mu)^2=
\sum(X_i-\overline{X})^2+\sum(\overline{X}-\mu)^2+
2\sum(X_i-\overline{X})(\overline{X}-\mu).
</math>
 
The third term is zero because it is equal to a constant times
 
:<math>\sum(\overline{X}-X_i)=0,</math>
 
and the second term has just ''n'' identical terms added together. Thus
:<math>
\sum(X_i-\mu)^2=
\sum(X_i-\overline{X})^2+n(\overline{X}-\mu)^2 ,
</math>
 
and hence
 
:<math>
\sum\left(\frac{X_i-\mu}{\sigma}\right)^2=
\sum\left(\frac{X_i-\overline{X}}{\sigma}\right)^2
+n\left(\frac{\overline{X}-\mu}{\sigma}\right)^2
=Q_1+Q_2.
</math>
 
Now the rank of ''Q''<sub>2</sub> is just 1 (it is the square of just one linear combination of the standard normal variables).  The rank of ''Q''<sub>1</sub> can be shown to be ''n'' &minus; 1, and thus the conditions for Cochran's theorem are met.
 
Cochran's theorem then states that ''Q''<sub>1</sub> and ''Q''<sub>2</sub> are independent, with chi-squared distributions with ''n'' &minus; 1 and 1 degree of freedom respectively. This shows that the sample mean and [[sample variance]] are independent.  This can also be shown by [[Basu's theorem]], and in fact this property ''characterizes'' the normal distribution – for no other distribution are the sample mean and sample variance independent.<ref>{{cite journal
|doi=10.2307/2983669
|first=R.C. |last=Geary |authorlink=Roy C. Geary
|year=1936
|title=The Distribution of the "Student's" Ratio for the Non-Normal Samples
|journal=Supplement to the Journal of the Royal Statistical Society
|volume=3 |issue=2 |pages=178–184
|jfm=63.1090.03
|jstor=2983669
}}</ref>
 
===Distributions===
 
The result for the distributions is written symbolically as
:<math>
\sum\left(X_i-\overline{X}\right)^2  \sim \sigma^2 \chi^2_{n-1}.
</math>
:<math>
n(\overline{X}-\mu)^2\sim \sigma^2 \chi^2_1,
</math>
 
Both these random variables are proportional to the true but unknown variance σ<sup>2</sup>. Thus their ratio does not depend on σ<sup>2</sup> and, because they are statistically independent. The distribution of their ratio is given by
 
:<math>
\frac{n\left(\overline{X}-\mu\right)^2}
{\frac{1}{n-1}\sum\left(X_i-\overline{X}\right)^2}\sim \frac{\chi^2_1}{\frac{1}{n-1}\chi^2_{n-1}}
  \sim F_{1,n-1}
</math>
 
where ''F''<sub>1,''n''&nbsp;&minus;&nbsp;1</sub> is the [[F-distribution]] with 1 and ''n''&nbsp;&minus;&nbsp;1 degrees of freedom (see also [[Student's t-distribution]]). The final step here is effectively the definition of a random variable having the F-distribution.
 
=== Estimation of variance ===
To estimate the variance σ<sup>2</sup>, one estimator that is sometimes used is the [[maximum likelihood]] estimator of the variance of a normal distribution
 
:<math>
\widehat{\sigma}^2=
\frac{1}{n}\sum\left(
X_i-\overline{X}\right)^2. </math>
 
Cochran's theorem shows that
 
:<math>
\frac{n\widehat{\sigma}^2}{\sigma^2}\sim\chi^2_{n-1}
</math>
 
and the properties of the chi-squared distribution show that the expected value of <math>\widehat{\sigma}^2</math> is σ<sup>2</sup>(''n'' &minus; 1)/''n''.
 
==Alternative formulation==
The following version is often seen when considering linear regression.{{Citation needed|date=July 2011}} Suppose that <math>Y\sim N_n(0,\sigma^2I_n)</math> is a standard [[Multivariate normal distribution|multivariate normal]] [[random vector]] (here <math>I_n</math> denotes the n-by-n [[identity matrix]]), and if <math>A_1,\ldots,A_k</math> are all n-by-n [[symmetric matrices]] with <math>\sum_{i=1}^kA_i=I_n</math>.  Then, on defining <math>r_i=Rank(A_i)</math>, any one of the following conditions implies the other two:
 
* <math>\sum_{i=1}^kr_i=n ,</math>
* <math>Y^TA_iY\sim\sigma^2\chi^2_{r_i}</math>  (thus the <math>A_i</math> are [[positive semidefinite]])
* <math>Y^TA_iY</math> is independent of <math>Y^TA_jY</math> for <math>i\neq j .</math>
 
== See also ==
* [[Cramér's theorem]], on decomposing normal distribution
* [[Infinite divisibility (probability)]]
 
{{refimprove|date=July 2011}}


==References==
==References==
<references/>
{{reflist}}


{{Experimental design|state=expanded}}
==External links==
* [http://www.nernstgoldman.physiology.arizona.edu/ Nernst/Goldman Equation Simulator]
* [http://www.physiologyweb.com/calculators/nernst_potential_calculator.html Nernst Equation Calculator]
* [http://www.physiologyweb.com/calculators/ghk_equation_calculator.html Goldman-Hodgkin-Katz Equation Calculator]
* [http://www.physiologyweb.com/calculators/electrochemical_driving_force_calculator.html Electrochemical Driving Force Calculator]


{{DEFAULTSORT:Cochran's Theorem}}
[[Category:Membrane biology]]
[[Category:Statistical theorems]]
[[Category:Characterization of probability distributions]]

Revision as of 09:51, 12 August 2014

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In a biological membrane, the reversal potential (also known as the Nernst potential) of an ion is the membrane potential at which there is no net (overall) flow of that particular ion from one side of the membrane to the other. In the case of post-synaptic neurons, the reversal potential is the membrane potential at which a given neurotransmitter causes no net current flow of ions through that neurotransmitter receptor's ion channel.[1]

In a single-ion system, reversal potential is synonymous with equilibrium potential; their numerical values are identical. The two terms refer to different aspects of the difference in membrane potential. Equilibrium refers to the fact that the net ion flux at a particular voltage is zero. That is, the outward and inward rates of ion movement are the same; the ion flux is in equilibrium. Reversal refers to the fact that a change of membrane potential on either side of the equilibrium potential reverses the overall direction of ion flux.[1]

The reversal potential is often called the "Nernst potential", as it can be calculated from the Nernst equation. Ion channels conduct most of the flow of simple ions in and out of cells. When a channel type that is selective to one species of ion dominates within the membrane of a cell (because other ion channels are closed, for example) then the voltage inside the cell will equilibrate (i.e. become equal) to the reversal potential for that ion (assuming the outside of the cell is at 0 volts). For example, the resting potential of most cells is close to the K+ (potassium ion) reversal potential. This is because at resting potential, potassium conductance dominates. During a typical action potential, the small resting ion conductance mediated by potassium channels is overwhelmed by the opening of a large number of Na+ (sodium ion) channels, which brings the membrane potential close to the reversal potential of sodium.

The relationship between the terms "reversal potential" and "equilibrium potential" only holds true for single-ion systems. In multi-ion systems, there are areas of the cell membrane where the summed currents of the multiple ions will equal zero. While this is a reversal potential in the sense that membrane potential reverses direction, it is not an equilibrium potential because not all (and in some cases, none) of the ions are in equilibrium and thus have net fluxes across the membrane. When a cell has significant permeabilities to more than one ion, the cell potential can be calculated from the Goldman-Hodgkin-Katz equation rather than the Nernst equation.

Mathematical models

The term driving force is related to equilibrium potential, and is likewise useful in understanding the current in biological membranes. Driving force refers to the difference between the actual membrane potential and an ion's equilibrium potential. It is defined by the following equation:

where is the Driving Force.

In words, this equation says that: the ionic current (Iion) is equal to that ion's conductance (gion) multiplied by the driving force, which is represented by the difference between the membrane potential and the ion's equilibrium potential (i.e. Vm-Eion). Note that the ionic current will be zero if the membrane is impermeable (gion = 0) to the ion in question, regardless of the size of the driving force.

A related equation (which is derived from the more general equation above) determines the magnitude of an end plate current (EPC), at a given membrane potential, in the neuromuscular junction:

where EPC is the end plate current, gACh is the ionic conductance activated by acetylcholine, Vm is the membrane potential, and Erev is the reversal potential. When the membrane potential is equal to the reversal potential, Vm-Erev is equal to 0 and there is no driving force on the ions involved.[1]

Use in research

When Vm is at the reversal potential (Vm-Erev is equal to 0), the identity of the ions that flow during an EPC can be deduced by comparing the reversal potential of the EPC to the equilibrium potential for various ions. For instance several excitatory ionotropic ligand-gated neurotransmitter receptors including glutamate receptors (AMPA, NMDA, and kainate), nicotinic acetylcholine (nACh), and serotonin (5-HT3) receptors are nonselective cation channels that pass Na+ and K+ in nearly equal proportions, giving an equilibrium potential close to 0 mV. The inhibitory ionotropic ligand-gated neurotransmitter receptors that carry Cl–, such as GABAA and glycine receptors, have equilibrium potentials close to the resting potential (approximately –70 mV) in neurons.[1]

This line of reasoning led to the development of experiments (by Akira Takeuchi and Noriko Takeuchi in 1960) that proved that acetylcholine-activated ion channels are approximately equally permeable to Na+ and K+ ions. The experiment was performed by lowering the external Na+ concentration, which lowers (more negative) the Na+ equilibrium potential and produces a negative shift in reversal potential. Conversely, increasing the external K+ concentration raises (more positive) the K+ equilibrium potential and produces a positive shift in reversal potential.[1]

See also

References

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