Correlation

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Correlation

 

Several sets of (x, y) points, with the Pearson correlation coefficient of x and y for each set. The correlation reflects the noisiness and direction of a linear relationship (top row), but not the slope of that relationship (middle), nor many aspects of nonlinear relationships (bottom). N.B.: the figure in the center has a slope of 0 but in that case, the correlation coefficient is undefined because the variance of Y is zero.

In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics it usually refers to the degree to which a pair of variables are linearly related. Familiar examples of dependent phenomena include the correlation between the height of parents and their offspring, and the correlation between the price of a good and the quantity the consumers are willing to purchase, as it is depicted in the so-called demand curve.

Correlations are useful because they can indicate a predictive relationship that can be exploited in practice. For example, an electrical utility may produce less power on a mild day based on the correlation between electricity demand and weather. In this example, there is a causal relationship, because extreme weather causes people to use more electricity for heating or cooling. However, in general, the presence of a correlation is not sufficient to infer the presence of a causal relationship (i.e., correlation does not imply causation).

Formally, random variables are dependent if they do not satisfy a mathematical property of probabilistic independence. In informal parlance, correlation is synonymous with dependence. However, when used in a technical sense, correlation refers to any of several specific types of mathematical operations between the tested variables and their respective expected values. Essentially, correlation is the measure of how two or more variables are related to one another. There are several correlation coefficients, often denoted ${\displaystyle \rho }$ or ${\displaystyle r}$, measuring the degree of correlation. The most common of these is the Pearson correlation coefficient, which is sensitive only to a linear relationship between two variables (which may be present even when one variable is a nonlinear function of the other). Other correlation coefficients – such as Spearman's rank correlation – have been developed to be more robust than Pearson's, that is, more sensitive to nonlinear relationships. Mutual information can also be applied to measure dependence between two variables.

Pearson's product-moment coefficient

Main article: Pearson product-moment correlation coefficient

 

Example scatterplots of various datasets with various correlation coefficients.

The most familiar measure of dependence between two quantities is the Pearson product-moment correlation coefficient (PPMCC), or "Pearson's correlation coefficient", commonly called simply "the correlation coefficient". It is obtained by taking the ratio of the covariance of the two variables in question of our numerical dataset, normalized to the square root of their variances. Mathematically, one simply divides the covariance of the two variables by the product of their standard deviations. Karl Pearson developed the coefficient from a similar but slightly different idea by Francis Galton.

A Pearson product-moment correlation coefficient attempts to establish a line of best fit through a dataset of two variables by essentially laying out the expected values and the resulting Pearson's correlation coefficient indicates how far away the actual dataset is from the expected values. Depending on the sign of our Pearson's correlation coefficient, we can end up with either a negative or positive correlation if there is any sort of relationship between the variables of our data set.

The population correlation coefficient ${\displaystyle {\rho }_{X,Y}}$ between two random variables ${\displaystyle X}$ and ${\displaystyle Y}$ with expected values ${\displaystyle {\mu }_X}$ and ${\displaystyle {\mu }_Y}$ and standard deviations ${\displaystyle {\sigma }_X}$ and ${\displaystyle {\sigma }_Y}$ is defined as:

$${\displaystyle {\rho }_{X,Y}=\mathrm{corr}(X,Y)=\frac{\mathrm{cov}(X,Y)}{{\sigma }_X{\sigma }_Y}=\frac{\mathrm{E}[(X-{\mu }_X)(Y-{\mu }_Y)]}{{\sigma }_X{\sigma }_Y},\text{if}{\sigma }_X{\sigma }_Y>0.}$$

where ${\displaystyle \mathrm{E}}$ is the expected value operator, ${\displaystyle \mathrm{cov}}$ means covariance, and ${\displaystyle \mathrm{corr}}$ is a widely used alternative notation for the correlation coefficient. The Pearson correlation is defined only if both standard deviations are finite and positive. An alternative formula purely in terms of moments is:

$${\displaystyle {\rho }_{X,Y}=\frac{\mathrm{E}(XY)-\mathrm{E}(X)\mathrm{E}(Y)}{\sqrt{\mathrm{E}(X^2)-\mathrm{E}(X{)}^2}\cdot \sqrt{\mathrm{E}(Y^2)-\mathrm{E}(Y{)}^2}}}$$

Correlation and independence

It is a corollary of the Cauchy–Schwarz inequality that the absolute value of the Pearson correlation coefficient is not bigger than 1. Therefore, the value of a correlation coefficient ranges between −1 and +1. The correlation coefficient is +1 in the case of a perfect direct (increasing) linear relationship (correlation), −1 in the case of a perfect inverse (decreasing) linear relationship (anti-correlation), and some value in the open interval ${\displaystyle (-1,1)}$ in all other cases, indicating the degree of linear dependence between the variables. As it approaches zero there is less of a relationship (closer to uncorrelated). The closer the coefficient is to either −1 or 1, the stronger the correlation between the variables.

If the variables are independent, Pearson's correlation coefficient is 0, but the converse is not true because the correlation coefficient detects only linear dependencies between two variables.

$${\displaystyle \begin{array}{rl}X,Y\text{ independent}& \Rightarrow {\rho }_{X,Y}=0(X,Y\text{ uncorrelated})\\ {\rho }_{X,Y}=0(X,Y\text{ uncorrelated})& \nRightarrow X,Y\text{ independent}\end{array}}$$

For example, suppose the random variable ${\displaystyle X}$ is symmetrically distributed about zero, and ${\displaystyle Y=X^2}$. Then ${\displaystyle Y}$ is completely determined by ${\displaystyle X}$, so that ${\displaystyle X}$ and ${\displaystyle Y}$ are perfectly dependent, but their correlation is zero; they are uncorrelated. However, in the special case when ${\displaystyle X}$ and ${\displaystyle Y}$ are jointly normal, uncorrelatedness is equivalent to independence.

Even though uncorrelated data does not necessarily imply independence, one can check if random variables are independent if their mutual information is 0.

Sample correlation coefficient

Given a series of ${\displaystyle n}$ measurements of the pair ${\displaystyle (X_i,Y_i)}$ indexed by ${\displaystyle i=1,\dots ,n}$, the sample correlation coefficient can be used to estimate the population Pearson correlation ${\displaystyle {\rho }_{X,Y}}$ between ${\displaystyle X}$ and ${\displaystyle Y}$. The sample correlation coefficient is defined as

${\displaystyle r_{xy}\stackrel{\underset{\mathrm{d}\mathrm{e}\mathrm{f}}{}}{=}\frac{\sum _{i=1}^n(x_i-\overline{x})(y_i-\overline{y})}{(n-1)s_x{s}_y}=\frac{\sum _{i=1}^n(x_i-\overline{x})(y_i-\overline{y})}{\sqrt{\sum _{i=1}^n(x_i-\overline{x}{)}^2\sum _{i=1}^n(y_i-\overline{y}{)}^2}},}$

where ${\displaystyle \overline{x}}$ and ${\displaystyle \overline{y}}$ are the sample means of ${\displaystyle X}$ and ${\displaystyle Y}$, and ${\displaystyle s_x}$ and ${\displaystyle s_y}$ are the corrected sample standard deviations of ${\displaystyle X}$ and ${\displaystyle Y}$.

Equivalent expressions for ${\displaystyle r_{xy}}$ are

${\displaystyle \begin{array}{rl}{r}_{xy}& =\frac{\sum x_i{y}_i-n\overline{x}\overline{y}}{n{s}_x^{\prime }{s}_y^{\prime }}\\ & =\frac{n\sum x_i{y}_i-\sum x_i\sum y_i}{\sqrt{n\sum x_i^2-(\sum x_i{)}^2}\sqrt{n\sum y_i^2-(\sum y_i{)}^2}}.\end{array}}$

where ${\displaystyle s_x^{\prime }}$ and ${\displaystyle s_y^{\prime }}$ are the uncorrected sample standard deviations of ${\displaystyle X}$ and ${\displaystyle Y}$.

If ${\displaystyle x}$ and ${\displaystyle y}$ are results of measurements that contain measurement error, the realistic limits on the correlation coefficient are not −1 to +1 but a smaller range. For the case of a linear model with a single independent variable, the coefficient of determination (R squared) is the square of ${\displaystyle r_{xy}}$, Pearson's product-moment coefficient.

Example

Consider the joint probability distribution of X and Y given in the table below.

${\displaystyle \mathrm{P}(X=x,Y=y)}$

y x	−1	0	1
0	0	1/3	0
1	1/3	0	1/3

For this joint distribution, the marginal distributions are:

${\displaystyle \mathrm{P}(X=x)=\{\begin{array}{ll}\frac{1}{3}& \text{for }x=0\\ \frac{2}{3}& \text{for }x=1\end{array}}$

${\displaystyle \mathrm{P}(Y=y)=\{\begin{array}{ll}\frac{1}{3}& \text{for }y=-1\\ \frac{1}{3}& \text{for }y=0\\ \frac{1}{3}& \text{for }y=1\end{array}}$

This yields the following expectations and variances:

${\displaystyle {\mu }_X=\frac{2}{3}}$

${\displaystyle {\mu }_Y=0}$

${\displaystyle {\sigma }_X^2=\frac{2}{9}}$

${\displaystyle {\sigma }_Y^2=\frac{2}{3}}$

Therefore:

${\displaystyle \begin{array}{rl}{\rho }_{X,Y}& =\frac{1}{{\sigma }_X{\sigma }_Y}\mathrm{E}[(X-{\mu }_X)(Y-{\mu }_Y)]\\ & =\frac{1}{{\sigma }_X{\sigma }_Y}\sum _{x,y}(x-{\mu }_X)(y-{\mu }_Y)\mathrm{P}(X=x,Y=y)\\ & =\left(1-\frac{2}{3}\right)(-1-0)\frac{1}{3}+\left(0-\frac{2}{3}\right)(0-0)\frac{1}{3}+\left(1-\frac{2}{3}\right)(1-0)\frac{1}{3}=0.\end{array}}$

Rank correlation coefficients

Main articles: Spearman's rank correlation coefficient and Kendall tau rank correlation coefficient

Rank correlation coefficients, such as Spearman's rank correlation coefficient and Kendall's rank correlation coefficient (τ) measure the extent to which, as one variable increases, the other variable tends to increase, without requiring that increase to be represented by a linear relationship. If, as the one variable increases, the other decreases, the rank correlation coefficients will be negative. It is common to regard these rank correlation coefficients as alternatives to Pearson's coefficient, used either to reduce the amount of calculation or to make the coefficient less sensitive to non-normality in distributions. However, this view has little mathematical basis, as rank correlation coefficients measure a different type of relationship than the Pearson product-moment correlation coefficient, and are best seen as measures of a different type of association, rather than as an alternative measure of the population correlation coefficient.

To illustrate the nature of rank correlation, and its difference from linear correlation, consider the following four pairs of numbers ${\displaystyle (x,y)}$:

(0, 1), (10, 100), (101, 500), (102, 2000).

As we go from each pair to the next pair ${\displaystyle x}$ increases, and so does ${\displaystyle y}$. This relationship is perfect, in the sense that an increase in ${\displaystyle x}$ is always accompanied by an increase in ${\displaystyle y}$. This means that we have a perfect rank correlation, and both Spearman's and Kendall's correlation coefficients are 1, whereas in this example Pearson product-moment correlation coefficient is 0.7544, indicating that the points are far from lying on a straight line. In the same way if ${\displaystyle y}$ always decreases when ${\displaystyle x}$ increases, the rank correlation coefficients will be −1, while the Pearson product-moment correlation coefficient may or may not be close to −1, depending on how close the points are to a straight line. Although in the extreme cases of perfect rank correlation the two coefficients are both equal (being both +1 or both −1), this is not generally the case, and so values of the two coefficients cannot meaningfully be compared. For example, for the three pairs (1, 1) (2, 3) (3, 2) Spearman's coefficient is 1/2, while Kendall's coefficient is 1/3.

Other measures of dependence among random variables

See also: Pearson product-moment correlation coefficient § Variants

The information given by a correlation coefficient is not enough to define the dependence structure between random variables. The correlation coefficient completely defines the dependence structure only in very particular cases, for example when the distribution is a multivariate normal distribution. (See diagram above.) In the case of elliptical distributions it characterizes the (hyper-)ellipses of equal density; however, it does not completely characterize the dependence structure (for example, a multivariate t-distribution's degrees of freedom determine the level of tail dependence).

Distance correlation was introduced to address the deficiency of Pearson's correlation that it can be zero for dependent random variables; zero distance correlation implies independence.

The Randomized Dependence Coefficient is a computationally efficient, copula-based measure of dependence between multivariate random variables. RDC is invariant with respect to non-linear scalings of random variables, is capable of discovering a wide range of functional association patterns and takes value zero at independence.

For two binary variables, the odds ratio measures their dependence, and takes range non-negative numbers, possibly infinity: ${\displaystyle [0,+\mathrm{\infty }]}$. Related statistics such as Yule's Y and Yule's Q normalize this to the correlation-like range ${\displaystyle [-1,1]}$. The odds ratio is generalized by the logistic model to model cases where the dependent variables are discrete and there may be one or more independent variables.

The correlation ratio, entropy-based mutual information, total correlation, dual total correlation and polychoric correlation are all also capable of detecting more general dependencies, as is consideration of the copula between them, while the coefficient of determination generalizes the correlation coefficient to multiple regression.

Sensitivity to the data distribution

Further information: Pearson product-moment correlation coefficient § Sensitivity to the data distribution

The degree of dependence between variables X and Y does not depend on the scale on which the variables are expressed. That is, if we are analyzing the relationship between X and Y, most correlation measures are unaffected by transforming X to a + bX and Y to c + dY, where a, b, c, and d are constants (b and d being positive). This is true of some correlation statistics as well as their population analogues. Some correlation statistics, such as the rank correlation coefficient, are also invariant to monotone transformations of the marginal distributions of X and/or Y.

Pearson/Spearman correlation coefficients between X and Y are shown when the two variables' ranges are unrestricted, and when the range of X is restricted to the interval (0,1).

Most correlation measures are sensitive to the manner in which X and Y are sampled. Dependencies tend to be stronger if viewed over a wider range of values. Thus, if we consider the correlation coefficient between the heights of fathers and their sons over all adult males, and compare it to the same correlation coefficient calculated when the fathers are selected to be between 165 cm and 170 cm in height, the correlation will be weaker in the latter case. Several techniques have been developed that attempt to correct for range restriction in one or both variables, and are commonly used in meta-analysis; the most common are Thorndike's case II and case III equations.

Various correlation measures in use may be undefined for certain joint distributions of X and Y. For example, the Pearson correlation coefficient is defined in terms of moments, and hence will be undefined if the moments are undefined. Measures of dependence based on quantiles are always defined. Sample-based statistics intended to estimate population measures of dependence may or may not have desirable statistical properties such as being unbiased, or asymptotically consistent, based on the spatial structure of the population from which the data were sampled.

Sensitivity to the data distribution can be used to an advantage. For example, scaled correlation is designed to use the sensitivity to the range in order to pick out correlations between fast components of time series. By reducing the range of values in a controlled manner, the correlations on long time scale are filtered out and only the correlations on short time scales are revealed.

Correlation matrices

The correlation matrix of ${\displaystyle n}$ random variables ${\displaystyle X_1,\dots ,X_n}$ is the ${\displaystyle n\times n}$ matrix ${\displaystyle C}$ whose ${\displaystyle (i,j)}$ entry is

${\displaystyle c_{ij}:=\mathrm{corr}(X_i,X_j)=\frac{\mathrm{cov}(X_i,X_j)}{{\sigma }_{X_i}{\sigma }_{X_j}},\text{if}{\sigma }_{X_i}{\sigma }_{X_j}>0.}$

Thus the diagonal entries are all identically one. If the measures of correlation used are product-moment coefficients, the correlation matrix is the same as the covariance matrix of the standardized random variables ${\displaystyle X_i/\sigma (X_i)}$ for ${\displaystyle i=1,\dots ,n}$. This applies both to the matrix of population correlations (in which case ${\displaystyle \sigma }$ is the population standard deviation), and to the matrix of sample correlations (in which case ${\displaystyle \sigma }$ denotes the sample standard deviation). Consequently, each is necessarily a positive-semidefinite matrix. Moreover, the correlation matrix is strictly positive definite if no variable can have all its values exactly generated as a linear function of the values of the others.

The correlation matrix is symmetric because the correlation between ${\displaystyle X_i}$ and ${\displaystyle X_j}$ is the same as the correlation between ${\displaystyle X_j}$ and ${\displaystyle X_i}$.

A correlation matrix appears, for example, in one formula for the coefficient of multiple determination, a measure of goodness of fit in multiple regression.

In statistical modelling, correlation matrices representing the relationships between variables are categorized into different correlation structures, which are distinguished by factors such as the number of parameters required to estimate them. For example, in an exchangeable correlation matrix, all pairs of variables are modeled as having the same correlation, so all non-diagonal elements of the matrix are equal to each other. On the other hand, an autoregressive matrix is often used when variables represent a time series, since correlations are likely to be greater when measurements are closer in time. Other examples include independent, unstructured, M-dependent, and Toeplitz.

In exploratory data analysis, the iconography of correlations consists in replacing a correlation matrix by a diagram where the “remarkable” correlations are represented by a solid line (positive correlation), or a dotted line (negative correlation).

Nearest valid correlation matrix

In some applications (e.g., building data models from only partially observed data) one wants to find the "nearest" correlation matrix to an "approximate" correlation matrix (e.g., a matrix which typically lacks semi-definite positiveness due to the way it has been computed).

In 2002, Higham formalized the notion of nearness using the Frobenius norm and provided a method for computing the nearest correlation matrix using the Dykstra's projection algorithm, of which an implementation is available as an online Web API.

This sparked interest in the subject, with new theoretical (e.g., computing the nearest correlation matrix with factor structure) and numerical (e.g. usage the Newton's method for computing the nearest correlation matrix) results obtained in the subsequent years.

Uncorrelatedness and independence of stochastic processes

Similarly for two stochastic processes ${\displaystyle {\left\{X_t\right\}}_{t\in \mathcal{T}}}$ and ${\displaystyle {\left\{Y_t\right\}}_{t\in \mathcal{T}}}$: If they are independent, then they are uncorrelated. The opposite of this statement might not be true. Even if two variables are uncorrelated, they might not be independent to each other.

Common misconceptions

Correlation and causality

Main article: Correlation does not imply causation

See also: Normally distributed and uncorrelated does not imply independent

The conventional dictum that "correlation does not imply causation" means that correlation cannot be used by itself to infer a causal relationship between the variables. This dictum should not be taken to mean that correlations cannot indicate the potential existence of causal relations. However, the causes underlying the correlation, if any, may be indirect and unknown, and high correlations also overlap with identity relations (tautologies), where no causal process exists. Consequently, a correlation between two variables is not a sufficient condition to establish a causal relationship (in either direction).

A correlation between age and height in children is fairly causally transparent, but a correlation between mood and health in people is less so. Does improved mood lead to improved health, or does good health lead to good mood, or both? Or does some other factor underlie both? In other words, a correlation can be taken as evidence for a possible causal relationship, but cannot indicate what the causal relationship, if any, might be.

Simple linear correlations

Anscombe's quartet: four sets of data with the same correlation of 0.816

The Pearson correlation coefficient indicates the strength of a linear relationship between two variables, but its value generally does not completely characterize their relationship. In particular, if the conditional mean of ${\displaystyle Y}$ given ${\displaystyle X}$, denoted ${\displaystyle \mathrm{E}(Y\mid X)}$, is not linear in ${\displaystyle X}$, the correlation coefficient will not fully determine the form of ${\displaystyle \mathrm{E}(Y\mid X)}$.

The adjacent image shows scatter plots of Anscombe's quartet, a set of four different pairs of variables created by Francis Anscombe. The four ${\displaystyle y}$ variables have the same mean (7.5), variance (4.12), correlation (0.816) and regression line (y = 3 + 0.5x). However, as can be seen on the plots, the distribution of the variables is very different. The first one (top left) seems to be distributed normally, and corresponds to what one would expect when considering two variables correlated and following the assumption of normality. The second one (top right) is not distributed normally; while an obvious relationship between the two variables can be observed, it is not linear. In this case the Pearson correlation coefficient does not indicate that there is an exact functional relationship: only the extent to which that relationship can be approximated by a linear relationship. In the third case (bottom left), the linear relationship is perfect, except for one outlier which exerts enough influence to lower the correlation coefficient from 1 to 0.816. Finally, the fourth example (bottom right) shows another example when one outlier is enough to produce a high correlation coefficient, even though the relationship between the two variables is not linear.

These examples indicate that the correlation coefficient, as a summary statistic, cannot replace visual examination of the data. The examples are sometimes said to demonstrate that the Pearson correlation assumes that the data follow a normal distribution, but this is only partially correct. The Pearson correlation can be accurately calculated for any distribution that has a finite covariance matrix, which includes most distributions encountered in practice. However, the Pearson correlation coefficient (taken together with the sample mean and variance) is only a sufficient statistic if the data is drawn from a multivariate normal distribution. As a result, the Pearson correlation coefficient fully characterizes the relationship between variables if and only if the data are drawn from a multivariate normal distribution.

Bivariate normal distribution

If a pair ${\displaystyle (X,Y)}$ of random variables follows a bivariate normal distribution, the conditional mean ${\displaystyle \mathrm{E}(X\mid Y)}$ is a linear function of ${\displaystyle Y}$, and the conditional mean ${\displaystyle \mathrm{E}(Y\mid X)}$ is a linear function of ${\displaystyle X}$. The correlation coefficient ${\displaystyle {\rho }_{X,Y}}$ between ${\displaystyle X}$ and ${\displaystyle Y}$, along with the marginal means and variances of ${\displaystyle X}$ and ${\displaystyle Y}$, determines this linear relationship:

${\displaystyle \mathrm{E}(Y\mid X)=\mathrm{E}(Y)+{\rho }_{X,Y}\cdot {\sigma }_Y\frac{X-\mathrm{E}(X)}{{\sigma }_X},}$

where ${\displaystyle \mathrm{E}(X)}$ and ${\displaystyle \mathrm{E}(Y)}$ are the expected values of ${\displaystyle X}$ and ${\displaystyle Y}$, respectively, and ${\displaystyle {\sigma }_X}$ and ${\displaystyle {\sigma }_Y}$ are the standard deviations of ${\displaystyle X}$ and ${\displaystyle Y}$, respectively.

The empirical correlation ${\displaystyle r}$ is an estimate of the correlation coefficient ${\displaystyle \rho }$. A distribution estimate for ${\displaystyle \rho }$ is given by

$${\displaystyle \pi (\rho |r)=\frac{\mathrm{\Gamma }(\nu +1)}{\sqrt{2\pi }\mathrm{\Gamma }(\nu +\frac{1}{2})}(1-r^2{)}^{\frac{\nu -1}{2}}\cdot (1-{\rho }^2{)}^{\frac{\nu -2}{2}}\cdot (1-r\rho {)}^{\frac{1-2\nu }{2}}F\left(\frac{3}{2},-\frac{1}{2};\nu +\frac{1}{2};\frac{1+r\rho }{2}\right)}$$

where ${\displaystyle F}$ is the Gaussian hypergeometric function and ${\displaystyle \nu =N-1>1}$ . This density is both a Bayesian posterior density and an exact optimal confidence distribution density.

Neuroeffector junction

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Neuroeffector_junction 

A neuroeffector junction is a site where a motor neuron releases a neurotransmitter to affect a target—non-neuronal—cell. This junction functions like a synapse. However, unlike most neurons, somatic efferent motor neurons innervate skeletal muscle, and are always excitatory. Visceral efferent neurons innervate smooth muscle, cardiac muscle, and glands, and have the ability to be either excitatory or inhibitory in function. Neuroeffector junctions are known as neuromuscular junctions when the target cell is a muscle fiber.

Non-synaptic transmission is characteristic of autonomic neuroeffector junctions. The structure of the autonomic neuromuscular junction consists of several essential features including that: the terminal portions of autonomic nerve fibers are varicose and mobile, transmitters being released 'en passage' from varying distances from the effector cells; while there is no structural post-junctional specialization on effector cells, receptors for neurotransmitters accumulate on cell membranes at close junctions. Muscle effectors are bundles rather than single smooth muscle cells that are connected by gap junctions which allow electrotonic spread of activity between cells. A multiplicity of transmitters are utilized by autonomic nerves, and co-transmission occurs often involving synergistic actions of the co-transmitters, although pre- and post-junctional neuromodulation of neurotransmitter release also take place. It is suggested that autonomic neural control of immune, epithelial and endothelial cells also involves non-synaptic transmission.

These are tight junctions, but in the autonomic nervous system and enteric nervous system the connecting junctions become much "looser", allowing for easier diffusion. This looseness allows for a wider signal receiving whereas in tighter junctions, more neurotransmitters get metabolized or broken down. In skeletal muscles, the junctions are mostly of the same distance and size because they innervate such definite structures of muscle fibers. In the Autonomic Nervous System however, these neuromuscular junctions are much less well defined.

Analysis of non-noradrenergic/non-cholinergic (NANC) transmission at single varicosities or swellings indicates that individual synapses possess different probabilities for the secretion of transmitter as well as different complements of autoreceptors and mixtures of post-junctional receptor subunits. There is then a local determination of the quantitative properties of single synapses.

Nerve terminals are the terminal part of the axon filled with neurotransmitters and are the location from which neurotransmitters are released. Nerve terminals may take different forms in different tissues. Nerve terminals appear like a button in the CNS, end plates in striated muscle and varicosities in many tissues including the gut. Buttons, endplates or varicosities all function to store and release neurotransmitters. In many peripheral tissues, the varicose axon branches in its proximal course and carries a covering of Schwann sheath, which is interrupted and finally lost in its most terminal part. The unmyelinated, preterminal axons with very long varicose branches are present in small axon bundles and varicose terminal axons are present as single isolated axons. The small axon bundles run parallel to and between muscle bundles and the "en passage" varicose axons are the main sources of innervations to the gut smooth muscle bundles.

Nonsynaptic post-junctional receptors are mostly G-protein coupled metabotropic receptors that produce a slower response. They include metabotropic receptors for the classical neurotransmitters, monoamines, norepinephrine, purines and peptide transmitters. Post-junctional receptors also include some ionotropic receptors such as nicotinic receptors in the central nervous system (CNS) as well as the autonomic nervous system (ANS).

Nonsynaptic junctional transmission is the only mode of transmission involving the varicosities that show no synaptic contacts that includes almost all nerve terminals whose target is not a neuron. Most smooth muscles exhibit both fast and slow junction potentials typically mediated by different classes of metabotropic receptors with different kinetics.

The close junctional neurotransmission is characterized by synapse like close contact between the pre-junctional release site and the post-junctional receptors. However, unlike the synapse, the junctional space is open to the extravascular space; the pre-junctional release site lacks the distinguishing features of the presynaptic active zone and release of the soluble transmitters; and the post junctional receptors include metabotropic receptors or slower acting ionotropic receptors.

Almost all tissues that exhibit close junctional neurotransmission also show wide junctional neurotransmission. Thus, wide junctional transmission has been described in many smooth muscles such as vas deferens, urinary bladder, blood vessels, gut as well as the nervous systems including ENS, autonomic ganglia and the CNS.

Control of gastrointestinal (GI) movements by enteric motoneurons is critical for orderly processing of food, absorption of nutrients and elimination of wastes. Neuroeffector junctions in the tunica muscularis might consist of synaptic-like connectivity with specialized cells, and contributions from multiple cell types in integrated post-junctional responses. Interstitial cells of Cajal (ICC) – non-muscular cells of mesenchymal origin—were proposed as potential mediators in motor neurotransmission. Neuromuscular junctions in GI smooth muscles may reflect innervation of, and post-junctional responses in, all three classes of post-junctional cells. Transduction of neurotransmitter signals by ICC cells and activation of ionic conductances would be conducted electronically via gap junctions to surrounding smooth muscle cells and influence the excitability of tissues.

Neuromuscular junction. 1. Axon innervating muscle fibers; 2. Junction between axon and muscle fiber; 3. Muscle; 4. Muscle fiber

Discovery

In the peripheral nervous system, local junctional transmission was recognized in the late 1960s and early 1970s. Until then, all chemical neurotransmission was thought to involve synapses and the innervations of tissue were considered synonymous with the existence of a synapse. Later, it was observed that at smooth muscle neuromuscular junctions in the gut and other peripheral autonomic neuroeffector junctions, neurotransmission takes place in the absence of any synapses and it was suggested that at these sites, neurotransmission involved non-synaptic transmission. Accordingly, nerve endings release their neurotransmitters in extracellular space in a manner similar to paracrine secretion. Target cells affected by a locally released transmitter even though located several hundreds to thousands of nanometers away from the release site are considered as being innervated.

The varicose axons were first visualized for adrenergic terminals using fluorescence histochemistry described by Falck and colleagues.

These varicose axons resemble strings of beads with varicosities 0.5–2.0 μ in diameter and 1 to 3 μ in length and separated by inter-varicosity axon 0.1 to 0.2 μ in diameter. The varicosities occur at 2–10 μm intervals and it has been estimated that a single adrenergic axon may have over 25,000 varicosities on its terminal part. There are also two types of contacts. These contacts are called large and small contacts, respectively. In the large contacts, the bare varicosities and the smooth muscles were separated by ~60 nm and in the small contacts the two were separated by ~400 nm. Overall, non-synaptic junctional space between the neural release site and the post-junctional receptors may show variable degrees of separation between the release site on the pre-junctional nerve terminal and the post-junctional receptors on the target cell.

The discovery of NANC inhibitory and excitatory transmission as well as the fact that such transmission has to be considered as occurring to smooth muscle cells coupled together in an electrical Autonomic postganglionic nerves terminate in systems syncytium and that the excitatory NANC transmission of collateral branches, each of which possesses of the order gives rise to a calcium-dependent action potential.

Research

Neuromuscular junctions in gastrointestinal (GI) smooth muscles may reflect innervation of, and post-junctional responses in, all three classes of post-junctional cells. Transduction of neurotransmitter signals by ICC cells and activation of ionic conductances would be conducted electronically via gap junctions to surrounding smooth muscle cells and influence excitability.

Studies do not exclude the possibility of parallel excitatory neurotransmission to ICC-DMP (deep muscular plexus) and smooth muscle cells. Different cells may utilize different receptors and signaling molecules. ICC are innervated and transmitters reach high enough concentration to activate post-junctional signaling pathways in ICC. If ICC are important intermediaries in motor neurotransmission, then loss of these cells could reduce communication between the enteric nervous system and the smooth muscle syncytium, resulting in reduced neural regulation of motility.

In pioneering studies it was shown unequivocally that the innervation of smooth muscles is by varicose nerve terminals. However, it was not until the advent of the electron microscope that we were able to provide us with a comprehensive view of the relationship between these varicose endings and smooth muscle.

Besides activation of K+ channels by NO, some authors have suggested that Ca2+-activated Cl− channels, which are active under basal conditions, can be suppressed as part of the post-junctional response to NO. These studies do not exclude the possibility of parallel excitatory neurotransmission to ICC-DMP and smooth muscle cells. Different cells may utilize different receptors and signaling molecules. These findings make the point that ICC are innervated and transmitters reach high enough concentration to activate post-junctional signaling pathways in ICC. There is no reason to assume a priori that responses to neurotransmitters released from neurons and exogenous transmitter substances are mediated by the same cells, receptors or post-junctional (transduction) signaling pathways. Neurotransmitters released from varicosities may be spatially limited to specific populations of receptors, whereas transmitters added to organ baths may bind to receptors on a variety of cells.

Structure and function

Non-synaptic transmission is characteristic of autonomic neuroeffector junctions. The essential features are that: the terminal portions of autonomic nerve fibers are varicose and mobile; transmitters are released from varicosities at varying distances from the effector cells; and while there is no structural post-junctional specialization on effector cells, receptors for neurotransmitters accumulate on cell membranes at close junctions. Besides smooth muscle, autonomic neural control of immune, epithelial, and endothelial cells also involves nonsynaptic transmission. Smooth muscle effectors are bundles rather than single cells, that are connected by gap junctions which allow electrotonic spread of activity between cells. Many smooth muscle cells in a transverse section through a muscle bundle show regions of very close apposition to adjacent cells at which connexins form junctions between the cells. Unlike in cardiac muscle, where gap junctions are confined to the ends of cardiac myocytes, smooth muscle gap junctions occur along the length of the muscle cells as well as towards their ends. There are small bundles of three to seven varicose axons, partially or wholly enveloped in Schwann-cell sheath, both on the surface of the muscle as well as in the body of smooth muscle bundles. In addition, single varicose axons can be found on the surface and in the muscle bundles, and become divested of Schwann cells in the region of apposition between the varicosities and smooth muscle cells.

The active zone of individual sympathetic varicosities, delineated by a high concentration of syntaxin, occupies an area on the pre-junctional membrane of about 0.2 μm²; this gives a junctional gap between the pre-junctional active zone and post-junctional membranes that varies between about 50 and 100 nm. The post-junctional membrane beneath the varicosity can possess a patch about 1 μm² of purinergic P2X1 receptors in high density, although this is not always the case. A nerve impulse gives rise to a transient increase in calcium concentration in every varicosity, primarily due to the opening of N-type calcium channels, as well as to a smaller increase in the intervaricose regions. The probability of secretion from a varicosity may depend on the number of secretosomes that the varicosity possesses, where a secretosome is a complex of syntaxin, synaptotagmin, an N-type calcium channel, and a synaptic vesicle.

A multiplicity of transmitters are utilized by autonomic nerves, and cotransmission occurs, often involving synergistic actions of the cotransmitters, although pre- and post-junctional neuromodulation of neurotransmitter release also take place. Cotransmission without co-storage occurs in parasympathetic nerves, where terminals staining for the vesicular acetylcholine transporter can also contain nitric oxide synthase, suggesting that they release NO as a gaseous neurotransmitter.

Neuroeffector Ca²⁺ transients (NCT) have been used to detect the packeted release of the neurotransmitter ATP acting on post-junctional P2X receptors to cause the Ca²⁺ influx. ATP released from varicosities is modulated by the concomitant release of noradrenaline that acts on the varicosities through α2-adrenoceptors to decrease the influx of calcium ions that accompanies the nerve impulse. NCT can also be used to detect the local effects of noradrenaline through its α2-adrenoceptor-mediated pre-junctional autoinhibitory effects on nerve terminal Ca²⁺ concentration and the probability of exocytosis (measured by counting NCTs). There is evidence that exocytosis from sympathetic varicosities depends on their history, and that the release of a packet of ATP transiently suppresses (or predicts the transient suppression of) subsequent release. The poverty of NCTs occurring within 5s of one another indicates that exocytosis from a varicosity transiently suppresses the probability of release from that varicosity. This could arise by autoinhibition (by the pre-junctional action of noradrenaline or purines) or due to a transient shortage of vesicles readily available for release.

ATP release (hence noradrenaline release, if there is strict corelease) is highly intermittent at these junctions (Brain et al. 2002), with a probability that a given action potential will evoke the release from a given varicosity of only 0.019. If there are n varicosities within the diffusion range of a particular varicosity, we can consider the number of such varicosities that might need to be present in order that, on average (using P =0.5 to give the median value), neurotransmitter will be released locally. During a five-impulse train, assuming that the last impulse in the train cannot autoinhibit the Ca²⁺ influx during the train, the expectation value of n can be found by solving [(1 − 0.019)⁴ⁿ] =(1 − 0.5), i.e. the probability that there will be no local release, given n varicosities within the diffusion range. This is n = [ln(0.5)/ln(0.981)]/4, or n≈9. If the density of varicosities is around 2.2 per 1000 μm³, this number of varicosities should occur within an average range (radius) of about 10 μm (noting that within such a radius there is a tissue volume of about 4200 μm³). Therefore, even in the presence of highly intermittent noradrenaline release, one would expect the average varicosity in this organ to be within 10 μm of a released packet of noradrenaline at some time during a five-impulse stimulus train (excluding the last impulse).

Junctional transmission is measured in seconds to minutes. The time course of the junctional potential has been divided into two most frequently observed time courses representing 'close' and 'wide' junctional transmissions. The "close" junctional transmission is associated with fast junction potential and the "wide" junctional transmission is associated with slow junction potential. The slow electrical potentials reach a peak in about 150 ms and then decline with a time constant between 250 and 500 ms. These responses typically last several seconds to minutes and may be depolarizing and excitatory, or hyperpolarizing and inhibitory, and have been called slow EJP or slow IJP, respectively.

Interstitial cells of Cajal

Over the past 20 years, many studies have given evidence that Interstitial cells of Cajal (ICC): (i) serve as pacemaker cells with unique ionic currents that generate electrical slow waves in GI muscles; (ii) provide a pathway for active slow wave propagation in GI organs; (iii) express receptors, transduction mechanisms and ionic conductances allowing them to mediate post-junctional responses to enteric motor neurotransmission; (iv) regulate smooth muscle excitability by contributing to resting potential and affecting syncytial conductance; and (v) manifest stretch-receptor functions regulating excitability and regulating slow wave frequency.

If this channel is open, conductance changes in cell are reflected in smooth muscle; post-junctional integrated responses are triggered by neuroeffector junctions and interstitial cells.

Based on anatomic location and function, two main types of ICC have been described: myenteric ICC (ICC-MY) and intramuscular ICC (ICC-IM). ICC-MY are present around the myenteric plexus and thought to be pacemaker cells for slow waves in the smooth muscle cells. Calcium imaging studies in the colon have shown that ICC-MY is innervated by nitrergic and cholinergic nerve terminals, though the nature of the contacts has not been well defined. ICC-IM is located in between the smooth muscle cells. Enteric nerves have been reported to make synaptic contacts with ICC-IM. These contacts include areas of electron dense lining on the inner aspect of the varicosity membrane without any postsynaptic density on the membrane of ICC. Such contacts were not reported between the nerves and the smooth muscles. If ICC are important intermediaries in motor neurotransmission, then loss of these cells could reduce communication between the enteric nervous system and the smooth muscle syncytium, resulting in reduced neural regulation of motility.

Classical excitatory and inhibitory neurotransmitters are concentrated and released from neurovesicles located in enteric nerve terminals or varicose regions of motor nerves, whereas nitric oxide is probably synthesized de novo as calcium concentration increases in nerve terminals upon membrane depolarization. Enteric nerve terminals make intimate synapses with ICC-IM, which are situated between the nerve terminals and neighbouring smooth muscle cells. ICC-IM play a critical role in the reception and transduction of cholinergic excitatory and nitrergic inhibitory neurotransmission. ICC-IM form gap junctions with smooth muscle cells and post-junctional electrical responses generated in ICC are conducted to the smooth muscle syncytium. By this contact, ICC can regulate the neuromuscular responses observed throughout the GI tract. Recent morphological evidence using anterograde tracing methods, has shown close apposition between vagal and spinal afferents and ICC-IM within the stomach wall (Fig. 5) and their absence in mutant animals that lack ICC-IM also supports a role for ICC-IM as possible integrators for in-series stretch-dependent changes in this organ.