formulasearchengine - User contributions [en]

Extreme physical information

2014-07-17T01:54:10Z

150.135.249.13: Made a few clarifications as to the meaning and implementation of the EPI approach.

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Beta distribution

2014-03-04T04:50:57Z

150.135.210.66:

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Quadrature (mathematics)

2013-08-30T23:33:54Z

150.135.222.128: /* See also */ Added a link to Tanh-sinh quadrature. It felt related. I torn whether to keep the list alphabetical or place it next to Gaussian quadrature.

In [[mathematics]] and [[economics]], the '''arc elasticity''' is the [[Elasticity (mathematics)|elasticity]] of one variable with respect to another between two given points. It is the ratio of the percentage change of one of the variables between the two points to the percentage change of the other variable. It contrasts with the ''point elasticity'', which is the limit of the arc elasticity as the distance between the two points approaches zero and which hence is defined at a single point rather than for a pair of points.

==Formula==

The ''y'' arc elasticity of ''x'' is defined as:

:<math>E_{x,y} = \frac{\% \mbox{ change in } x}{\% \mbox{ change in } y}</math>

where the percentage change in going from point 1 to point 2 is usually calculated relative to the midpoint:

:<math>\% \mbox{ change in } x = \frac{x_2 - x_1}{(x_2 + x_1)/2};</math>

:<math>\% \mbox{ change in } y = \frac{y_2 - y_1}{(y_2 + y_1)/2}.</math>

The use of the midpoint arc elasticity formula (with the midpoint rather than the initial point (''x''1, ''y''1) used for calculating percentages) was advocated by [[R. G. D. Allen]] for use when ''x'' refers to the quantity of a good demanded or supplied and ''y'' refers to its price, due to the following properties: (1) it is symmetric with respect to the two prices and quantities, (2) it is independent of the units of measurement, and (3) it yields a value of unity if the total revenues (price times quantity) at the two points are equal.<ref>{{cite journal |first=R. G. D. |last=Allen |year=1933 |title=The Concept of Arc Elasticity of Demand |journal=[[Review of Economic Studies]] |volume=1 |issue=3 |pages=226–229 |jstor=2967486 }}</ref>

The arc elasticity is used when there is not a general function for the relationship of two variables, but two points on the relationship are known. in contrast, calculation of the point elasticity requires detailed knowledge of the functional relationship and can be calculated wherever the function is defined.

For comparison, the ''y'' point elasticity of ''x'' is given by
:<math>E_{x,y} = \frac{\partial x}{\partial y} \cdot \frac{y}{x} = \frac{\partial \ln x}{\partial \ln y}</math>

==Application in economics==

The arc elasticity of quantity demanded (or quantity supplied) Q with respect to price P, also known as the arc price elasticity of demand (or supply), is calculated as
:<math>(\% \mbox{ change in }Q)/(\%\mbox{ change in }P)</math>

===Example===

Suppose that two points on a demand curve, <math>(Q_1, P_1)</math> and <math>(Q_2, P_2)</math>, are known. (Nothing else might be known about the demand curve.) Then the arc elasticity is obtained using the formula 
:<math>E_p =\frac{\frac{Q_2-Q_1}{(Q_1+Q_2)/2}}{\frac{P_2-P_1}{(P_1+P_2)/2}}.</math>

Suppose the quantity of hot dogs demanded at halftime of football games is measured at two different games: at one measurement it is 80 units, and at the other measurement it is 120 units. The percent change, measured against the average, would be (120-80)/((120+80)/2))=40%.

In contrast, if the percentage change were measured against the initial value, the result would be (120-80)/80= 50%. The percent change for the opposite trend, 120 units to 80 units, would be (80-120)/120 = -33.3%. The midpoint formula has the benefit that a movement from A to B is measured as the same as a movement from B to A in absolute value. (In this case, the movement from 120 to 80 would be measured as -40%.)

Suppose that the change in the price of hot dogs, which led to this change in quantity demanded from 80 to 120, was from $3 to $1. The percent change in price measured against the midpoint would be (1-3)/2 = -100%, so the [[price elasticity of demand]] is 40%/(-100%) or -40%. It is common to refer to the ''absolute value'' of the price elasticity as simply price elasticity, since for a normal (decreasing) demand curve the elasticity is always negative and so the "minus" part can be made implicit. Thus the demand of the football fans for hot dogs has 40% elasticity.

==See also==
* [[Elasticity of a function]]
* [[Elasticity (economics)]]

==References==
<references/>

[[Category:Elasticity (economics)]]

Activation function

2013-08-30T23:13:14Z

150.135.222.128: /* See also */ Added a link to the softmax activation function. It felt related.

'''BCM theory''', '''BCM synaptic modification''', or the '''BCM rule''', named for [[Elie Bienenstock]], [[Leon Cooper]], and Paul Munro, is a physical theory of learning in the [[visual cortex]] developed in 1981. Due to its successful experimental predictions, the theory is arguably the most accurate model of [[synaptic plasticity]] to date{{Citation needed|date=March 2009}}.

"The BCM model proposes a sliding threshold for [[Long-term potentiation]] or [[Long-term depression]] induction and states that synaptic plasticity is stabilized by a dynamic adaptation of the time-averaged postsynaptic activity. According to the BCM model, reducing the postsynaptic activity decreases the LTP threshold and increases the LTD threshold. The opposite applies to the increase in postsynaptic activity."<ref>Bologna, M., Agostino, R., Gregori, B., Belvisi, D., Manfredi, M., & Berardelli, A. (2010). Metaplasticity of the human trigeminal blink reflex M. Bologna et al. Metaplasticity of the blink reflex. European Journal of Neuroscience, 32(10), 1707-1714. {{doi|10.1111/j.1460-9568.2010.07446.x}}.</ref>

== Development ==
In 1949, [[Donald Hebb]] proposed a working mechanism for memory and computational adaption in the brain now called [[Hebbian learning]], or the maxim that ''cells that fire together, wire together''. This law formed the basis of the brain as the modern [[neural network]], theoretically capable of [[Turing complete]] computational complexity{{Citation needed|date=July 2008}}, and thus became a standard materialist model for the mind.

However, Hebb's rule has problems, namely that it has no mechanism for connections to get weaker and no upper bound for how strong they can get. In other words, the model is unstable, both theoretically and computationally. Later modifications gradually improved Hebb's rule, normalizing it and allowing for decay of synapses, where no activity or unsynchronized activity between neurons results in a loss of connection strength. New biological evidence brought this activity to a peak in the 1970s, where theorists formalized various approximations in the theory, such as the use of firing frequency instead of potential in determining neuron excitation, and the assumption of ideal and, more importantly, linear synaptic integration of signals. That is, there is no unexpected behavior in the adding of input currents to determine whether or not a cell will fire.

These approximations resulted in the basic form of BCM below in 1979, but the final step came in the form of mathematical analysis to prove stability and computational analysis to prove applicability, culminating in Bienenstock, Cooper, and Munro's 1982 paper.

Since then, experiments have shown evidence for BCM behavior in both the [[visual cortex]] and the [[hippocampus]], the latter of which plays an important role in the formation and storage of memories. Both of these areas are well-studied experimentally, but both theory and experiment have yet to establish conclusive synaptic behavior in other areas of the brain. Furthermore, a biological mechanism for [[synaptic plasticity]] in BCM has yet to be established.<ref>{{cite journal |last=Cooper |first=L.N. |authorlink=Leon Cooper |year=2000 |month= |title=Memories and memory: A physicist's approach to the brain |journal=International Journal of Modern Physics A |volume=15 |issue=26 |pages=4069–4082 |id= |url=http://physics.brown.edu/physics/researchpages/Ibns/Lab%20Publications%20(PDF)/memoriesandmemory.pdf |accessdate= 2007-11-11 |quote= }}</ref>

== Theory ==
{{Confusing|date=January 2012}}
The basic BCM rule takes the form

:<math>\,\frac{d m_j(t)}{d t} = \phi(\textbf{c}(t))d_j(t)-\epsilon m_j(t),</math>

where <math>m_j</math> is the synaptic weight of the <math>j</math>th synapse, <math>d_j</math> is that synapse's input current, <math>\textbf{c}</math> is the weighted presynaptic output vector, <math>\phi</math> is the postsynaptic [[activation function]] that changes sign at some output threshold <math>\theta_M</math>, and <math>\epsilon</math> is the (often negligible) time constant of uniform decay of all synapses.

This model is merely a modified form of the [[Hebbian learning]] rule, <math>\dot{m_j}=c d_j</math>, and requires a suitable choice of activation function, or rather, the output threshold, to avoid the Hebbian problems of instability. This threshold was derived rigorously in BCM noting that with <math>c(t)=\textbf{m}(t)\cdot\textbf{d}(t)</math> and the approximation of the average output <math>\bar{\textbf{c}}(t) \approx \textbf{m}(t)\bar{d}(t)</math>, for one to have stable learning it is sufficient that

:<math>\,\sgn\phi(c,\bar{c}) = \sgn\left(c-\left(\frac{\bar{c}}{c_0}\right)^p\bar{c}\right) ~~ \textrm{for} ~ c>0, ~ \textrm{and}</math>
:<math>\,\phi(0,\bar{c}) = 0 ~~ \textrm{for} ~ \textrm{all} ~ \bar{c},</math>

or equivalently, that the threshold <math>\theta_M(\bar{c}) = (\bar{c}/c_0)^p\bar{c}</math>, where <math>p</math> and <math>c_0</math> are fixed positive constants.<ref>{{cite journal |last=Bienenstock |first=Elie L. |authorlink=Elie Bienenstock |coauthors=[[Leon Cooper]], [[Paul Munro]] |date=January 1982|title=Theory for the development of neuron selectivity: orientation specificity and binocular interaction in visual cortex |journal=The Journal of Neuroscience |volume=2 |issue=1 |pages=32–48 |id= |url=http://www.physics.brown.edu/physics/researchpages/Ibns/Cooper%20Pubs/070_TheoryDevelopment_82.pdf |accessdate= 2007-11-11 |quote= |pmid=7054394 }}</ref>

When implemented, the theory is often taken such that

:<math>\,\phi(c,\bar{c}) = c(c-\theta_M) ~~~ \textrm{and} ~~~ \theta_M = \langle c^2 \rangle = \frac{1}{\tau}\int_{-\infty}^t c^2(t^\prime)e^{-(t-t^\prime)/\tau}d t^\prime,</math>

where angle brackets are a time average and <math>\tau</math> is the time constant of selectivity.

The model has drawbacks, as it requires both [[long-term potentiation]] and [[long-term depression]], or increases and decreases in synaptic strength, something which has not been observed in all cortical systems. Further, it requires a variable activation threshold and depends strongly on stability of the selected fixed points <math>c_0</math> and <math>p</math>. However, the model's strength is that it incorporates all these requirements from independently-derived rules of stability, such as [[Normalisable wave function|normalizability]] and a decay function with time proportional to the square of the output.<ref>{{cite web |url=http://www.cs.tau.ac.il/~nin/Courses/NC05/BCM.ppt |title=The BCM theory of synaptic plasticity |accessdate=2007-11-11 |last=Intrator |first=Nathan |coauthors= |date=2006-2007 |work=Neural Computation |publisher=School of Computer Science, Tel-Aviv University }}</ref>

== Experiment ==
The first major experimental confirmation of BCM came in 1992 in investigating [[Long-term potentiation|LTP]] and [[Long-term depression|LTD]] in the [[hippocampus]]. The data showed qualitative agreement with the final form of the BCM activation function.<ref>{{cite journal |last=Dudek |first=Serena M. |authorlink=Serena Dudek |coauthors=[[Mark F. Bear|Mark Bear]] |year=1992 |month= |title=Homosynaptic long-term depression in area CA1 of hippocampus and effects of N-methyl-D-aspartate receptor blockade |journal=Proc. Natl. Acad. Sci. |volume=89 |issue= 10|pages=4363–4367 |id= |url=http://www.pnas.org/cgi/reprint/89/10/4363.pdf |accessdate= 2007-11-11 |quote=|doi=10.1073/pnas.89.10.4363 |pmid=1350090 |pmc=49082 }}</ref> This experiment was later replicated in the [[visual cortex]], which BCM was originally designed to model.<ref>{{cite journal |last=Kirkwood |first=Alfredo |authorlink=Alfredo Kirkwood |coauthors=[[Marc G. Rioult]], [[Mark F. Bear]] |year=1996 |month= |title=Experience-dependent modification of synaptic plasticity in rat visual cortex |journal=[[Nature (journal)|Nature]] |volume=381 |issue= 6582|pages=526–528 |id= |url=http://www.nature.com/nature/journal/v381/n6582/abs/381526a0.html |accessdate= 2007-11-11 |quote=|doi=10.1038/381526a0 |pmid=8632826 }}</ref> This work provided further evidence of the necessity for a variable threshold function for stability in Hebbian-type learning (BCM or others).

Experimental evidence has been non-specific to BCM until Rittenhouse ''et al.'' confirmed BCM's prediction of synapse modification in the visual cortex when one eye is selectively closed. Specifically,

:<math>\log\left(\frac{m_{\rm closed}(t)}{m_{\rm closed}(0)}\right) \sim -\overline{n^2}t,</math>

where <math>\overline{n^2}</math> describes the variance in spontaneous activity or noise in the closed eye and <math>t</math> is time since closure. Experiment agreed with the general shape of this prediction and provided an explanation for the dynamics of monocular eye closure ([[monocular deprivation]]) versus binocular eye closure.<ref>{{cite journal |last=Rittenhouse |first=Cynthia D. |authorlink= |coauthors=Harel Z. Shouval, Michael A. Paradiso, Mark F. Bear |year=1999 |month= |title=Monocular deprivation induces homosynaptic long-term depression in visual cortex |journal=[[Nature (journal)|Nature]] |volume=397 |issue= 6717|pages=347–50 |id= |url=http://www.nature.com/nature/journal/v397/n6717/abs/397347a0.html |accessdate= 2007-11-11 |quote=|doi=10.1038/16922 |pmid=9950426 }}</ref> The experimental results are far from conclusive, but so far have favored BCM over competing theories of plasticity.

== Applications ==
While the algorithm of BCM is too complicated for large-scale [[parallel distributed processing]], it has been put to use in lateral networks with some success.<ref>{{cite web |url=http://www.cs.tau.ac.il/~nin/Courses/NC05/bcmppr.pdf |title=BCM Learning Rule, Comp Issues |accessdate=2007-11-11 |last=Intrator |first=Nathan |coauthors= |date=2006-2007 |work=Neural Computation |publisher=School of Computer Science, Tel-Aviv University }}</ref> Furthermore, some existing computational network learning algorithms have been made to correspond to BCM learning.<ref>{{cite journal |last=Baras |first=Dorit |authorlink= |coauthors=Ron Meir |year=2007 |month= |title=Reinforcement Learning, Spike-Time-Dependent Plasticity, and the BCM Rule |journal=Neural Computation |volume= 19|issue=19 |pages=2245–2279 |id=2561 |url=http://eprints.pascal-network.org/archive/00002561/01/RL-STDP_Final.pdf |accessdate=2007-11-11 |quote=|doi=10.1162/neco.2007.19.8.2245 |pmid=17571943 }}</ref>

== References ==
{{reflist}}

== External links ==
* [http://www.scholarpedia.org/article/BCM_rule Scholarpedia article]

[[Category:Biophysics]]
[[Category:Computational neuroscience]]
[[Category:Neuroscience]]

Optical coherence tomography

2012-07-25T21:59:08Z

150.135.97.207:

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Talk:Differintegral

2011-05-07T03:53:32Z

150.135.222.80: /* Question about Riemann-Liouville differintegral */ new section

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Schrödinger equation

2006-05-23T21:52:15Z

150.135.248.180: Shortened the first equation, deleted erroneous assertion that the Schrodinger wave equation "defines the energy" of a system.

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