Mean Squared Displacement - Revision history

Bradley Monk at 04:08, 17 November 2018

2018-11-17T04:08:52Z

← Older revision		Revision as of 21:08, 16 November 2018
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	==Notes==		==Notes==
	Finding the MSD of the simulation is ([http://en.wikipedia.org/wiki/Mean_squared_displacement a trivial task]) using a Matlab toolbox (see video below). The target extrasynaptic MSD was '''0.1 <sup>µm²</sup>&frasl;<sub>s</sub>''' (from Choquet {{Fig\|[[File:ChoquetDiffusionRate1.png]]}}). Scaling the model PSD size and distance between synapses to real-world values based on the target MSD is not a trivial task, and may actually be related only indirectly. Just because a particle moves quickly, does not mean it moves from A to B quickly - it depends on how much stuff there is to collide with. Linear estimates have shown that glycine receptor movements along dendrites at a speed of 1–2 µm/min (.016-.008 µm/s) {{Fig\|[[File:Choquet MSD2.png]]}}		Finding the MSD of the simulation is ([http://en.wikipedia.org/wiki/Mean_squared_displacement a trivial task]) using a Matlab toolbox (see video below). The target extrasynaptic MSD was '''0.1 <sup>µm²</sup>&frasl;<sub>s</sub>''' (from Choquet {{Fig\|[[File:ChoquetDiffusionRate1.png]]}}). Scaling the model PSD size and distance between synapses to real-world values based on the target MSD is not a trivial task, and may actually be related only indirectly. Just because a particle moves quickly, does not mean it moves from A to B quickly - it depends on how much stuff there is to collide with. Linear estimates have shown that glycine receptor movements along dendrites at a speed of 1–2 µm/min (.016-.008 µm/s)
			{{Fig\|[[File:Choquet MSD2.png]]}}

	First, the randomly generated step-size was scaled to produce an MSD of '''0.1 <sup>units²</sup>&frasl;<sub>step</sub>''' using the Matlab Brownian motion toolbox. Second, the dimensions of the model need to be scaled. It was found that an XY random step-size of µ{{=}}0.4 (σ{{=}}.2) units produced an MSE ≈ '''0.1 <sup>units²</sup>&frasl;<sub>step</sub>'''. Next, the arbitrary 0.4 units need to be given meaning...		First, the randomly generated step-size was scaled to produce an MSD of '''0.1 <sup>units²</sup>&frasl;<sub>step</sub>''' using the Matlab Brownian motion toolbox. Second, the dimensions of the model need to be scaled. It was found that an XY random step-size of µ{{=}}0.4 (σ{{=}}.2) units produced an MSE ≈ '''0.1 <sup>units²</sup>&frasl;<sub>step</sub>'''. Next, the arbitrary 0.4 units need to be given meaning...
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	*[http://www.mathworks.com/matlabcentral/fileexchange/40692-mean-square-displacement-analysis-of-particles-trajectories Matlab toolbox]		*[http://www.mathworks.com/matlabcentral/fileexchange/40692-mean-square-displacement-analysis-of-particles-trajectories Matlab toolbox]

			<!--
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Bradley Monk at 09:08, 24 November 2014

2014-11-24T09:08:11Z

← Older revision		Revision as of 02:08, 24 November 2014
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			{{Box\|font=150%\|width=50%\|float=left\|text=12px\|boarder=solid #aaa 1px\|Note to Visitors\|<big>See the page on [[Brownian Motion]] for a more complete description of MSD</big>}}
			{{Clear}}

	MEAN SQUARED DISPLACEMENT EQUATIONS
			===MEAN SQUARED DISPLACEMENT EQUATIONS===

	<big>		<big>
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	~~What the fuck~~!!		No! No! No!
	~~[[File:WTF.gif]]~~

Bradley Monk at 05:20, 10 August 2014

2014-08-10T05:20:40Z

← Older revision		Revision as of 22:20, 9 August 2014
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	}}<!-- END ARTICLE -->		}}<!-- END ARTICLE -->

	[[Category:ReDiClus]]		[[Category:ReDiClus]][[Category:Neurobiology]]

Bradley Monk at 06:44, 12 August 2013

2013-08-12T06:44:22Z

← Older revision		Revision as of 23:44, 11 August 2013
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			MEAN SQUARED DISPLACEMENT EQUATIONS

			<big>
			* Δ = 2d*D
			* k = √(d * D)
			* µⁿ = k * √2 / √π
			* λ = √(2d * D) or (2d * D)<sup>0.5</sup>
			* λᵃ = λ/√2
			* Λᵈ = 1/√10
			</big>

			:* Δ mean squared displacement (MSD)
			:* d dimensions
			:* D diffusion rate
			:* k σ of sampling distribution that yields D (µ=0)
			:* µⁿ mean of half normal distribution with σ = k
			:* λ mean length of the diagonal XY step
			:* λᵃ mean X & Y step size components of λ
			:** 2a² = λ²
			:** λ = √2*a
			:** a = λ/√2
			:** also: λᵃ * .797 = µⁿ
			:* Λᵈ scalar constant multiplied with k, λ, or λᵃ converting D to 0.D
			:** Λᵈ = reciprocal of √10
			:** 1.0/√10 = .31623 = 0.D
			:** 0.1/√10 = .031623 = 0.0D


			;* FOR D = 0.5
			* Δ = 2.0
			* k = 1.0
			* µⁿ = 0.7979
			* λ = 1.4142
			* λᵃ = 1.0

			;* FOR D = 5.0
			* Δ = 20 (/10 = 2)
			* k = 3.1623 (* .31623 = 1.0)
			* µⁿ = 2.5231
			* λ = 4.4721 (* .31623 = 1.4142)
			* λᵃ = 3.1623 (* .31623 = 1.0)


			;* FOR D = 0.3
			* Δ = 1.2
			* k = 0.7746
			* µⁿ = 0.6180
			* λ = 1.0954
			* λᵃ = 0.7746

			;* FOR D = 3.0
			* Δ = 12 (/10 = 1.2)
			* k = 2.4495 (* .31623 = .7746)
			* µⁿ = 1.9544
			* λ = 3.4641 (* .31623 = 1.0955)
			* λᵃ = 2.4495 (* .31623 = 0.7746)



	==MSD Explained==		==MSD Explained==

Bradley Monk: /* MSD Explained */

2013-08-10T06:06:53Z

MSD Explained

Show changes

Bradley Monk at 02:28, 9 August 2013

2013-08-09T02:28:29Z

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 <big><code>λ = k * √2 / √π</code></big>
 * <code>mS = k*sqrt(2)/sqrt(pi)</code>
 ==Notes==

Bradley Monk: /* MSD Explained */

2013-08-09T00:48:25Z

MSD Explained

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	[http://bradleymonk.com/matlab/msd/MSDTuto.html]		[http://bradleymonk.com/matlab/msd/MSDTuto.html]
	}}		}}

			[[File:Brownian.gif\|thumb\|500px]]

	Albert Einstein in a study of Brownian motion and he showed that the mean square of the distance traveled by particle following a random walk is proportional to the time elapsed. This relationship can be written as:		Albert Einstein in a study of Brownian motion and he showed that the mean square of the distance traveled by particle following a random walk is proportional to the time elapsed. This relationship can be written as:

Bradley Monk: /* MSD Explained */

2013-08-09T00:43:37Z

MSD Explained

@@ Line 4: / Line 4: @@
 In statistical mechanics, the mean squared displacement (MSD or average squared displacement) is the most common measure of the spatial extent of random motion; one can think of MSD as the amount of the system "explored" by a random walker.
 The [[Normal Distribution|Probability Density Function]] (PDF) for a particle in one dimension is found by solving the one-dimensional Diffusion equation. (This equation states that the position probability density diffuses out over time - this is the method used by Einstein to describe a Brownian particle.
@@ Line 10: / Line 12: @@
 <code><big>r²{{=}}4•D•t</big></code>
 *r² MSD
-*D Diffusion Coefficient (diffusion rate)
+*d dimensions
-*t Time step
+*D diffusion coefficient (diffusion rate)
 ==Notes==

Bradley Monk at 23:47, 8 August 2013

2013-08-08T23:47:59Z

← Older revision		Revision as of 16:47, 8 August 2013
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	The [[Normal Distribution\|Probability Density Function]] (PDF) for a particle in one dimension is found by solving the one-dimensional Diffusion equation. (This equation states that the position probability density diffuses out over time - this is the method used by Einstein to describe a Brownian particle.		The [[Normal Distribution\|Probability Density Function]] (PDF) for a particle in one dimension is found by solving the one-dimensional Diffusion equation. (This equation states that the position probability density diffuses out over time - this is the method used by Einstein to describe a Brownian particle.


			As a particle travels, the molecule is jostled by collisions with other molecules which prevent it from following a straight line. If the path is examined in close detail, it will be seen to be a good approximation to a random walk. Mathematically, a random walk is a series of steps, one after another, where each step is taken in a completely random direction from the one before. This kind of path was famously analyzed by Albert Einstein in a study of Brownian motion and he showed that the mean square of the distance traveled by particle following a random walk is proportional to the time elapsed. In two dimensions this relationship can be written as:

			<code><big>r²{{=}}4•D•t</big></code>

			*r² MSD
			*D Diffusion Coefficient (diffusion rate)
			*t Time step


	==Notes==		==Notes==

Bradley Monk: /* MSD Explained */

2013-08-08T23:33:51Z

MSD Explained

← Older revision		Revision as of 16:33, 8 August 2013
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	==MSD Explained==		==MSD Explained==

	xx		In statistical mechanics, the mean squared displacement (MSD or average squared displacement) is the most common measure of the spatial extent of random motion; one can think of MSD as the amount of the system "explored" by a random walker.



			The [[Normal Distribution\|Probability Density Function]] (PDF) for a particle in one dimension is found by solving the one-dimensional Diffusion equation. (This equation states that the position probability density diffuses out over time - this is the method used by Einstein to describe a Brownian particle.

	==Notes==		==Notes==