<< Chapter < Page Chapter >> Page >

Computational method

To solve for v ( t ) computationally, we first look at the times with no input spikes ( k I < t < ( k + 1 ) I ). Integrating both sides of equation [link] from t - d t to t and using the trapezoid rule, we find

τ ( v ( t ) - v ( t - d t ) ) = v r d t - v ( t ) + v ( t - d t ) 2 , which can be rearranged as v ( t ) = 2 d t 2 τ + 1 · v r + 2 τ - 1 2 τ + 1 · v ( t - d t ) .

When there is an input spike, we add w i n p to v ( t ) , which is shown in

v i n p ( t ) = v ( t ) + w i n p .

Analytic method

To solve for v ( t ) analytically, we first look at v ( t ) between input spikes. From equation [link] , we get

τ v ' ( t ) = ( v r - v ( t ) ) .

Solving this ordinary differential equation gives us

v ( t ) = v r + c e - t / τ ,

where c is the constant of integration. We know we want v ( 0 ) = v r + w i n p , so c must equal w i n p . Thus, we have

v ( t ) = v r + w i n p e - t / τ , where 0 t < I ,

which simply tells us that after one input spike at t = 0 , w i n p decays so that v ( t ) approaches v r . Consider the following calculations of v ( t ) for up to three input spikes.

At t = I , we have a second input spike, and at I < t < 2 I , we decay the input to find

v ( I t < 2 I ) = v r + w i n p e - t / τ + w i n p e - ( t - I ) / τ .

Finally, at t = 2 I , we have a third input spike and see

v ( 2 I ) = v r + w i n p e - 2 I / τ + w i n p e - I / τ + w i n p .

To determine when the voltage reaches threshold and the cell spikes, we need only examine the peak values of v , which are when t = k I , 0 k n - 1 . Thus, we use the following generalized formula to calculate v ( ( n - 1 ) I ) when there are n total input spikes:

n , v ( ( n - 1 ) I ) = v r + w i n p k = 0 n - 1 e - I / τ k .

[link] shows that in the absence of spikes, the peak voltages approach an asymptote. This asymptote can be calculated by

v = lim n v ( ( n - 1 ) I ) = v r + w i n p k = 0 e - I / τ k = v r + w i n p 1 1 - e - I / τ .

If v < v t h , then the cell will never spike.

Voltage as a function of time as calculated by equation [link] . The peak voltages are denoted by asterisks. Here we set v t h = - 52 m V . ( AnpeakV.m )

Problems and results

Minimum input weight for activity

Computational vs. analytic method

We found the minimum input weight w i n p necessary for the cell to spike at least once as a function of the input time interval I when given a sufficiently long simulation.

Let the interspike interval I and input weights w i n p satisfy 2 I 30 and 2 w i n p 20 .

In the computational method, the Matlab program compW.m calculates v ( t ) according to equations [link] and [link] . In AnalysisW.m , the minimum w i n p is calculated by

w i n p = ( v t h - v r ) ( 1 - e - I / τ ) ,

which was obtained by setting v of equation [link] to v v t h where

v t h v r + w i n p 1 1 - e - I / τ .

[link] shows that as the input time interval increases, greater input weight is necessary for the cell to spike at least once ( AnalysisW.m ). We note on the graph the value of w i n p = 10 . 11 at I = 20 because these two values will be put to use in the next section.

Comparison of w i n p from computation and analysis as a function of I . v ( t ) is calculated by equations [link] and [link] in compW.m . w i n p is calculated by equation [link] in AnalysisW.m . (Plotted in AnalysisW.m )

Number of input spikes versus input weight

Computational vs. analytic method

We determine the minimum number of input spikes necessary for the cell to spike as a function of input weight.

We use I = 20 and consider only the weights that produce at least one spike with sufficient simulation, starting with w i n p = 10 . 2 as shown in [link] . Let n 1 denote the minimum number of input spikes of weight w i n p necessary for v ( t ) to reach v t h . We see that

Questions & Answers

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where is the latest information on a no technology how can I find it
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now

Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'The art of the pfug' conversation and receive update notifications?