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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

Is there any normative that regulates the use of silver nanoparticles?
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research.net
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Introduction about quantum dots in nanotechnology
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
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Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
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Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
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CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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s. Reply
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Graphene has a hexagonal structure
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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