<< 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

what is the coefficient of -4×
Mehri Reply
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
I'm 13 and I understand it great
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
how do they get the third part x = (32)5/4
kinnecy Reply
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
is it a question of log
I rally confuse this number And equations too I need exactly help
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
Commplementary angles
Idrissa Reply
im all ears I need to learn
right! what he said ⤴⤴⤴
greetings from Iran
salut. from Algeria
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!

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?