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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

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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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