8.7 Mathematical modeling of hippocampal spatial memory with place  (Page 5/6)

Computational method

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

When there is an input spike, we add $w_{inp}$ to $v\left(t\right)$ , which is shown in

Analytic method

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

Solving this ordinary differential equation gives us

where $c$ is the constant of integration. We know we want $v\left(0\right)=v_r+w_{inp}$ , so $c$ must equal $w_{inp}$ . Thus, we have

which simply tells us that after one input spike at $t=0$ , $w_{inp}$ decays so that $v\left(t\right)$ approaches $v_r$ . Consider the following calculations of $v\left(t\right)$ for up to three input spikes.

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

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

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

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

If $v_{\infty }<v_{th}$ , then the cell will never spike.

Problems and results

Minimum input weight for activity

Computational vs. analytic method

We found the minimum input weight $w_{inp}$ 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 $w_{inp}$ satisfy $2\le I\le 30$ and $2\le w_{inp}\le 20$ .

In the computational method, the Matlab program compW.m calculates $v\left(t\right)$ according to equations [link] and [link] . In AnalysisW.m , the minimum $w_{inp}$ is calculated by

which was obtained by setting $v_{\infty }$ of equation [link] to $v_{\infty }\ge v_{th}$ where

[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 $w_{inp}=10.11$ at $I=20$ because these two values will be put to use in the next section.

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 $w_{inp}=10.2$ as shown in [link] . Let $n_1$ denote the minimum number of input spikes of weight $w_{inp}$ necessary for $v\left(t\right)$ to reach $v_{th}$ . We see that