- #1

- 815

- 70

Does anyone know if it is possible to develop a fully analytical solution for a leaky integrate and fire neuron driven by arbitrary time-varying current? Here's what I have so far (setting as many possible constants to 0 and 1):

## \dot{V} = - V + I(t) ## and if ##V(t) = 1## then ## \lim_{\epsilon\rightarrow 0} V(t + |\epsilon|) = 0 ##

Let ##\{ t^{(1)}, t^{(2)}, ...\} = \{ t: V(t) = 1\} ##

We can solve when not spiking as ## V(t) = V(t_0) e^{-t} + \int_{t_0}^t e^{-t'} I(t-t') dt' ##.

So the solution in general for ##t^{(i-1)} < t < t^{(i)} ## is ## V(t) = \int_{t^{(i-1)}}^{t} e^{-t'} I(t-t') dt' ##.

We can also write the general equation as ## \dot{V} = - V + I(t) - \sum_i \delta(t-t^{(i)}) ##.

It looks almost possible to solve these equations for the spike times ##t^{(i)}## but I'm not sure whether it can be done. We can write ## 1 = \int_{t^{(i-1)}}^{t^{(i)}} e^{-t'} I(t-t') dt' ##, but I don't think spike times can be found from this analytically for arbitrary ##I(t)##. I guess what I'd really like is way to plug something for ##t^{(i)}## into the equation ## \dot{V} = - V + I(t) - \sum_i \delta(t-t^{(i)}) ## and make things a bit more explicit, even if it ends up being a self-consistent equation.

*The equations:*## \dot{V} = - V + I(t) ## and if ##V(t) = 1## then ## \lim_{\epsilon\rightarrow 0} V(t + |\epsilon|) = 0 ##

*Some further notation:*Let ##\{ t^{(1)}, t^{(2)}, ...\} = \{ t: V(t) = 1\} ##

*My attempt at a solution:*We can solve when not spiking as ## V(t) = V(t_0) e^{-t} + \int_{t_0}^t e^{-t'} I(t-t') dt' ##.

So the solution in general for ##t^{(i-1)} < t < t^{(i)} ## is ## V(t) = \int_{t^{(i-1)}}^{t} e^{-t'} I(t-t') dt' ##.

We can also write the general equation as ## \dot{V} = - V + I(t) - \sum_i \delta(t-t^{(i)}) ##.

It looks almost possible to solve these equations for the spike times ##t^{(i)}## but I'm not sure whether it can be done. We can write ## 1 = \int_{t^{(i-1)}}^{t^{(i)}} e^{-t'} I(t-t') dt' ##, but I don't think spike times can be found from this analytically for arbitrary ##I(t)##. I guess what I'd really like is way to plug something for ##t^{(i)}## into the equation ## \dot{V} = - V + I(t) - \sum_i \delta(t-t^{(i)}) ## and make things a bit more explicit, even if it ends up being a self-consistent equation.

Last edited: