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I'm looking for a general analytical solution to a particular ODE that comes up in neuroscience a lot. My feeling is that such a solution can't be obtained, otherwise someone would have presented it by now, but I don't have a good understanding of why it is so hard to solve.

Let ##\dot{V} = \frac{dV}{dt}## and ##I(t)## be some function (if it helps you to find a solution, you could assume certain conditions on ##I(t)##, but I'm trying to find the solution for the general case). The model is defined by the following two equations: 1) ## \dot{V} = - V + I(t) ## while ##V(t) < 1##, and 2) if ##V(t) = 1## then ## \lim_{\epsilon\rightarrow 0} V(t + |\epsilon|) = 0 ##

Let ##\{ t^{(1)}, t^{(2)}, ...\} = \{ t: V(t) = 1\} ##. The key problem is to identify this set.

Some

The equation is piecewise linear. We can solve for any interval ##\left[t_0,t\right] \subset \left(t^{(i-1)}, t^{(i)}\right)## 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' ##. So the times ##t^{(i)}## could be identified by solving the equation ## 1 = \int_{t^{(i-1)}}^{t^{(i)}} e^{-t'} I(t^{(i)}-t') dt' ## - I've tried Taylor expanding ##I(t)## which allows the integral to be solved as a series but then I got stuck.

We can also compress the two usual rules of the model into a single equation ## \dot{V} = - V + I(t) - \sum_i \delta(t-t^{(i)}) ##, then we can identify the general solution ##V(t) = e^{-t}V(0) - \sum_i e^{-(t-t^{(i)})} \Theta(t-t^{(i)}) + \int_0^t e^{-t'}I(t-t')dt'## where ##\Theta(x)## is the Heaviside step function, but of course we don't know the ##t^{(i)}##s. Equivalently, we can write as ## \dot{V} = - V + I(t) - |\dot{V}|\delta(V-1) ## (I believe this is correct, I used a rule for the composition of a function with a delta function). Then because ##|\dot{V}| \ge 0 ## at the point ##V=1## we have ##|\dot{V}| = \dot{V}## and so ##(1+\delta(V-1)) \dot{V} = -V+I(t)##. This equation is nice and compact, but looks quite tricky to solve.

I'd be curious to see if anyone can provide some further insights!

*The equations as they are usually given:*Let ##\dot{V} = \frac{dV}{dt}## and ##I(t)## be some function (if it helps you to find a solution, you could assume certain conditions on ##I(t)##, but I'm trying to find the solution for the general case). The model is defined by the following two equations: 1) ## \dot{V} = - V + I(t) ## while ##V(t) < 1##, and 2) 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\} ##. The key problem is to identify this set.

Some

*attempts at a solution:*The equation is piecewise linear. We can solve for any interval ##\left[t_0,t\right] \subset \left(t^{(i-1)}, t^{(i)}\right)## 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' ##. So the times ##t^{(i)}## could be identified by solving the equation ## 1 = \int_{t^{(i-1)}}^{t^{(i)}} e^{-t'} I(t^{(i)}-t') dt' ## - I've tried Taylor expanding ##I(t)## which allows the integral to be solved as a series but then I got stuck.

We can also compress the two usual rules of the model into a single equation ## \dot{V} = - V + I(t) - \sum_i \delta(t-t^{(i)}) ##, then we can identify the general solution ##V(t) = e^{-t}V(0) - \sum_i e^{-(t-t^{(i)})} \Theta(t-t^{(i)}) + \int_0^t e^{-t'}I(t-t')dt'## where ##\Theta(x)## is the Heaviside step function, but of course we don't know the ##t^{(i)}##s. Equivalently, we can write as ## \dot{V} = - V + I(t) - |\dot{V}|\delta(V-1) ## (I believe this is correct, I used a rule for the composition of a function with a delta function). Then because ##|\dot{V}| \ge 0 ## at the point ##V=1## we have ##|\dot{V}| = \dot{V}## and so ##(1+\delta(V-1)) \dot{V} = -V+I(t)##. This equation is nice and compact, but looks quite tricky to solve.

I'd be curious to see if anyone can provide some further insights!

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