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madness
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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.
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!
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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