I LIF Neuron Equation Solution for arbitrary time-dependent current (Neural Dynamics)

gigorina
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In the book Neural Dynamics: https://neuronaldynamics.epfl.ch/online/Ch1.S3.html

There is a solution to the following differential equation (LIF Neuron) for arbitrary time-dependent current. I was trying to figure out the steps the author took to get to the solution.

Original Equation:
Screenshot 2024-01-07 at 21.49.10.png

Solution:
Screenshot 2024-01-07 at 21.49.02.png
 
Thread 'Direction Fields and Isoclines'

Thread 'Direction Fields and Isoclines'

I sketched the isoclines for $$ m=-1,0,1,2 $$. Since both $$ \frac{dy}{dx} $$ and $$ D_{y} \frac{dy}{dx} $$ are continuous on the square region R defined by $$ -4\leq x \leq 4, -4 \leq y \leq 4 $$ the existence and uniqueness theorem guarantees that if we pick a point in the interior that lies on an isocline there will be a unique differentiable function (solution) passing through that point. I understand that a solution exists but I unsure how to actually sketch it. For example, consider a...
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