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amnsbr
 2 years ago
How this equation:
I_{s}=\sum_{b=1}^{Nu}w_{b}\int_{\infty }^{t}d\tau K_{s}(t\tau)u_{b}(\tau)
differentiates to this:
\tau_{s}\frac{\mathrm{d} I_{s} }{\mathrm{d} t}=I_{s}+\sum_{b=1}^{Nu}w_{b}u_{b}
given that Ks(t)=exp(tau/t)/tau
amnsbr
 2 years ago
How this equation: I_{s}=\sum_{b=1}^{Nu}w_{b}\int_{\infty }^{t}d\tau K_{s}(t\tau)u_{b}(\tau) differentiates to this: \tau_{s}\frac{\mathrm{d} I_{s} }{\mathrm{d} t}=I_{s}+\sum_{b=1}^{Nu}w_{b}u_{b} given that Ks(t)=exp(tau/t)/tau

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amnsbr
 2 years ago
Best ResponseYou've already chosen the best response.0\[I_{s}=\sum_{b=1}^{Nu}w_{b}\int\limits_{\infty }^{t}d\tau K_{s}(t\tau)u_{b}(\tau)\] \[\tau_{s}\frac{\mathrm{d} I_{s} }{\mathrm{d} t}=I_{s}+\sum_{b=1}^{Nu}w_{b}u_{b}\]

SithsAndGiggles
 2 years ago
Best ResponseYou've already chosen the best response.0Let me just rewrite this question so it's easier to read: \[\large I_s=\sum_{b=1}^\nu w_b\int_{\infty}^t\frac{\exp\left[\frac{\tau}{t\tau}\right]}{\tau}u_b(\tau)~d\tau\] And you're asking how the derivative (with respect to t, apparently) is \[\large \tau_s\frac{dI_s}{dt}=I_s+\sum_{b=1}^\nu w_bu_b\] Am we to assume \(\large u_b\) is the Heaviside step function?

amnsbr
 2 years ago
Best ResponseYou've already chosen the best response.0ub() is convolution of pb() with a boxcar filter and pb() is a sum of dirac functions i don't know what is Heaviside step function, sry!

SithsAndGiggles
 2 years ago
Best ResponseYou've already chosen the best response.0I don't remember much about convolution, and I've never heard of the boxcar filter you mention. Sorry I can't help! By the way, the Heaviside step function is \[u_t(x)=u(xt)=\begin{cases}1&\text{for }x\ge t\\0&\text{for }x<t\end{cases}\] You might know it by some other name.

amnsbr
 2 years ago
Best ResponseYou've already chosen the best response.0u(t) is the firing rate of a neuron and is continous in time,i think you can assume that it's just an arbitrary function of time! About Heaviside,yes it is similar to Delta dirac function btw, boxcar filter is a simple filter which is 1 in [a,b] and zero otherwise

SithsAndGiggles
 2 years ago
Best ResponseYou've already chosen the best response.0@phi, it's the given as exponential function in the integral. \[\large K_s(t)=\frac{\exp\left[\frac{\tau}{t}\right]}{\tau}\]

phi
 2 years ago
Best ResponseYou've already chosen the best response.0Do we know anything about \( w_b(t)\) ? in the meantime, I think we can use this http://mathmistakes.info/facts/CalculusFacts/learn/doi/doi.html
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