A community for students. Sign up today!
Here's the question you clicked on:
 0 viewing
 one year 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
 one year 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

This Question is Open

amnsbr
 one year 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
 one year 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
 one year 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
 one year 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
 one year 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
 one year 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
 one year 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
Ask your own question
Ask a QuestionFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.