Recent content by Lucid Dreamer

Yes and how does sticking an electrode within the neuron achieve this. I understand the concept of depolarization, but I am more interested in the particular details leading to the state of depolarization. The neuron contains a salt solution of sodium ions, potassium ions, calcium ions...

Yeah, it's usually connected to a DC source or battery. But my questions has more to do with the actual mechanism of how electrons in the electrodes can be used to positively or negatively polarize a salt solution

Yeah, the only way I can think of right now is that it creates a local depolarization. So if the electrode were charged positive, it would attract many negatively charged ions, thus leaving the surroundings positively polarized. But I have a feeling that's wrong. I know that electrodes are...

Hello, I'm not sure if this is the right place for this question, but I am trying to understand how placing electrodes in the brain elicits action potentials. In a neuron (brain cell), there exists a solution of sodium and potassium ions (among other things). When the brain wants to...

Hello, I'm not sure if this is the right place for this question, but I am trying to understand how placing electrodes in the brain elicits action potentials. In a neuron (brain cell), there exists a solution of sodium and potassium ions (among other things). When the brain wants to produce an...

Hi Guys, I am just starting readings on machine learning and came across ways that the error can be used to learn the target function. The way I understand it, Error: e = f(\vec{x}) - y* Loss: L(\vec{x}) = \frac{( f(\vec{x}) - y* )^2}{2} Empirical Risk: R(f) = \sum_{i=o}^{m} \frac{(...

I am trying to use Sow/Reap in Mathematica to replace Append in my code. The problem is that I need to define the tags within my code, but doing so doesn't reap the desired results. A simplified version of my code is Clear[tag]; Reap[tag = {a,b}; Sow[1, a];Sow[2,b];, tag] However, if I...

Hello, I am looking at a derivation that involves (note x is a column vector) \frac {d(\vec{x}^T\vec{x})} {d\vec{x}} = \vec{x}^{T} So I convert to summation notation and evaluate as follows \sum_{i,j} \frac {d(x_{i}x^{i})} {dx^{j}} \sum_{i,j} \frac {dx_{i}} {dx^{j}} x^{i} + \sum_{i,j}...

Hello, I was looking at ways I could optimize a simulation I wrote in Mathematica and came across the command compile. From what I understand, it defines a function that expects a numerical value for the variable. But I don't understand how this would be any different from just ordinarily...

I came across this integral of a vector valued function. \int \mathbf A(t) \vec{w(t)} dt = \int \mathbf B(t) . I want to isolate \vec{w(t)} and so I multiply by \left (\int \mathbf A(t) dt \right)^{-1} on both sides. \left (\int \mathbf A(t) dt \right)^{-1} \int \mathbf A(t) \vec{w(t)} dt...

f: \mathbb{R}^m \rightarrow \mathbb{R}^n is represented by a n \times m matrix. I don't see how \frac{df}{d\vec{x}} is also represented by a n \times m matrix. Would you be able to provide a reference text for vector calculus that also does a fair treatment of matricies?

I'm not sure if this is right, but here's what I think. Suppose \frac{df}{d\vec{x}}: \mathbb{R}^m \rightarrow \mathbb{R}^n Then \frac{df}{d\vec{x}} is an n \times m matrix. Let \vec{x} \epsilon \mathbb{R}^m so that \vec{x} is a m \times 1 column vector. In the special case...

I am trying to understand summation notation and there are a few inconsistencies in my head that I would like to clear up. Suppose C is an m*n matrix and \vec{x} is a 1*m row vector. Then, \vec{x}C = \sum_{i} x_{i}C_{ij} = \sum_{i} C_{ij}x_{i} = \sum_{i} {C_{ji}}^Tx_{i} = C^T \vec{x}...

Yeah. that does make sense, thanks. But does anybody know why my original line of thought is wrong?

I don't see how it is a scalar quantity. We proved that for \vec{y}=A\vec{x}, \frac {d\vec{y}}{d\vec{x}}=A using the definition \frac {d\vec{y}}{d\vec{x}}=[(\frac {d}{d\vec{x}})^T (\vec{y}^T)]^T . This yields a matrix for \frac {d\vec{y}}{d\vec{x}} and does as well for \frac...