A few years ago, Jo Etzel and I got into a brief but useful investigation of the effects of apparent head motion in fMRI data collected with SMS-EPI. The shorter TR (and smaller voxels) afforded by SMS-EPI generated a spiky appearance in the six motion parameters (three translations, three rotations) produced by a rigid body realignment algorithm for motion correction, such as MCFLIRT in FSL. The apparent head motion is caused by magnetic susceptibility variations of the subject's chest as he/she breathes, leading to a change in the magnetic field across the head which, in turn, adds a ..read more

  Most of the biology we need to learn can be treated orthogonal to the mathematics, whereas the mathematics underlies all the physics and engineering to come. As a change of pace, then, I'm going to start covering some of the biology so I can jump back and forth between two separate tracks. One track will involve Mathematics, then Physics, then Engineering, the other will be Cell Biology, Anatomy, Physiology and then Biochemistry.   Let's begin with a simple overview of cell structure:  https://www.youtube.com/watch?v=0xe1s65IH0w The owner prohibits embedding this video in othe ..read more

 A new podcast on YouTube We all know the best science at a conference happens either during the coffee breaks or in the pub afterwards. This being the case, practiCal fMRI and a guest sit down for coffee (or something stronger) to discuss some aspect of functional neuroimaging in what we hope is an illuminating, honest fashion. It's not a formal presentation. It's not even vaguely polished. It’s simply a frank, open discussion like you might overhear during a conference coffee break. In the inaugural Coffee Break, I sit down with Ravi Menon to discuss two recent papers refuting the exis ..read more

  A13. Eigenvectors and Eigenvalues Let's end this section on linear algebra with a brief exploration of eigenvectors and their eigenvalues. An eigenvector is simply one which is unchanged by a linear transformation except to be scaled by some constant. The constant factor (scalar) by which the eigenvector is scaled is called its eigenvalue. If the eigenvalue is negative then the direction of the vector is reversed as well as scaled.  Curious about the terminology? Eigen means "proper" or "characteristic" in German. So if you're struggling to understand or remember what eigenvectors ..read more

  With some understanding of basic matrix manipulations, we're ready to begin using matrices to solve systems of linear equations. In this post, you'll learn a few standard tools for solving small systems - system defined by a small number of equations - by hand. Naturally, larger systems as found in fMRI will use computers to solve the equations, but you should understand what's going on when you push the buttons. A11. Elementary row operations and elimination   This is just your standard algebraic manipulation to solve multiple simultaneous equations, e.g. dividing both sides of a ..read more

  Before getting back to the lectures from 3Blue1Brown, try this part review, part preview: Now let's get back into the meaning with a little more detail.   A9. The dot (or scalar) product  The dot product is a way to estimate how much two vectors interact in a common dimension. If the vectors are orthogonal to each other, they don't interact in a common dimension so their dot product is zero. This is like asking how much north-south movement is involved in an east-west heading: none. But if two vectors are perfectly parallel then this is equivalent to the two vectors lying o ..read more

  Now we start to think about transformations between dimensions, e.g. taking a 2D vector into a 3D space. Non-square matrices come up frequently in engineering and research applications, including fMRI analysis, so you'll want a good understanding of their meaning.     A8. Non-square matrices Let's look at a simple physical interpretation of changing the number of dimensions. We previously saw how to invert a square matrix. But how do we invert a non-square matrix?   ________________ ..read more

Continuing the series on linear algebra using the lectures from 3Blue1Brown, we are getting into some of the operations that will become mainstays of fMRI processing later on. It's entirely possible to do the processing steps in rote fashion as an fMRI practitioner, but understanding the foundations should help you recognize the limits of different approaches. 4. Matrix multiplication as composition In this video we see how to treat more than one transformation on a space, and how the order of transformations is important.   Q: While brains come in all shapes and sizes, we often seek ..read more

  What is linear algebra? To get us going, I'm going to use the excellent lecture series by 3Blue1Brown and do my best to add some MRI-related questions after each video. Hopefully the connections won't be too cryptic. Don't worry if you can't answer my questions. It's more important that you understand the lectures. No doubt you'll find other material on YouTube and web pages to clarify things. Let's start with a couple of definitions. While you'll find many examples online, for our purposes we can assume that a linear system is one where the size of the output or outputs scales in propo ..read more

  Make coffee, fire up YouTube, click, watch, go about your day. Not so fast! To actually learn the material you'll see, you will need a minimum of the lecture itself, some sort of reading around the lecture (which could be reviewing a transcript or supporting documents), and then answer some questions on that material. So, as this excellent didactic lecture from an anesthesiologist makes clear, questioning is key:     https://www.youtube.com/watch?v=d7IPiNE4_QE   I don't have banks of questions ready to pepper you with at the end of each video, I'm afraid, although I will ..read more