### Archives For Neuroscience

Would you accept US\$1,000,000 to solve a maths problem? Apparently, not everyone would say yes. A new prize of this amount was recently announced, in an attempt to prove Beal’s Conjecture. Originally offered with a prize of US\$5,000 in 1997, Beal’s conjecture remains unsolved. Today, the Beal Prize has been increased to one million dollars, according to an announcement from the American Mathematical Society.

But what is Beal’s Conjecture? Let’s instead start with something more well known, Fermat’s Last Theorem. Pythagoras originally proposed a formula for the right-angle triangle, where a^2 + b^2 = c^2. This equation has an infinite number of natural number, or positive integer, solutions. However, Fermat claimed that any system with integer exponents greater than 2 (as in Pythagoras’ Theorem) has no integer solutions in a, b, and c. Fermat was kind enough to solve his conjecture for an integer exponent of 4, but he left the rest unsolved. In 1995, Sir Andrew Wiles released a (then-flawed) solution to Fermat’s Conjecture, which included over 100 pages of work over the course of seven years. His story, and the story of the theorem, is a fantastic one, and I recommend reading more on it.

In 1993, two years prior to the solution of Fermat’s Conjecture and five years into Wiles’ quarantine, Andrew Beal proposed another conjecture. It is an extension of the aforementioned theorem. He claimed that the system a^x + b^y = c^z with a, b, c, x, y, and z being positive integers may only have an integer solution for x, y, z > 2 if a, b, and c have a common factor. As mentioned above, he promised US\$5,000 to one who could provide a proof or counterexample of his conjecture. Put less abstractly, if we say that a=7, b=7, x=6, and y=7, then we have 7^6 + 7^7 = 941,192. Solving this, we have 7^6 + 7^7 = 98^3 = 941,192. Note that x, y, and z are all integers greater than 2. Thus, Beal would claim that a, b, and c must have a common factor. In this case they do, considering that 98 is divisible by 7 (or 98/7 = 14). There are many (possibly infinite) examples like this, but we still need a proof or counterexample of the conjecture. To date, it remains unsolved, and a solution will be reward with one million dollars.

In addition to the Beal Prize, the Clay Mathematics Institute offers US\$1,000,000 for a solution to any of seven listed problems. As of this post, only one has been solved, and no money has yet been accepted. These Millennium Prize Problems continue to baffle mathematicians. It is fascinating to consider that there are so many open problems in mathematics, including those integral to number theory, such as Hilbert’s eighth problem.

Not everyone reading this post is a mathematician. Many of us, including myself, think visually. We like pictures, and we like problems we can solve, or at least ones that currently have solutions. So I’ll introduce one! I’m going to turn this post now to a classic problem that began to lay the foundations for graph theory and topologyFor a related post in topology, I recommend my post, Diving through Dimensions. Some of you may be aware of this problem, and I hope I do it justice. Let us begin by traveling to the capital of Prussia, Königsberg. The city (now Kaliningrad) was set on opposite sides of the Pregel River. On this river sat two islands. Thus, we have four land masses. Connecting these regions were seven bridges, as laid out below in red:

The people of Königsberg posed a question: Is it possible to traverse the city, crossing all bridges once and only once? Let us assume that one cannot dig under the ground, fly through the sky, cross the water by river, or use teleportation technology. One may only access each land mass by crossing bridges. Additionally, one may begin and end at any point. The only requirement is that each bridge must be crossed and that it cannot be crossed more than once.

Leonhard Euler proposed a solution to this problem in 1735. He began by first reducing each land mass to a point. The only relevant information is found in the bridges and where they connect, with the areas of land masses being irrelevant. This combination of nodes (points) and edges (lines) is commonly used in graph theory. Euler noticed that when when reaches a node by one bridge, one must leave that node by another bridge, resulting in an even number of bridges during a full pass-through over a node. Thus, all nodes that are passed over (that is, they are not the beginning nor the end) must have an even number of edges. There may be at most two nodes with an odd number of edges, and these nodes must serve as the beginning and/or the end of our journey.

Now, take a look at our bridges as Euler may have drawn them:

In this case, we see that the top and bottom nodes on the left each have three bridges, the rightmost node has three bridges, and the middle node on the left has five bridges. In other words, all four nodes have an odd number of edges. This violates our requirement that no more than two nodes may have an odd number of edges. As a result, Euler demonstrated that there is no way to traverse each of the Prussian bridges once and only once. This requirement can be applied to any drawing similar to the one above. I recommend trying it out and testing Euler’s proposal. It is quite rewarding.

If you are really interested, take a gander at the current layout on Google Maps:

It seems that the people of Kaliningrad demolished two of our bridges! The Königsberg bridge problem now has a solution. A part of me likes to think that the bridges were demolished for no other reason than to provide a solution!

As mentioned above, Euler’s solution laid the framework for what we call graph theory. Graph theory, or the study of graphs like the one shown above, has myriad applications. It is used in computer science to model networking. Linguists take advantage of graphs to examine semantic structure. Chemists represent atoms and bonds using graphs. Social network analysis is described in the same terminology, using the same theory, where each person or group may be a node. In biology, we use it to model migration or the spread of disease. More relevant to my work, weighted graphs are used in network analysis, and computational neuroscientists may recognize graphs like the one above when modeling neural networks.

What we thus see is something fantastic. Abstract open problems like the one Euler solved and those proposed by Beal and the Clay Mathematics Institute provide foundational tools that can (and often do) advance our knowledge in numerous fields. Euler’s work propelled us into graph theory. A solution to the Navier-Stokes open problem will advance our understanding of fluid flow. Even if the abstract does not become practical, the journey is delightful.

For the first part of this series and to learn a bit more about 3D reconstruction of computed tomography (CT) slices, check out NEURODOME I: Introduction and CT Reconstruction. Our Kickstarter is now LIVE!

“As I stand out here in the wonders of the unknown at Hadley, I sort of realize there’s a fundamental truth to our nature. Man must explore. And this is exploration at its greatest.” – Cdr. David Scott, Apollo 15

It is official. Our Kickstarter for NEURODOME has launched. I have already described a bit about my role in the project and described CT reconstruction. Future posts will delve into fMRI imaging and reconstruction, along with additional imaging modalities and perhaps a taste of medical imaging in space. You might be surprised at the number of challenges astronauts had to take while aboard rockets, shuttles, and the ISS. All of this will be part of the NEURODOME series.

With our launch, we hope to raise enough funds to develop a planetarium show that illustrates our desire to explore. To do so, real data will be used in the fly-throughs. Our first video, The Journey Inward, provides a basic preview of what you might expect.

Those who work closely with me know that I am part of a project entitled Neurodome (www.neurodome.org). The concept is simple. To better understand our motivations to explore the unknown (e.g. space), we must look within. To accomplish this, we are creating a planetarium show using real data: maps of the known universe, clinical imaging (fMRI, CT), and fluorescent imaging of brain slices, to name a few. From our web site:

Humans are inherently curious. We have journeyed into space and have traveled to the bottom of our deepest oceans. Yet no one has ever explained why man or woman “must explore.” What is it that sparks our curiosity? Are we hard-wired for exploration? Somewhere in the brain’s compact architecture, we make the decision to go forth and explore.

The NEURODOME project is a planetarium show that tries to answer these questions. Combining planetarium production technology with high-resolution brain imaging techniques, we will create dome-format animations that examine what it is about the brain that drives us to journey into the unknown. Seamlessly interspersed with space scenes, the NEURODOME planetarium show will zoom through the brain in the context of cutting edge of astronomical research. This project will present our most current portraits of neurons, networks, and regions of the brain responsible for exploratory behavior.

To embark upon this journey, we are launching a Kickstarter campaign next week, which you will be able to find here. Two trailers and a pitch video showcase our techniques and our vision. For now, you can see our “theatrical” trailer, which combines some real data with CGI, below. Note that the other trailer I plan to embed in a later post will include nothing but real data.

I am both a software developer and curator of clinical data in this project. This involves acquisition of high-resolution fMRI and CT data, followed by rendering of these slices into three-dimension objects that can be used for our dome-format presentation. How do we do this? I will begin by explaining how I reconstructed a human head from sagittal sections of CT data. In a later post, I will describe how we can take fMRI data of the brain and reconstruct three-dimensional models by a process known as segmentation.

How do we take a stack of images like this:

(click to open)

and convert it into three-dimensional objects like these:

These renders allow us to transition, in a large-scale animation, from imagery outside the brain to fMRI segmentation data and finally to high-resolution brain imaging. The objects are beneficial in that they can be imported into most animation suites. To render stacks of images, I created a simple script in MATLAB. A stack of 131 saggital sections, each with 512×512 resolution, was first imported. After importing the data, the script then defines a rectangular grid in 3D space. The pixel data from each of these CT slices is interpolated and mapped to the 3D mesh. For example, we can take the 512×512 two-dimensional slice and interpolate it so that the new resolution is 2048×2048. Note that this does not create new data, but instead creates a smoother gradient between adjacent points. If there is interest, I can expand upon the process of three-dimensional interpolation in a later post.

I then take this high-resolution structure mapped to the previously-defined three-dimensional grid and create an isosurface. The function takes volume data in three dimensions and a certain isovalue. An isovalue in this case corresponds to a particular intensity of our CT data. The script searches for all of these isovalues in three dimensions and connects the dots. In doing so, a surface in which all of the points have the same intensity is mapped. These vertices and faces are sent to a “structure” in our workspace. The script finally converts this structure to a three-dimensional “object” file (.obj). Such object files can then be used in any animation suites, such as Maya or Blender. Using Blender, I was able to create the animations shown above. Different isovalues correspond to different parts of the image. For example, a value/index of ~1000 corresponds to skin in the CT data, and a value/index of ~2400 corresponds to the bone intensity. Thus, we can take a stack of two-dimensional images and create beautiful structures for exploration in our planetarium show.

In summary the process is as follows:

1. A stack of saggital CT images is imported into MATLAB.
2. The script interpolates these images to increase the image (but not data) resolution.
3. A volume is created from the stack of high-resolution images.
4. The volume is “sliced” into a surface corresponding to just one intensity level.
5. This surface is exported to animations suites for your viewing pleasure.

This series will continue in later posts. I plan to describe more details of the project, and I will delve into particulars of each post if there is interest. You can find more information on this project at http://www.neurodome.org.

Every year, I read an article written in 1972 by P.W. Anderson, More is Different. This exercise provides two functions. On one hand, it is a kind of ritualistic experience through which I can reflect on the past year. On the other, it allows me to revisit the paper with an expanded knowledge base. The paper revisits an age-old discussion in science: Are less fundamental fields of research simply applied versions of their counterparts?

In 1965, V.F. Weisskopf, in an essay entitled In Defence of High Energy Physics, delineated two types of fields. One, which he called intensive, sought after fundamental laws. The other, extensive, used these fundamental laws to explain various phenomena. In other words, extensive research is simply applied intensive research. In many ways, various fields are closer in proximity to fundamental laws than others. Most of neuroscience is more fundamental than psychology, in that it is reduced to smaller scales and focuses on simpler parts of a more complex system. This psychology, however, is closer to its fundamental laws than the social sciences. Again, where psychology focuses on the workings of individual and small-group dynamics, social sciences use many of these laws to explain their work. Molecular biology is seen as more fundamental than cell biology. Chemistry is less fundamental than many-body physics, which is less fundamental than particle physics. The argument by Weisskopf seems to be in favor when discussing fields in terms of their size scale.

Changes between size scale, however, leads us to a discussion of symmetry. Anderson begins his discussion with the example of ammonia. This molecule forms a pyramid, with one nitrogen at its ‘peak’ and three hydrogrens forming the base. A problem arises, however. When discussing a nucleus, we see that there is no dipole moment, or no net direction of charge. However, the negative nitrogen and positive hydrogens form a structure that disobeys this law, or so one might think. It actually turns out that symmetry is preserved through tunneling of the nitrogen, flipping the structure and creating a net dipole moment of zero. Simply put, symmetry is preserved. Weisskopf’s argument continues to hold, even with the scale change.

However, when the molecule becomes very large, such as sugars made by living systems, this inversion no longer occurs, and the symmetry is broken. The fundamental laws applied at the level of the nucleus now no longer hold. Additionally, one can ask: Knowing only what we learned about the symmetry of a nucleus, could we then infer the behavior of ammonia, glucose, crystal lattices, or other complex structures? The fundamental laws, while still applied to the system, do not capture the behavior at this new scale. On top of that, very large systems break the symmetry entirely.

Andersen goes on to discuss a number of other possibilities. In addition to structure, he analyses time dependence, conductivity, and the transfer of information. In particular, consider the crystal that carries information in living systems: DNA. Here, we have a structure that need not be symmetric, and new laws of information transfer arise from this structure and its counterparts that would not be predicted from particle physics or many-body physics alone.  Considering the DNA example, we must then ask ourselves: Can questions in social sciences, psychology, and biology be explained by DNA alone? We are often tempted, and rightfully so, to reduce these complex systems to changes in our DNA structure. This much is true. However, can we predictably rebuild the same social psychology from such a simple code? With the addition of epigenetics, we are trying to do so, but I argue that we are not yet there. In fact, I argue that we never will be there.

The message here is that larger, more complex systems, while built upon the fundamental laws of their reduced counterparts, display unique phenomena of their own. We can continue to reduce complex systems to smaller scales. In doing so, complexities and phenomena of the larger systems are lost. Starting only with knowledge from the fundamental laws, can we predict all of the phenomena of the larger scales, without prior knowledge of those phenomena? Probably not. This is another kind of broken symmetry, where traverse fields in an intensive direction will lead one in the formulation of a fundamental laws, but traversing in the extensive direction from those fundamental laws will lead to more and more possibilities that need not be the one from where we started. As scales grow, so too does the probability of broken symmetry.

Thus, when stating that “X is just applied Y,” remember ammonia.