The Human Brain & Graph Theory

Transcription

1 The Human Brain & Graph Theory

2 Graph Theory A graph is a collection of vertices (or points) that are connected by edges (or lines) Edges may overlap Graphs do not need edges Graphs can be directed with an arrow, denoting the direction of the edge "Graph Theory World of Mathematics." World of Mathematics.

3 Leonhard Euler ( ) Founder of Graph Theory Came up with a way to show if a graph can be drawn without lifting the pencil off the paper or drawing the same line twice If a graph has more than 2 vertices with an odd number of edges attached to it, then the above is not possible

4 Bridges of Königsberg There are four land masses, connected by seven bridges Is it possible to cross every bridge only once? "Graph Theory World of Mathematics." World of Mathematics.

5 Bridges of Königsberg The answer is, of course, no When drawn out, it s easy to tell that here are more that two vertices that have an odd number of edges coming off of them "Graph Theory World of Mathematics." World of Mathematics.

6 The Traveling Salesman Problem A salesman is trying to find the shortest possible route, while reaching every house on his stop In order to find this route, he could just try every possible combination until he finds the best one, but this is very impractical There are more than 5 million combinations for just 10 locations As of now there is no definitive answer to this problem due to it s sheer scale, but right now the thinking is to compute the best route from the first couple thousand reasonable possibilities, then work your way down

7 Four Color Theory In 1852, Francis Guthrie proposed that any map of shapes could be filled in with only four colors The only stipulation being that no color can be directly adjacent to itself This problem has only ever been proven by a computer, and even then, the validity of the proof was called into question due to the fact that all the code was written by humans Currently this theory has yet to be explicitly disproven, but has also yet to be explicitly proven

8 Four Color Theory to Graph Theory If each shape is converted into a vertex, and every border an edge, the color map becomes a graph "Graph Theory World of Mathematics." World of Mathematics.

9 Euler s Formula In addition to graph theory, Euler also came up with a Planar Graph, and a formula to define it A Planar Graph is a graph that has no overlapping edges, or open faces Faces + Vertices = Edges + 1 This formula holds true for every Planar Graph The formula works on 3-Dimensional shapes as well, but the + 1 is changed to a + 2 "Graph Theory World of Mathematics." World of Mathematics.

10 Spanning Tree The spanning tree is a type of graph described in graph theory A spanning tree graph is simply a graph that has vertices connected by edges When the edges loop around to connect, it forms what is known as a circuit Two famous circuits are Euler s and Hamilton s Euler s circuit uses every edge of a graph exactly once Similar to his bridge problem Hamilton s circuit uses each vertex exactly once *Edges are often weighted in a Spanning Tree Wikipedia. Wikimedia Foundation

11 Spanning Tree Examples Hamilton circuit Euler circuit "Hamiltonian Cycle." -- from Wolfram MathWorld. An. "Euler Paths and Euler Circuits."

12 Minimum Spanning Tree A spanning tree graph that uses the smallest number of edges possible to connect all vertices This ends up being exactly one less edge than the number of vertices Logically, this means that there are no circuits, as that would produce at least one redundancy "15110 Fall 2011 [Cortina/von Ronne]." Principles of Computing, Fall '11, Programming Assignment 9

13 The Human Brain The human brain contains roughly 86 billion nerve cells (neurons; vertices) The neurons are connected by trillions of connections (synapses; edges) The sheer scale of a graph of the human brain would be enough to say that graphing it in one continuous line would be impossible, but what about reaching every neuron once? "Typical and Atypical Development of Functional Human Brain Networks: Insights from Restingstate FMRI." Frontiers.

14 Spanning Tree & Brain Synapses The most obvious way to graph any function of the brain is to use a spanning tree graph In the brain, electrons are fired along paths to and from neurons In a graph like this, the electron s paths would act as the edges, and the neurons the vertices "Scalable Neuroscience: Plenty of Room at the Bottom for Accelerated Returns." Scalable Neuroscience: Plenty of Room at the Bottom for Accelerated Returns.

15 Blood Flow in the Brain-Spanning Trees Blood flow into the brain begins to branch off from the spinal column at the basilar artery From there, blood is pumped through a series of veins all throughout the brain Walker Chieffe s study put forth the idea of cutting the smallest number of blood vessels as possible, so as to not cause any damage to the brain With a spanning tree graph it becomes possible to map the brain s blood vessels and easily locate the safest locations for incision Purves, Dale. The Blood Supply of the Brain and Spinal Cord. U.S. National Library of Medicine,

16 Blood Flow in the Brain-Spanning Trees Naturally the edges of the graph would be the blood vessels themselves The vertices would be located at each fork in the blood vessel s path A minimum spanning tree would also be useful, highlighting the vessels most essential to the brain s survival If any vessel from the minimum tree is severed, the portion of the brain receiving that blood would die "Scalable Neuroscience: Plenty of Room at the Bottom for Accelerated Returns." Scalable Neuroscience: Plenty of Room at the Bottom for Accelerated Returns.

17 Blood Flow in the Brain-Spanning Trees The thickness and/or the vitality of any given vessel would determine its weight The weight number would tell the surgeon specifically how important any given vessel is and give them a better understanding of which vessels should be avoided over others "Scalable Neuroscience: Plenty of Room at the Bottom for Accelerated Returns." Scalable Neuroscience: Plenty of Room at the Bottom for Accelerated Returns.

18 Other Possible Applications More efficient roadways A spanning tree graph could be used to map out all the current roadways and the distances of the roads that connect everything. The distances could then be reworked to create a faster, more efficient map Faster motherboards Traveling Salesman describes getting from one point to the next in the shortest possible path, this fits in perfectly with motherboards and getting the electricity from one capacitor to the next quickly More efficient wireless networks Radio and cell towers could be placed at the most efficient distances away from one another by possibly using the Traveling Salesman problem again to determine the fastest way for a signal to get from A to B starting at any point on a map

19 Other Possible Applications Internet A spanning tree graph could be used to make a theoretical map of the internet by using all the webpages as vertices and having edges represented by links going back and forth between them Social media closeness This is really just a cool little statistic. On average, everyone on facebook is about six people away from anyone else on the planet. This was found by applying the spanning tree graph to a simulated network of facebook users

20 Information Slide expressions Student s Name: Jacob Lauer and Abigail Fikter expressions Student s School: Kenston High School Title of Submission: The Brain and Graph Theory Name of expressions Teacher: Mr. Koltas Science Intern s Name and Research Project Title: Walker Chieffe--A Model for Neurosurgical Exposure to Minimize Wound Healing Time and Infection expressionist Statement: Our project outlines the ability to map the human brain s blood flow and constructing a network of its synapses using graph theory. We defined graph theory using multiple examples, breaking it down into its most basic forms.

21 Works Cited "15110 Fall 2011 [Cortina/von Ronne]." Principles of Computing, Fall '11, Programming Assignment 9. N.p., n.d. Web. 17 Nov An. "Euler Paths and Euler Circuits." (n.d.): n. pag. Web. "At UPS, the Algorithm Is the Driver." WSJ. N.p., n.d. Web. 17 Nov "Graph Theory World of Mathematics." World of Mathematics. N.p., n.d. Web. 17 Nov "Hamiltonian Cycle." -- from Wolfram MathWorld. N.p., n.d. Web. 17 Nov "Minimum Spanning Trees." Discrete Mathematics and Its Applications Spanning Trees and Optimization Problems (2004): Web.

22 Works Cited Continued Produce, Difficult To, Georges Gonthier Is A Senior Researcher At Microsoft, and Research Cambridge. Formal Proof The Four- Color Theorem (n.d.): n. pag. Web. Purves, Dale. The Blood Supply of the Brain and Spinal Cord. U.S. National Library of Medicine, n. d. Web. 17 Nov "Scalable Neuroscience: Plenty of Room at the Bottom for Accelerated Returns." Scalable Neuroscience: Plenty of Room at the Bottom for Accelerated Returns. N.p., n.d. Web. 17 Nov "Typical and Atypical Development of Functional Human Brain Networks: Insights from Restingstate FMRI." Frontiers. N.p., n.d. Web. 17 Nov Wikipedia. Wikimedia Foundation, n.d. Web. 17 Nov