Mathematics and the brain

Summary: The brain is studied through models and through algorithm-dependent medical technology. The neurology of mathematical thought is a vibrant field.

The applications of mathematics to the study and understanding of the brain have been varied and widespread. They include models to predict the start of seizures using dynamical systems; maps of the brain using projective, hyperbolic, or other geometries, as well as graph theory; applications of morphometrics, which is the statistical study of shapes, to schizophrenic brains; and dynamic simulations and visualizations of electrochemical activity in neurons. Other models are used to study how electrical signals propagate along nerve cells and the way in which electrical discharges in nerve cells tend to synchronize and form waves. Medical technology used in brain treatment and studies uses mathematical algorithms; for example, to create and process computer-generated images of brain cells, as well as to measure functions like blood flow, glucose consumption, and electrical activity. Mathematics is also important in modern medical devices that involve nerve fibers within or leading to the brain, such as cochlear implants. How mathematical thought arises in the brain—from arithmetic to abstract thinking—is also of great interest. Mathematics and the Brain was the theme of Mathematics Awareness Month in 2007.

Brain Composition and Structure

Before proceeding with some applications of mathematics in the study of the brain, it is important to have an idea of brain composition and structure. In humans, this complex organ consists of perhaps 100 billion nerve cells (or neurons), with roughly a total of 100 trillion connections between neurons (or synapses). Although some nerve cells do regenerate, and new connections between nerve cells are made, overall these numbers tend to decline after birth. Even with advances in computer processing and storage, the sheer number of neurons and connections hints at the enormous scope of the problem inherent in understanding the brain. By comparison, the nematode Caenorhabditis elegans has 959 cells in the entire organism, 302 being nerve cells, which result in over 5000 connections between neurons. Even for something of this vastly smaller scale, the nematode’s neural connections were initially mapped after more than 10 years of effort by the mid-1980s, and earned a Nobel Prize for Sydney Brenner—who famously called C. elegans “nature’s gift to science.” Those results have since been updated.

A single neuron generally consists of: a main cell body (or soma); many filamentous dendrites, which are where signals from other neurons are usually received; and a single axon, which typically communicates to the dendrites of other neurons. The electrical voltage across the neuron’s cell membrane varies as the concentrations of calcium, sodium, potassium, and chloride ions fluctuate, producing a fluctuating electrical signal. The electrical signal is transferred from one neuron’s axon to another’s dendrite across a gap known as a “synapse.” Some synapses, known as “electrical synapses,” involve a direct channel that connects the two cells’ cytoplasm and allows for very fast electrical transmission. By contrast, chemical synapses involve molecules known as “neurotransmitters,” which mediate signal transmission. The human brain utilizes more than 100 types of neurotransmitters. However, just two of these types arise at the vast majority of synapses; namely, glutamate and gamma aminobutyric acid. In addition to neurons, glial cells serve various support functions for neurons. One important function of special kinds of glial cells—namely, special kinds of oligodendrocytes—is the myelinization of axons. Myelin, a fatty substance, essentially electrically insulates neurons. Because they are pinkish white, and white when stored in formaldehyde, bundles of myelinized axons make up what is known as “white matter” in the brain. On the other hand, “grey matter,” as seen on the surface of the cerebral cortex in a typical brain slice image, comprises of the soma, dendrites, and other kinds of glial cells, such as astrocytes. While C. Elegans has fewer than 60 glial cells, the human brain likely has at least as many glial cells as neurons, although the ratio varies widely in different brain regions.

Applications of Neural Networks

How neurons collectively convey information is also of much interest to researchers. Interestingly, attempts to model so-called “artificial neural networks” have led to highly useful algorithms used in many areas of mathematics, science, and engineering in their own right, having nothing to do with the study of the brain. Computers are often “trained” with data sets using such neural networks to help process data. Neural networks can be found in software used in fields as varied as financial analysis and fraud detection, robotics, handwriting analysis, and voice recognition. As another example, much mathematics is used in processing and analyzing the enormous amount of neuron image data, and neural network algorithms are now being used to help automate that processing to help computers track neural connectivity.

Brain Mapping and Study

Mathematics also has been used to help in producing accurate maps of the cortex of various parts of the brain. The extensive folding in the human brain in the cerebral cortex, which produces peaks or ridges (or gyri) and valleys or furrows (or sulchi) makes it difficult to compare two different brain surfaces. A calculus-based geometry is used to find effective maps. As another example of an application, mathematics is used extensively in devices such as cochlear implants, useful to deaf individuals who still have a functioning auditory nerve. In humans, as many as 30,000 individual nerve cells in the inner ear pass through the auditory nerve to the brain. Different sound frequencies innervate different nerve cells; roughly speaking, lower frequencies innervate nerve cells in the basilar membrane closer to the beginning of the cochlea, as opposed to higher frequencies innervating cells further along. But the precise mapping of which cells are affected by which frequencies follows a logarithmic mathematical pattern, as a function of distance in the cochlea. Using various radio signal technologies, external sounds are transmitted to a receiver in the inner ear, which connects to implanted electrodes for nerve innervation. There, mathematics is used in the computational processing to convert the received frequencies into the appropriate electrical innervations, so that only certain nerve cells are stimulated for certain frequencies.

Examples of mathematics applied to the study of the brain abound in the five-year, National Institutes of Health–funded Human Connectome Project. This project, somewhat analogous to the Human Genome Project, was funded in 2010 for approximately $40 million. Mapping all the connections between neurons in the human brain in a meaningful way is the goal of the Connectome project. One component involves constructing connection data from 1200 individuals, including numerous twins. Developing effective ways to collect the data set, as well as analyze the results, involves several areas of mathematics in crucial ways. First, instruments must be able to create high-resolution images of the brain tissue of living humans in a completely noninvasive way. Next, the enormous image data must yield to automated computer analysis that can determine the actual neural connections within the brain. Finally, the connection data set must be amenable to meaningful analysis by researchers interested in understanding normal brain processing as well as diseases. At each stage, mathematics plays a crucial role.

Brain Imaging Technologies

Magnetic resonance imaging (MRI) is commonly used today for noninvasive imaging of the internal structure of the human body; for example, to help determine if knee surgery or back surgery is warranted. In standard MRIs, a powerful magnetic field changes rapidly, and, by doing so, it manipulates the minute magnetic fields produced by protons in water molecules inside the body—a weak signal can be detected externally from the protons being flipped around by the strong magnetic fields. From these weak and indirect measurements, solving the inverse problem using mathematics related to calculus is used to create what appear as two-dimensional slices through the body. In the case of brain studies, the resolution of standard MRIs is adequate to see tumors but is too crude to see individual neurons or even to effectively track bundles of neurons. Since myelin is a fatty substance, water outside neurons will generally not diffuse into axons; rather, this water will tend to diffuse along the length of axons—the water percolates along axons or white matter.

Diffusion tensor imaging (DTI) uses a variation of a standard MRI to determine the diffusion direction, and hence determine bulk nerve fibers. While DTI can produce high-resolution images of nerve fibers, difficulties arise when fibers cross. Water diffusion in this case can now take multiple paths at the crossing points, and it is thus difficult to track nerve fibers at these crossing points. Diffusion spectrum imaging (DSI) involves more mathematics that determines more precisely how water diffuses and is not limited to thinking that water diffuses in only one direction. Roughly speaking, the mathematics is a mixture of calculus and statistical ideas, and it is interesting that two-dimensional ellipses and three-dimensional ellipsoids play a role in the mathematics of DTIs and DSIs. The resulting images of nerve fibers are visually striking.

Not all techniques for imaging neurons rely on such indirect approaches as conventional MRI, or the MRI-based DSI and DTI. Recall that those techniques are used primarily for imaging nerve fibers, not individual neurons. Techniques for higher-resolution imaging of actual neurons are somewhat direct. Jeff Lichtman and others developed the use of genes encoding three proteins that fluoresce in, essentially, the colors red, blue, and green. Genetically modifying mice with these genes, as well as an enzyme that randomly arranges the genes amongst neurons, allows for mice neurons to appear in one of now approximately 150 colors, creating what Lichtman has termed a “brainbow.” Another approach relies on a genetically modified version of the rabies virus, which ordinarily is well suited for traveling from neuron to neuron on its journey to the brain. By tagging the modified virus with a fluorescent molecule, one obtains bright images of neurons connecting to just one other neuron to further aid in understanding neural connectivity. All in all, exceptionally striking images are displayed in many places including on the Internet. While the imaging of individual neurons is often more direct and makes less use of mathematics than the MRI approaches to imaging nerve fibers, much mathematics is subsequently used in automating the process of tracking neurons and nerve fibers and, ultimately, the connections found might be described and analyzed by an area of mathematics known as “graph theory.”

The approaches to imaging nerve fibers discussed above, such as DTI and DSI, rely on MRI instruments; that is, they rely on indirect methods involving minute variations in very small magnetic fields from protons that are assailed by powerful externally generated magnetic fields. They use indirect information coming from throughout the local environment inside the brain, and mathematics is crucial to inverting the recorded data to recover what is going on at a particular location in the brain. But not all imaging inside the brain focuses just on the connections between individual neurons or bundles of neurons. Other areas of interest include determining which parts of the brain are stimulated at which times by which activities. Here, other inversion processes are used to see what is happening in the brain, including functional MRI (fMRI), positron-emission tomography (PET), and electroencephalograms (EEG). Blood oxygen level dependence (BOLD) uses MRI technology that takes into account the very slight differences in the magnetic fields from water molecules in blood, depending on whether the blood is carrying oxygen. Hemoglobin bound to oxygen is diamagnetic, essentially repelled by a magnetic field. Hemoglobin without bound oxygen is paramagnetic, or attracted to a magnetic field. In either case, it affects the overall magnetic fields from the water molecules. This effect leads to functional MRI (fMRI), which is used to examine oxygenated blood in the brain. The principle is that high neural activity is probably associated with increased blood flow.

PET, another imaging approach, uses an analog of glucose with a radioactive fluorine atom attached. When it decays, it produces a particle known as a positron that is quickly annihilated upon encountering an electron, and two photons stream out in opposite directions. Photomultiplier detectors essentially notice the two photons, and mathematics is used to invert this problem and determine where the annihilation occurred, which presumably is near where the brain was consuming the glucose-like food. A single-photon to PETs, Single Photon Emission Computed Tomography (SPECT), is also utilized.

As the final example of imaging approaches, EEGs focus on using electrical activity recorded on the scalp to see what voltages are created by bulk neurons extending over somewhat larger regions of the brain, as neurons synchronize their electrical signaling. EEGs thus have less spatial resolution than some of the other imaging approaches. Magnetoencephalography (MEG) is a magnetic analog of the EEG in that it is also a noninvasive procedure. Rather than using electrodes attached to a person’s scalp for measurements, as in an EEG, very precise superconducting quantum interference devices (SQUIDs) detect weak magnetic fields directly arising from electrical brain activity. A mathematical inverse process makes the externally obtained magnetic data usable and converts it to internal electrical activity. MEG typically offers greater resolution, so it can localize the electrical activity more precisely, than EEGs.

Much mathematics is used to model the flow of electrical impulses in the brain. Wave phenomena in the brain arise in varied contexts, from the propagation of signals down a neuron, to collective behavior of many neurons resulting in rhythmic activity. More specifically, the Nernst equation, named for Walther Nernst, and its generalization, the Goldman equation, named for David Goldman, help relate ion concentrations to voltages. How those voltages change in a neuron as it is stimulated by other neurons is modeled by the Hodgkin–Huxley set of equations, which are a calculus-based set of differential equations that resulted in a Nobel Prize for Alan Hodgkin and Andrew Huxley. Next, an area of mathematics known as “dynamical systems” helps model how the firing of individual neurons can naturally become synchronized and produce wave behavior at different frequencies, which are ultimately recorded on EEGs. Normal rhythmic activity is important in activities such as sleeping, breathing, or walking. Abnormal rhythmic activity is manifested in various diseases; for example, forms of schizophrenia, Parkinson’s disease, and epilepsy demonstrate deviations from what is considered typical rhythmical behavior.

Neurology of Learning Mathematic

How the brain learns mathematics is another area of interest to researchers. Psychology and other social sciences bring light to bear on this subject but so too does the study of various neural pathologies. As an example, dyscalculia, which has been called a form of “number blindness” (by analogy to color blindness), is a pathology wherein individuals cannot acquire arithmetic skills. For instance, individuals fully capable of language communication who cannot tell if one whole number is larger than another or are unable to do 2-digit computations are considered “number blind.” For other examples, there are cases of individuals with increasing difficulties with speech—primarily because of atrophy in the temporal lobes leading to dementia—having highly reduced vocabulary including an inability to name common objects, yet whose arithmetic abilities remained virtually flawless. Similarly, there are instances of autistic individuals essentially unable to speak or understand speech, who nevertheless can perform computations. Infants can notice when the number of objects in a display changes or when a number of objects are hidden behind a screen.

Finally, there are instances of stroke victims who have fully intact language but lack numerical skills, such as not being able to count past 4, or say how many days are in a week. These examples indicate that language is not crucial for arithmetic computations, and, further, language may not be necessary for learning to calculate. Generally speaking, computations seem to be localized to the parietal lobe at the top of the brain, whereas key language areas, such as Broca’s (frontal lobe), named for Paul Broca, and Wernicke’s (temporal lobe), named for Carl Wernicke, reside elsewhere. However, there are ongoing debates among neuroscientists regarding what the highlighted areas on images mean with regard to brain functionality.

A related issue is how mathematical thinking beyond the level of simple arithmetic evolved in humans, including its relationship with the development of language and increasingly abstract reasoning. There are different and intriguing hypotheses regarding why language evolved roughly 200,000 years ago, whereas various forms of numerical and algorithmic abstraction evolved within the past few thousand years.

Bibliography

Bookstein, Fred. “Morphometrics.” Math Horizons 3 (February 1996).

Joint Policy Board for Mathematics. “Mathematics Awareness Month, April 2007: Mathematics and the Brain.” http://www.mathaware.org/mam/07/announcement.html.

Martindale, Diane. “Road Map for the Mind: Old Mathematical Theorems Unfold the Human Brain.” Scientific American 285, no. 2 (August 2001).

Schoonover, Carl. Portraits of the Mind: Visualizing the Brain from Antiquity to the 21st Century. New York: Abrams, 2010.

Sousa, David. How the Brain Learns Mathematics. Thousand Oaks, CA: Corwin Press, 2008.