Mathematical research and the nervous system

SUMMARY: Mathematicians use a variety of mathematical modeling techniques to map and analyze nervous systems.

Human beings and many animals have two systems that are responsible for regulating and coordinating the activities of the body: the nervous system and the endocrine system. The first provides extremely fast responses, like reacting to touching a hot stove. The second responds more slowly and continuously, such as regulating blood sugar after a meal. Both systems work by detecting internal and external variations, such as shapes, odors, or temperature, to maintain the balance of body functions. Neuroscience is the study of the nervous system, including the brain, and its functions. It is an interdisciplinary field that draws concepts and methods from many fields, such as mathematics, psychology, biology, physics, and medicine. The Hodgkin–Huxley equations, named for Alan Hodgkin and Andrew Huxley, are fundamental to the development of mathematical models and simulations that have long been the basis of many experiments to study the nervous system. Many neuroscience researchers and teachers use the open source NEURON computer simulation system, which incorporates systems of equations and computational algorithms to mathematically model and display the behavior of individual neurons or networks of neurons in a dynamic way that is often difficult or impossible to achieve in traditional laboratory experiments.

Nervous System Processes

Everyday situations can highlight the complex action of the nervous system. For example, in a soccer match, players anticipate the opportunity to act. At the exact moment the ball is thrown in a player’s direction, thousands of nerve connections start to become active. In milliseconds, the player begins to use sensory memories and visual information to immediately decide the best course of action, such as to kick the ball to another player or directly to the goal. The central nervous system consists of the brain and spinal cord. The brain is the control central of the nervous system. The spinal cord conducts electrical signals between the brain and various nerves throughout the body, and controls some reflex functions. Neurons are cells that propagate the electrical impulses in the nervous system, and glial cells help maintain parts of the nervous system. For example, they produce myelin, which coats many neurons like insulation in electrical wiring. The neurons have important properties, such as excitability and conductivity, and act similarly to an electric current transmitted along a wire. This phenomenon occurs because of permeation of ions, such as sodium and potassium, through the neural membrane, which generate an electrical signal that propagates between neurons via its branched structure, consisting of thousands of small extensions.

Early Research

Nerve impulse propagation and the nervous system processes have been researched for many years using theories and techniques from genetics, molecular biology, physiology, psychology, and mathematics, among others. In the 1950s, physiologists and biophysicists Alan Hodgkin and Andrew Huxley experimented on the nervous systems of squids, specifically on a structure known as the “giant axon.” An axon transmits electrical impulses in the nervous system, and a squid’s giant axon can be up to 1 millimeter in diameter, much larger than most axons and visible to the naked eye. These experiments led to the development of the Hodgkin–Huxley equations, which are nonlinear ordinary differential equations that describe or approximate the electrical characteristics of neurons and other electrically excitable cells, such as those in the heart. They involve concepts like gates (channels that allow the ions to flow), voltage thresholds, and conductances, which act together to determine if and when a neuron “fires” an electrical burst. They are very similar to electric circuit theory, and some models of nervous systems look very much like electrical circuit diagrams.

Other Mathematical Connections

Hodgkin and Huxley won a Nobel Prize for their experimental and mathematical work, which has since led to other mathematical explorations of the nervous system. The nervous system in mammals is a very complex dynamic system, with many interconnected components. Periodic rhythms are found in some types of movement-related behaviors that are governed by the nervous system, like walking and breathing. They are also related to sensation and cognition. Studies of all these various substructures involve not only understanding how each structure behaves on its own but also how they interconnect and communicate with one another. Because of the vast degree of intercorrelation among various nervous system structures, from individual neurons to larger structures like the brain and spinal cord, one challenge facing mathematical modelers is creating systems of equations that optimize the ability of the equations to realistically represent neuronal systems and their behaviors while making them tractable for computation and interpretation. One of the interesting mathematical phenomena that researchers study is called “gamma and beta rhythms.” These brain waves have been connected to so-called “higher” mental activity, like perception and consciousness, as well as to synchronous activity that may help link various sensory inputs into a single mental construction of an object. However, many questions remain. Techniques such as graphs, circuits, networks, clustering, geometry, and simulation all play a role in investigation of nervous system properties and functions.

One additional important advance in neuroscience is the neurochip. It can be used to help link biological neurons and semiconductor materials, which may one day help to create prosthetics that integrate fully into the body’s own neural system. They may also facilitate treatments for neurological conditions such as Alzheimer’s and Parkinson’s Disease.

Scientists are using data learned from the study of human neural networks and applying them to new technologies. A recent development has been the development of artificial neural networks. These are machine-learning models that can be trained to complete many tasks. Their architecture was modeled on the manner neurons process information in the human brain. 

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