Dynamical neuroscience

The dynamical systems approach to neuroscience helps scientists understand how the brain works by using math to model complex patterns and changes. This branch of mathematical biology looks at how the nervous system behaves through something called nonlinear dynamics, which means that small changes can lead to big differences in how things work. In these systems, all possible states are shown in a phase space, and they can shift dramatically in a process called bifurcation, sometimes even showing unpredictable behavior known as chaos.

Dynamical neuroscience studies these patterns at many levels of the brain, from the activity of single nerve cells all the way up to complex cognitive processes, sleep states, and large groups of neurons working together. For many years, scientists have modeled neurons as these special nonlinear systems, but these ideas can apply to other parts of the nervous system too. For example, chemical reactions, like those in the Gray–Scott model, can also show rich and chaotic dynamics.

Another interesting part of this field is how information moves inside the brain. Information theory connects with thermodynamics to create infodynamics, which studies how information flows in nonlinear systems, especially in the brain. This helps us understand how different parts of the nervous system talk to each other and work together.

History

One of the earliest models of the neuron was the integrate-and-fire model, developed in 1907. Later, in 1952, Alan Hodgkin and Andrew Huxley created the Hodgkin–Huxley model using studies of the squid giant axon. More models followed, like the FitzHugh–Nagumo model in 1962 and the Morris–Lecar model in 1981.

Computers helped scientists study these complex models better. They made it possible to solve hard math problems related to neurons. In 2007, a book by Eugene Izhikivech, Dynamical Systems in Neuroscience, helped make this topic more popular in schools and research. Today, these models are still used in biophysics and computational neuroscience.

Neuron dynamics

Main article: biological neuron model

Neurons are amazing cells that can send signals to each other. Scientists study how neurons work using math and physics ideas. One important idea is how the inside of a neuron changes over time. When certain parts of the neuron open, ions move in and out, changing the neuron's state. This creates a back-and-forth effect that helps neurons send signals.

Neurons can be thought of as having a “trigger point.” When they reach this point, they send a signal. This is similar to a ball sitting at the top of a hill. With a small push, the ball rolls down, goes around a loop, and returns to the top. In neurons, the “push” comes from tiny electric forces, and the “loop” is the way the neuron’s parts are connected. This lets neurons talk to each other and control many body functions. Some neurons can even keep sending signals over and over, like a heart’s natural pacemaker.

Global neurodynamics

The global behavior of a network of neurons depends on several key factors: how each neuron works, how neurons connect to each other, the overall layout of the network, and outside influences like temperature changes. Scientists can create models of these networks by choosing how neurons behave and how they interact.

These models can show different types of stable patterns, called attractors. For example, some patterns help with memory, while others help control eye movements or recognize smells. Understanding these patterns helps scientists learn how the brain processes information.

Main article: Attractor network

Beyond neurons

While neurons are key players in how our brain works, scientists now know they depend a lot on their surroundings. Right around neurons, there is a special space filled with other cells called glial cells. These glial cells aren't just background players—they help control how active neurons are.

Neurons also rely on many tiny molecular reactions, using chemicals like G-proteins, neurotransmitters, and energy from ATP. These reactions help neurons send signals and stay alive, showing just how complex each tiny cell can be.

Cognitive neuroscience

The computational approaches to theoretical neuroscience use artificial neural networks to study how groups of neurons work together, rather than focusing on individual neurons. These networks, also linked to artificial intelligence, help scientists understand how the brain processes information and remembers things.

Hopfield networks are a special kind of artificial neural network. They use a mathematical tool called the Lyapunov function to study how stable systems stay balanced. These networks are important for understanding how memories are triggered by cues, and they reflect a stable state in biological systems known as homeostasis.