Dynamical neuroscience

The dynamical systems approach to neuroscience is a special way to study how living things think and feel using math. It looks at how tiny parts of our bodies, like cells in the brain, work together in complex ways. These tiny parts can change in surprising ways depending on different conditions.

Dynamical neuroscience helps us understand how the brain works at many levels, from single cells to big thoughts and actions. It shows how the brain can switch between different states, like waking up or sleeping, and how neurons talk to each other in big groups.

Neurons have been studied as these special systems for many years, but dynamical systems can also appear in other parts of the nervous system. For example, chemicals in the brain can behave in unpredictable ways, and the flow of fluids around neurons also plays a role. By using ideas from information theory and thermodynamics, scientists can learn even more about how the brain processes information in complex patterns.

History

One of the earliest ways scientists studied how neurons work was through math and physics. In 1907, they created a simple model called the integrate-and-fire model. Later, in 1952, two scientists named Alan Hodgkin and Andrew Huxley used a special part of a squid to make an even better model, called the Hodgkin–Huxley model. Other scientists made simpler versions of these models in the years that followed.

As computers became more powerful in the late 20th century, they helped scientists study neurons in new ways. Computers could solve very complicated math problems that were too hard to solve by hand. This led to the creation of a field called computational neuroscience. In 2007, a book called Dynamical Systems in Neuroscience by Eugene Izhikivech helped many people learn about this exciting area of study.

Neuron dynamics

Main article: biological neuron model

Neurons are amazing cells that help our brains send messages. Scientists study how neurons work using math and physics. One way they do this is by looking at how the neuron's energy changes and how tiny doors in the neuron open and close.

When a neuron gets enough energy, special doors open to let tiny particles in or out. This changes the neuron's energy again and again, like a circle. This helps scientists understand how neurons talk to each other.

One famous example is called the Morris–Lecar model. It uses two main things: the neuron's energy (V) and a helper number (N) that tells how likely a door is open. These two things change over time based on each other. This helps scientists see how neurons can fire or send signals.

Neurons can act like balls in a lake. Normally, they stay still. But if something pushes them hard enough, they "fire" and send a message, then go back to rest. This is called excitability and helps neurons share information. Sometimes, neurons can also act like heart cells that keep beating on their own.

Global neurodynamics

The way groups of neurons work together depends on a few important things: how each neuron behaves, how they connect to each other, the layout of these connections, and outside influences like temperature changes.

We can create models of these networks by choosing how each neuron acts and how they interact. These models help us understand complex behaviors in the brain, like remembering things or recognizing smells. Some networks can show steady patterns, while others can change in more unpredictable ways.

Beyond neurons

While neurons are very important for how our brain works, scientists now know that neurons depend a lot on what’s around them. Right outside neurons, there is a special space that contains other cells called glial cells. These cells were once thought to just support neurons, but they actually play a big role in how neurons behave.

Neurons also rely on many tiny chemical reactions to function. Inside each neuron, small parts called organelles work together using chemicals like G-proteins and neurotransmitters, and they need energy from ATP. These chemicals help neurons send signals and stay active, showing just how complex our brain truly is.

Cognitive neuroscience

The computational approaches to theoretical neuroscience use artificial neural networks to study how the brain works. These networks simplify individual neurons to look at how groups of neurons behave together. Even though neural networks are often linked to artificial intelligence, they help us understand how the mind processes information.

Hopfield networks are a special type of neural network. They use a mathematical tool called the Lyapunov function to study stability in systems. These networks are important for understanding how memories work, especially how cues can trigger memories. Stability in living systems is known as homeostasis.