Dynamical neuroscience

The motivation for a dynamical approach to neuroscience stems from an interest in the physical complexity of neuron behavior. As an example, consider the coupled interaction between a neuron's membrane potential and the activation of ion channels throughout the neuron. As the membrane potential of a neuron increases sufficiently, channels in the membrane open up to allow more ions in or out. The ion flux further alters the membrane potential, which further affects the activation of the ion channels, which affects the membrane potential, and so on. This is often the nature of coupled nonlinear equations. A nearly straight forward example of this is the Morris–Lecar model:

${\displaystyle {\begin{aligned}C{dV \over dt}&=g_{Ca}M_{ss}(V-V_{C}a)-g_{K}N(V-V_{K})-g_{L}(V-V_{L})+I_{\text{app}}\\[6pt]{dN \over dt}&={{N_{ss}-N} \over {\tau _{N}}}\end{aligned}}}$

See the Morris–Lecar paper[10] for an in-depth understanding of the model. A more brief summary of the Morris Lecar model is given by Scholarpedia.[11]

In this article, the point is to demonstrate the physiological basis of dynamical neuron models, so this discussion will only cover the two variables of the equation:

• ${\displaystyle V}$ represents the membrane's current potential
• ${\displaystyle N}$ is the so-called "recovery variable", which gives us the probability that a particular potassium channel is open to allow ion conduction.

Most importantly, the first equation states that the change of ${\displaystyle V}$ with respect to time depends on both ${\displaystyle V}$ and ${\displaystyle N}$, as does the change in ${\displaystyle N}$ with respect to time. ${\displaystyle M_{ss}}$ and ${\displaystyle N_{ss}}$ are both functions of ${\displaystyle V}$. So we have two coupled functions, ${\displaystyle g(V,N)}$ and ${\displaystyle g(V,N)}$.

Different types of neuron models utilize different channels, depending on the physiology of the organism involved. For instance, the simplified two-dimensional Hodgkins–Huxley model considers sodium channels, while the Morris–Lecar model considers calcium channels. Both models consider potassium and leak current. Note, however, that the Hodgkins–Huxley model is canonically four-dimensional.[12]

One of the predominant themes in classical neurobiology is the concept of a digital component to neurons. This concept was quickly absorbed by computer scientists where it evolved into the simple weighting function for coupled artificial neural networks. Neurobiologists call the critical voltage at which neurons fire a threshold. The dynamical criticism of this digital concept is that neurons don't truly exhibit all-or-none firing and should instead be thought of as resonators.[13]

In dynamical systems, this kind of property is known as excitability. An excitable system starts at some stable point. Imagine an empty lake at the top of a mountain with a ball in it. The ball is in a stable point. Gravity is pulling it down, so it's fixed at the lake bottom. If we give it a big enough push, it will pop out of the lake and roll down the side of the mountain, gaining momentum and going faster. Let's say we fashioned a loop-de-loop around the base of the mountain so that the ball will shoot up it and return to the lake (no rolling friction or air resistance). Now we have a system that stays in its rest state (the ball in the lake) until a perturbation knocks it out (rolling down the hill) but eventually returns to its rest state (back in the lake). In this example, gravity is the driving force and spatial dimensions x (horizontal) and y (vertical) are the variables. In the Morris Lecar neuron, the fundamental force is electromagnetic and ${\displaystyle V}$ and ${\displaystyle N}$ are the new phase space, but the dynamical picture is essentially the same. The electromagnetic force acts along ${\displaystyle V}$ just as gravity acts along ${\displaystyle y}$. The shape of the mountain and the loop-de-loop act to couple the y and x dimensions to each other. In the neuron, nature has already decided how ${\displaystyle V}$ and ${\displaystyle N}$ are coupled, but the relationship is much more complicated than the gravitational example.

This property of excitability is what gives neurons the ability to transmit information to each other, so it is important to dynamical neuron networks, but the Morris Lecar can also operate in another parameter regime where it exhibits oscillatory behavior, forever oscillating around in phase space. This behavior is comparable to pacemaker cells in the heart, that don't rely on excitability but may excite neurons that do.

Biological neural networks can be modeled by choosing an appropriate biological neuron model to describe the physiology of the organism and appropriate coupling terms to describe the physical interactions between neurons (forming the network). Other global considerations must be taken into consideration, such as the initial conditions and parameters of each neuron.

In terms of nonlinear dynamics, this requires evolving the state of the system through the functions. Following from the Morris Lecar example, the alterations to the equation would be:

${\displaystyle {\begin{aligned}C{dV_{i} \over dt}&=g_{Ca}M_{ss}(V_{i}-V_{Ca})-g_{K}N_{i}(V_{i}-V_{K})-g_{L}(V_{i}-V_{L})+I_{\text{app}}+D(V_{i})\\[6pt]{dN_{i} \over dt}&={{N_{ss}-N_{i}} \over {\tau _{N}}}\end{aligned}}}$

where ${\displaystyle V}$ now has the subscript ${\displaystyle i}$, indicating that it is the ith neuron in the network and a coupling function has been added to the first equation. The coupling function, D, is chosen based on the particular network being modeled. The two major candidates are synaptic junctions and gap junctions.

• Point attractors – memory, pattern completion, categorizing, noise reduction
• Line attractors – neural integration: oculomotor control
• Ring attractors – neural integration: spatial orientation
• Plane attractors – neural integration: (higher dimension of oculomotor control)
• Cyclic attractors – central pattern generators
• Chaotic attractors – recognition of odors and chaos is often mistaken for random noise.

Please see Scholarpedia's page for a formal review of attractor networks.[14]

Glia, once considered a mere support system for neurons, have been found to serve a significant role in the brain.[16][17] The subject of how the interaction between neuron and glia have an influence on neuron excitability is a question of dynamics.[18]

Like any other cell, neurons operate on an undoubtedly complex set of molecular reactions. Each cell is a tiny community of molecular machinery (organelles) working in tandem and encased in a lipid membrane. These organelles communicate largely via chemicals like G-proteins and neurotransmitters, consuming ATP for energy. Such chemical complexity is of interest to physiological studies of the neuron.

Neurons in the brain live in an extracellular fluid, capable of propagating both chemical and physical energy alike through reaction-diffusion and bond manipulation that leads to thermal gradients. Volume transmission has been associated with thermal gradients caused by biological reactions in the brain.[19] Such complex transmission has been associated with migraines.[20]

The Lyapunov function is a nonlinear technique used to analyze the stability of the zero solutions of a system of differential equations. Hopfield networks were specifically designed such that their underlying dynamics could be described by the Lyapunov function. Stability in biological systems is called homeostasis. Particularly of interest to the cognitive sciences, Hopfield networks have been implicated in the role of associative memory (memory triggered by cues).[22]