(intro needed here)
Electrophysiology of the neuron 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: : \begin{align} C {dV \over dt} & = g_{Ca} M_{ss} (V-V_Ca) - 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{align} See the Morris–Lecar paper for an in-depth understanding of the model. A more brief summary of the Morris Lecar model is given by Scholarpedia. 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: • V represents the membrane's current potential • 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 V with respect to time depends on both V and N, as does the change in N with respect to time. M_{ss} and N_{ss} are both functions of V. So we have two coupled functions, g(V,N) and 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.
Excitability of neurons 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. 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 V and N are the new
phase space, but the dynamical picture is essentially the same. The electromagnetic force acts along V just as gravity acts along 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 V and 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. ==Global neurodynamics==