Mathematical modeling | Nonlinear systems
Theta model
The theta model, or Ermentrout–Kopell canonical model, is a biological neuron model originally developed to model neurons in the animal Aplysia, and later used in various fields of computational neuroscience. The model is particularly well suited to describe neuron bursting, which are rapid oscillations in the membrane potential of a neuron interrupted by periods of relatively little oscillation. Bursts are often found in neurons responsible for controlling and maintaining steady rhythms. For example, breathing is controlled by a small network of bursting neurons in the brain stem. Of the three main classes of bursting neurons (square wave bursting, parabolic bursting, and ), the theta model describes parabolic bursting. Parabolic bursting is characterized by a series of bursts that are regulated by a slower external oscillation. This slow oscillation changes the frequency of the faster oscillation so that the frequency curve of the burst pattern resembles a parabola. In the original paper, the model consists of one fast variable and an arbitrary number of slow variables, where the fast variable describes the membrane voltage of a neuron and the slow variable(s) allow the membrane potential to transition between spiking and quiescent states. However, the theta model as popularly known consists of only one fast variable with all slow variables replaced by a constant or time-dependent scalar function. In contrast, the Hodgkin–Huxley model consists of four state variables (one voltage variable and three gating variables) and the Morris–Lecar model is defined by two state variables (one voltage variable and one gating variable). The single state variable of the theta model, and the elegantly simple equations that govern its behavior allow for analytic, or closed-form solutions (including an explicit expression for the phase response curve). The dynamics of the model take place on the unit circle, and are governed by two cosine functions and a real-valued input function. Similar models include the quadratic integrate and fire (QIF) model, which differs from the theta model by only by a change of variables and , which consists of Hodgkin–Huxley type equations and also differs from the theta model by a series of coordinate transformations. Despite its simplicity, the theta model offers enough complexity in its dynamics that it has been used for a wide range of theoretical neuroscience research as well as in research beyond biology, such as in artificial intelligence. (Wikipedia).
