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 elliptic bursting), 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. 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 elliptic bursting), 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. The model has just one state variable which describes the membrane voltage of a neuron. 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 Plant's model, 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. Bursting is 'an oscillation in which an observable of the system, such as voltage or chemical concentration, changes periodically between an active phase of rapid spike oscillations (the fast sub-system) and a phase of quiescence'. Bursting comes in three distinct forms: square wave bursting, parabolic bursting, and elliptic bursting. There exist some models that do not fit neatly into these categories by qualitative observation, but it is possible to sort such models by their topology (i.e. such models can be sorted 'by the structure of the fast subsystem'). All three forms of bursting are capable of beating and periodic bursting. Periodic bursting (or just bursting) is of more interest because many phenomena are controlled by, or arise from, bursting. For example, bursting due to a changing membrane potential is common in various neurons, including but not limited to cortical chattering neurons, thalamacortical neurons, and pacemaker neurons. Pacemakers in general are known to burst and synchronize as a population, thus generating a robust rhythm that can maintain repetitive tasks like breathing, walking, and eating. Beating occurs when a cell bursts continuously with no periodic quiescent periods, but beating is often considered to be an extreme case and is rarely of primary interest. Bursting cells are important for motor generation and synchronization. For example, the pre-Bötzinger complex in the mammalian brain stem contains many bursting neurons that control autonomous breathing rhythms. Various neocortical neurons (i.e. cells of the neocortex) are capable of bursting, which 'contribute significantly to network behavior '. The R15 neuron of the abdominal ganglion in Aplyisa, hypothesized to be a cell (i.e. a cell that produces hormones), is known to produce bursts characteristic of neurosecretory cells. In particular, it is known to produce parabolic bursts. Since many biological processes involve bursting behavior, there is a wealth of various bursting models in scientific literature. For instance, there exist several models for interneurons and cortical spiking neurons. However, the literature on parabolic bursting models is relatively scarce. Parabolic bursting models are mathematical models that mimic parabolic bursting in real biological systems. Each burst of a parabolic burster has a characteristic feature in the burst structure itself – the frequency at the beginning and end of the burst is low relative to the frequency in the middle of the burst. A frequency plot of one burst resembles a parabola, hence the name 'parabolic burst'. Furthermore, unlike elliptic or square-wave bursting, there is a slow modulating wave which, at its peak, excites the cell enough to generate a burst and inhibits the cell in regions near its minimum. As a result, the neuron periodically transitions between bursting and quiescence.