The theta model, or Ermentrout–Kopell canonical model, is a biological neuron model originally developed to mathematically describe neurons in the animal Aplysia. The model is particularly well-suited to describe neural bursting, which is characterized by periodic transitions between rapid oscillations in the membrane potential followed by quiescence. This bursting behavior is often found in neurons responsible for controlling and maintaining steady rhythms such as breathing, swimming, and digesting. Of the three main classes of bursting neurons (square wave bursting, parabolic bursting, and elliptic bursting), the theta model describes parabolic bursting, which is characterized by a parabolic frequency curve during each burst.
The model consists of one variable that describes the membrane potential of a neuron along with an input current. The single variable of the theta model obeys relatively simple equations, allowing for analytic, or closed-form solutions, which are useful for understanding the properties of parabolic bursting neurons. In contrast, other biophysically accurate neural models such as the Hodgkin–Huxley model and Morris–Lecar model consist of multiple variables that cannot be solved analytically, requiring numerical integration to solve.
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.
Background and history
Bursting is "an oscillation in which an observable [part] 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 [the] network behavior [of neocortical neurons]". The R15 neuron of the abdominal ganglion in Aplyisa, hypothesized to be a neurosecretory 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.