Theta model

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 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. 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. Parabolic bursting has been studied most extensively in the R15 neuron, which is one of six types of neurons of the Aplysia abdominal ganglion and one of thirty neurons comprising the abdominal ganglion. The Aplysia abdominal ganglion was studied and extensively characterized because its relatively large neurons and proximity of the neurons to the surface of the ganglion made it an ideal and "valuable preparation for cellular electrophysical studies". Early attempts to model parabolic bursting were for specific applications, often related to studies of the R15 neuron. This is especially true of R. E. Plant and Carpenter, whose combined works comprise the bulk of parabolic bursting models prior to Ermentrout and Kopell's canonical model. Though there was no specific mention of the term "parabolic bursting" in Plant's papers, Plant's model(s) do involve a slow, modulating oscillation which control bursting in the model(s). This is, by definition, parabolic bursting. Both of Plant's papers on the topic involve a model derived from the Hodgkin–Huxley equations and include extra conductances, which only add to the complexity of the model. Carpenter developed her model primarily for a square wave burster. The model was capable of producing a small variety of square wave bursts and produced parabolic bursts as a consequence of adding an extra conductance. However, the model applied to only spatial propagation down axons and not situations where oscillations are limited to a small region in space (i.e. it was not suited for "space-clamped" situations). The lack of a simple, generalizable, space-clamped, parabolic bursting model motivated Ermentrout and Kopell to develop the theta model.

Characteristics of the model

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