Theta model
[1] 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,[2] swimming,[3] and digesting.
[7] The model consists of one variable that describes the membrane potential of a neuron along with an input current.
[8] 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.
[18] 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".
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.
[20] For example, the pre-Bötzinger complex in the mammalian brain stem contains many bursting neurons that control autonomous breathing rhythms.
[30] 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".
[31] Early attempts to model parabolic bursting were for specific applications, often related to studies of the R15 neuron.
Derivation of the Ermentrout and Kopell canonical model begins with the general form for parabolic bursting, and notation will be fixed to clarify the discussion.
describe the membrane potential and ion channels, typical of many conductance-based biological neuron models.
must produce a circle in phase space that is invariant, meaning it does not change under certain transformations.
is some slow wave which can be both negative and positive, the system is capable of producing parabolic bursts.
represents the perturbation current, a closed form solution of the phase response curve (PRC) does not exist.
However, the theta model is a special case of such an oscillator and happens to have a closed-form solution for the PRC.
At first glance, the mechanisms of bursting in both systems are very different: In Plant's model, there are two slow oscillations – one for conductance of a specific current and one for the concentration of calcium.
In the process, Soto-Trevino, et al. discovered that the theta model was more general than originally believed.
The quadratic integrate-and-fire (QIF) model was created by Latham et al. in 2000 to explore the many questions related to networks of neurons with low firing rates.
[12] It was unclear to Latham et al. why networks of neurons with "standard" parameters were unable to generate sustained low frequency firing rates, while networks with low firing rates were often seen in biological systems.
According to Gerstner and Kistler (2002), the quadratic integrate-and-fire (QIF) model is given by the following differential equation: where
) surpasses its firing threshold and rises rapidly (indeed, it reaches arbitrarily large values in finite time); this represents the peak of the action potential.
To simulate the recovery after the action potential, the membrane voltage is then reset to a lower value
To avoid dealing with arbitrarily large values in simulation, researchers will often set an upper limit on the membrane potential, above which the membrane potential will be reset; for example Latham et al. (2000) reset the voltage from +20 mV to −80 mV.
Though the theta model was originally used to model slow cytoplasmic oscillations that modulate fast membrane oscillations in a single cell, Ermentrout and Kopell found that the theta model could be applied just as easily to systems of two electrically coupled cells such that the slow oscillations of one cell modulates the bursts of the other.
[7] Such cells serve as the central pattern generator (CPG) of the pyloric system in the lobster stomatograstic ganglion.
[36] In such a system, a slow oscillator, called the anterior burster (AB) cell, modulates the bursting cell called the pyloric dilator (PD), resulting in parabolic bursts.
[7] A group led by Boergers,[16] used the theta model to explain why exposure to multiple simultaneous stimuli can reduce the response of the visual cortex below the normal response from a single (preferred) stimulus.
Their computational results showed that this may happen due to strong stimulation of a large group of inhibitory neurons.
McKennoch et al. (2008) derived a steepest gradient descent learning rule based on theta neuron dynamics.
[18] Their model is based on the assumption that "intrinsic neuron dynamics are sufficient to achieve consistent time coding, with no need to involve the precise shape of postsynaptic currents..." contrary to similar models like SpikeProp and Tempotron, which depend heavily on the shape of the postsynaptic potential (PSP).
