On a population of labile synthesized relaxation oscillators

Neuronal electrical rhythms described by composite mapped clock oscillators

The Mapped Clock Oscillator (MCO) model is a representation of omnipresent transmembrane voltage oscillations in excitable cells. We present a generalized version of the MCO that can model neuronal electrical oscillations, both labile and omnipresent, entirely within the framework of a system of ordinary differential equations. The previously described MCO was a second-order system, whereas the model presented here, which we call the composite MCO (cMCO) is a fourth-order system. Furthermore, we show how this cMCO can also be adapted to describe a pair of cells that forms a functional unit, as illustrated here by a model of the CA3 pyramidal cell and its basket cell interneuron feedback loop. The model was able to reproduce the high frequencies (super gamma) and possibly chaotic dynamics observed in the biological system.

A synaptic input portal for a mapped clock oscillator model of neuronal electrical rhythmic activity

Neuronal electrical oscillations play a central role in a variety of situations, such as epilepsy and learning. The mapped clock oscillator (MCO) model is a general model of transmembrane voltage oscillations in excitable cells. In order to be able to investigate the behaviour of neuronal oscillator populations, we present a neuronal version of the model. The neuronal MCO includes an extra input portal, the synaptic portal, which can reflect the biological relationships in a chemical synapse between the frequency of the presynaptic action potentials and the postsynaptic resting level, which in turn affects the frequency of the postsynaptic potentials. We propose that the synaptic input-output relationship must include a power function in order to be able to reproduce physiological behaviour such as resting level saturation. One linear and two power functions (Butterworth and sigmoidal) are investigated, using the case of an inhibitory synapse. The linear relation was not able to produce physiologically plausible behaviour, whereas both the power function examples were appropriate. The resulting neuronal MCO model can be tailored to a variety of neuronal cell types, and can be used to investigate complex population behaviour, such as the influence of network topology and stochastic resonance.