Bumps, breathers, and waves in a neural network with spike frequency adaptation

In this Letter we introduce a continuum model of neural tissue that include the effects of so-called spike frequency adaptation (SFA). The basic model is an integral equation for synaptic activity that depends upon the non-local network connectivity, synaptic response, and firing rate of a single neuron. A phenomenological model of SFA is examined whereby the firing rate is taken to be a simple state-dependent threshold function. As in the case without SFA classical Mexican-Hat connectivity is shown to allow for the existence of spatially localized states (bumps). Importantly an analysis of bump stability using recent Evans function techniques shows that bumps may undergo instabilities leading to the emergence of both breathers and traveling waves. Moreover, a similar analysis for traveling pulses leads to the conditions necessary to observe a stable traveling breather. Direct numerical simulations both confirm our theoretical predictions and illustrate the rich dynamic behavior of this model, including the appearance of self-replicating bumps.

Voltage and current spectra for matrix power converters

Matrix power converters are used for transforming one alternating-current power supply to another, with different peak voltage and frequency. There are three input lines, with sinusoidally varying voltages which are 120◦ out of phase one from another, and the output is to be delivered as a similar three-phase supply. The matrix converter switches rapidly, to connect each output line in sequence to each of the input lines in an attempt to synthesize the prescribed output voltages. The switching is carried out at high frequency and it is of practical importance to know the frequency spectra of the output voltages and of the input and output currents. We determine in this paper these spectra using a new method, which has significant advantages over the prior default method (a multiple Fourier series technique), leading to a considerably more direct calculation. In particular, the determination of the input current spectrum is feasible here, whereas it would be a significantly more daunting procedure using the prior method instead.