Deterministic and stochastic dynamics of the diffusive memristor

Memristive devices and circuits are rapidly gaining interest for their utility in computing applications, more specifically in neuromorphic computing in which novel memristive circuits modelled into an artificial neuron are starting to mimic the dynamics and behaviours of biological neurons. The dynamics at centre stage are the spiking phenomena which modern artificial neuron circuits are capable of producing that are starting to mimic the spike dynamics observed in biological neurons. The diffusive memristor, a new form of memristive device, is of keen interest due to their ability to generate a wide range of spiking regimes. This could lead to them being the ideal device for applications in neuromorphic computing systems, with applications in artificial intelligence and other forms of unconventional computing methods.

This thesis is a collection of four connected papers along with the background and context which has motivated this work. The papers which will be presented are: (i) Deterministic mechanisms of spiking in diffusive memristors, (ii) Stochastic instabilities of the diffusive memristor, (iii) Deterministic modeling of the diffusive memristor, and (iv) Coexistence of spiking regimes in the diffusive memristor.

In the first paper we investigate an artificial neuron circuit which utilises a diffusive memristor. We eliminate noise from the device and perform a bifurcation analysis on the circuit. We uncover the dominant mechanisms of spiking in the device along with discovering that our model can produce both forms of negative differential resistance which is necessary for the device to generate its unique pulses. We compare our theoretical results to the experimental devices to show that the model is capable of modelling the real world phenomena observed in the diffusive memristor. This analysis allows us to justify the usage of our set of differential equations as they are shown to be able to reproduce the results seen in experimental work.

The second paper makes a mild modification to the original model by changing the potential shape which is motivated by the spike dynamics we can observe in experimental devices. We perform a bifurcation analysis on this circuit which reveals that the device without the influence of noise is only capable of generating a single spiking regime. We reintroduce noise into the system and analyse what parameter values are necessary for the emergence of a second spiking regime. We also analyse the threshold dynamics observed in experimental devices. This understanding of what parameter regions can produce spike dynamics is crucial when it comes to utilising these devices for computational purposes.

The third paper seeks to find if it is possible to generate the noise induced dynamics observed in the second paper in an entirely deterministic setting. We make modifications to the governing Fokker-Planck equation and are able to produce a set of coupled differential equations which arise from averaging out the dynamics of the stochastic equations that are formed out of the Fokker-Planck. We perform a bifurcation analysis on this new set of equations and show that it is capable of producing two separate spiking regimes in the deterministic setting, successfully yielding what was once an entirely noise driven event in a noiseless setting. We also observe dynamical behaviour in the new model which is reminiscent to that of chaotic phenomena.

In the final paper we add an additional particle into the equation system which governs the dynamics of the diffusive memristor. This is motivated by the observed coexistence of two separate forms of spiking in experimental devices. We introduce a new particle into the model which is associated with an additional conductive pillar. The model is able to produce two separate forms of spiking which can exist at the same external voltage. We perform a bifurcation analysis to show that both forms of spiking appear from the same underlying mechanism, before conducting a further analysis in which we utilise the inherent stochasticity of the diffusive memristor to produce noise driven transitions between these two distinct forms of spiking. The theoretical findings are then compared to experimental results; we illustrate that both the theoretical model and experimental devices can produce two forms of spiking that both have unique statistical features.