Dynamical behaviour of a population-based neuronal firing rate model with adaptation current

Anita Windisch, 4th year PhD, Eötvös Loránd University, Budapest, Hungary

BACKGROUND:

Modelling and studying the behaviour of neural networks are important for neurobiological and medical applications, as well as for artificial intelligence. It brings us closer to understanding how the human brain works by detecting the mechanism of how certain phenomena occur. Studying different types of neuron models is useful to understand the development of several diseases, such as epilepsy, and to build artificial networks for industrial use as well.

One of the two main types of neuron models is the spiking neuron model. A well-known example is the integrate-and-fire model, which consists of equations describing the change in membrane potential and components generating the spikes. The other type is the rate model, which enables us to describe the behaviour of neurons with similar properties together, considering them as one population.

The main aim of my research as a mathematician is to provide conditions for the occurrence of different types of oscillations, e.g., sharp wave-ripple and gamma oscillations, and also for the appearance of bistability in the system.

METHODOLOGY:

For my PhD project, I am investigating a 3-dimensional version of the Cowan-Wilson model, which is a rate model consisting of an excitatory and an inhibitory population recurrently connected. Firing adaptation is also added to the system, which appears as negative feedback for the excitatory cells. A rectified linear unit activation function is applied for each population, shifted by the effective threshold.

Bifurcation theory is used to analyse how the solutions change as some parameters are varied in the system. Local bifurcations, where a significant change in the dynamical behaviour occurs in a neighbourhood of an equilibrium point, can be calculated analytically since the model is piecewise linear. Explaining global phenomena where a qualitative change in the phase portrait can be detected close to a global shape, e.g., a periodic orbit, requires the use of numerical tools too.

RESULTS:

I investigated the model while changing the parameter of the adaptation and a control parameter, which is used to scale the strength of the synaptic connections together. Local bifurcations were determined and the dynamical behaviour in each domain created by the bifurcation curves was characterized. I detected the birth of gamma oscillations via Hopf-like bifurcation and in some domains ripple oscillation was also found.

I have started to work out a general description of the model where the number of steady states is characterized without fixing any parameter value.

Fig: (a) Stable periodic orbit and equilibrium (blue). Their attracting area is separated by an unstable periodic orbit (red). (b) Sharp wave-ripple oscillation as a stable periodic orbit (black), and an unstable steady state (red).

FUTURE WORK:

Currently, I am working on completing the full characterization with the stability of equilibria. Shortly, my goal is to give conditions for the birth of ripple oscillations which is a global phenomenon in the investigated model.

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