On Time-Delayed Coupled Hindmarsh-Rose Neurons: Stability of Equilibria and Synchronisation

Neurons play an important role in the processing of information in the brain. Many models are developed over the years that exhibit the dynamical behaviour of these neurons. The complex Hodgkin-Huxley model is widely the most used model in computational science, while the Hindmarsh-Rose model is nowadays a very good and relatively more simple model of a neuron. Furthermore, an important property of neurons is that in some cases their dynamical behaviour synchronises. The objective of this research is to investigate the stability of the equilibrium and the synchronisation of a network of coupled Hindmarsh-Rose neurons in the case that a timedelayed coupling is applied. First a closer examination is done of the recently new parametrised Hindmarsh-Rose model. A stability analysis is performed on the equilibrium point and simulations are performed to investigate which dynamical behaviour and bifurcations occurs, both for varying value of the externally applied current. Furthermore, a specially designed software package for stability and bifurcation analysis of time-delayed coupled systems, DDE-BIFTOOL, is used for this model to verify the obtained results. Stabilisability of the equilibrium of the network of four unidirectionally time-delayed coupled Hindmarsh-Rose neurons is investigated by means of stabilisability techniques for linear time-delay systems. These techniques, and the use of DDE-BIFTOOL, give a strong foundation that the equilibrium of the network is not stabilisable. Furthermore, a condition is derived for (partial) synchronisation of this network. This is done by investigating the global stability of the zero solution of the error dynamics of the network by using a Lyapunov-Krasovskii functional. A sector condition is used to deal with the nonlinear terms in the model. Formulating the problem as an eigenvalue optimisation problem gives the maximal time-delay that is allowed at a specific coupling strength for which the synchronisation still holds. For two different coupling strengths, the stability diagram of the invariant manifolds that are present in this network are determined by means of simulations. Therefore a slightly weaker condition for synchronisation is used, so called practical synchronisation. Finally, some parameter values of the new parametrised Hindmarsh-Rose model are changed in such a way that the model possesses three equilibria for a larger range of the externally applied current than the 1984 model. Again a stability analysis is done of the equilibria and compared with the results of DDE-BIFTOOL. The little research on this subject showed that it is hard to find realistic parameter values for which three equilibria exist for varying value of the externally applied current and for which interesting dynamical behaviour occurs such as bursting and chaotic behaviour.