Neural Fields with Rebound Currents: Novel Routes to Patterning

The understanding of how spatio-temporal patterns of neural activity may arise in the cortex of the brain has advanced with the development and analysis of neural field models. To replicate this success for sub-cortical tissues, such as the thalamus, requires an extension to include relevant ionic currents that can further shape firing response. Here we advocate for one such approach that can accommodate slow currents. By way of illustration we focus on incorporating a T-type calcium current into the standard neural field framework. Direct numerical simulations are used to show that the resulting tissue model has many of the properties seen in more biophysically detailed model studies, and most importantly the generation of oscillations, waves, and patterns that arise from rebound firing. To explore the emergence of such solutions we focus on one-and two-dimensional spatial models and show that exact solutions describing homogeneous oscillations can be constructed n the limit that the firing rate nonlinearity is a Heaviside function. A linear stability analysis, using techniques from non-smooth dynamical systems, is used to determine the points at which bifurcations from synchrony can occur. Furthermore, we construct periodic travelling waves and investigate their stability with the use of an appropriate Evans function. The stable branches of the dispersion curve for periodic travelling waves are found to be in excellent agreement with simulations initiated from an unstable branch of the synchronous solution