Neural field models are now in common usage in mathematical neuroscience to describe the coarse-grained activity of cortical tissue [1]. For mathematical convenience they often assume that anatomical connectivity is homogenous. However, this is far from the truth. For example, in the primary visual cortex (V1) it is known that there are maps reflecting the fact that neurons respond preferentially to stimuli with particular features. The classic example is that of orientation preference (OP), whereby cells respond preferentially to lines and edges of a particular orientation. The OP map changes continuously as a function of cortical location, except at singularities or pinwheels. The underlying periodicity in the microstructure of V1 is approximately 1mm, the domain of which corresponds to the so-called cortical hypercolumn. Other anatomical evidence suggests that longer-range, patchy horizontal connections link neurons in different hypercolumns provided that they have similar orientation preferences. This project will consider a field of hypercolumns that respects this biological reality. The mathematical model will be that of an integro-differential equation for V1 activity, with V1 viewed as a fiber bundle that associates to every point of the cortex (or retina by the retino-cortical map) a copy of the unit circle [2]. 

The project will focus on combining realistic retino-cortical maps [3] with next generation neural field models [4] and state-of the art numerical methods [5] to understand not only mechanisms for visual illusions, but also basic notions of how biological tissue can perform visual computations for image completion. The project will involve a mix of high performance scientific computation, nonlinear dynamics, differential geometry, and an enthusaism for learning about visual neuroscience. 

References 

1. S Coombes, P beim Graben and R Potthast, 2014. Tutorial on Neural Field Theory, Neural Fields, Ed. S Coombes, P beim Graben, R Potthast and J J Wright, Springer Verlag 

2. P C Bressloff and J D Cowan, 2003. The functional geometry of local and horizontal connections in a model of V1. Journal of Physiology-Paris, 97:221TH236. 

3. A Johnston 1989 The geometry of the topographic map in striate cortex. Vision Research, 29, 1493-1500 

4. A Byrne, D Avitabile and S Coombes, 2017. A next generation neural field model: The evolution of synchrony within patterns and waves, preprint 

5. J Rankin, D Avitabile, J Baladron, G Faye, DJB Lloyd, 2014. Continuation of localized coherent structures in nonlocal neural field equations. SIAM Journal on Scientific Computing 36 (1), B70-B93.