| Summary: | Explorations of visual hallucinations, and in particular those of Billock and Tsou [V. A. Billock and B. H. Tsou, Proceedings of the National Academy of Sciences USA, 104 (2007), pp. 8490--8495], show that annular rings with a background flicker can induce visual hallucinations in humans that take the form of radial fan shapes. The well-known retinocortical map tells us that the corresponding patterns of neural activity in the primary visual cortex for rings and arms in the retina are orthogonal stripe patterns. The implication is that cortical forcing by spatially periodic input can excite orthogonal modes of neural activity. Here we show that a simple scalar neural field model of primary visual cortex with state-dependent spatial forcing is capable of modelling this phenomenon. Moreover, we show that this occurs most robustly when the spatial forcing has a 2:1 resonance with modes that would otherwise be excited by a Turing instability. By utilising a weakly nonlinear multiple-scales analysis we determine the relevant amplitude equations for uncovering the parameter regimes which favour the excitation of patterns orthogonal to sensory drive. In combination with direct numerical simulations we use this approach to shed further light on the original psychophysical observations of Billock and Tsou.
Homogeneous connectivity profiles are commonly used as standard in neural field models. However, connectivity is known to be patchy and organised by a roughly hexagonal periodicity. We use a periodically modulated connectivity profile in the neural field model to explore this patchiness. Turing analysis and direct numerical simulations allow us to determine the consequences of this on pattern formation. The orientation preference map in primary visual cortex also has roughly periodic structure organised around pinwheels. We show that a multi-layered neural field model with patchy connectivity is capable of generating a realistic orientation preference map.
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