Bifurcations of Stationary Solutions in an Interacting Pair of E-I Neural Fields

Persistent activity has been identified as a neural correlate of working memory [J. M. Fuster and G. E. Alexander, Science, 173 (1971), pp. 652–654; S. Funahashi, C. J. Bruce, and P. S. GoldmanRakic, J. Neurophysiol., 61 (1989), pp. 331–349; P. S. Goldman-Rakic, Neuron, 14 (1995), pp. 477–485]. In neural field theory, stationary bumps are localized states of neural activity that have been used to model this persistent activity. In [S. E. Folias and G. B. Ermentrout, Phys. Rev. Lett., 107 (2011), 228103], we proposed that the persistent activity may be modeled by interacting neural field layers which support persistent activity when each layer in isolation cannot. In this paper, we study the existence, linear stability, and bifurcations of various stationary bumps in a pair of Amari neural fields and in a pair of excitatory-inhibitory (E-I) neural fields on one-dimensional spatial domains. Both support stationary bumps composed of a bump in each layer with (i) identical profiles, (ii) identical centers but different profiles, and (iii) spatially offset centers, and we identify a direct relationship between the two models through the spatial structure of the eigenfunctions for the linearization of the neural field equations about a specific type of stationary bump. Traveling bumps, breathers, and other spatiotemporal phenomena are also found.