Our research in mathematical neuroscience has focused on the spatio-temporal dynamics of continuum neural fields with particular applications to vision. Neural fields model the large-scale dynamics of spatially structured biological neural networks in terms of nonlinear integro-differential equations, whose associated integral kernels represent the spatial distribution of neuronal synaptic connections. They provide an important example of spatially extended excitable systems with nonlocal interactions, and exhibit a wide range of spatially coherent dynamics, including traveling waves, oscillations and Turing-like patterns.

P. C. Bressloff. Spatiotemporal dynamics of continuum neural fields: Invited Topical review. *J. Phys. A* **45** 033001 (2012).

P. C. Bressloff and S. R. Carroll. Stochastic neural fields as gradient dynamical systems. *Phys. Rev. E* **100** 012402 (2019).

P. C. Bressloff. Stochastic neural field theory of wandering bumps on a sphere. *Physica D* **399** 138-152 (2019).

P. C. Bressloff. Stochastic neural field model of stimulus-dependent neural variability. *PLoS Comp. Biol.* **15** e1006755 (2019).

P. C. Bressloff and Z. P. Kilpatrick. Nonlinear Langevin equations for the wandering of fronts in stochastic neural fields. *SIAM J. Appl. Dyn. Syst.* **14** 305-334 (2015).

P. C. Bressloff. From invasion to extinction in heterogeneous neural fields. *J. Math. Neurosci.* **2** 6 (2012).

P. C. Bressloff and M. A. Webber. Front Propagation in stochastic neural fields. *SIAM J. Appl. Dyn. Syst. * **11** 708-740 (2012).

P. C. Bressloff. Stochastic neural field theory and the system-size expansion. *SIAM J. Appl. Math.* **70** 1488--1521 (2009).

P. C. Bressloff and Z. Kilpatrick. Two-dimensional bumps in a piecewise smooth neural field model with synaptic depression. *SIAM J. Appl. Math* ** 71** 379-408 (2011).

Z. Kilpatrick and P. C. Bressloff. Stability of bumps in piecewise smooth neural fields with nonlinear adaptation. *Physica D* **239** 1048-1060 (2010).

Z. Kilpatrick and P. C. Bressloff. Effects of synaptic depression and adaptation on spatiotemporal dynamics of an excitatory neuronal network. *Physica D* **239** 547-560 (2010).

Z. Kilpatrick and P. C. Bressloff. Spatially structured oscillations in a two--dimensional neuronal network with synaptic depression. *J. Comp. Neurosci.* **28** 193-209 (2010).

Z. Kilpatrick, S. E. Folias and P. C. Bressloff. Traveling pulses and wave propagation failure in an inhomogeneous neural network. *SIAM J. Appl. Dyn. Syst.* **7** 161-185 (2008).

P. C. Bressloff. Weakly interacting pulses in synaptically coupled excitable neural media. *SIAM J. Appl. Math.* **66** 57-81 (2006).

S. E. Folias and P. C. Bressloff. Stimulus-locked traveling waves and breathers in an excitatory neural network. *SIAM J. Appl. Math.* **65** 2067-2092 (2005).

S. E. Folias and P. C. Bressloff. Breathers in two-dimensional neural media. *Phys. Rev. Lett.* **95** 208107 (2005).

P. C. Bressloff and S. E. Folias. Front bifurcations in an excitatory neural network *SIAM J. Appl. Math.* **65** 131-151 (2005).

S. E. Folias and P. C. Bressloff. Breathing pulses in an excitatory neural network *SIAM J. Dyn. Syst.* **3** 378-407 (2004).

P. C. Bressloff, S. Folias, A Prat and Y-X Li. Oscillatory waves in inhomogeneous neural media *Phys. Rev. Lett.* **91** 178101 (2003).

P. C. Bressloff. Traveling fronts and wave propagation failure in an inhomogeneous neural network. *Physica D* **155**
83-100 (2001).

S. Carroll and P. C. Bressloff. Binocular rivalry waves in directionally selective neural field models. *Physica D* **285** 8-17 (2014).

M. A. Webber and P. C. Bressloff. The effects of noise on binocular rivalry waves: a stochastic neural field model. *J. Stat. Mech.* **3** P03001 (2013).

P. C. Bressloff and M. A. Webber. Neural field model of binocular rivalry waves. *J. Comput. Neurosci.* **32** 233-252 (2012).

Z. P. Kilpatrick and P. C. Bressloff. Binocular rivalry in a competitive neural network with synaptic depression. *SIAM J. Appl. Dyn. Syst.* **9** 1303-1347 (2010).

S. Carroll and P. C. Bressloff. Symmetric bifurcations in a neural field model for encoding the direction of spatial contrast gradients. *SIAM J. Appl. Dyn. Syst.* **17** 1-51 (2018).

S. Carroll and P. C. Bressloff. Phase equation for patterns of orientation selectivity in a neural field model of visual cortex. *SIAM J. Appl. Dan. Syst.* **15** 60-83 (2016).

P. C. Bressloff and S. Carroll. Laminar neural field model of laterally propagating waves of orientation selectivity. *PLoS Comput. Biol.* **11** e1004545 (2015).

P. C. Bressloff and S. M. Carroll. Spatio-temporal dynamics of neural fields on product spaces. * SIAM J. Appl. Dyns. Syst.* **13** 1620-1653 (2014).

P. C. Bressloff and J. D. Cowan. A spherical model for orientation and
spatial-frequency tuning in a cortical hypercolumn. *Phil. Trans. Roy. Soc. Lond. B* **357** 1643-1667 (2003).

P. C. Bressloff and J. D. Cowan. An amplitude approach to contextual effects in primary visual cortex. *Neural
Comput.* **14** 493-525 (2002).

P. C. Bressloff. Spatially periodic modulation of cortical patterns by long-range horizontal connections. *Physica D* ** 185** 131-157 (2003).

P. C. Bressloff and J. D. Cowan. Functional geometry of local and horizontal connections in a model of V1. *J. Physiol. (Paris)* **97** 221-236
(2003).

P. C. Bressloff. Bloch waves, periodic feature maps and cortical pattern formation. *Phys. Rev. Lett.* **89** 088101 (2002).

P. C. Bressloff and J. D. Cowan. The visual cortex as a crystal. *Physica D* **173** 226-258 (2002).

P. C. Bressloff, J. D. Cowan, M. Golubitsky and P. J. Thomas. Scalar and pseudoscalar bifurcations: pattern formation
on the visual cortex. *Nonlinearity* **14** 739-775 (2001).

P. C. Bressloff, J. D. Cowan, M. Golubitsky, P. J. Thomas and M. Wiener. Geometric visual hallucinations, Euclidean
symmetry and the functional architecture of striate cortex. *Phil. Trans. Roy. Soc. Lond. B* **356** 299-330 (2001).

A. M. Oster and P. C. Bressloff. A developmental model of ocular dominance formation on a growing cortex. *Bull. Math Biol.* **68** 73-98 (2006).

P. C. Bressloff. Spontaneous symmetry breaking in self-organizing neural fields. *Biol. Cybern.* **93** 256-274 (2005).