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).