Jack D. Cowan, PhD

Professor

             MATH/CNS 32000

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Research Summary

Research in my group is concerned with understanding how circuits in the visual cortex process information. We use mathematics as our main tool to investigate such problems.

Some 35 years ago, I introduced a set of nonlinear differential equations to describe neural net activity, in which I approximated the nonlinear response of a neuron to injected current by a smooth “squashing” nonlinearity. Five years later I developed a neural field theory (with H.R. Wilson) and then showed (with G.B. Ermentrout) how to use nonlinear stability theory (i.e. bifurcation theory and symmetry groups) to analyze how patterns of stable activity could arise in neural networks. About 10 years ago, the neuroscience community began to take seriously our early suggestion that recurrent excitation in cortical circuits was an important mechanism underlying the tuning properties seen in the responses of most cortical neurons to external stimulation. Over the past decade, I have worked with a series of graduate students and collaborators (M.C. Wiener, T. Mundel, A. Dimitrov, P. J. Thomas; and (recently) P. C. Bressloff and M. Golubitsky) to sharpen and elaborate on this suggestion, and to develop a detailed theory of the action of the visual cortex. This work continues with a new group of graduate students.

Some Selected Papers

S. Winograd and J.D. Cowan (1963). Reliable Computation in the Presence of Noise. MIT Press.

J.D. Cowan (1968). Statistical Mechanics of Nervous Nets. In: "Proceedings of 1967 NATO Conference on Neural Networks", Ed. E.R. Caianiello, Springer-Verlag, pp. 181-188.

H.R. Wilson and J.D. Cowan (1972). Excitatory and Inhibitory Interactions in Localized Populations of Model Neurons. Biophysical J., 12: 1-24.

H.R. Wilson and J.D. Cowan (1973). A Mathematical Theory of the Functional Dynamics of Cortical and Thalamic Nervous Tissue. Kybernetik, 13: 55-80.

E.G. Butz and J.D. Cowan (1974). Transient Potentials in Neurons with Arbitrary Geometry. Biophysical J., 14: 1-22.

G.B. Ermentrout and J.D. Cowan (1979). A Mathematical Theory of Visual Hallucination Patterns. Biological Cybernetics, 34: 137-150.

V.A. Whitelaw and J.D. Cowan (1981). Specificity and Plasticity of Retinotectal Connections: A Computational Model. J. Neuroscience, 1: 1369-1387.

J.D. Cowan and A.E. Friedman (1991). Simple Spin Models for the Development of Ocular Dominance Columns and Iso-orientation Patches. Advances in Neural Information Processing Systems, 3: 26-31.

C. Weber, H Ritter, J. Cowan and K. Obermayer (1997). A Model for Intrinsic and Activity Dependent Mechanisms underlying the Development and Regeneration of the Retinotectal Map in Goldfish. Phil.Trans.Roy.Soc. (Lond) B, 352: 1603-1623.

T. Mundel, A. Dimitrov, and J.D. Cowan (1997). Visual Cortex Circuitry and Orientation Tuning. Advances in Neural Information Processing Systems, 9: 887-893.

A. Dimitrov, and J.D. Cowan (1998). Spatial Decorrelation in Orientation-Selective Cortical Cells. Neural Computation, 10: 1779-1795.

P.C. Bressloff, N.W. Bressloff, and J.D. Cowan (2000). Dynamical Mechanism for Sharp Orientation Tuning in an Integrate-and-Fire Model of a Cortical Hypercolumn. Neural Computation, 12: 2473-2512. 

P. C. Bressloff, J.D. Cowan, M. Golubitsky, P. J. Thomas and M.C. Wiener (2001). Geometric Visual Hallucinations, Euclidean Symmetry, and the Functional Architecture of Striate Cortex. Phil.Trans.Roy.Soc. (Lond) B, 356: 1-32.

Updated 9/17/04.