PAUL BRESSLOFF
Paul Bressloff
email: bressloff@math.utah.edu		 Professor of Mathematics

The Bressloff Lab
Brain and Behavior

RESEARCH:

Theoretical neuroscience

The main focus of our research is to use mathematical modeling and analysis to understand how the brain functions as a complex dynamical system. This work involves many different levels of description from the molecular basis of memory and learning at an individual synapse, to the physiological mechanisms underlying neuronal firing, to the properties of local circuits processing elements of a visual scene for example, to the large-scale structure of cortex responsible for higher cognitive function.

Mathematical modeling of primary visual cortex and its long-range circuitry

The primary visual cortex (V1) is the first cortical area to receive visual information transmitted by ganglion cells of the retina via the lateral geniculate nucleus (LGN) of the thalamus to the back of the brain. A fundamental property of the functional architecture of V1 is an orderly retinotopic mapping of the visual field onto its surface, with the left and right halves of the visual field mapped onto the right and left V1 cortices respectively.
The retino-cortical mapping

Superimposed upon this retinotopic map are a number of additional (approximately) periodic feature maps reflecting the fact that neurons respond preferentially to various aspects of a stimulus such as its orientation and left/right eye (ocular) dominance. Orientation preference changes continuously as a function of cortical location except at singularities or pinwheels that tend to be located at the center of ocular dominance columns. Around each orientation pinwheel is a ring of orientation selective cells.
LEFT: Distribution of orientation patches revealed by optical imaging [Courtesy of Gary Blasdel] RIGHT: Orientation pinwheels

The laterally spreading connectional fields made by pyramidal cells in superficial layers of V1 are broken into regularly spaced patches. These patchy lateral connections, which can extend up to 6mm in cortex, appear to link neurons with common functional properties such as orientation preference. Ongoing work by Dr. Alessandra Angelucci and Dr. Jennifer Lund at the Moran Eye Center, University of Utah, has revealed that feedback connections from higher cortical areas such as V2, V3 and MT appear to obey similar rules.
The patchy nature of lateral and feedback long-range connections in primate V1. [Courtesy of Dr. Alessandra Angelucci and Dr. Jennifer Lund (University of Utah)]

The long-range lateral and feedback connections are thought to provide a substrate for center-surround interactions. The surround fields are regions beyond the classical receptive field of a neuron where stimuli do not drive the cell, but modulate its response to stimuli placed within its classical receptive field. One common experimental paradigm is to investigate the response to stimuli consisting of circular (center) and annular (surround) gratings of differing contrasts, orientations and diameters. When both center and surround are stimulated with high contrast gratings close to the orientation of the unit's peak tuning response, the surround typically suppresses the center unit's tuning response. However, the degree of suppression tends to decrease, and sometimes becomes facilitatory, for surround stimulations at orientations sufficiently dissimilar to the preferred orientation of the center or at low contrasts.
Center-surround modulation

We are currently analyzing a large-scale dynamical model of primary visual cortex in order to understand the role of long-range lateral and feedback connections in modulating both the classical receptive field, which expands at low contrasts, and the inhibitory surround. [In collaboration with Dr. Jack Cowan, University of Chicago]. These processes are extremely important, since they provide a substrate for V1 to integrate spatial information across significant parts of the visual field, thus contributing to the creation of a global visual percept from local features. Such integration has traditionally been attributed to higher visual areas that are known to have larger classical receptive fields.

Geometric visual hallucinations

We have recently developed a theory of geometric visual hallucinations based on the original idea of Ermentrout and Cowan that some disturbance such as a drug or flickering light can destabilize the visual part of the brain inducing a spontaneous pattern of cortical activity. The geometry of the resulting hallucination thus reflects the intrinsic architecture of the visual cortex. So analyzing such patterns should help in deepening our understanding of how the brain processes images in normal vision.
LEFT: Spiral hallucination RIGHT: Spiral hallucination

Our work has focused on a continuum model of V1 in which cells signal both the position and orientation of a local stimulus. Using symmetric bifurcation theory, we have shown that the cortical activity patterns underlying common visual hallucinations can be accounted for in terms of certain symmetry properties of the lateral connections, specifically, that they are invariant under the action of the planar Euclidean group - the group of rigid motions in the plane - rotations, reflections and translations. The resulting representation is twisted due to an anisotropy in the lateral connections, which tends to favor directions that are correlated with the orientation preferences of the interacting cells. [In collaboration with Dr. Jack Cowan (University of Chicago) and Dr. Martin Golubitsky (University of Houston)]
LEFT: Pressure phosphene RIGHT: Honeycomb hallucination

In order to account for more complex hallucinations, we will need to extend our theory to incorporate additional properties of images such as color, depth and motion perception.
LEFT: Lattice tunnel hallucination RIGHT: Tunnel hallucination

Traveling waves in excitable neural media

A number of experimental studies have observed waves of excitation propagating in cortical slices when stimulated appropriately. The propagation velocity of such waves is of order 0.06 meters per sec, which is much slower than the typical speed of 0.5 meters per sec found for action potential propagation along axons. Traveling waves of electrical activity have also been observed in vivo. Often these traveling waves are found to occur during periods without sensory stimulation with the subsequent presentation of a stimulus inducing a switch to synchronous oscillatory behavior. Hence, determining the conditions under which cortical wave propagation occurs could be important for understanding the processing of sensory stimuli as well as more pathological forms of behaviour such as epileptic seizures and migraines.

The dependence of wave velocity on the underlying pattern of synaptic connectivity has been analysed in the case of homogeneous networks. However, certain care must be taken in interpreting these results since traveling wave solutions are sensitive to the degree of homogeneity in the connectivity pattern, particularly in the case of slowly moving waves. This is a consequence of the fact that traveling wavefronts are not structurally stable solutions, that is, they correspond to heteroclinic orbits of an equivalent dynamical system. We have recently used homegenization theory to show that in a simplified one-dimensional cortical model, inhomogeneities induced by long-range lateral connections can lead to propagation failure. Moreover, the conditions for propagation failure are sensitive to the precise structure of the weight distribution.

Coupled neural oscillators

The dynamics of coupled oscillator arrays has been the subject of much recent experimental and theoretical interest. Example systems include Josephson junctions, lasers, oscillatory chemical reactions, heart pacemaker cells, central pattern generators and cortical neural oscillators. Most work in this area has been concerned with smoothly coupled oscillators under the assumption of weak interactions so that averaging methods can be used to reduce the system to a phase model. On the other hand, many oscillators in nature communicate via pulses. Examples include neural oscillators, fireflies, digital phase-locked loops and certain models of self-organized criticality.   In collaboration with Dr Steve Coombes, we have recently developed a theory of strong coupling instabilities in networks of integrate-and-fire (IF) neurons. An IF neuron fires a spike whenever its state variable reaches some threshold, and immediately after firing the state variable is reset to some zero resting level - the dynamics can thus be reduced to a nonlinear mapping of the firing times (threshold-crossing times) of the neurons. For sufficiently slow synaptic interactions the network behaviour is compatible with a corresponding firing rate model obtained by performing a short-term time-average of the IF dynamics. Our theory has been applied to a number of neural systems including swimming locomotion in Xenopus, orientation tuning in primary visual cortex, and wave propagation in excitable neural tissue. We have also been developing integrate-and-fire models of cellular processes such as the intracellular propagation of calcium spikes, and the propagation of waves along dendrites with active spines.
Arnold tongues for a sinusoidally forced integrate-and-fire neuron

Selected Publications

Bressloff, P. C., and Cowan, J. D. The visual cortex as a crystal. Submitted to Physica D (2002). pdf

Bressloff, P. C. Bloch waves, periodic feature maps and cortical pattern formation. Phys. Rev. Lett. (2002). In press. pdf

Lund, J. S., Angelucci, A., and Bressloff, P. C. (2003) Anatomical substrates for the functional column in macaque primary visual cortex. Cerebral Cortex, 12:15-24.

Bressloff, P. C., and Cowan, J. D. Pattern Formation, Neural. In: Handbook of brain theory and neural networks (2nd edition); M. Arbib editor (MIT Press, 2002). In press.

Bressloff, P. C., and Cowan, J. D. Spontaneous pattern formation in primary visual cortex. In: Nonlinear dynamics: where do we go from here? S. J. Hogan, A. Champneys and B. Krauskopf editors (Institute of Physics: Bristol, 2002). In press. pdf

Bressloff, P. C., and Cowan, J. D. (2003) Spherical model of orientation and spatial frequency tuning in a cortical hypercolumn. Phil. Trans. Roy. Soc. B , 358:1643-1667. pdf

Bressloff, P. C., and Cowan, J. D. SO(3) symmetry breaking mechanism for orientation and spatial frequency tuning in visual cortex. Phys. Rev. Lett. (2002) 88:078102. pdf

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

Bressloff, P. C., Cowan, J. D., Golubitsky, M., Thomas, P. J., and Wiener, M. What geometric visual hallucinations tell us about the visual cortex. Neural Comput. (2002) 14:473-491. pdf

Bressloff, P. C., Cowan, J. D., Golubitsky, M., Thomas, P. J., and Wiener, M. Geometric visual hallucinations, Euclidean symmetry and the functional architecture of striate cortex. Phil. Trans. Roy. Soc. B (2001) 40:299-330. pdf

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

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

Bressloff, P. C., and Coombes, S. Mathematical reduction techniques for modeling biophysical neural networks. In: Biophysical Neural networks: Foundations of analytical neuroscience (edited by R. R. Poznanski); Mary Ann Liebert, Inc., New York. (2001) pp. 215-269.

Bressloff, P. C. Traveling waves and pulses in a one-dimensional network of integrate-and-fire neurons. J. Math. Biol. (2000) 40:169-183. pdf

Bressloff, P. C., and Coombes, S. Dynamical theory of spike train dynamics in networks of integrate-and-fire oscillators. SIAM J. Appl. Math. (2000) 60:828-841. pdf

Coombes, S., and Bressloff, P. C. Solitary waves in a model of dendritic cable with active spines. SIAM J. Appl. Math. (2000) 61:432-453. pdf

Bressloff, P. C., and Coombes, S. Dynamics of strongly coupled spiking neurons. Neural Comput. (2000) 12:91-129. pdf

Roper, P. N., Bressloff, P. C., and Longtin, A. A temperature-dependent phase model of mammalian cold receptors. Neural Comput. (2000) 12:1067-1093. pdf

Bressloff, P. C., Bressloff, N. W., and Cowan, J. D. Dynamical mechanism for sharp orientation tuning in an integrate-and-fire model of cortical hypercolumns. Neural Comput. (2000) 12:2473-2511. pdf

For more information contact Paul C. Bressloff, 5-1633
E-mail:
bressloff@math.utah.edu

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