zbMATH — the first resource for mathematics

Spatially periodic modulation of cortical patterns by long-range horizontal connections. (English) Zbl 1030.92006
Summary: We analyze spontaneous pattern formation in a continuum model of primary visual cortex that incorporates spatially periodic inhomogeneities in the distribution of long-range horizontal connections. These inhomogeneities reflect the underlying crystalline-like structure of the cortex, as exemplified by the distribution of cytochrome oxidase blobs.
We first solve the linear eigenvalue problem for a primary Turing instability, and show how the resulting activity pattern can lock to the underlying cortical lattice when certain resonance conditions are satisfied. This result is then extended to the weakly nonlinear regime by performing a multiple scale perturbation expansion of the nonlocal integro-differential equation that determines cortical activity. The resulting amplitude equation describes the effects of long-wavelength modulations of a primary roll pattern in the presence of periodic inhomogeneities, and is shown to exhibit a commensurate-incommensurate transition analogous to that found in convective fluid systems with external periodic forcing. Extensions of our theory to models of cortical development are briefly discussed.

92C20 Neural biology
45K05 Integro-partial differential equations
45M10 Stability theory for integral equations
Full Text: DOI
[1] N.W. Ashcroft, N.D. Mermin, Solid State Physics, Saunders, London, 1976.
[2] Alloway, K.D.; Crist, J.; Mutic, J.J.; Roy, S.A., Corticostriatal projections from rat barrel cortex have an anisotropic organization that correlates with vibrissal whisking behavior, J. neurosci., 19, 10908-10922, (1999)
[3] Angelucci, A.; Levitt, J.B.; Walton, E.J.S.; Hupe, J.-M.; Bullier, J.; Lund, J.S., Circuits for local and global signal integration in primary visual cortex, J. neurosci., 22, 8633-8646, (2002)
[4] Bak, P., Commensurate phases, incommensurate phases and the devil’s staircase, Rep. prog. phys., 45, 587-629, (1982)
[5] Blasdel, G.G., Orientation selectivity, preference, and continuity in monkey striate cortex, J. neurosci., 12, 3139-3161, (1992)
[6] Blasdel, G.G.; Campbell, D., Orientation selectivity, preference, and continuity in monkey striate cortex, J. neurosci., 21, 8286-8301, (2001)
[7] Bosking, W.H.; Zhang, Y.; Schofield, B.; Fitzpatrick, D., Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex, J. neurosci., 17, 2112-2127, (1997)
[8] Bressloff, P.C.; Cowan, J.D.; Golubitsky, M.; Thomas, P.J.; Wiener, M., Geometric visual hallucinations, Euclidean symmetry and the functional architecture of striate cortex, Phil. trans. roy. soc. lond. B, 356, 299-330, (2001)
[9] Bressloff, P.C.; Cowan, J.D.; Golubitsky, M.; Thomas, P.J., Scalar and pseudoscalar bifurcations: pattern formation on the visual cortex, Nonlinearity, 14, 739-775, (2001) · Zbl 1017.37025
[10] Bressloff, P.C., Bloch waves, periodic feature maps and cortical pattern formation, Phys. rev. lett., 89, 088101, (2002)
[11] P.C. Bressloff, J.D. Cowan, The visual cortex as a crystal, Physica D 173 (2002) 226-258. · Zbl 1017.92009
[12] P.C. Bressloff, J.D. Cowan, Spontaneous pattern formation in primary visual cortex, in: S.J. Hogan, A. Champneys, B. Krauskopf (Eds.), Nonlinear Dynamics and Chaos: Where Do We Go From Here? Institute of Physics, Bristol, 2002.
[13] Calloway, E.M.; Katz, L.C., Experience and refinement of clustered horizontal connections in cat striate cortex, J. neurosci., 10, 1134-1154, (1990)
[14] Chiu, C.; Weliky, M., Relationship of correlated spontaneous activity to functional ocular dominance columns in the developing visual cortex, Neuron, 35, 1123-1134, (2002)
[15] Coullet, P., Commensurate – incommensurate transition in nonequilibrium systems, Phys. rev. lett., 56, 724-727, (1986)
[16] Cowan, J.D., Spontaneous symmetry breaking in large scale nervous activity, Int. J. quant. chem., 22, 1059-1082, (1982)
[17] Cross, M.C.; Newell, A.C., Convection patterns in large aspect ration systems, Physica D, 10, 299-328, (1984) · Zbl 0592.76052
[18] Cross, M.C.; Hohenberg, P.C., Pattern formation outside of equilibrium, Rev. mod. phys., 65, 851-1111, (1993) · Zbl 1371.37001
[19] Ermentrout, G.B.; Cowan, J.D., A mathematical theory of visual hallucination patterns, Biol. cybernet., 34, 137-150, (1979) · Zbl 0409.92008
[20] Ermentrout, G.B.; Cowan, J.D., Secondary bifurcation in neuronal nets, SIAM J. appl. math., 39, 323-340, (1980) · Zbl 0453.92007
[21] Ermentrout, G.B., Neural networks as spatial pattern forming systems, Rep. prog. phys., 61, 353-430, (1998)
[22] Fitzpatrick, D., Seeing beyond the receptive field in primary visual cortex, Curr. opt. neurobiol., 10, 438-443, (2000)
[23] Gilbert, C.D.; Wiesel, T.N., Clustered intrinsic connections in cat visual cortex, J. neurosci., 3, 1116-1133, (1983)
[24] Hirsch, J.D.; Gilbert, C.D., Synaptic physiology of horizontal connections in the cat’s visual cortex, J. physiol. lond., 160, 106-154, (1992)
[25] Horton, J.C., Cytochrome oxidase patches: a new cytoarchitectonic feature of monkey visual cortex, Phil. trans. roy. soc. lond. B, 304, 199-253, (1984)
[26] S. LeVay, S.B. Nelson, Columnar organization of the visual cortex, in: A.G. Leventhal (Ed.), The Neural Basis of Visual Function, CRC Press, Boca Raton, 1991, pp. 266-315.
[27] Levitt, J.B.; Lewis, D.A.; Yoshioka, T.; Lund, J.S., Topography of pyramidal neuron intrinsic connections in macaque prefrontal cortex, J. comp. neurol., 338, 360-376, (1993)
[28] Livingstone, M.S.; Hubel, D.H., Specificity of intrinsic connections in primate primary visual cortex, J. neurosci., 4, 2830-2835, (1984)
[29] Lowe, M.; Gollub, J.P., Solitons and the commensurate – incommensurate transition in a convective nematic fluid, Phys. rev. A, 31, 3893-3896, (1985)
[30] Lund, J.S.; Angelucci, A.; Bressloff, P.C., Anatomical substrates for functional columns in macaque monkey primary visual cortex, Cereb. cortex, 12, 15-24, (2003)
[31] Malach, R.; Amir, Y.; Harel, M.; Grinvald, A., Relationship between intrinsic connections and functional architecture revealed by optical imaging and in vivo targeted biocytin injections in primate striate cortex, Proc. natl. acad. sci., 90, 10469-10473, (1993)
[32] Melchitzky, D.S.; Sesack, S.R.; Pucak, M.L.; Lewis, D.A., Synaptic targets of pyramidal neurons providing intrinsic horizontal connections in monkey prefrontal cortex, J. comp. neurol., 390, 211-224, (1998)
[33] Murphy, K.M.; Jones, D.G.; Fenstemaker, S.B.; Pegado, V.D.; Kiorpes, L.; Movshon, J.A., Spacing of cytochrome oxidase blobs in visual cortex of normal and strabismic monkeys, Cereb. cortex, 8, 237-244, (1998)
[34] Murphy, K.M.; Duffy, K.R.; Jones, D.G.; Mitchell, D.E., Development of cytochrome oxidase blobs in visual cortex of normal and visually deprived cats, Cereb. cortex, 11, 237-244, (2001)
[35] Newell, A.C.; Whitehead, J.A., Finite bandwidth, finite amplitude convection, J. fluid mech., 38, 279, (1969) · Zbl 0187.25102
[36] Obermayer, K.; Blasdel, G., Geometry of orientation and ocular dominance columns in monkey striate cortex, J. neurosci., 13, 4114-4129, (1993)
[37] Read, H.L.; Winer, J.A.; Schreiner, C.E., Modular organization of intrinsic connections associated with spectral tuning in cat auditory cortex, Proc. natl. acad. sci. USA, 98, 8042-8047, (2001)
[38] Rockland, K.S.; Lund, J.S., Intrinsic laminar lattice connections in primate visual cortex, J. comp. neurol., 216, 303-318, (1983)
[39] Ruthazer, E.S.; Stryker, M.P., The role of activity in the development of long-range horizontal connections in area 17 of the ferret, J. neurosci., 16, 7253-7269, (1996)
[40] Segel, L.A., Distant side-walls cause slow amplitude modulation of cellular convection, J. fluid mech., 38, 203, (1969) · Zbl 0179.57501
[41] Sincich, L.C.; Horton, J.C., Divided by cytochrome oxidase: a map of the projections from V1 to V2 in macaques, Science, 295, 1734-1737, (2002)
[42] Swindale, N.V., A model for the formation of ocular dominance stripes, Proc. roy. soc. B, 208, 243-264, (1980)
[43] Swindale, N.V., The development of topography in visual cortex: a review of models, Network, 7, 161-247, (1996) · Zbl 0903.92012
[44] Toth, L.J.; Rao, S.C.; Kim, D.; Somers, S.; Sur, M., Subthreshold facilitation and suppression in primary visual cortex revealed by intrinsic signal imaging, Proc. natl. acad. sci., 93, 9869-9874, (1996)
[45] Trepel, C.; Duffy, K.R.; Pegado, V.D.; Murphy, K.M., Patchy distribution of NMDAR1 subunit immunoreactivity in developing visual cortex, J. neurosci., 18, 3404-3415, (1998)
[46] D. Walgraef, Spatio-Temporal Pattern Formation, Springer, Berlin, 1997. · Zbl 0871.76003
[47] Werner, H.; Richter, T., Circular stationary solutions in two-dimensional neural fields, Biol. cybernet., 85, 211-217, (2001) · Zbl 1160.92323
[48] Wilson, H.R.; Cowan, J.D., A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue, Kybernetik, 13, 55-80, (1973) · Zbl 0281.92003
[49] Yoshioka, T.; Blasdel, G.G.; Levitt, J.B.; Lund, J.S., Relation between patterns of intrinsic lateral connectivity, ocular dominance, and cytochrome oxidase-reactive regions in macaque monkey striate cortex, Cereb. cortex, 6, 297-310, (1996)
[50] Yabuta, N.H.; Callaway, E.M., Cytochrome-oxidase blobs and intrinsic horizontal connections of layer 2/3 pyramidal neurons in primate V1, Visual neurosci., 15, 1007-1027, (1998)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.