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Two-bump solutions of Amari-type models of neuronal pattern formation. (English) Zbl 1011.92007

Summary: We study a partial integro-differential equation defined on a spatially extended domain that arises in the modeling of pattern formation in neuronal networks. For a one-dimensional domain we develop criteria for the existence and stability of equal-width “2-bump” solutions under the assumption that the firing rate function is the Heaviside function. We apply these criteria to an example for which the connectivity is of lateral inhibition type (i.e., the coupling function has one positive zero) and find that families of 2-bump solutions exist, but none of the solutions is stable. Extensive numerical searches suggest that this is true for all coupling functions of this form. However, for a large class of coupling functions which have three positive zeros, we find the coexistence of both stable and unstable 2-bump solutions. We also extend our investigation to two spatial dimensions and give numerical evidence for the coexistence of 1-bump and 2-bump solutions. Our results imply that lateral inhibition type coupling is not sufficient to produce stable patterns that are more complex than single isolated patches of high activity.

MSC:

92B20 Neural networks for/in biological studies, artificial life and related topics
45K05 Integro-partial differential equations
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[1] Amari, S., Dynamics of pattern formation in lateral-inhibition type neural fields, Biol. cybern., 27, 77-87, (1977) · Zbl 0367.92005
[2] Asaad, W.F.; Rainer, G.; Miller, E.K., Task-specific neural activity in the primate prefrontal cortex, J. neurophysiol., 84, 451-459, (2000)
[3] 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)
[4] Bressloff, P.C., Traveling fronts and wave propagation failure in an inhomogeneous neural network, Physica D, 155, 83-100, (2001) · Zbl 1004.92005
[5] Bressloff, P.C., Bloch waves, periodic feature maps, and cortical pattern formation, Phys. rev. lett., 89, 088101, (2002)
[6] 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. B, 40, 299-330, (2001)
[7] Bressloff, P.C.; Cowan, J.D.; Golubitsky, M.; Thomas, P.J.; Wiener, M., What geometric visual hallucinations tell us about the visual cortex, Neural comput., 14, 473-491, (2002) · Zbl 1037.91083
[8] Compte, A.; Brunel, N.; Goldman-Rakic, P.; Wang, X.-J., Synaptic mechanisms and network dynamics underlying spatial working memory in a cortical network model, Cereb. cortex, 10, 910-923, (2000)
[9] S. Coombes, G.L. Lord, M.R. Owen, Waves and bumps in neuronal networks with axo-dendritic synaptic interactions, Physica D 178 (2003) 219-241. · Zbl 1013.92006
[10] Durstewitz, D.; Seamans, J.K.; Sejnowski, T.J., Neurocomputational models of working memory, Nat. neurosci., 3, 1184-1191, (2000)
[11] Ermentrout, G.B., Neural networks as spatio-temporal pattern forming systems, Rep. prog. phys., 61, 4, 353-430, (1998)
[12] M.A. Giese, Dynamic Neural Field Theory for Motion Perception, Kluwer Academic Publishers, Boston, 1998.
[13] Gutkin, B.; Ermentrout, G.B.; O’Sullivan, J., Layer 3 patchy recurrent connections may determine the spatial organization of sustained activity in the primate frontal cortex, Neurocomputing, 32-33, 391-400, (2000)
[14] Gutkin, B.S.; Laing, C.R.; Chow, C.C.; Ermentrout, G.B.; Colby, C.R., Turning on and off with excitation: the role of spike-timing asynchrony and synchrony in sustained neural activity, J. comput. neurosci., 11, 2, 121-134, (2001)
[15] D. Hansel, H. Sompolinsky, Modeling feature selectivity in local cortical circuits, in: C. Koch, I. Segev (Eds.), Methods in Neuronal Modeling, 2nd ed., MIT Press, Cambridge, MA, 1998. · Zbl 0948.68154
[16] Kishimoto, K.; Amari, S., Existence and stability of local excitations in homogeneous neural fields, J. math. biol., 7, 303-318, (1979) · Zbl 0406.92010
[17] Laing, C.R.; Chow, C.C., Stationary bumps in networks of spiking neurons, Neural comp., 13, 7, 1473-1494, (2001) · Zbl 0978.92004
[18] Laing, C.R.; Troy, W.C.; Gutkin, B.; Ermentrout, G.B., Multiple bumps in a neuronal model of working memory, SIAM J. appl. math., 63, 62-97, (2002) · Zbl 1017.45006
[19] Levitt, J.B.; Lewis, D.A.; Yoshioka, T.; Lund, J.S., Topography of pyramidal neuron intrinsic connections in macaque monkey prefrontal cortex (areas 9 and 46), J. comp. neurol., 338, 360-376, (1993)
[20] Lewis, D.A.; Anderson, S.A., The functional architecture of the prefrontal cortex and schizophrenia, Psychol. med., 25, 5, 887-894, (1995)
[21] 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. USA, 90, 10469-10473, (1993)
[22] Miller, E.K.; Erickson, C.A.; Desimone, R., Neural mechanisms of visual working memory in prefrontal cortex of the macaque, J. neurosci., 16, 16, 5154-5167, (1996)
[23] Pouget, A.; Snyder, L.H., Computational approaches to sensorimotor transformations, Nat. neurosci., 3, 1192-1198, (2000)
[24] Rainer, G.; Asaad, W.F.; Miller, E.K., Memory fields of neurons in the primate prefrontal cortex, Proc. natl. acad. sci. USA, 95, 15008-15013, (1998)
[25] Roerig, B.; Chen, B., Relationships of local inhibitory and excitatory circuits to orientation preference maps in ferret visual cortex, Cereb. cortex, 12, 187-198, (2002)
[26] Stringer, S.M.; Trappenberg, T.P.; Rolls, E.T.; de Araujo, I.E.T., Self-organizing continuous attractor networks and path integration: one-dimensional models of head direction cells, Network-comp. neural, 13, 217-242, (2002) · Zbl 1025.92004
[27] Taylor, J.G., Neural bubble dynamics in two dimensions: foundations, Biol. cybern., 80, 393-409, (1999) · Zbl 0984.92009
[28] E. Thelen, G. Schöner, C. Scheier, L.B. Smith, The dynamics of embodiment: a field theory of infant perseverative reaching, Behav. Brain Sci. 24 (1) (2000). http://www.cogsci.soton.ac.uk/bbs/Archive/bbs.thelen.html.
[29] Wang, X.J., Synaptic reverberation underlying mnemonic persistent activity, Trends neurosci., 24, 8, 455-463, (2001)
[30] Werner, H.; Richter, T., Circular stationary solutions in two-dimensional neural fields, Biol. cybern., 85, 117-211, (2001) · Zbl 1160.92323
[31] Wilson, H.R.; Cowan, J.D., A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue, Kybernetic, 13, 55-80, (1973) · Zbl 0281.92003
[32] Zhang, K., Representation of spatial orientation by the intrinsic dynamics of the head-direction cell ensemble: a theory, J. neurosci., 16, 2112-2126, (1996)
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