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Topographic organization of nerve fields

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Abstract

The vertebrate nervous system has topographic interconnections in many parts, known for example as retinotopy, somatotopy, etc. It is plausible that modifiable synapses play an important role in forming and refining these connections together with the sensory experiences. To elucidate the mechanism of topographic organization, we propose a simple model consisting of two nerve fields connected by modifiable excitatory synapses. The model also includes modifiable inhibitory synapses. The behavior of the model is described by a set of simultaneous non-linear integro-differential equations. By analyzing the equations, we obtain the equilibrium solution of topographic connections. It is also proved that a part of the presynaptic field which is frequently stimulated comes to be mapped on a large area of the postsynaptic field so that it has a good resolution.

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Literature

  • Arbib, M. A. 1972.The Metaphorical Brain. New York: Wiley.

    MATH  Google Scholar 

  • Amari, S. 1972. “Characteristics of Random Nets of Analog Neuron-like Elements.”IEEE Trans., Systems, Man and Cybernetics.SMC-2, 643–657.

    Article  MATH  Google Scholar 

  • — 1974. “A Method of Statistical Neurodynamics.”Kybernetik 14, 201–215.

    MATH  MathSciNet  Google Scholar 

  • — 1977a. “Dynamics of Pattern Formation in Lateral-inhibition Type Neural Fields.”Biol. Cybernet. 27, 77–87.

    Article  MATH  MathSciNet  Google Scholar 

  • — 1977b. “Neutral Theory of Association and Concept-formation.”Biol. Cybernet. 26, 175–185.

    Article  MATH  MathSciNet  Google Scholar 

  • —, and Arbib, M. A. 1977. “Competition and Cooperation in Neural Nets.”Systems Neuroscience Ed. J. Metzler, 119–165. New York: Academic Press.

    Google Scholar 

  • —, and Takeuchi, A. 1978. “Mathematical Theory on Formation of Category Detecting Nerve Cells.”Biol. Cybernet. 29, 127–136.

    Article  MATH  MathSciNet  Google Scholar 

  • —, Yoshida, K. and Kanatani, K. 1977. “A Mathematical Foundation for Statistical Neurodynamics.”SIAM J. Appl. Math. 33, 95–126.

    Article  MATH  MathSciNet  Google Scholar 

  • Gaze, R. M. 1970.The Formation of Nerve Connections. London: Academic Press.

    Google Scholar 

  • —, Jacobson, M. and Szekely, G. 1965. “On the Formation of Connections by Compound Eyes inXenopus”.J. Physiol. 176, 409–417.

    Google Scholar 

  • — and Keating, M. J. 1972. “The Visual System and ‘Neuronal Specificity’.”Nature, Lond. 237, 375–378.

    Article  Google Scholar 

  • Geman, S. 1979. “Some Averaging and Stability Results for Random Differential Equations.”SIAM J. Appl. Math. 36, 86–105.

    Article  MATH  MathSciNet  Google Scholar 

  • Hope, R. A., Hammond, B. J. and Gaze, R. M. 1976. “The Arrow Model—Retinotectal Specificity and Map Formation in the Goldfish Visual System.”Proc. R. Soc. B 194, 447–466.

    Google Scholar 

  • Kishimoto, K. and Amari, S. 1979. “Existence and Stability of Local Excitations in Neural Fields.”J. Math. Biol. 7, 303–318.

    Article  MATH  MathSciNet  Google Scholar 

  • Kohonen, T. and Oja, E. 1976. “Fast Adaptive Formation of Orthogonalizing Filters and Associative Memory in Recurrent Networks of Neuron-like Elements.”Biol. Cybernet. 21, 85–95.

    Article  MATH  MathSciNet  Google Scholar 

  • Malsburg, C. von der. 1973. “Self-organization of Orientation Sensitive Cells in the Striate Cortex.”Kybernetik 14, 85–100.

    Article  Google Scholar 

  • Prestige, M. C. and Willshaw, D. J. 1975. “On a Role for Competition in the Formation of Patterned Neural Connections.”Proc. R. Soc. B 190, 77–98.

    Article  Google Scholar 

  • Sharma, S. C. 1972. “Reformation of Retinotectal Connections after Various Tectal Ablations in Adult Goldfish.”Exp. Neurol. 34, 171–182.

    Article  Google Scholar 

  • Sperry, R. W. 1951. “Mechanism of Neural Maturation.”Handbook of Experimental Psychology, Ed. S. S. Steven, pp. 236–280. New York: Wiley.

    Google Scholar 

  • Takeuchi, A. and Amari, S. “Formation of Topographic Maps and Columnar Microstructures in Nerve Fields.”Biol. Cybernet. (in press).

  • Willshaw, D. J. and Malsburg, C. von der. 1976. “How Patterned Neural Connections can be Set up by Self Organization.”Proc. R. Soc. B 194, 431–445.

    Google Scholar 

  • Wilson, H. R. 1975. “A Synaptic Model for Spatial Frequency Adaptation.”J. Theoret. Biol. 50, 327–352.

    Article  Google Scholar 

  • Yoon, M. 1971. “Reorganization of Retinotectal Projection Following Surgical Operations on the Optic Tectum in Goldfish.”Expl. Neurol. 33, 395–411.

    Article  Google Scholar 

  • —, 1972. “Transposition of the Visual Projection from the Nasal Hemi-retina onto the Foreign Rostral Zone of the Optic Tectum in Goldfish.”Expl. Neurol. 37, 451–462.

    Article  Google Scholar 

  • Zeeman, C. 1962. “The topology of the Brain and Visual Perception.”Topology of 3-Manifolds and Related Topics, Ed. M. K. Fort, pp. 240–256, Englewood Cliffs: Prentice-Hall.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Engineering and Instrumentation Physics, University of Tokyo, 113, Tokyo, Japan

    Shun-Ichi Amari

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  1. Shun-Ichi Amari
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Amari, SI. Topographic organization of nerve fields. Bltn Mathcal Biology 42, 339–364 (1980). https://doi.org/10.1007/BF02460791

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  • DOI: https://doi.org/10.1007/BF02460791

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