Formation of topographic maps and columnar microstructures in nerve fields

Topographic connections are found in many parts of the vertebrate nervous systems, known for example as retinotopy. The self-organizing ability of Hebb type modifiable synapses plays an important role in forming, at least in refining, the topographic connections. We present a mathematical analysis of a revised version of the Willshaw-Malburg model of topographic formation, solving the equations of synaptic self-organization coupled with the field equation of neural excitations. The equilibrium solutions are obtained and their stability is studied. It is proved that two cases exist depending on parameters. In one case, the smooth topographic organization is obtained as a stable equilibrium of the equations. In the other case, this solution becomes unstable, and instead the topographic organization with columnar microstructures appears. This might explain the columnar structures in the cerebrum. The theory is confirmed by computer simulated experiments.

[1]  S. Amari,et al.  Mathematical theory on formation of category detecting nerve cells , 1978, Biological Cybernetics.

[2]  S. Amari.  Dynamics of pattern formation in lateral-inhibition type neural fields , 1977, Biological Cybernetics.

[3]  D. Willshaw,et al.  On a role for competition in the formation of patterned neural connexions , 1975, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[4]  J. Szentágothai.  The Ferrier Lecture, 1977 The neuron network of the cerebral cortex: a functional interpretation , 1978, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[5]  S. Geman.  SOME AVERAGING AND STABILITY RESULTS FOR RANDOM DIFFERENTIAL EQUATIONS , 1979 .

[6]  S.-I. Amari,et al.  Neural theory of association and concept-formation , 1977, Biological Cybernetics.

[7]  S. Sharma,et al.  Reformation of retinotectal projections after various tectal ablations in adult goldfish. , 1972, Experimental neurology.

[8]  V. Mountcastle.  Modality and topographic properties of single neurons of cat's somatic sensory cortex. , 1957, Journal of neurophysiology.

[9]  D. Hubel,et al.  Receptive fields of single neurones in the cat's striate cortex , 1959, The Journal of physiology.

[10]  R. M. Gaze,et al.  The Visual System and “Neuronal Specificity” , 1972, Nature.

[11]  R. M. Gaze,et al.  The arrow model: retinotectal specificity and map formation in the goldfish visual system , 1976, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[12]  S. Amari.  Topographic organization of nerve fields , 1979, Neuroscience Letters.

[13]  Roman Bek,et al.  Discourse on one way in which a quantum-mechanics language on the classical logical base can be built up , 1978, Kybernetika.

[14]  M. Yoon.  Transposition of the visual projection from the nasal hemiretina onto the foreign rostral zone of the optic tectum in goldfish. , 1972, Experimental neurology.

[15]  C. Malsburg,et al.  How patterned neural connections can be set up by self-organization , 1976, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[16]  S. Amari,et al.  Existence and stability of local excitations in homogeneous neural fields , 1979, Journal of mathematical biology.

[17]  C. Malsburg.  Self-organization of orientation sensitive cells in the striate cortex , 2004, Kybernetik.

[18]  A. Towe.  Notes on the hypothesis of columnar organization in somatosensory cerebral cortex. , 1975, Brain, behavior and evolution.