Abstract
Spatial characteristics of brain matter affect dynamics of informational flow. It seems important to investigate into the topology of neural information to better understand biological neural nets as well as for their computer science analogs. Mathematical braids are proposed as tool for modeling the neuronal topology. Neurological basis of neuronal path is reviewed. We demonstrate mathematical algorithms for path description and transformation. A simulation environment for neural braid construction and transformation is implemented. Experimental evaluation of 1310719 braid-defined neural topologies illustrates how neural path intersections affect information processing and memory recall. The mathematical representation of synaptic pruning is proposed. Pruning of neural nets shows the applicability of the approach to the simplification of neural graphs for computational resource saving.
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Bhambhani, V., Valbuena-Reyes, L., and Tanner, H., Spatially distributed cellular neural networks, Int. J. Intelligent Comput. Cybernetics, 2011, vol. 4, no. 4, pp. 465–486.
Sporns, O., Structure and function of complex brain networks, Dialogues Clin. Neurosci., 2013, vol. 15, no. 3, pp. 247–262.
Coddington, L.T., Nietz, A.K., and Wadiche, J.I., The contribution of extrasynaptic signaling to cerebellar information processing, Cerebellum, 2014, vol. 13, no. 4, pp. 513–520.
Hebb, D., The Organization of Behavior: A Neurophysiological Theory, New York: Wiley, 1949.
Yoder, J. and Yaeger, L., Evaluating topological models of neuromodulation in polyworld, in Artificial Life 14: Proceedings of the Fourteenth International Conference on the Synthesis and Simulation of Living Systems, Cambridge, MA: MIT Press, 2014, pp. 916–923.
Kassel, C. and Turaev, V., Braid Groups, Graduate Texts in Mathematics, 2007.
Tower, D.B., Structural and functional organization of mammalian cerebral cortex: The correlation of neurone density with brain size, Cortical neurone density in the fin whale (Balaenoptera Physalus L.) with a note on the cortical neuronedensity in the Indian elephant, J. Comp. Neurol., 1954, vol. 101, pp. 19–51.
Tonegawa, S., Liu, X., Ramirez, S., and Redondo, R., Memory engram cells have come of age, Neuron, 2015, vol. 87, no. 5, pp. 918–931.
Josselyn, S.A., Kohler, S., and Frankland, P.W., Finding the engram, Nat. Rev. Neurosci., 2015, vol. 16, pp. 521–534.
Izhikevich, E.M., Polychronization: Computation with spikes, Neural Comput., 2006, vol. 18, pp. 245–282.
Fernando, C., Vasas, V., Szathmáry, E., and Husbands, P., Evolvable neuronal paths: A novel basis for information and search in the brain, PLoS ONE, 2011, vol. 6, no. 8, p. e23534.
De-Miguel, F.F. and Fuxe, K., Extrasynaptic neurotransmission as a way of modulating neuronal functions, Front. Physio., 2012, vol. 3, p. 16.
Chistopolsky, I.A., Vorontsov, D.D., and Sakharov, D.A., Monitoring of neuroactive factors released from a pattern-generating network, Acta Biol. Hungarica, 2008, vol. 59, pp. 29–31.
Staras, K., Kemenes, G., and Benjamin, P.R., Pattern-generating role for motoneurons in a rhythmically active neuronal network, J. Neurosci., 1998, vol. 18, pp. 3669–3688.
De Pittà, M., Volman, V., Berry, H., and Ben-Jacob, E., A tale of two stories: Astrocyte regulation of synaptic depression and facilitation, PLoS Comput Biol., 2011, vol. 7, no. 12, p. e1002293.
Schafer, D.P., Lehrman, E.K., Kautzman, A.G., Koyama, R., Mardinly, A.R., Yamasaki, R., Ransohoff, R.M., Greenberg, M.E., Barres, B.A., and Stevens, B., Microglia sculpt postnatal neural circuits in an activity and complement-dependent manner, Neuron, 2012, vol. 74, no. 4, pp. 691–705.
Chechik, G., Meilijson, I., and Ruppin, E., Synaptic pruning in development: A computational account, Neural Computation, 1998, vol. 10, no. 7, pp. 1759–1777.
Sossinsky, A., Knots, Mathematics with a Twist, Harvard University Press, 2002.
Dehornoy, P., Efficient solutions to the braid isotopy problem, Disc. Appl. Math, 2008, vol. 156, pp. 3094–3112.
Dehornoy, P., A fast method for comparing braids, Adv. Math., 1997, vol. 125, pp. 200–235.
Dehornoy, P., Dynnikov, I., Rolfsen, D., and Wiest, B., Why are braids orderable? Panoramas et Synthèses, Soc. Math., 2002, vol. 14.
Balaban, P.M., Cellular mechanisms of behavioral plasticity in terrestrial snail, Neurosci. Biobehav. Rev., 2002, vol. 26, no. 5, pp. 597–630.
Herrmann, A. and Gertner, W., Noise and the PSTH response to current transients, I: General theory and application to the integrate-and-fire neuron, J. Comput. Neurosci., 2001, vol. 11, pp. 135–151.
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Lukyanova, O., Nikitin, O. Neuronal topology as set of braids: Information processing, transformation and dynamics. Opt. Mem. Neural Networks 26, 172–181 (2017). https://doi.org/10.3103/S1060992X17030043
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DOI: https://doi.org/10.3103/S1060992X17030043