# American Institute of Mathematical Sciences

2014, 11(2): 343-361. doi: 10.3934/mbe.2014.11.343

## Distributed, layered and reliable computing nets to represent neuronal receptive fields

 1 Facultad de Informática, Universidad Politécnica de Madrid (UPM), Spain 2 Instituto Universitario de Ciencias y Tecnologías Cibernéticas, Universidad de Las Palmas de Gran Canaria, Spain, Spain

Received  September 2012 Revised  April 2013 Published  October 2013

Receptive fields of retinal and other sensory neurons show a large variety of spatiotemporal linear and non linear types of responses to local stimuli. In visual neurons, these responses present either asymmetric sensitive zones or center-surround organization. In most cases, the nature of the responses suggests the existence of a kind of distributed computation prior to the integration by the final cell which is evidently supported by the anatomy. We describe a new kind of discrete and continuous filters to model the kind of computations taking place in the receptive fields of retinal cells. To show their performance in the analysis of different non-trivial neuron-like structures, we use a computer tool specifically programmed by the authors to that effect. This tool is also extended to study the effect of lesions on the whole performance of our model nets.
Citation: Arminda Moreno-Díaz, Gabriel de Blasio, Moreno-Díaz Jr.. Distributed, layered and reliable computing nets to represent neuronal receptive fields. Mathematical Biosciences & Engineering, 2014, 11 (2) : 343-361. doi: 10.3934/mbe.2014.11.343
##### References:
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Xu, Extensive disinhibitory region beyond the classical receptive field of cat retinal ganglion cells,, Vision Res., 32 (1992), 219. Google Scholar [13] M. London and M. Häusser, Dendritic computation,, Annu. Rev. Neurosci., 28 (2005), 503. doi: 10.1146/annurev.neuro.28.061604.135703. Google Scholar [14] P. Marmarelis and K. I. Naka, Non-linear analysis and synthesis of receptive field responses in the catfish retina. I. Horizontal cell-ganglion chains,, J. Neuriphysiol., 36 (1973), 605. Google Scholar [15] P. Marmarelis and K. I. Naka, Non-linear analysis and synthesis of receptive field responses in the catfish retina. II. One input white noise analysis,, J. Neuriphysiol., 36 (1973), 619. Google Scholar [16] D. Marr, Vision,, W. H. Freeman and Company, (1982). Google Scholar [17] W. S. McCulloch, Embodiments of Mind,, MIT Press, (1988). Google Scholar [18] R. Moreno-Díaz, An Analytical Model of the Group 2 Ganglion Cell in the Frog'S Retina,, Report, (1965), 33. Google Scholar [19] R. Moreno-Díaz and G. de Blasio, Systems methods in visual modelling,, Systems Analysis Modelling Simulation, 43 (2003), 1159. doi: 10.1080/02329290310001600255. Google Scholar [20] R. Moreno-Díaz and G. de Blasio, Systems and computational tools for neuronal retinal models,, in Computer Aided Systems Theory - EUROCAST 2003, (2003), 494. doi: 10.1007/978-3-540-45210-2_45. Google Scholar [21] R. Moreno-Díaz, Jr., Computación Paralela y Distribuida: Relaciones Estructura-Función en Retinas,, Ph.D thesis, (1993). Google Scholar [22] R. Moreno-Díaz, Jr. and K. N. Leibovic, On some methods in neuromathematics (or the development of mathematical methods for the description of structure and function in neurons),, in From Natural to Artificial Neural Computation, (1995), 209. Google Scholar [23] C. L. Passaglia, D. K. Freeman and J. B. Troy, Effects of remote stimulation on the modulated activity of cat retinal ganglion cells,, The Journal of Neuroscience, 29 (2009), 2467. doi: 10.1523/JNEUROSCI.4110-08.2009. Google Scholar [24] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. Flannery, Numerical Recipes: The Art of Scientific Computing,, $3^{rd}$ edition, (2007). Google Scholar [25] R. W. Rodieck, Quantitative analysis of cat retinal ganglion cell response to visual stimuli,, Vision Res., 5 (1965), 583. doi: 10.1016/0042-6989(65)90033-7. Google Scholar [26] R. W. Rodieck and J. Stone, Response of cat retinal ganglion cells to moving visual patterns,, J. Neurophysiol., 28 (1965), 819. Google Scholar [27] G. Schweigart and U. T. Eysel, Activity-dependent receptive field changes in the surround of adult cat visual cortex lesions,, European Journal of Neuroscience, 15 (2002), 1585. doi: 10.1046/j.1460-9568.2002.01996.x. Google Scholar [28] I. Segev, What do dendrites and their synapses tell the neuron?, J. Neurophysiol., 95 (2006), 1295. doi: 10.1152/classicessays.00039.2005. Google Scholar [29] T. Shou, W. Wang and H. Yu, Orientation biased extended surround of the receptive field of car retinal ganglion cells,, Neuroscience, 98 (2000), 207. Google Scholar [30] P. Sterling, The ganglion receptive field,, in The Retinal Basis of Vision (eds. J. Toyoda, (1999), 163. Google Scholar [31] G. Szegö, Orthogonal Polynomials,, American Mathematical Society Colloquium Publications, (1959). Google Scholar [32] Y. Tokutake and M. A. Freed, Retinal ganglion cells - spatial organization of the receptive field reduces temporal redundancy,, European Journal of Neuroscience, 28 (2008), 914. doi: 10.1111/j.1460-9568.2008.06394.x. Google Scholar [33] J. B. Troy and T. Shou, The receptive fields of cat retinal ganglion cells in physiological and pathological states: where we are after half a century of research,, Progress in Retinal and Eye Research, 21 (2002), 263. doi: 10.1016/S1350-9462(02)00002-2. Google Scholar [34] M. Van Wyk, W. R. Taylor and D. I. Vaney, Local edge detectors: A substrate for fine spatial vision at low temporal frequencies in rabbit retina,, The Journal of Neurosci., 26 (2006), 13250. Google Scholar [35] M. Van Wyk, H. Wässle and W. R. Taylor, Receptive field properties of ON- and OFF-ganglion cells in the mouse retina,, Visual Neurosci., 26 (2009), 297. Google Scholar [36] F. Werblin, A. Jacobs and J. Teeters, The computational eye,, Spectrum IEEE, 33 (1996), 30. doi: 10.1109/6.490054. Google Scholar

show all references

##### References:
 [1] H. B. Barlow, Summation and inhibition in the frog's retina,, J. Physiol., 119 (1953), 69. Google Scholar [2] G. de Blasio, A. Moreno-Díaz and R. Moreno-Díaz, Bioinspired computing nets for directionality in vision,, Computing, 94 (2012), 449. doi: 10.1007/s00607-012-0186-z. Google Scholar [3] G. de Blasio, A. Moreno-Díaz, R. Moreno-Díaz, Jr. and R. Moreno-Díaz, New biomimetic neural structures for artificial neural nets,, in Computer Aided Systems Theory - EUROCAST 2011: 13th International Conference, (2011), 6. doi: 10.1007/978-3-642-27549-4_4. Google Scholar [4] W. Feller, An Introduction to Probability Theory and its Applications. Vol. I,, Third edition, (1968). Google Scholar [5] S. B. Frost, S. Barbay, K. M. Friel, E. J. Plautz and R. J. Nudo, Reorganization of remote cortical regions after ischemic brain injury: A potential substrate for stroke recovery,, J. Neuriphysiol., 89 (2003), 3205. Google Scholar [6] P. Hammond, Contrasts in spatial organization of receptive fields at geniculate and retinal levels: Centre-surround and outer-surround,, J. Physiol., 228 (1973), 115. Google Scholar [7] H. Hochstadt, The Functions in Mathematical Physics,, Second edition, (1986). Google Scholar [8] D. H. Hubel and T. N. Wiesel, Anatomical demonstration of columns in the monkey striate cortex,, Nature, 221 (1969), 747. Google Scholar [9] H. Kolb, How the retina works,, American Scientist, 91 (2003), 28. Google Scholar [10] S. W. Kuffler, Discharge patterns and functional organization of mammalian retina,, J. Neurophysiol., 16 (1953), 37. Google Scholar [11] K. N. Leibovic, Principles of brain function: Information processing in convergent and divergent pathways,, in Progress in Cybernetics and Systems, (1982), 91. Google Scholar [12] C. Y. Li, Y. X. Zhou, X. Pei, F. T. Qiu, C. Q. Tang and X. Z. Xu, Extensive disinhibitory region beyond the classical receptive field of cat retinal ganglion cells,, Vision Res., 32 (1992), 219. Google Scholar [13] M. London and M. Häusser, Dendritic computation,, Annu. Rev. Neurosci., 28 (2005), 503. doi: 10.1146/annurev.neuro.28.061604.135703. Google Scholar [14] P. Marmarelis and K. I. Naka, Non-linear analysis and synthesis of receptive field responses in the catfish retina. I. Horizontal cell-ganglion chains,, J. Neuriphysiol., 36 (1973), 605. Google Scholar [15] P. Marmarelis and K. I. Naka, Non-linear analysis and synthesis of receptive field responses in the catfish retina. II. One input white noise analysis,, J. Neuriphysiol., 36 (1973), 619. Google Scholar [16] D. Marr, Vision,, W. H. Freeman and Company, (1982). Google Scholar [17] W. S. McCulloch, Embodiments of Mind,, MIT Press, (1988). Google Scholar [18] R. Moreno-Díaz, An Analytical Model of the Group 2 Ganglion Cell in the Frog'S Retina,, Report, (1965), 33. Google Scholar [19] R. Moreno-Díaz and G. de Blasio, Systems methods in visual modelling,, Systems Analysis Modelling Simulation, 43 (2003), 1159. doi: 10.1080/02329290310001600255. Google Scholar [20] R. Moreno-Díaz and G. de Blasio, Systems and computational tools for neuronal retinal models,, in Computer Aided Systems Theory - EUROCAST 2003, (2003), 494. doi: 10.1007/978-3-540-45210-2_45. Google Scholar [21] R. Moreno-Díaz, Jr., Computación Paralela y Distribuida: Relaciones Estructura-Función en Retinas,, Ph.D thesis, (1993). Google Scholar [22] R. Moreno-Díaz, Jr. and K. N. Leibovic, On some methods in neuromathematics (or the development of mathematical methods for the description of structure and function in neurons),, in From Natural to Artificial Neural Computation, (1995), 209. Google Scholar [23] C. L. Passaglia, D. K. Freeman and J. B. Troy, Effects of remote stimulation on the modulated activity of cat retinal ganglion cells,, The Journal of Neuroscience, 29 (2009), 2467. doi: 10.1523/JNEUROSCI.4110-08.2009. Google Scholar [24] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. Flannery, Numerical Recipes: The Art of Scientific Computing,, $3^{rd}$ edition, (2007). Google Scholar [25] R. W. Rodieck, Quantitative analysis of cat retinal ganglion cell response to visual stimuli,, Vision Res., 5 (1965), 583. doi: 10.1016/0042-6989(65)90033-7. Google Scholar [26] R. W. Rodieck and J. Stone, Response of cat retinal ganglion cells to moving visual patterns,, J. Neurophysiol., 28 (1965), 819. Google Scholar [27] G. Schweigart and U. T. Eysel, Activity-dependent receptive field changes in the surround of adult cat visual cortex lesions,, European Journal of Neuroscience, 15 (2002), 1585. doi: 10.1046/j.1460-9568.2002.01996.x. Google Scholar [28] I. Segev, What do dendrites and their synapses tell the neuron?, J. Neurophysiol., 95 (2006), 1295. doi: 10.1152/classicessays.00039.2005. Google Scholar [29] T. Shou, W. Wang and H. Yu, Orientation biased extended surround of the receptive field of car retinal ganglion cells,, Neuroscience, 98 (2000), 207. Google Scholar [30] P. Sterling, The ganglion receptive field,, in The Retinal Basis of Vision (eds. J. Toyoda, (1999), 163. Google Scholar [31] G. Szegö, Orthogonal Polynomials,, American Mathematical Society Colloquium Publications, (1959). Google Scholar [32] Y. Tokutake and M. A. Freed, Retinal ganglion cells - spatial organization of the receptive field reduces temporal redundancy,, European Journal of Neuroscience, 28 (2008), 914. doi: 10.1111/j.1460-9568.2008.06394.x. Google Scholar [33] J. B. Troy and T. Shou, The receptive fields of cat retinal ganglion cells in physiological and pathological states: where we are after half a century of research,, Progress in Retinal and Eye Research, 21 (2002), 263. doi: 10.1016/S1350-9462(02)00002-2. Google Scholar [34] M. Van Wyk, W. R. Taylor and D. I. Vaney, Local edge detectors: A substrate for fine spatial vision at low temporal frequencies in rabbit retina,, The Journal of Neurosci., 26 (2006), 13250. Google Scholar [35] M. Van Wyk, H. Wässle and W. R. Taylor, Receptive field properties of ON- and OFF-ganglion cells in the mouse retina,, Visual Neurosci., 26 (2009), 297. Google Scholar [36] F. Werblin, A. Jacobs and J. Teeters, The computational eye,, Spectrum IEEE, 33 (1996), 30. doi: 10.1109/6.490054. Google Scholar
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