A neural model of how the brain represents and compares multi-digit numbers: Spatial and categorical processes

Neural Networks

Volumn 16, Issue 8, 2003, Pages 1107-1140

  Grossberg, Stephen ^a   Repin, Dmitry V ^a  

^a BOSTON UNIVERSITY   (United States)

Author keywords

Inferior parietal cortex; Learning; Number; Place value system; Spatial map; Speech category; What and Where cortical streams

Indexed keywords

BRAIN; ERROR ANALYSIS; NUMBER THEORY; SEMANTICS;

NEURAL MODELS;

NEURAL NETWORKS;

ANALYTICAL ERROR; ARTICLE; ARTIFICIAL NEURAL NETWORK; AUDITORY NERVOUS SYSTEM; BRAIN CORTEX; BRAIN FUNCTION; BRAIN MAPPING; CALCULATION; LANGUAGE; MATHEMATICAL ANALYSIS; MENTAL TASK; PARIETAL LOBE; PRIORITY JOURNAL; SIMULATION; TASK PERFORMANCE; VISUAL NERVOUS SYSTEM;

EID: 0042334608     PISSN: 08936080     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0893-6080(03)00193-X     Document Type: Review

References (86)

1. Abbott L.F., Varela K., Sen K., Nelson S.D. Synaptic depression and cortical gain control. Science. 275:1997;220-223.
2. Andersen R.A., Essick G.K., Siegel R.M. Enclosing of spatial location by posterior parietal neurons. Science. 230:1985;456-458.
3. Ashcraft M.H. Children's knowledge of simple arithmetic: A developmental model and simulation. Bisanz J., Brainerd C.J., Kali R. Formal methods in developmental psychology: Progress in cognitive development research. 1987;Springer, New York, NY.
4. Ashcraft M.H. Cognitive arithmetic: a review of data and theory. Cognition. 44:1992;75-106.
5. Baloch A.A., Grossberg S. A neural model of high-level motion processing: line motion and for motion dynamics. Vision Research. 37:1997;3037-3059.
6. Baloch A.A., Grossberg S., Mingolla E., Nogueira C.A.M. A neural model of first-order and second-order motion perception and magnocellular dynamics. Journal of the Optical Society of America. 16:1999;953-978.
7. Brysbaert M. Arabic number reading: on the nature of the numerical scale and the origin of phonological recording. Journal of Experimental Psychology: General. 124:(4):1995;434-452.
8. Buchanan G.M., Bitterman M.E. Learning in honeybees as a function of amount and frequency of reward. Animal Learning and Behavior. 16:(3):1998;247-255.
9. Buck B.H., Black S.E., Behrmann M., Caldwell C., Bronskill M.J. Spatial- and object-based attentional deficits in Alzheimer's disease. Relationship to HMPAO-SPECT measures of parietal perfusion. Brain. 120:(7):1997;1229-1244.
10. Cajory F. A history of mathematical notation. 1928;Open Court Publishing Company, Chicago, IL.
11. Campbell J.I.D., Clark J.M. Cognitive number processing: An encoding-complex perspective. Campbell J.I.D. The nature and origin of mathematical skills. 1992;Elsevier, Amsterdam.
12. Chey J., Grossberg S., Mingolla E. Neural dynamics of motion processing and speed discrimination. Vision Research. 38:1998;2769-2786.
13. Church R.M., Meck W.H. The numerical attribute of stimuli. Roitblat H.L., Bever T.G., Terrace H.J. Animal cognition. 1984;Erlbaum, Hillsdale, NJ.
14. Clark J.M., Campbell J.I.D. Integrated versus modular theories of number skills and acalculia. Brain and Cognition. 17:1991;204-239.
15. Cohen M., Grossberg S. Parallel auditory filtering by sustained and transient channels separates coarticulated vowels and consonants. IEEE Transactions on Speech and Audio Processing. 5:1997;301-318.
16. Courtney S.M., Ungerleider L.G., Keil K., Haxby J.V. Object and spatial visual working memory activate separate neural systems in human cortex. Cerebral Cortex. 6:(1):1996;39-49.
17. Dehaene S. Varieties of numerical abilities. Cognition. 44:1992;1-42.
18. Dehaene S. The number sense: How the mind creates mathematics. 1997;Oxford University Press, New York, NY.
19. Dehaene S., Bossini S., Giraux P. The mental representation of parity and number magnitude. Journal of Experimental Psychology: General. 122:(3):1993;371-396.
20. Dehaene S., Changeux J.-P. Development of elementary numerical abilities: a neuronal model. Journal of Cognitive Neuroscience. 5:(4):1993;390-407.
21. Dehaene S., Cohen L. Dissociable mechanisms of subitizing and counting: neuropsychological evidence from simultanagnosic patients. Journal of Experimental Psychology: Human Perception and Performance. 20:(5):1994;958-975.
22. Dehaene S., Cohen L. Cerebral pathways for calculation: double dissociation between rote verbal and quantitative knowledge of arithmetic. Cortex. 33:(2):1997;219-250.
23. Dehaene S., Dupoux E., Mehler J. Is numerical comparison digital? Analogical and symbolic effects in two-digit number comparison. Journal of Experimental Psychology: Human Perception and Performance. 16:(3):1990;626-641.
24. Dehaene S., Spelke E., Pinel P., Stanescu R., Tsivkin S. Sources of mathematical thinking: behavioral and brain-imaging evidence. Science. 284:1999;970-974.
25. Dehaene S., Tzourio N., Frak V., Raynaud L., Cohen L., Mehler J., Mazoyer B. Cerebral activations during number multiplication and comparison: a PET study. Neuropsychologia. 34:(11):1996;1097-1106.
26. Deibert E., Kraut M., Kremen S., Hart J. Jr. Neural pathways in tactile object recognition. Neurology. 52:(7):1999;1413-1417.
27. Emmerton J., Lohmann A., Niemann J. Pigeons' serial ordering of numerosity with visual arrays. Animal Learning and Behavior. 25:(2):1997;234-244.
28. Fairbank B.A. Jr. Experiments on the temporal aspects of number perception. Dissertation Abstracts International. 30:(1B):1969;403.
29. Fiala J.C., Grossberg S., Bullock D. Metabotropic glutamate receptor activation in cerebellar Purkinje cells as substrate for adaptive timing of the classically conditioned eye blink response. Journal of Neuroscience. 16:1996;3760-3774.
30. Francis G., Grossberg S. Cortical dynamics of form and motion integration: persistence, apparent motion, and illusory contours. Vision Research. 36:1996;149-173.
31. Francis G., Grossberg S., Mingolla E. Cortical dynamics of feature binding and reset: control of visual persistence. Vision Research. 34:1994;1089-1104.
32. Gallistel C.R., Gelman R. Preverbal and verbal counting and computation. Cognition. 44:1992;43-74.
33. Gancarz G., Grossberg S. A neural model of saccadic eye movement control explains task-specific adaptation. Vision Research. 39:1999;3123-3143.
34. Gielen I., Bryzbaert M., Dhont A. The syllable-length effect in number processing is task-dependent. Perception and Psychophysics. 50:(5):1991;449-458.
35. Goodale M.A., Milner A.D. Separate visual pathways for perception and action. Trends in Neurosciences. 15:(1):1992;20-25.
36. Grabowski T.J., Damasio H., Damasio A.R. Premotor and prefrontal correlates of category-related lexical retrieval. Neuroimage. 7:(3):1998;232-243.
37. Grossberg S. Neural pattern discrimination. Journal of Theoretical Biology. 27:1970;291-337.
38. Grossberg S. A neural theory of punishment and avoidance. II. Quantitative theory. Mathematical Biosciences. 15:1972;253-285.
39. Grossberg S. Contour enhancement, short term memory, and constancies in reverberating neural networks. Studies in Applied Mathematics. 52:1973;217-257.
40. Grossberg S. How does a brain build a cognitive code? Psychological Review. 87:1980;1-51.
41. Grossberg S. Nonlinear neural networks: principles, mechanisms, and architectures. Neural Networks. 1:1988;17-61.
42. Grossberg S. How is a moving target continuously tracked behind occluding cover? Watanabe T. High level motion processing: Computational, neurobiological, and psychophysical perspectives. 1999;3-52 MIT Press, Cambridge, MA.
43. Grossberg S. Pitch-based streaming in auditory perception. Griffith N., Todd P. Musical networks: Parallel distributed perception and performance. 1999;117-140 MIT Press, Cambridge, MA.
44. Grossberg S., Kelly F.J. Neural dynamics of binocular brightness perception. Vision Research. 39:1999;3796-3816.
45. Grossberg S., Kuperstein M. Neural dynamics of adaptive sensory-motor control. 1986;Pergamon Press, Elmsford, NY.
46. Grossberg S., Mingolla E., Viswanathan L. Neural dynamics of motion integration and segmentation within and across apertures. Vision Research. 41:2001;2521-2553.
47. Grossberg S., Rudd M.E. A neural architecture for visual motion perception: group and element apparent motion. Neural Networks. 2:1989;421-450.
48. Grossberg S., Rudd M.E. Cortical dynamics of visual motion perception: short-range and long-range apparent motion. Psychological Review. 99:(1):1992;78-121.
49. Gruber O., Kleinschmidt A., Binkofski F., Steinmetz H., von Cramon D.Y. Cerebral correlates of working memory for temporal information. Neuroreport. 11:(8):2000;1689-1693.
50. den Heyer K., Briand K. Priming single digit numbers: automatic spreading activation dissipates as a function of semantic distance. American Journal of Psychology. 99:(3):1986;315-340.
51. Hinrichs J.V., Yurko D.S., Hu J.-M. Two-digit number comparison: use of place information. Journal of Experimental Psychology: Human Perception and Performance. 7:(4):1981;890-901.
52. Link S.W. Modeling imageless thought: the relative judgment theory of numerical comparisons. Journal of Mathematical Psychology. 34:(1):1990;2-41.
53. Link S.W. The wave theory of difference and similarity (Scientific Psychology Series). 1992;Erlbaum, Hillsdale, NJ.
54. Luschei E.S., Fuchs A.F. Activity of brain stem neurons during eye-movement of alert monkeys. Journal of Neurophysiology. 35:1972;445-461.
55. Maguire E.A., Burgess N., Donnett J.G., Frackowiak R.S.J., Frith C.D., O'Keefe J. Knowing where and getting there: a human navigation network. Science. 280:1998;921-924.
56. Mandler G., Shebo B.J. Subitizing: an analysis of its component processes. Journal of Experimental Psychology: General. 111:(1):1982;1-22.
57. Markram H., Tsodyks M. Redistribution of synaptic efficacy between neocortical pyramidal neurons. Nature. 382:1996;807-810.
58. McCloskey M. Cognitive mechanisms in numerical processing: evidence from acquired dyscalculia. Cognition. 44:1992;107-157.
59. McCloskey M., Macaruso P. Representing and using numerical information. American Psychologist. 50:(5):1995;351-363.
60. Mechner F. Sequential dependencies of the lengths of consecutive response runs. Journal of the Experimental Analysis of Behavior. 1:1958;229-233.
61. Meck W.H., Church R.M. A mode control model of counting and timing processes. Journal of Experimental Psychology: Animal Behavior Processes. 9:(3):1983;320-334.
62. Menninger K. Number words and number symbols: A cultural history of numbers. 1969;MIT Press, Cambridge, MA.
63. Mishkin M., Ungerleider L.G., Macko K.A. Object vision and spatial vision: two cortical pathways. Trends in Neurosciences. 6:(10):1983;414-417.
64. Naccache L., Dehaene S. The priming method: imaging unconscious repetition priming reveals an abstract representation of number in the parietal lobes. Cerebral Cortex. 11:2001;966-974.
65. Nishitani N., Uutela K., Shibasaki H., Hari R. Cortical visuomotor integration during eye pursuit and eye-finger pursuit. Journal of Neuroscience. 19:(7):1999;2647-2657.
66. Ögmen H., Gagné S. Neural network architecture for motion perception and elementary motion detection in the fly visual system. Neural Networks. 3:1990;487-506.
67. Parkman J.M. Temporal aspects of digit and letter inequality judgments. Journal of Experimental Psychology. 91:(2):1971;191-205.
68. Pesenti M., Thioux M., Seron X., De Volder A. Neuroanatomical substrates of arabic number processing, numerical comparison, and simple addition: a PET study. Journal of Cognitive Neuroscience. 12:2000;461-479.
69. Poltrock S.E., Schwartz D.R. Comparative judgments of multidigit numbers. Journal of Experimental Psychology: Learning, Memory, and Cognition. 10:1984;32-45.
70. Pinel P., Le Clec G., van de Moortele P.-F., Naccache L., Le Bihan D., Dehaene S. Event-related fMRI analysis of the cerebral circuit for number comparison. NeuroReport. 10:1999;1473-1479.
71. Rainer G., Asaad W.F., Miller E.K. Selective representation of relevant information by neurons in the primate prefrontal cortex. Nature. 393:1998;577-579.
72. Ramachandran V.S., Hubbard E.M. Psychophysical investigations into the neural basis of synaesthesia. Proceedings of the Royal Society of London B. 268:2001;979-983.
73. Ramachandran V.S., Hubbard E.M. Synaesthesia - a window into perception, thought and language. Journal of Consciousness Studies. 8:2001;3-34.
74. Repin, D. V., & Grossberg, S (1999). How does the brain represent numbers. Paper presented at the third international conference on cognitive and neural systems. Boston: Boston University.
75. Repin, D. V., & Grossberg, S (1999). A neural model of how the brain represents numbers. Paper presented at the 21st international summer school of brain research. Amsterdam.
76. Restle F. Speed of adding and comparing numbers. Journal of Experimental Psychology. 83:1970;274-278.
77. Rickard T.C., Romero S.G., Basso G., Wharton C., Flitman S., Grafman J. The calculating brain: an fMRI study. Neuropsychologia. 38:2000;325-335.
78. Rilling M., McDiarmid C. Signal detection in fixed-ratio schedules. Science. 148:1965;526-527.
79. Robinson D.A. Oculormotor unit behavior in the monkey. Journal of Neurophysiology. 35:1970;393-404.
80. Schiller P.H. The discharge characteristics of single units in the oculomotor and abducens nuclei of the unanesthetized monkey. Experimental Brain Research. 10:1970;347-362.
81. Seron X., Pesenti M., Noel M.-P., Deloche G. Images of numbers, or 'When 98 is upper left and 6 is sky blue'. Cognition. 44:1992;159-196.
82. Starkey P., Spelke E.S., Gelman R. Detection of intermodal numerical correspondences by human infants. Science. 222:1983;179-181.
83. Stein B.E., Meredith M.A. The merging of the senses. 1993;MIT Press, Cambridge, MA.
84. Ullsperger P., Grune K. Processing of multi-dimensional stimuli: P300 component of the event-related brain potential during mental comparison of compound digits. Biological Psychology. 40:1995;17-31.
85. Washburn D.A., Rumbaugh D.M. Ordinal judgments of numerical symbols by macaques (Macaca mulatta). Psychological Science. 2:(3):1991;190-193.
86. Wynn K. Psychological foundations of number: numerical competence in human infants. Trends in Cognitive Sciences. 2:(8):1998;296-303.