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Can brains generate random numbers?

joint work with. Mark Goldsmith. Can brains generate random numbers?. Canada Research Chairs. Communication Guidelines for Chairholders.

yoshio-maldonado
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Can brains generate random numbers?

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  1. joint work with Mark Goldsmith Can brains generate random numbers?

  2. Canada Research Chairs Communication Guidelines for Chairholders In all professional publications, presentations and conferences, we ask you to identify yourself as a Canada Research Chair and acknowledge the contribution of the program to your research. In 2000, the Government of Canada created a permanent program to establish 2000 research professorships—Canada Research Chairs—in eligible degree-granting institutions across the country.

  3. seizure post-ictal period pre-ictal period ictal period

  4. I. PAR TIAL (FOCAL, LOCAL) SEIZURES A.Simple partial seizuresB.Complex partial seizuresC.Simple partial seizures evolving to secondarily generalized seizures II. GENERALIZED SEIZURES (CONVULSIVE OR NOT) A.Absence seizuresB.Myoclonic seizuresC.Clonic seizuresD.Tonic seizuresE.Tonic-clonic seizuresF.Atonic seizures 1981 scheme for classification of epileptic seizures (International League Against Epilepsy)

  5. post-ictal period pre-ictal period ictal period irregular disorderly erratic chaotic random irregular disorderly erratic chaotic random rhytmic organized synchronized

  6. Digression: Conway’s Game of Life 1. Any live cell with fewer than two live neighbours dies, as if caused by under-population. 2. Any live cell with more than three live neighbours dies, as if by overcrowding. 3. Any dead cell with exactly three live neighbours becomes a live cell, as if by reproduction. 4. All other cells remain as they are. 2 1 8 x 3 7 6 5 4

  7. Nithum Thain Seminar at Concordia, July 2009 Is there an initial conguration that causes Conway's Game of Life to evolve in a way resembling a partial seizure, proceeding from an erratic flutter of apparently unpredictable patterns to sustained rhythmic changes that would begin in a small part of the grid and gradually spread, synchronized, over a larger area before subsiding to give way to the initial erratic mode?

  8. A logical calculus of the ideas immanent in nervous activity, Bulletin of Mathematical Biophysics5 (1943) 115 --133. Warren Sturgis McCulloch (1898 – 1969) Walter Harry Pitts, Jr. (1923 – 1969)

  9. axon synapse axon axon soma (and its dendrites) synapse synapse axon neuron McCulloch-Pitts neuron

  10. synaptic weights: -1 +1 +1 0

  11. Nithum Thain’s question: Our variation: Is there a McCulloch-Pitts network that evolves in a way resembling a partial seizure, proceeding from an erratic flutter of apparently unpredictable patterns to sustained rhythmic changes that would begin in a small part of the grid and gradually spread, synchronized, over a larger area before subsiding to give way to the initial erratic mode? An easier question: Is there a McCulloch-Pitts network that evolves in a way resembling the pre-ictal period of a partial seizure, an erratic flutter of apparently unpredictable patterns? n First step towards the easier question: Are there a McCulloch-Pitts networks with n neurons and period 2 ? Is there an initial conguration that causes Conway's Game of Life to evolve in a way resembling a partial seizure, proceeding from an erratic flutter of apparently unpredictable patterns to sustained rhythmic changes that would begin in a small part of the grid and gradually spread, synchronized, over a larger area before subsiding to give way to the initial erratic mode?

  12. Theorem: For every positive integer n there is a McCulloch-Pitts network with n neurons and period 2 . n Related previous work: P.C. McGuire, H. Bohr, J.W. Clark, R. Haschke, C.L. Pershing, J. Rafelski, Threshold disorder as a source of diverse and complex behavior in random nets, Neural Networks15 (2002), 1243--1258. R. Legenstein, W. Maass, What makes a dynamical system computationally powerful? In: S. Haykin, J.C. Principe, T. Sejnowski, J. McWhirter (Eds.), New Directions in Statistical Signal Processing: From Systems to Brain, pp. 127154, The MIT Press, Cambridge, 2005. Any other references ???

  13. synaptic weights: +1 +1 -1 -1 +1

  14. Trajectory of our 4-neuron McCulloch-Pitts network: Trajectory of another 4-neuron McCulloch-Pitts network: the last bit flips only twice! the second bit flips 10 times; all other bits flip 6 times

  15. The number of McCulloch-Pitts networks with n neurons and period 2 : n 2 with 2 neurons; 1 isomorphism class 48 with 3 neurons; 2 isomorphism classes 9984 with 4 neurons; 56 isomorphism classes

  16. Trajectory of our 4-neuron McCulloch-Pitts network as a generator of uniform random numbers in the interval [0,1): the last bit flips only twice but the last bit is negligible: it contributes only 0.062 to the total buckets [0,0.5) and [0.5,1) the same bucket is never repeated three times in a row

  17. Batteries of statistical tests for sequences of uniform random numbers in the interval [0,1). P. L'Ecuyer, R. Simard, TestU01: A C library for empirical testing of random number generators, ACM Transactions on Mathematical Software, 33 (2007), Article 22, 40 pages. The least stringent of these batteries: SmallCrush Is there a McCulloch-Pitts random number generator which passes all ten tests of SmallCrush?

  18. http://arxiv.org/abs/1208.6451

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