Can brains generate random numbers? joint work with Mark Goldsmith

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seizure ictal periodpre-ictal period post-ictal period

I. PARTIAL (FOCAL, LOCAL) SEIZURES II. GENERALIZED SEIZURES (CONVULSIVE OR NOT) A. Simple partial seizures B. Complex partial seizures C. Simple partial seizures evolving to secondarily generalized seizures A. Absence seizures B. Myoclonic seizures C. Clonic seizures D. Tonic seizures E. Tonic-clonic seizures F. Atonic seizures 1981 scheme for classification of epileptic seizures (International League Against Epilepsy)

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

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 x

Nithum Thain 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? Seminar at Concordia, July 2009

Walter Harry Pitts, Jr. (1923 – 1969) Warren Sturgis McCulloch (1898 – 1969) Bulletin of Mathematical Biophysics 5 (1943) A logical calculus of the ideas immanent in nervous activity,

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

synaptic weights:

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? Nithum Thain’s question: 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? Our variation: 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? An easier question: First step towards the easier question: Are there a McCulloch-Pitts networks with n neurons and period 2 ? n

Theorem: For every positive integer n there is a McCulloch-Pitts network with n neurons and period 2. n Related previous work: Any other references ??? 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 Networks 15 (2002), 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 , The MIT Press, Cambridge, 2005.

+1 synaptic weights: +1

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

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

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

Batteries of statistical tests for sequences of uniform random numbers in the interval [0,1). Is there a McCulloch-Pitts random number generator which passes all ten tests of SmallCrush ? 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