EUROPHYSICS LETTERS 15 March 1998 Europhys. Lett., 41 (6), pp. 599-604 (1998) Ordered and disordered dynamics in random networks L. Glass 1 ( * ) and C. Hill 2,3 1 Department of Physiology, McGill University 3655 Drummond Street, Montreal, Quebec, Canada H3G 1Y6 2 Department of Physics, McGill University - Montreal, Quebec, Canada 3 Department of Physics, Cornell University - Ithaca, NY, USA (received 20 October 1997; accepted in final form 29 January 1998) PACS. 05.45+b – Theory and models of chaotic systems. PACS. 05.50+q – Lattice theory and statistics; Ising problems. PACS. 87.22Jb – Muscle contraction, nerve conduction, synaptic transmission, memorization, and other neurophysiological processes (excluding perception processes and speech). Abstract. – Random Boolean networks that model genetic networks show transitions between ordered and disordered dynamics as a function of the number of inputs per element, K, and the probability, p, that the truth table for a given element will have a bias for being 1, in the limit as the number of elements N →∞. We analyze transitions between ordered and disordered dynamics in randomly constructed ordinary differential equation analogues of the random Boolean networks. These networks show a transition from order to chaos for finite N . Qualitative features of the dynamics in a given network can be predicted based on the computation of the mean dimension of the subspace admitting outflows during the integration of the equations. Genetic networks have been modeled by random Boolean networks in which time is discrete and each element computes a Boolean function based on the values of inputs to that element [1]. Since the number of human genes is of the order of 100,000, and each gene is idealized as either on (1) or off (0), the state space for the human gene activity is huge. An order-disorder transition has been described for random Boolean networks in the limit that the number of variables, N →∞, as a function of the number of inputs per variable, K, and the probability, p, that the truth table for a given element will have a bias for being 1 [2]-[8]. The order-disorder boundary is given by K c = 1 2p c (1 - p c ) , (1) where K c and p c represent the values of K and p on the boundary [2], [4], [6]. Kauffman has argued that for a network to be biologically meaningful, it should have relatively few attractors, ( * ) E-mail: glass@cnd.mcgill.ca c  EDP Sciences