Biol. Cybern. 64, 353-356 (1991) Biological Cybernetics 9 Springer-Vcdag 1991 An artificial hysteresis binary neuron: a model suppressing the oscillatory behaviors of neural dynamics Y. Takefuji and K. C. Lee Department of Electrical Engineering and Applied Physics, Case Western Reserve University; Cleveland, OH 44106, USA Received April 24, 1990/Accepted in revised form November 7, 1990 Abstract. A hysteresis binary McCulloch-Pitts neuron model is proposed in order to suppress the complicated oscillatory behaviors of neural dynamics. The artificial hysteresis binary neural network is used for scheduling time-multiplex crossbar switches in order to demon- strate the effects of hysteresis. Time-multiplex crossbar switching systems must control traffic on demand such that packet blocking probability and packet waiting time are minimized. The system using n x n processing elements solves an n x n crossbar-control problem with O(1) time, while the best existing parallel algorithm requires O(n) time. The hysteresis binary neural net- work maximizes the throughput of packets through a crossbar switch. The solution quality of our system does not degrade with the problem size. Introduction In time-multiplex communication systems, crossbar packet switches route traffic from the input to output where a message packet is transmitted from the source to the destination. The randomly incoming traffic must be controlled and scheduled to eliminate conflict at the crossbar switch where the conflict is that two or more users may simultaneously access to a single output. The goal of the traffic-scheduling for time-multiplex cross- bar switches is not only to maximize the throughput of packets through a crossbar switch but also to minimize packet blocking probability and packet waiting time. A request for packet transmission through an n x n crossbar is described by an n x n traffic matrix T. In the traffic matrix T, each element tg represents a request of packets from input i to output j. For example, tij = 0 means that there is no packet to be transmitted on the jth output line from the ith input line. tij = 1 means that at least one packet on the ith input line should be transmitted on the jth output line of the crossbar. In 1979 Inukai at COMSAT lab. proposed the O(n 2) sequential algorithm for the n x n crossbar switch problem (Inukai 1979). In 1989 Rose at AT&T Bell lab. presented the O(n) parallel algorithm based on a cellu- lar automaton where n 2 processing elements are used for solving an n x n traffic matrix problem (Rose 1989). Chen, Mavor, Denyer, and Renshaw proposed the O(n 2) sequential algorithm of traff• routing problems for the multiprocessor system (Chen et al. 1990). They proved that the problem is NP-complete (Chen et al. 1990). In 1989 Marrakchi and Troudet at Bellcore proposed the n x n neural network algorithm based on Hopfield network model (Marrakchi and Troudet 1989). However with the Hopfield neural network, the state of the system is forced to converge to the local minimum. In other words, the solution quality drasti- cally degrades with the problem size. Takefuji and Lee have successfully used the Hopfield neural network with McCulloch-Pitts binary neurons for solving the graph planarization problem (Takefuji and Lee 1989) and the tiling problem (Takefuji and Lee 1990a) where the state of the system converges to the near-global minimum in O(1) time. They proved that the state of the binary neural network system is guaranteed to converge to the local minimum (Takefuji and Lee 1990b). In 1986 Hoffman and Benson introduced sigmoid neurons with hysteresis for learning, where any changes in synaptic connection strengths are replaced by hysteresis (Hoffman and Benson 1986). Due to the hysteresis associated with each neuron, the system tends to stay in the region of phase space where it is located. They proposed the theory on a role for sleep in learning (Hoffman and Benson 1986). Dynamic and static hysteresis in Crayfish stretch receptors was reported by Segundo and Martinez in 1985 (Segundo and Martinez 1985). They stated that hysteresis may be more widespread than suspected in sensory and perhaps other system. In 1989 Keeler, Pichler, and Ross presented the effects of hysteresis in pattern recognition and learning for improving the signal-to-noise ratio (Keeler et al. 1989). In this paper the hysteresis property is exploited in order to reduce the complicated oscillatory behaviors of neural dynam- ics for solving combinatorial optimization problems. The hysteresis with each neuron enhances the state of