Heartbeat Zeeman model

Then x variable plays in Zeeman s model the role of length of a fibre of the cardiac muscle while the b variable corresponds to the electrochemical control (contraction of the cardiac muscle is triggered by a biochemically generated electric impulse). A stable stationary point E may occur near the point B which is infinitely sensitive to perturbations. To transfer the system from the stable stationary point E to B, a perturbation of the system is required if E is located close to B the perturbation can be small. The mechanism of switching the heart from the state of equilibrium E (lack of heartbeat) to the state of action involves removing the system from the state E to B by way of stimulation, for example by an electric impulse. On reaching the state B the model system imitates the heartbeat — this is the trajectory BB CC E. A subsequent cycle requires the repeated stimulation at the point E. 

The method used by Zeeman to model the heartbeat and the process of nerve impulse transmission by means of differential equations (see Section 3.5) is frequently employed in chemical kinetics (where it is called the Tikhonov method or the method of stationary concentrations). Application of the method to chemical kinetics equations will be discussed in Chapter 4. 

The heartbeat model and the model of nerve impulse transmission by the nerve axon are described in Zeeman s paper. A description of ecological and population models can be found in a paper by Auslander and Huffaker, as well as in a paper by May. A more comprehensive treatment of sequential models is found in a paper by Otto. The cited paper of Ulam and von Neumann has been ahead of its time — the authors have examined deterministic chaos generating sequential processes using the first computers available. 

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