7.18. The problem concerns a model by McCarley and Hobson [99] that is intended to explain regular oscillations in cat n

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7.18. The problem concerns a model by McCarley and Hobson [99] that is intended to explain regular oscillations in cat neuron activity associated with the sleep cycle. Activity is measured in the number of electrical discharges per second. Two groups of cells are considered. If x(t) and y(t) are the activity levels of the two cell groups, the authors propose the equations dx = ax -bxy, dt dy dt =-cy+dxy, (7.44) where a, b, c, and d are positive constants.

(a) The following are excerpts from the paper that explain why the particular equa- tions were chosen. A: "Evidence that the rate of change of activity levels... is proportional to the current level of activity." B: "Nonlinear interaction was to be expected. We model this effect... in accord with the reasonable phys- iological postulate that the effect of an excitatory or inhibitory input to the two populations will be proportional to the current level of discharge activ- ity." Which terms are justified by part A, and which by part B? Which are the excitatory influences, and which the inhibitory? (b) Show that equations (7.44) have a steady state solution in which neither x nor y is zero. Show further that if m(t) and n(t) are the departures of x(t) and y(t) from steady state values, then linearized equations for m and n have the form dm = -an, dn dt = Bm. (7.45) dt Express a and B in terms of the original constants a, b, c, and d. (c) Find the general solution to (7.45). The solution is oscillatory. Find the period of oscillation in terms of a and B.