Figure 2.

Parameter estimation for Process L. The variable L was assumed to increase during 10-s epochs of wakefulness and rapid eye movement sleep and decrease during 10-s epochs of SWS according to Equations 4 and 5. (A) UA and LA are functions of time due to the fact that the dynamic range of the lactate sensor attenuated over the 43 h of recording time. To compute UA(t) and LA(t) we constructed the relative frequency histogram of lactate for each moving 2-h window in the dataset. The 99% level of the lactate histogram was chosen as the upper asymptote (UA(t)) and the 1% level of the lactate histogram was chosen as the lower asymptote (LA(t)). (B) Upper left inset shows the histogram for the lactate signal at t = 8 h and the inset to the right shows the histogram for the lactate signal at time t = 38 h indicating that the 1% level and 99% level have changed over the course of the experiment. This change is taken into account in the change of the dashed lines. (C) To determine the optimal values for the parameters τ_i and τ_d in brute force fashion we performed an exhaustive search over a reasonable range of values for these two parameters and computed the sum of squares error for each combination. (D) Best fit of the model to the data from (B) using the optimal parameters found from exhaustive search (τ_i = 1.20 and τ_d = 0.28). (E) Successive guesses for the optimal choices of τ_i and τ_d using the Nelder-Mead method. The iterative method converges to (τ_i = 1.20 and τ_d = 0.28) using just 32 function evaluations. (F) Best fit of the model to the data from (B) using the optimal parameters found from Nelder-Mead. Black horizontal bars on the tops of (B,D,F) indicate the 12 h dark periods and the gray horizontal bar indicates when sleep deprivation occurred.