We further explored the space of these two parameters (kfi   and kfr  ) by measuring the impulse response at different contrasts for many different parameter values, thereby mapping the effects of kfi   and kfr   on changes in gain, temporal response, and the biphasic temporal response. Changes in gain resulted when either fast inactivation or recovery were slow compared to activation, thus leading to depletion of the resting state during increased activation ( Figure 8C). Considering a simplified three-state system at equilibrium, the inflow and outflow of all states are the same (i.e., R

 ∞u  ∞ka   = A  ∞ki   = I  ∞kr BMN 673 ic50  ), where u  ∞ is a steady input to the kinetics block. The equilibrium occupancy of the resting state can then be solved as equation(Equation 3) R∞=(1+u∞c1)−1,R∞=(1+u∞c1)−1,where c1=(ka/ki+ka/kr)c1=(ka/ki+ka/kr). Thus, when either ki or kr are small compared to ka, c1 becomes large and weights the effect of the input u∞ more heavily. This changes the resting state occupancy and, therefore, the gain (see Equation 2) significantly

with contrast. This relationship allows the adaptive change in gain to be approximated analytically directly from the rate constants Trametinib supplier of the model ( Figure S5A). Contrast-dependent changes in temporal filtering occurred when fast inactivation (kfi  ) was prolonged but such changes were unaffected by the rate constant of fast recovery (kfr  ) ( Figure 8D). Because of the lack of dependence on kfr  , we considered a simplified system of three states with no return pathway, R→ukaA→kfiI. We can derive that the impulse response of this system is a weighted sum of two exponentials (see Supplemental Experimental Procedures), one with a time constant, u∞(σ)kau∞(σ)ka, that depends on the contrast (σ  ), and one with time constant, kfi  , that is independent

very of contrast. The weighting between these two exponentials is set by a constant that depends on the contrast and the inactivation rate such that when kfi/kakfi/ka is small, the variable exponential is weighted more heavily. We can use this understanding to predict the adaptive change in temporal filtering directly from the rate constants of the model ( Figure S5B). Finally, the change in differentiation of the temporal filter was produced primarily by fast recovery, with some dependence on fast inactivation as well (Figure 8E). By comparing the state occupancies to the impulse response, Fk, we saw that Fk was more biphasic when the increase in the inactivated state I1 exceeded the depletion of the resting state ( Figure S5C). Consequently, when recovery was slow, as compared to the steps of activation and inactivation, there was transiently a higher level of inactivation, causing an undershoot in the level of activation.