Proof: For any active GC, the following equation is valid: equation(Equation 14) ∂L∂ai=−∑mWmirm+θi=0. Assume that more than M   GCs are active. Then, we have at least M   + 1 such equations for M   unknowns rm  . Such a system in the general case (if M   + 1 corresponding vectors Ω→i are independent) is inconsistent Osimertinib molecular weight and has no solution. Thus,

the number of coactive GCs cannot exceed M. Note:   Consider the case of small but nonzero firing thresholds of GCs θ  . In this case, two regimes can be distinguished. If vector x→ can be expanded in terms of vectors W→i with positive coefficients, the firing rates of M   GCs are generally ∼1, but the responses of MCs are small (∼θ  ). This is the regime of sparse overcomplete representations. If the glomerular input vector x→ cannot be represented as a superposition of GCs weights W→i with positive coefficients (incomplete representation), the responses of cells are essentially (ignoring contributions ∼θ) given by the solution of homogeneous problem ( Equation 12), which explains our attention to this problem. In this case, according to theorem 1, fewer than M GCs have large firing rates, and only one has a small firing rate

(∼θ). Here, we suggest that the presence of a large threshold for GC firing (Figure 5A) can lead to inaccurate representations of odorants, similar to the nonnegativity of the GC firing rates (Figure 6). This observation Cabozantinib molecular weight allows us to explain the transition between the awake and anesthetized responses. We use a simplified model of a bulbar network containing only one GC (Figure 2). This network has the advantage that an exact solution can be found even when a finite threshold for firing is present for the activation of the GCs. Consider the input configuration shown in Figure 2C. Assume for simplicity that all of the nonzero weights and MC inputs have unit strengths. Then, the Lyapunov function for the activity of the single GC a is equation(Equation 15) L(a)=K2(1−a)2+θa. Here,

we have to assume that a≥0a≥0; K   is the number of nonzero weights for GCs (K   = 3 in Figure 2). By minimizing the Lyapunov function, we obtain a   = 1–θ /   K, for θ≤Kθ≤K and zero otherwise. The activity Carnitine palmitoyltransferase II of the first MC ( Figure 2C, Figure S1) cannot be affected by the GC, because the GC makes no synapses onto this cell. The activities of MCs 2, 3, and 5 are given by equation(Equation 16) r2,3,5=1−a=θKfor θ≤Kθ≤K. MCs increase their firing rate to activate the GC. The amount of increase is equal to the threshold for activation of the GC divided by the number of MCs contributing to the input current; i.e., K. The activity of the GC is assumed to rise fast above the threshold so that it suppresses all significant increases of inputs above the value given by Equation 16. The responses of MCs as functions of the threshold θ are shown in Figure S1. For large firing thresholds θ, all MCs that receive receptor inputs respond to the odorant.