To satisfy Equation 4, vector x→ should be represented as a sum of N   basis vectors W→i with coefficients ai  . The latter are unknown coefficients, which correspond to the activity of GCs. The number of unknowns is therefore equal to the number of GCs. The number

of conditions that the unknowns have Antidiabetic Compound Library supplier to satisfy is equal to the number of MCs for which the balance between excitation and inhibition is sought. Because there are more unknowns (GCs) than constraints (MCs), the problem specified by Equation 4 has many solutions; i.e., it is overcomplete. This means that several combinations of firing patterns of GCs are consistent with any given set of glomerular inputs x→. This also means that, in the absence of constraints, GCs can accurately represent any MC beta-catenin inhibitor input. Thus, constraints imposed by the second term in the Lyapunov function lead to inaccurate representations of the glomerular inputs by the GCs and, consequently, nonvanishing responses of the MCs to odorants. If GCs represent the excitatory

inputs of the MCs exactly, then MCs are unresponsive as a result of the exact balance between excitation and inhibition. Therefore, GCs must fail to represent glomerular inputs; i.e., the GC code must be incomplete. Among several reasons for the incompleteness of the GC code, the nonnegativity of the GC firing rates is the most straightforward. Indeed, any M-dimensional vector (MC receptor input) can be accurately represented as a linear sum of M independent vectors (GC-to-MC synaptic weights) if the coefficients in this representation (GC firing

rates) are allowed to be both positive and negative. This is certainly true if more than M basis vectors are available (i.e., N >> M). However, because the GC firing rates cannot be negative, the representation cannot be always performed accurately, which leads to substantial MC responses. The impact of nonnegative GC firing rates is illustrated for the olfactory bulb network with three MCs and eight GCs (Figure 6). A particular odorant input is shown as a vector x→ in three-dimensional space, where each dimension corresponds to the receptor input to one of the MCs. Each GC is shown by the blue basis vector (Figure 6A). not The number of basis vectors is given by the number of GCs; e.g., eight in this example. The components of each basis vector determine the strength of the inhibitory dendrodendritic synapse from a given GC to all of the MCs. The olfactory bulb is therefore expected to represent the input vector x→ as a superposition of eight synaptic weight vectors W→i(i=1..8). The input vectors within the convex cone enveloping the weight vectors can be obtained from the weight vectors by mixing them with positive coefficients (GC firing rates). The input vectors outside the cone cannot be represented by the GC with positive coefficients.