This nearest vector forms the imperfect representation of the odorant by the GC. The difference between GC representation and real vector of inputs x→−x˜→ is the error of the representation; i.e., MC odorant response r→ that is transmitted to the olfactory cortex. In the case of incomplete representations (Figure 6B), the GCs encode a Selleckchem Osimertinib point x˜→ on the boundary of the enveloping cone. Not

all GCs are simultaneously active. Indeed, in Figure 6B, only two GCs on the boundary (red weight vectors) are active, while the others contribute to the representation with zero coefficients (firing rates). The number of coactive GCs is one less than the dimensionality of the input space determined by the number of the MCs M. Thus, in Figure 6B, two GCs are contributing to the representation. In Experimental Procedures, we prove that the number of coactive GCs in the model described is less than the number of MCs. Because the number of GCs in the olfactory bulb is substantially larger than the number of MCs, only a small fraction of the GCs is coactive. Therefore, our model predicts sparse responses of GCs. For www.selleckchem.com/products/byl719.html a large number of MCs and a random set of network weights, the representations of odorants

by GCs are typically incomplete (see Supplemental Information available online). Hence, for a large network, the region inside of the cone of completeness (see Figure 6) is expected to shrink. This implies that it becomes almost impossible to expand a random input vector to the basis containing vectors with positive components by using only nonnegative coefficients. In the Supplemental Information, we show that the number of coactive GCs for random

binary inputs with M   MCs is ∼M. Because an exact representation of the M  -dimensional random input requires M   vectors, this result implies that the representation of odorants by GCs is typically imprecise. The GC code is therefore incomplete. We also show that for sparse GC-to-MC connectivity, when only K<Non-specific serine/threonine protein kinase i.e., M. Therefore, GCs cannot represent MC inputs precisely in the case of random connectivity, which implies ubiquity of incomplete representations. So far, we have discussed the responses of MCs and GCs in the stationary state established after odorant onset. We found that the responses of MCs may be spatially or combinatorially sparse in the steady state. This means that a small fraction of MCs carries sustained responses to odorants. Here, we address the responses during the transitional period immediately following the odorant onset. Within this model, many MCs should display sharp activity transients that are followed by exponentially decaying responses. The responses of most of the MCs are temporally sparse.