While our model predicts temporarily precise, strong, and brief MC response to odors, more complex Selleckchem Trichostatin A temporal responses observed in Shusterman et al., 2011 and Cury and Uchida, 2010 are beyond accurate description by our model, mainly due to oversimplified temporal profile of the stimulus. In a more realistic situation, the sniff dynamics controls the raise of the odorant concentration at the epithelium, and different receptors are activated at different phases of the sniffing cycle. The mechanism suggested here could be implemented in other parts of the nervous system and in other species. Thus, the inhibitory interneurons of the insect antennal lobe

(Assisi et al., 2011) could form representations of odorant-dependent inputs of the projection neurons

JQ1 mw in a similar way. Our mechanisms could also be implemented on the basis of axon-dendrite synapses if the symmetry in the synaptic strength between excitatory and inhibitory neurons (Equation 1) is approximately satisfied. This model could therefore be implemented by cortical networks. We theoretically address the experimental evidence of spatially sparse odor codes carried by MCs in awake rodents. We propose a novel role for the GCs of the olfactory bulb in which they collectively build an incomplete representation of the odorants in the inhibitory currents returned to the MCs. These inhibitory currents lead to the balance between excitation and inhibition and suppression of responses however for most of the MCs. Because the representation formed by GCs is incomplete, a small number of MCs can carry information to the cortex, leading to the sparse olfactory codes. Our model predicts sharp transient responses in a large population of MCs. This function is facilitated by the network architecture that includes bidirectional dendrodendritic MC-to-GC synapses. Our model is based on the following equations describing the responses of MCs and GCs, rm and ai: equation(Equation 5) rm=xm−∑i=1NWmiai, equation(Equation 6) u˙i=−ui+∑m=1MW˜imrm,and equation(Equation 7) ai=g(ui),ai=g(ui),where

ui   is the membrane voltage of the GC and rm=ym−y¯m. Here, ym   and y¯m are instantaneous, and average activity of the MC number m  , u˙=du/dt. We assume therefore that the dynamics of MC responses is fast enough to reflect the values of inputs instantly. The description in terms of differential (Equation 5), (Equation 6) and (Equation 7) is equivalent to the minimization of the Lyapunov function: equation(Equation 8) L(a→)=12ε∑m=1M(xm−∑iWmiai)2+∑i=1NC(ai),where C(a) the cost function is defined as equation(Equation 9) C(a)=∫0ag−1(a′)da′. The temporal behavior of the system can be viewed as a form of gradient descent; i.e., u˙i=−∂L/∂ai. The time derivative of the Lyapunov function LL is always negative: equation(Equation 10) L˙=∑i∂L∂aia˙i=∑i(−u˙i)a˙i=−∑i[g−1(ai)]′[a˙i]2≤0.