Deconstructing Odorant Identity via Primacy in Dual Networks

Abstract

In the olfactory system, odor percepts retain their identity despite substantial variations in concentration, timing, and background. We study a novel strategy for encoding intensity-invariant stimulus identity that is based on representing relative rather than absolute values of stimulus features. For example, in what is known as the primacy coding model, odorant identities are represented by the conditions that some odorant receptors are activated more strongly than others. Because, in this scheme, odorant identity depends only on the relative amplitudes of olfactory receptor responses, identity is invariant to changes in both intensity and monotonic nonlinear transformations of its neuronal responses. Here we show that sparse vectors representing odorant mixtures can be recovered in a compressed sensing framework via elastic net loss minimization. In the primacy model, this minimization is performed under the constraint that some receptors respond to a given odorant more strongly than others. Using duality transformation, we show that this constrained optimization problem can be solved by a neural network whose Lyapunov function represents the dual Lagrangian and whose neural responses represent the Lagrange coefficients of primacy and other constraints. The connectivity in such a dual network resembles known features of connectivity in olfactory circuits. We thus propose that networks in the piriform cortex implement dual computations to compute odorant identity with the sparse activities of individual neurons representing Lagrange coefficients. More generally, we propose that sparse neuronal firing rates may represent Lagrange multipliers, which we call the dual brain hypothesis. We show such a formulation is well suited to solve problems with multiple interacting relative constraints.

Figure 1.

Using relative values of receptor responses can solve the problem of recovering sparse concentration vector $\overrightarrow{x}$. (A) An example concentration vector x alongside its reconstruction (blue) using only relative information that a group of receptors P respond more strongly than the rest. (B) Correlation between the reconstructed and true concentration vectors for different sparsity (K) values and number of primary receptors (p). (C) Correlation between reconstruction and stimulus with various levels of noise injected on input signal$\overrightarrow{y}$(blue) and input neuron response$\overrightarrow{r}$(red).

Figure 2.

The solutions provided by soft and hard primacy conditions are similar. (A) An example concentration vector $\overrightarrow{x}$ alongside its hard primacy reconstruction (blue) and the soft primacy reconstruction (orange). (B) The correlation between the hard and soft primacy solutions for various sparsity K and primacy number p.

Figure 3.

The dual network model described by equations (2.22) through (2.25). (A) The structure of the network. α cells (light green) implement the dual representation of the concentration vector. β cells (dark green) implement the non-negativity constraints on the concentration vector. (B) Example firing rates of alpha cells. (C) Example firing rates of beta cells. (D) Example of the firing rate model’s reconstruction (orange) of the concentration vector compared with the original concentration vector (black). (E) Correlation between the firing rate reconstruction of the concentration vector and the true concentration vector for a range of stimulus and network parameters, K and p.

Figure 4.

Network diagram of the simplicial dual network.

Figure 5.

(A) The structure of the dual network implementing SIR model. α cells (light green) implement the dual representation of the concentration vector. ${\overline{\beta }}^{*}$ cells (dark green) represent the reconstruction of the concentration vector. (B) Example firing rates of α cells. (C) Example of the sparse firing rates of ${\overline{\beta }}^{*}$ cells. (D) Comparison of the simplicial SIR model’s reconstruction (orange) and the concentration vector. (E) Correlation between simplicial SIR model’s reconstruction of the concentration vector and the true concentration vector for a range of stimulus and network parameters, K and p.

Figure 6.

Possible mapping of our simplicial dual network implementing the primacy model to the known olfactory circuitry. The neurons of our network are depicted with circles and the corresponding brain region we suggest for those neurons are shown in the surrounding box. The suggested analogy with specific cell types are given outside each circle. AON, PC, and OSNs stand for anterior olfactory nucleus, piriform cortex, and olfactory sensory neurons respectively.