Sparse incomplete representations: a potential role of olfactory granule cells

Abstract

Mitral/tufted cells of the olfactory bulb receive odorant information from receptor neurons and transmit this information to the cortex. Studies in awake behaving animals have found that sustained responses of mitral cells to odorants are rare, suggesting sparse combinatorial representation of the odorants. Careful alignment of mitral cell firing with the phase of the respiration cycle revealed brief transient activity in the larger population of mitral cells, which respond to odorants during a small fraction of the respiration cycle. Responses of these cells are therefore temporally sparse. Here, we propose a mathematical model for the olfactory bulb network that can reproduce both combinatorially and temporally sparse mitral cell codes. We argue that sparse codes emerge as a result of the balance between mitral cells' excitatory inputs and inhibition provided by the granule cells. Our model suggests functional significance for the dendrodendritic synapses mediating interactions between mitral and granule cells.

Figure 1

MC–GC network model. MCs (triangles) receive excitatory inputs from glomeruli (large circles). Active/inactive glomeruli are shown by the full/empty circles, respectively. The combinatorial pattern of glomerular activation represents the olfactory stimulus. The MC output is sent to the downstream parts of the brain for further processing via lateral olfactory tract (LOT). MCs and GCs (blue circles) are connected by reciprocal dendrodendritic synapses shown only for one GC. The GCs receive the excitatory inputs from the MCs (red arrow) and MCs are inhibited by GCs (blue arrow).

Figure 2

Balance between excitation and inhibition. The olfactory bulb network includes only one GC for simplicity. (A) If activity of MCs is weak, they do not drive the GC above firing threshold. The inputs from receptor neurons (red arrows) determine the activity of MCs directly. (B) If the inputs from MCs to the GC are strong enough to activate it, for example, due to additional inputs from the cortex (green), the GC substantially reduces the activation of the MCs through the inhibitory feedback (blue arrows) provided by the reciprocal dendrodendritic synapses, thus implementing a balance between excitation and inhibition for each MC. The activity of MCs (gray) is reduced to the level sufficient to drive the GC, i.e. vanish in the limit of a large number of MCs. (C) Strong responses of some MCs are possible if the pattern of inhibition is incomplete. Only the leftmost MC responds substantially, despite receptor inputs into the other cells. The MC responses are sparse, similarly to observed experimentally in awake behaving animals.

Figure 3

Redundancy reduction in the MC code. (A) Two strongly overlapping glomerular activations lead to dissimilar patterns of MC activity. (B) If a different GC is brought close to firing threshold by the cortical inputs, the overlap is eliminated for a different pair of odorants.

Figure 4

GCs form representations of MC glomerular inputs. (A) Several GCs may form a more complete representation. (B) GCs compete for better representation. The cell with a larger overlap with the glomerular inputs (left) can remove the excitation from the inputs of the GC with smaller overlap (right), rendering this cell inactive.

Figure 5

GC individual cost function C(a) . (A) input-output relationship (u and a are the GC's input current and the resulting firing rate) (B) The cost function prohibits negative firing rates. For positive firing rates, the cost function is approximately linear and proportional to the firing threshold θ.

Figure 6

Non-negativity of the GC firing rates leads to incomplete odorant representations. The olfactory bulb includes three MCs (3D input space) and eight GCs. The inputs into MCs from receptor neurons are represented by the black arrows. The synaptic weights from eight GCs to the MCs are shown by the blue arrows. (A) If inputs lie within the cone restricted by the weight vectors, the input vector can be represented as an exact superposition (linear sum) of the weight vectors with positive coefficients. The GC representation of MC inputs is exact, and the MCs are expected to respond weakly due to an exact balance between inhibition and excitation. (B) If the input vector is outside the cone of the GC weights, the GCs cannot represent the inputs exactly. This is because the GC firing rates (coefficients of expansion) cannot be negative. The best approximation of the inputs $\overrightarrow{\tilde{x}}$ is the nearest vector to the input vector x⃗ on the surface of the cone. $\overrightarrow{\tilde{x}}$ is formed by two GCs (red). Because x⃗ is different from $\overrightarrow{\tilde{x}}$, the GC code is incomplete. MC response is $\overrightarrow{r}=\overrightarrow{x}-\overrightarrow{\tilde{x}}$.

Figure 7

The transient responses of MCs. (A) Four classes of MCs can be identified depending on their relationship to the GC connectivity and glomerular inputs. These classes include I - cells that receive glomerular inputs but are not inhibited by the GC set corresponding to given odorant, II – cells in which excitation is balanced by inhibition almost precisely, III – cells that receive inhibitory inputs only, and IV – cells that receive no inputs. Because responses of MCs are defined with respect to their spontaneous activity, they can be negative as indicated by the color of Type III cell. (B) Firing rates (FR) of cells in four classes as a function of time after a sharp odorant onset (gray region). Type II cells exhibit sharp activity transients that decay to undetectable levels as a function of time.

Figure 8

Possible functional significance of dendro-dendritic synapses. (A) Networks of inhibitory neurons implementing sparse overcomplete representations decomposes the input pattern into the linear sum of dictionary elements (Rozell et al., 2008). The response of each cell in the recurrent layer indicates whether the given dictionary element is present in the input. The feedforward excitatory weights of a recurrent layer neuron number k contain the dictionary element represented by this neuron d_km. The recurrent inhibitory weights are proportional to the overlap between dictionary elements, making similar input patterns compete more strongly. The biological implementation of this constraint is unclear. (B) In networks with dendrodendritic synapses, the connectivity between GCs is inhibitory. Because every two GCs are two synapses away from each other, the strength of pairwise inhibition contains an overlap between dendrodendritic weights thus automatically satisfying the constraint on the recurrent inhibitory weights shown in (A).