Non-topographical contrast enhancement in the olfactory bulb

Fragmented maps and the dimensionality of stimulus representations

All auditory stimuli can be mapped as spectra along a single axis of frequency. Retinotopic images of visual stimuli can be unambiguously represented as two-dimensional patterns of ganglion cell activation. However, substantially higher-dimensional spaces are required to map the analogous sensory space in which odor stimuli are distributed [11,19-21]. Heuristically, this reflects the large number of different ways that one can gradually alter the molecular structure and charge distributions of odorous molecules or the component ratios of multicomponent odors; more concretely, it reflects the large number of independent sensors (i.e., types of odorant receptor) that provide the raw material for olfactory representations. Each glomerulus represents a convergent population of olfactory sensory neurons expressing the same odorant receptor protein, which determines its molecular receptive range; due to this convergence, glomeruli can be treated as selective, low-noise chemosensory units [22]. In mice, there are roughly 1000 such receptor types and hence roughly 1000 chemoreceptively distinct sets of glomeruli [23-25]. Odorant stimulation activates characteristic, odor-specific groups of glomeruli, and perceptual similarities in odor quality correlate with the similarities in these odor-evoked patterns of glomerular activation [26,27]. As each glomerulus is capable of being independently activated to differing degrees, whether by natural or artificial stimuli [28], there are roughly 1000 potential dimensions in mice along which odor quality representations may be gradually varied. Consequently, in order for nearest-neighbor relationships to reliably reflect odorant similarity in mice, a topographical map of odor representations would need to occupy an approximately 1000-dimensional similarity space. In layered neural structures, however, only two physical dimensions are available for topographical mapping.

As the olfactory sensory neuron complement converges onto the glomerular layer of the olfactory bulb, these high-dimensional representations of odorant similarity are projected onto the two physical dimensions of the olfactory bulb input layer, a process facilitated by the glomerular architecture of that layer. Mathematically, whenever high-dimensional representational patterns are projected onto lower-dimensional spaces, they form discrete maps, that is, patchy maps with embedded discontinuities (Figure 2). Neural network algorithms such as the self-organizing map [29,30], which perform such transformations adaptively, are able to retain local similarity relationships up to certain limits imposed by the topology of projection, at which point discontinuities are inevitable. One familiar example of this phenomenon in the nervous system is the Penfield-Rasmussen somatosensory homunculus [31,32], essentially a one-dimensional representation of the two-dimensional external body surface mapped coronally along the posterior parietal cortex. While body surface contiguity is maintained within discrete segments of this map, there are also sharp discontinuities mandated by its projection onto a one-dimensional surface as well as the presence of anisotropic elaborations (such as limbs) that disrupt contiguity relationships. For example, thumb and neck are represented in adjacent cortical regions, as are toes and genitalia, along with many more subtle examples, while representations of the arms and hands are interposed between those of the face and the rest of the head. The existence of such discontinuities in the neural representation is inevitable, as illustrated in Figure 2; furthermore, the frequency of the discontinuities will increase along with the difference between the intrinsic dimensionality of the representation and that of the neural substrate. Consequently, as the high-dimensional odor space defined by an animal's olfactory receptor complement and current chemical environment projects onto the two physical dimensions of the olfactory bulb glomerular layer, the chemotopic similarity map expressed thereon will be unavoidably fragmented, incorporating many embedded discontinuities. This is exactly what has been observed in numerous studies of odor representations across the glomerular layer of the olfactory bulb, although these commonalities have not always been emphasized by the different laboratories authoring these studies.

Substantial efforts have been made by several laboratories to elucidate an underlying order to the distribution of glomeruli across the surface of the olfactory bulb. While the multiple glomeruli activated by a given odorant stimulus are typically scattered, imaging studies of odorant-specific glomerular activation patterns have observed a broad tendency for the centroids of these activated groups to migrate to progressively more ventral regions as the odorant molecules become larger – e.g., straight-chain aldehydes of increasing aliphatic chain lengths – and more hydrophobic [33-35]. This phenomenon has been attributed to the differential sorption of odorants along the nasal inspiratory path (reviewed in [36]). Similarly, within the dorsal region of the bulb (the aspect most accessible to in vivo imaging studies), longer chain lengths have been associated with centroids of activation positioned more rostrally and laterally [34,37-39], though concentration artifacts may have influenced this finding [37]. Further studies have established ad hoc modules or clusters within the olfactory bulb that exhibit greater densities of glomeruli sensitive to particular odorant moieties than would be expected by chance [40-44]. However, these broad chemotopic profiles across the bulbar surface must not be confused with a reliable topography of feature similarity across the olfactory bulb. The chemotopic maps that have been described do not extend to the finer scale of individual glomeruli upon which contrast enhancement mechanisms operate [37,45]. In each of the imaging studies cited above, many glomeruli responding to similar odorant features are located at considerable distances from one another, and there are innumerable examples of strongly activated glomeruli located adjacent to unresponsive glomeruli in response to any given odorant – a pattern indicative of a highly fragmented map of odor quality. While both of these phenomena have been consistently observed in bulbar imaging studies, they have rarely attracted substantial discussion. The important consequence of these data is that, while some neighboring glomeruli are similarly tuned for odorant stimuli, others are not; hence, the physical proximity of glomeruli is not a reliable indicator of similarity in their molecular receptive ranges. For center-surround lateral inhibition to reliably mediate contrast enhancement, each of the odorant stimuli that substantially activate any given glomerulus must also activate all of its physical neighbors, albeit to a greater or lesser degree. This is not the case in the olfactory bulb.

Glomerular circuitry implements non-topographical contrast enhancement (NTCE)

Contrast enhancement can be functionally defined as a process of competition between neurons proportional to the similarity of the information that they mediate; this definition is, of course, agnostic to mechanism. The olfactory bulb input layer provides precisely the architecture necessary to implement a form of non-topographical contrast enhancement that is not limited by its anatomically two-dimensional architecture. The NTCE algorithm described here depends upon two interacting computational mechanisms derived from established bulbar circuitry. The first mechanism operates independently within each glomerulus at the synaptic triad between OSN axons, periglomerular cell spines, and mitral cell apical dendrites [46-48]. Direct sensory input from OSN axonal arborizations activates both periglomerular neurons and mitral cell primary dendrites, while the periglomerular cells in turn inhibit local mitral cell primary dendrites via dendrodendritic synapses (Figure 1). This architecture creates a contrast enhancement generator element within each glomerulus that transforms mitral cell activity along a half-hat function (Equation 2, Figure 3; see below) such that mitral cells connected to moderately activated glomeruli – i.e., the "edges" of the high-dimensional odor representation – are specifically inhibited out of the active ensemble. When all glomeruli are combined, their collective output activity reflects the input pattern of glomerular activation filtered through a Mexican hat function with a maximum dimensionality equal to the number of glomeruli and a topology of similarity inherited from the odor environment. In other words, NTCE generates a pattern of inhibition equivalent to that which would be generated by a competitive lateral inhibitory network in which the strength of inhibition between any two glomeruli was proportional to the similarity in their molecular receptive ranges. This mechanism is illustrated below in an abstract model and in a uniglomerular compartmental model.

The second mechanism of NTCE is a global negative feedback loop utilizing the lateral excitatory network of external tufted and short-axon cells (the ET/SA network; Figure 1). This network feeds the average level of bulbar activity back upon all mitral cells as inhibition, which is necessary in order to moderate the influence of odor concentration and maintain the first NTCE mechanism within its effective dynamic range. This mechanism is illustrated below in a multiglomerular compartmental model.

Computational principles: illustrative model

Irrespective of modality or dimensionality, successful contrast enhancement implies that the single or few most strongly activated units will yield robust output in response to a given stimulus, while output activity in more modestly activated units will be specifically inhibited and minimally-activated units will remain inactive. That is, output activity as a function of input activity yields a half-hat function (Figure 3, Mi_out; see Methods), the generator of the familiar Mexican hat function (Figure 3, inset) and a signature function of contrast enhancement. As shown in a simple illustrative model (see Methods), this half-hat function can be generated independently within each glomerulus by driving parallel sigmoidal excitatory and inhibitory processes with the same sensory input, given that the inhibitory process both is more sensitive to that input than the parallel excitatory process and saturates at a lower activity level. Both of these conditions are favored by the much smaller volume and higher input resistance of periglomerular cell spines compared with mitral cell dendrites. Under these conditions, each glomerulus inhibits its own mitral cell-mediated output in scaled proportion to its sensory input, such that its net mitral output level exhibits a half-hat function along the axis of ligand-receptor affinity (Figure 3, Mi_out). Hence, mitral cells innervating glomeruli that are the most strongly tuned to a given odorant feature will be activated, while mitral cells innervating moderately well-tuned glomeruli will be inhibited. Because of this natural attunement to relative degrees of glomerular activation irrespective of their location or interconnectivity, NTCE naturally inherits the intrinsic topology of the external chemosensory environment (as filtered by the animal's complement of primary odorant receptors). This is a critical property for a system likely to encounter environments in which the distribution of relevant odorants is unpredictable.

Uniglomerular compartmental model

In order to determine whether the established electrophysiological and cytoarchitectonic properties of olfactory glomerular circuitry could indeed mediate these computations, we constructed a compartmental model of olfactory glomerular circuitry using the simulation language NEURON [49-51]. We first implemented a uniglomerular model in order to demonstrate the first of the two mechanisms underlying NTCE. The model included representations of mitral cells [52,53], periglomerular cells constrained to available anatomical and biophysical data and exhibiting appropriate single-cell physiological responses to inputs [2,47,54-58], and olfactory sensory neuron terminals designed to convey the average activation level of a coactivated and highly convergent olfactory sensory neuron population. To better illustrate the NTCE phenomenon, we limited the number of unconstrained parameters in our model and entirely omitted post-glomerular circuitry such as granule cells and mitral-granule reciprocal connectivity. Accordingly, we made no effort to model mitral cell bistability or complex spike patterning; the temporal window of interest in this model is the period preceding the first mitral cell spike, determining whether and when a mitral cell will first fire and hence preceding the recruitment of granule cells into the olfactory response. Mitral cell spike rates are used herein solely as a convenient measure of the degree of neural activation.

First, identical levels of olfactory sensory input were delivered to periglomerular dendritic spines and mitral distal dendritic tufts via glutamatergic synapses in a uniglomerular version of the model. We did not alter the relative sensitivities of the two neuron types to sensory input other than that afforded by the periglomerular spine's greater input resistance and smaller volume, though such a sensitivity difference could have been readily generated by manipulating postsynaptic receptor densities. The periglomerular cell spine also inhibited the mitral cell dendrite via a thresholded, graded mechanism. Stimulation of the OSN terminals of this network with low-affinity odorant stimuli generated no activity in mitral cells (Figure 4Aviii), while the mitral cell activation evoked by medium-affinity odorants was dissipated by periglomerular inhibition before spike initiation (Figure 4Avi–vii). Stimulation with the highest-affinity odorants, however, evoked activity in mitral cells that overcame PG-mediated inhibition and produced action potentials (Figure 4Av). Plotting mitral cell activation as a function of odor ligand-receptor affinity generated a half-hat function (Figure 4B) corresponding to that in the illustrative model (Figure 3, Mi_out), demonstrating a successful, classical contrast enhancement generator element within a single glomerulus. In contrast, mitral cells exhibited monotonically increasing activation in response to presentation of higher-affinity odorants when PG-mediated inhibition was absent (Figure 4Aiv–i).

In our analysis, the established biophysical characteristics of glomerular neurons and synapses materially contribute to the effectiveness of NTCE. The synaptic triad between OSNs, periglomerular cell spines, and mitral cells [46-48] is optimal for enabling periglomerular inhibition to effectively shunt the synaptic excitation of mitral cells, owing to the close proximity of excitatory and inhibitory inputs [59-61]. The high input resistance, thin dendritic processes, and tiny spines of PG neurons effect a a a rapid local depolarization of membrane upon excitation by sensory input (Figure 4Aix–x). This enables the fast release of GABA onto mitral cell primary dendrites before the concomitant excitation of these larger, leakier dendrites can evoke an action potential in the electrotonically distant soma or somewhat nearer primary dendrite (Figure 4Aiii–iv; [52,53,62]. Indeed, recordings from mitral cells in vivo have shown fast inhibitory postsynaptic potentials (ipsps) preceding mitral cell spike initiation even in the most rapidly-responding mitral cells [58,63,64]. While this chloride-mediated shunt inhibition may often be insufficiently hyperpolarizing to evoke clearly visible ipsps, it is highly effective at blocking spike initiation by multiplicatively decrementing the cumulative effect of excitatory sensory input (Figure 4Avi–vii;[61,65]; see also [66]). Furthermore, the activation of low-threshold T-type calcium currents in PG cells [55], while not essential to contrast enhancement, enhances its efficacy by potentiating the effect of small inputs on local PG inhibitory output (Figure 4Aix). Finally, preintegrative mechanisms such as NTCE, in which inhibition is directed to mitral cell inputs prior to their integration into a spiking response, enjoy theoretical computational advantages that postintegrative lateral inhibitory mechanisms (such as those mediated by mitral-granule interactions) cannot provide, including an improved capacity to represent arbitrary combinations of overlapping inputs and to recognize partial input patterns [67,68].

Multiglomerular compartmental model

The second mechanism required by the NTCE algorithm is essentially a global inhibitory feedback loop. This mechanism, based on the ET/SA lateral excitatory network and inhibitory PG cells (Figure 1), mitigates the effects of absolute odor concentration, thus enabling mitral cell activity patterns to better reflect relative odor ligand-receptor affinities and efficacies and hence preserve the representation of odor quality across a reasonable concentration range. We constructed a multiglomerular version of this model in order to directly illustrate non-topographical contrast enhancement among multiple glomeruli differentially activated by the same odorant, and to assess the ability of the olfactory bulb ET/SA network to normalize bulbar responses to different odor concentrations via global feedback inhibition (see Methods). In the present model, ET/SA network activity was simulated by projecting slow synaptic inhibition from each of the other glomeruli onto mitral cell primary dendrites (Figure 4, inset), effectively translating the NTCE half-hat function of each glomerulus on its abscissa in proportion to average bulbar input intensity such that only a small and consistent population of the most strongly activated mitral cells overcame this activity-dependent inhibition. Ten glomeruli were interconnected via the ET/SA network such that all mitral cells received identically-weighted inhibitory inputs from each of the other nine glomeruli. The ten glomeruli were assigned different affinities for each test odorant such that each odorant representation constituted a unique ten-dimensional vector. Addition of intraglomerular inhibition mediated by periglomerular cells and global normalizing inhibition mediated by the ET/SA network strongly disambiguated the mitral cell response profiles evoked by two similar odorants (Figure 5A,B).

Mechanisms of concentration compensation

The persistence of odor quality across concentrations is a difficult and continuing problem in olfaction. The most plausible explanation for this property is that common elements are retained among the representations of different concentrations of the same odorant, suggesting that some sort of normalization for concentration is likely to occur within the olfactory bulb. Indeed, such a normalization process has been proposed for the olfactory bulb based on imaging data showing that normalized odor-specific glomerular activity maps are relatively similar across stimulus concentrations [69,70], and that the degree of similarity in these normalized maps predicts similarities in perceived odor quality across concentrations more reliably than do non-normalized maps [69]. Consistent with these proposals, the NTCE circuit mechanism inherently compensates for concentration as the primary glomerular representation, measured predominantly from OSN presynaptic arbors, is transformed into the secondary representation based on patterns of activated mitral cells. Specifically, global feedback inhibition mediated via the ET/SA network serves to improve stability in odor quality representations among mitral cells across reasonable concentration ranges. When a given model odorant was applied at a series of concentrations (with synaptic weights unchanged), the higher concentrations recruited increasingly broad distributions of (presynaptically) activated glomeruli, as has been observed in imaging studies [38,69,71]. With intraglomerular inhibition intact, but in the absence of interglomerular inhibition, mitral cell activation patterns also broadened with increased odorant concentrations, as these higher concentrations recruited more weakly-tuned glomeruli into the ensemble of excited mitral cells (Figure 5C). However, with an intact interglomerular ET/SA network, each odorant concentration in the series could evoke comparable levels of mitral cell activity in a consistent, odor-specific subpopulation of mitral cells, conferring a degree of concentration-independence upon odor-specific mitral cell response profiles (Figure 5D). While the synaptic strengths underlying interglomerular inhibition could be tuned so as to produce either gradual recruitment or gradual elimination of mitral cells from the ensemble as odor concentration increased, a relative constancy in the ensemble of activated mitral cells is conservative in that odor-evoked mitral cell spike counts do not vary consistently with changing odorant concentrations [72-74], and certainly do not scale with the monotonically increasing presynaptic activation of olfactory glomeruli as shown in imaging studies. More specifically, given realistic odors that present multiple, diverse ligands for olfactory receptors, this constancy is only an average; individual mitral cells' responses would be expected to change qualitatively as their relative degrees of input activation change with respect to the population average. Indeed, mitral cell response profiles measured across concentration series have presented appropriately complex results; for example, nonresponsive mitral cells often become inhibited as odor concentration increases [72-74], while other mitral cells change from a net inhibitory response to a net excitatory response [73].

The normalization of olfactory representations for concentration does not imply that concentration information is lost. Model mitral cells excited by an odor stimulus exhibited shorter spike latencies as concentration increased (Figure 4), consistent with experimental data [63,75]. Concentration may also be represented in the degree of stimulus-evoked spike synchronization among mitral cells, as has been suggested for the insect antennal lobe [76]. Finally, middle and deep tufted cells, as well as the trigeminal system, may contribute to the representation and perception of odor intensity [77]. NTCE does not eliminate concentration information, but simply mitigates its impact on odor quality representations.

Contrast enhancement tested with systematically varying odor stimuli

The clearest demonstration of contrast enhancement in the olfactory bulb has been provided by Yokoi and colleagues [10]. Briefly, these authors showed that an individual mitral cell could respond to systematically varied odorant stimuli with a classical Mexican hat function: e.g., one featured mitral cell exhibited no response to a 3-carbon n-aldehyde, inhibition to a 4-carbon n-aldehyde, excitation to 5-, 6-, and 7-carbon n-aldehydes, inhibition to an 8-carbon n-aldehyde, and no response to 9-, 10-, or 11-carbon n-aldehydes. Furthermore, administration of the GABA_A antagonist bicuculline appeared to broaden this Mexican hat response, transforming an inhibitory mitral cell odor response into an excitatory response. While the authors attributed this effect to the blockade of granule cell-mediated inhibition, it is also consistent with a blockade of GABAergic periglomerular cell-mediated inhibition and with NTCE. We modeled these authors' data by presenting a systematically varying odorant series to our ten-glomerulus model (see Methods). As NTCE is independent of specific lateral projections, the model did not have to be adjusted to perform contrast enhancement along the particular axes of variation used to model the homologous series of odorants. NTCE was able to replicate the contrast enhancement function within model mitral cells based on periglomerular inhibition alone (Figure 6A; compare with Figure 2A in reference [10]). Furthermore, blockade of this periglomerular inhibition transformed an inhibitory mitral cell odor response into an excitatory response (Figure 6B; compare with Figure 5D in reference [10]).

Interpretations of existing data

Contrast enhancement among the representations of structurally similar odorants has been widely attributed to a process of reciprocal inhibition between mitral cells mediated via mutual dendrodendritic synapses with the dendrites of inhibitory granule cells [10,78-81], though lateral interactions via periglomerular neurons have also been proposed to mediate this function [2,82,83]. Lateral projections and their putative formation of an inhibitory surround have been both explicitly and implicitly presumed to reflect the two physical dimensions afforded by the columnar structure of the olfactory bulb such that physically neighboring mitral cells mutually inhibit one another more strongly than do more distantly neighboring cells [2,10,38,79,80,84,85], perhaps by analogy with the retina [86]. However, as explained above, physical proximity-based solutions such as decremental lateral inhibition cannot effectively mediate similarity-dependent computations such as contrast enhancement in a high-dimensional modality such as olfaction.

This problem has been specifically investigated in the analogous honeybee antennal lobe, using calcium imaging of odor-evoked glomerular activity among convergent OSNs and projection neurons (PNs; analogous to vertebrate mitral cells). Increasing odorant concentrations increased glomerular activation levels and recruited additional glomeruli, as in the vertebrate OB, while PNs tended to become more inhibited as odorant concentrations increased, particularly within reasonable concentration ranges [87]. Furthermore, in a systematic study of the transformation between OSN and PN odor-evoked activity patterns, PN activity patterns were significantly better fit by a model using a functionally-based pattern of inhibition (comparable to that generated by NTCE) than by a model employing nearest-neighbor lateral inhibition [11]. Indeed, the best fits based on nearest-neighbor inhibitory projections were similar to control fits based on random interglomerular projections.

Even within the vertebrate olfactory bulb, the hypothesis of proximity-based representations of odor similarity, while popular, is contraindicated in the literature. Mitral cells innervating neighboring glomeruli exhibit unrelated odorant response profiles [88], in contrast to mitral cells innervating the same glomerulus, which have very similar profiles [89]. Furthermore, as odor quality is gradually changed, using common, behaviorally validated models for the systematic variation of odor quality [10,27,90,91], the glomeruli recruited and lost from the gradually shifting odor representation are broadly distributed across the bulb rather than being consistently adjacent [42,45,71]. Quantitative measurements of response-correlated maps have revealed broad regions of odor-evoked inhibition among mitral cells that do not directly sample from strongly activated glomerular domains [92], a result consistent with numerous studies suggesting that a net inhibition is a common response of mitral cells to odor or olfactory nerve stimulation [54,63,72,74,75,89,92]. While this result alone does not exclude the possibility of (broad) center-surround inhibition, it is more consistent with a widespread, locally-induced, nontopographical pattern of inhibition such as that produced by NTCE. Indeed, in support of the latter hypothesis, strong odor-evoked inhibition of mitral cells has been clearly observed at locations far removed from the region(s) of peak activation evoked by that odor [92], and in vivo recordings coupled with calcium imaging data have suggested that action potential amplitudes and evoked calcium transients in mitral cell lateral dendrites do not decline in amplitude with increasing distance from the soma, but instead are propagated nondecrementally, suggesting an axon-like mechanism [93]. Both of these results argue against a decremental, topographically localized projection of lateral inhibition. Finally, while granule cell-mediated lateral inhibition can affect spike timing in mitral cells [94], these inhibitory synaptic inputs, widely distributed along mitral cell lateral dendrites, may not be capable of preventing spike initiation in mitral cells, particularly given that mitral cell spikes may be initiated in the primary dendrite and even potentially the glomerular tuft [52,62,95]. This implicates glomerular circuitry, rather than external plexiform layer circuitry, as the primary determinant of whether a given mitral cell will respond to a given odor input with action potentials or with inhibition. However, the long projections of mitral cell lateral dendrites and their dense interconnectivity with granule cell dendrites are likely to underlie a network of mutual inhibition among mitral cells approaching all-to-all connectivity, and hence are likely to be capable of performing secondary computations upon arbitrarily high-dimensional maps. The mitral-granule network is therefore potentially well suited to transform glomerular output in accordance with odor learning, presumably through the learned modification of mitral-granule reciprocal synapses and learning-correlated granule cell neurogenesis [96-99], and perhaps involving the generation of field oscillations and the synchronization of mitral cell spikes [94,100-103]. Indeed, a contrast enhancement transformation based on local field oscillations and spike synchronization has been modeled in antennal lobe PNs, analogous to mitral cells [76,104].

In the interests of simplicity, some known glomerular circuit elements were omitted from the present model. First, periglomerular cells deliver GABA_B-ergic and D2 dopaminergic presynaptic inhibition onto the axon terminals of convergent OSNs [105-109]. This feedback inhibition regulates OSN glutamate release and has been proposed to help normalize OSN synaptic output with respect to concentration; such an effect would be expected to facilitate NTCE. Second, mutual excitation among mitral cells innervating the same glomerulus [110-113] would be expected to underlie cooperativity in the glomerular output response, sharpening the distinction between highly activated glomeruli in which mitral cells generate spikes and more modestly-activated glomeruli in which incoming OSN activity is dissipated without evoking mitral spikes. Third, self- (and potentially mutual) inhibition among PG cells, while hindering PG cell spike initiation, actually depolarizes PG cells and hence may have a potentiating effect on their graded inhibition of mitral cell apical dendrites [114]. This putative positive feedback loop may further contribute to the efficacy of PG inhibition of mitral cell spike initiation. Finally, some PG cells have axons that project to nearby glomeruli; while sparse, these projections have served as a basis for some models of lateral inhibition [82,83]. The utility of these projections is not clear; they potentially may serve to improve the functional differentiation of neighboring glomeruli with very similar molecular receptive ranges, such as might be generated via activity-dependent segregation [16].

The compartmentalized architecture of olfactory glomeruli may reflect a specific adaptation for the representation and processing of high-dimensional computational spaces within the physical constraints of two-dimensional cortical layers. If true, this would explain the observation that a functionally similar glomerular architecture has evolved independently several times in diverse phyla [115], and indeed is a nearly universal characteristic of complex olfactory systems. The limited diversity observed in these analogous olfactory structures (olfactory bulb, antennal lobe) may even suggest fundamental constraints on the effective processing of olfactory information above a certain level of complexity. This principle may also extend to other brain regions possessing glomerular architectures, notably the cerebellar cortex, implying that these circuits are also involved in the processing of high-dimensional representations.

Illustrative model of NTCE

NTCE depends upon two interacting computational mechanisms, each of which is consistent with available anatomical and physiological data. The first of these mechanisms, based on the rapid shunt inhibition of activated mitral cells by coactivated periglomerular spines, narrows the population of activated mitral cells with respect to the population of activated glomeruli such that only the most strongly activated subset of activated glomeruli generates action potentials in its associated mitral cells. Crucially, this contrast enhancement mechanism is independent of interglomerular projections, depending only upon relative degrees of glomerular activation and hence naturally inheriting the topology of odor similarity that is generated by the olfactory receptor complement. The second mechanism, essentially a global negative feedback loop, mitigates the effects of absolute odor concentration, thus enabling mitral cell activity patterns to reflect relative odor ligand-receptor affinities and efficacies and hence preserve the representation of odor quality across a reasonable concentration range.

The model depicted in Figure 3 was composed in Matlab (Mathworks, Natick, MA) and illustrates the first of these two mechanisms. For simplicity of illustration, odor concentrations are held constant. Mitral cells (Mi_in) and periglomerular cells (PG) within a given glomerulus independently receive identical levels of direct sensory input (depending on the odor ligand-receptor affinity of their OSNs; abscissa), although they respond to this input differently based on their passive membrane properties (see Results and Discussion). As neuronal activation levels in this illustration (ordinate) depend upon ligand-receptor binding between the odorant and the olfactory receptor species projecting to the glomerulus in question, the direct activations of both Mi and PG cells vary sigmoidally as a function of odor ligand-receptor affinity:

$\{{\text{Mi}}_{\text{in}},\text{PG}\}=[\text{K}(1/1+{(\frac{{\text{Y}}_{1/2}}{\text{A}})}^{\text{m}})]\left(1\right)$

where K is a scaling constant determining relative excitability (due to input resistance), Y_1/2 is the odor ligand-receptor affinity at which half-maximal output activity would be evoked in the mitral or PG cell via direct sensory input, A denotes the odor ligand-receptor affinity variable (abscissa), and m is the cooperativity (Hill equivalent; [116]) of each process. Net mitral cell output (Mi_out) – i.e., the generation of spiking activity – depends on both direct mitral cell activation by sensory input (Mi_in) and on the secondary inhibitory influence of periglomerular cell activity (PG), resulting in a net activation function for mitral cell output that depends on the difference between these two sigmoids:

${\text{Mi}}_{\text{out}}=[{\text{K}}_{\text{e}}(1/1+{(\frac{{\text{Y}}_{1/2(\text{e})}}{\text{A}})}^{{\text{m}}_{\text{e}}})]-[{\text{K}}_{\text{i}}(1/1+{(\frac{{\text{Y}}_{1/2(\text{i})}}{\text{A}})}^{{\text{m}}_{\text{i}}})]\left(2\right)$

in which subscript e denotes the parameters of mitral cell direct excitation (Mi_in) and subscript i denotes the inhibitory effect of periglomerular cell activity (PG). In the example shown in Figure 3, K_e = 1, K_i = 0.6, Y_1/2(e) = 10^-5 M, Y_1/2(i) = 10^-4 M, and both m terms are unity.

Contrast enhancement is typically described with an on-center/inhibitory surround, or Mexican hat, function [117]; Figure 3, inset). Traditional Mexican hat functions in one dimension are typically modeled as a difference of Gaussians, or as the difference between two differences of sigmoids. Differences between sigmoids of equal maxima (K_e, K_i) and cooperativities (m_e, m_i) yield bell-like curves; the difference between two bell-like curves (whether gaussians or differences of sigmoids) generates a Mexican hat function given that the excitatory difference is of greater maximum magnitude and the inhibitory difference is of greater breadth. The corresponding function in NTCE is one half of a Mexican hat, modeled as a difference between sigmoids in which K_e > K_i and Y_1/2(e) < Y_1/2(i) and referred to as a half-hat function (Figure 3, Mi_out; equation 2). Mexican hat functions of arbitrary dimensionality can be generated by rotating this half-hat function around the coordinate of its maximum activation level in a given context. When K_e > K_i and Y_1/2(e) < Y_1/2(i) – both conditions which are favored by the smaller volume and higher input resistance of periglomerular cell spines compared with mitral cell dendrites [118] – the output process Mi_out (equation 2) exhibits contrast enhancement along the axis of odor ligand-receptor affinity A (see Results and Discussion).

Compartmental model of NTCE

The compartmental model depicted in Figures 4 and 5 was composed in the neural simulation language NEURON [49-51]. Three cell types were explicitly modeled: mitral cells, periglomerular cells, and olfactory sensory neuron (OSN) axonal arbors. To minimize unconstrained parameters, the ET/SA network [2] was modeled as a set of delayed inhibitory synapses from each OSN to all mitral cells sampling from glomeruli other than the one to which that OSN projected, hence abstracting the projection of interglomerular inhibition mediated through the chain composed of local external tufted cells, short-axon cells, and periglomerular cells in other glomeruli. Granule cells and other post-glomerular circuitry were omitted. For simplicity, AMPA and NMDA glutamatergic synapses were combined into a single mechanism. All synapses were thresholded and graded so as to reflect either genuinely graded synaptic communication in vivo (e.g., PG to Mi synapses) or the spike density functions of large neuronal populations (e.g., convergent OSN inputs to PG and Mi cells), and approached steady-state exponentially.