481.03SFN 2018
Optimal encoding of odor concentration for olfactory navigation is approximated by the Hill nonlinearityJonathan D. Victor1, Sebastian D. Boie1, Erin G. Connor2, John P. Crimaldi2, G. Bard Ermentrout3, Katherine I. Nagel4
1Feil Family Brain and Mind Research Institute, Weill Cornell Med.College, New York, NY;2Civil, Environmental and Architectural Engin., Univ. of Colorado, Boulder, CO; 3Mathematics, Univ. of Pittsburgh, Pittsburgh, PA
4Neurosci. Institute, NYU Langone Med. School, New York, NY
Results
Introduction
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A: Odor concentration is measured at multiple grid locations (triangles) within adynamic odor plume.B: Since the odor concentration varies with time, each location yields a distributionof odor concentrations. The large histogram shows the distribution of odorconcentrations across all grid points; the two smaller histograms show thedistribution of odor concentrations at two example grid points.
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Overview
C-E: Evaluation of schemes encoding odor concentration. C: Locations across the gridpoints are assigned equal a priori probability. D: An odor sample is obtained at arandomly-chosen grid point, and encoded into a code word that represents a discreterange of odor concentrations. Several alternative discretizations are considered. E: Foreach code word, the a posteriori probability of location within the grid is computed viaBayes Theorem. For each discretization, we then compute the Shannon mutualinformation between location and code word. Modified from (Boie, Connor et al. 2018).
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c1/2 = mean x 8
optimal, compressedhistogram equalization
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optimal, no compressionoptimal, compressedhistogram equalizationHill; c1/2=meanA
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Information transmitted about location by the histogram equalization (HE) code is farfrom optimal; the Hill nonlinearity code is close to optimal. Circles: informationtransmitted by the optimized code as a function of log2(M), where M is the number ofcode words (filled symbols) or as a function of the entropy of the code (open symbols).Hexagrams: information transmitted by the HE code. For HE, entropy is equal tolog2M. Open squares: performance of codes in which the Hill nonlinearity is followedby uniform segmentation into M code words. Positions of the two squares along theabscissa indicate log2M and the entropy of the code word distribution. A: 10 cm/secunbounded environment sampled with full grid; B: five environments and three grids.
Information transmitted about location by the Hill nonlinearity is maximized whenthe semi-saturation constant c1/2 is near the mean concentration in the environment.Black: c1/2 equal to the mean across grid points. Blue: c1/2 larger than mean. Yellow: c1/2smaller than mean. Filled circles: optimal code for location. Hexagrams: HE code.Abscissa: entropy of code word distribution. A: 10 cm/sec unbounded environmentsampled with full grid; B: five environments and three grids.
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Olfactory navigation is a sensorimotor behavior that is critical to the survival of a wide range of organisms. It is madecomputationally challenging by the turbulent nature of natural olfactory plumes. Evolutionarily successful organismsaccomplish olfactory navigation by making navigation decisions on a moment-by-moment basis. These decisions arenecessarily based on a limited knowledge of the odor plume. Limitations arise not only because measurements arerestricted to sensor’s locations, but also because the sensors have limited accuracy and bandwidth. Our focus here is howthese limited resources are best used. We take an information-theoretic approach: how can odor concentration be encodedinto a fixed number of bits in a way that maximizes information about location within a plume?
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2 3 4 6 8 12 16 24 32 histogram equalization
The key observation is that in an optimaldiscretization of an interval, any sub-intervalis optimally discretized. This holds because ofthe chain rule for entropy. The optimaldiscretization of the entire interval can thenbe built from a library of optimaldiscretizations of sub-intervals. Initially, alibrary of optimal discretizations of [0 x] into2 segments is constructed (blue and redsymbols). Iteratively, each library is used tobuild a library of optimal discretizationscontaining one additional segment.
Dynamic programmingstrategy for determiningthe discretization of arange into M code wordsthat maximizes theShannon informationabout an underlyingvariable (location).
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Experimental Methods
Five olfactory environments. Average odorintensity (first column), and snapshots of odorintensity on the first and last data frames (lasttwo columns). The color scale is logarithmic,ranging 0.003 to 1 (equal to the inletconcentration). For the bounded dataset, afalse floor was placed just under the releasepoint. For the obstacle dataset, the obstacle isindicated by the solid gray square; a portionof the plume could not be imaged because ofobstacle’s shadow (hatched parallelogram).
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An odor surrogate (acetone madeneutrally buoyant by mixing with air andhelium) was isokinetically released into awind tunnel at the center of its entrance.Turbulence was induced by an entrancegrid (6.4 mm diameter rods and a 25.5mm mesh spacing, followed by a 15 cmlong honeycomb section). Fluorescencewas induced with a 1 mm thick light sheetfrom a Nd:YAG 266nm pulsed laser.Fluorescence, proportional to acetoneconcentration, was imaged using a high-efficiency sCMOS camera. Modified fromConnor et al., 2018.
Spatiotemporal Measurement of Odor Concentrations in a Turbulent Air Plume
Summary & ConclusionsWe used a combined experimental and
theoretical approach to analyze optimalcoding strategies for the purpose of olfactorynavigation.
Planar laser-induced fluorescence wasused to measure spatiotemporalcharacteristics of turbulent plumes in air.
A new dynamic programming algorithmwas used to identify optimal codingschemes.
Histogram equalization, the optimal strategyfor transmitting information aboutconcentration, is sub-optimal for transmittinginformation about location. For location,gentler nonlinearities yield greaterinformation per code word.
The advantage of a more gently saturatingnonlinearity is even greater whencompressibility of the code word stream istaken into account.
Optimal behavior is approximated by a Hillreceptor binding nonlinearity, with bindingconstant c1/2 at the mean odor concentration.
Optimal segmentations for the 10 cm/sec unbounded environment, full grid. A: Positions of the cutpoints, for 2 to 32 codewords (M). For each value of M, there are M-1 cutpoints, which separate the concentration range into segmentscorresponding to the code words. B: stepwise nonlinearities corresponding to selected values of M. As shown by the arrowsfor M=2 (black) and M=3 (blue), the nonlinearities in B have a step increment of height 1/M at the cutpoints in A. Histogramequalization corresponds to the diagonal. C: As in B, but plotted as a function of normalized concentration, rather quantile.
ReferencesBoie, S. D., E. G. Connor, et al. (2018). "Information-theoretic analysis of realistic odor plumes: What cues areuseful for determining location?" PLoS ComputationalBiology 14(7): e1006275.
Connor, E. G., M. McHugh, et al. (2018). "Quantificationof airborne odor plumes using planar laser-inducedfluorescence." Experiments in Fluids 59: 137.
Collaboration Supported by The NSF IdeasLab Initiative:JDV, SDB - NSF IOS 1555891; JPC, EGC - NSF PHY 1555862
GBE - NSF PHY 1555916; KIN – NSF IOS 1555933
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[ X ]: odorant concentration
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Hill Nonlinearity
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Sampling Grids. Three two-dimensional grids (below), superimposed oncontour lines corresponding to average odor concentrations of 0.1, 0.03, and0.01 times the inlet concentration. One-dimensional sampling grids (right) inX and Y directions. Analyses at these grids yielded similar results (not shown).
