JV SFN2018 FINAL - jdvicto/vps/sfn18_vibice.pdf · 481.03 SFN 2018 Optimal encoding of odor...
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481.03 SFN 2018 Optimal encoding of odor concentration for olfactory navigation is approximated by the Hill nonlinearity Jonathan D. Victor 1 , Sebastian D. Boie 1 , Erin G. Connor 2 , John P. Crimaldi 2 , G. Bard Ermentrout 3 , Katherine I. Nagel 4 1 Feil Family Brain and Mind Research Institute, Weill Cornell Med.College, New York, NY; 2 Civil, Environmental and Architectural Engin., Univ. of Colorado, Boulder, CO; 3 Mathematics, Univ. of Pittsburgh, Pittsburgh, PA 4 Neurosci. Institute, NYU Langone Med. School, New York, NY Results Introducon posterior probability B 0 1 0 1 0 1 0 1 p(c) p(c) A 0 0 1 1 p(c) 15 cm 20 cm 25 cm 10 cm 5 cm source C m p ( m) m 1 m 2 m p ( m) m 1 m 2 x y p(l) alternative encodings c p(c) odor distribution p ( l | m 1 ) p ( l | m 2 ) p ( l | m 2 ) x y p ( l | m 1 ) A: Odor concentraon is measured at mulple grid locaons (triangles) within a dynamic odor plume. B: Since the odor concentraon varies with me, each locaon yields a distribuon of odor concentraons. The large histogram shows the distribuon of odor concentraons across all grid points; the two smaller histograms show the distribuon of odor concentraons at two example grid points. D E Overview C-E: Evaluaon of schemes encoding odor concentraon. C: Locaons across the grid points are assigned equal a priori probability. D: An odor sample is obtained at a randomly-chosen grid point, and encoded into a code word that represents a discrete range of odor concentraons. Several alternave discrezaons are considered. E: For each code word, the a posteriori probability of locaon within the grid is computed via Bayes Theorem. For each discrezaon, we then compute the Shannon mutual informaon between locaon and code word. Modified from (Boie, Connor et al. 2018). wide grid narrow grid full grid 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0 0.2 0.4 0 0.2 0.4 0 0.2 0.4 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0 1 2 0 1 2 0 1 2 entropy (bits) 0 1 2 3 4 5 0 0.2 0.4 0.6 0 1 2 3 4 5 0 0.2 0.4 0.6 0 1 2 3 0 0.2 0.4 0.6 4 5 informaon (bits) 20cm/s obstacle 20cm/s unbounded 5cm/s unbounded 10cm/s unbounded 10cm/s bounded Algorithm 0 1 2 3 4 5 log 2 (M) or entropy (bits) informaon (bits) 0 .2 .4 .6 0 1 2 3 4 5 entropy (bits) informaon (bits) 0 .2 .4 .6 c 1/2 = mean x 0.125 c 1/2 = mean x 0.25 c 1/2 = mean x 0.5 c 1/2 = mean c 1/2 = mean x 2 c 1/2 = mean x 4 c 1/2 = mean x 8 opmal, compressed histogram equalizaon number of code words (M) 2 3 4 6 8 12 16 24 32 opmal, no compression opmal, compressed histogram equalizaon Hill; c 1/2 =mean A A B Informaon transmied about locaon by the histogram equalizaon (HE) code is far from opmal; the Hill nonlinearity code is close to opmal. Circles: informaon transmied by the opmized code as a funcon of log 2 (M), where M is the number of code words (filled symbols) or as a funcon of the entropy of the code (open symbols). Hexagrams: informaon transmied by the HE code. For HE, entropy is equal to log 2 M. Open squares: performance of codes in which the Hill nonlinearity is followed by uniform segmentaon into M code words. Posions of the two squares along the abscissa indicate log 2 M and the entropy of the code word distribuon. A: 10 cm/sec unbounded environment sampled with full grid; B: five environments and three grids. Informaon transmied about locaon by the Hill nonlinearity is maximized when the semi-saturaon constant c 1/2 is near the mean concentraon in the environment. Black: c 1/2 equal to the mean across grid points. Blue: c 1/2 larger than mean. Yellow: c 1/2 smaller than mean. Filled circles: opmal code for locaon. Hexagrams: HE code. Abscissa: entropy of code word distribuon. A: 10 cm/sec unbounded environment sampled with full grid; B: five environments and three grids. log 2 (M) or entropy (bits) informaon (bits) 20cm/s obstacle 20cm/s unbounded 5cm/s unbounded 10cm/s unbounded 10cm/s bounded B Olfactory navigaon is a sensorimotor behavior that is crical to the survival of a wide range of organisms. It is made computaonally challenging by the turbulent nature of natural olfactory plumes. Evoluonarily successful organisms accomplish olfactory navigaon by making navigaon decisions on a moment-by-moment basis. These decisions are necessarily based on a limited knowledge of the odor plume. Limitaons arise not only because measurements are restricted to sensor’s locaons, but also because the sensors have limited accuracy and bandwidth. Our focus here is how these limited resources are best used. We take an informaon-theorec approach: how can odor concentraon be encoded into a fixed number of bits in a way that maximizes informaon about locaon within a plume? A B C 0 0.25 0.5 0.75 1 0 .25 .5 .75 1 0 0.25 0.5 0.75 1 0 .25 .5 .75 1 2 4 8 16 32 0 0.25 0.5 0.75 1 number of code words (M) Quanle Quanle Concentraon coding quanle coding quanle number of code words (M) 2 3 4 6 8 12 16 24 32 histogram equalizaon The key observaon is that in an opmal discrezaon of an interval, any sub-interval is opmally discrezed. This holds because of the chain rule for entropy. The opmal discrezaon of the enre interval can then be built from a library of opmal discrezaons of sub-intervals. Inially, a library of opmal discrezaons of [0 x] into 2 segments is constructed (blue and red symbols). Iteravely, each library is used to build a library of opmal discrezaons containing one addional segment. Dynamic programming strategy for determining the discrezaon of a range into M code words that maximizes the Shannon informaon about an underlying variable (locaon). M = 2 M = 3 M = 4 M = 5 M = 6 M = 7 M = 8 0 1 normalized concentraon 1 0.1 0.01 10 cm average first frame last frame 20cm/s obstacle 20cm/s unbounded 20cm/s bounded 10cm/s unbounded 5cm/s unbounded Experimental Methods Five olfactory environments. Average odor intensity (first column), and snapshots of odor intensity on the first and last data frames (last two columns). The color scale is logarithmic, ranging 0.003 to 1 (equal to the inlet concentraon). For the bounded dataset, a false floor was placed just under the release point. For the obstacle dataset, the obstacle is indicated by the solid gray square; a poron of the plume could not be imaged because of obstacle’s shadow (hatched parallelogram). Camera Turbulence Grid Laser T est Secon Laser Sheet Acetone Seeded Flow Honeycomb Flow Carrier Gas Water Bath Liquid Acetone Cylindrical Lens Fan 0.3m 1m An odor surrogate (acetone made neutrally buoyant by mixing with air and helium) was isokinecally released into a wind tunnel at the center of its entrance. Turbulence was induced by an entrance grid (6.4 mm diameter rods and a 25.5 mm mesh spacing, followed by a 15 cm long honeycomb secon). Fluorescence was induced with a 1 mm thick light sheet from a Nd:YAG 266nm pulsed laser. Fluorescence, proporonal to acetone concentraon, was imaged using a high- efficiency sCMOS camera. Modified from Connor et al., 2018. Spaotemporal Measurement of Odor Concentraons in a Turbulent Air Plume Summary & Conclusions We used a combined experimental and theorecal approach to analyze opmal coding strategies for the purpose of olfactory navigaon. Planar laser-induced fluorescence was used to measure spaotemporal characteris cs of turbulent plumes in air. A new dynamic programming algorithm was used to idenfy opmal coding schemes. Histogram equalizaon, the opmal strategy for transming informaon about concentraon, is sub-opmal for transming informaon about locaon. For locaon, gentler nonlinearies yield greater informaon per code word. The advantage of a more gently saturang nonlinearity is even greater when compressibility of the code word stream is taken into account. Opmal behavior is approximated by a Hill receptor binding nonlinearity, with binding constant c 1/2 at the mean odor concentraon. Opmal segmentaons for the 10 cm/sec unbounded environment, full grid. A: Posions of the cutpoints, for 2 to 32 code words (M). For each value of M, there are M-1 cutpoints, which separate the concentraon range into segments corresponding to the code words. B: stepwise nonlinearies corresponding to selected values of M. As shown by the arrows for M=2 (black) and M=3 (blue), the nonlinearies in B have a step increment of height 1/M at the cutpoints in A. Histogram equalizaon corresponds to the diagonal. C: As in B, but ploed as a funcon of normalized concentraon, rather quanle. References Boie, S. D., E. G. Connor, et al. (2018). "Informaon- theorec analysis of realisc odor plumes: What cues are useful for determining locaon?" PLoS Computaonal Biology 14(7): e1006275. Connor, E. G., M. McHugh, et al. (2018). "Quanficaon of airborne odor plumes using planar laser-induced fluorescence." Experiments in Fluids 59: 137. Collaboraon Supported by The NSF IdeasLab Iniave: JDV, SDB - NSF IOS 1555891; JPC, EGC - NSF PHY 1555862 GBE - NSF PHY 1555916; KIN – NSF IOS 1555933 wide grid 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0 1 2 3 4 5 0 0.2 0.4 0.6 0 1 2 3 4 5 0 0.2 0.4 0.6 0 1 2 3 4 5 0 0.2 0.4 0.6 0 0.2 0.4 0 0.2 0.4 0 0.2 0.4 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0 1 2 0 1 2 0 1 2 narrow grid full grid [ X ]: odorant concentraon f ([ X ]): bound fracon c : semisaturaon constant 1/2 [ ] ([ ]) [ ] X f X c X = + 1/2 Hill Nonlinearity wide grid narrow grid full grid 5cms unbounded 10cms unbounded 10cms bounded 20cms unbounded 20cms obstacle X grids Y grids prior probability concentration (c) concentration (c) concentration (c) x y x y x y Sampling Grids. Three two-dimensional grids (below), superimposed on contour lines corresponding to average odor concentraons of 0.1, 0.03, and 0.01 mes the inlet concentraon. One-dimensional sampling grids (right) in X and Y direcons. Analyses at these grids yielded similar results (not shown).
Transcript of JV SFN2018 FINAL - jdvicto/vps/sfn18_vibice.pdf · 481.03 SFN 2018 Optimal encoding of odor...
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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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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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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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).
