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).