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Biomimetic searching strategies Massimo Vergassola CNRS, URA 2171 Institut Pasteur, Unit “In Silico Genetics ”

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2m wind Male Moth released Zigzag Casting: Extended crosswind Source Odors Direction and velocity of the wind are determined by air currents and visual clues. Zigzagging and casting (J.S. Kennedy, e.g. in Physiological Entomology,1983)

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Sniffers Olfactory robots with applications to the detection of chemical leaks, drugs, bombs, land and/or sea mines. D. Martinez “On the right scent” Nature, 445, 371-372, 2007 (N&V).

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Micro vs macro-organisms: the role of size and transport

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Chemotaxis of living organisms Temporal or spatial gradients are sensed and either climbed or descended. Crucial that the chemoattractant field be smooth and the concentration high enough to be measurable. Gradients ought to provide a reliable local cue.

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Physical constraints on concentration measurements (Berg & Purcell, Biophys. J., 1977) Smoluchowski’s diffusion-limited rate of encounters Reliable measurement of concentration requires: Measured hits in the time T int >> fluctuations: Bottomline: Chemotaxis requires exponential integration times for exponentially small concentrations

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Searches by macroorganisms Responses times are O(ms) Away from the source, gradients are not effectively traceable and do not always point to the source. Odor encounters are sparse and sporadic. Yet, birds respond Km’s away and moths locate females hundreds of meters away.

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Existing sniffers rely on micro-organism mimetic strategies Chemotactic methods, e.g. Ishida et al. (1996); Kuwana et al. (1999); the robolobster by Grasso & Atema et al. (2000); Russell et al. (2003). Plume-tracking, e.g. Belanger & Arbas (1998); Li, Farrell, Cardé (2001); Farrell, Pang, Li (2003)&(2005); Ishida et al. (2005); Pang, Farrell (2006). Effective in dense conditions (relatively close to the source)

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Sniffer front view

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Sniffer in action

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Strategies for searches starting far away from the source, in dilute conditions? M.V., E. Villermaux, B.I. Shraiman Infotaxis as a strategy for searching without gradients. Nature, 445: 406- 9, 2007.

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In a nutshell Concentration is not a good local clue in dilute conditions. What else could we track in the “desert”, when nothing is detected? 1. Build a map of probability for the source position on the basis of the history of receptions. 2. Move locally to make the map sharp as fast as possible, i.e. maximize the rate of entropy reduction.

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The message of odor encounters The source emits particles that are transported in the (random) environment. Consider them as a message sent to the searcher. Message in a random medium. Use the trace of odor encounters experienced by the searcher to infer the position of the source. r 1, t 1 r 2, t 2 r 3, t 3

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Decoding the message Hit rate at position r if source located at r 0. As in message decoding, construct the posterior distribution P t (r 0 ) for the position of the source r 0 from the trace ((r 1,t 1 ),(r 2,t 2 ),…,(r H,t H )) of the hits.

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A simple model of random medium “Particles” are patches of odors where mixing has not dissipated them below the detectibility threshold. Particles emitted at rate R, advected by a mean wind V, having a finite lifetime and diffused with diffusivity D. After some algebra

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General problem: How should we exploit the posterior and deal with its uncertainties? The “unusual” feature is that the field cannot be quite trusted and is continuously updated. ML is not suitable.

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Search time-entropy relationship N points to visit. Probability at the j-th visited point is p j and neighborhood constraints dismissed. Gibbs distribution reducing to (T>>1)

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Search times vs entropy Note the exponential dependence on S, contrary to the “standard” optimal code length inequality: The reason is that the “search alphabet” is degenerate, i.e. made of a single letter. Words are discriminated by their length only ( no coalescence as in Huffman coding )

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Infotaxis Choose the local direction of motion maximizing the rate of information acquired: Maximum expected reduction of the entropy of the field P t (r 0 ). With the expected hit rate

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Exploitation vs exploration Gradients of concentration in chemotaxis Rate of acquisition of information, i.e. reduction of entropy of the posterior field P t (r 0 ). Exploration: passive gathering of information. Exploitation: maximum likelihood. RS Sutton, AG Barto Reinforcement Learning MIT Press, 1998.

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Infotactic trajectories

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pM sperm responding (sea urchin) Kaupp et al., Nature Cell Biology, 2003

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Search time statistics

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Infotaxis is the most robust and rapid among a set of alternative strategies

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Robustness to inaccuracies in the model of the environment Independent detection model in a real jet flow

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Spatial maps in animal brains Microstructure of a spatial map in the entorhinal cortex Nature, 2005 and following papers by E.I. Moser and colls. Spatial cues are transmitted to the hippocampus J. O’Keefe, J. Dostrovsky Brain Research 1971 discovery of place cells in hippocampus (see also The Hippocampus as a Cognitive Map, 1978)

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In collaboration with Boris Shraiman (Kavli Inst. Theor. Phys., UCSB) Emmanuel Villermaux (IRPHE, Marseille)

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A simple possible way to account for time correlations A model where consecutive detections have a space-independent rate give: Consecutive detections are counted just once

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Learning about the source and the medium Start the searcher with rough estimates of the parameters which make the rate function R(r|r 0 ) flatter than in reality not stuck. The searcher will get to the source slowly but steadily. Once there, infer from its odor encounter trace the parameters of the medium and the source.

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Learning about the source and the medium Log- likelihood of the experienced series of odor encounters.

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ENTROPY Entropy measures the uncertainty in a random experiment. Let X be a discrete random variable with range S X = { 1,2,3,... k} and pmf p k = P X.

ENTROPY Entropy measures the uncertainty in a random experiment. Let X be a discrete random variable with range S X = { 1,2,3,... k} and pmf p k = P X.

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