The corresponding EID is overlaid in magenta a darker color indicates higher expected information should the animal take a new sensory measurement in the corresponding location. Higher values in the y direction represents higher confidence of the target at the given x location. ( D) In the top panel, we show an idealization of the moth’s belief about the flower’s location when the flower reaches the center point (blue line Gaussian distribution above the moth). We simulate the tracking of the flower using the ergodic information harvesting algorithm, our implementation of energy-constrained proportional betting. ( D–F) An illustration of an animal tracking an object constrained to movement in a line, in this case a hypothetical moth using visual signals to track a flower it is feeding from while the flower sways in a breeze in a manner approximated by a 1-D sinusoid-a natural behavior ( Sponberg et al., 2015). Black dots along the trajectory indicate samples at fixed time intervals (longer distances between dots indicate higher speed). The natural trade-off between exploration and exploitation that emerges leads to localization of the target and rejection of the distractor (adapted from Figure 3 of Miller et al., 2016). In contrast, energy-constrained proportional betting ( C) samples the EID proportionate to its density and is balanced by the cost of movement. ( B) Trajectories generated by information maximization (entropy minimization) locally maximizes the expected information density (EID) at every step, which here commands a path straight to the nearby distractor. Because the peak expected information is typically not at the same location as the object, we illustrate the target peak as the point of maximum target visibility. ( A) The heat map represents the expected information density. Similarly, in behaviors where animals sample discretely over time, animals vary their sampling frequency or the location at which samples are taken, as observed in bats, rats, beaked whales, humans, and pulse electric fish ( Yovel et al., 2010 Mitchinson et al., 2007 Hartmann, 2001 Kothari et al., 2018 Caputi et al., 2003 Pluta and Kawasaki, 2008 Nelson and MacIver, 2006 Schnitzler et al., 2003 Madsen et al., 2005 Yang et al., 2016 Hoppe and Rothkopf, 2019).
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For example, weakly electric fish will track and stay near a moving refuge, but in addition to the large motions needed to stay near the refuge, there are small whole-body oscillations-an electrosensory analog to microsaccades ( Video 1 and Figure 1-figure supplement 1 Stamper et al., 2012). These movements appear unrelated to the movements that are necessary to achieve the task at hand. This theory, energy-constrained proportional betting, predicts the small and seemingly extraneous movements that sensory organs or animals undergo as they near or track a target of interest (see Figure 1-figure supplement 1 Martin, 1965 Basil et al., 2000 Ferner and Weissburg, 2005 Webb et al., 2004 Willis and Avondet, 2005 Porter et al., 2007 Louis et al., 2008 Duistermars et al., 2009 Yovel et al., 2010 Khan et al., 2012 Stamper et al., 2012 Catania, 2013 Sponberg et al., 2015 Lockey and Willis, 2015 Rucci and Victor, 2015 Stöckl et al., 2017). We have developed a theory that unifies these two dimensions of information acquisition and can be applied across sensory modalities and species.
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Because movement is energetically costly, there is likely a balance between the benefits of increased sensory information and energetic costs for obtaining that information ( MacIver et al., 2010). Movement can be used to obtain information that is unevenly distributed in the environment.
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It predicts sense organ movements in animals and can prescribe sensor motion for robots to enhance performance. Our theory unifies the metabolic cost of motion with information theory. Trajectories generated in this way show good agreement with measured trajectories of fish tracking an object using electrosense, a mammal and an insect localizing an odor source, and a moth tracking a flower using vision.
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We propose a new theory for these movements called energy-constrained proportional betting, where the probability of moving to a location is proportional to an expectation of how informative it will be balanced against the movement’s predicted energetic cost. While multiple theories for these movements exist-in that they support infotaxis, gain adaptation, spectral whitening, and high-pass filtering-predicted trajectories show poor fit to measured trajectories. While animals track or search for targets, sensory organs make small unexplained movements on top of the primary task-related motions.