breed [preys prey] breed [larvae larva] breed [sensors sensor] sensors-own [spike-prob spike-count] patches-own [prior posterior likelihood PrSghere] globals [normalise larvae-size dt dummy conv-count times-to-convergence max-then max-now] to setup ;; (for this model to work with NetLogo's new plotting features, ;; __clear-all-and-reset-ticks should be replaced with clear-all at ;; the beginning of your setup procedure and reset-ticks at the end ;; of the procedure.) __clear-all-and-reset-ticks set-default-shape turtles "circle" set-patch-size 16 resize-world -15 15 -15 15 set dt .001 set larvae-size .2 * (max-pxcor - min-pxcor) create-larvae 1 [ set color red set size larvae-size * 2 ] create-sensors S [ set color black set size 1 layout-circle sensors larvae-size ] setup-simulation end to setup-simulation ask patches [set pcolor 102] create-preys 1 [ set color red set size patch-size / 10 set heading theta fd larvae-size + r * 10 ] ask patches with [distance one-of larvae > larvae-size] [ ;initial prior is Poisson. set prior (1 / count patches with [distance one-of larvae > larvae-size]) set pcolor 102 + 50 * prior set PrSghere [] ;what is the probability that each sensor will spike, if the prey were on this patch? foreach sort-by [[who] of ?1 > [who] of ?2] sensors [ let d distance ? / 10 let sigma (1 / d ^ 2) let intensity (g * sigma + n) set PrSghere fput (1 - e ^ (- intensity * dt)) PrSghere ] ] ;what is the probability that each sensor will spike, given prey location? ask sensors [ set spike-count 0 ;get the distance to prey let r1 (sum [distance myself] of preys) / 10 ;calculate the signals let sigma1 1 / r1 ^ 2 ;expected number of spikes in time interval 1 let intensity1 (g * sigma1 + n) ;given prey location, this sensor fires with probability spike-prob. set spike-prob (1 - e ^ (- intensity1 * dt)) ] end to go tick generate-spikes calculate-likelihood update-prior end to generate-spikes ask sensors [ ;draw a random number and see if sensor spikes ifelse random-float 1 < spike-prob [set color white set spike-count (spike-count + 1)] [set color black] ] end to calculate-likelihood ask patches with [distance one-of larvae > larvae-size] [ ;since sensors are independent, Pr(spike set) = product of individual sensors spiking. ;what is the probability of the observed spike set if a prey were on this patch? set likelihood 1 foreach sort-by [[who] of ?1 > [who] of ?2] sensors [ ifelse [color] of ? = white [set likelihood (likelihood * item ([who] of ? - 1) PrSghere)] [set likelihood (likelihood * (1 - item ([who] of ? - 1) PrSghere) ) ] ] set posterior (likelihood * prior) ] end to update-prior set normalise (sum [posterior] of patches) ask patches with [distance one-of larvae > larvae-size] [ set posterior (posterior / normalise) set pcolor 102 + 50 * posterior ;set pcolor 102 + 1000 * posterior if pcolor >= 110 [set pcolor 109.9] set prior posterior ] end @#$#@#$#@ GRAPHICS-WINDOW 705 27 1211 554 15 15 16.0 1 10 1 1 1 0 0 0 1 -15 15 -15 15 1 1 1 ticks 30.0 BUTTON 234 250 300 283 NIL setup NIL 1 T OBSERVER NIL NIL NIL NIL 1 SLIDER 308 143 428 176 S S 0 50 25 1 1 NIL HORIZONTAL BUTTON 357 455 420 488 NIL go T 1 T OBSERVER NIL NIL NIL NIL 1 SLIDER 529 134 644 167 g g 1 10 1 1 1 NIL HORIZONTAL SLIDER 529 172 644 205 n n 0 100 100 1 1 NIL HORIZONTAL SLIDER 44 128 223 161 r r .01 .5 0.1 .01 1 NIL HORIZONTAL SLIDER 44 165 224 198 theta theta 0 360 0 1 1 NIL HORIZONTAL TEXTBOX 15 251 201 284 2) Click 'setup' 24 0.0 1 TEXTBOX 16 40 293 75 1) Choose parameters 24 0.0 1 TEXTBOX 284 114 457 132 Number of sensors 18 0.0 1 TEXTBOX 56 100 206 122 Location of prey 18 0.0 1 TEXTBOX 517 81 658 124 Sensor gain and (uniform) noise 18 0.0 1 TEXTBOX 14 334 697 428 3) Run the model. The posterior is represented by the color of the blue background; white = highly probable prey is there, dark blue = highly improbable prey is there. 24 0.0 1 TEXTBOX 243 455 346 483 Click once to run, again to pause. 11 0.0 1 TEXTBOX 147 494 543 537 To adjust the speed of the simulation, there is a slider at the top of the screen. Slow down the simulation to see individual neurones responding, then speed up the simulation to watch the posterior evolve more quickly. 11 0.0 1 TEXTBOX 208 13 493 31 Note: see the information tab for more model details 11 0.0 1 @#$#@#$#@ ## WHAT IS IT? This model illustrates how spatial distributions of spiking sensors (small black circles) can, in principle, use Bayesian inference about some state of the world. In this model, the state of the world considered is the location of a prey (small red circle) with respect to a predator (large red circle). Watch how spatial distributions of spiking sensors can represent a probability distribution of prey location that converges on its actual location ## HOW IT WORKS The predator has S neurones (S is user-defined) that are equally spaced around the circumference of the predator. Each sensor fires a spike on a time step with probability Pr(S|r,theta), where (r, theta) is the position of the prey with respect to the predator, as outlined in the manuscript. Given this conditional probability and a prior distribution, we can infer the prey location by Bayes' rule with the individual spikes and non-spikes of the individual neurones. Since the posterior distribution is in two dimensions, the probability that the prey is on a certain square patch is indicated by the colour of that patch. If the patch is blue, then the prey is unlikely to be on that patch, given the output of the sensors. If the patch is white, then the prey is highly likely to be on that patch, given the output of the sensors. ## HOW TO USE IT To start, set the parameters of the model to your desired values. Choose the location of the prey using the `distance-from-predator' and 'angle' sliders and the number of sensors S that the predator possesses. Choose the gain and noise of each sensor; sensors are defined to be equivalent, so every sensor will have the same gain and noise parameter values. Next, click the 'setup' button and then click 'go.' Click the 'go' button once to start running a simulation, and click again to pause it. Use the speed slider at the top of the screen to control the speed of the simulation. Every time step is one millisecond in duration. When a neurone spikes in a time step, that neurone flashes white, and when a neuron does not spike on a time step, that neurone is black. The posterior distribution of prey location is modified accordingly (see the manuscript) and the simulation proceeds to the next time step. ## THINGS TO TRY Investigate how the gain/noise ratio affects the speed of convergence of the posterior to the actual prey location. The higher the gain/noise ratio, the fewer time steps required for the posterior to accurately estimate prey location. Investigate how the proximity of the prey to the predator affects the speed of convergence of the posterior to the actual prey location. The smaller r is, the fewer time steps required for the posterior to accurately estimate prey location. Investigate how the number of sensors S affects the speed of convergence of the posterior to the actual prey location. Generally speaking, the higher S is, the fewer time steps required for the posterior to accurately estimate prey location. ## EXTENDING THE MODEL To relax the assumption that prey is stationary, we need to utilise some basic theory from particle filtering (sequential Monte Carlo approximation algorithms). @#$#@#$#@ default true 0 Polygon -7500403 true true 150 5 40 250 150 205 260 250 airplane true 0 Polygon -7500403 true true 150 0 135 15 120 60 120 105 15 165 15 195 120 180 135 240 105 270 120 285 150 270 180 285 210 270 165 240 180 180 285 195 285 165 180 105 180 60 165 15 arrow true 0 Polygon -7500403 true true 150 0 0 150 105 150 105 293 195 293 195 150 300 150 box false 0 Polygon -7500403 true true 150 285 285 225 285 75 150 135 Polygon -7500403 true true 150 135 15 75 150 15 285 75 Polygon -7500403 true true 15 75 15 225 150 285 150 135 Line -16777216 false 150 285 150 135 Line -16777216 false 150 135 15 75 Line -16777216 false 150 135 285 75 bug true 0 Circle -7500403 true true 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