breed [predators predator] breed [preys prey] breed [s1s s1] breed [s0s s0] breed [posteriors posterior] breed [neurones neurone] breed [spikes spike] breed [ghosts ghost] breed [bars bar] breed [avgs avg] breed [std-ups std-up] breed [std-downs std-down] s1s-own [ps1] s0s-own [ps0] posteriors-own [actual] ghosts-own [actual] std-ups-own [actual] std-downs-own [actual] avgs-own [actual] globals [dt scale scale-avg marker marker-avg draw S resolution ghost-trail expect standard convergence-time] 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 resize-world -16 16 -18 16 set convergence-time [] reset end to reset clear-turtles set-patch-size 15 ask patches [set pcolor white] set resolution 400 create-s1s resolution [set shape "dot" set size .5 set heading 90 set color black setxy (-14 + who * 28 / count s1s) -18] create-s0s resolution [set shape "dot" set size .5 set color black setxy (-14 + (who - resolution) * 28 / resolution) -18] create-posteriors resolution [ set shape "dot" set size .3 set color 0 setxy (-14 + (who - 2 * resolution) * 28 / resolution) 0 ] ask turtles with [(breed = s1s or breed = s0s or breed = posteriors) and (xcor < -12.3)] [die] if prior? = "normal" [ let mu .25 let sigma .1 let A (1 / (sigma * sqrt(2 * pi))) ask posteriors [ set actual (A * e ^ (-1 * ((([xcor] of self + 14) / 28) - (mu + 14) / 28) ^ 2 / (2 * sigma ^ 2)) ) ] ] if prior? = "dirac-delta" [ ask min-one-of posteriors [abs xcor] [set actual resolution] ] if prior? = "uniform" [ ask posteriors [set actual resolution / count s1s] ] set dt .001 ask s0s [let I (g * (([xcor] of self + 14) / 56) ^ (-2) + n) * dt set ps0 e ^ (-1 * I) set ycor (ycor + 4 * e ^ (-1 * I))] ask s1s [let I (g * (([xcor] of self + 14) / 56) ^ (-2) + n) * dt set ps1 1 - e ^ (-1 * I) set ycor (ycor + 4 * (1 - e ^ (-1 * I))) ] create-predators 1 [setxy -14 15 set size 2.5 set shape "circle" set color red] create-neurones 1 [setxy -14 15 set size 1 set shape "circle" set color black] create-preys 1 [setxy (-14 + D * 28 * 2) 15 set size 1 set shape "circle" set color blue] ask patch -13 -14 [set plabel-color black set plabel "Pr(S=1|D)"] ask patch -13 -18 [set plabel-color black set plabel "Pr(S=0|D)"] set scale 14 / (2 * floor ([actual] of one-of posteriors with-max [actual])) set marker round (14 / scale) ask patch -15 12 [set plabel-color black set plabel marker] ask patch -15 -13 [set plabel-color black set plabel 0] ask patch 13 -13 [set plabel-color black set plabel "red = posterior mean, blue = +/- 3 posterior std. devs."] ask posteriors [set ycor (0 + scale * actual)] if prior? = "dirac-delta" [ask posteriors with [xcor = 0] [set shape "line" set heading 0 set ycor ycor - 3.5 set size scale * actual]] ;get probabilities of spiking let signal (1 / d ^ 2) let intensity (g * signal + n) set draw (1 - e ^ (- intensity * dt)) create-bars 1 [set color black set shape "line" set size 32 set heading 90 setxy 0 13.5] create-bars 1 [set color black set shape "line" set size 32 set heading 90 setxy 0 -.5] create-bars 1 [set color black set shape "line" set size 32 set heading 90 setxy 0 -1.5] create-bars 1 [set color black set shape "line" set size 32 set heading 90 setxy 0 -13.5] end to go ask spikes [set xcor xcor - .5] set S random-float 1 ifelse S < draw [ ask neurones [set color white] display wait .05 ask neurones [set color black] create-spikes 1 [set shape "line" set size .7 setxy 14 -1.1 set color black set heading 0] display ] [ask neurones [set color black] ] ask spikes with [xcor < -15] [die] if (prior? = "normal" or prior? = "uniform") [update-posterior] tick end to update-posterior ask ghosts [set color color + .9] ask ghosts with [color > 9.9] [die] ask posteriors [hatch-ghosts 1 [ set size .3 set color 1 set shape "dot"]] ;update the posterior ifelse S < draw [ask posteriors [set actual (actual * item 0 [ps1] of s1s with [xcor = [xcor] of myself])]] [ask posteriors [set actual (actual * item 0 [ps0] of s0s with [xcor = [xcor] of myself])]] ;normalise new posterior let normalise (sum [actual] of posteriors) ask posteriors [set actual (actual * resolution / normalise)] plot-smart end to plot-smart ;does the plot need to be rescaled? let top max [ycor] of turtles with [breed = posteriors or breed = ghosts] ifelse (top > 11 or top < 6) [set scale 13 / (2 * floor ([actual] of one-of turtles with [breed = posteriors or breed = ghosts] with-max [actual])) set marker round (13 / scale) if (((marker < 50) and ((prior? = "uniform") or (prior? = "normal"))) or (marker < 500 and prior? = "dirac-delta")) [ ask patch -15 12 [set plabel-color black set plabel marker] ask ghosts [set ycor (0 + scale * actual)]] ask posteriors [set ycor (0 + scale * actual)] ] [ask ghosts [set ycor (0 + scale * actual)] ask posteriors [set ycor (0 + scale * actual)]] ;get mean and spread of posterior and plot it ask avgs [set xcor (xcor - .5)] ask std-ups [set xcor (xcor - .5)] ask std-downs [set xcor (xcor - .5)] set expect sum [actual * (xcor + 14) / 56] of posteriors / resolution set standard (((sum [actual * ((xcor + 14) / 56) ^ 2] of posteriors / resolution) - expect ^ 2) ^ (1 / 2)) * 3 create-avgs 1 [set color red set shape "dot" set size .5 set actual expect setxy 14 actual] create-std-ups 1 [set color blue set shape "dot" set size .3 set actual (expect + standard) setxy 14 actual] ifelse (expect - standard < 0) [create-std-downs 1 [set color blue set shape "dot" set size .3 set actual 0 setxy 14 actual]] [create-std-downs 1 [set color blue set shape "dot" set size .3 set actual (expect - standard) setxy 14 actual]] let upper max [ycor] of std-ups let lower min [ycor] of std-downs ifelse (upper > -4 or upper < -8) [set scale-avg 11 / ([actual] of one-of std-ups with-max [actual]) set marker-avg (round (1000 * 11 / scale-avg)) / 1000 ask patch -15 -2 [set plabel-color black set plabel marker-avg] ask avgs [set ycor (-13 + scale-avg * actual)] ask std-ups [set ycor (-13 + scale-avg * actual)] ask std-downs [set ycor (-13 + scale-avg * actual)] ] [ask avgs [set ycor (-13 + scale-avg * actual)] ask std-ups [set ycor (-13 + scale-avg * actual)] ask std-downs [set ycor (-13 + scale-avg * actual)]] ask avgs with [xcor < -15] [die] ask std-ups with [xcor < -15] [die] ask std-downs with [xcor < -15] [die] end @#$#@#$#@ GRAPHICS-WINDOW 366 10 871 566 16 -1 15.0 1 10 1 1 1 0 1 1 1 -16 16 -18 16 1 1 1 ticks 30.0 BUTTON 146 352 212 385 NIL setup NIL 1 T OBSERVER NIL NIL NIL NIL 1 SLIDER 129 123 221 156 D D .01 .3 0.1 .01 1 NIL HORIZONTAL SLIDER 129 164 221 197 g g 1 15 6 1 1 NIL HORIZONTAL SLIDER 129 204 221 237 n n 0 100 100 1 1 NIL HORIZONTAL CHOOSER 108 249 246 294 prior? prior? "normal" "uniform" "dirac-delta" 1 BUTTON 145 481 213 514 NIL go T 1 T OBSERVER NIL NIL NIL NIL 1 TEXTBOX 64 52 313 112 1) Set distance between predator and prey, parameters of the sensor, and initial prior 16 0.0 1 TEXTBOX 124 322 274 342 2) Click setup 16 0.0 1 TEXTBOX 81 423 296 464 3) Click go; click once to run, click again to pause 16 0.0 1 TEXTBOX 26 20 358 38 Note: see Information tab for more details and observations 11 0.0 1 TEXTBOX 67 535 310 564 A speed slider can be found at the top of the screen to adjust the speed of the simulation 11 0.0 1 @#$#@#$#@ ## WHAT IS IT? This model illustrates how individual spikes and non-spikes can be interpreted as measurements about some state of the world, and how we can perform Bayesian inference on those measurements to infer about the world. In this model, the state of the world considered is the distance D from a predator (red) to a prey (blue). Given a probabilistic model of spiking given D, Pr(S|D), Bayes' rule states how to calculate Pr(D|S) given the outputs of the neurone. ## HOW IT WORKS The predator has a neurone that fires a spike on a time step with probability Pr(S|D), as outlined in the manuscript. Given this conditional probability, which is plotted at the bottom of the screen, and a prior distribution Pr(D), we can infer the prey location Pr(D|S) by Bayes' rule. By letting the posterior distribution Pr(D|S) become the prior on the next time step, we can perform Bayes' rule recursively in time. ## HOW TO USE IT To start, set the parameters of the model to your desired values. Choose whether the prior is a normal, uniform, or Dirac-delta distribution. Click the `setup' button, and you're ready to run simulations. Next, click the `go' button. Click the button once to start running a simulation, and click again to pause it. The model is run at a sufficiently slow speed so that the viewer can see how individual spikes (and non-spikes) can be used to perform Bayesian inference about D. Every time step is one millisecond in duration. When the predator's neurone spikes in a time step, the predator's neurone flashes white and makes a pulse of sound. By Bayes' rule, the posterior Pr(D|S) shifts to the left. When the neurone doesn't spike in a time step, the posterior decays to the right. The observed spike train is streamed below. The sensor's conditional probabilities of spiking are in the lowest plot. ## THINGS TO NOTICE Pr(D|S) converges quickly to the observed prey location when D is small, often in less than 500 time steps (one-half of one second in real-time), and even when the gain/noise ratio for the sensor is extremely small. When D is small (say 0.05 or so), then Pr(D|S) quickly converges on D = d, in as little as 20 or 30 timesteps, despite low gain-to-noise ratios. The convergence of the posterior distribution to the observed prey location is insensitive to the prior distribution of the prey, so long as the prior has reasonably heavy tails. To verify this, run the model for a uniform, normal, and dirac-delta prior by changing the chooser labeled 'prior?' The posterior Pr(D|S) will converge on the observed prey location D = d, so long as the prior is not inconsistent with the observed prey location. ## THINGS TO TRY Investigate how the gain-to-noise ratio affects the speed of convergence of Pr(D|S) to the observed value of D = d. Investigate how the observed value of D = d affects the speed of convergence of Pr(D|S). The larger D = d is, the longer it takes Pr(D|S) to converge to D = d. ## EXTENDING THE MODEL Spatial distributions of these sensors can provide information on the location of other organisms in higher dimensions. This is consistent with the notion of "skin brains" (Holland 2003). To deal with the more general case of prey moving, we need to introduce the conditional probability Pr(X(t)|X(t-1)) (a dynamic prior). 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