Motivation Background Neural Code C.elegans tap withdrawal simulation Parallel parking Conclusion Neural Programs: Towards Adaptive Control in Cyber-Physical Systems Konstantin Selyunin 1 , Denise Ratasich 1 Ezio Bartocci 1 , M.A. Islam 2 Scott A. Smolka 2 , Radu Grosu 1 1 Vienna University of Technology Cyber-Physical Systems Group E182-1 Institut of Computer Engineering 2 Stony Brook University, NY, USA
Motivation Background Neural Code C.elegans tap withdrawal simulation Parallel parking Conclusion Motivation • Programs are not robust • Case studies: neural circuit simulation & parallel parking 0.016 AVA AVB 0.018 0.020 Potential, V 0.022 0.024 0.026 0.028 0.030 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 Time, s • Parameter synthesis: plateaus are bad for optimizations
Motivation Background Neural Code C.elegans tap withdrawal simulation Parallel parking Conclusion Motivation II This presentation: • How to incorporate “smooth” decisions in CPS to make systems more robust using neural circuits and GBN • Technique to learn parameters of a model • Application to two case studies and the relation between them
Motivation Background Neural Code C.elegans tap withdrawal simulation Parallel parking Conclusion Background D I • Bayesian Networks • express probabilistic dependencies between variables G S • are represented as DAGs • allow compact representation using CPDs L
Motivation Background Neural Code C.elegans tap withdrawal simulation Parallel parking Conclusion Background D I • Bayesian Networks • express probabilistic dependencies between variables G S • are represented as DAGs • allow compact representation using CPDs L • Gaussian Distributions • Univariate and Multivariate Gaussian distributions
Motivation Background Neural Code C.elegans tap withdrawal simulation Parallel parking Conclusion Background D I • Bayesian Networks • express probabilistic dependencies between variables G S • are represented as DAGs • allow compact representation using CPDs L • Gaussian Distributions • Univariate and Multivariate Gaussian distributions • Step function vs. sigmoid
Motivation Background Neural Code C.elegans tap withdrawal simulation Parallel parking Conclusion Background • Passing random variables through conditions q ~ N(0 , 0.5) if ( q > 0.15) { q; } p ~ N(4 , 0.7) if ( p <= 3.0) { p; }
Motivation Background Neural Code C.elegans tap withdrawal simulation Parallel parking Conclusion Towards the nif statement Our setting: • Program operates random variables (RVs) • RVs are mutually dependent Gaussians
Motivation Background Neural Code C.elegans tap withdrawal simulation Parallel parking Conclusion Towards the nif statement Our setting: • Program operates random variables (RVs) • RVs are mutually dependent Gaussians Questions: • How to incorporate uncertainty of making a decision and make decisions “smooth”? • How to avoid cutting distributions when passing a variable through a condition or a loop?
Motivation Background Neural Code C.elegans tap withdrawal simulation Parallel parking Conclusion Towards the nif statement Our setting: • Program operates random variables (RVs) • RVs are mutually dependent Gaussians Questions: • How to incorporate uncertainty of making a decision and make decisions “smooth”? • How to avoid cutting distributions when passing a variable through a condition or a loop? We propose to use nif s instead of traditional if statements.
Motivation Background Neural Code C.elegans tap withdrawal simulation Parallel parking Conclusion Neural if The nif statement: nif( x # y, σ 2 ) • Inequality relation {≥ , >, <, ≤} • Variance (represents our confidence of making a decision) Example: nif( x >= a, σ 2 ) S1 else S2
Motivation Background Neural Code C.elegans tap withdrawal simulation Parallel parking Conclusion nif( x # a, σ 2 ) : Evaluation 1. Compute the difference between x, a x - a − ǫ if # is >, if # is ≥ , x - a diff(x,a) = a - x − ǫ if # is <, if # is ≤ . a - x
Motivation Background Neural Code C.elegans tap withdrawal simulation Parallel parking Conclusion nif( x # a, σ 2 ) : Evaluation 1. Compute the difference between x, a x - a − ǫ if # is >, if # is ≥ , x - a diff(x,a) = a - x − ǫ if # is <, if # is ≤ . a - x 2. Compute quantiles of the probability density function 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 q1 (0,0.4) q2 (0,0.4) 0.4 0.4 * ** q1 (0, ) 0.3 0.3 q2 (0, ) 0.2 0.2 q1 (0,4) q2 (0,4) 0.1 0.1 diff(1,0) 0.0 0.0 -5 0 5 -5 0 5 diff(x,a) x
Motivation Background Neural Code C.elegans tap withdrawal simulation Parallel parking Conclusion nif( x # a, σ 2 ) : Evaluation 1. Compute the difference between x, a x - a − ǫ if # is >, if # is ≥ , x - a diff(x,a) = a - x − ǫ if # is <, if # is ≤ . a - x 2. Compute quantiles of the probability density function 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 q1 (0,0.4) q2 (0,0.4) 0.4 0.4 * ** q1 (0, ) 0.3 0.3 q2 (0, ) 0.2 0.2 q1 (0,4) q2 (0,4) 0.1 0.1 diff(1,0) 0.0 0.0 -5 0 5 -5 0 5 diff(x,a) x 3. Check if a random sample is within the interval
Motivation Background Neural Code C.elegans tap withdrawal simulation Parallel parking Conclusion nif : Example if ( x > 0.15) { No. of samples x ~ N(0 , 0.1) x; } No. of samples -0.4 -0.2 0.0 0.2 0.4 x -0.4 -0.2 0.0 0.2 0.4 No. of samples x nif ( x > 0.15, 0.1) { x; } -0.4 -0.2 0.0 0.2 0.4 x
Motivation Background Neural Code C.elegans tap withdrawal simulation Parallel parking Conclusion Limit case σ 2 → 0 nif( x >= a, σ 2 ) S1 else S2 • For the case with “no uncertainty” ( σ 2 → 0) the PDF is expressed as the Dirac function: • δ ( x ) = + ∞ if x = 0 else 0 • � ∞ −∞ δ ( x ) dx = 1
Motivation Background Neural Code C.elegans tap withdrawal simulation Parallel parking Conclusion Limit case σ 2 → 0 nif( x >= a, σ 2 ) S1 else S2 • For the case with “no uncertainty” ( σ 2 → 0) the PDF is expressed as the Dirac function: • δ ( x ) = + ∞ if x = 0 else 0 • � ∞ −∞ δ ( x ) dx = 1 • σ 2 → 0 : the nif statement is equivalent to the if condition
Motivation Background Neural Code C.elegans tap withdrawal simulation Parallel parking Conclusion nwhile Extension of a traditional while statement that incorporates uncertainty nwhile ( x # a , σ 2 ) { P 1 }
Motivation Background Neural Code C.elegans tap withdrawal simulation Parallel parking Conclusion nwhile Extension of a traditional while statement that incorporates uncertainty nwhile ( x # a , σ 2 ) { P 1 } Evaluation: 1. Compute diff(x,a) , obtain quantiles q 1 and q 2 2. Check if a random sample is within the interval 3. If sample within the interval, execute P1 and go to 1, else exit
Motivation Background Neural Code C.elegans tap withdrawal simulation Parallel parking Conclusion Case study 1: C.elegans C.elegans • a 1-mm round worm • each adult individual has exactly 302 neurons • extensively studied in evolutional- and neurobiology Tap withdrawal response • apply stimulus to mechanosensory (input) neurons • observe the behavior: forward / backward movement Goal • express the behavior using neural program
Motivation Background Neural Code C.elegans tap withdrawal simulation Parallel parking Conclusion Case study 1: C.elegans C.elegans • a 1-mm round worm • each adult individual has exactly 302 neurons • extensively studied in evolutional- and neurobiology Tap withdrawal response • apply stimulus to mechanosensory (input) neurons • observe the behavior: forward / backward movement Goal • express the behavior using neural program
Motivation Background Neural Code C.elegans tap withdrawal simulation Parallel parking Conclusion Case study 1: C.elegans C.elegans • a 1-mm round worm • each adult individual has exactly 302 neurons • extensively studied in evolutional- and neurobiology Tap withdrawal response • apply stimulus to mechanosensory (input) neurons • observe the behavior: forward / backward movement Goal • express the behavior using neural program
Motivation Background Neural Code C.elegans tap withdrawal simulation Parallel parking Conclusion Neural connections 101 E (ij) w gap w syn V (i) V (j) V (i) V (j) g gap g syn (V (i) ) V (j) V (i) V (i) V (j) Synaptic connection Gap junction connection • chemical nature • instantaneous resistive connection • either active or not • linear combination of inputs • synaptic weight w syn • gap junction weight w gap • use nif to model each synaptic connection
Motivation Background Neural Code C.elegans tap withdrawal simulation Parallel parking Conclusion C.elegans Tap withdrawal circuit PLM ¡ REV ¡ 2 ¡ 2 ¡ FWD ¡ AVM ¡
Motivation Background Neural Code C.elegans tap withdrawal simulation Parallel parking Conclusion C.elegans Tap withdrawal circuit REV ¡ 56 ¡ PLM ¡ AVA ¡ AVD ¡ 2 ¡ 2 ¡ AVB ¡ PVC ¡ AVM ¡ 28 ¡ FWD ¡ Why ¡the ¡synapses ¡and ¡neurons ¡AVD, ¡PVC? ¡
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