Zoomable feedback amplifiers

Transcription

1 Zoomable feedback amplifiers How to make a behavior zoomable? Rodolphe Sepulchre University of Cambridge MA workshop on Distributed Control and Decision Making Over Networks, September 2015 and the possible relevance of this question for Distributed Control and Decision Making Over Networks 2 The European Extremely Large Telescope 6000 sensors at the nanoscale resolution provide little information on long range deformations m! (A. Sarlette RS, Automatica 2014)

2 A multiresolution electrical behavior Multiresolution sensing with one population of (bursting) sensors Organ level Function DBS electrodes [mm] LFPs 500µ Circuit level 1000ms Optrodes [µm] Action potentials Cellular level 10µ 10ms (Krahe, 2004) Pharmacology [nm] on channels Molecular level 1pA 1ms 5 Localization at the core of multiresolution A localization principle Part. A feedback principle for localization Part. Localization across scales Upscaling & downscaling (An inspiration from neuronal behaviors) 8

3 The amplifier : a localized behavior Effect (y) Local window of linearity Sensitivity (dy/du) Cause (u) Local window of sensitivity The feedback amplifier : a feedback localized behavior Cause (u) With the proper lens, every behavior is an amplifier Black principle: negative feedback linearizes Positive feedback quantizes 1 1 K 1 1 (K large) 1 1 K K O(K) K O(K) = sat 1 ( K ) = sat 1 1K () = sat 1 ( K ) = 1 1 K 1 apple K 1 Sensitivity domain is spread by positive feedback Sensitivity domain is spread by negative feedback (The essence of control theory) Hysteretic behavior: memory, ONOFF devices (The essence of digital technology) 11 12

4 The success of negative feedback turned positive feedback into history. Feedback : the great absent of mathematical science Occurrences of the word feedback are exceptional throughout physics, mathematics, and computer science. Usually associated to positive feedback (autocatalysis,...) revealing statistics: (Tucker, 1972) 3 occurrences (positive feedback) 3 occurrences (positive feedback) Positive and negative feedback amplifiers The feedback principle of localization Negative feedback linearizes Continuous behavior Positive feedback quantizes OnOff behavior put Behavior is linear analog output O(k) k K K Behavior is resonant output 1 0 Behavior is switch digital Analog technology Digital technology k large k small k large exogenous (output primarily reflects the ) endogenous (output primarily reflects memory of the past) sensitivity is exogenous low uniform sensitivity is high localized sensitivity is endogenous low uniform

5 Balanced feedback localizes K linear )))))))))))))))))))))) localized )))))))))))))))))))) memor/ ) k K K k k K ( large) ( small) ( large) The principle of robust resonant behavior: first positive, then negative feedback fast lag amplifier output output output 1 slow lag O(k) k K K 0 High frequency behavior: Low frequency behavior: Necessary localization in some frequency range! A pervasive principle : first, then Gain local global Amplitude: activation near rest / inactivation away from rest The localized feedback amplifier Time : Space : Network: fast activation / slow adaptation short range excitation / long range inhibition local sync / global heterogeneity Feedback localizes, i.e. controls the window of sensitivity. The excitable behavior is a bridge between the world of positive feedback (endogenous, discrete) and the world of negative feedback (exogenous, continuous). Localized behaviors are tractable.

6 Analysis : excitable behaviors are tractable An inspiration from neuroscience... HodgkinHuxley model (1952) is a localized feedback amplifier ON n general, interconnecting a mixture of positive and negative amplifiers makes the behavior intractable. Localization makes the behavior tractable, because it calls for a local analysis. Singularity theory classifies all possible robust behaviors near a singular behavior. OFF L O W FAST H G H THEN Part. A feedback principle for localization The takehome message is : 1. f localization matters, then positive feedback should be rehabilitated in control theory. Part. A feedback motif to localize across scales 2. The feedback localized amplifier is a tractable nonlinear behavior

7 The setup : a network of feedback localized amplifiers... finegrain = individual = cellular Homogenization acts as negative feedback coarsegrain = population = network <> <> Timeaveraging Population averaging Space averaging (Mean field theory) Note: the finegrain positive feedback is the only thing that makes the coarsegrain behavior distinct from noise Coupling (or sync or consensus ) is the network correlate of positive feedback No coarsegrain exogenous behavior without finegrain endogenous behavior Localization of the network behavior: a finegrain AND a coarsegrain source interconnectivity exogenous, continuous endogenous, discrete heterogeneity 28

8 WilsonCowan theory Longterm consequences of WilsonCowan theory From the coarsegrain behavior <> <> Today, the E motif is studied at every scale of biological organization. BUT little understanding of the crosstalking between scales. start again... Modulation? Robustness? Control? Adaptation? Plasticity? finegrain or coarse grain? E E finegrain or coarse grain? A widespread model of the network behavior A feedback motif to localize across scales E coarse grain (i.e. interconnectivity) FAST heterogeneity finegrain and coarse grain interconnectivity ULTRA frequency network This model is flawed : it does not respect the first, then requirement of localization The motif contains a finegrained source of positive feedback for the coarsegrained behavior state

9 A tworesolution finegrained behavior high (upstate) low (downstate) An inspiration from neuroscience: ALL (?) endogenous bursters have two distinct ionic sources of positive feedback FAST high f fast f slow f low f up down ULTRA fast excitability high state fastslow excitability highlow state hyperpolarized lowstate A twomode cellular behavior A feedback motif to localize across scales FAST heterogeneity interconnectivity fast excitable spiker slow slow excitable burster ULTRA time network state finegrained behavior is exogenous finegrained behavior is endogenous Population of twolevel individual behaviors interconnected in the slow time scale: Anything between a heterogenous population of spiking individuals and a homogenous population of synchronized bursters

10 Localization across scales Part. A feedback motif to localize across scales The takehome message is : 1. the source of endogenous behavior at the fine scale is what controls the exogenous behavior at the coarse scale. Red = synchronized network oscillations leading to LFPs 2. The finegrained source of coarsegrained positive feedback is what makes connectivity lively and fun Blue = no LFP despite similar spiking activity in network Conclusions The switchlet proposal No multiresolution without localization. Feedback is central to localization. Local windows of positive feedback surrounded by negative feedback are key to selective, modulable, and robust amplification. Localization is a source of tractability. Neurobiology is an underexploited source of inspiration for systems and control. Predigital cybernetics teaches us a fundamental property of feedback : localization No localization in a discrete and endogenous world ruled by positive feedback only, nor in a continuous and exogenous world ruled by negative feedback only. Localization is the essence of multiresolution behaviors 40

11 Multiresolution behaviors A multidisciplinary journey with many collaborators, including: Guillaume Drion, Alessio Franci, Julie Dethier, Tim O Leary, incent Seutin. A review: our CDC 2015 tutorial paper Neuronal behaviors: a control perspective