Positive feedback regulation Rodolphe Sepulchre -- University of - PowerPoint PPT Presentation

LCCC is a positive environment (i.e. the metric of positive systems) Positive feedback regulation Rodolphe Sepulchre -- University of Cambridge LCCC workshop on Learning and Adaptation for Sensorimotor Control - Lund - October 2018 ! 2 Regulation across scales Take-home message Positive feedback is essential to regulation across scales Control across scales by positive and negative feedback, R.S., Alessio Franci, Guillaume Drion. Annual Reviews of Control, Robotics, and Autonomous Systems. In press. At low speed, the regulation is across scales . ! 3

A key concept Contents • Positive and negative feedback regulation • Positive feedback regulation of bursting • Positive feedback regulation of the half-center oscillator • Positive feedback regulation of central pattern generators S-behavior Infra-sensitive Ultra-sensitive (localised gain) (linear-like) (switch-like) (continuous) (discrete) ! 6 Excitability as mixed feedback amplification Excitability as mixed feedback amplification ‘instantaneous’ ‘fast’ ‘slow’ ‘First’ positive feedback : ultra-sensitivity, threshold, fast switch. inactivation of activation of inward current inward current + activation of outward current ‘Then’ negative feedback : infra-sensitivity, refractoriness, slow repolarization. (Hodgkin-Huxley, 1952) ‘ = ’ spike : discrete event triggered by continuous input ! 7

Positive feedback = negative conductance A simplified model of excitability I = g ( · )( V − E ) (Nagumo circuit) (Relay-feedback system) local gain : δ I = g ( · ) δ V + δ g ( · )( V − E ) dynamic ! I V + the (variational) conductance can be transiently negative if - activation of an inward current ( V < E ) ( δ g ( · ) > 0) I 0 = kV − V 3 b 3 − n + I ( δ g ( · ) < 0) of an outward current ( V > E ) or inactivation n τ s + 1 τ ˙ n = − n + bV The capacitor is neglected and the fast positive feedback is approximated as instantaneous An electrical model across scales Excitability as mixed feedback amplification micro-scale macro-scale ‘First’ positive ‘then’ negative feedback ‘ = ’ spike meso-scale No ultra-sensitivity without positive feedback Does this scale up ? Active nodal currents provide positive or negative feedback. ! 11 ! 12 Active network currents are excitatory or inhibitory.

Bursting as two mode excitability Contents • Positive and negative feedback regulation • Positive feedback regulation of bursting • Positive feedback regulation of the half-center oscillator Two independent positive feedback loops mean two independent thresholds : high/fast and low/slow • Positive feedback regulation of central pattern generators A burst is a spike of spikes. Two independent negative feedback loops mean independent regulation of intra-burst refractoriness and inter-burst refractoriness. Input-output behavior is spike excitable or burst excitable depending on the neuron polarization. A (widely accepted) textbook model of bursting A burster is fragile without slow positive feedback STG R15 PßC TC CA1 CA1+ Izhikevich, Chapter 9 Control Terman and Ermentrout, Chapter 5 Keener and Sneyd, Chapter 9 1.2 x g 0.8 x g Variability in mean spike height Variability in burst period Variability in spikes per burst STG R15 PßC TC CA1 CA1+ STG R15 PßC TC CA1 CA1+ STG R15 PßC TC CA1 CA1+ 1.6 0.4 1.6 0 1 0 0.5 -1.0 -1.5 Five published models of bursting. The red ones lack slow positive feedback. (Izhikevich, 2008, p.330) The model CA1+ is the model CA1 with slower calcium activation. Bursting = negative feedback adaptation of spiking. (Franci, Drion, RS, 2018)

The slow positive feedback is the key regulator of A burster is rigid without slow positive feedback transitions between “on” and “off” modes (McCormick & Bal, 1997) A cellular regulation fundamental to brain ‘states’ (arousal, attention, …) A key target for neuromodulation. tunable rigid ! 18 (Franci, Drion, RS, 2018) Positive feedback regulation of bursting Contents No distinction between high/fast and low/slow threshold without two independent positive feedback loops The low/slow positive feedback is essential to make bursting • Positive and negative feedback regulation ! robust (with respect to parameter uncertainty) ! tunable (many types of bursters) ! neuromodulable (transitions between spiking and bursting) • Positive feedback regulation of bursting ! tractable (three time-scale analysis) • Positive feedback regulation of the half-center oscillator • Positive feedback regulation of central pattern generators

A long debated question The half-center oscillator: a fundamental motif of clock control (Brown, 1911 ! ) Which currents contribute to the post-inhibitory rebound ? In particular, versus ? I h I Ca,T cellular behavior network behavior The on-off control is through the maximal conductance of the slow positive feedback current only . No change in (synaptic) coupling parameters. (Dethier, Drion, Franci, RS, 2015) The PIR is fragile without positive feedback The feedback properties of the two currents differ strikingly Only the slow activation of contributes to I Ca,T the low/slow positive feedback regulation of the behavior A hidden example of positive feedback regulation (Dethier, Drion, Franci, RS, 2015) (Dethier, Drion, Franci, RS, 2015)

Cellular positive feedback is essential to network behavior Positive feedback regulation of the half-center oscillator A cellular mechanism for network control. ! robust to noise, parameter uncertainty, and network heterogeneity Fundamental to tunability, robustness, and control of the network behavior. ! tunable by synaptic coupling (e.g. network frequency) An example of regulation across scales. ! 25 (Dethier, Drion, Franci, RS, 2015) Central pattern generators as interconnected half-center oscillators Circuit con fj guration Circuit rhythms Functional connectivity Contents No rhythm 1 1 1 2 3 3 2 4 2 3 4 4 5 5 5 1 s Modulated neuron Coexistence of fast (1,2,3) and slow (3,4,5) rhythms • Positive and negative feedback regulation 1 1 2 3 2 3 4 4 5 5 • Positive feedback regulation of bursting Slow three-neurons rhythm (3,4,5) 1 1 2 • Positive feedback regulation of the half-center oscillator 3 2 3 4 4 5 5 Fast three-neurons rhythm (1,2,3) • Positive feedback regulation of central pattern generators 1 1 2 3 2 3 4 4 5 5 Global rhythm (1,2,3,4,5) 1 1 2 3 2 3 4 4 5 5 Cellular control of functional connectivity. No synaptic tuning involved. (Drion, Franci, RS, 2018)

Regulation across scales is lost without the cellular positive feedback Positive feedback regulation of central pattern generators One step closer to a tractable model of one of the most extensively studied central pattern Circuit rhythms in A Isolated HCO’s Circuit rhythms in generators : co-regulation of pyloric and gastric rhythms in the STG. the original STG model the restorative variant 1 1 2 2 3 4 3 4 NMD 5 5 1 s B Full circuit 1 1 2 3 2 3 4 4 5 5 1 s 1 1 2 2 3 4 3 4 NMD 5 5 ! 30 (Drion, Franci, RS, 2019) Conclusions • Positive feedback is essential to regulation across scales. • Why? because it regulates ultra-sensitivity and thresholds. • The role of positive feedback regulation is poorly understood and often neglected both in control and in neurophysiology. • No learning across scales without positive feedback ?