Excitable behaviors What is an excitable behavior? R. Sepulchre University of Cambridge Collective dynamics, control and maging, nstitute for Theoretical Studies, ETH, June 2017 2 From google From google 3 4
From google A behavioral property (t) (t) pulses spike A family of trajectories characterised by current pulses and allornone voltage spikes 5 6 A threshold phenomenon : localised sensitivity analogdigital conversion Excitable behaviors Neuronal networks are interconnections of neurons and synapses. n neurons, the current is the input. n synapses, the voltage is the input. The allornone nature of the spike makes the behavior nonlinear and hybrid. ntractable? Excitable behaviors have a characteristic scale, or resolution. Tractable? 7 8
The switchlet project: a system theory of excitability A statespace paradigm? What is an excitable behavior? How is it regulated? How can we study interconnections of excitable systems? What makes those nonlinear systems tractable? 2007 2010 2014 What makes those systems worth studying beyond their relevance in neurophysiology? Great for computations but limited for system theoretic questions Tractability of highdimensional NL models? spatiotemporal modeling? stochastic modeling? nterconnections? Robustness? Modulation? A historical hint The behavioral approach to system theory The typical regulator system can frequently be described, in essentials, by differential equations of no more than perhaps the second, third or fourth order. n contrast, the order of the set of differential equations describing the typical negative feedback amplifier used in telephony is likely to be very much greater. As a matter of idle curiosity, once counted to find out what the order of the set of equations in an amplifier had just designed would have been, if had worked with the differential equations directly. t turned out to be 55. Henrik Bode, Feedback: the history of an idea, 1960 Bode developed loopshaping analysis to overcome the intractability of sensitivity analysis of electrical circuits aimed at signal transmission See J. Willems, CSM 2007 for more 11 12
The textbook picture Modelling excitability 13 A oneport circuit. Neurons are electrical circuits classified according to their step response. The voltage clamp experiment : What Hodgkin and Huxley did (1) A circuit model identifying a system through its inverse The action potential 15 = g(,t) nput conductance = step response of the inverse = local gain
A voltagedependent transfer function What Hodgkin and Huxley did (2) e.g. = G(s; ) K( ) ( )s 1 = Na K Tearing apart two distinct ionic currents Admittance identified from the step response at an operating point of the circuit.! NOT from the statespace model ( ) dg Z dt = g K(x)dx 18 Modelling conductances Fitting a state space model Delayed SLOW firstorder response (POSTE conductance) Delayed FAST firstorder response (NEGATE conductance) SLOW firstorder response (POSTE conductance) 19 20
A differential model of excitability 1 2 passive switch regulator 1 2 3 3 Differential behavioral theory backbone fast negative resistance device slow resistive device How much of a behavior can be inferred from a local description around specific trajectories? Kirchoff law: = 1 2 3 = 1 = 2 = 3 in particular: from linearised models around equilibrium trajectories? R(s; w) w =0 F (w) =0 log! negative real positive real 21 Many antecedents: Kalman vs Aizerman conjecture Contraction analysis vs Lyapunov analysis Differential positivity vs monotonicity Differential dissipativity vs dissipativity Singularity theory 22 The general ansatz A differential behavior can be analysed at different resolutions, e.g. Z Z R(s 0 ; w 0 )= R(s; w)h(s s 0 ; w w 0 )dwds C W h : resolution kernel Analysing excitability The data dictate the relevant resolutions log! 23
The two resolutions of an excitable behavior The two resolutions of an excitable behavior g f ( ) g s ( ) g(,t) g f ( )e t f gs ( )e t s G(s; ) log! g l 1 g f ( ). f s 1. 25 1 g s ( ). s s 1. 26 A quasistatic model A statespace representation C = l ( ) f ( f ) s ( s ) f f = f s s = s Z ( )= g l g f (x)dx A hysteretic switch in the fast time scale Fitzhugh Nagumo model : C = 0; l ( )= 3 3 ; f ( )=k f ; s ( )= k s Z ( )= g l g f (x)g s (x)dx A monotone resistor in the slow time scale
A mixed feedback representation Balanced feedback localizes passive RC circuit K K fast lag linear localized memor/ k K k k ( large) ( small) ( large) K slow lag output output output 1 High frequency behavior: Low frequency behavior: Necessary localization in some frequency range! input O(k) input 0 input k K K Benefits of a differential approach Ongoing research Modelling / Analysis / Synthesis is faithful to the data Transfer functions at a resolution A realm of tractable methodologies tools, e.g. from LT system theory and singularity theory Sensitivity analysis, regulation, and synthesis of excitable circuits Extensions are straightforward : e.g. spatiotemporal excitability replaces LT by LTS Spatiotemporal excitability, network excitability,
Bursting as interconnection of excitable systems burst excitable (t) burst nterconnecting excitable behaviors (t) fast excitable 1 slow 2 1 excitable 2 = 1 2 = 1 = 2 33 34 Bursting, an essential component of neuronal signalling A novel theory of bursting log! A. Franci, G. Drion, R. Sepulchre. An organizing center in a planar model of neuronal excitability SAM Journal on Applied Dynamical Systems, 11(4), pp. 16981722, 2012 G. Drion, A. Franci,. Seutin, R. Sepulchre. A Balance Equation Determines a Switch in Neurona Excitability, PLoS Computational Biology 9(5) : e1003040, 2013. A. Franci, G. Drion, R. Sepulchre. Modeling the modulation of neuronal bursting: a singularity theory approach. SAM J. Appl. Dyn. Syst. 132 (2014), pp. 798829 35 G. Drion, A. Franci, J. Dethier, R. Sepulchre. Dynamic input conductances shape neuronal spiking. eneuro, DO: 10.1523/ENEURO.003114. 2015. 36
A twostate automaton ( two switches) continuously regulated ( two regulators) 37 38 The dominant bursting model of neurodynamics SLOW FAST adaptation Should we care? SLOW FAST S L O W ULTRASLOW adaptation Endogenous bursting : Slow negative feedback (adaptation) provides the driving oscillating input to the excitable model zhikevich, Chapter 9 Terman and Ermentrout, Chapter 5 Keener and Sneyd, Chapter 9 ULTRASLOW No modulation (no route to burst) No robustness (fragile to noise and time scale separation) No interconnections Classification based on bifurcations the slow negative conductance controls the modulation between spike and burst The motif is as robust as the spiking motif nterconnection based approach No classification ; loop shaping regulation 39
Benefits of a differential approach Ongoing research Modelling / Analysis / Synthesis is faithful to the data Synthesis of excitable and bursting circuits A realm of tractable methodologies tools, e.g. from LT system theory and singularity theory ntegrate and fire models of excitability and bursting Extensions are straightforward : e.g. spatiotemporal excitability replaces LT by LTS Spatiotemporal excitable networks Conclusions Conclusions What is an excitable behavior? A relationship between current pulses and voltage spikes characterised by a window of ultrasensitivity at a given scale. A continuous behavior with a discrete readout. How to model excitability? Differential modelling: The data only provide a local model around specific (e.g. equilibrium) trajectories of the parts. What is an excitable behavior? How to model excitability? A relationship between current pulses and voltage spikes characterised by a window of ultrasensitivity at a given scale. A continuous behavior with a discrete readout. Differential modelling: The data only provide a local model around equilibrium trajectories of the parts. How to analyse and design excitable behaviors? ntegrate the differential models at different resolutions Switchlets are to systems what wavelets are to signals nterconnecting excitable behaviors How to analyse and design excitable behaviors? Analyse the differential models at different resolutions Switchlets are to systems what wavelets are to signals nterconnecting excitable behaviors nterconnecting two excitable systems provides a system theory of bursting 43 (J. Willems, CSM, 2007) nterconnecting two excitable systems provides a system theory of bursting 44
Collaborators Dr Alessio Franci Dr Felix Miranda Luka Ribar Dr Timothy O Leary lario Cirillo Dr Marko Seslija Thiago Burghi Dr Guillaume Drion Tomas van Pottelbergh Dr Fulvio Forni