Excitable behaviors. What is an excitable behavior? From google. R. Sepulchre -- University of Cambridge

Size: px
Start display at page:

Download "Excitable behaviors. What is an excitable behavior? From google. R. Sepulchre -- University of Cambridge"

Transcription

1 Excitable behaviors What is an excitable behavior? R. Sepulchre University of Cambridge Collective dynamics, control and maging, nstitute for Theoretical Studies, ETH, June From google From google 3 4

2 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

3 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? 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

4 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

5 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

6 A differential model of excitability 1 2 passive switch regulator 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 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

7 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 g s ( ). s s 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

8 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,

9 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 = 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 , 2012 G. Drion, A. Franci,. Seutin, R. Sepulchre. A Balance Equation Determines a Switch in Neurona Excitability, PLoS Computational Biology 9(5) : e , A. Franci, G. Drion, R. Sepulchre. Modeling the modulation of neuronal bursting: a singularity theory approach. SAM J. Appl. Dyn. Syst. 132 (2014), pp G. Drion, A. Franci, J. Dethier, R. Sepulchre. Dynamic input conductances shape neuronal spiking. eneuro, DO: /ENEURO

10 A twostate automaton ( two switches) continuously regulated ( two regulators) 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

11 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

12 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

Basic elements of neuroelectronics -- membranes -- ion channels -- wiring

Basic elements of neuroelectronics -- membranes -- ion channels -- wiring Computing in carbon Basic elements of neuroelectronics -- membranes -- ion channels -- wiring Elementary neuron models -- conductance based -- modelers alternatives Wires -- signal propagation -- processing

More information

Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting

Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting Eugene M. Izhikevich The MIT Press Cambridge, Massachusetts London, England Contents Preface xv 1 Introduction 1 1.1 Neurons

More information

From neuronal oscillations to complexity

From neuronal oscillations to complexity 1/39 The Fourth International Workshop on Advanced Computation for Engineering Applications (ACEA 2008) MACIS 2 Al-Balqa Applied University, Salt, Jordan Corson Nathalie, Aziz Alaoui M.A. University of

More information

Math 345 Intro to Math Biology Lecture 20: Mathematical model of Neuron conduction

Math 345 Intro to Math Biology Lecture 20: Mathematical model of Neuron conduction Math 345 Intro to Math Biology Lecture 20: Mathematical model of Neuron conduction Junping Shi College of William and Mary November 8, 2018 Neuron Neurons Neurons are cells in the brain and other subsystems

More information

An Introductory Course in Computational Neuroscience

An Introductory Course in Computational Neuroscience An Introductory Course in Computational Neuroscience Contents Series Foreword Acknowledgments Preface 1 Preliminary Material 1.1. Introduction 1.1.1 The Cell, the Circuit, and the Brain 1.1.2 Physics of

More information

Neural Excitability in a Subcritical Hopf Oscillator with a Nonlinear Feedback

Neural Excitability in a Subcritical Hopf Oscillator with a Nonlinear Feedback Neural Excitability in a Subcritical Hopf Oscillator with a Nonlinear Feedback Gautam C Sethia and Abhijit Sen Institute for Plasma Research, Bhat, Gandhinagar 382 428, INDIA Motivation Neural Excitability

More information

Neural Modeling and Computational Neuroscience. Claudio Gallicchio

Neural Modeling and Computational Neuroscience. Claudio Gallicchio Neural Modeling and Computational Neuroscience Claudio Gallicchio 1 Neuroscience modeling 2 Introduction to basic aspects of brain computation Introduction to neurophysiology Neural modeling: Elements

More information

LIMIT CYCLE OSCILLATORS

LIMIT CYCLE OSCILLATORS MCB 137 EXCITABLE & OSCILLATORY SYSTEMS WINTER 2008 LIMIT CYCLE OSCILLATORS The Fitzhugh-Nagumo Equations The best example of an excitable phenomenon is the firing of a nerve: according to the Hodgkin

More information

arxiv: v3 [math.ds] 27 Nov 2013

arxiv: v3 [math.ds] 27 Nov 2013 Modeling the modulation of neuronal bursting: a singularity theory approach arxiv:1305.7364v3 [math.ds] 27 Nov 2013 Alessio Franci 1,, Guillaume Drion 2,3, & Rodolphe Sepulchre 2,4 1 INRIA Lille-Nord Europe,

More information

Neuroscience applications: isochrons and isostables. Alexandre Mauroy (joint work with I. Mezic)

Neuroscience applications: isochrons and isostables. Alexandre Mauroy (joint work with I. Mezic) Neuroscience applications: isochrons and isostables Alexandre Mauroy (joint work with I. Mezic) Outline Isochrons and phase reduction of neurons Koopman operator and isochrons Isostables of excitable systems

More information

Implementing robust neuromodulation in neuromorphic circuits

Implementing robust neuromodulation in neuromorphic circuits Implementing robust neuromodulation in neuromorphic circuits Fernando Castaños Centro de Investigación y de Estudios Avanzados Instituto Politécnico Nacional, México Email: castanos@ieee.org Alessio Franci

More information

Equivalent Circuit Model of the Neuron

Equivalent Circuit Model of the Neuron Generator Potentials, Synaptic Potentials and Action Potentials All Can Be Described by the Equivalent Circuit Model of the Membrane Equivalent Circuit Model of the Neuron PNS, Fig 211 The Nerve (or Muscle)

More information

Modeling Action Potentials in Cell Processes

Modeling Action Potentials in Cell Processes Modeling Action Potentials in Cell Processes Chelsi Pinkett, Jackie Chism, Kenneth Anderson, Paul Klockenkemper, Christopher Smith, Quarail Hale Tennessee State University Action Potential Models Chelsi

More information

Bo Deng University of Nebraska-Lincoln UNL Math Biology Seminar

Bo Deng University of Nebraska-Lincoln UNL Math Biology Seminar Mathematical Model of Neuron Bo Deng University of Nebraska-Lincoln UNL Math Biology Seminar 09-10-2015 Review -- One Basic Circuit By Kirchhoff's Current Law 0 = I C + I R + I L I ext By Kirchhoff s Voltage

More information

Bursting and Chaotic Activities in the Nonlinear Dynamics of FitzHugh-Rinzel Neuron Model

Bursting and Chaotic Activities in the Nonlinear Dynamics of FitzHugh-Rinzel Neuron Model Bursting and Chaotic Activities in the Nonlinear Dynamics of FitzHugh-Rinzel Neuron Model Abhishek Yadav *#, Anurag Kumar Swami *, Ajay Srivastava * * Department of Electrical Engineering, College of Technology,

More information

Electrophysiology of the neuron

Electrophysiology of the neuron School of Mathematical Sciences G4TNS Theoretical Neuroscience Electrophysiology of the neuron Electrophysiology is the study of ionic currents and electrical activity in cells and tissues. The work of

More information

Modelling biological oscillations

Modelling biological oscillations Modelling biological oscillations Shan He School for Computational Science University of Birmingham Module 06-23836: Computational Modelling with MATLAB Outline Outline of Topics Van der Pol equation Van

More information

Basic elements of neuroelectronics -- membranes -- ion channels -- wiring. Elementary neuron models -- conductance based -- modelers alternatives

Basic elements of neuroelectronics -- membranes -- ion channels -- wiring. Elementary neuron models -- conductance based -- modelers alternatives Computing in carbon Basic elements of neuroelectronics -- membranes -- ion channels -- wiring Elementary neuron models -- conductance based -- modelers alternatives Wiring neurons together -- synapses

More information

1 Introduction and neurophysiology

1 Introduction and neurophysiology Dynamics of Continuous, Discrete and Impulsive Systems Series B: Algorithms and Applications 16 (2009) 535-549 Copyright c 2009 Watam Press http://www.watam.org ASYMPTOTIC DYNAMICS OF THE SLOW-FAST HINDMARSH-ROSE

More information

Texas A & M University Department of Mechanical Engineering MEEN 364 Dynamic Systems and Controls Dr. Alexander G. Parlos

Texas A & M University Department of Mechanical Engineering MEEN 364 Dynamic Systems and Controls Dr. Alexander G. Parlos Texas A & M University Department of Mechanical Engineering MEEN 364 Dynamic Systems and Controls Dr. Alexander G. Parlos Lecture 5: Electrical and Electromagnetic System Components The objective of this

More information

NONLINEAR AND ADAPTIVE (INTELLIGENT) SYSTEMS MODELING, DESIGN, & CONTROL A Building Block Approach

NONLINEAR AND ADAPTIVE (INTELLIGENT) SYSTEMS MODELING, DESIGN, & CONTROL A Building Block Approach NONLINEAR AND ADAPTIVE (INTELLIGENT) SYSTEMS MODELING, DESIGN, & CONTROL A Building Block Approach P.A. (Rama) Ramamoorthy Electrical & Computer Engineering and Comp. Science Dept., M.L. 30, University

More information

6.3.4 Action potential

6.3.4 Action potential I ion C m C m dφ dt Figure 6.8: Electrical circuit model of the cell membrane. Normally, cells are net negative inside the cell which results in a non-zero resting membrane potential. The membrane potential

More information

Decoding. How well can we learn what the stimulus is by looking at the neural responses?

Decoding. How well can we learn what the stimulus is by looking at the neural responses? Decoding How well can we learn what the stimulus is by looking at the neural responses? Two approaches: devise explicit algorithms for extracting a stimulus estimate directly quantify the relationship

More information

Tracking the State of the Hindmarsh-Rose Neuron by Using the Coullet Chaotic System Based on a Single Input

Tracking the State of the Hindmarsh-Rose Neuron by Using the Coullet Chaotic System Based on a Single Input ISSN 1746-7659, England, UK Journal of Information and Computing Science Vol. 11, No., 016, pp.083-09 Tracking the State of the Hindmarsh-Rose Neuron by Using the Coullet Chaotic System Based on a Single

More information

Single neuron models. L. Pezard Aix-Marseille University

Single neuron models. L. Pezard Aix-Marseille University Single neuron models L. Pezard Aix-Marseille University Biophysics Biological neuron Biophysics Ionic currents Passive properties Active properties Typology of models Compartmental models Differential

More information

Lecture 10 : Neuronal Dynamics. Eileen Nugent

Lecture 10 : Neuronal Dynamics. Eileen Nugent Lecture 10 : Neuronal Dynamics Eileen Nugent Origin of the Cells Resting Membrane Potential: Nernst Equation, Donnan Equilbrium Action Potentials in the Nervous System Equivalent Electrical Circuits and

More information

Passivity-based Control of Euler-Lagrange Systems

Passivity-based Control of Euler-Lagrange Systems Romeo Ortega, Antonio Loria, Per Johan Nicklasson and Hebertt Sira-Ramfrez Passivity-based Control of Euler-Lagrange Systems Mechanical, Electrical and Electromechanical Applications Springer Contents

More information

Two Port Network. Ram Prasad Sarkar

Two Port Network. Ram Prasad Sarkar Two Port Ram Prasad Sarkar 0 Two Post : Post nput port Two Post Fig. Port Output port A network which has two terminals (one port) on the one side and another two terminals on the opposite side forms a

More information

Wave Pinning, Actin Waves, and LPA

Wave Pinning, Actin Waves, and LPA Wave Pinning, Actin Waves, and LPA MCB 2012 William R. Holmes Intercellular Waves Weiner et al., 2007, PLoS Biology Dynamic Hem1 waves in neutrophils Questions? How can such waves / pulses form? What molecular

More information

Notes for course EE1.1 Circuit Analysis TOPIC 10 2-PORT CIRCUITS

Notes for course EE1.1 Circuit Analysis TOPIC 10 2-PORT CIRCUITS Objectives: Introduction Notes for course EE1.1 Circuit Analysis 4-5 Re-examination of 1-port sub-circuits Admittance parameters for -port circuits TOPIC 1 -PORT CIRCUITS Gain and port impedance from -port

More information

Analog Electronics Mimic Genetic Biochemical Reactions in Living Cells

Analog Electronics Mimic Genetic Biochemical Reactions in Living Cells Analog Electronics Mimic Genetic Biochemical Reactions in Living Cells Dr. Ramez Daniel Laboratory of Synthetic Biology & Bioelectronics (LSB 2 ) Biomedical Engineering, Technion May 9, 2016 Cytomorphic

More information

RF Upset and Chaos in Circuits: Basic Investigations. Steven M. Anlage, Vassili Demergis, Ed Ott, Tom Antonsen

RF Upset and Chaos in Circuits: Basic Investigations. Steven M. Anlage, Vassili Demergis, Ed Ott, Tom Antonsen RF Upset and Chaos in Circuits: Basic Investigations Steven M. Anlage, Vassili Demergis, Ed Ott, Tom Antonsen AFOSR Presentation Research funded by the AFOSR-MURI and DURIP programs OVERVIEW HPM Effects

More information

9 Generation of Action Potential Hodgkin-Huxley Model

9 Generation of Action Potential Hodgkin-Huxley Model 9 Generation of Action Potential Hodgkin-Huxley Model (based on chapter 2, W.W. Lytton, Hodgkin-Huxley Model) 9. Passive and active membrane models In the previous lecture we have considered a passive

More information

Parameters for Minimal Model of Cardiac Cell from Two Different Methods: Voltage-Clamp and MSE Method

Parameters for Minimal Model of Cardiac Cell from Two Different Methods: Voltage-Clamp and MSE Method Parameters for Minimal Model of Cardiac Cell from Two Different Methods: oltage-clamp and MSE Method Soheila Esmaeili 1, * and Bahareh beheshti 1 Department of Biomedical engineering, ran University of

More information

Single-Cell and Mean Field Neural Models

Single-Cell and Mean Field Neural Models 1 Single-Cell and Mean Field Neural Models Richard Bertram Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University Tallahassee, Florida 32306 The neuron

More information

Lyapunov functions on nonlinear spaces S 1 R R + V =(1 cos ) Constructing Lyapunov functions: a personal journey

Lyapunov functions on nonlinear spaces S 1 R R + V =(1 cos ) Constructing Lyapunov functions: a personal journey Constructing Lyapunov functions: a personal journey Lyapunov functions on nonlinear spaces R. Sepulchre -- University of Liege, Belgium Reykjavik - July 2013 Lyap functions in linear spaces (1994-1997)

More information

Finite-Time Thermodynamics of Port-Hamiltonian Systems

Finite-Time Thermodynamics of Port-Hamiltonian Systems Finite-Time Thermodynamics of Port-Hamiltonian Systems Henrik Sandberg Automatic Control Lab, ACCESS Linnaeus Centre, KTH (Currently on sabbatical leave at LIDS, MIT) Jean-Charles Delvenne CORE, UC Louvain

More information

M. Dechesne R. Sepulchre

M. Dechesne R. Sepulchre Systems and models in chronobiology A delay model for the circadian rhythm M. Dechesne R. Sepulchre Department of Electrical Engineering and Computer Science Monteore Institute University of Liège 24th

More information

Nonlinear systems, chaos and control in Engineering

Nonlinear systems, chaos and control in Engineering Nonlinear systems, chaos and control in Engineering Module 1 block 3 One-dimensional nonlinear systems Cristina Masoller Cristina.masoller@upc.edu http://www.fisica.edu.uy/~cris/ Schedule Flows on the

More information

Chimera states in networks of biological neurons and coupled damped pendulums

Chimera states in networks of biological neurons and coupled damped pendulums in neural models in networks of pendulum-like elements in networks of biological neurons and coupled damped pendulums J. Hizanidis 1, V. Kanas 2, A. Bezerianos 3, and T. Bountis 4 1 National Center for

More information

Basic. Theory. ircuit. Charles A. Desoer. Ernest S. Kuh. and. McGraw-Hill Book Company

Basic. Theory. ircuit. Charles A. Desoer. Ernest S. Kuh. and. McGraw-Hill Book Company Basic C m ш ircuit Theory Charles A. Desoer and Ernest S. Kuh Department of Electrical Engineering and Computer Sciences University of California, Berkeley McGraw-Hill Book Company New York St. Louis San

More information

Final Project Descriptions Introduction to Mathematical Biology Professor: Paul J. Atzberger. Project I: Predator-Prey Equations

Final Project Descriptions Introduction to Mathematical Biology Professor: Paul J. Atzberger. Project I: Predator-Prey Equations Final Project Descriptions Introduction to Mathematical Biology Professor: Paul J. Atzberger Project I: Predator-Prey Equations The Lotka-Volterra Predator-Prey Model is given by: du dv = αu βuv = ρβuv

More information

Νευροφυσιολογία και Αισθήσεις

Νευροφυσιολογία και Αισθήσεις Biomedical Imaging & Applied Optics University of Cyprus Νευροφυσιολογία και Αισθήσεις Διάλεξη 5 Μοντέλο Hodgkin-Huxley (Hodgkin-Huxley Model) Response to Current Injection 2 Hodgin & Huxley Sir Alan Lloyd

More information

Consider the following spike trains from two different neurons N1 and N2:

Consider the following spike trains from two different neurons N1 and N2: About synchrony and oscillations So far, our discussions have assumed that we are either observing a single neuron at a, or that neurons fire independent of each other. This assumption may be correct in

More information

Dynamical modelling of systems of coupled oscillators

Dynamical modelling of systems of coupled oscillators Dynamical modelling of systems of coupled oscillators Mathematical Neuroscience Network Training Workshop Edinburgh Peter Ashwin University of Exeter 22nd March 2009 Peter Ashwin (University of Exeter)

More information

Structure and Measurement of the brain lecture notes

Structure and Measurement of the brain lecture notes Structure and Measurement of the brain lecture notes Marty Sereno 2009/2010!"#$%&'(&#)*%$#&+,'-&.)"/*"&.*)*-'(0&1223 Neurons and Models Lecture 1 Topics Membrane (Nernst) Potential Action potential/voltage-gated

More information

Problem Set Number 02, j/2.036j MIT (Fall 2018)

Problem Set Number 02, j/2.036j MIT (Fall 2018) Problem Set Number 0, 18.385j/.036j MIT (Fall 018) Rodolfo R. Rosales (MIT, Math. Dept., room -337, Cambridge, MA 0139) September 6, 018 Due October 4, 018. Turn it in (by 3PM) at the Math. Problem Set

More information

Introduction and the Hodgkin-Huxley Model

Introduction and the Hodgkin-Huxley Model 1 Introduction and the Hodgkin-Huxley Model Richard Bertram Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University Tallahassee, Florida 32306 Reference:

More information

Figure Circuit for Question 1. Figure Circuit for Question 2

Figure Circuit for Question 1. Figure Circuit for Question 2 Exercises 10.7 Exercises Multiple Choice 1. For the circuit of Figure 10.44 the time constant is A. 0.5 ms 71.43 µs 2, 000 s D. 0.2 ms 4 Ω 2 Ω 12 Ω 1 mh 12u 0 () t V Figure 10.44. Circuit for Question

More information

A simple electronic circuit to demonstrate bifurcation and chaos

A simple electronic circuit to demonstrate bifurcation and chaos A simple electronic circuit to demonstrate bifurcation and chaos P R Hobson and A N Lansbury Brunel University, Middlesex Chaos has generated much interest recently, and many of the important features

More information

Nature-inspired Analog Computing on Silicon

Nature-inspired Analog Computing on Silicon Nature-inspired Analog Computing on Silicon Tetsuya ASAI and Yoshihito AMEMIYA Division of Electronics and Information Engineering Hokkaido University Abstract We propose CMOS analog circuits that emulate

More information

Nonlinear Observer Design and Synchronization Analysis for Classical Models of Neural Oscillators

Nonlinear Observer Design and Synchronization Analysis for Classical Models of Neural Oscillators Nonlinear Observer Design and Synchronization Analysis for Classical Models of Neural Oscillators Ranjeetha Bharath and Jean-Jacques Slotine Massachusetts Institute of Technology ABSTRACT This work explores

More information

Construction of Classes of Circuit-Independent Chaotic Oscillators Using Passive-Only Nonlinear Devices

Construction of Classes of Circuit-Independent Chaotic Oscillators Using Passive-Only Nonlinear Devices IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 48, NO. 3, MARCH 2001 289 Construction of Classes of Circuit-Independent Chaotic Oscillators Using Passive-Only Nonlinear

More information

Dynamical systems in neuroscience. Pacific Northwest Computational Neuroscience Connection October 1-2, 2010

Dynamical systems in neuroscience. Pacific Northwest Computational Neuroscience Connection October 1-2, 2010 Dynamical systems in neuroscience Pacific Northwest Computational Neuroscience Connection October 1-2, 2010 What do I mean by a dynamical system? Set of state variables Law that governs evolution of state

More information

Presented by Sarah Hedayat. Supervised by Pr.Cappy and Dr.Hoel

Presented by Sarah Hedayat. Supervised by Pr.Cappy and Dr.Hoel 1 Presented by Sarah Hedayat Supervised by Pr.Cappy and Dr.Hoel Outline 2 Project objectives Key elements Membrane models As simple as possible Phase plane analysis Review of important Concepts Conclusion

More information

Stochastic differential equations in neuroscience

Stochastic differential equations in neuroscience Stochastic differential equations in neuroscience Nils Berglund MAPMO, Orléans (CNRS, UMR 6628) http://www.univ-orleans.fr/mapmo/membres/berglund/ Barbara Gentz, Universität Bielefeld Damien Landon, MAPMO-Orléans

More information

REAL-TIME COMPUTING WITHOUT STABLE

REAL-TIME COMPUTING WITHOUT STABLE REAL-TIME COMPUTING WITHOUT STABLE STATES: A NEW FRAMEWORK FOR NEURAL COMPUTATION BASED ON PERTURBATIONS Wolfgang Maass Thomas Natschlager Henry Markram Presented by Qiong Zhao April 28 th, 2010 OUTLINE

More information

The Significance of Reduced Conductance in the Sino-Atrial Node for Cardiac Rythmicity

The Significance of Reduced Conductance in the Sino-Atrial Node for Cardiac Rythmicity The Significance of Reduced Conductance in the Sino-Atrial Node for Cardiac Rythmicity Will Nesse May 14, 2005 Sino-Atrial Node (SA node): The Pacemaker of the Heart 1 The SA node and the Atria: How the

More information

DYNAMICS and CONTROL

DYNAMICS and CONTROL DYNAMICS and CONTROL MODULE 1I (III) Models of Systems & Signals Formalism Presented by Pedro Albertos Professor of Systems Engineering and Control - UPV 1 Modules: Examples of systems and signals Models

More information

Mathematical Foundations of Neuroscience - Lecture 3. Electrophysiology of neurons - continued

Mathematical Foundations of Neuroscience - Lecture 3. Electrophysiology of neurons - continued Mathematical Foundations of Neuroscience - Lecture 3. Electrophysiology of neurons - continued Filip Piękniewski Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, Poland

More information

Some of the different forms of a signal, obtained by transformations, are shown in the figure. jwt e z. jwt z e

Some of the different forms of a signal, obtained by transformations, are shown in the figure. jwt e z. jwt z e Transform methods Some of the different forms of a signal, obtained by transformations, are shown in the figure. X(s) X(t) L - L F - F jw s s jw X(jw) X*(t) F - F X*(jw) jwt e z jwt z e X(nT) Z - Z X(z)

More information

Index. Index. More information. in this web service Cambridge University Press

Index. Index. More information.  in this web service Cambridge University Press A-type elements, 4 7, 18, 31, 168, 198, 202, 219, 220, 222, 225 A-type variables. See Across variable ac current, 172, 251 ac induction motor, 251 Acceleration rotational, 30 translational, 16 Accumulator,

More information

Electrical Circuits Lab Series RC Circuit Phasor Diagram

Electrical Circuits Lab Series RC Circuit Phasor Diagram Electrical Circuits Lab. 0903219 Series RC Circuit Phasor Diagram - Simple steps to draw phasor diagram of a series RC circuit without memorizing: * Start with the quantity (voltage or current) that is

More information

1 Hodgkin-Huxley Theory of Nerve Membranes: The FitzHugh-Nagumo model

1 Hodgkin-Huxley Theory of Nerve Membranes: The FitzHugh-Nagumo model 1 Hodgkin-Huxley Theory of Nerve Membranes: The FitzHugh-Nagumo model Alan Hodgkin and Andrew Huxley developed the first quantitative model of the propagation of an electrical signal (the action potential)

More information

Sophomore Physics Laboratory (PH005/105)

Sophomore Physics Laboratory (PH005/105) CALIFORNIA INSTITUTE OF TECHNOLOGY PHYSICS MATHEMATICS AND ASTRONOMY DIVISION Sophomore Physics Laboratory (PH5/15) Analog Electronics Active Filters Copyright c Virgínio de Oliveira Sannibale, 23 (Revision

More information

FUZZY CONTROL OF CHAOS

FUZZY CONTROL OF CHAOS International Journal of Bifurcation and Chaos, Vol. 8, No. 8 (1998) 1743 1747 c World Scientific Publishing Company FUZZY CONTROL OF CHAOS OSCAR CALVO CICpBA, L.E.I.C.I., Departamento de Electrotecnia,

More information

Dynamics and complexity of Hindmarsh-Rose neuronal systems

Dynamics and complexity of Hindmarsh-Rose neuronal systems Dynamics and complexity of Hindmarsh-Rose neuronal systems Nathalie Corson and M.A. Aziz-Alaoui Laboratoire de Mathématiques Appliquées du Havre, 5 rue Philippe Lebon, 766 Le Havre, FRANCE nathalie.corson@univ-lehavre.fr

More information

Delayed Feedback and GHz-Scale Chaos on the Driven Diode-Terminated Transmission Line

Delayed Feedback and GHz-Scale Chaos on the Driven Diode-Terminated Transmission Line Delayed Feedback and GHz-Scale Chaos on the Driven Diode-Terminated Transmission Line Steven M. Anlage, Vassili Demergis, Renato Moraes, Edward Ott, Thomas Antonsen Thanks to Alexander Glasser, Marshal

More information

Lecture 11 : Simple Neuron Models. Dr Eileen Nugent

Lecture 11 : Simple Neuron Models. Dr Eileen Nugent Lecture 11 : Simple Neuron Models Dr Eileen Nugent Reading List Nelson, Biological Physics, Chapter 12 Phillips, PBoC, Chapter 17 Gerstner, Neuronal Dynamics: from single neurons to networks and models

More information

Topics in Neurophysics

Topics in Neurophysics Topics in Neurophysics Alex Loebel, Martin Stemmler and Anderas Herz Exercise 2 Solution (1) The Hodgkin Huxley Model The goal of this exercise is to simulate the action potential according to the model

More information

Lecture IV: LTI models of physical systems

Lecture IV: LTI models of physical systems BME 171: Signals and Systems Duke University September 5, 2008 This lecture Plan for the lecture: 1 Interconnections of linear systems 2 Differential equation models of LTI systems 3 eview of linear circuit

More information

Stability and Control of dc Micro-grids

Stability and Control of dc Micro-grids Stability and Control of dc Micro-grids Alexis Kwasinski Thank you to Mr. Chimaobi N. Onwuchekwa (who has been working on boundary controllers) May, 011 1 Alexis Kwasinski, 011 Overview Introduction Constant-power-load

More information

9 Generation of Action Potential Hodgkin-Huxley Model

9 Generation of Action Potential Hodgkin-Huxley Model 9 Generation of Action Potential Hodgkin-Huxley Model (based on chapter 12, W.W. Lytton, Hodgkin-Huxley Model) 9.1 Passive and active membrane models In the previous lecture we have considered a passive

More information

The Spike Response Model: A Framework to Predict Neuronal Spike Trains

The Spike Response Model: A Framework to Predict Neuronal Spike Trains The Spike Response Model: A Framework to Predict Neuronal Spike Trains Renaud Jolivet, Timothy J. Lewis 2, and Wulfram Gerstner Laboratory of Computational Neuroscience, Swiss Federal Institute of Technology

More information

arxiv: v1 [physics.bio-ph] 2 Jul 2008

arxiv: v1 [physics.bio-ph] 2 Jul 2008 Modeling Excitable Systems Jarrett L. Lancaster and Edward H. Hellen University of North Carolina Greensboro, Department of Physics and Astronomy, Greensboro, NC 27402 arxiv:0807.0451v1 [physics.bio-ph]

More information

Bursting Oscillations of Neurons and Synchronization

Bursting Oscillations of Neurons and Synchronization Bursting Oscillations of Neurons and Synchronization Milan Stork Applied Electronics and Telecommunications, Faculty of Electrical Engineering/RICE University of West Bohemia, CZ Univerzitni 8, 3064 Plzen

More information

Compartmental Modelling

Compartmental Modelling Modelling Neurons Computing and the Brain Compartmental Modelling Spring 2010 2 1 Equivalent Electrical Circuit A patch of membrane is equivalent to an electrical circuit This circuit can be described

More information

NE 204 mini-syllabus (weeks 4 8)

NE 204 mini-syllabus (weeks 4 8) NE 24 mini-syllabus (weeks 4 8) Instructor: John Burke O ce: MCS 238 e-mail: jb@math.bu.edu o ce hours: by appointment Overview: For the next few weeks, we will focus on mathematical models of single neurons.

More information

Linearization of F-I Curves by Adaptation

Linearization of F-I Curves by Adaptation LETTER Communicated by Laurence Abbott Linearization of F-I Curves by Adaptation Bard Ermentrout Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A. We show that negative

More information

3 Action Potentials - Brutal Approximations

3 Action Potentials - Brutal Approximations Physics 172/278 - David Kleinfeld - Fall 2004; Revised Winter 2015 3 Action Potentials - Brutal Approximations The Hodgkin-Huxley equations for the behavior of the action potential in squid, and similar

More information

Computational Neuroscience. Session 4-2

Computational Neuroscience. Session 4-2 Computational Neuroscience. Session 4-2 Dr. Marco A Roque Sol 06/21/2018 Two-Dimensional Two-Dimensional System In this section we will introduce methods of phase plane analysis of two-dimensional systems.

More information

Exploring experimental optical complexity with big data nonlinear analysis tools. Cristina Masoller

Exploring experimental optical complexity with big data nonlinear analysis tools. Cristina Masoller Exploring experimental optical complexity with big data nonlinear analysis tools Cristina Masoller Cristina.masoller@upc.edu www.fisica.edu.uy/~cris 4 th International Conference on Complex Dynamical Systems

More information

Advanced Models of Neural Networks

Advanced Models of Neural Networks Advanced Models of Neural Networks Gerasimos G. Rigatos Advanced Models of Neural Networks Nonlinear Dynamics and Stochasticity in Biological Neurons 123 Gerasimos G. Rigatos Unit of Industrial Automation

More information

CS 436 HCI Technology Basic Electricity/Electronics Review

CS 436 HCI Technology Basic Electricity/Electronics Review CS 436 HCI Technology Basic Electricity/Electronics Review *Copyright 1997-2008, Perry R. Cook, Princeton University August 27, 2008 1 Basic Quantities and Units 1.1 Charge Number of electrons or units

More information

Dynamical phase transitions in periodically driven model neurons

Dynamical phase transitions in periodically driven model neurons Dynamical phase transitions in periodically driven model neurons Jan R. Engelbrecht 1 and Renato Mirollo 2 1 Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA 2 Department

More information

Dynamical Systems in Neuroscience: Elementary Bifurcations

Dynamical Systems in Neuroscience: Elementary Bifurcations Dynamical Systems in Neuroscience: Elementary Bifurcations Foris Kuang May 2017 1 Contents 1 Introduction 3 2 Definitions 3 3 Hodgkin-Huxley Model 3 4 Morris-Lecar Model 4 5 Stability 5 5.1 Linear ODE..............................................

More information

TWO PORT NETWORKS Introduction: A port is normally referred to a pair of terminals of a network through which we can have access to network either for a source for measuring an output We have already seen

More information

Controlo Switched Systems: Mixing Logic with Differential Equations. João P. Hespanha. University of California at Santa Barbara.

Controlo Switched Systems: Mixing Logic with Differential Equations. João P. Hespanha. University of California at Santa Barbara. Controlo 00 5 th Portuguese Conference on Automatic Control University of Aveiro,, September 5-7, 5 00 Switched Systems: Mixing Logic with Differential Equations João P. Hespanha University of California

More information

Problem Solving 8: Circuits

Problem Solving 8: Circuits MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics OBJECTIVES Problem Solving 8: Circuits 1. To gain intuition for the behavior of DC circuits with both resistors and capacitors or inductors.

More information

Voltage-clamp and Hodgkin-Huxley models

Voltage-clamp and Hodgkin-Huxley models Voltage-clamp and Hodgkin-Huxley models Read: Hille, Chapters 2-5 (best) Koch, Chapters 6, 8, 9 See also Clay, J. Neurophysiol. 80:903-913 (1998) (for a recent version of the HH squid axon model) Rothman

More information

Synchronization and control in small networks of chaotic electronic circuits

Synchronization and control in small networks of chaotic electronic circuits Synchronization and control in small networks of chaotic electronic circuits A. Iglesias Dept. of Applied Mathematics and Computational Sciences, Universi~ of Cantabria, Spain Abstract In this paper, a

More information

Canonical Neural Models 1

Canonical Neural Models 1 Canonical Neural Models 1 Frank Hoppensteadt 1 and Eugene zhikevich 2 ntroduction Mathematical modeling is a powerful tool in studying fundamental principles of information processing in the brain. Unfortunately,

More information

Cybergenetics: Control theory for living cells

Cybergenetics: Control theory for living cells Department of Biosystems Science and Engineering, ETH-Zürich Cybergenetics: Control theory for living cells Corentin Briat Joint work with Ankit Gupta and Mustafa Khammash Introduction Overview Cybergenetics:

More information

A Mathematical Study of Electrical Bursting in Pituitary Cells

A Mathematical Study of Electrical Bursting in Pituitary Cells A Mathematical Study of Electrical Bursting in Pituitary Cells Richard Bertram Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University Collaborators on

More information

Dissipative Systems Analysis and Control

Dissipative Systems Analysis and Control Bernard Brogliato, Rogelio Lozano, Bernhard Maschke and Olav Egeland Dissipative Systems Analysis and Control Theory and Applications 2nd Edition With 94 Figures 4y Sprin er 1 Introduction 1 1.1 Example

More information

46th IEEE CDC, New Orleans, USA, Dec , where θ rj (0) = j 1. Re 1 N. θ =

46th IEEE CDC, New Orleans, USA, Dec , where θ rj (0) = j 1. Re 1 N. θ = Proceedings of the 46th IEEE Conference on Decision and Control New Orleans, LA, USA, Dec. 12-14, 2007 INVITED SESSION NUMBER: 157 Event-Based Feedback Control of Nonlinear Oscillators Using Phase Response

More information

Homework 1 solutions

Homework 1 solutions Electric Circuits 1 Homework 1 solutions (Due date: 2014/3/3) This assignment covers Ch1 and Ch2 of the textbook. The full credit is 100 points. For each question, detailed derivation processes and accurate

More information

FUZZY CONTROL OF CHAOS

FUZZY CONTROL OF CHAOS FUZZY CONTROL OF CHAOS OSCAR CALVO, CICpBA, L.E.I.C.I., Departamento de Electrotecnia, Facultad de Ingeniería, Universidad Nacional de La Plata, 1900 La Plata, Argentina JULYAN H. E. CARTWRIGHT, Departament

More information

Passive Membrane Properties

Passive Membrane Properties Passive Membrane Properties Communicating through a leaky garden hose... Topics I Introduction & Electrochemical Gradients Passive Membrane Properties Action Potentials Voltage-Gated Ion Channels Topics

More information

The Physics of the Heart. Sima Setayeshgar

The Physics of the Heart. Sima Setayeshgar The Physics of the Heart Sima Setayeshgar Department of Physics Indiana University Indiana Unversity Physics REU Seminar, July 27 2005 1 Stripes, Spots and Scrolls Indiana Unversity Physics REU Seminar,

More information