Dynamical systems in neuroscience. Pacific Northwest Computational Neuroscience Connection October 1-2, 2010
|
|
- Sheryl Johnston
- 6 years ago
- Views:
Transcription
1 Dynamical systems in neuroscience Pacific Northwest Computational Neuroscience Connection October 1-2, 2010
2 What do I mean by a dynamical system? Set of state variables Law that governs evolution of state variables in time Often takes the form of differential equations ex: dv dw = V ( a V )( V 1) w + I = bv cw
3 What do I mean by a dynamical system? Set of state variables Law that governs evolution of state variables in time Often takes the form of differential equations ex: dv = V ( a V )( V 1) w + I state variables dw = bv cw parameters
4 Dynamical systems arise as models for single neurons ex: Hodgkin-Huxley equations C dv dn = n dm dh = I g K n 4 ( V E ) K g Na m 3 h( V E ) Na g ( L V E ) L ( ( V ) n ) / τ n V ( ) = ( m ( V ) m) / τ ( m V ) = h ( ( V ) h) / τ ( h V )
5 Hodgkin and Huxley s circuit model of a neuronal membrane: 10
6 Hodgkin and Huxley s circuit model of a neuronal membrane: Voltage response to input current 10µA/cm 2 V(mV) t (ms) 50 11
7 Hodgkin and Huxley s circuit model of a neuronal membrane: f(hz) 12
8 C dv dn = n dm dh = I g K n 4 ( V E ) K g Na m 3 h( V E ) Na g ( L V E ) L ( ( V ) n ) / τ n V ( ) = ( m ( V ) m) / τ ( m V ) = h ( ( V ) h) / τ ( h V ) dv = F( V) ( ) V = V,n, m,h Basic question: can I get info about ( ) solutions by analyzing F V?
9 Phase plane analysis dv dw = V ( a V )( V 1) w + I = bv cw
10 Phase plane analysis: Nullclines 0.2 dw = dv = dv dw = V ( a V )( V 1) w + I = bv cw
11 Phase plane analysis: Equilibria and stability dv dw = V ( a V )( V 1) w + I = bv cw a=.1, b=.01, c=.02, I=0
12 Phase plane analysis: Equilibria and stability dv dw = V ( a V )( V 1) w + I = bv cw a=.2, b=.01, c=.02, I=0
13 Phase plane analysis: Bistability dv dw = V ( a V )( V 1) w + I = bv cw a=.1, b=.01, c=.07, I=0
14 Bifurcations: a qualitative change in 0.2 behavior dv dw = V ( a V )( V 1) w + I = bv cw I = 0
15 Bifurcations: a qualitative change in behavior dv dw = V ( a V )( V 1) w + I = bv cw I = 1
16 Bifurcation diagrams: an example V V I_0 I0
17 Two types of bifurcations: Type I vs. Type II (or Class 1 vs. Class 2) saddle node on invariant circle Class 1 excitability subcritical Hopf: Class 2 excitability firing rate firing rate Mean input I Mean input I
18 Some simplified models Phase model Integrate-and-fire Firing rate models
19 Reduction of 1. Brain recordings: voltage spikes via ionic currents neurons to phases: 2. Nonlinear oscillator eqn. for each neuron V q t V = voltage q = conductance e.g Rose and Hindmarsh, Hodgkin and Huxley, etc. 3. Coordinate change phase V θ fire V 0 2π θ Winfree 74, Guckenheimer 75, 25
20 Reduction of neurons to Nonlinear oscillator eqn. for each neuron phases: V = voltage q = conductance V q t 3. Coordinate change phase V θ fire V 0 2π θ Winfree 74, Guckenheimer 75, 25
21 Goal: simple phase description natural frequency original neural 26
22 Goal: simple phase description natural frequency original neural J(x,t) 27
23 Goal: simple phase description natural frequency original neural J(x,t) perturbation J(x,t) J(t) J(x,t) J(x,t) 28
24 Finding θ / V = z(θ), the phase response curve:, parameterized by θ NOTE: this technique often used in lab experiments! perturb with 5 mv stim. Thanks to Jeff Moehlis 30
25 Phase response curves for different neurons look very different! [Ermentrout and Kopell, Van-Vreeswick, Bressloff, Izhikevich, Moehlis, Holmes, S-B] Hodgkin-Huxley Leaky Integrate and Fire 33
26 With phase dynamics natural frequency phase response curve (phase sensitivity curve) π θ fire θ=0 31
27 We can study synchrony in network of two coupled neurons I syn (t) 32
28 For example, Hodgkin-Huxley... C dv dn = n dm dh = I g K n 4 ( V E ) K g Na m 3 h( V E ) Na g ( L V E ) L ( ( V ) n) / τ n V ( ) = ( m ( V ) m) / τ ( m V ) = h ( ( V ) h) / τ ( h V ) 37
29 The Hodgkin-Huxley model PRC z(θ) 38
30 The Hodgkin-Huxley model PRC z(θ) Moral: Fast excitatory coupling can synchronize HH neurons 39
31 Analyze via Poincare map between firing times of θ 1 Nancy Kopell, Bard Ermentrout, -- weak coupling theory See: synchronized state θ 12 =0 is stable fixed point for map 41
32 Let s try Hodgkin-Huxley + A-current C dv dn = n dm dh = I g K n 4 ( V E ) K g Na m 3 h( V E ) Na g ( L V E ) L g A a 3 b( V E ) A ( ( V ) n) / τ n V ( ) = ( m ( V ) m) / τ ( m V ) = h da = a db = b ( ( V ) h ) / τ h V ( ) ( ( V ) a ) / τ a V ( ) ( ( V ) b) / τ b V ( ) 42
33 The Hodgkin Huxley plus A current model PRC z(θ) 44
34 The Hodgkin Huxley plus A current model PRC z(θ) Moral: Excitatory coupling actually DEsynchronizes HH neurons with A currents Stable anti-synchronized state However, inhibition does synchronize! When inhibtion, not excitation, synchronizes Van Vreeswijk et al
35 Beyond impulse coupling Brain has gap junctions, Kuramoto, Kopell, Ermentrout -average coupling functions: as well as slow chemical synapses. get a system depending on phase differences only 46
36 Integrate-and-fire models Hodgkin- Huxley V(t) Claim: conductances approx. constant in this range Above value V thresh, stereotyped spike takes over In range [V reset,v thresh ], following rescaling, V thresh V reset INTEGRATE AND FIRE MODEL OF LAPICQUE Gerstner, Abbott, others relate to HH V(t)
37 Many variations... dv = V + av 2 ; V < V thresh quadratic integrate and fire (QIF) dv = 1 τ m ( ( E V + Δ e V V T )/Δ T ) L T exponential integrate and fire (EIF) V(t)
38 Firing rate models David Tank 56
39 How does firing rate f depends on input current I? via different firing rate vs. current ( f-i ) curves Hodgkin-Huxley Integrate + Fire f(hz) f(hz) I I f(hz) Piecewise linear f(hz) Sigmoidal, gain g I I 57
40 Dynamics? Think of neural units described by firing rates y which approach equilibrium rates f(input) with time constant τ m. nonlinearity of input-output function f allows amazing array of neural network computations Hopfield, Grossberg, Cohen associative memory: recall and learning can encode complex input-output functions 58
41 Computational tools for dynamical systems XPP AUTO MATCONT
42 References Check website later!
Neuronal Dynamics: Computational Neuroscience of Single Neurons
Week 4 part 5: Nonlinear Integrate-and-Fire Model 4.1 From Hodgkin-Huxley to 2D Neuronal Dynamics: Computational Neuroscience of Single Neurons Week 4 Recing detail: Two-dimensional neuron models Wulfram
More informationBasic 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 informationReduction of Conductance Based Models with Slow Synapses to Neural Nets
Reduction of Conductance Based Models with Slow Synapses to Neural Nets Bard Ermentrout October 1, 28 Abstract The method of averaging and a detailed bifurcation calculation are used to reduce a system
More informationDynamical 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 informationChapter 24 BIFURCATIONS
Chapter 24 BIFURCATIONS Abstract Keywords: Phase Portrait Fixed Point Saddle-Node Bifurcation Diagram Codimension-1 Hysteresis Hopf Bifurcation SNIC Page 1 24.1 Introduction In linear systems, responses
More informationSimple models of neurons!! Lecture 4!
Simple models of neurons!! Lecture 4! 1! Recap: Phase Plane Analysis! 2! FitzHugh-Nagumo Model! Membrane potential K activation variable Notes from Zillmer, INFN 3! FitzHugh Nagumo Model Demo! 4! Phase
More informationNeural 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 informationDynamical 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 informationCanonical 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 informationPhase Locking. 1 of of 10. The PRC's amplitude determines which frequencies a neuron locks to. The PRC's slope determines if locking is stable
Printed from the Mathematica Help Browser 1 1 of 10 Phase Locking A neuron phase-locks to a periodic input it spikes at a fixed delay [Izhikevich07]. The PRC's amplitude determines which frequencies a
More informationSingle 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 informationNeural 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 informationFrom 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 informationSynchronization and Phase Oscillators
1 Synchronization and Phase Oscillators Richard Bertram Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University Tallahassee, Florida 32306 Synchronization
More informationEvent-driven simulations of nonlinear integrate-and-fire neurons
Event-driven simulations of nonlinear integrate-and-fire neurons A. Tonnelier, H. Belmabrouk, D. Martinez Cortex Project, LORIA, Campus Scientifique, B.P. 239, 54 56 Vandoeuvre-lès-Nancy, France Abstract
More informationIN THIS turorial paper we exploit the relationship between
508 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 10, NO. 3, MAY 1999 Weakly Pulse-Coupled Oscillators, FM Interactions, Synchronization, Oscillatory Associative Memory Eugene M. Izhikevich Abstract We study
More informationOn the Response of Neurons to Sinusoidal Current Stimuli: Phase Response Curves and Phase-Locking
On the Response of Neurons to Sinusoidal Current Stimuli: Phase Response Curves and Phase-Locking Michael J. Schaus and Jeff Moehlis Abstract A powerful technique for analyzing mathematical models for
More informationWhen Transitions Between Bursting Modes Induce Neural Synchrony
International Journal of Bifurcation and Chaos, Vol. 24, No. 8 (2014) 1440013 (9 pages) c World Scientific Publishing Company DOI: 10.1142/S0218127414400136 When Transitions Between Bursting Modes Induce
More informationMANY scientists believe that pulse-coupled neural networks
IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 10, NO. 3, MAY 1999 499 Class 1 Neural Excitability, Conventional Synapses, Weakly Connected Networks, and Mathematical Foundations of Pulse-Coupled Models Eugene
More informationNonlinear Dynamics of Neural Firing
Nonlinear Dynamics of Neural Firing BENG/BGGN 260 Neurodynamics University of California, San Diego Week 3 BENG/BGGN 260 Neurodynamics (UCSD) Nonlinear Dynamics of Neural Firing Week 3 1 / 16 Reading Materials
More informationOn the Dynamics of Delayed Neural Feedback Loops. Sebastian Brandt Department of Physics, Washington University in St. Louis
On the Dynamics of Delayed Neural Feedback Loops Sebastian Brandt Department of Physics, Washington University in St. Louis Overview of Dissertation Chapter 2: S. F. Brandt, A. Pelster, and R. Wessel,
More informationPhase Response Curves, Delays and Synchronization in Matlab
Phase Response Curves, Delays and Synchronization in Matlab W. Govaerts and B. Sautois Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281-S9, B-9000 Ghent, Belgium
More informationNeuroscience 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 informationThis document is downloaded from DR-NTU, Nanyang Technological University Library, Singapore.
This document is downloaded from DR-NTU, Nanyang Technological University Library, Singapore. Title Small-signal neural models and their applications Author(s) Basu, Arindam Citation Basu, A. (01). Small-signal
More informationWaves and oscillations in networks of coupled neurons
Waves and oscillations in networks of coupled neurons Bard Ermentrout Department of Mathematics University of Pittsburgh Pittsburgh, PA 1526. April 25, 26 1 Introduction Neural systems are characterized
More informationSingle-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 information46th 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 informationDynamical 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 informationTitle. Author(s)Fujii, Hiroshi; Tsuda, Ichiro. CitationNeurocomputing, 58-60: Issue Date Doc URL. Type.
Title Neocortical gap junction-coupled interneuron systems exhibiting transient synchrony Author(s)Fujii, Hiroshi; Tsuda, Ichiro CitationNeurocomputing, 58-60: 151-157 Issue Date 2004-06 Doc URL http://hdl.handle.net/2115/8488
More informationElectrophysiology 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 informationMathematical Analysis of Bursting Electrical Activity in Nerve and Endocrine Cells
Mathematical Analysis of Bursting Electrical Activity in Nerve and Endocrine Cells Richard Bertram Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University
More informationUNIVERSITY OF CALIFORNIA SANTA BARBARA. Neural Oscillator Identification via Phase-Locking Behavior. Michael J. Schaus
UNIVERSITY OF CALIFORNIA SANTA BARBARA Neural Oscillator Identification via Phase-Locking Behavior by Michael J. Schaus A thesis submitted in partial satisfaction of the requirements for the degree of
More informationFactors affecting phase synchronization in integrate-and-fire oscillators
J Comput Neurosci (26) 2:9 2 DOI.7/s827-6-674-6 Factors affecting phase synchronization in integrate-and-fire oscillators Todd W. Troyer Received: 24 May 25 / Revised: 9 November 25 / Accepted: November
More informationThe Effects of Voltage Gated Gap. Networks
The Effects of Voltage Gated Gap Junctions on Phase Locking in Neuronal Networks Tim Lewis Department of Mathematics, Graduate Group in Applied Mathematics (GGAM) University of California, Davis with Donald
More informationDetermining the contributions of divisive and subtractive feedback in the Hodgkin-Huxley model
JComputNeurosci(2014)37:403 415 DOI 10.1007/s10827-014-0511-y Determining the contributions of divisive and subtractive feedback in the Hodgkin-Huxley model Sevgi Şengül & Robert Clewley & Richard Bertram
More information9 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 informationHybrid Integrate-and-Fire Model of a Bursting Neuron
LETTER Communicated by John Rinzel Hybrid Integrate-and-Fire Model of a Bursting Neuron Barbara J. Breen bbreen@ece.gatech.edu William C. Gerken wgerken@ece.gatech.edu Robert J. Butera, Jr. rbutera@ece.gatech.edu
More informationPhase Oscillators. and at r, Hence, the limit cycle at r = r is stable if and only if Λ (r ) < 0.
1 Phase Oscillators Richard Bertram Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University Tallahassee, Florida 32306 2 Phase Oscillators Oscillations
More informationBasic 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 informationAn 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 informationEntrainment and Chaos in the Hodgkin-Huxley Oscillator
Entrainment and Chaos in the Hodgkin-Huxley Oscillator Kevin K. Lin http://www.cims.nyu.edu/ klin Courant Institute, New York University Mostly Biomath - 2005.4.5 p.1/42 Overview (1) Goal: Show that the
More informationVoltage-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 informationDecoding. 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 informationNonlinear dynamics vs Neuronal Dynamics
Nonlinear dynamics vs Neuronal Dynamics Cours de Methodes Mathematiques en Neurosciences Brain and neuronal biological features Neuro-Computationnal properties of neurons Neuron Models and Dynamical Systems
More informationChimera 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 informationMultistability in Bursting Patterns in a Model of a Multifunctional Central Pattern Generator.
Georgia State University ScholarWorks @ Georgia State University Mathematics Theses Department of Mathematics and Statistics 7-15-2009 Multistability in Bursting Patterns in a Model of a Multifunctional
More informationVoltage-clamp and Hodgkin-Huxley models
Voltage-clamp and Hodgkin-Huxley models Read: Hille, Chapters 2-5 (best Koch, Chapters 6, 8, 9 See also Hodgkin and Huxley, J. Physiol. 117:500-544 (1952. (the source Clay, J. Neurophysiol. 80:903-913
More informationSynchrony in Neural Systems: a very brief, biased, basic view
Synchrony in Neural Systems: a very brief, biased, basic view Tim Lewis UC Davis NIMBIOS Workshop on Synchrony April 11, 2011 components of neuronal networks neurons synapses connectivity cell type - intrinsic
More informationLecture 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 informationLinearization 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 informationBursting 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 informationThreshold Curve for the Excitability of Bidimensional Spiking Neurons
Threshold Curve for the Excitability of Bidimensional Spiking Neurons Arnaud Tonnelier INRIA-Grenoble 655 avenue de l Europe, Montbonnot 38334 Saint Ismier, France (Dated: July 29, 214) We shed light on
More informationCISC 3250 Systems Neuroscience
CISC 3250 Systems Neuroscience Systems Neuroscience How the nervous system performs computations How groups of neurons work together to achieve intelligence Professor Daniel Leeds dleeds@fordham.edu JMH
More informationLocalized activity patterns in excitatory neuronal networks
Localized activity patterns in excitatory neuronal networks Jonathan Rubin Amitabha Bose February 3, 2004 Abstract. The existence of localized activity patterns, or bumps, has been investigated in a variety
More informationThe Theory of Weakly Coupled Oscillators
The Theory of Weakly Coupled Oscillators Michael A. Schwemmer and Timothy J. Lewis Department of Mathematics, One Shields Ave, University of California Davis, CA 95616 1 1 Introduction 2 3 4 5 6 7 8 9
More information9 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 informationChapter 2 The Hodgkin Huxley Theory of Neuronal Excitation
Chapter 2 The Hodgkin Huxley Theory of Neuronal Excitation Hodgkin and Huxley (1952) proposed the famous Hodgkin Huxley (hereinafter referred to as HH) equations which quantitatively describe the generation
More informationConsider 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 informationTime Delays in Neural Systems
Time Delays in Neural Systems Sue Ann Campbell 1 Department of Applied Mathematics, University of Waterloo, Waterloo ON N2l 3G1 Canada sacampbell@uwaterloo.ca Centre for Nonlinear Dynamics in Physiology
More informationPhase Response Properties and Phase-Locking in Neural Systems with Delayed Negative-Feedback. Carter L. Johnson
Phase Response Properties and Phase-Locking in Neural Systems with Delayed Negative-Feedback Carter L. Johnson Faculty Mentor: Professor Timothy J. Lewis University of California, Davis Abstract Oscillatory
More informationMathematical 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 informationarxiv: v1 [q-bio.nc] 9 Oct 2013
1 arxiv:1310.2430v1 [q-bio.nc] 9 Oct 2013 Impact of adaptation currents on synchronization of coupled exponential integrate-and-fire neurons Josef Ladenbauer 1,2, Moritz Augustin 1, LieJune Shiau 3, Klaus
More informationPatterns of Synchrony in Neural Networks with Spike Adaptation
Patterns of Synchrony in Neural Networks with Spike Adaptation C. van Vreeswijky and D. Hanselz y yracah Institute of Physics and Center for Neural Computation, Hebrew University, Jerusalem, 9194 Israel
More informationThe Phase Response Curve of Reciprocally Inhibitory Model Neurons Exhibiting Anti-Phase Rhythms
The Phase Response Curve of Reciprocally Inhibitory Model Neurons Exhibiting Anti-Phase Rhythms Jiawei Zhang Timothy J. Lewis Department of Mathematics, University of California, Davis Davis, CA 9566,
More informationA 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 informationModeling 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 information1 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 informationDissecting the Phase Response of a Model Bursting Neuron
SIAM J. APPLIED DYNAMICAL SYSTEMS ol. 9, No. 3, pp. 659 703 c 2010 Society for Industrial and Applied Mathematics Dissecting the Phase Response of a Model Bursting Neuron William Erik Sherwood and John
More informationMathematical Foundations of Neuroscience - Lecture 9. Simple Models of Neurons and Synapses.
Mathematical Foundations of Neuroscience - Lecture 9. Simple Models of and. Filip Piękniewski Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, Poland Winter 2009/2010
More informationLIMIT 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 informationDynamics of the Exponential Integrate-and-Fire Model with Slow Currents and Adaptation
Noname manuscript No. (will be inserted by the editor) Dynamics of the Exponential Integrate-and-Fire Model with Slow Currents and Adaptation Victor J. Barranca Daniel C. Johnson Jennifer L. Moyher Joshua
More informationComputational 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 information1. Introduction - Reproducibility of a Neuron. 3. Introduction Phase Response Curve. 2. Introduction - Stochastic synchronization. θ 1. c θ.
. Introduction - Reproducibility of a euron Science (995 Constant stimuli led to imprecise spike trains, whereas stimuli with fluctuations produced spike trains with timing reproducible to less than millisecond.
More informationDynamics 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 informationActivity Driven Adaptive Stochastic. Resonance. Gregor Wenning and Klaus Obermayer. Technical University of Berlin.
Activity Driven Adaptive Stochastic Resonance Gregor Wenning and Klaus Obermayer Department of Electrical Engineering and Computer Science Technical University of Berlin Franklinstr. 8/9, 187 Berlin fgrewe,obyg@cs.tu-berlin.de
More informationSupporting Online Material for
www.sciencemag.org/cgi/content/full/319/5869/1543/dc1 Supporting Online Material for Synaptic Theory of Working Memory Gianluigi Mongillo, Omri Barak, Misha Tsodyks* *To whom correspondence should be addressed.
More informationOPTIMAL INPUTS FOR PHASE MODELS OF SPIKING NEURONS
Proceedings of,,, OPTIMAL INPUTS FOR PHASE MODELS OF SPIKING NEURONS Jeff Moehlis Dept. of Mechanical Engineering University of California, Santa Barbara Santa Barbara, CA 936 Email: moehlis@engineering.ucsb.edu
More informationPresented 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 informationNeural Spike Train Analysis 1: Introduction to Point Processes
SAMSI Summer 2015: CCNS Computational Neuroscience Summer School Neural Spike Train Analysis 1: Introduction to Point Processes Uri Eden BU Department of Mathematics and Statistics July 27, 2015 Spikes
More informationMATH 3104: THE HODGKIN-HUXLEY EQUATIONS
MATH 3104: THE HODGKIN-HUXLEY EQUATIONS Parallel conductance model A/Prof Geoffrey Goodhill, Semester 1, 2009 So far we have modelled neuronal membranes by just one resistance (conductance) variable. We
More informationEffects of Noisy Drive on Rhythms in Networks of Excitatory and Inhibitory Neurons
Effects of Noisy Drive on Rhythms in Networks of Excitatory and Inhibitory Neurons Christoph Börgers 1 and Nancy Kopell 2 1 Department of Mathematics, Tufts University, Medford, MA 2155 2 Department of
More informationNeuronal Dynamics: Computational Neuroscience of Single Neurons
Nonlinear Integrate-and-Fire Model Neronal Dynamics: Comptational Neroscience of Single Nerons Nonlinear Integrate-and-fire (NLIF) - Definition - qadratic and expon. IF - Extracting NLIF model from data
More informationFast neural network simulations with population density methods
Fast neural network simulations with population density methods Duane Q. Nykamp a,1 Daniel Tranchina b,a,c,2 a Courant Institute of Mathematical Science b Department of Biology c Center for Neural Science
More informationComputational Modeling of Neuronal Systems (Advanced Topics in Mathematical Physiology: G , G )
Computational Modeling of Neuronal Systems (Advanced Topics in Mathematical Physiology: G63.2855.001, G80.3042.004) Thursday, 9:30-11:20am, WWH Rm 1314. Prerequisites: familiarity with linear algebra,
More informationOn the Phase Reduction and Response Dynamics of Neural Oscillator Populations
LETTER Communicated by Bard Ermentrout On the Phase Reduction and Response Dynamics of Neural Oscillator Populations Eric Brown ebrown@math.princeton.edu Jeff Moehlis jmoehlis@math.princeton.edu Program
More informationNumerical Simulation of Bistability between Regular Bursting and Chaotic Spiking in a Mathematical Model of Snail Neurons
International Journal of Theoretical and Mathematical Physics 2015, 5(5): 145-150 DOI: 10.5923/j.ijtmp.20150505.08 Numerical Simulation of Bistability between Regular Bursting and Chaotic Spiking in a
More informationIntroduction 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 information3 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 informationSingle-Compartment Neural Models
Single-Compartment Neural Models BENG/BGGN 260 Neurodynamics University of California, San Diego Week 2 BENG/BGGN 260 Neurodynamics (UCSD) Single-Compartment Neural Models Week 2 1 / 18 Reading Materials
More informationPhase Response. 1 of of 11. Synaptic input advances (excitatory) or delays (inhibitory) spiking
Printed from the Mathematica Help Browser 1 1 of 11 Phase Response Inward current-pulses decrease a cortical neuron's period (Cat, Layer V). [Fetz93] Synaptic input advances (excitatory) or delays (inhibitory)
More informationBiological Modeling of Neural Networks
Week 3 part 1 : Rection of the Hodgkin-Huxley Model 3.1 From Hodgkin-Huxley to 2D Biological Modeling of Neural Netorks - Overvie: From 4 to 2 dimensions - MathDetour 1: Exploiting similarities - MathDetour
More informationCoupling in Networks of Neuronal Oscillators. Carter Johnson
Coupling in Networks of Neuronal Oscillators Carter Johnson June 15, 2015 1 Introduction Oscillators are ubiquitous in nature. From the pacemaker cells that keep our hearts beating to the predator-prey
More informationFRTF01 L8 Electrophysiology
FRTF01 L8 Electrophysiology Lecture Electrophysiology in general Recap: Linear Time Invariant systems (LTI) Examples of 1 and 2-dimensional systems Stability analysis The need for non-linear descriptions
More informationBalance of Electric and Diffusion Forces
Balance of Electric and Diffusion Forces Ions flow into and out of the neuron under the forces of electricity and concentration gradients (diffusion). The net result is a electric potential difference
More informationAnalysis of burst dynamics bound by potential with active areas
NOLTA, IEICE Paper Analysis of burst dynamics bound by potential with active areas Koji Kurose 1a), Yoshihiro Hayakawa 2, Shigeo Sato 1, and Koji Nakajima 1 1 Laboratory for Brainware/Laboratory for Nanoelectronics
More informationJohn Rinzel, x83308, Courant Rm 521, CNS Rm 1005 Eero Simoncelli, x83938, CNS Rm 1030
Computational Modeling of Neuronal Systems (Advanced Topics in Mathematical Biology: G63.2852.001, G80.3042.001) Tuesday, 9:30-11:20am, WWH Rm 1314. Prerequisites: familiarity with linear algebra, applied
More informationSynchronization of Elliptic Bursters
SIAM REVIEW Vol. 43,No. 2,pp. 315 344 c 2001 Society for Industrial and Applied Mathematics Synchronization of Elliptic Bursters Eugene M. Izhikevich Abstract. Periodic bursting behavior in neurons is
More informationMath 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 informationLocalized Excitations in Networks of Spiking Neurons
Localized Excitations in Networks of Spiking Neurons Hecke Schrobsdorff Bernstein Center for Computational Neuroscience Göttingen Max Planck Institute for Dynamics and Self-Organization Seminar: Irreversible
More informationNE 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