Verfügbarkeit: Mathematics as a laboratory tool

Mathematics as a laboratory tool: dynamics, delays and noise

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Bibliographische Detailangaben
Hauptverfasser:	Milton, John (VerfasserIn), Ohira, Toru 1963- (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	New York, NY [u.a.] Springer 2014
Schlagworte:	Biowissenschaftlerin Neurologie Mathematik Cytologie
Online-Zugang:	Inhaltsverzeichnis Klappentext
Beschreibung:	XXV, 500 S. Ill., graph. Darst.
ISBN:	9781461490951

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Datensatz im Suchindex

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adam_text	Contents Preface ............................................................ vii Notation........................................................... xv Tools..............................................................xvii 1 Science and the Mathematics of Black Boxes ...................... 1 1.1 The Scientific Method ....................................... 3 1.2 Dynamical Systems ......................................... 3 1.2.1 Variables ........................................... 4 1.2.2 Measurements ....................................... 6 1.2.3 Units .............................................. 7 1.3 Input-Output Relationships .................................. 8 1.3.1 Linear Versus Nonlinear Black Boxes ................... 9 1.3.2 The Neuron as a Dynamical System ..................... 10 1.4 Interactions Between System and Surroundings ................. 13 1.5 What Have We Learned? .................................... 14 1.6 Exercises for Practice and Insight ............................. 15 2 The Mathematics of Change ..................................... 17 2.1 Differentiation ............................................. 18 2.2 Differential Equations ....................................... 19 2.2.Î Population Growth ................................... 20 2.2.2 Time Scale of Change ................................ 22 2.2.3 Linear ODEs with Constant Coefficients ................. 23 2.3 Black Boxes ............................................... 25 2.3. 1 Nonlinear Differential Equations ....................... 27 2.4 Existence and Uniqueness ................................... 29 2.5 What Have We Learned? .................................... 30 2.6 Exercises for Practice and Insight ............................. 30 SIX xx Contents 3 Equilibria and Steady States .................................... 33 3.1 Law of Mass Action ........................................ 34 3.2 Closed Dynamical Systems .................................. 36 3.2.1 Equilibria: Drag Binding .............................. 36 3.2.2 Transient Steady States: Enzyme Kinetics ................ 39 3.3 Open Dynamical Systems .................................... 42 3.3.1 Water Fountains ..................................... 43 3.4 The Steady-State Approximation * ............................ 45 3.4.1 Steady State: Enzyme-Substrate Reactions ............... 45 3.4.2 Steady State: Consecutive Reactions .................... 48 3.5 Existence of Fixed Points .................................... 52 3.6 What Have We learned? ..................................... 54 3.7 Exercises for Practice and Insight ............................. 55 c Stability ....................................................... 57 4.1 Landscapes in Stability ...................................... 58 4.1.1 Postural Stability .................................... 62 4.1.2 Perception of Ambiguous Figures ...................... 62 4.1.3 Stopping Epileptic Seizures ............................ 63 4.2 Fixed-Point Stability ........................................ 64 4.3 Stability of Second-Order ODEs .............................. 66 4.3.1 Real Eigenvalues .................................... 68 4.3.2 Complex Eigenvalues ................................. 70 4.3.3 Phase-Plane Representation ........................... 72 4.4 Illustrative Examples ........................................ 74 4.4.1 The Lotka- Volterra Equation .......................... 74 4.4.2 The van der Pol Oscillator ............................. 78 4.4.3 Computer: Friend or Foe? ............................. 80 4.5 Lyapunov s Insight ......................................... 81 4.5.1 Conservative Dynamical Systems ....................... 81 4.5.2 Lyapunov s Direct Method ............................ 83 4.6 What Have We Learned? .................................... 85 4.7 Exercises for Practice and Insight ............................. 86 Fixed Points: Creation and Destruction ........................... 91 5.1 Saddle-Node Bifurcation .................................... 93 5.1.1 Neuron Bistability ................................... 94 5.2 Transcritical Bifurcation ..................................... 96 5.2.1 Postponement of Instability ............................ 98 5.3 Pitchfork Bifurcation .......................................100 5.3.1 Finger-Spring Compressions ...........................101 5.4 Near the Bifurcation Point ...................................104 5.4.1 The Slowing-Down Phenomenon .......................105 5.4.2 Critical Phenomena ..................................106 5.5 Bifurcations at the Benchtop .................................107 Contents xxi 5.6 What Have We Learned? ....................................108 5.7 Exercises for Practice and Insight .............................109 6 Transient Dynamics ............................................ Ill 6.1 Step Functions .............................................113 6.2 Ramp Functions ............................................115 6.3 Impulse Responses .........................................116 6.3.1 Measuring the Impulse Response .......................119 6.4 The Convolution Integral ....................................120 6.4.1 Summing Neuronal Inputs .............................123 6.5 Transients in Nonlinear Dynamical Systems ....................126 6.5. і Excitability ......................................... 1 27 6.5.2 Bounded Ті me-Dependent States .......................129 6.6 What Have We Learned? ....................................131 6.7 Exercises for Practice and Insight .............................132 7 Frequency Oomain I: Bode Plots and Transfer Functions ...........137 7.1 Low-Pass Filters ...........................................138 7.2 Laplace Transform Toolbox ..................................142 7.2.1 Tool 8: Eulers Formula ...............................142 7.2.2 Tool 9: The Laplace Transform .........................146 7.3 Transfer Functions ..........................................149 7.4 Biological Filters ...........................................153 7.4.1 Crayfish Photoreceptor ...............................154 7.4.2 Osmoregulation in Yeast ..............................156 7.5 Bode-Plot Cookbook ........................................157 7.5.1 Constant Term (Gain) ................................158 7.5.2 Integral or Derivative Term ............................159 7.5.3 Lags and Leads ......................................161 7.5.4 Time Delays ........................................163 7.5.5 Complex Pair: Amplification ...........................163 7.6 Interpreting Biological Bode Plots ............................166 7.6.1 Crayfish Photoreceptor Transfer Function ................166 7.6.2 Yeast Osmotic Stress Transfer Function ................. Î 67 7.7 What Have We Learned? ....................................169 7.8 Exercises for Practice and Insight .............................169 8 Frequency Domain II: Fourier Analysis and Power Spectra .........175 8.1 Laboratory Applications .....................................177 8.1.1 Creating Sinusoidal Inputs ............................177 8. і .2 Electrical Interference ................................ J 79 8.1.3 The Electrical World .................................180 8.2 Fourier Analysis Toolbox ....................................181 8.2.1 Tool 10: Fourier Series ................................182 8.2.2 Tool 1 1 : The Fourier Transform ........................185 xiî Contents 8.3 Applications of Fourier Analysis ..............................189 8.3.1 Uncertainties in Fourier Analysis .......................189 8.3.2 Approximating a Square Wave ......................... 1 92 8.3.3 Power Spectrum .....................................194 8.3.4 Fractal Heartbeats ....................................196 8.4 Digital Signals .............................................198 8.4Л Low-Pass Filtering ...................................198 8.4.2 introducing the FFT ..................................201 8.4.3 Power Spectrum: Discrete Time Signals .................204 8.4.4 Using FFTs in the Laboratory ..........................209 8.4.5 Power Spectral Density Recipe .........................213 8.5 What Have We Learned? ....................................215 8.6 Exercises for Practice and Insight .............................216 9 Feedback and Control Systems ..................................219 9.1 Feedback Control ..........................................221 9.2 Pupil Light Reflex (PLR) ....................................223 9.3 Linear^Feedback .........................................225 9.3.1 Proportional-Integral-Derivative (PID) Controllers ........227 9.4 Delayed Feedback ..........................................229 9.4.1 Example: Wright s Equation ...........................232 9.5 Production-Destruction .....................................235 9.5.1 Negative Feedback: PLR ..............................236 9.5.2 Negative Feedback: Gene Regulatory Systems ............241 9.5.3 Stability: General Forms ..............................243 9.5.4 Positive Feedback: Cheyne-Stokes Respiration ...........244 9.6 Electronic Feedback ........................................247 9.6.1 The Clamped PLR ...................................247 9.7 intermittent Control ........................................250 9.7.1 Virtual Balancing ....................................250 9.8 Integrating DDEs ...........................................252 9.9 What Have We Learned? ....................................252 9.10 Exercises for Practice and insight .............................253 10 Oscillations ....................................................259 10. ! Neural Oscillations .........................................260 JO. 1 .1 Hodgkin-Huxley Neuron Model .......................260 10.2 Hopf Bifurcations ..........................................269 10.2.1 PLR: Supercritical Hopf Bifurcation ....................271 10.2.2 HH Equation: Hopf Bifurcations ....................... 272 10.3 Analyzing Oscillations ......................................273 10.4 Poincaré Sections ..........................................274 10.4. 1 Gait Stride Variability ................................275 10.5 Phase Resetting ............................................276 10.5.1 Stumbling ..........................................281 Contents xxiii 10.6 Interacting Oscillators .......................................282 10.6.1 Periodic Forcing .....................................283 10.6.2 Coupled Oscillators ..................................290 10.7 What Have We Learned? ....................................291 10.8 Exercises for Practice and Insight .............................292 11 Beyond Limit Cycles ...........................................295 11.1 Chaos and Parameters .......................................296 11.2 Dynamical Diseases and Parameters ...........................301 11.3 Chaos in the Laboratory .....................................304 11.3.1 Flour Beetle Cannibalism .............................304 11.3.2 Clamped PLR with Mixed Feedback ..................307 11.4 Complex Dynamics: The Human Brain ........................310 1 1.4.І Reduced Hodgkin-Huxley Models ......................310 11.4.2 Delay-Induced Transient Oscillations ...................314 11.5 What Have We Learned? ....................................316 11.6 Exercises for Practice and Insight .............................317 12 Random Perturbations .........................................321 12.1 Noise and Dynamics ........................................322 12.1.1 Stick Balancing at the Fingertip ........................323 12.1.2 Noise and Thresholds .................................324 12.1.3 Stochastic Gene Expression ...........................324 12.2 Stochastic Processes Toolbox ................................325 12.2.1 Random Variables and Their Properties ..................325 12.2.2 Stochastic Processes ..................................329 12.2.3 Statistical Averages ..................................332 12.3 Laboratory Evaluations ......................................336 12.3.1 Intensity ............................................336 12.3.2 Estimation of the Probability Density Function ...........336 12.3.3 Estimation of Moments ...............................339 12.3.4 Broad-Shouldered Probability Density Functions ..........340 12.4 Correlation Functions .......................................342 1 2.4.1 Estimating Autocorrelation ............................344 12.5 Power Spectrum of Stochastic Signals .........................347 12.6 Examples of Noise .........................................350 12.7 Cross-Correlation Function ..................................351 12.7.1 Impulse Response ....................................353 12.7.2 Measuring Time Delays ...............................353 12.7.3 Coherence ..........................................355 12.8 What Have We Learned? ....................................355 12.9 Problems for Practice and Insight .............................356 xxiv Contents 13 Noisy Dynamical Systems .......................................359 1 3.1 The Langevin Equation: Additive Noise ......................360 13.1.1 The Retina As a Recording Device ...................364 13.1.2 Time-Delayed Langevin Fquation ....................367 13.1.3 Skill Acquisition ...................................368 13.1.4 Stochastic Gene Expression .........................369 13.1.5 Numerical Integration of Langevin Equations ..........372 13.2 Noise and Thresholds .....................................373 13.2.1 Linearization ......................................373 13.2.2 Dithering .........................................375 13.2.3 Stochastic Resonance ...............................376 13.3 Parametric Noise .........................................379 13.3.1 On-Off Intermittency: Quadratic Map .................380 13.3.2 Langevin Equation with Parametric Noise .............382 13.3.3 Stick Balancing at the Fingertip ......................383 13.4 What Have We Learned? ..................................384 13.5 Problems for Practice and Insight ...........................385 14 Random Walks ................................................389 14.1 A Simple Random Walk ...................................391 14.1.1 Walking Molecules as Cellular Probes .................395 14.2 Anomalous Random Walks ................................397 14.3 Correlated Random Walks .................................397 14.3. 1 Random Walking While Standing Still ................398 14.3.2 Gait Stride Variability: DPA.........................401 14.3.3 Walking on the DNA Sequence ......................402 14.4 Delayed Random Walks ...................................402 14.5 Portraits in Diffusion .....................................404 14.5.1 Finding a Mate ....................................405 14.5.2 Facilitated Diffusion on the Double Helix ..............407 14.6 Optimal Search Patterns: Levy Flights .......................408 14.7 Probability Density Functions ..............................410 14.7.1 Master Equation ...................................411 14.8 Tool 1 2: Generating Functions .............................412 14.8.1 Approaches Using the Characteristic Function ..........416 14.9 What Have We Learned? ..................................420 14.10 Problems for Practice and Insight ...........................421 15 Thermodynamic Perspectives ....................................425 15.1 Equilibrium .............................................427 15.1.1 Mathematical Background ..........................427 15.1.2 First and Second Laws of Thermodynamics ............430 15.1.3 Spontaneity .......................................432 15.2 Nonequilibrmm .....,....................................434 Contents xxv 1 5.3 Dynamics and Temperature ................................436 15.3.1 Near Equilibrium ..................................438 15.3.2 Principle of Detailed Balancing ......................441 15.4 Entropy Production .......................................443 15.4.1 Minimum Entropy Production .......................443 15.4.2 Microscopic Entropy Production .....................444 15.5 Far from Equilibrium .....................................447 15.5.1 The Sandpile Paradigm .............................447 15.5.2 Power Laws ......................................449 15.5.3 Measuring Power Laws .............................451 15.5.4 Epileptic Quakes ..................................452 15.6 What Have We Learned? ..................................454 15.7 Exercises for Practice and Insight ...........................456 16 Concluding Remarks ...........................................459 References .........................................................461 Index .............................................................489 John Milton· ToruOhira Mathematics as a Laboratory Tool Dynamics, Delays and Noise The importance of mathematics in the undergraduate biology curriculum is ever increasing, as is the importance of biology within the undergraduate applied mathematics curriculum. This ambitious forward thinking book strives to make concrete connections between the two fields at the undergraduate level, bringing in a wide variety of mathematical methods such as signal processing, systems identification, and stochastic differential equations to an undergraduate audience interested in biological dynamics. The presentation stresses a practical hands-on approach: important concepts are introduced using linear first- or second-order differential equations that can be solved using pencil and paper ; next, these are extended to real world applications through the use of computer algorithms written in Scientific Python or similar software. This book developed from a course taught by Professor John Milton at the University of Chicago and developed and continued over many years with Professor Toru Ohira at the Claremont Colleges. The tone of the book is pedagogical, engaging, accessible, with lots of examples and exercises. The authors attempt to tread a line between accessibility of the text and mathematical exposition. Online laboratories are provided as a teaching aid. At the beginning of each chapter a number of questions are posed to the reader, and then answered at the conclusion of the chapter. Milton and Ohiras book is aimed at an undergraduate audience, makes close ties to the laboratory, and includes a range of biological applications, favoring physiology. This makes it a unique contribution to the literature. This book will be of interest to quantitatively inclined undergraduate biologists, biophysicists and bioengineers and in addition through its focus on techniques actually used by biologists, the authors hope this text will help shape curricula in biomathematics education going forward.
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publisher	Springer
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spelling	Milton, John Verfasser aut Mathematics as a laboratory tool dynamics, delays and noise John Milton ; Toru Ohira New York, NY [u.a.] Springer 2014 XXV, 500 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Biowissenschaftlerin (DE-588)4698137-8 gnd rswk-swf Neurologie (DE-588)4041888-1 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Cytologie (DE-588)4070177-3 gnd rswk-swf Mathematik (DE-588)4037944-9 s Neurologie (DE-588)4041888-1 s Cytologie (DE-588)4070177-3 s DE-604 Biowissenschaftlerin (DE-588)4698137-8 s 1\p DE-604 Ohira, Toru 1963- Verfasser (DE-588)1059084139 aut Erscheint auch als Online-Ausgabe 978-1-4614-9096-8 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027608287&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027608287&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk
spellingShingle	Milton, John Ohira, Toru 1963- Mathematics as a laboratory tool dynamics, delays and noise Biowissenschaftlerin (DE-588)4698137-8 gnd Neurologie (DE-588)4041888-1 gnd Mathematik (DE-588)4037944-9 gnd Cytologie (DE-588)4070177-3 gnd
subject_GND	(DE-588)4698137-8 (DE-588)4041888-1 (DE-588)4037944-9 (DE-588)4070177-3
title	Mathematics as a laboratory tool dynamics, delays and noise
title_auth	Mathematics as a laboratory tool dynamics, delays and noise
title_exact_search	Mathematics as a laboratory tool dynamics, delays and noise
title_full	Mathematics as a laboratory tool dynamics, delays and noise John Milton ; Toru Ohira
title_fullStr	Mathematics as a laboratory tool dynamics, delays and noise John Milton ; Toru Ohira
title_full_unstemmed	Mathematics as a laboratory tool dynamics, delays and noise John Milton ; Toru Ohira
title_short	Mathematics as a laboratory tool
title_sort	mathematics as a laboratory tool dynamics delays and noise
title_sub	dynamics, delays and noise
topic	Biowissenschaftlerin (DE-588)4698137-8 gnd Neurologie (DE-588)4041888-1 gnd Mathematik (DE-588)4037944-9 gnd Cytologie (DE-588)4070177-3 gnd
topic_facet	Biowissenschaftlerin Neurologie Mathematik Cytologie
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027608287&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027608287&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT miltonjohn mathematicsasalaboratorytooldynamicsdelaysandnoise AT ohiratoru mathematicsasalaboratorytooldynamicsdelaysandnoise

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