TUTORIALS IN MATHEMATICAL BIOSCIENCES I: MATHEMATICAL NEUROSCIENCE

By EMMA WILLIAMS

CONTENTS INTRODUCTION TO NEURONS AVNER FRIEDMAN 1 THE STRUCTURE OF CELLS 2 NERVE CELLS 3 ELECTRICAL CIRCUITS AND THE HODGKIN-HUXLEY MODEL 4 THE CABLE EQUATION REFERENCES AN INTRODUCTION TO DYNAMICAL SYSTEMS AND NEURONAL DYNAMICS DAVID TERMAN 1 INTRODUCTION 2 ONE DIMENSIONAL EQUATIONS 2 1 THE GEOMETRIC APPROACH 2 2 BIFURCATIONS 2 3 BISTABILITY AND HYSTERESIS 3 TWO DIMENSIONAL SYSTEMS 3 1 THE PHASE PLANE 3 2 AN EXAMPLE 3 3 OSCILLATIONS 3 4 LOCAL BIFURCATIONS 3 5 GLOBAL BIFURCATIONS 3 6 GEOMETRIC SINGULAR PERTURBATION THEORY 4 SINGLE NEURONS 4 1 SOME BIOLOGY 4 2 THE HODGKIN-HUXLEY EQUATIONS 4 3 REDUCED MODELS 4 4 BURSTING OSCILLATIONS 4 5 TRAVELING WAVE SOLUTIONS 5 TWO MUTUALLY COUPLED CELLS 5 1 INTRODUCTION 5 2 SYNAPTIC COUPLING 5 3 GEOMETRIC APPROACH 5 4 SYNCHRONY WITH EXCITATORY SYNAPSES 5 5 DESYNCHRONY WITH INHIBITORY SYNAPSES 6 ACTIVITY PATTERNS IN THE BASAL GANGLIA 6 1 INTRODUCTION 6 2 THE BASAL GANGLIA 6 3 THE MODEL 6 4 ACTIVITY PATTERNS 6 5 CONCLUDING REMARKS NEURAL OSCILLATORS BARD ERMENTROUT 2 HOW DOES RHYTHMICITY ARISE 3 PHASE-RESETTING AND COUPLING THROUGH MAPS 4 DOUBLETS, DELAYS, AND MORE MAPS 5 AVERAGING AND PHASE MODELS 5 1 LOCAL ARRAYS 6 NEURAL NETWORKS 6 1 SLOW SYNAPSES 6 2 ANALYSIS OF THE REDUCED MODEL 6 3 SPATIAL MODELS PHYSIOLOGY AND MATHEMATICAL MODELING OF THE AUDITORY SYSTEM ALLA BORISYUK 1 1 AUDITORY SYSTEM AT A GLANCE 1 2 SOUND CHARACTERISTICS 2 PERIPHERAL AUDITORY SYSTEM 2 1 OUTER EAR 2 2 MIDDLE EAR 2 3 INNER EAR COCHLEA HAIR CELLS 2 4 MATHEMATICAL MODELING OF THE PERIPHERAL AUDITORY SYSTEM 3 AUDITORY NERVE (AN) 3 1 AN STRUCTURE 3 2 RESPONSE PROPERTIES 3 3 HOW IS AN ACTIVITY USED BY BRAIN? 3 4 MODELING OF THE AUDITORY NERVE 4 COCHLEAR NUCLEI 4 1 BASIC FEATURES OF THE CN STRUCTURE 4 2 INNERVATION BY THE AUDITORY NERVE FIBERS 4 3 MAIN CN OUTPUT TARGETS 4 4 CLASSI?CATIONS OF CELLS IN THE CN 4 5 PROPERTIES OF MAIN CELL TYPES 4 6 MODELING OF THE COCHLEAR NUCLEI 5 SUPERIOR OLIVE SOUND LOCALIZATION, JE?RESS MODEL 5 1 MEDIAL NUCLEUS OF THE TRAPEZOID BODY (MNTB) 5 2 LATERAL SUPERIOR OLIVARY NUCLEUS (LSO) 5 3 MEDIAL SUPERIOR OLIVARY NUCLEUS (MSO) 5 4 SOUND LOCALIZATION COINCIDENCE DETECTOR MODEL 6 MIDBRAIN 6 1 CELLULAR ORGANIZATION AND PHYSIOLOGY OF MAMMALIAN IC 6 2 MODELING OF THE IPD SENSITIVITY IN THE INFERIOR COLLICULUS 7 THALAMUS AND CORTEX