THE COLLECTED MATHEMATICAL WORKS OF GEORGE WILLIAM HILL

By JESSICA MOORE

CONTENTS INTRODUCTION NEURONS WHAT IS A SPIKE? WHERE IS THE THRESHOLD? WHY ARE NEURONS DI?ERENT, AND WHY DO WE CARE? BUILDING MODELS DYNAMICAL SYSTEMS PHASE PORTRAITS BIFURCATIONS HODGKIN CLASSI?CATION NEUROCOMPUTATIONAL PROPERTIES BUILDING MODELS (REVISITED) REVIEW OF IMPORTANT CONCEPTS BIBLIOGRAPHICAL NOTES ELECTROPHYSIOLOGY OF NEURONS IONS NERNST POTENTIAL IONIC CURRENTS AND CONDUCTANCES EQUIVALENT CIRCUIT RESTING POTENTIAL AND INPUT RESISTANCE VOLTAGE-CLAMP AND I-V RELATION CONDUCTANCES VOLTAGE-GATED CHANNELS ACTIVATION OF PERSISTENT CURRENTS INACTIVATION OF TRANSIENT CURRENTS HYPERPOLARIZATION-ACTIVATED CHANNELS THE HODGKIN-HUXLEY MODEL HODGKIN-HUXLEY EQUATIONS ACTION POTENTIAL PROPAGATION OF THE ACTION POTENTIALS DENDRITIC COMPARTMENTS SUMMARY OF VOLTAGE-GATED CURRENTS EXERCISES ONE-DIMENSIONAL SYSTEMS ELECTROPHYSIOLOGICAL EXAMPLES I-V RELATIONS AND DYNAMICS LEAK + INSTANTANEOUS INA,P GEOMETRICAL ANALYSIS EQUILIBRIA STABILITY EIGENVALUES UNSTABLE EQUILIBRIA ATTRACTION DOMAIN THRESHOLD AND ACTION POTENTIAL BISTABILITY AND HYSTERESIS TOPOLOGICAL EQUIVALENCE LOCAL EQUIVALENCE AND THE HARTMAN-GROBMAN THEOREM SADDLE-NODE (FOLD) BIFURCATION SLOW TRANSITION BIFURCATION DIAGRAM BIFURCATIONS AND I-V RELATIONS QUADRATIC INTEGRATE-AND-FIRE NEURON TWO-DIMENSIONAL SYSTEMS PLANAR VECTOR FIELDS NULLCLINES TRAJECTORIES LIMIT CYCLES RELAXATION OSCILLATORS LOCAL LINEAR ANALYSIS EIGENVALUES AND EIGENVECTORS LOCAL EQUIVALENCE CLASSI?CATION OF EQUILIBRIA EXAMPLE: FITZHUGH-NAGUMO MODEL BISTABILITY AND ATTRACTION DOMAINS STABLE/UNSTABLE MANIFOLDS HOMOCLINIC/HETEROCLINIC TRAJECTORIES SADDLE-NODE BIFURCATION ANDRONOV-HOPF BIFURCATION CONDUCTANCE-BASED MODELS AND THEIR REDUCTIONS MINIMAL MODELS AMPLIFYING AND RESONANT GATING VARIABLES INA,P+IK-MODEL INA,T-MODEL INA,P+IH-MODEL IH+IKIR-MODEL IK+IKIR-MODEL IA-MODEL CA2+-GATED MINIMAL MODELS REDUCTION OF MULTIDIMENSIONAL MODELS HODGKIN-HUXLEY MODEL EQUIVALENT POTENTIALS NULLCLINES AND I-V RELATIONS REDUCTION TO SIMPLE MODEL EQUILIBRIUM (REST STATE) SADDLE-NODE (FOLD) SADDLE-NODE ON INVARIANT CIRCLE SUPERCRITICAL ANDRONOV-HOPF SUBCRITICAL ANDRONOV-HOPF LIMIT CYCLE (SPIKING STATE) FOLD LIMIT CYCLE HOMOCLINIC OTHER INTERESTING CASES THREE-DIMENSIONAL PHASE SPACE CUSP AND PITCHFORK BOGDANOV-TAKENS RELAXATION OSCILLATORS AND CANARDS BAUTIN SADDLE-NODE HOMOCLINIC ORBIT HARD AND SOFT LOSS OF STABILITY NEURONAL EXCITABILITY EXCITABILITY HODGKIN’S CLASSI?CATION CLASSES 1 AND 2 CLASS 3 RAMPS, STEPS, AND SHOCKS BISTABILITY CLASS 1 AND 2 SPIKING INTEGRATORS VS RESONATORS FAST SUBTHRESHOLD OSCILLATIONS FREQUENCY PREFERENCE AND RESONANCE FREQUENCY PREFERENCE IN VIVO THRESHOLDS AND ACTION POTENTIALS THRESHOLD MANIFOLDS RHEOBASE POSTINHIBITORY SPIKE INHIBITION-INDUCED SPIKING SPIKE LATENCY 7 2 10 FLIPPING FROM AN INTEGRATOR TO A RESONATOR 7 2 11 TRANSITION BETWEEN INTEGRATORS AND RESONATORS SLOW MODULATION SPIKE FREQUENCY MODULATION I-V RELATION SLOW SUBTHRESHOLD OSCILLATION REBOUND RESPONSE AND VOLTAGE SAG AHP AND ADP SIMPLE MODELS SIMPLEST MODELS INTEGRATE-AND-FIRE RESONATE-AND-FIRE QUADRATIC INTEGRATE-AND-FIRE SIMPLE MODEL OF CHOICE CANONICAL MODELS CORTEX REGULAR SPIKING (RS) NEURONS INTRINSICALLY BURSTING (IB) NEURONS MULTI-COMPARTMENT DENDRITIC TREE CHATTERING (CH) NEURONS LOW-THRESHOLD SPIKING (LTS) INTERNEURONS FAST SPIKING (FS) INTERNEURONS LATE SPIKING (LS) INTERNEURONS DIVERSITY OF INHIBITORY INTERNEURONS THALAMUS THALAMOCORTICAL (TC) RELAY NEURONS RETICULAR THALAMIC NUCLEUS (RTN) NEURONS THALAMIC INTERNEURONS HIPPOCAMPAL CA1 PYRAMIDAL NEURONS SPINY PROJECTION NEURONS OF NEOSTRIATUM AND BASAL GANGLIA MESENCEPHALIC V NEURONS OF BRAINSTEM STELLATE CELLS OF ENTORHINAL CORTEX MITRAL NEURONS OF THE OLFACTORY BULB BURSTING ELECTROPHYSIOLOGY EXAMPLE: THE INA,P+IK+IK(M)-MODEL FAST-SLOW DYNAMICS CENTRAL PATTERN GENERATORS AND HALF-CENTER OSCILLATORS GEOMETRY FAST-SLOW BURSTERS AVERAGING EQUIVALENT VOLTAGE HYSTERESIS LOOPS AND SLOW WAVES BIFURCATIONS “RESTING ?BURSTING ?TONIC SPIKING” CLASSI?CATION FOLD/HOMOCLINIC CIRCLE/CIRCLE SUBHOPF/FOLD CYCLE FOLD/FOLD CYCLE FOLD/HOPF FOLD/CIRCLE NEUROCOMPUTATIONAL PROPERTIES HOW TO DISTINGUISH? BURSTS AS A UNIT OF NEURONAL INFORMATION CHIRPS SYNCHRONIZATION 10 SYNCHRONIZATION SOLUTIONS TO EXERCISES REFERENCES