Models of the stochastic activity of neurones. Lecture notes in biomathematics; 12.

These notes have grown from a series of seminars given at Leeds between 1972 and 1975. They represent an attempt to gather together the different kinds of model which have been proposed to account for the stochastic activity of neurones, and to provide an introduction to this area of mathematical biology. A striking feature of the electrical activity of the nervous system is that it appears stochastic: this is apparent at all levels of recording, ranging from intracellular recordings to the electroencephalogram. The chapters start with fluctuations in membrane potential, proceed through single unit and synaptic activity and end with the behaviour of large aggregates of neurones: L have chgaen this seque~~e\/~~';uggest that the interesting behaviourr~f :the nervous system - its individuality, variability and dynamic forms - may in part result from the stochastic behaviour of its components. I would like to thank Dr. Julio Rubio for reading and commenting on the drafts, Mrs. Doris Beighton for producing the final typescript and Mr. Peter Hargreaves for preparing the figures.

1. Stochastic fluctuations in membrane potential.- 1.1 Thermal or Johnson-Nyquist Noise.- 1.2 Shot Noise.- 1.3 Flicker Noise.- 1.4 Conductance fluctuations.- 1.5 References.- 2. Quantal Fluctuations in Generator Potential.- 2.1 Psychophysical Evidence for Quantal Sensitivity in Vision.- 2.2 Some Neural Machines.- 2.3 Fluctuations in the Limulus Eccentric Cell Generator Potential.- 2.4 References.- 3. Models of Action Potential Initiation.- 3.1 The Logical Neurone.- 3.2 The Perfect Integrator Model.- 3.3 The Leaky Integrator Model.- 3.4 Two-Time Constant Models.- 3.5 The Hodgkin-Huxley Equations.- 3.6 Reduced Hodgkin-Huxley Equations.- 3.7 References.- 4. Fluctuations in Excitability.- 4.1 Single Time Constant Models with Fluctuating Threshold.- 4.2 Two-Time Constant Models with Fluctuating Threshold.- 4.3 Fluctuations in Hodgkin-Huxley variables.- 4.4 Models with Time-varying Threshold.- 4.5 References.- 5. Statistical Properties of a Spike Train.- 5.1 Stationarity.- 5.2 Interval Densities and Distributions.- 5.3 Spectral Densities of Point Processes.- 5.4 Input-Output Relations.- 5.5 References.- 6. Random Walk Models.- 6.1 The Random Walk Model.- 6.2 The Probability of First Passage.- 6.3 The Interval Probability Density.- 6.4 A Diffusion Approximation.- 6.5 Introduction of a Reflecting Barrier.- 6.6 Generalization of the Random Walk as a Birth and Death Process.- 6.7 Birth and Death Process Model of the Leaky Integrator.- 6.8 More Complex Random Walks.- 6.9 Some Comments on the Random Walk Models.- 6.10 References.- 7. Diffusion Models.- 7.1 The Input Process.- 7.2 The Wiener Process.- 7.3 The Ornstein-Uhlenbeck Process.- 7.4 The Diffusion Equations.- 7.5 The First Passage Time Distribution.- 7.6 The SIPIT Model.- 7.7 The SILIT Model.- 7.8 The Inverse Approach to Diffusion Models.- 7.9 References.- 8. Superposition Models.- 8.1 Some General Properties of Superposed Processes.- 8.2 Superposition of Regular Generator Trains.- 8.4 Some Applications of Superposition Theory.- 8.5 References.- 9. Collision Models.- 9.1 The Mechanism of Collision.- 9.2 The Geometry of Collision.- 9.3 Mutual Interaction Between Two Non-adjacent Generator Sites.- 9.4 The Biology of Collision.- 9.5 References.- 10. Gating and Selective Interaction Models.- 10.1 Compound Exponential Distributions and Gating Models.- 10.2 Selective Interaction Models.- 10.3 Models with Sub-threshold Interactions.- 10.4 Neural Micro-nets with Reciprocal Inhibition.- 10.5 References.- 11. Models of Synaptic Transmission.- 11.1 The neuro-muscular junction.- 11.2 Spontaneous miniature end-plate potentials.- 11.3 Presynaptic depolarization enhanced release.- 11.4 Models of facilitation and depression.- 11.5 Transmitter induced fluctuations in conductance.- 11.6 Dynamics of signal transfer across a synapse.- 11.7 References.- 12. Models of the stochastic activity of neural aggregates.- 12.1 Nets of logical neurones.- 12.2 Networks continuous in time and space: a field approach.- 12.3 Temporal behaviour of nets of excitatory and inhibitory neurones.- 12.4 Spatially distributed excitatory and inhibitory subpopulations.- 12.5 Some qualitative speculations.- 12.6 References.- 13. Information transmission by model neurones.- 13.1 The axon as a channel.- 13.2 Information preservation in a neural chain.- 13.3 Signal detection.- 13.4 Tranmission of steady signals using a rate code.- 13.5 Information transmission of periodic signals.- 13.6 Transmission of stochastic, time-varying signals.- 13.7 Transinformations of different codes.- 13.8 References.