Abstract
The spike train activity of neurones is considered as a point process, and methods of analysing and interpreting recorded spike trains are considered. The generation of a continuous process (membrane noise) from interacting point processes is described.
Similar content being viewed by others
Literature
Begenisich, T. and C. F. Stevens. 1975. “How many Conductance States do Potassium Channels have?.”Biophys. J. 15, 843–846.
Brillinger, D. R. 1978. “Comparative Aspects of the Study of Ordinary Time Series and Point Processes.” InDevelopments in Statistics, Ed. P. R. Krinshnaiah, Vol. 1, pp. 33–133. New York: Academic Press.
Chen, Y. 1976. “Differentiation of Channel Models by Noise Analysis”Biophys. J. 16, 965–971.
— and T. L. Hill. 1973. “Fluctuations and Noise in Kinetic Systems. Application to K+ Channels in the Squid Axon.”Biophys. J. 13, 1276–1295.
Çinlar, E. 1972. “Superposition of Point Processes.” InStochastic Point Processes: Statistical Analysis, Theory and Applications, Ed. P. A. W. Lewis, 549–606. New York: Wiley.
Conti, F. and E. Neher. 1980. “Single Channel Recordings of K+ Currents in Squid Axons.”Nature, Lond 285, 140–143.
— and E. Wanke. 1975. “Channel Noise in Nerve Membrane and Lipid Bilayers.”Q. Rev. Biophys. 8, 451–506.
Cox, F. R. 1962.Renewal Theory. London: Methuen.
— and P. A. W. Lewis. 1966.The Statistical Analysis of Series of Events. London: Methuen.
Daley, D. J. and D. Vere-Jones. 1972. “A Summary of the Theory of Stochastic Point Processes.” InStochastic Point Processes—Statistical Analysis, Theory and Applications, Ed. P. A. W. Lewis, pp. 299–383. New York: Wiley Interscience.
Enright, J. T. 1967. “The Spontaneous Neuron Subject to Tonic Stimulation.”J. theor. Biol. 16, 54–77.
Fatt, P. and B. Katz. 1952. “Spontaneous Subthreshold Activity at Motor Nerve Endings.”J. Physiol., Lond 117, 109–128.
Fienberg, S. E. 1974. “Stochastic Models for Single Neuron Firing Trains: a Survey.”Biometrics 30, 399–427.
French, A. S. and A. V. Holden. 1971. “Alias-free Sampling of Neuronal Spike Trains.”Kybernetik 8, 165–171.
Gerstein, G. L. and B. Mandelbrot. 1964. “Random Walk Models for the Spike Activity of a Single Neuron.”Biophys. J. 4, 41–68.
Gnedenko, B. V. and A. N. Kolmogorov. 1966.Limit Distributions for Sums of Independent Random Variables. Trans. K.L. Chung. Reading, MA: Addison Wesley.
Heiden, C. 1969. “Power Spectrum of Stochastic Pulse Sequences with Correlation between Parameters.”Phys. Rev. 188, 319–326.
Hill, T. L. and Y. Chen. 1972. “On the Theory of Ion Transport across the Nerve Membrane, IV. Noise from Open-Close Kinetics of K+ Channels.”Biophys. J. 12, 948–959.
Hodgkin, A. L. and A. F. Huxley. 1952. “A Quantitative Description of Membrane Current and its Application to Conduction and Excitation in Nerve.”J. Physiol., Lond. 117, 500–544.
Holden, A. V. 1975. “A Note on Convolution and Stable Distributions in the Nervous System.”Biol. Cybernet. 20, 171–174.
—. 1976a.Lecture Notes in Biomathematics 12 Models of the Stochastic Activity of Neurones. Berlin: Springe-Verlag.
—. 1976b. “Flicker Noise and Structural Changes in Nerve Membrane.”J. theor. Biol. 57, 243–246.
—. 1978. “Information Transfer in a Chain of Model Neurones.” InProgress in Cybernetics and Systems Research III. General Systems Methodology, Fuzzy Mathematics and Fuzzy Systems, Biocybernetics and Theoretical Neurobiology, Ed. R. Trappl, G. J. Klir and L. Ricciardi, pp. 558–569. Washington: Hemisphere.
—. 1982a. “The Mathematics of Excitation.” InMathematics Studies 58. Biomathematics in 1980, Ed. L. M. Ricciardi and A. C. Scott, pp. 15–47. Amsterdam: North Holland.
—. 1982b. “On the Variability of Pacemaker Activity.”Progress in Cybernetics and Systems Research IX. Cybernetics in Biology and Medicine, Cognition and Learning Health Care Systems, Ed. R. Trappl, L. M. Ricciardi and G. Pask, pp. 139–145. Washington: Hemisphere.
—. 1982c. “Stochastic, Quantal Membrane Conductances and Neuronal Function.” InLecture Notes in Biomathematics 45. Competition and Cooperation in Neutral Nets, Ed. S. I. Amari and M. Arbib, pp. 121–140. Berlin: Springer-Verlag.
— and S. M. Ramadan. 1979. “Stable Distributions of Interspike Interval obtained from Neurones of the Pond-snail.”J. Physiol., Lond 290, 26–29P.
— and —. 1980. “Identification of Endogenous and Exogenous Activity in a Molluscan Neurone by Spike Train Analysis.”Biol. Cybernet. 37, 107–114.
— and J. E. Rubio. 1976. “A Model for Flicker Noise in Nerve Membranes.”Biol. Cybernet. 24, 227–236.
— and —. 1978. “Retardation Currents in Excitable Membranes and Models of Flicker Noise.”Biol. Cybernet. 30, 45–54.
Khintchine, A. Ya. and Levy, P. 1936. “Sur les Lois Stable.”C. r. hebd. Séanc Acad. Sci. Paris 202, 374–382.
Lawrence, A. J. 1971. “Selective Interaction of a Poisson and Renewal Process: the Dependence Structure of the Intervals between Responses.”J. appl. Prob. 8, 170–183.
Levy, P. 1940. “Sur Certain Processes Stochastiques Homogenes.”Comp. Math. 7, 283.
Lukes, T. 1961. “The Statistical Properties of Sequences of Stochastic Pulses.”Proc. Phys. Soc. Lond. 78, 153–168.
Matthes, K. 1972. “Infinitely Divisible Point Processes.” InStochastic Point Processes—Statistical Analysis, Theory and Applications, Ed. P. A. W. Lewis, pp. 384–404. New York: Wiley.
—, J. Kerstan and J. Mecke. 1978.Infinitely Divisible Point Processes. Chichester Wiley.
Matthews, P. B. C. and R. B. Stein. 1969. “The Regularity of Primary and Secondary Muscle Spindle Afferent Discharges.”J. Physiol. 202, 59–82.
Miller, H. D.. 1967. “A Note on Passage Times and Infinitely Divisible Distributions.”J. appl. Prob. 4, 402–405.
—. 1979. “Infinite Divisibility in Stochastic Processes.”Ann. Probabil. 7, 406–417.
Murthy, V. K., 1974.The General Point Process. Reading, MA: Addison Wesley.
Nakahama, H., N. Ishii, and M. Yamamoto 1972. “Markov Process of Maintained Impulse Activity in Central Single Neurones.”Kybernetik 11, 61–72.
—. and H. Fujii. 1974. “Statistical Inference on Markov Process of Neuronal Impulse Sequences.”Kybernetik 15 47–64.
—. and R. Obata. 1975. “Statistical Dependence as a Measure to Evaluate Markov Properties of Stochastic Point Processes.”Biol. Cybernet. 18, 191–208.
Neher, E. and B. Sakmann. 1976. “Noise Analysis of Drug Induced Voltage Clamp Currents in Denervated Frog Muscle Fibres.”J. Physiol. 258, 705–729.
Neher, E. and C. F. Stevens, 1977. “Conductance Fluctuations and Ionic Pores in Membranes.”Ann. Rev. Biophys. Bioeng. 6, 345–381.
Nelsen, D. E. 1964. “Calculations of Power Density Spectra for a Class of Randomly Jittered Waveforms.”M.I.T.-Q.P.R.-R.L.E. 74, 168–179.
Nyquist, H. 1928. “Thermal Agitation of Electric Charge in Conductors.”Phys. Rev. 32, 110–113.
Ricciardi, L. M. 1977.Lecture Notes in Biomathematics 14. Diffusion Processes and Related Topics in Biology. Berlin: Springer-Verlag.
Rodieck, R. W., N.Y-S. Kiang and G.L. Gerstein. 1962. “Some Quantitative Methods for the Study of Spontaneous Activity of Single Neurons.”Biophys. J. 2, 351–368.
Rushton, W. A. H. 1961. “Peripheral Coding in the Nervous System.” InSensory Communication, Ed. W. A. Rosenblith, pp. 169–182. Cambridge, MA: M.I.T. Press.
Sampath, G. and S. K. Srinivasan. 1977.Lecture Notes in Biomathematics 16. Stochastic Models for Spike Trains of Single Neurons. Berlin: Springer-Verlag.
Schick, K. L. 1974. “Power Spectra of Pulse Sequences and Implications for Membrane Fluctuations.”Acta biotheor. 23, 1–17.
Scott, A. C. 1975. “The Electrophysics of a Nerve Fiber.”Rev. Mod. Phys. 47, 487–533.
Shapley, R. 1969. “Fluctuations in the Response to Light of Visual Neurones inLimulus.”Nature, Lond. 221 437–440.
Srinivasan, S. K. 1974.Stochastic Point Processes and their Applications. London: Griffin.
Stevens, C. F. 1972. “Inferences about Membrane Properties from Electrical Noise Measurements.”Biophys. J. 12, 1028–1047.
Verveen, A. A. and L. J. DeFelice. 1974. “Membrane Noise.”Prog. Biophys. mol. Biol. 28, 189–265.
Walløe, L. 1970. “On the Transmission of Information through Sensory Neurones.”Biophys. J. 10, 745–763.
— J. K. S. Jansen and K. Nygaard. 1969. “A Computer Simulated Model of a Secondorder Sensory Neurone.”Kybernetik 6, 130–141.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Holden, A.V. Stochastic processes in neurophysiology: Transformation from point to continuous processes. Bltn Mathcal Biology 45, 443–465 (1983). https://doi.org/10.1007/BF02459582
Issue Date:
DOI: https://doi.org/10.1007/BF02459582