Skip to main content
Log in

Improved neuronal models for studying neural networks

Kybernetik Aims and scope Submit manuscript

Abstract

Previous neuronal models used for the study of neural networks are considered. Equations are developed for a model which includes: 1) a normalized range of firing rates with decreased sensitivity at large excitatory or large inhibitory input levels, 2) a single rate constant for the increase in firing rate following step changes in the input, 3) one or more rate constants, as required to fit experimental data for the adaptation of firing rates to maintained inputs. Computed responses compare well with the types of neuronal responses observed experimentally. Depending on the parameters, overdamped increases and decreases, damped oscillatory or maintained oscillatory changes in firing rate are observed to step changes in the input. The integrodifferential equations describing the neuronal models can be represented by a set of first-order differential equations. Steady-state solutions for these equations can be obtained for constant inputs, as well as the stability of the solutions to small perturbations. The linear frequency response function is derived for sufficiently small time-varying inputs. The linear responses are also compared with the computed solutions for larger non-linear responses.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  • Agin,D.: Hodkin-Huxley equations: logarithmic relation between membrane current and frequency of repetitive activity. Nature (Lond.) 201, 625–626 (1964)

    Google Scholar 

  • Andersen,P., Andersson,S.A.: Physiological basis of the alpha rhythm. New York: Appleton-Century-Croft 1968

    Google Scholar 

  • Brown,M.C., Stein,R.B.: Quantitative studies on the slowly adapting stretch receptors of the crayfish. Kybernetik 3, 175–185 (1966)

    Google Scholar 

  • Bush,B.M.H., Roberts,A.: Resistance reflexes from a crab muscle receptor without impulses. Nature (Lond.) 218, 1171–1173 (1968)

    Google Scholar 

  • Calvin,W.H., Schwindt,P.C.: Steps in production of motoneuron spikes during rhythmic firing. J. Neurophysiol. 35, 297–310 (1972)

    Google Scholar 

  • Chandler,W.K., Meves,H.: Evidence for two types of sodium conductance in axons perfused with sodium fluoride solution. J. Physiol. (Lond.) 211, 653–678 (1970)

    Google Scholar 

  • Chen,C.F., Baumgarten,R. von, Takeda,R.: Pacemaker properties of completely isolated neurones in Aplysia California. Nature (Lond.) New Biol. 233, 27–29 (1971)

    Google Scholar 

  • Chorafas,D.N.: Systems and simulation. New York: Academic Press 1965

    Google Scholar 

  • Connor,J.A., Stevens,C.F.: Prediction of repetitive firing behaviour from voltage clamp data on an isolated neurone soma. J. Physiol. (Lond.) 213, 31–54 (1971)

    Google Scholar 

  • Cowan, J.D.: Ph. D. Thesis, University of London 1967

  • Cowan, J.D.: A statistical mechanics of neurons activity. In: Gerstenhaber, M., Providence, R.I. Eds.: Some mathematical questions in biology, pp. 1–58. American Mathematical Society 1970

  • D'Azzo,J.J., Houpis,C.H.: Feedback control systems analysis and synthesis. New York: McGraw-Hill 1966

    Google Scholar 

  • Frankenhaeuser,B., Huxley,A.F.: The action potential in the myelinated nerve fiber of Xenopus laevis as computed on the basis of voltage clamp data. J. Physiol. (Lond.) 171, 302–215 (1964)

    Google Scholar 

  • Gainer,H.: Electrophysiological behavior of an endogenously active neurosecretory cell. Brain Res. 39, 403–418 (1972)

    Google Scholar 

  • Griffith,J.S.: On the stability of brain-like structures. Biophys. J. 3, 229–308 (1963)

    Google Scholar 

  • Hodgkin,A.L.: The local electrical changes associated with repetitive action in a non-medullated axon. J. Physiol. (Lond.) 107, 165–181 (1948)

    Google Scholar 

  • Hodgkin,A.L., Horowicz,P.: Potassium contractures in single muscle fibres. J. Physiol. (Lond.) 153, 386–403 (1960)

    Google Scholar 

  • Hodgkin,A.L., Huxley,A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. (Lond.) 117, 500–544 (1952)

    Google Scholar 

  • Hodgman,C.D. (Ed.): C.R.C. Standard mathematical tables. Cleveland: Chemical Rubber Publishing Company 1972

    Google Scholar 

  • Jack,J.J.B., Redman,S.J.: An electrical description of the motoneurone, and its application to the analysis of synaptic potentials. J. Physiol. (Lond.) 215, 321–352 (1972)

    Google Scholar 

  • Kernell,D.: The adaptation and the relation between discharge frequency and current strength of cat lumbosacral motoneurones stimulated by longlasting injected currents. Acta physiol. scand. 65, 65–73 (1965)

    Google Scholar 

  • Knight,B.W., Toyoda,J.-I., Dodge,F.A.: A quantitative description of the dynamics of excitation and inhibition in the eye of Limulus. J. gen. Physiol. 56, 421–431 (1970)

    Google Scholar 

  • Leung, K.V., Mangeron, D., Oğuztöreli, M.N., Stein, R.B.: On the stability and numerical solutions of two neural models. Utilitas mathematica (in press, 1973)

  • Leung, K.V., Mangeron, D., Oğuztöreli, M.N., Stein, R. B.: On a class of non-linear integro-differential equations. I. Rend. Accad. Naz. Lincei (in press, 1973)

  • Leung, K.V., Mangeron, D., Oğuztöreli, M.N., Stein, R.B.: On a class of non-linear integro-differential equations. II. Rend. Accad. Naz. Lincei (in press, 1973)

  • Lippold,O.C.J.: Oscillations in the stretch reflex arc and the origin of the rhythmical 8–12 c/s component of physiological tremor. J. Physiol. (Lond.) 206, 359–382 (1970)

    Google Scholar 

  • Magleby,K.L., Stevens,C.F.: A quantitative description of end-plate currents. J. Physiol. (Lond.) 233, 151–172 (1972)

    Google Scholar 

  • Matthews,P.B.C., Stein,R.B.: The sensitivity of muscle spindle afferents to small sinusoidal changes of length. J. Physiol. (Lond.) 200, 723–743 (1969)

    Google Scholar 

  • McCulloch,W.S., Pitts,W.H.: A logical calculus of the ideas immanent in nervous activity. Bull. math. Biophys. 5, 115–133 (1943)

    Google Scholar 

  • Mendelson,M.: Oscillator neurons in crustacean ganglia. Science 171, 1170–1173 (1971)

    Google Scholar 

  • Milsum,J.H.: Biological control systems analysis. New York: McGraw-Hill 1966

    Google Scholar 

  • Moore,G.P., Perkel,D.H., Segundo,J.P.: Statistical analysis and functional interpretation of neuronal spike data. Ann. Rev. Physiol. 28, 443–522 (1966)

    Google Scholar 

  • Morishita,I., Yajima,A.: Analysis and simulation of networks of mutually inhibiting neurons. Kybernetik 11, 154–165 (1972)

    Google Scholar 

  • Oğuztöreli,M.N.: On the neural equations of Cowan and Stein. Utilitas Mathematica 2, 305–317 (1972)

    Google Scholar 

  • Purple,R.L., Dodge,F.A.: Self-inhibition in the eye of Limulus. In: Bernhard,C.G. (Ed.): Functional organization of the compound eye. Oxford: Pergamon Press 1966

    Google Scholar 

  • Rall,W.: Time constants and electronic length of membrane cylinders and neurons. Biophys. J. 9, 1483–1508 (1969)

    Google Scholar 

  • Ruch,T.C, Patton,H.D.: Physiology and biophysics. Philadelphia: Saunders 1966

    Google Scholar 

  • Shapiro,B.I., Lenherr,F.K.: Hodgkin-Huxley axon: increased modulation and linearity of response to constant current stimulus. Biophys. J. 12, 1145–1158 (1972)

    Google Scholar 

  • Sokolnikoff,I.S., Redheffer,R.M.: Mathematics of physics and modern engineering. New York: McGraw-Hill 1958

    Google Scholar 

  • Sokolove,P.: Computer simulation of after-inhibition in crayfish slowly adapting stretch receptor. Biophys. J. 12, 1429–1451 (1972)

    Google Scholar 

  • Stein,R.B.: The frequency of nerve action potentials generated by applied currents. Proc. Roy. Soc. (Lond.) Ser. B. 167, 64–86 (1967)

    Google Scholar 

  • Stein,R.B.: The stochastic properties of spike trains recorded from nerve cells. In: Lewis,P.A.W. (Ed.): Stochastic point processes pp. 700–731. New York: Wiley 1972

    Google Scholar 

  • Stein,R.B., Leung,K.V., Oğuztöreli,M.N., Williams,D.W.: Properties of small neural networks. Kybernetik 14, 223–230 (1974)

    Google Scholar 

  • Strumwasser,F.: Types of information stored in single neurons. In: Wiersma,C.A.G. (Ed.): Invertebrate nervous systems. Chicago: University of Chicago Press 1967

    Google Scholar 

  • Takahashi,K., Kubota,K., Udo,M.: Recurrent facilitation in cat pyramidal tract cells. J. Neurophysiol. 30, 22–34 (1967)

    Google Scholar 

  • Thomas,R.C.: Membrane current and intracellular sodium changes in a snail neurone during extrusion of injected sodium. J. Physiol. (Lond.) 201, 495–514 (1969)

    Google Scholar 

  • Werblin,F.S., Dowling,J.E.: Organization of the retina of the mudpuppy, Necturus maculosus. II. Intracellular recording. J. Neurophysiol. 32, 339–355 (1969)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Stein, R.B., Leung, K.V., Mangeron, D. et al. Improved neuronal models for studying neural networks. Kybernetik 15, 1–9 (1974). https://doi.org/10.1007/BF00270755

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00270755

Keywords

Navigation