Skip to main content
Log in

Models of classical conditioning

  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

Classical conditioning is a basic form of associative learning in the animal kingdom. Many paradigmatic features of classical conditioning appear to be conserved throughout species and phyla and are independent of stimulus nature. This paper presents an analysis of trial-based and real-time models of classical conditioning which are mathematical abstractions of the underlying processing principles. Various models are reviewed and in a formal analysis, their capability of simulating and explaining classical conditioning is investigated. Since every existing model fails to simulate some particular conditioning phenomena and since some modelling approaches are not appropriate for detailed mathematical analysis, new model components will be introduced that overcome most of the weaknesses observed in the other models.

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.

Similar content being viewed by others

References

  • Atkinson, R. C. and W. K. Estes (1963). Stimulus sampling theory, in Handbook of Mathematical Psychology, R. D. Luce, R. B. Bush and E. Galanter (Eds), Vol. 3, New York: Wiley, pp. 121–268.

    Google Scholar 

  • Barto, A. G., R. S. Sutton and C. W. Anderson (1983). Neuronlike adaptive elements that can solve difficult learning control problems. IEEE Trans. Syst., Man Cybern. 13, 834–846.

    Google Scholar 

  • Bitterman, M. E., R. Menzel, A. Fietz and S. Schäfer (1983). Classical conditioning of proboscis extension in honeybees (Apis mellifera). J. Comp. Psychol. 97, 107–119.

    Article  Google Scholar 

  • Byrne, J. H. and K. J. Gingrich (1989). Mathematical model of cellular and molecular processes contributing to associative and nonassociative learning in aplysia, in Neural Models of Placticity: Experimental and Theoretical Approaches, J. H. Byrne and W. O. Berry (Eds), London: Academic Press, Chap. 4, pp. 58–72.

    Google Scholar 

  • Donegan, N. H., M. A. Gluck and R. F. Thompson (1989). Intergrating behavioural and biological models of classical conditioning, in Computational Models of Learning in Simple Neural Systems, Hawkins, R. D. and G. H. Bower, (Eds) (The Psychology of Learning and Motivation Vol. 23), New York: Academic Press, pp. 109–156.

    Google Scholar 

  • Gelperin, A., J. J. Hopfield and D. W. Tank (1986). The logic of limax learning, in Model Neural Networks and Behavior, A. I. Selverston (Ed.), New York: Plenum Press, Chap. 13, pp. 237–261.

    Google Scholar 

  • Grossberg, S. (1987). Competitive learning: from interactive activation to adaptive resonance. Cognit. Sci. 11, 23.

    Article  Google Scholar 

  • Hammer, M. and R. Menzel (1995). Learning and memory in the honeybee. J. Neurosci. 15, 1617–1630.

    Google Scholar 

  • Hawkins, R. D. (1989). A simple circuit model for higher-order features of classical conditioning, in Neural models of plasticity, J. H. Byrne and W. O. Berry (Eds), Orlando: Academic Press, Chap. 5, pp. 73–92.

    Google Scholar 

  • Hebb, D. O. (1949). The Organization of Behavior, New York: Wiley.

    Google Scholar 

  • Hellstern, F., R. Malaka and M. Hammer (1997). Backward inhibitory learning in honeybees. Manuscript for Learning and Memory.

  • Hopfield, J. J. (1982). Neural networks and physical systems with emergent collective computational abilities. Proc. Natl Acad. Sci. USA 79, 2554.

    Article  MathSciNet  Google Scholar 

  • Klein, J. (1993). Lernabhängiges Timing der konditionierten Reaktion bei der olfaktorischen Konditionierung der Honigbiene, Apis mellifera. Diplomarbeit, FU Berlin.

    Google Scholar 

  • Klopf, A. H. (1989). Classical conditioning: phenomena predicted by a drive-reinforcement model of neural function, in Neural Models of Placticity: Experimental and Theoretical Approaches, J. H. Byrne and W. O. Berry (Eds), London: Academic Press, Chap. 6, pp. 94–103.

    Google Scholar 

  • Klopf, A. H. and J. S. Morgan (1990). The role of time in natural intelligence: implications of classical and instrumental conditioning for neuronal and neural-network modeling, in Learning and Computational Neuroscience: Foundations and Adaptive Networks, A. Gabriel and J. Moore (Eds), Cambridge: MIT Press, Chap. 11, pp. 463–495.

    Google Scholar 

  • Macintosh, N. J. (1974). The Psychology of Animal Learning. New York, Oxford: Oxford University Press.

    Google Scholar 

  • Macintosh, N. J. (1983). Conditioning and Associative Learning. New York, Oxford: Oxford University Press.

    Google Scholar 

  • Malaka, R. and M. Hammer (1996). Real-time models of classical conditioning, in Proceedings of the International Conference on Neural Networks (ICNN’96), Washington, Vol. 2, Piscataway, NJ: IEEE Press, pp. 768–773.

    Google Scholar 

  • Malaka, R., R. Lange and M. Hammer (1995a). A comparative study of real-time models for classical conditioning. In Learning and Memory: Proceedings of the 23rd Göttingen Neurobiology Conference, Vol. 1, N. Elser and R. Menzel (Eds), New York: Thieme-Verlag, Stuttgart, p. 74.

    Google Scholar 

  • Malaka, R., R. Lange and M. Hammer (1995b). A constant prediction model for classical conditioning, in Learning and Memory: Proceedings of the 23rd Göttingen Neurobiology Conference, Vol. 1, N. Elser and R. Menzel (Eds), New York: Thieme-Verlag, p. 75.

    Google Scholar 

  • Menzel, R. (1990). Learning, memory and ‘cognition’ in honey bees, in Neurobiology of Comparative Cognition, R. P. Kesner and D. S. Olton (Eds), Hillsdale, NJ: Lawrence Erlbaum Ass. Publ.

    Google Scholar 

  • Menzel, R., M. Hammer, G. Braun, J. Mauelshagen and M. Sugawa (1991). Neurobiology of learning and memory in honeybees, in The Behaviour and Physiology of Bees, L. J. Goodman and R. C. Fisher (Eds), Oxon, U.K.: CAB International, Chap. 22, pp. 323–353.

    Google Scholar 

  • Montague, P. R. and T. J. Sejnowski (1994). The predictive brain: temporal coincidence and temporal order in synaptic learning mechanisms. Learning and Memory 1, 1–33.

    Google Scholar 

  • Moore, J. W., N. E. Berthier and D. E. J. Blazis (1990). Classical eye-blink conditioning: brain systems and implementation of a computational model, in Learning and Computational Neuroscience: Foundations and Adaptive Networks, A. Gabriel and J. Moore (Eds), London: MIT Press, Chap. 8, pp. 359–387.

    Google Scholar 

  • Moore, J. W. and D. E. J. Blazis (1989). Simulation of a classical conditioned response: a cerebellar neural network implementation of the Sutton-Barto-Desmond model, in Neural Models of Placticity: Experimental and Theoretical Approaches, J. H. Byrne and W. O. Berry (Eds), London: Academic Press, Chap. 11, pp. 187–207.

    Google Scholar 

  • Pavlov, I. P. (1927). Conditioned Reflexes. London: Oxford University Press.

    Google Scholar 

  • Pearce, J. M. (1987). A model for stimulus generalization in Pavlovian conditioning. Psychol. Rev. 94, 61–73.

    Article  Google Scholar 

  • Pearce, J. M. and G. Hall (1980). A model for Pavlovian learning: variations in the effectiveness of conditioned but not of unconditioned stimuli. Psychol. Rev. 87, 532–552.

    Article  Google Scholar 

  • Rescorla, R. A. (1977). Pavlovian second-order conditioning: some implications for instrumental behaviour, in Operant Pavlovian Interactions, H. Davis and H. M. B. Hurwitz (Eds), Hillsdale, NJ: Erlbaum.

    Google Scholar 

  • Rescorla, R. A. (1988). Behavioral studies of Pavlovian conditioning. Ann. Rev. Neuroscience 11, 329–352.

    Article  Google Scholar 

  • Rescorla, R. A. and A. R. Wagner (1972). A theory of Pavlovian conditioning: variations in the effectiveness of reinforcement and non-reinforcement, in Classical conditioning: Current Theory and Research, Vol. 2, A. H. Black and W. F. Prokasy (Eds), Appleton-Century-Crofts, pp. 64–99.

  • Rumelhart, D. E., G. E. Hinton and R. J. Williams (1986). Learning internal represantations by error propagation, in Parallel Distributed Processing: Explorations in the Microstructures of Cocnition, Vol. 1, D. E. Rumelhart and J. L. McClelland (Eds), London: MIT Press, Chap. 8, pp. 318–362.

    Google Scholar 

  • Sejnowski, T. J. and G. Tesauro (1989). The Hebb rule for synaptic plasticity: algorithms and implementations, in Neural Models of Placticity: Experimental and Theoretical Approaches, J. H. Byrne and W. O. Berry (Eds), London: Academic Press, Chap. 6, pp. 94–103.

    Google Scholar 

  • Smith, B. H., C. I. Abramson and T. R. Tobin (1991). Conditional withholding of proboscis extension in honey bees (apis mellifera) during discriminative punishment. J. Comp. Psychol. 105, 345–356.

    Article  Google Scholar 

  • Sutton, R. S. and A. G. Barto (1981). Toward a modern theory of adaptive networks: expectation and prediction. Psychol. Rev. 88, 135–170.

    Article  Google Scholar 

  • Sutton, R. S. and A. G. Barto (1990). Time-derivative models of Pavlovian reinforcement, in Learning and Computational Neuroscience: Foundations and Adaptive Networks, A. Gabriel and J. Moore (Eds), London: MIT Press, Chap. 12, pp. 497–537.

    Google Scholar 

  • Sutton, R. S., A. G. Barto and R. J. Williams (1991). Reinforcement learning is direct adaptive optimal control. to appear in Proceedings of the 1991 American Control Conference.

  • Tesauro, G. (1990). Neural models of classical conditioning: A theoretical viewpoint, in Connectionist Modelling and Brain Function: The Developing Interface, S. J. Hanson and C. R. Olson (Eds), London: MIT Press, Chap. 4, pp. 74–104.

    Google Scholar 

  • Thompson, R. F. (1989). Neural circuit for classical conditioning of eyelid closure response, in Neural Models of Placticity: Experimental and Theoretical Approaches, J. H. Byrne and W. O. Berry (Eds), London: Academic Press, Chap. 9, pp. 160–177.

    Google Scholar 

  • Wagner, A. R. (1981). SOP: A model of automatic memory processing in animal behaviour, in Information Processing in Animals: Memory Mechanisms, Vol. 85, N. E. Spear and R. R. Miller (Eds), Hillsdale, NJ: Erlbaum, Chap. 1, pp. 5–44.

    Google Scholar 

  • Wagner, A. R. and N. H. Donegan (1989). Some relationships between a computational model (SOP) and a neural circuit for Pavlovian (rabbit eyeblink) conditioning, in Computational Models of Learning in Simple Neural Systems, (The Psychology of Learning and Motivation Vol. 23), R. D. Hawkins and G. H. Bower (Eds), London: Academic Press, pp. 157–203.

    Google Scholar 

  • Wagner, A. R. and M. B. Larew (1985). Opponent processes and Pavlovian inhibition, in Information Processing in Animals: Conditioned Inhibition, R. R. Miller and N. E. Spear (Eds), Hillsdale, NJ: Erlbaum, pp. 233–265.

    Google Scholar 

  • Wagner, A. R. and R. A. Rescorla (1972). Inhibition in Pavlovian conditioning: application of a theory, in Inhibition and Learning, R. A. Boakes and M. S. Halliday (Eds), London: Academic Press, pp. 301–336.

    Google Scholar 

  • Widrow, G. and M. E. Hoff (1960). Adaptive switching circuits, in Western Electronic Show and Convention, Convention Record, Vol. 4, pp. 96–194.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Malaka, R. Models of classical conditioning. Bull. Math. Biol. 61, 33–83 (1999). https://doi.org/10.1006/bulm.1998.9998

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1006/bulm.1998.9998

Keywords

Navigation