Spike-timing-dependent plasticity for neurons with recurrent connections


The dynamics of the learning equation, which describes the evolution of the synaptic weights, is derived in the situation where the network contains recurrent connections. The derivation is carried out for the Poisson neuron model. The spiking-rates of the recurrently connected neurons and their cross-correlations are determined self- consistently as a function of the external synaptic inputs. The solution of the learning equation is illustrated by the analysis of the particular case in which there is no external synaptic input. The general learning equation and the fixed-point structure of its solutions is discussed.

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

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 199

This is the net price. Taxes to be calculated in checkout.


  1. Bi GQ, Poo MM (2001) Synaptic modification by correlated activity: Hebb’s postulate revisited. Annu Rev Neurosci 24:139–166

  2. Burkitt AN, van Hemmen JL (2003) How synapses in the auditory system wax and wane: theoretical perspectives. Biol Cybern 89:318–332

  3. Gerstner W, Kempter R, van Hemmen JL, Wagner H (1996) A neuronal learning rule for sub-millisecond temporal coding. Nature 383: 76–78

  4. Hebb DO (1949) The organization of behavior. Wiley, New York

  5. Hawkes AG (1971) Point spectra of some mutually exciting point processes. J Roy Stat Soc B 33:438–443

  6. Kempter R, Gerstner W, van Hemmen JL, Wagner H (1998) Extracting oscillations: neuronal coincidence detection with noisy periodic spike input. Neural Comput 10:1987–2017

  7. Kempter R, Gerstner W, van Hemmen JL (1999) Hebbian learning and spiking neurons. Phys Rev E 59:4498–4514

  8. Kempter R, Leibold C, Wagner H, van Hemmen JL (2001b) Formation of temporal feature maps by axonal propagation of synaptic learning. Proc Natl Acad Sci USA 98(7):4166–4171

  9. Kistler WM, van Hemmen JL (2000) Modeling synaptic plasticity in conjunction with the timing of pre- and postsynaptic action potentials. Neural Comput 12:385–405

  10. Lamperti J (1966) Probability, 2nd edn. Benjamin, New York. (1996) Wiley, New York

  11. Leibold C, Kempter R, van Hemmen JL (2001) Temporal map formation in the barn owl’s brain. Phys Rev Lett 87:248101

  12. Leibold C, Kempter R, van Hemmen JL (2002) How spiking neurons give rise to a temporal-feature map: from synaptic plasticity to axonal selection. Phys Rev E 65:051915

  13. Markram H, Lübke J, Forscher M, Sakmann B (1997) Regulation of synaptic efficacy by coincidence of postsynaptic APs and EPSPs. Science 275:213–215

  14. Meffin H, Besson J, Burkitt AN, Grayden DB (2006) Learning the structure of correlated synaptic subgroups using stable and competitive spike-timing-dependent plasticity. Phys Rev E73:041911

  15. Sanders JA, Verhulst F (1985) Averaging methods in nonlinear dynamical systems. Springer, Berlin

  16. Song F, Miller KD, Abbott LF (2000) Competitive Hebbian learning through spike-timing-dependent synaptic plasticity. Nature Neurosci 3:919-926

  17. Van Hemmen JL (2001) Theory of synaptic plasticity. In: Moss F, Gielen S (eds) Handbook of biophysics, vol 4. Elsevier, Amsterdam, pp 771–823

Download references

Author information

Correspondence to A. N. Burkitt.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Burkitt, A.N., Gilson, M. & van Hemmen, J.L. Spike-timing-dependent plasticity for neurons with recurrent connections. Biol Cybern 96, 533–546 (2007).

Download citation


  • Synaptic Weight
  • Recurrent Network
  • Hebbian Learning
  • Recurrent Connection
  • Output Spike