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Continuous neural network with windowed Hebbian learning


We introduce an extension of the classical neural field equation where the dynamics of the synaptic kernel satisfies the standard Hebbian type of learning (synaptic plasticity). Here, a continuous network in which changes in the weight kernel occurs in a specified time window is considered. A novelty of this model is that it admits synaptic weight decrease as well as the usual weight increase resulting from correlated activity. The resulting equation leads to a delay-type rate model for which the existence and stability of solutions such as the rest state, bumps, and traveling fronts are investigated. Some relations between the length of the time window and the bump width is derived. In addition, the effect of the delay parameter on the stability of solutions is shown. Also numerical simulations for solutions and their stability are presented.

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  1. Abbott LF, Nelson SB (2000) Synaptic plasticity: taming the beast. Nat Neurosci 3:1178–1183

    Article  CAS  PubMed  Google Scholar 

  2. Amari S (1977) Dynamics of pattern formation in lateral-inhibition type neural fields. Biol Cybern 27:77–87

    Article  CAS  PubMed  Google Scholar 

  3. Baladron J, Fasoli D, Faugeras OD, Touboul J (2011) Mean field description of and propagation of chaos in recurrent multipopulation networks of Hodgkin–Huxley and Fitzhugh–Nagumo neurons. arXiv:1110.4294

  4. Bienenstock EL, Cooper LN, Munro PW (1982) Theory for the development of neuron selectivity: orientation specificity and binocular interaction in visual cortex J Neurosci 2:32–48

  5. Bressloff PC (2009) Lectures in mathematical neuroscience. Mathematical biology, IAS/Park City mathematical series 14, pp 293–398

  6. Bressloff PC (2012) Spatiotemporal dynamics of continuum neural fields. J Phys A Math Theor 45(3):033001

  7. Bressloff PC, Coombes S (2013) Neural bubble dynamics revisited. Cogn Comput 5:281–294

  8. Connell L (2007) Representing object colour in language comprehension. Cognition 102(3):476–485

  9. Coombes H, Schmidt S, Bojak I (2012) Interface dynamics in planar neural field models. J Math Neurosci 2(1):1–27

  10. Coombes P, Beim Graben S, Potthast R (2012b) Tutorial on neural field theory. Springer, Berlin

    Google Scholar 

  11. Coombes S (2005) Waves, bumps, and patterns in neural field theories. Biol Cybern 93:91–108

    Article  CAS  PubMed  Google Scholar 

  12. Coombes S, Owen MR (2007) Exotic dynamics in a firing rate model of neural tissue with threshold accommodation. In: Botelho F, Hagen T, Jamison J (eds) Fluids and waves: recent trends in applied analysis, vol 440. AMS Contemporary Mathematics, pp 123–144

  13. Dayan P, Abbott LF (2003) Theoretical neuroscience: computational and mathematical modeling of neural systems. J Cogn Neurosci 15(1):154–155

  14. Ermentrout GB, Cowan JD (1979) A mathematical theory of visual hallucination patterns. Biol Cybern 34:137–150

    Article  CAS  PubMed  Google Scholar 

  15. Földiák P (1991) Learning invariance from transformation sequences. Neural Comput 3(2):194–200

    Article  Google Scholar 

  16. Galtier MN, Faugeras OD, Bressloff PC (2011) Hebbian learning of recurrent connections: a geometrical perspective. Neural Comput 24(9):2346–2383

  17. Gerstner W, Kistler WK (2002) Mathematical formulations of Hebbian learning. Biol Cybern 87:404–415

    Article  PubMed  Google Scholar 

  18. Goldman-Rakic PS (1995) Cellular basis of working memory. Neuron 14:477–485

    Article  CAS  PubMed  Google Scholar 

  19. Golomb D, Amitai Y (1997) Propagating neuronal discharges in neocortical slices: computational and experimental study. J Neurophysiol 78(3):1199–1211

  20. Hebb DO (1949) The organization of behavior; a neuropsychological theory

  21. Huang X (2004) Spiral waves in disinhibited mammalian neocortex. J Neurosci 24:9897–9902

    Article  PubMed Central  CAS  PubMed  Google Scholar 

  22. Itskov D, Hansel V, Tsodyks M (2011) Short-term facilitation may stabilize parametric working memory trace. Front Comput Neurosci 5(40)

  23. Lu Y, Sato Y, Amari S (2011) Traveling bumps and their collisions in a two-dimensional neural field. Neural Comput 23:1248–1260

    Article  PubMed  Google Scholar 

  24. Miller KD, MacKayt DJC (1994) The role of constraints in Hebbian learning. Neural Comput 6:100–126

    Article  Google Scholar 

  25. Oja E (1982) Simplified neuron model as a principal component analyzer. J Math Biol 15(3):267–273

    Article  CAS  PubMed  Google Scholar 

  26. Olshausen BA, Field DJ (1997) Sparse coding with an overcomplete basis set: a strategy employed by v1? Vis Res 37(23):3311–3325

    Article  CAS  PubMed  Google Scholar 

  27. Pinto DJ (2005) Initiation, vitro involve distinct mechanisms. J Neurosci 25:8131–8140

    Article  CAS  PubMed  Google Scholar 

  28. Pinto DJ, Ermentrout GB (2001) Spatially structured activity in synaptically coupled neuronal networks: I. Traveling fronts and pulses. SIAM J Appl Math 62:206–225

    Article  Google Scholar 

  29. Rao RPN, Sejnowski TJ (2001) Spike-timing-dependent Hebbian plasticity as temporal difference learning. Neural Comput 13(10):2221–2237

    Article  CAS  PubMed  Google Scholar 

  30. Robinson PA (2011) Neural field theory of synaptic plasticity. J Theor Biol 285:156–163

    Article  CAS  PubMed  Google Scholar 

  31. Sandstede B (2007) Evans functions and nonlinear stability of traveling waves in neuronal network models. Int J Bifurc Chaos 17:2693–2704

    Article  Google Scholar 

  32. Sejnowski TJ (1977) Statistical constraints on synaptic plasticity. J Theor Biol 69(2):385–389

    Article  CAS  PubMed  Google Scholar 

  33. Wallis G, Baddeley R (1997) Optimal, unsupervised learning in invariant object recognition. Neural Comput 9:883–894

    Article  CAS  PubMed  Google Scholar 

  34. Wilson HR, Cowan JD (1972) Excitatory and inhibitory interactions in localized populations of model neurons. Biophys J 12:1–24

    Article  PubMed Central  CAS  PubMed  Google Scholar 

  35. Wilson HR, Cowan JD (1973) A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetika 13(2):55–80

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It is pleasure for the authors to thank A. Abbasian for his useful comments about the biological aspects of the problem. This research was in part supported by a grant from IPM (No. 91920410). The third author was also partially supported by National Elites Foundation.

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Correspondence to M. Sharifitabar.

Appendix: Proof of Theorem 1

Appendix: Proof of Theorem 1


We claim that in (9), if \({\mathrm {Re}}\lambda \ge 0\), we must have \({\mathrm {Im}}\lambda =0\). If it is not the case, assume that \(\lambda =r+{\mathrm {i}}m\) and \(m\ne 0\) and \(r\ge 0\). Let \(\theta =\alpha \gamma \tau ^{-1} f(\overline{u})^2f'(\overline{u})(W+\widehat{w}_m(\xi ))\) and considering imaginary parts of (9),

$$\begin{aligned} m-\theta \frac{r{\mathrm {e}}^{-\delta r}\sin (\delta m)-m\left( 1-{\mathrm {e}}^{-\delta r}\cos (\delta m)\right) }{r^2+m^2}=0. \end{aligned}$$

Since \(m\ne 0\),

$$\begin{aligned} r^2+m^2=\theta \left( r{\mathrm {e}}^{-\delta r}\frac{\sin (\delta m)}{m}+{\mathrm {e}}^{-\delta r}\cos (\delta m)-1\right) . \end{aligned}$$

Therefore we obtain

$$\begin{aligned} r^2<\theta \left( (\delta r+1){\mathrm {e}}^{-\delta r}-1\right) \le 0, \end{aligned}$$

which is a contradiction (Note that \(\theta \ge 0\)).

The left-hand side of (9) is an increasing function of \(\lambda \in [0,\infty ]\). Therefore, its value at \(\lambda =0\) must be positive for stability, i.e.,

$$\begin{aligned}&1\!-\!(1\!-\!\alpha )f'(\overline{u})\widehat{w}_m(\xi )\!-\!\alpha \gamma \delta f(\overline{u})^2f'(\overline{u})(W+\widehat{w}_m(\xi ))>0, \\&\quad \text {for every}\,\xi \in {\mathbb {R}}. \end{aligned}$$

which concludes the results in the desired cases. (Note that \(\widehat{w}_m(\xi )\le W\) when the synaptic weight kernel is positive everywhere and also in the other case, Mexican-hat kernel \(w_m(x)=\frac{1}{4}(1-|x|){\mathrm {e}}^{-|x|}\), we have \(W=0\) and \(\widehat{w}_m(\xi )\le \frac{1}{4}\)). Also in the first case, \(\overline{u}\) is a root of \(\big (1-\kappa {\mathrm {e}}^{-\gamma \delta f(\overline{u})^2}\big )Wf(\overline{u})-\overline{u}\) and the stability condition reads that the derivative of this function with respect to \(\overline{u}\) should be negative at \(\overline{u}\) and this is the case for the extreme possible values of \(\overline{u}\) (which are of course positive) since the value of this function is positive at zero and negative at infinity. In fact in general, the possible constant steady states are alternatively stable and unstable. \(\square \)

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Fotouhi, M., Heidari, M. & Sharifitabar, M. Continuous neural network with windowed Hebbian learning. Biol Cybern 109, 321–332 (2015).

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  • Neural field
  • Continuous network
  • Bump
  • Traveling front
  • Delay equation
  • Existence
  • Stability

Mathematics Subject Classification

  • 35B35
  • 35C07
  • 45K05
  • 92B20
  • 92C20