Journal of Computational Neuroscience

, Volume 20, Issue 3, pp 321–348 | Cite as

On the sensitive dependence on initial conditions of the dynamics of networks of spiking neurons

  • Arunava BanerjeeEmail author


We have previously formulated an abstract dynamical system for networks of spiking neurons and derived a formal result that identifies the criterion for its dynamics, without inputs, to be “sensitive to initial conditions”. Since formal results are applicable only to the extent to which their assumptions are valid, we begin this article by demonstrating that the assumptions are indeed reasonable for a wide range of networks, particularly those that lack overarching structure. A notable aspect of the criterion is the finding that sensitivity does not necessarily arise from randomness of connectivity or of connection strengths, in networks. The criterion guides us to cases that decouple these aspects: we present two instructive examples of networks, one with random connectivity and connection strengths, yet whose dynamics is insensitive, and another with structured connectivity and connection strengths, yet whose dynamics is sensitive. We then argue based on the criterion and the gross electrophysiology of the cortex that the dynamics of cortical networks ought to be almost surely sensitive under conditions typically found there. We supplement this with two examples of networks modeling cortical columns with widely differing qualitative dynamics, yet with both exhibiting sensitive dependence. Next, we use the criterion to construct a network that undergoes bifurcation from sensitive dynamics to insensitive dynamics when the value of a control parameter is varied. Finally, we extend the formal result to networks driven by stationary input spike trains, deriving a superior criterion than previously reported.


Dynamical systems Sensitive dependence Spiking neurons 


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  1. Amit DJ, Brunel N (1997) Model of global spontaneous activity and local structured activity during delay periods in the cerebral cortex. Cerebral Cortex 7: 237–252.Google Scholar
  2. Banerjee A (2001) On the phase-space dynamics of systems of spiking neurons: I. model and experiments. Neural Computation 13: 161–193.Google Scholar
  3. Banerjee A (2001) On the phase-space dynamics of systems of spiking neurons: II. formal analysis. Neural Computation 13: 195–225.Google Scholar
  4. Bi G-q, Poo M-m (1998) Synaptic modifications in cultured hippocampal neurons: dependence on spike timing, synaptic strength, and postsynaptic cell type. Journal of Neuroscience 18: 10464–10472.Google Scholar
  5. Braitenberg V, Schüz A (1991) Anatomy of the Cortex: Statistics and Geometry. Springer-Verlag, Berlin Heidelberg New York.Google Scholar
  6. Britten KH, Shadlen MN, Newsome WT, Movshon JA (1993) Responses of neurons in macaque MT to stochastic motion signals. Vision Neuroscience 10: 1157–1169.Google Scholar
  7. Brunel N, Hakim V (1999) Fast global oscillations in networks of integrate-and-fire neurons with low firing rates. Neural Computation 11: 1621–1671.Google Scholar
  8. Brunel N (2000) Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. Journal of Computational Neuroscience 8: 183–208.Google Scholar
  9. Burns BD, Webb AC (1976) The spontaneous activity of neurones in the cat’s cerebral cortex. Proceedings of the Royal Society of London, B, Biological Sciences 194: 211–233.Google Scholar
  10. Cazelles B, Ferriere RH (1992) How predictable is chaos. Nature 355: 25–26.Google Scholar
  11. Chow CC (1998) Phase-locking in weakly heterogeneous neuronal networks. Physica D 118: 343–370.Google Scholar
  12. Cox CL, Denk W, Tank DW, Svoboda K (2000) Action potentials reliably invade axonal arbors of rat neocortical neurons. Proceedings of the National Academy of Sciences U.S.A. 97: 9724–9728.Google Scholar
  13. Freeman WJ, Skarda C (1985) Spatial EEG patterns, nonlinear dynamics and perception: The non-Sherringtonian view. Brain Research Reviews 10: 147–175.Google Scholar
  14. Froemke RC, Dan Y (2002) Spike timing dependent synaptic modification induced by natural spike trains. Nature 416: 433–437.Google Scholar
  15. Gerstner W, van Hemmen JL (1992) Associative memory in a network of spiking neurons. Network 3: 139–164.Google Scholar
  16. Gerstner W, van Hemmen JL, Cowan JD (1996) What matters in neuronal locking. Neural Computation 8: 1653–1676.Google Scholar
  17. Grassberger P, Procaccia I (1983) Measuring the strangeness of strange attractors. Physica D 9: 189–208.Google Scholar
  18. Gray RM (1988) Probability, Random Processes, and Ergodic Properties. Springer-Verlag, New York Berlin Heidelberg.Google Scholar
  19. Hessler NA, Shirke AM, Malinow R (1993) The probability of transmitter release at a mammalian central synapse. Nature 366: 569–572.Google Scholar
  20. Jacobs AL, Werblin FS (1998) Spatiotemporal patterns at the retinal output. Journal of Neurophysiology 80: 447–451.Google Scholar
  21. Kingman JFC (1973) The ergodic theory of subadditive stochastic processes. Annals of Probability 1: 883–909.Google Scholar
  22. Latham PE, Richmond BJ, Nelson PG, Nirenberg S (2000) Intrinsic dynamics in neuronal networks. I. theory. Journal of Neurophysiology 83: 808–827.Google Scholar
  23. MacGregor RJ, Lewis ER (1977) Neural Modeling, Plenum Press, New York.Google Scholar
  24. Mainen ZF, Sejnowski T (1995) Reliability of spike timing in neocortical neurons. Science 268: 1503–1507.Google Scholar
  25. Markram H, Lübke J, Frotscher M, Sakmann B (1997) Regulation of synaptic efficacy by coincidence of postsynaptic APs and EPSPs. Science 275: 213–215.Google Scholar
  26. Meister M, Berry MJ (1999) The neural code of the retina. Neuron 22: 435–450.Google Scholar
  27. Nowak LG, Sanchez-Vives MV, McCormick DA (1997) Influence of low and high frequency inputs on spike timing in visual cortical neurons. Cerebral Cortex 7: 487–501.Google Scholar
  28. Osborne AR, Provenzale A (1989) Finite correlation dimension for stochastic systems with power-law spectra. Physica D 35: 357–381.Google Scholar
  29. Raastad M, Storm JF, Andersen P (1992) Putative single quantum and single fibre excitatory post synaptic currents show similar amplitude range and variability in rat hippocampal slices. European Journal of Neuroscience 4: 113–117.Google Scholar
  30. Reinagel P, Reid RC (2000) Temporal coding of visual information in the thalamus. Journal of Neuroscience 20:14: 5392–5400.Google Scholar
  31. Rosenmund C, Clements JD, Westbrook GL (1993) Non-uniform probability of glutamate release at a hippocampal synapse. Science 262: 754–757.Google Scholar
  32. Schüz A (1992) Randomness and constraints in the cortical neuropil, In Information Processing in the Cortex Aertsen, V, Braitenberg V (Eds.), Springer-Verlag, Berlin Heidelberg New York, pp. 3–21.Google Scholar
  33. Seung HS, Lee DD, Reis BY, Tank DW (2000) Stability of the memory of eye position in a recurrent network of conductance-based model neurons. Neuron 26: 259–271.Google Scholar
  34. Shepherd GM (1998) The Synaptic Organization of the Brain, Oxford University Press, New York.Google Scholar
  35. Snowden RJ, Treue S, Andersen RA (1992) The response of neurons in areas V1 and MT of the alert rhesus monkey to moving random dot patterns. Experimental Brain Research 88: 389–400.Google Scholar
  36. Tolhurst DJ, Movshon JA, Dean AF (1983) The statistical reliability of signals in single neurons in cat and monkey visual cortex. Vision Research 23: 775–785.Google Scholar
  37. Tomko G, Crapper D (1974) Neuronal variability: non-stationary responses to identical visual stimuli. Brain Research 79: 405–418.Google Scholar
  38. van Vreeswijk C, Sompolinsky H (1996) Chaos in neuronal networks with balanced excitatory and inhibitory activity. Science 274: 1724–1726.Google Scholar
  39. van Vreeswijk C, Sompolinsky H (1998) Chaotic balanced state in a model of cortical circuits. Neural Computation 10: 1321–1372.Google Scholar

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© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.Computer and Information Science and Engineering DepartmentUniversity of FloridaGainesville

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