Coding and Information Processing in Neural Networks

  • Wulfram Gerstner
  • J. Leo van Hemmen
Part of the Physics of Neural Networks book series (NEURAL NETWORKS)


This paper reviews some central notions of the theoretical biophysics of neural networks, viz., information coding through coherent firing of the neurons and spatio-temporal spike patterns. After an introduction to the neural coding problem we first turn to oscillator models and analyze their dynamics in terms of a Lyapunov function. The rest of the paper is devoted to spiking neurons, a pulse code. We review the current neuron models, introduce a new and more flexible one, the spike response model (SRM), and verify that it offers a realistic description of neuronal behavior. The corresponding spike statistics is considered as well. For a network of SRM neurons we present an analytic solution of its dynamics, analyze the possible asymptotic states, and check their stability. Special attention is given to coherent oscillations. Finally we show that Hebbian learning also works for low activity spatio-temporal spike patterns. The models which we study always describe globally connected networks and, thus, have a high degree of feedback. We only touch upon functional feedback, that is, feedback between areas that have different tasks. Information processing in conjunction with functional feedback is treated explicitly in a companion paper [94].


Firing Rate Spike Train Model Neuron Interspike Interval Gain Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Abbott LF, Kepler TB (1990) Model neurons: From Hodgkin Huxley to Hopfield. In: Statistical Mechanics of Neural Networks, L. Garrido (Ed.), Lecture Notes in Physics 368 (Springer, Berlin) pp. 5–18Google Scholar
  2. [2]
    Abbott LF (1991) Realistic synaptic inputs for model neural networks. Network 2:245–258MATHGoogle Scholar
  3. [3]
    Abbott LF, van Vreeswijk C (1993) Asynchronous states in a network of pulse-coupled oscillators. Phys. Rev. E 48:1483–1490ADSGoogle Scholar
  4. [4]
    Abeles M, Lass Y (1975) Transmission of information by the axon. Biol. Cy-bern. 19:121–125Google Scholar
  5. [5]
    Abeles M (1982) Local Cortical Circuits (Springer, Berlin)Google Scholar
  6. [6]
    Abeles M (1991) Corticonics: Neural Circuits of the Cerebral Cortex (Cam-bridge University Press, Cambridge)Google Scholar
  7. [7]
    Abeles M, Prut Y, Bergman H, Vaadia E, Aertsen A (1993) Intergration, synchronicity, and periodicity. In: Brain Theory, A. Aertsen (Ed.) (Elsevier, Amsterdam)Google Scholar
  8. [8]
    Abeles M (1994) Firing rates and well-timed events in the cerebral cortex. This volume, Ch. 3Google Scholar
  9. [9]
    Abramowitz M, Stegun IA (1965) Handbook of Mathematical Functions (Dover, New York)Google Scholar
  10. [10]
    Adrian ED (1926) The impulses produced by sensory nerve endings. J. Physiol. (London) 61:49–72Google Scholar
  11. [11]
    Amit DJ, Gutfreund H, Sompolinsky H (1985) Spin-glass models of neural networks. Phys. Rev. A 32:1007–1032MathSciNetADSGoogle Scholar
  12. [12]
    Amit DJ, Gutfreund H, Sompolinsky H (1987) Statistical mechanics of neural networks near saturation. Ann. Phys. (NY) 173:30–67ADSGoogle Scholar
  13. [13]
    Amit DJ, Tsodyks MV (1991) Quantitative study of attractor neural networks retrieving at low spike rates. I. Substrate-spike rates and neuronal gain. Network 3:259–274Google Scholar
  14. [14]
    Bauer HU, Pawelzik K (1993) Alternating oscillatory and stochastic dynamics in a model for a neuronal assembly. To appear in Physica D 69Google Scholar
  15. [15]
    Bernander Ö, Douglas RJ, Martin KAC, Koch C (1991) Synaptic background activity influences spatio-temporal integration in pyramidal cells. Proc. Natl. Acad. Sci. U.S.A. 88:11,569–11,573Google Scholar
  16. [16]
    Bialek W, Rieke P, de Ruyter van Stevenick RR, Warland D (1991) Reading a neural code. Science 252:1854–1857ADSGoogle Scholar
  17. [17]
    Bindman L, Christofi G, Murphy K, Nowicky A (1991) Long-term potenti-ation (LTP) and depression (LTD) in the neocortex and hippocampus: An overview. In: Aspects of Synaptic Transmission, T.W. Stone (Ed.) (Taylor and Francis, London), Vol. 1Google Scholar
  18. [18]
    Brown TH, Johnston D (1983) Voltage-clamp analysis of mossy fiber synaptic input to hippocampal neurons. J. Neurophysiol. 50:487–507Google Scholar
  19. [19]
    Brown TH, Ganong AH, Kairiss EW, Keenan CL, Kelso SR (1989) Long-term potentation in two synaptic systems of the hippocampal brain slice. In: Neural Models of Plasticity, J.H. Byrne and W.O. Berry (Eds.) (Academic Press, San Diego, CA), pp. 266–306Google Scholar
  20. [20]
    Buhmann J, Schulten K (1986) Associative recognition and storage in a model network with physiological neurons. Biol. Cybern. 54:319–335MATHGoogle Scholar
  21. [21]
    Bush P, Douglas RJ (1991) Synchronization of bursting action potential discharge. Neural Comput. 3:19–30Google Scholar
  22. [22]
    Choi MY (1988) Dynamic model of neural networks. Phys. Rev. Lett. 61:2809–2812MathSciNetADSGoogle Scholar
  23. [23]
    Connors BW, Gutnick MJ, Prince DA (1982) Electrophysiological properties of neocortical neurons in vitro. J. Neurophysiol. 48:1302–1320Google Scholar
  24. [24]
    Cox DR (1962) Renewal Theory (Methuen, London)MATHGoogle Scholar
  25. [25]
    Davis JL, Eichenbaum H (Eds.) (1991) Olfaction. A Model System for Computational Neuroscience (MIT Press, Cambridge, Mass.)Google Scholar
  26. [26]
    Dinse HRO, Krüger K, Best J (1991) Temporal structure of cortical information processing: Cortical architecture, oscillations, and nonseparability of spatio-temporal receptive field organization. In: Neuronal Cooperativity, J. Krüger (Ed.) (Springer, Berlin) pp. 68–104Google Scholar
  27. [27]
    Eckhorn R, Grüsser OJ, Kröller J, Pellnitz K, Pöpel B (1976) Efficiency of different neural codes: Information transfer calculations for three different neural systems. Biol. Cybern. 22:49–60MATHGoogle Scholar
  28. [28]
    Eckhorn R, Bauer R, Jordan W, Brosch M, Kruse W, Munk M, Reit-boeck HJ (1988) Coherent oscillations: A mechanism of feature linking in the visual cortex? Biol. Cybern. 60:121–130Google Scholar
  29. [29]
    Eckhorn R, Krause F, Nelson JI (1993) The RF cinematogram: A cross-correlation technique for mapping several visual fields at once. Biol. Cy-bern. 69:37–55Google Scholar
  30. [30]
    Ekeberg Ö, Wallen P, Lansner A, Traven H, Brodin L, Grillner S (1991) A computer based model for realistic simulations of neural networks. Biol. Cy-bern. 65:81–90Google Scholar
  31. [31]
    Eggermont JJ (1990) The Correlative Brain (Springer, Berlin)Google Scholar
  32. [32]
    van Enter ACD and van Hemmen JL (1984) Statistical-mechanical formalism for spin glasses. van Phys. Rev. A 29:355–365ADSGoogle Scholar
  33. [33]
    Ermentrout GB (1985) Synchronization in a pool of mutually coupled oscillators with random frequencies. J. Math. Biol. 22:1–9MathSciNetMATHGoogle Scholar
  34. [34]
    Ermentrout GB (1985) The behavior of rings of coupled oscillators. J. Math. Biol. 23:55–74MathSciNetMATHGoogle Scholar
  35. [35]
    Ermentrout GB (1990) Oscillator death in populations of “all to all” coupled nonlinear oscillators. Physica D 41:219–231MathSciNetADSMATHGoogle Scholar
  36. [36]
    Ermentrout GB (1992) Stable periodic solutions to discrete and continuum arrays of weakly coupled nonlinear oscillators. SIAM J. Appl. Math. 52:1665–1687MathSciNetMATHGoogle Scholar
  37. [37]
    Ermentrout GB, Kopell N (1990) Oscillator death in systems of coupled neural oscillators. SIAM J. Appl. Math. 50:125–146, and references quoted thereinMathSciNetMATHGoogle Scholar
  38. [38]
    Ermentrout GB, Kopell N (1991) Multiple pulse interactions and averaging in systems of coupled neural oscillators. J. Math. Biol. 29:195–217MathSciNetMATHGoogle Scholar
  39. [39]
    Eskandar EN, Richmond BJ, Hertz JA, Optican LM, Troels K (1992) Decoding of neuronal signals in visual pattern recognition. In: Advances in Neural Information Processing 4, J-E. Moody et al. (Eds.) (Morgan Kaufman, San Mateo, CA), pp. 356–363Google Scholar
  40. [40]
    Fitz Hugh R (1961) Impulses and physiological states in theoretical models of nerve membranes. Biophys. J. 1:445–66Google Scholar
  41. [41]
    Freeman WJ (1975) Mass Action in the Nervous System (Academic Press, New York)Google Scholar
  42. [42]
    Gardner E (1988) The space of interactions in neural network models. J. Phys. A: Math. Gen. 21:257–270ADSGoogle Scholar
  43. [43]
    Gerstner W (1990) Associative memory in a network of “biological” neurons. In: Advances in Neural Information Processing Systems 3, R.P. Lipp-mann, J.E. Moody, D.S. Touretzky (Eds.) (Morgan Kaufmann, San Mateo, CA), pp. 84–90Google Scholar
  44. [44]
    Gerstner W, van Hemmen JL (1992a) Associative memory in a network of spiking neurons. Network 3:139–164MATHGoogle Scholar
  45. [45]
    Gerstner W, van Hemmen JL (1992b) Universality in neural networks: The importance of the mean firing rate. Biol. Cybern. 67:195–205MATHGoogle Scholar
  46. [46]
    Gerstner W, Ritz R, van Hemmen JL (1993a) A biologically motivated and analytically soluble model of collective oscillations in the cortex. I. Theory of weak locking. Biol. Cybern. 68:363–374MATHGoogle Scholar
  47. [47]
    Gerstner W, Ritz R, van Hemmen JL (1993b) Why spikes? Hebbian learning and retrieval of time-resolved excitation patterns. Biol. Cy-bern. 69:503–515MATHGoogle Scholar
  48. [48]
    Gerstner W (1993) Kodierung und Signalübertragung in neuronalen Systemen — Assoziative Netzwerke mit stochastisch feuernden Neuronen (Verlag Harri Deutsch, Frankfurt), Reihe Physik, Bd. 15Google Scholar
  49. [49]
    Gerstner W, van Hemmen JL (1993) Coherence and incoherence in a globally coupled ensemble of pulse-emitting units. Phys. Rev. Lett. 71:312–315ADSGoogle Scholar
  50. [50]
    Grensing D, Kühn R (1986) Random site spin glass models J. Phys. A: Math. Gen. 19:L1153–L1157ADSGoogle Scholar
  51. [51]
    Gray CM, Singer W (1989) Stimulus-specific neuronal oscillations in orientation columns of cat visual cortex. Proc. Natl. Acad. Sci. U.S.A. 86:1698–1702ADSGoogle Scholar
  52. [52]
    Gray CM, König P, Engel AK, Singer W (1989) Oscillatory responses in cat visual cortex exhibit intercolumnar synchronization which reflects global stimulus properties. Nature 338:334–337ADSGoogle Scholar
  53. [53]
    Guckenheimer J, Holmes P (1983) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, Berlin)Google Scholar
  54. [54]
    Hassard B, Wan BY (1978) Bifurcation formulae derived from center manifold theory. J. Math. Anal. Appl. 63:297–312MathSciNetMATHGoogle Scholar
  55. [55]
    Hebb DO (1949) The Organization of Behavior (Wiley, New York)Google Scholar
  56. [56]
    van Hemmen JL, Kühn R (1986) Nonlinear neural networks. Phys. Rev. Lett. 57:913–916ADSGoogle Scholar
  57. [57]
    van Hemmen JL, Grensing D, Huber A, Kühn R (1986) Elementary solu tion of classical spin glass models. Z. Phys. B 65:53–63MathSciNetADSGoogle Scholar
  58. [58]
    van Hemmen JL, Grensing D, Huber A, Kühn R (1988) Nonlinear neural networks I and II. J. Stat. Phys. 50:231–257 and 259-293ADSMATHGoogle Scholar
  59. [59]
    van Hemmen JL, Gerstner W, Herz AVM, Kühn R, Sulzer B, Vaas M (1990) Encoding and decoding of patterns which are correlated in space and time. In: Konnektionismus in Artificial Intelligence und Kognitions-forschung, G. Dorffner (Ed.) (Springer-Verlag, Berlin), pp. 153–162Google Scholar
  60. [60]
    van Hemmen JL, Ioffe LB, Kühn R, Vaas M (1990) Increasing the efficiency of a neural network through unlearning. Physica A 163:386–392ADSGoogle Scholar
  61. [61]
    van Hemmen JL, Kühn R (1991) Collective phenomena in neural networks. In: Models of Neural Networks, E. Domany, J.L. van Hemmen, K. Schulten (Eds.) (Springer-Verlag, Berlin), pp. 1–105Google Scholar
  62. [62]
    van Hemmen JL, Wreszinski WF (1993) Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators J. Stat. Phys. 72:145–166ADSMATHGoogle Scholar
  63. [63]
    Hertz J, Krogh A, Palmer RG (1991) Introduction to the Theory of Neural Computation (Addison-Wesley, Redwood City, CA)Google Scholar
  64. [64]
    Herz AVM, Sulzer B, Kühn R, van Hemmen JL (1988) The Hebb rule: Storing static and dynamic objects in an associative neural network. Europhys. Lett. 7:663–669.ADSGoogle Scholar
  65. [65]
    Herz AVM, Sulzer B, Kühn R, van Hemmen JL (1989) Hebbian learning reconsidered: Representation of static and dynamic objects in associative neural nets. Biol. Cybern. 60:457–467MATHGoogle Scholar
  66. [66]
    Herz AVM, Li Z, van Hemmen JL (1991) Statistical mechanics of temporal association in neural networks with transmission delays. Phys. Rev. Lett. 66:1370–1373MathSciNetADSMATHGoogle Scholar
  67. [67]
    Hirsch MW, Smale S (1974) Differential Equations, Dynamical Systems, and Linear Algebra (Academic Press, New York), Chap. 9MATHGoogle Scholar
  68. [68]
    Hodgkin AL, Huxley AF (1952) A quantitative description of ion currents and its applications to conduction and excitation in nerve membranes. J. Physiol. (London) 117:500–544Google Scholar
  69. [69]
    Hopfield JJ (1982) Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. U.S.A. 79:2554–2558MathSciNetADSGoogle Scholar
  70. [70]
    Hopfield JJ (1984) Neurons with graded response have computational properties like those of two-state neurons. Proc. Natl. Acad. Sci. U.S.A. 81:3088–3092ADSGoogle Scholar
  71. [71]
    Horn D, Usher M (1989) Neural networks with dynamical thresholds. Phys. Rev. A 40:1036–1044ADSGoogle Scholar
  72. [72]
    Hubel DH, Wiesel TN (1977) Functional architecture of macaque monkey visual cortex. Proc. R. Soc. London Ser. B 198:1–59ADSGoogle Scholar
  73. [73]
    Iooss G, Joseph DD (1980) Elementary Stability and Bifurcation Theory (Springer, Berlin), Chap. V; to be fair, this book is explicit but not “elementary.”Google Scholar
  74. [74]
    Jack JJB, Noble D, Tsien RW (1975) Electric Current Flow in Excitable Cells (Clarendon Press, Oxford)Google Scholar
  75. [75]
    Kandel ER, Schwartz JH (1985) Principles of Neural Science, 2nd Ed. (Elsevier, Amsterdam)Google Scholar
  76. [76]
    Kelso SR, Ganong AH, Brown TH (1986) Hebbian synapses in hippocampus. Proc. Natl. Acad. Sci. U.S.A. 83:5326–5330ADSGoogle Scholar
  77. [77]
    König P, Schulen TB (1991) Stimulus-dependent assembly formation of oscillatory responses: I. Synchronization. Neural Comput. 3:155–166Google Scholar
  78. [78]
    Konishi M (1986) Centrally synthesized maps of sensory space. Trends in Neurosci. 9:163–168Google Scholar
  79. [79]
    Kopell N (1986) Phase methods for coupled oscillators and related topics: An annnotated bibliography J. Stat. Phys. 44:1035–1042Google Scholar
  80. [80]
    Krauth W, Mézard M (1987) Learning algorithms with optimal stability in neural networks. J. Phys. A: Math. Gen. 20:L745–L752ADSGoogle Scholar
  81. [81]
    Krüger J (1983) Simultaneous individual recordings from many cerebral neurons: Techniques and results. Rev. Physiol. Biochem. Pharmacol. 98:177–233Google Scholar
  82. [82]
    Krüger J, Aiple F (1988) Multielectrode investigation of monkey striate cortex: Spike train correlations in the infragranular layers. J. Neurophysiol. 60:798–828Google Scholar
  83. [83]
    Krüger J, Becker JD (1991) Recognizing the visual stimulus from neuronal discharges. Trends in Neurosci. 14:282–286Google Scholar
  84. [84]
    Kuffler SW, Nicholls JG, Martin AR (1984) From Neuron to Brain, 2nd Ed. (Sinauer, Sunderland, Mass.)Google Scholar
  85. [85]
    Kuramoto Y (1975) Self-entrainment of a population of coupled nonlinear oscillators. In: International Symposium on Mathematical Problems in Theoretical Physics, H. Araki (Ed.) (Springer, Berlin), pp. 420–422Google Scholar
  86. [86]
    Kuramoto Y (1984) Cooperative dynamics of oscillator community. Progr. Theor. Phys. Suppl. 79:223–240ADSGoogle Scholar
  87. [87]
    Kuramoto Y (1984) Chemical Oscillations, Waves, and Turbulence (Springer, Berlin), pp. 68–77Google Scholar
  88. [88]
    Kuramoto Y, Nishikawa I (1987) Statistical macrodynamics of large dynamical systems. Case of a phase transition in oscillator communities. J. Stat. Phys. 49:569–605MathSciNetADSMATHGoogle Scholar
  89. [89]
    Lamperti J (1966) Probability (Benjamin, New York), Chap. 7MATHGoogle Scholar
  90. [90]
    Lancaster B, Adams PR (1986) Calcium-dependent current generating the afterhyperpolarization of hippocampal neurons. J. Neurophysiol. 55:1268–1282Google Scholar
  91. [91]
    Larson J, Lynch G (1986) Induction of synaptic potentiation in hippocampus by patterned stimulation involves two events. Science 232:985–988ADSGoogle Scholar
  92. [92]
    Little WA, Shaw GL (1978) Analytical study of the memory storage capacity of a neural network. Math. Biosci. 39:281–290MathSciNetMATHGoogle Scholar
  93. [93]
    MacKay DM, McCulloch WS (1952) The limiting information capacity of a neuronal link. Bull, of Math. Biophy. 14:127–135Google Scholar
  94. [94]
    von der Malsburg C, Buhmann J (1992) Sensory segmentation with coupled neural oscillators. Biol. Cybern. 67 233–242MATHGoogle Scholar
  95. [95]
    Matthews PC, Strogatz SH (1990) Phase diagram for the collective behavior of limit-cycle oscillators. Phys. Rev. Lett. 65:1701–1704MathSciNetADSMATHGoogle Scholar
  96. [96]
    Mirollo RE, Strogatz SH (1990) Jump bifurcation and hysteresis in an infinite-dimensional dynamical system of coupled spins. SIAM J. Appl. Math. 50:108–124MathSciNetMATHGoogle Scholar
  97. [97]
    Mirollo RE, Strogatz SH (1990) Synchronization of pulse coupled biological oscillators. SIAM J. Appl. Math. 50:1645–1662MathSciNetMATHGoogle Scholar
  98. [98]
    Nagumo J, Arimoto S, Yoshizawa S (1962) An active pulse transmission line simulating nerve axon. Proc. IRE 50:2061–2070Google Scholar
  99. [99]
    Neven H, Aertsen A (1992) Rate coherence and event coherence in the visual cortex: A neuronal model of object recognition. Biol. Cybern. 67:309–322Google Scholar
  100. [100]
    Niebur E, Kammen DM, Koch C, Rudermann D, Schuster HG (1991) Phase-coupling in two-dimensional networks of interacting oscillators. In: Advances in Neural Information Processing Systems 3, R.P. Lippmann, J.E. Moody, D.S. Touretzky (Eds.) (Morgan Kaufmann, San Mateo, CA), pp. 123–127Google Scholar
  101. [101]
    Optican LM, Richmond BJ (1987) Temporal encoding of two-dimensional patterns by single units in primate inferior cortex: III. Information theoretic analysis. J. Neurophysiol. 57:162–178Google Scholar
  102. [102]
    Perkel DH, Gerstein GL, Moore GP (1967) Neuronal spike trains and stochastic point processes. I. The single spike train. Biophys. J. 7:391–418Google Scholar
  103. [103]
    van der Pol B (1927) Forced oscillations in a circuit with nonlinear resistance (reception with reactive triode). The London, Edinburgh, and Dublin Philos. Mag. and J. Sci. 3:65–80Google Scholar
  104. [104]
    Rall W (1964) Theoretical significance of dendritic trees for neuronal input-output relations. In: Neural Theory and Modeling, R.F. Reiss (Ed.) (Stanford University Press), pp. 73–97Google Scholar
  105. [105]
    Rapp M, Yarom Y, Segev I (1992) The impact of parallel fiber background activity on the cable properties of cerebellar Purkinje cells. Neural Comput. 4:518–533Google Scholar
  106. [106]
    Reitboeck HJA (1983) A multielectrode matrix for studies of temporal signal correlations within neural assemblies. In: Synergetics of the Brain, E. Basar et al. (Eds.) (Springer, Berlin), pp. 174–182Google Scholar
  107. [107]
    Richmond BJ, Optican LM, Podell M, Spitze H (1987) Temporal encoding of two-dimensional patterns by single units in primate inferior cortex: I. Response characteristics. J. Neurophysiol. 57:132–146Google Scholar
  108. [108]
    Ritz R, Gerstner W, van Hemmen JL (1994) Associative binding and segregation in a network of spiking neurons. This volume, Ch. 5Google Scholar
  109. [109]
    Rotter S, Heck D, Aertsen A, Vaadia E (1993) A stochastic model for networks of spiking cortical neurons: Time-dependent description on the basis of membrane currents. In: Gene, Brain, Behavior, H. Eisner and M. Heisenberg (Eds.) (Thieme, Stuttgart), p. 491Google Scholar
  110. [110]
    Rudin W (1974) Real and Complex Analysis (McGraw-Hill, New York), p. 63MATHGoogle Scholar
  111. [111]
    de Ruyter van Steveninck RR, Bialek W (1988) Real-time performance of a movement-sensitive neuron in the blowfly visual system: Coding and information transfer in short spike sequences. Proc. R. Soc. London Ser. B 234:379–414ADSGoogle Scholar
  112. [112]
    Sakaguchi H, Shinomoto S, Kuramoto Y (1987) Local and global self-entrainments in oscillator lattices. Progr. Theor. Phys. 77:1005–1010ADSGoogle Scholar
  113. [113]
    Sakaguchi H, Shinomoto S, Kuramoto Y (1988) Mutual entrainaient in oscillator lattices with nonvariational-type interactions. Progr. Theor. Phys. 79:1069–1079MathSciNetADSGoogle Scholar
  114. [114]
    Schillen TB, König P (1991) Stimulus-dependent assembly formation of oscillatory responses. II. Desynchronization. Neural Comput. 3:167–177Google Scholar
  115. [115]
    Schuster HG and Wagner P (1990a) A model for neuronal oscillations in the visual cortex: 1. Mean-field theory and derivation of the phase equations. Biol. Cybern. 64:77–82MATHGoogle Scholar
  116. [116]
    Schuster HG and Wagner P (1990b) A model for neuronal oscillations in the visual cortex: 2. Phase description and feature dependent synchronization. Biol. Cybern. 64:83–85MATHGoogle Scholar
  117. [117]
    Singer W (1991) The formation of cooperative cell assemblies in the visual cortex. In: Neural Cooperativity, J. Krüger (Ed.) (Springer, Berlin), pp. 165–183Google Scholar
  118. [118]
    Singer W (1994) The role of synchrony in neocortical processing and synap-tic plasticity. This volume, Ch. 4Google Scholar
  119. [119]
    Sompolinsky H, Golomb D, and Kleinfeld D (1990) Global processing of visual stimuli in a neural network of coupled oscillators. Proc. Natl. Acad. Sci. U.S.A. 87:7200–7204ADSGoogle Scholar
  120. [120]
    Sompolinsky H, Golomb D, Kleinfeld D (1991) Cooperative dynamics in visual processing. Phys. Rev. A 43:6990–7011ADSGoogle Scholar
  121. [121]
    Stein RB (1967) The information capacity of nerve cells using a frequency code. Biophys. J. 7:797–826Google Scholar
  122. [122]
    Stein RB (1967) The frequency of nerve action potentials generated by applied currents. Proc. R. Soc. London Ser. B167:64–86ADSGoogle Scholar
  123. [123]
    Strogatz SH, Mirollo RE (1988) Collective synchronization in lattices of nonlinear oscillators with randomness. J. Phys. A: Math. Gen. 21:L699–L705MathSciNetADSGoogle Scholar
  124. [124]
    Strogatz SH, Mirollo RE (1991) Stability of incoherence in a population of coupled oscillators. J. Stat. Phys. 63:613–635MathSciNetADSGoogle Scholar
  125. [125]
    Strogatz SH, Mirollo RE, Matthews PC (1992) Coupled nonlinear oscillators below the synchronization threshold: Relaxation be generalized Landau damping. Phys. Rev. Lett. 68:2730–2733MathSciNetADSMATHGoogle Scholar
  126. [126]
    Stuart GJ, Sakmann B (1994) Active propagation of somatic action potentials into neocortical pyramidal cell dendrites. Nature 367:69–72ADSGoogle Scholar
  127. [127]
    Traub RD, Wong RKS, Miles R, Michelson H (1991) A model of a CA3 hippocampal pyramidal neuron incorporating voltage-clamp data on intrinsic conductances. J. Neurophysiol.66:635–Google Scholar
  128. [128]
    Tsodyks M, Mitkov I, Sompolinsky H (1993) Patterns of synchrony in inhomogeneous networks of oscillators with pulse interaction. Phys. Rev. Lett. 71:1281–1283ADSGoogle Scholar
  129. [129]
    Usher M, Schuster HG, Niebur E (1993) Dynamics of populations of integrate-and-fire neurons, partial synchronization and memory. Neural Comput. 5:570–586Google Scholar
  130. [130]
    Varadhan SRS (1980) Diffusion Problems and Partial Differential Equations (Springer, Berlin), p. 266 et seq.Google Scholar
  131. [131]
    Wang D, Buhmann J, von der Malsburg C (1990) Pattern segmentation in associative memory. Neural Comput.2:94–Google Scholar
  132. [132]
    Wilson HR, Cowan JD (1972) Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J. 12:1–24ADSGoogle Scholar
  133. [133]
    Wilson MA, Bhalla US, Uhley JD, Bower JM (1989) GENESIS: A system for simulating neural networks. In: Advances in Neural Information Processing Systems, D. Touretzky (Ed.) (Morgan Kaufmann, San Mateo, CA), pp. 485–492Google Scholar
  134. [134]
    Wong RKS, Prince DA, Basbaum AI (1979) Intradendritic recordings from hippocampal neurons. Proc. Natl. Acad. Sci. U.S.A. 76:986–990ADSGoogle Scholar
  135. [135]
    Yamada WM, Koch C, Adams PR (1989) Multiple channels and calcium dynamics. In: Methods in Neuronal Modeling, from Synapses to Networks, C. Koch and I. Segev (Eds.) (MIT Press, Cambridge, Mass.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1994

Authors and Affiliations

  • Wulfram Gerstner
    • 1
  • J. Leo van Hemmen
    • 1
  1. 1.Physik-Department der TU MünchenGarching bei MünchenGermany

Personalised recommendations