Abstract
Recurrent neural networks with full symmetric connectivity have been extensively studied as associative memories and pattern recognition devices. However, there is considerable evidence that sparse, asymmetrically connected, mainly excitatory networks with broadly directed inhibition are more consistent with biological reality. In this paper, we use the technique of return maps to study the dynamics of random networks with sparse, asymmetric connectivity and nonspecific inhibition. These networks show three qualitatively different kinds of behavior: fixed points, cycles of low period, and extremely long cycles verging on aperiodicity. Using statistical arguments, we relate these behaviors to network parameters and present empirical evidence for the accuracy of this statistical model. The model, in turn, leads to methods for controlling the level of activity in networks. Studying random, untrained networks provides an understanding of the intrinsic dynamics of these systems. Such dynamics could provide a substrate for the much more complex behavior shown when synaptic modification is allowed.
Similar content being viewed by others
References
Amaral DG, Ishizuka N, Claiborne B (1990) Neurons, numbers and the hippocampal networks. Prog Brain Res, 83:1–11
Amari S (1971) Characteristics of randomly connected threshold-element networks and network systems. Proc IEEE 59:35–47
Amari S (1972) Learning patterns and pattern sequences by self-organizing nets of threshold elements. IEEE Trans Comput C-21:1197–1206
Amari S (1974) A method of statistical neurodynamics. Kybernetik 14:201–215
Amari S (1989) Characteristics of sparsely encoded associative memory. Neural Networks 2:451–457
Amari S, Maginu K (1988) Statistical neurodynamics of associative memory. Neural Networks 1:63–73
Amari S, Yoshida K, Kanatani K (1977) A mathematical foundation for statistical neurodynamics. SIAM J Appl Math 33:95–126
Ambrose-Ingerson J, Granger, R, Lynch G (1990) Simulation of paleocortex performs hierarchical clustering. Science 247:1344–1348
Bauer K, Krey U (1990) On learning and recognition of temporal sequences of correlated patterns. Z Phys B Condensed Matter 79:461–475
Buhmann J, Schulten K (1988) Storing sequences of biased patterns in neural networks with stochastic dynamics. In: Eckmiller R, Malsburg C von der (eds) Neural computers. (NATO ASI series, vol F41) Springer, Berlin Heidelberg New York, pp 231–242
Cohen MA, Grossberg S (1983) Absolute stability of global pattern formation and parallel memory storage by competitive neural networks. IEEE Trans Syst Man Cybern 13:815–826
Crisanti A, Sompolinsky H (1987) Dynamics of spin systems with randomly asymmetric bonds: Langevin dynamics and a spherical model. Phys Rev A36:4922–4939
Derrida B, Pomeau Y (1986) Random networks of automata: a simple annealed approximation. Europhys Lett 1:45–49
Derrida B, Gardner E, Zippelius A (1987) An exactly solvable asymmetric neural network model. Europhys Lett 4:167–173
Eichenbaum H, Otto T (1992) The hippocampus — what does it do? Behav Neural Biol 57:2–36
Fukushima K (1973) A model of associative memory in the brain. Kybernetik 12:58–63
Furman GG (1965) Comparison of models for subtractive and shunting lateral-inhibition in receptor neurons. Kybernetik 2:257–274
Gardner-Medwin AR (1976) The recall of events through the learning of associations between their parts. Proc R Soc Lond [Biol] 194:375–402
Gardner-Medwin AR (1989) Doubly modifiable synapses: a model of short and long term auto-associative memory. Proc R Soc Lond [Biol] 238:137–154
Gibson WG, Robinson J (1992) Statistical analysis of the dynamics of a sparse associative memory. Neural Networks 5:645–661
Grossberg S (1988) Nonlinear neural networks: principles, mechanisms, and architectures. Neural Networks 1:17–61
Gutfreund H, Mèzard M (1988) Processing of temporal sequences in neural networks. Phys Rev Lett 61:235–238
Gutfreund H, Reger JD, Young AP (1988) The nature of attractors in an asymmetric spin glass with deterministic dynamics. J Phys A Math Nucl Gen 21:2775–2797
Hemmen JL van, Kühn R (1991) Collective phenomena in neural networks. In: Domany E, Hemmen JL van, Schulten K (eds) Models of neural networks. Springer, Berlin Heidelberg New York
Hertz A, Grinstein G, Solla SA (1987) Irreversible spin glasses and neural networks. In: Hemmen JL van, Morgenstern I (eds) Heidelberg colloquium on glassy dynamics. Springer, Berlin Heidelberg New York, pp 538–546
Hertz J, Krogh A, Palmer RG (1991) Introduction to the theory of neural computation. Addison-Wesley, Redwood City, Calif
Herz A, Sulzer B, Kühn R, Hemmen JL van (1989) Hebbian learning reconsidered: representation of static and dynamic objects in associative neural nets. Biol Cybern 60:457–467
Heskes TM, Gielen S (1992) Retrieval of pattern sequences at variable speeds in a neural network model. Neural Networks 5:145–152
Hopfield JJ (1982) Neural networks and physical systems with emergent collective computational abilities. Proc Natl Acad Sci USA 79:2554–2558
Ishizuka N, Weber J, Amaral DG (1990) Organization of intrahippocampal projections originating from CA3 pyramidal cells in the rat. J Comp Neurol 295:580–623
Kleinfeld D (1986) Sequential state generation by model neural networks. Proc Natl Acad Sci USA 83:9469–9473
Kree R, Zippelius A (1991) Asymmetrically diluted neural networks. In: Domany E, Hemmen JL van, Schulten K (eds) Models of neural networks. Springer, Berlin Heidelberg New York, pp 193–212
Kürten KE (1988) Critical phenomena in model neural networks. Phys Lett A129:157–160
Levy WB (1989) A computational approach to hippocampal function. In: Hawkins RD, Bower GH (eds) Computational models of learning in simple neural systems. (The psychology of learning and motivation, vol 23). Academic Press, San Diego, pp 243–305
Marr D (1971) Simple memory: a theory for archicortex. Philos Trans R Soc Lond [Biol] 262:23–81
McNaughton BL, Nadel L (1989) Hebb-Marr networks and the neurobiological representation of action in space. In: Gluck MA, Rumelhart D (eds) Neuroscience and connectionist theory. Erlbaum, Hillsdale, NJ, pp 1–63
Nakao M, Watanabe K, Takahashi T, Mizutani Y, Yamamoto M (1992) Structural properties of network attractor associated with neuronal dynamics transition. Proceedings of International Joint Conference on Neural Networks, Baltimore, vol 3, pp 529–534
Nützel K (1991) The length of attractors in asymmetric random neural networks with deterministic dynamics. J Phys A Math Nucl Gen 24:L151–157
O'Keefe J, Nadel L (1978) The hippocampus as a cognitive map. Oxford University Press, Oxford
Parisi G (1986) Asymmetric neural networks and the process of learning. J Phys A Math Nucl Gen 19:L675-L680
Reiss M, Taylor JG (1991) Storing temporal sequences, Neural Networks 4:773–787
Riedel U, Kühn R, van Hemmen JL (1988) Temporal sequences and chaos in neural nets. Phys Rev A 28: 1105–1108
Rieger H, Schreckenberg M, Zittartz J (1989) Glauber dynamics of the asymmetric SK-model. Z Phys B Condensed Matter 74:527–538
Rolls ET (1989) Functions of neuronal networks in the hippocampus and neocortex in memory. In: Byrne JH, Berry WO (eds) Neural models of plasticity. Academic Press, New York, pp 240–265
Rolls ET, Treves A (1990) The relative advantages of sparse versus distributed encoding for associative neuronal networks in the brain. Network 1:407–421
Segal M (1990) Serotonin attenuates a slow inhibitory postsynaptic potential in rat hippocampal neurons. Neuroscience 36:631–641
Sompolinsky H, Kanter I (1986) Temporal association in asymmetric neural networks. Phys Rev Lett 57:2861–2864
Sompolinsky H, Crisanti A, Sommers HJ (1988) Chaos in random neural networks. Phys Rev Lett 61:259–262
Spitzner P, Kinzel W (1989a) Freezing transition in asymmetric random neural networks with deterministic dynamics. Z Phys B Condensed Matter 77:511–517
Spitzner P, Kinzel W (1989b) Hopfield network with directed bonds. Z Phys B Condensed Matter 74:539–545
Squire LR, Zola-Morgan S (1991) The medial temporal lobe memory system. Science 253:1380–1386
Swanson LW, Köhler C, Björklund A (1987) The limbic region. I. The septohippocampal system. In: Björklund A, Hökfelt T, Swanson LW (eds) Integrated Systems of the CNS, part 1. (Handbook of chemical neuroanatomy, vol 5) Elsevier, Amsterdam, pp 125–277
Treves A, Rolls ET (1992) Computational constraints suggest the need for two distinct input systems to the hippocampal CA3. Hippocampus 2:189–200
Tsuda I (1992) Dynamic link of memory — chaotic memory map in nonequilibrium neural networks. Neural Networks 5:313–326
Willshaw DJ, Buckingham JT (1990) An assessment of Marr's theory of the hippocampus as a temporary memory store. Philos Trans R Soc Lond [Biol] 329:205–215
Yao Y, Freeman WJ (1990) Model of biological pattern recognition with spatially chaotic dynamics. Neural Networks 3:153–170
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Minai, A.A., Levy, W.B. The dynamics of sparse random networks. Biol. Cybern. 70, 177–187 (1993). https://doi.org/10.1007/BF00200831
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00200831