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Permitted and forbidden sets in symmetric threshold-linear networks.

Richard H R Hahnloser1, H Sebastian Seung, Jean-Jacques Slotine

  • 1Howard Hughes Medical Institute, Department of Brain and Cognitive Sciences, MIT E25-210, Cambridge, MA 02139, U.S.A. rhahnloser@mit.edu

Neural Computation
|March 7, 2003
PubMed
Summary

Symmetric threshold-linear networks exhibit stable states if their matrix is copositive. This research defines minimal conditions for convergence and multistability in neural networks, offering a new view of long-term memory.

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Area of Science:

  • Computational neuroscience
  • Theoretical neuroscience
  • Neural network dynamics

Background:

  • Recurrent cortical circuits are crucial for biological computation.
  • Previous studies identified sufficient conditions for bounded dynamics and multistability in threshold-linear networks.
  • Minimal conditions for convergence and multistability remained unclear.

Purpose of the Study:

  • To determine the necessary and sufficient conditions for convergence and multistability in symmetric threshold-linear networks.
  • To explore the relationship between network properties and the storage of information.
  • To generalize the concept of memory in neural networks.

Main Methods:

  • Analysis of symmetric threshold-linear networks.
  • Investigation of network matrix properties (copositivity, positive semidefiniteness).

Related Experiment Videos

  • Characterization of permitted and forbidden neuron coactivation states.

    • Main Results:

    • Convergence to attractive fixed points is equivalent to matrix copositivity.
    • Nonconnected attractive fixed points (multiattractivity) correspond to non-positive semidefinite matrices.
    • Permitted neuron coactivation sets exhibit clustering properties.

    Conclusions:

    • Establishes copositivity as the necessary and sufficient condition for convergence in these networks.
    • Provides a generalized framework for understanding long-term memory as stored in synaptic connections.
    • Highlights the link between network dynamics and information storage in biological computation.