Elsevier

Neurocomputing

Volumes 32–33, June 2000, Pages 741-748
Neurocomputing

Delay adaptation in the nervous system

https://doi.org/10.1016/S0925-2312(00)00239-3Get rights and content

Abstract

Time delays are ubiquitous in the nervous system. Empirical findings suggest that time delays are adapted when considering the synchronous activity of neurons. We introduce a framework for studying the dynamics of self-organized delay adaptation in systems which optimize coincidence of inputs. The framework comprises two families of delay adaptation mechanisms, delay shift and delay selection. For the important case of periodically modulated input we derive conditions for the existence and stability of solutions which constrain learning rules for reliable delay adaptation. Delay adaptation is also applicable in the case of several spatio-temporal neuronal input patterns.

Introduction

Interactions in the nervous system are associated with time delays. Postsynaptic potentials have a finite rise time, delays arise from signal integration in the dendritic tree, and there is a considerable conduction time for action potentials running down an axon. A precise neuronal signal integration for the purpose of target localization requires an adaptation of such time delays. Examples include the auditory system of barn owls, echolocation in bats, and the lateral line system of weakly electric fish [9], [1]. Time delays and their putative adaptation have to be considered also for synchronization phenomena associated with the binding of sensory information in the neocortex [2], [3]. A number of observations suggest that time delays in the nervous system are adaptive: time delays in the optic nerve are equalized [11], signals in visual callosal axons arrive simultaneously at all axonal endings [8], internodal distances in the barn owl auditory system are short resulting in a slow signal conduction [1], and neurons in vitro can inhibit the formation of a myelin sheet by firing at a low frequency [12].

Two mechanisms have been proposed for the self-organized adaptation of transmission delays in the nervous system. One mechanism (“delay shift”) assumes that the transmission delays themselves are altered [7], [5]. This mechanism is possible because transmission velocities in the nervous system can be altered, for example, by changing the length and thickness of dendrites and axons, the extent of myelination of axons, or the density and type of ion channels. The second mechanism (“delay selection”) supposes that a range of delay lines are present in the beginning from which during development appropriate subsets become selected [6]. Here we introduce a novel framework to describe the dynamics of self-organized delay adaptation expressed in the form of integro-differential equations which permit the mechanisms of delay adaptation to be explored in a precise manner.

Section snippets

Model

Consider a neural network consisting of a large number of presynaptic neurons and one postsynaptic neuron which receives its input via delay lines, τi (Fig. 1a). The input I of the postsynaptic neuron at time t reads I(t)=∑i,kωiδ(t−(kT+τi))E(t), where ωi denotes the efficacy of synapse i,δ is the Dirac delta distribution, and ∘ denotes the convolution with an excitatory postsynaptic potential (EPSP), E(t). For our analysis, we assume that the presynaptic neurons fire synchronously at times kT

Delay shift

In this case, the weights are not modified and the source term, Q(τ,t), on the right-hand side of (1) vanishes. The dynamics are governed by (2), where the drift velocity, v=dτ/dt, of the delays realizes the Hebbian adaptation,v(τ,t)≔γτ−∞Wτ(τ−τ′)P(τ′,t)dτ′and γτ denotes the learning rate. For delays τ where ρ(τ,0)≠0 we assume ω(τ,0)=1 without loss of generality, and , imply that ω(τ,t)=1 for all t if ρ(τ,t)≠0.

For the distribution of spike times we assume a linear neural response, P(τ,t)≅βJ(τ,

Delay selection

For pure delay selection, the drift velocity of the delays, v(τ,t), vanishes and the total input of the postsynaptic neuron is not conserved. , result inρ(τ,t)∂ω(τ,t)∂t=Q(τ,t).From a straightforward generalization of the Hebb rule we obtain the source densityQ(τ,t)=γωω(τ,t)ρ(τ,t)−∞Wω(τ−τ′)P(τ′,t)dτ′with γω denoting the corresponding learning rate. Without loss of generality we assume ρ(τ,0)≡1 which implies ρ(τ,t)≡1 for arbitrary t because v(τ,t)≡0. Eq. (4) has an equilibrium solution ω(τ,t)=ω

Discussion

Due to the delay adaptation mechanism, the postsynaptic neuron becomes sensitive to a certain spatio-temporal input pattern. The idea of an ensemble coding in the nervous system comprises the notion that neurons are involved in multiple tasks. For the temporal coding this requires that they be sensitive to more than one spatio-temporal pattern. In a numerical study we now demonstrate that the delay shift learning rule is capable of adjusting two input patterns for the same postsynaptic neuron.

Christian W. Eurich got his Ph.D. in Theoretical Physics in 1995 from the University of Bremen (Germany). As a postdoc, he worked with John Milton and Jack Cowan at the University of Chicago, and he spent some time at the Max-Planck-Institute for Fluid Dynamics in Göttingen (Germany). In 1997, he returned to the Department of Theoretical Neurophysics at the University of Bremen. His research interests include neural networks with time delays, visuomotor behavior in amphibians, information

References (13)

  • C.E. Carr

    Processing of temporal information in the brain

    Annu. Rev. Neurosci.

    (1993)
  • R. Eckhorn et al.

    Coherent oscillations: a mechanism of feature linking in the visual cortex?

    Biol. Cybernet.

    (1988)
  • C.M. Gray et al.

    Oscillatory responses in cat visual cortex exhibit inter-columnar synchronisation which reflects global stimulus properties

    Nature

    (1989)
  • R. Eckhorn et al.

    Feature linking via synchronization among distributed assemblies: simulations of results from cat visual cortex

    Neural Comput.

    (1990)
  • C.W. Eurich et al.

    Dynamics of self-organized delay adaptation

    Phys. Rev. Lett.

    (1999)
  • W. Gerstner et al.

    A neuronal learning rule for sub-millisecond temporal coding

    Nature

    (1996)
There are more references available in the full text version of this article.

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Christian W. Eurich got his Ph.D. in Theoretical Physics in 1995 from the University of Bremen (Germany). As a postdoc, he worked with John Milton and Jack Cowan at the University of Chicago, and he spent some time at the Max-Planck-Institute for Fluid Dynamics in Göttingen (Germany). In 1997, he returned to the Department of Theoretical Neurophysics at the University of Bremen. His research interests include neural networks with time delays, visuomotor behavior in amphibians, information processing in neural populations, avalanche phenomena in neural networks, and motor control problems such as balancing tasks and postural sway.

Klaus Pawelzik finished his Ph.D. in Theoretical Physics in 1990 at the J-W-Goethe University (Frankfurt, Germany). He became fellow post-doc at the Max-Planck-Institute for Brain Research in 1991 and joined the Nonlinear Dynamics Group of Prof. Theo Geisel at the Institute for Theoretical Physics in Frankfurt. He worked at the Computational Neurobiology Lab headed by Terry Sejnowski at the Salk Institute, San Diego in 1994/1995. He continued his work on theoretical aspects of dynamics and coding in neural systems in 1996 at the Max-Planck-Institut für Strömungsforschung (Göttingen, Germany) until in 1998 he became Professor for Theoretical Physics and Biophysics at the University of Bremen. The range of his interests includes models of the visual system and the hippocampus, networks of spiking neurons, dynamics of synapses, neural coding, data analysis, artificial neural networks, and robotics.

Udo Ernst is currently finishing his Ph.D. in Theoretical Physics at the J-W-Goethe University (Frankfurt, Germany). Since 1997 he also works at the Max-Planck-Institut für Strömungsforschung (Göttingen, Germany). His interests cover temporal coding and nonlinear dynamics in neuronal systems, synchronization and oscillation phenomena, and dynamics and organization of receptive fields in the visual cortex.

Andreas Thiel studied Physics at the University of Marburg (Germany). In 1998, he finished his Diploma thesis about self-organizing connections between orientation detectors. Since 1999, he is a Ph.D. student at the University of Bremen.

Jack D. Cowan finished his Ph.D. in Electrical Engineering in 1967 at the Imperial College of Science and Technology in London. In 1967, he became Professor of Mathematical Biology at the University of Chicago. Since then, he has held several professoral positions there, including positions at the Collegiate Divison of Biology, the Department of Biophysics and Theoretical Biology and the Department of Neurology. In 1989, he became External Professor at Santa Fé Institute. Jack Cowan currently is Professor of Applied Mathematics and Theoretical Biology at the University of Chicago. His research in neurobiology focuses on the development and regeneration of eye–brain connections, the architecture of primate visual cortex, on hallucinations, epilepsies and visual migraines. His interests in applied mathematics include local bifurcation theory, bifurcation in the presence of symmetry and stochastic nonlinear processes with applications to neurobiology.

John G. Milton got his Ph.D. in Biophysical Chemistry in 1975 from McGill University (Montreal, Canada). After working in Japan and France as a postdoc, he received his MDCM from McGill University in 1982. From 1987 until 1988 he was Assistant Professor at the Department of Physiology at McGill University. After various guest faculties in Canada and the USA, in 1989 he became Adjunct Professor at the Center of Nonlinear Dynamics in Physiology and Medicine at McGill University. Since 1996, John Milton is also Associate Professor at the Department of Neurology at the University of Chicago. His interests in research include biophysical systems, especially those with delays like the pupil light reflex or postural sway, dynamical diseases and spiking neurons.

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