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A dynamic neural field model of continuous input integration

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Abstract

The ability of neural systems to turn transient inputs into persistent changes in activity is thought to be a fundamental requirement for higher cognitive functions. In continuous attractor networks frequently used to model working memory or decision making tasks, the persistent activity settles to a stable pattern with the stereotyped shape of a “bump” independent of integration time or input strength. Here, we investigate a new bump attractor model in which the bump width and amplitude not only reflect qualitative and quantitative characteristics of a preceding input but also the continuous integration of evidence over longer timescales. The model is formalized by two coupled dynamic field equations of Amari-type which combine recurrent interactions mediated by a Mexican-hat connectivity with local feedback mechanisms that balance excitation and inhibition. We analyze the existence, stability and bifurcation structure of single and multi-bump solutions and discuss the relevance of their input dependence to modeling cognitive functions. We then systematically compare the pattern formation process of the two-field model with the classical Amari model. The results reveal that the balanced local feedback mechanisms facilitate the encoding and maintenance of multi-item memories. The existence of stable subthreshold bumps suggests that different to the Amari model, the suppression effect of neighboring bumps in the range of lateral competition may not lead to a complete loss of information. Moreover, bumps with larger amplitude are less vulnerable to noise-induced drifts and distance-dependent interaction effects resulting in more faithful memory representations over time.

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Acknowledgements

The work received financial support from FCT through the PhD fellowship PD/BD/128183/2016 the project “Neurofield” (PTDC/MAT-APL/31393/2017) and the research centre CMAT within the project UID/MAT/00013/2020.

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Authors and Affiliations

  1. Research Centre of Mathematics, University of Minho, Guimarães, Portugal

    Weronika Wojtak & Wolfram Erlhagen

  2. Research Centre Algoritmi, University of Minho, Guimarães, Portugal

    Weronika Wojtak & Estela Bicho

  3. Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, UK

    Stephen Coombes

  4. Department of Mathematics, Vrije Universiteit, Amsterdam, The Netherlands

    Daniele Avitabile

  5. MathNeuro Team, Inria Sophia Antipolis Méditerranée Research Centre, Sophia Antipolis, France

    Daniele Avitabile

Authors
  1. Weronika Wojtak
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  2. Stephen Coombes
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  3. Daniele Avitabile
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  4. Estela Bicho
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  5. Wolfram Erlhagen
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Correspondence to Weronika Wojtak.

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Example codes implemented in MATLAB are available at https://github.com/w-wojtak/A-dynamic-neural-field-model-of-continuous-input-integration.

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Communicated by Benjamin Lindner.

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A Numerical simulations

A Numerical simulations

Numerical simulations of the model were done in MATLAB using a forward Euler method with parameters \(L=30\), \(N=12000\), \(\mathrm{dx}=2L/N=0.005\), \(T=100\), \(M=10000\), \(\mathrm{dt}= T/M=0.01\), unless stated otherwise in specific examples. Numerical simulations of the stochastic model were done using the Euler-Maruyama method. To compute the spatial convolution of w and f we employ a fast Fourier transform (FFT), using MATLAB’s in-built functions fft and ifft to perform the Fourier transform and the inverse Fourier transform, respectively. Periodic boundary conditions are used. By choosing a sufficiently large domain size, we make sure that the localized patterns evolve sufficiently far from the boundaries.

For performing numerical continuation, we use the method described in (Rankin et al. 2014) and adapt MATLAB code available in (Avitabile 2016). The main advantage of this method is that it can be applied directly to the full integral model. This is possible due to the usage of Newton-GMRES solvers combined with a fast Fourier transform (FFT) employed for computing the convolution term (Rankin et al. 2014).

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Wojtak, W., Coombes, S., Avitabile, D. et al. A dynamic neural field model of continuous input integration. Biol Cybern 115, 451–471 (2021). https://doi.org/10.1007/s00422-021-00893-7

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