Abstract
We study multifrequency Hebbian plasticity by analyzing phenomenological models of weakly connected neural networks. We start with an analysis of a model for single-frequency networks previously shown to learn and memorize phase differences between component oscillators. We then study a model for gradient frequency neural networks (GrFNNs) which extends the single-frequency model by introducing frequency detuning and nonlinear coupling terms for multifrequency interactions. Our analysis focuses on models of two coupled oscillators and examines the dynamics of steady-state behaviors in multiple parameter regimes available to the models. We find that the model for two distinct frequencies shares essential dynamical properties with the single-frequency model and that Hebbian learning results in stronger connections for simple frequency ratios than for complex ratios. We then compare the analysis of the two-frequency model with numerical simulations of the GrFNN model and show that Hebbian plasticity in the latter is locally dominated by a nonlinear resonance captured by the two-frequency model.
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Notes
The amount of change in connection phases \(\theta _{ij}\) after a perturbation depends on connection amplitudes \(A_{ij}\). Once the connections grow strong enough compared to the magnitude of perturbation, and when learning is slow (with small \(\gamma \) and \(\kappa \)), the plastic connections act like fixed coupling and are not altered significantly by sporadic perturbations of small amplitudes. Accordingly, the oscillators are attracted back to the previous relative phase after a small perturbation. Thus, plastic connections which are neutrally stable on a long timescale constitute an attractor on a short timescale. For the purpose of demonstrating neutral stability, the simulation shown in Fig. 2 used fast learning and a strong perturbation.
We chose the quintic coefficient to be \(-1\) because here we want to examine the stabilization of amplitude dynamics without altering phase dynamics. In fully expanded models (13) and (24), the quintic coefficient \(d_i = \beta _{2i}+\text {i}\delta _{2i}\) has both amplitude (radial) and phase (azimuthal) components.
Note that the oscillation frequency of plastic connection \({\dot{\theta }}_{ij} = -\frac{\kappa r_i r_j}{A_{ij}} \sin \psi _{ij}\) is proportional to \(\kappa \). When learning is slow (with small \(\gamma \) and \(\kappa \)), plastic connections oscillate at slow frequencies and behave like fixed coupling on a short timescale. See footnote 2 for a related discussion on the timescale of learning.
See Eq. (9), for example, where the intrinsic part of the learning equation takes a similar form as the oscillator equation, except the former does not have any imaginary terms like \(\text {i}\omega \), which can be interpreted as the natural frequency being zero. Thus, plastic connections resonate when the oscillators maintain a fixed phase difference (or phase-locked) because that is when the input term \(\kappa z_i {\bar{z}}_j\) is stationary.
Note that (13) has \(\epsilon \) in the intrinsic higher-order terms (with the coefficient \(d_i\)) as well as in the coupling terms (with \(c_{ij}\)). The original weakly connected system is considered \(\epsilon \)-perturbation of the uncoupled system, from which the canonical model is derived using averaging theory (Hoppensteadt and Izhikevich 1996a). Here, to capture resonance between distinct frequencies, the canonical model is expanded to include higher-order perturbation terms (see Hoppensteadt and Izhikevich 1997, p. 172). Hence, both the higher-order intrinsic terms and the coupling terms are expressed as powers of \(\epsilon \).
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Acknowledgements
The authors thank Parker Tichko, Karl Lerud, and two anonymous reviewers for their helpful comments on the previous versions of the paper.
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Early stages of this work were supported by National Science Foundation BCS-1027761 and Air Force Office of Scientific Research FA9550-12-10388.
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Communicated by Benjamin Lindner.
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Kim, J.C., Large, E.W. Multifrequency Hebbian plasticity in coupled neural oscillators. Biol Cybern 115, 43–57 (2021). https://doi.org/10.1007/s00422-020-00854-6
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DOI: https://doi.org/10.1007/s00422-020-00854-6