Abstract
Recurrent neural networks with associative memory properties are typically based on fixed-point dynamics, which is fundamentally distinct from the oscillatory dynamics of the brain. There have been proposals for oscillatory associative memories, but here too, in the majority of cases, only binary patterns are stored as oscillatory states in the network. Oscillatory neural network models typically operate at a single/common frequency. At multiple frequencies, even a pair of oscillators with real coupling exhibits rich dynamics of Arnold tongues, not easily harnessed to achieve reliable memory storage and retrieval. Since real brain dynamics comprises of a wide range of spectral components, there is a need for oscillatory neural network models that operate at multiple frequencies. We propose an oscillatory neural network that can model multiple time series simultaneously by performing a Fourier-like decomposition of the signals. We show that these enhanced properties of a network of Hopf oscillators become possible by operating in the complex-variable domain. In this model, the single neural oscillator is modeled as a Hopf oscillator, with adaptive frequency and dynamics described over the complex domain. We propose a novel form of coupling, dubbed “power coupling,” between complex Hopf oscillators. With power coupling, expressed naturally only in the complex-variable domain, it is possible to achieve stable (normalized) phase relationships in a network of multifrequency oscillators. Network connections are trained either by Hebb-like learning or by delta rule, adapted to the complex domain. The network is capable of modeling N-channel electroencephalogram time series with high accuracy and shows the potential as an effective model of large-scale brain dynamics.
Highlights
There are two prominent approaches to characterizing the neural code: the instantaneous rate or frequency at which the neuron fires action potential (“the spike frequency code” or the “rate code”) and the time of the occurrence of action potential (“spike time code”)
Since existing literature does not commit to the exact size of a “neural ensemble,” activity measured at different scales goes by different names, including local field potentials (LFPs), electrocorticograms (ECoGs), electroencephalograms (EEGs), etc
The unique contribution of the proposed network of complex neural oscillators is the notion of power coupling by which it becomes possible to achieve a stable normalized phase relationship between two oscillators with arbitrary natural frequencies
Summary
There are two prominent approaches to characterizing the neural code: the instantaneous rate or frequency at which the neuron fires action potential (“the spike frequency code” or the “rate code”) and the time of the occurrence of action potential (“spike time code”). The canonical model of Hopf oscillator without any external input is described as follows in complex state variable representation state variable : z. Is (to understand the vector notations refer to Figure 1): This motivates us to adapt the learning rule for the natural frequency of the oscillator, as proposed in eq 2c, by dropping the magnitude of oscillation because of the same reason as described by Righetti et al (2006). 2a-2c and observed that the proposed learning rule for the natural frequency of the oscillator allows it to adapt to the frequency of the input complex sinusoidal signal, as shown in the following Figure 2 We have simulated the eqs. 2a-2c and observed that the proposed learning rule for the natural frequency of the oscillator allows it to adapt to the frequency of the input complex sinusoidal signal, as shown in the following Figure 2
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
