Currently, in brain theory, there are two broad approaches to the problem of neural code. It is customary to assume that the fundamental unit of computation in the brain is the single neuron whose activity is in the form of a spike train. According to the rate code approach, the neural code is in the form of spike frequency computed over a convenient time window; whereas according to the spike time code, the neural code is in the form of spike timing and the exact time of occurrence of the spikes matter. There is, however, a third school of thought that considers not a single neuron but a neural ensemble as the fundamental unit of computation in the brain. The average activity of a neural ensemble then, is not a spike train but a smoother signal, an oscillation, whose analysis lends itself to tools from traditional signal processing. Thus, the brain may be viewed as a network of neural ensembles and brain function (sensory-motor, cognitive, affective, autonomic) can be expressed in terms of synchronized oscillations of the neural ensembles.
Nonlinear oscillators present themselves as convenient tools to model the synchronized oscillations of neural ensembles. However, nonlinear oscillator networks are typically used to model almost exclusively rhythmic functions of the brain like locomotion, oscillatory feature binding in perception, or general synchronization phenomena of the brain. General oscillatory network models that can describe a wide range of input/output behavior of the brain are for all practical purposes non-existent. To construct such models, it is necessary to invent general trainable oscillatory networks with universal computation properties.
The aim of the current thesis, therefore, is to create a general class of trainable oscillatory neural networks to model a wide range of neural phenomena and input/output behavior. To this end, we describe three studies: (i) a general trainable oscillatory neural network consisting of Hopf oscillators. In order to achieve a stable phase relationship among the oscillators, we proposed a new form of coupling among the oscillators known as the power coupling. We then show that a network of Hopf oscillators with power coupling is able to learn arbitrary time series by performing a Fourier-like decomposition of the target signal. Using this generative model, we show that we can reconstruct multidimensional EEG signals accurately. (ii) The previous study is an example of supervised learning applied to the proposed oscillatory network. In the second study, we describe an example of the oscillatory network trained by unsupervised learning. Specifically, we describe an oscillatory variation of the well-known unsupervised learning model the Self-Organizing Map, known as the Oscillatory Tonotopic Self-Organizing Map (OTSOM). The OTSOM describes the tonotopic organization observed in the primary auditory cortices of mammals. Extensive work has been done on the primary auditory cortices of bats. The brains of these animals which navigate by echolocation are endowed with a rich, hierarchical auditory system. OTSOM is modeled using a 2D array of adaptable Hopf oscillators, called the CAO (Cortical Array of Oscillators), which receives projections from a reference oscillator (SRO). We have shown that the modified power coupling synapse unilaterally connecting the CAO oscillators to the SRO can provide the CAO oscillators with a unique ability to encode the phase offset of the perturbing complex sinusoidal signal. While the frequencies of the CAO oscillators self-organize along one dimension linearly, the characterizing angles self-organize along the orthogonal dimension. (iii) In the third study, we describe a more general multilayer oscillatory neural network and develop it as a general model of locomotion rhythms known as, the Oscillatory Central Pattern Generator Network, (OCPGN) to model the limb oscillations of universal gait patterns observed in quadruped animals. The limb oscillations for a given gait pattern are characterized by the relative phases among them. The OCPGN model has two hidden layers of Hopf oscillators, connected through unidirectional power coupling connections. This network takes as input a static vector that represents the particular locomotor rhythm to be generated and produces the corresponding rhythm as the output.