Learning of spatiotemporal patterns in Ising-spin neural networks: analysis of storage capacity by path integral methods

Theoretical Study of Oscillator Neurons in Recurrent Neural Networks

Neurons in a network can be both active or inactive. Given a subset of neurons in a network, is it possible for the subset of neurons to evolve to form an active oscillator by applying some external periodic stimulus? Furthermore, can these oscillator neurons be observable, that is, is it a stable oscillator? This paper explores such possibility, finding that an important property: any subset of neurons can be intermittently co-activated to form a stable oscillator by applying some external periodic input without any condition. Thus, the existing of intermittently active oscillator neurons is an essential property possessed by the networks. Moreover, this paper shows that, under some conditions, a subset of neurons can be fully co-activated to form a stable oscillator. Such neurons are called selectable oscillator neurons. Necessary and sufficient conditions are established for a subset of neurons to be selectable oscillator neurons in linear threshold recurrent neuron networks. It is proved that a subset of neurons forms selectable oscillator neurons if and only if the real part of each eigenvalue of the associated synaptic connection weight submatrix of the network is not larger than one. This simple condition makes the concept of selectable oscillator neurons tractable. The selectable oscillator neurons can be regarded as memories stored in the synaptic connections of networks, which enables to find a new perspective of memories in neural networks, different from the equilibrium-type attractors.

Time-periodic lattice of spiral pairs in excitable media

The feasibility of a spiral-type solution, periodic both in time and in space, of a reaction-diffusion equation (specifically the FitzHugh-Nagumo system) in an excitable medium is numerically demonstrated. The solution consists of arrays of interacting spiral pairs, which repeatedly create by partial annihilation a system of residual portions (RPs). The latter behaves as a source to the next generation of the spiral-pair array. If basic (highest) translational symmetry is not conserved, pointwise perturbations, above a certain threshold, are shown to be able to destroy the pattern after a certain transient time by changing its symmetry. If the basic translational symmetry is preserved, such perturbations do not cause destruction unless occurring at the nearest vicinity of the RP site. Singular value decomposition methods are used to analyze the structure of the pattern, revealing the importance of the spiral pairs and the RPs.

Storage of Phase-Coded Patterns via STDP in Fully-Connected and Sparse Network: A Study of the Network Capacity

We study the storage and retrieval of phase-coded patterns as stable dynamical attractors in recurrent neural networks, for both an analog and a integrate and fire spiking model. The synaptic strength is determined by a learning rule based on spike-time-dependent plasticity, with an asymmetric time window depending on the relative timing between pre and postsynaptic activity. We store multiple patterns and study the network capacity. For the analog model, we find that the network capacity scales linearly with the network size, and that both capacity and the oscillation frequency of the retrieval state depend on the asymmetry of the learning time window. In addition to fully connected networks, we study sparse networks, where each neuron is connected only to a small number z ≪ N of other neurons. Connections can be short range, between neighboring neurons placed on a regular lattice, or long range, between randomly chosen pairs of neurons. We find that a small fraction of long range connections is able to amplify the capacity of the network. This imply that a small-world-network topology is optimal, as a compromise between the cost of long range connections and the capacity increase. Also in the spiking integrate and fire model the crucial result of storing and retrieval of multiple phase-coded patterns is observed. The capacity of the fully-connected spiking network is investigated, together with the relation between oscillation frequency of retrieval state and window asymmetry.