Learning-rate-dependent clustering and self-development in a network of coupled phase oscillators

Abstract

We investigate the role of the learning rate in a Kuramoto Model of coupled phase oscillators in which the coupling coefficients dynamically vary according to a Hebbian learning rule. According to the Hebbian theory, a synapse between two neurons is strengthened if they are simultaneously co-active. Two stable synchronized clusters in anti-phase emerge when the learning rate is larger than a critical value. In such a fast learning scenario, the network eventually constructs itself into an all-to-all coupled structure, regardless of initial conditions in connectivity. In contrast, when learning is slower than this critical value, only a single synchronized cluster can develop. Extending our analysis, we explore whether self-development of neuronal networks can be achieved through an interaction between spontaneous neural synchronization and Hebbian learning. We find that selfdevelopment of such neural systems is impossible if learning is too slow. Finally, we demonstrate that similar to the acquisition and consolidation of long-term memory, this network is capable of generating and remembering stable patterns.

Phase-Coupled Oscillators with Plastic Coupling: Synchronization and Stability

“…Now observe that matrix (49) is the Jacobian of system (37). It remains to show that this matrix can have only real eigenvalues.…”

“…In the above equations and further in the proof, diag(x), where vector x ∈ R m , denotes a diagonal matrix of size m × m with elements of x in its diagonal. The main idea of the proof is to show that at the equilibria of system (37), the Jacobian does not have eigenvalues with zero real part. To show this we first prove that the Jacobian is invertible at each equilibrium for almost all values of parameters α and q, and then demonstrate that all its eigenvalues are real.…”

“…and α ∈ O α . Notice that H(µ, K, q) = 0 only at the equilibria of system (37). Next, the Jacobian of H(µ, K, q) is…”

“…We observe that the second matrix in the product (50) is indeed symmetric since A T 2 = A 3 , and A 1 is symmetric. To summarize the results, for almost all randomly selected system parameters α ij and q ij , the Jacobian of system (37) at each equilibrium cannot have eigenvalues with zero real part, and system (37) converges to a stable equilibrium almost surely, so does system (2).…”

Synchronization in phase-coupled Kuramoto oscillator networks with axonal delay and synaptic plasticity

“…We implement a version of the Hebbian learning rule into the model that increases connectionstrengths between those oscillators whose phases are close in value [16,19,25]. Thus, we define a second set of differential equations,…”