Generalised Kuramoto models with time-delayed phase-resetting for k-dimensional clocks

Abstract

We consider a class of Kuramoto models, with an array of $N$ individual $k$-dimensional clocks $(k>1)$, coupled within a directed, range dependent, network. For each directed connection, a signal triggered at the sending clock incurs a (real valued) time delay before arriving at the receiving clock, where it induces an instantaneous phase reset affecting all $k$-phases. Instantaneous phase resetting maps (PRMs) for $k$-dimensional clocks have received little attention. The system may be treated as open and subject to periodic, or other types of, PRM forcing at any individual clock, as a result of external forcing stimuli. We show how the full system, with $Nk$ phase variables, responds to such stimuli, as the impact spreads across the entire network. Beyond simulations, we employ methods to reverse engineer the dynamical behaviour of the whole: estimating the intrinsic dimensions of the responses to different experiments; and by analysing pairwise comparisons between those responses. This shows that the system’s responses are governed by a hierarchy of internal dynamical modes, existing across both the $Nk$ phases and over time.

We argue that this Kuramoto system is a model for the human cortex, where each $k$-dimensional clock models the dynamics of a single neural column, which contains 10,000 densely inter-connected neurons. The Kuramoto model exploits the natural network of networks architecture of the human cortex. An array of $N=$1M such columns/clocks is at the 10B neuron scale of the human cortex. However its simulation is far more accessible than very large scale (VLS) simulations of neuron-to-neuron systems on supercomputers. The latent modes may have important implications for cognition (information processing) and for consciousness (the existence of internal phenomenological experiences). We argue that the existence of the latter plays a key role in preconditioning the former, reducing the decision sets and the cognitive load, and thus enabling a fast-thinking evolutionary advantage.

This is the first time that systems of $k$-dimensional clocks ($k>$ 1), coupled via time-lagged PRMs, within range dependent networks, have been deployed to demonstrate the basic internal phenomenological elements (of consciousness) and their potential role within immediate cognition.

Statement of Significance: We argue that this Kuramoto system is a model for the human cortex, where each of 1M k-dimensional clocks models the dynamics of a single neural column (containing 10,000 densely inter-connected neurons). This Kuramoto model exploits the natural network of networks architecture of the human cortex. Large scale human cortex simulations, with 10B neutrons, usually require a super computer. We show that similar results, using this model, can be obtained on a laptop. In particular we show that such dynamical can support internal phenomenological elements (of conscious experience) and we discuss their potential role in preconditioning immediate cognition, furnishing a “fast thinking” evolutionary advantage to the human brain.