Stabilization of causally and non-causally coupled map lattices

Abstract

Two-dimensional coupled map lattices have global stability properties that depend on the coupling between individual maps and their neighborhood. The action of the neighborhood on individual maps can be implemented in terms of "causal" coupling (to spatially distant past states) or "non-causal" coupling (to spatially distant simultaneous states). In this contribution we show that globally stable behavior of coupled map lattices is facilitated by causal coupling, thus indicating a surprising relationship between stability and causality. The influence of causal versus non-causal coupling for synchronous and asynchronous updating as a function of coupling strength and for different neighborhoods is analyzed in detail.

From the Dynamics of Coupled Map Lattices to the Psychological Arrow of Time

“…A specific class of such models, particularly proposed for brain studies by Kaneko and collaborators [10], is called coupled map lattices (CMLs). In the remainder of this section we will briefly introduce their main features and, in the following section, summarize recent results [11] of a surprising relation between their stability and the extent to which their internal interactions are causal in the sense that past events effectuate future events. Insofar as (1) this neurobiological causality is necessary for stable neuronal assemblies, and (2) the stability of neuronal assemblies is necessary for them to be correlates of mental representation, it will be argued that a psychological arrow of time emerges from the neurobiological level of description.…”

“…If the interaction can be regarded as instantaneous, ∆t ≈ 0, the situation can be approximated by α = γ = 0 and β = 1. Such a type of coupling, sometimes called "future coupling" [13], will be referred to as non-causal coupling [11] in the following, since the simultaneity of the interaction between vertex and neighbors makes the distinction of cause and effect impossible.…”

“…In this way, past states in the neighborhood of a vertex are considered to act on the present state of the vertex with limited signal speed, so that the effect of an interaction is delayed with respect to its cause. Such a type of coupling will therefore by denoted as causal coupling in the following [11].…”

“…Atmanspacher and Scheingraber [11,16] showed that unstable fixed points at individual vertices can be inherently stabilized as a consequence of their coupling to neighboring unstable fixed points. In the context of the present paper, this stabilization of fixed points is particularly interesting because fixed-point solutions are always both time-translation and time-reversal invariant.…”

“…Following up on earlier investigations of CMLs using the logistic map at r = 4 with synchronous and asynchronous updating for small neighborhoods, Atmanspacher and Scheingraber [11] presented numerical results for the stabilization of CMLs for different degrees of causal versus non-causal coupling. As mentioned above, the limiting cases are instantaneous interactions β = 1; non-causal coupling), finite-time interactions (α = 1; causal coupling), and backward-in-time interactions (γ = 1; anti-causal coupling).…”