Stomatal patchiness and task-performing networks

Abstract

Background: Patchy stomatal conductance is a poorly understood and little-studied phenomenon. It is relatively common, yet it appears to be detrimental to water-use efficiency under some conditions and has no immediately obvious physiological function of any kind. Much of the difficulty in studying patchy stomatal conductance is tied to its unpredictability, both in occurrence and in characteristics.

Scope and conclusions: Statistical analyses of the variability of stomatal patchiness reveal remarkable similarities to structures and behaviours found in locally connected networks of dynamic units that perform tasks. Such systems solve problems that reside at the level of the entire network despite the absence of a central processor or a mechanism for directly sharing information over the entire system. Frequently, task performance is emergent, in the sense that no unit independently performs the task. Because each unit in the network can communicate with only its immediate neighbours, problem solving is accomplished by the states of the individual units self-organizing into synchronized, collective patterns. In some cases, patches of states form and move coherently over the network, thus providing a means for distantly separated parts of the network to communicate. Often, exactly what form these patches take and how they move as the units synchronize is highly unpredictable. In analogy with such networks, it is suggested that stomatal patchiness may be a signature that plants optimize gas exchange in a more sophisticated and adaptive manner than if performed by their individual stomata independently.

Fig. 1.

Top row: images of chlorophyll fluorescence from a leaf of Xanthium strumarium following a sudden decrease in ambient humidity at time zero. The image represents a leaf area of approximately 4 cm². Brighter regions represent portions of the leaf with lower photosynthesis and therefore lower stomatal conductance. Conductance patches appear within the first hour and move about the leaf for several hours thereafter. Bottom row: pixels that tend to brighten over a period of a few minutes are coloured red, whereas those that tend to dim are coloured yellow. Pixels showing little change are black. These coloured structures propagate coherently over the leaf surface. The movement of the patches is best seen in the supplementary video clips available online at http://aob.oxfordjournals.org.

Fig. 2.

Diagrammatic representation of the majority classification task. The system consists of a two-dimensional network of units (A), represented in the diagram as squares, each of which (B) receives input from itself and its four nearest neighbours. Each unit can exist in one of two states, denoted as black and white in this diagram. The network starts with some distribution of states, and the goal of the dynamics is to determine which state is in the majority. The dynamic solution proceeds by each unit updating its state according to a simple rule that depends on its own state and the states of its neighbours (C). The task is performed successfully when all units assume the state (D) (possibly after many time steps) that was initially in the majority (white, in the example shown).

Fig. 3.

Majority classification by a locally connected network. The network consists of 512 × 512 units and operates as described in Fig. 2 and the text. With 15 % of the units initially in the ‘white’ state, the system shows essentially no patches and reaches the correct answer in fewer than ten time steps. With 40 % of the units initially in the white state, the system takes longer to reach the correct answer and stationary ‘patches’ of states appear before the process is complete. With 45 % of the units initially in the white state, complex moving patches emerge, and the system does not reach the solution until almost 70 time steps. A different random arrangement of initial states (still with 45 % white) yields longer-lasting and qualitatively different-looking behaviour.