The Daisystat: A model to explore multidimensional homeostasis James Dyke Max Planck Institute for Biogeochemistry, 07745 Jena, Germany [email protected] Abstract over aeons whereas the Homeostat was built from four de- commissioned Royal Air Force bomb aiming devices and The Homeostat was a physical device that demonstrated operated at millisecond speed) both systems exhibit very Ashby’s notion of ‘ultrastability’. The components interact in such a way as to maintain sets of essential variables to similar behaviour that can be observed in the Daisystat. within critical ranges in the face of an externally imposed In the following sections, the Homeostat and Daisyworld regime of perturbations. The Daisystat model is presented models will be described. The Daisystat is then presented that bears a number of similarities to Ashby’s Homeostat but and two sets of results shown. The first set shows how a which can also be considered as a higher dimensional version single-environmental-variable-Daisystat responds to a pro- of the Watson & Lovelock Daisyworld model that sought to explain how homeostasis operating at the planetary scale may gressive driving perturbation, the second set shows how a arise in the absence of foresight or planning. The Daisystat four environmental variable Daisystat responds to instanta- model features a population of diverse individuals that af- neous shocking perturbations. The establishment and main- fect and are affected by the environment in different ways. tenance of homeostasis in both cases is given in terms of The Daisystat model extends Daisyworld in that homeostasis ‘rein control’. It will be shown that the behaviour of the pop- is observed with systems comprised of four environmental variables and beyond. It is shown that the behaviour of the ulation is analogous to the behaviour of the electromechan- population is analogous to the ‘uniselector’ in the Homeostat ical Homeostat in that the volume of possible connections in that rapid changes in the population allows the system to between elements of the system is ‘searched’ until new feed- ‘search’ for stable states. This allows the system to find and back values are found that produce homeostatic states. Such recover homeostatic states in the face of externally applied a process is the result of natural selection operating on a perturbations. It is proposed that the Daisystat may afford insights into the evolution of increasingly complex systems population of diverse individuals. No notions of higher level such as the Earth system. selection, altruism or kin selection are required to explain the homeostatic behaviour of the system. The ‘law of requi- site variety’ (Ashby, 1956) is seen operating in the Daisystat Introduction in that there are lower bounds for the amount of genetic and This paper introduces a new model that demonstrates home- phenotypic diversity in the population in order for homeosta- ostasis in the face of external perturbations: the Daisys- sis to be established and maintained. It is proposed that the tat. The Daisystat is a hybrid of ‘Daisyworld’ and ‘Home- Daisystat can be used as a tool to explore the evolution and ostat’ as it shares salient features with both models. The emergence of real world complex systems such as the Earth Daisyworld model (Lovelock (1983); Watson and Love- system. lock (1983)) was initially intended as a cybernetic proof of concept for planetary homeostasis as formulated in The Homeostat Gaia Theory which proposed that the Earth system (where The Homeostat was an electromechanical device designed ‘Earth system’ is defined as the Earth’s atmosphere, oceans, and constructed by W. R. Ashby. The Homeostat consisted cryosphere, lithosphere and biota) was a homeostatic entity of four units. Each unit produced an output that was fed into that maintained conditions to within the range that allowed the inputs of the other units and back to itself via a recurrent widespread life (Lovelock, 1979). The Homeostat was a connection. Fig. 1 shows a schematic of the Homeostat units physical device that exhibited ultrastability - the ability to and their connections. The inputs into the ith unit, Ii, are the respond to a particular regime of perturbations in such ways sum of the outputs of the other units multiplied by a set of as to maintain certain essential variables to within essen- input weights: tial ranges (Ashby, 1960). While the spatial and temporal j=4 scales of Daisyworld and the Homeostat are very different X Ii = Oj!j;i (1) (Daisyworld considers self-regulation at a planetary scale j=1 1 Unit 1 Unit 2 0.5 0 Output Unit 3 Unit 4 −0.5 −1 0 50 100 150 200 250 300 Figure 1: Schematic of the Homeostat and the connections Time between units. Double arrow headed lines represent the two connections that link two units. Each unit has an output con- Figure 2: Output of the four Homeostat units. The third nection to the other three units and one recurrent connection unit (second from bottom) is perturbed at Time = 200 by to itself. decreasing its output by 1. This drives the unit outside of its essential range of [-0.5,0.5] and actuates the uniselector that where !j;i is the weight for the connection from the jth unit creates a new set of random weights. This produces large to the ith unit. A weight can either increase or decrease changes in all other units and actuation of their uniselectors a connection input. Each unit has a target value, T . The until a new stable state is achieved. unit’s output, O, is the difference between the input and tar- O = T − I get value: . This represents the first level of Daisyworld homeostatic control in the Homeostat. The second level of control is derived from the establishment of essential ranges While Daisyworld is a simple model of a planetary system, for the output of the units. If the output of a unit moves it is more complicated than the Homeostat with a number of outside of the essential range, then a uniselector component different feedback mechanisms that feature non-linear func- randomly generates connection weights for that unit until the tions. However, at its heart it is similar in that two units unit output moves back within the essential range. For ex- in the form of two species or type of plants (commonly re- ample, if the essential range is [-0.5,0.5] and O = 0:6 then ferred to as ‘daisies’) exert unidirectional effects on a regu- the uniselector would generate new weights for all connec- lated variable in the form of planetary temperature. These tions into that unit until the output moves back within the effects stem from the different albedo of the daisies. Albedo essential range. The Homeostat demonstrated ultrastability is a measure of the reflectivity of an object. Black daisies that was a consequence of Ashby’s law of requisite variety. have lower albedo than white daisies. Changing the relative In order for the Homeostat to maintain stable states in the proportion of black and white daisies will affect the plane- face of perturbations, it must be able to reconfigure itself in tary albedo and so the global temperature. The black and at least as many ways as these perturbations demand. Con- white daisies share the same parabolic growth response to sequently, the volume of possible connection weight values temperature. Both grow at maximum rates when their local must encompass all possible values that would be required temperatures are 22.5◦ Celsius with growth progressively to produce stable states. decreasing, until it is zero when the temperature is 5◦ or 40◦ Homeostat simulations start by having the uniselectors for Celsius. each unit create random weights. This produces initially Daisyworld simulations consist of seeding a grey planet chaotic behaviour whereby one unit drives another unit out that has an intermediate albedo of 0.5 with black and white of its essential range which responds with new uniselector daisy seeds. This planet orbits a star much like the sun values which may drive another unit of of its essential range which over geological time scales increases in luminosity or and so on. Given sufficient iterations of the uniselector pro- brightness. On a lifeless planet, as the star increases in lu- cess, a set of weights will be generated that proves to be minosity, the temperature increases approximately linearly stable in that the outputs of all units remain within their es- (the actual temperature response being a quartic function of sential ranges. An example Homeostat simulation is shown luminosity). The situation is markedly different when black in Fig. 2. The Homeostat finds a stable state and is then per- and white daisies are present in that the temperature rapidly turbed when Time = 200 by decreasing the output of one moves towards the maximum growth rate temperature and unit by 1. This leads to all units moving out of their es- then stays within the range that the daisies are able to grow sential range and a period of uniselector activity that creates over as luminosity increases. This demonstrates how plan- new random weights which produces a new attractor which etary regulation may emerge as a consequence of biologi- the system relaxes towards. cal activity that is not the result of intentional design and in ways compatible with natural selection. Fig. 3 shows a review. The Daisystat is intended to address one of these planetary temperature being regulated when both daisies are more important limitations that was succinctly identified by present and Fig. 4 show how this regulation is the result of J. Kirchner: the change in the proportional coverage of the black and white daisies. “Daisyworld is a one-feedback model; there is only one environmental variable and it is regulated by ex- tremely strong feedback with the simplest possible bio- 100 sphere.