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«Abstract. All biological organisms must be able to regulate certain essential internal variables, e.g. core body temperature in mammals, in order to ...»

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Hysteresis and the Limits of Homeostasis: From

Daisyworld to Phototaxis

James Dyke and Inman Harvey

Evolutionary and Adaptive Systems Group,

Centre for Computational Neuroscience and Robotics

Informatics, School of Science & Technology

University of Sussex, BN1 9QH, UK

j.g.dyke@sussex.ac.uk, inmanh@sussex.ac.uk

Abstract. All biological organisms must be able to regulate certain essential

internal variables, e.g. core body temperature in mammals, in order to survive.

Almost by definition, those that cannot are dead. Changes that result in a mammal being able to tolerate a wider range of core body temperatures make that organism more robust to external perturbations. In this paper we show that when internal variables are regulated via ‘rein control’ mechanisms, decreasing the range of tolerable values increases the area of observed hysteresis but does not decrease the limits of regulation. We present circumstances where increasing the range of tolerable values actually decreases robustness to external perturbation.

1 Introduction In a biological context, the term homeostasis is applied to the inherent tendency in an organism toward maintenance of physiological stability. For example, mammals must maintain core body temperature to within a certain range if they are to survive.

Mechanisms to maintain a minimum core body temperature could be shivering and reduction of blood circulation to the extremities. If core body temperature increases to the upper limits of this viability range, then sweating and dilation of capillaries will lower core body temperature. Following Ashby [1] we define such internal variables as essential variables. Furthermore we define the tolerance - the range of values that the essential variable must be maintained within - as the essential range. For example core body temperature in Homo sapiens must be maintained within the essential range of approximately 35-41 degrees Celsius.

In this paper we will argue that for systems that are regulated via ‘rein control’ (as discussed below), decreasing the essential range may not decrease the range over which homeostasis is performed. We will demonstrate that increasing the essential range may actually decrease robustness to external perturbation. Clynes [2] postulated that many physiological homeostatic processes operate on the basis of opposing control reins that each pull in a single direction in response to certain variables; in order to regulate for both upper and lower limits, two reins, two separate mechanisms M. Capcarrere et al. (Eds.): ECAL 2005, LNAI 3630, pp. 241 – 251, 2005.

© Springer-Verlag Berlin Heidelberg 2005 242 J. Dyke and I. Harvey are required. For a physiological application of rein control see Saunders et al [4] who employ Clynes’ rein control analysis to understand the mechanisms responsible for the regulation of blood sugar levels in humans.

We will show that altering the essential range changes the area of hysteresis.

Hysteresis will be observed in a system when the output or behaviour is bistable as an input parameter that represents some property of the system is changed through some range of values; when the input is increased through that range, the output is one function of the input, yet if the input decreases, the output is a different function of the input, thus tracing out a ‘hysteresis loop’.

In order to explore these issues, analysis will be carried out on the behaviour of a simple two-box ‘cable car’ model that performs phototaxis. This model is based on the radically simplified Daisyworld model, as detailed by Harvey in [3]. The original Watson & Lovelock Daisyworld model [5] was intended to demonstrate the homeostatic properties of a planet that is covered with varying proportions of black and white daisies. Watson & Lovelock employ the Stefan-Boltzmann law to determine the temperatures of the daisies and bare earth. While such an approach involves a non-linear change in temperature with respect to absorbed energy, the relationship between the albedo and the temperature of a body is straightforward;

given a fixed amount of short-wave radiation, the lower the albedo, the darker the body, the less radiation is reflected and so the higher the temperature. When the star that heats the planet is dim, the planet is cool. Black daisies, having a lower albedo than either white daisies or the bare earth absorb more of the radiated short-wave energy from the star and so will be warmer than either white daisies or bare earth. If the brightness or luminosity of the star steadily increases, then black daisies will begin to grow. As the proportional coverage of black daisies increases, the net albedo of the planet decreases. This raises the temperature and so increases the rate of daisy growth. The result of this positive feedback is a population explosion of black daisies and a sharp increase in the planetary and daisy temperatures. If luminosity continues to increase, the planet eventually becomes too warm for black daisies to be able to grow. Only white daisies are cool enough to survive as they reflect a greater proportion of the incoming solar radiation. In this way, the black and white daisies regulate the planetary temperature, keeping it within the essential range over a wider range of luminosities than would be the case with a bare lifeless planet.





Rather than formulating an

Abstract

model of a homeostatic system, we instead follow the precedent of Daisyworld and present the cable car model in the form of a ‘parable’. To that end, the simplification process begun in [3] is taken further. The cut-down ‘toy’ physics is reduced to simple linear responses to a light source position whilst the relationship between temperature and albedo is dispensed with. These further simplifications will aid investigation into the relationship between homeostasis, hysteresis, essential range values, and in particular demonstrate that increasing the essential range of the model decreases the area of observed hysteresis but does not increase the limits of self-regulation. Furthermore there are circumstances where increasing the range of tolerable values actually decreases the limits of self-regulation.

Hysteresis and the Limits of Homeostasis: From Daisyworld to Phototaxis 243

1.1 Organisation of Paper

The cable car model will be introduced and compared to Daisyworld in the following section. Both models are composed of two control reins, loosely coupled via their interaction with a shared external driving force. Results from the cable car model will be presented in Section 3. Section 4 will analyse the results. Section 5 concludes the paper.

2 The Cable Car Model The model is based on the cable cars found in San Francisco. Unlike the systems used in the Alps and other mountainous regions, the San Francisco system consists of cables that are located under the road surface and connect to tram like cars. In our model a photovoltaic cell – a ‘solar panel’ – is located on the roof and supplies current to an electric motor which instead of being located in a winding house, is carried within the cable car itself. As the motor turns, it pulls in a cable that moves the car up the side of a valley. The output of the solar panel, and therefore the force that the motor produces, changes linearly with varying inclination from a moveable light source. When the light source is directly overhead, maximum output is produced and so maximum motor output is achieved. Deviations left or right by either the cable car or light source result in decreasing energy production. The range of light source locations that produce current in the solar panel we call the activation range and is analogous to the essential range of viable daisy temperatures in Daisyworld. It is assumed that the light source is so far away (e.g. the sun) that the energy input depends solely on relative angle to vertical, and any distance change is irrelevant.

This does not make any substantive difference to the behaviour of the model but does allow easier analysis.

Fig. 1. As the light source enters the activation range of the solar panel, the motor rotates anticlockwise, pulling on the cable which moves the car to the left and so up the valley slope. The gradient of the slope can be understood to increase non-linearly, e.g. the valley has a ‘U’ shape and so the car experiences an increasing ‘resisting’ force due to gravity as it moves further from its starting position.

As the light moves into the left-hand side of the activation range, the solar panel will begin to produce current. The motor will turn anti-clockwise pulling in the cable 244 J. Dyke and I. Harvey and so move the car to the left. This will bring the light nearer the centre of the solar panel’s activation range and further increase the motor output, and move the car further to the left, higher up the valley slope. The cable car has a dimensionless mass of 1 unit. As the car moves further from its starting position, the gradient of the slope increases and so the ‘resisting’ force due to gravity pulling the car back to the bottom

of the valley, γ, increases:

–  –  –

Where X is the position of the car in dimensionless x-units, l is the length of the slope and η measures the rate of increase of the resisting force as the car travels higher up the slope. For the simulations presented in this paper η was set to 1 and so γ increases linearly from 0 when the car is at the bottom, to 1 when the car is at the top of the valley.

α is the force produced by the motor. This was set to vary linearly from 0 to a

maximum of 1 in response to the output of the solar panel:

–  –  –

where Xlight is the location of the light source and ϕ is the activation range of the solar panel. This produces a ‘witches hat’ shaped activation function that can be understood as a piece-wise linear version of the original Daisyworld parabolic growth function.

0.9 0.8

–  –  –

0.6 0.5 0.4 0.3 0.2 0.1

–  –  –

Fig. 2. The output of the solar panel is maximized and so motor output is greatest when the light is directly overhead. As the light moves away from this point, motor output decreases linearly.

The model is completed with the introduction of another cable car that moves up the opposite side of the valley.

The energy provided by the solar panel turns the motor clockwise and so the car moves to the right. A spring is attached between the cars. As the cars move apart, the spring is stretched and a force is exerted that pulls the cars back together. This force is

found with:

Hysteresis and the Limits of Homeostasis: From Daisyworld to Phototaxis 245

F = ς ( X right − X left ). (1.3)

Where ς is the ‘elasticity’ of the spring and is parameterized from 0 (infinitely elastic, giving F = 0), to 1 (completely rigid so that both cars move as a single unit). It is important to note that F is based on the horizontal distance between the cars as measured in x-units. This will differ from the ‘actual’ distance due to the changing gradient of the valley slope. Such a difference does not make any substantive difference to the results, but does allow simpler computations. Table 1 lists the parameters of the cable car and Daisyworld models and allows a comparison of the two.

Fig. 3. The left car motor pulls to the left, whilst the right car motor pulls to the right.

Depending on the strength of the spring, both cars will move independently or together. The solar panels remain pointing straight up irrespective of the orientation of the cable cars.

Table 1. A comparison of cable car model and Daisyworld parameters

–  –  –

3 Results Steady state values over a range of light source positions were found numerically by

employing the following algorithm:

1. Calculate the energy produced by the solar panels from the angle of inclination of the light source and therefore the force of rotation of the motors.

2. Calculate the resistance pulling both cars back to the bottom of the valley.

3. Calculate the car’s new positions as a sum of the motor output and resisting forces.

246 J. Dyke and I. Harvey

4. Calculate the force exerted by the coupling spring connecting both cars.

5. Move the cars towards a point midway between them in proportion to the coupling spring force.

6. Go back to 1.

For each time-step, the light source position remained fixed whilst steps 1-6 were iterated until changes in the position of the cable cars were vanishingly small as calculated at double floating point accuracy. In practice 10,000 iterations were sufficient. The width of the activation range was set to 20 x-units. The width of the valley was set to 100 x-units. The top of the left hand slope was located at x = 0, the bottom at x = 50 and the top of the right hand slope at x = 100. Motor output varied from 0 to a maximum of 1. η was set to 1 in order that γ varied linearly from 0, to a maximum resisting force of 1 when a car was at the top of either slope. The strength of the coupling spring was set to 5%, ς = 0.05.

–  –  –

Fig. 4. Solid lines show the car’s horizontal position (top plot) and motor output (bottom plot) when the light is moving left to right (forwards). Dashed lines show positions and motor output when the light is moving right to left (backwards). Arrows indicate the hysteresis loops.



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