«Abstract. All biological organisms must be able to regulate certain essential internal variables, e.g. core body temperature in mammals, in order to ...»
Hysteresis and the Limits of Homeostasis:
From Daisyworld to Phototaxis
James Dyke & Inman Harvey
Evolutionary and Adaptive Systems Group
Centre for Computational Neuroscience and Robotics
Informatics, School of Science & Technology
University of Sussex, BN1 9QH, UK
firstname.lastname@example.org & email@example.com
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  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  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 are required. For a physiological application of rein control see Saunders et al  who employ Clynes’ rein control analysis to understand the mechanisms responsible for the regulation of blood sugarlevels 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 . The original Watson & Lovelock Daisyworld model  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
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  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.
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 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:
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.
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.
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:
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 form 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.
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.
4. Calculate the force exerted by the coupling cable connecting both cars.
5. Move the cars towards a point midway between them in proportion to the coupling cable force.
6. Go back to 1.
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 output when the light is moving right to left (backwards). Arrows indicate the hysteresis loops.
As the light moves from left to right, both cars move immediately away from the bottom of the valley. As the light continues to move, the left car slowly moves back down the slope whilst the right car continues to move to the right until its motor output reaches its maximum. As the light moves further, it goes past the centre of the right car’s activation range and so solar panel and motor output decreases. This moves the car to the left and so further away from the light source. The right car continues to move back down the slope until it is at rest at the bottom of the valley.
As the light reverses direction the same situation occurs but with the right car steadily decreasing in motor output and the left car motor output steadily increasing to its maximum and then abruptly falling to zero. Such behaviour is similar to the growth curves and area of hysteresis observed in Daisyworld simulations in which there is initial rapid growth, then steady decline of black daisies whilst the white daisies population slowly increases, peaks and then suddenly collapse.
Fig. 5. Time is given in dimensionless units on the horizontal axis on both plots. The top plots shows both cars initially at rest at the bottom of the valley where x = 50. They are not activated by the light source until time step 210. The light source moves back and forth along the horizontal plane. The motor outputs and positions of the cars reflect this motion.
Hysteresis is observed when simulations are performed with the light source outside of the activation range of either cable car. Once the cable cars begin to move, they are able to track the light over the entire range of light source positions. If the light source begins within the activation range of either car, or if the right/left hand car is held at its maximum distance from the bottom of the valley, and the light is introduced from the right/left, then there is no period when the cars are inactive. This produces the perhaps counter-intuitive result that decreasing the activation range may not decrease the range of light positions over which phototaxis can be achieved.
For example, the hat function could be transformed into a ‘spike’ function with the result that the light source must be directly overhead in order to produce solar panel output and so motor force.