Norm-Establishing and Norm-Following in Autonomous Agency

Living agency is subject to a normative dimension (good-bad, adaptive-maladaptive) that is absent from other types of interaction. We review current and historical attempts to naturalize normativity from an organism-centered perspective, identifying two central problems and their solution: (1) How to define the topology of the viability space so as to include a sense of gradation that permits reversible failure, and (2) how to relate both the processes that establish norms and those that result in norm-following behavior. We present a minimal metabolic system that is coupled to a gradient-climbing chemotactic mechanism. Studying the relationship between metabolic dynamics and environmental resource conditions, we identify an emergent viable region and a precarious region where the system tends to die unless environmental conditions change. We introduce the concept of normative field as the change of environmental conditions required to bring the system back to its viable region. Norm-following, or normative action, is defined as the course of behavior whose effect is positively correlated with the normative field. We close with a discussion of the limitations and extensions of our model and some final reflections on the nature of norms and teleology in agency.


Introduction
The subject of biological agency and natural norms is attracting increasing attention [7,17,26,32,33,40,42,51,58,59], and artificial life is very well suited to make some conceptual progress on key aspects of agency and its origins. In fact, minimal models of agency have been a recurring topic in the field, from protocellular models to robotics-see [7] for references.
From bacteria to humans, the way in which living systems actively regulate their relationship with their environments strongly contrasts with inanimate objects. Whereas the behavior of living systems generally qualifies as agency, the inanimate world is a world of events. Part of this distinction between actions and events lies in the normative character of the former. Actions are subject to normative judgment: They are either good or bad, appropriate or inappropriate, adaptive or maladaptive. A bacterium might be described as failing to move up a sugar gradient, but it makes little (if any) sense to say that a planet has failed to follow an orbit. This normative dimension of agency is widespread in nature, and it continues to capture the attention of philosophers, theoretical biologists, psychologists, and roboticists alike, for it has proven to be a difficult property to define, naturalize, model, or synthesize.
A recent article [7] described normativity as one of the essential features of an agent, which is defined as "an autonomous organization capable of adaptively regulating its coupling with the environment according to the norms established by its own viability conditions" [7, p. 376]. The main goal of this article is to make explicit what is meant by the expression "according to the norms established by its own viability conditions." Similar expressions have been used [8,16,26,40,59], but no model has yet been developed to illustrate and describe in detail the meaning of this expression and others closely associated with it.
We shall start by introducing the problem of normativity and agency in more detail and identifying the two most prominent theories that are currently available for naturalizing a notion of proper, or nonderived, normativity: the evolutionary and the organizational (or organismic) accounts of normative function. We then focus on those developments of the organizational approach that have had more impact in the fields of cybernetics, artificial life, and adaptive behavior. Having set up the context and identified some key problems of previous approaches, we describe our minimal model. The model covers "norm-establishing" and "norm-following" aspects. First we describe a minimal metabolic system that allows us to define norm-establishing at the protocellular scale. A parametric analysis of changes in environmental conditions of metabolism allows us to define and quantify the notions of "precariousness," "viability," and "normativity" in terms of regions inside the viability space. Next, we move into norm-following by adding a chemotactic behavioral mechanism to the metabolic system just described. By projecting the effect of chemotactic behavior into the viability space, we can precisely define "normative behavior" and characterize an adaptive region that emerges from the relational dynamics between agent and environment. Then, we provide a set of generalized definitions that are applicable beyond our model. We end up with some possible extensions of our model and some discussion about the nature of norms and teleology in agency.
2 Agency, Adaptivity, and Normativity 2.1 The Framework: Events versus Actions, Descriptive versus Normative, and the Challenge of Natural Norms In a world of events that science describes as governed by necessary laws and principles, the concept of agency can appear to be in tension with the very method and assumptions of science. The notion of agency has been traditionally [24] formulated as behavior caused or motivated by reasons, desires, or intentions and as opposed to just antecedent events: To run home so as to protect yourself from a storm is not the same as being pushed by the wind. In a world of fundamental particles governed by laws there seems to be no room for agency-no way to formulate a concept of what processes or interactions ought to be, or are intended to be, as different from what they actually are; no room for norms that can be violated so that an agent might be said to fail. And yet we have the experience of acting, of following or failing to follow certain norms (be they biological, psychological, or social).
Ultimately, filling in the gap between fundamental physics and our own experience of agency might require the inclusion of concepts such as "will," "awareness," or "reasons" into the explanatory picture. But minimal forms of natural behavior (such as chemotaxis) encapsulate some of the most important properties of "higher" levels of agency [7], and their explanation can arguably be considered a prerequisite to ground and naturalize the very notion of agency and its more complex manifestations. It has long been argued that there is continuity between biological and human agency [15,17,33,42,56,58,59]: that human agency is a natural complexification of lower forms of agency, where "natural complexification" implies no fundamental discontinuity nor the addition of essentially different principles. Without explicitly attaching to a strong continuity thesis, however, it seems reasonable to assume that the modeling and conceptualization of biological agency (even of the simplest forms) constitutes a valuable conceptual and methodological resource for making progress toward higher forms of agency. Moreover, the case of biological agency seems more amenable to scientific experimentation, modeling, and formalization than psychological, ethical, or social forms. It therefore provides a solid departure point to investigate the nature of agency.
One of the key properties of agency that shows up across different levels is normativity, the dimension of behavior in which value comes into play. Actions are good or bad, adaptive or maladaptive, appropriate or inappropriate [8,17,21]. The issue of normativity has several manifestations and a long philosophical and theoretical history. One of its earlier formulations comes from David Hume, who noted the "impossibility" of deriving ought from is, prescriptive statements from descriptive ones. There seems to be no justified manner in which (no matter how many) statements about "X being or operating as A" can lead to a valid inference about "X having to be or having to operate as A." This has important consequences for the fixation of epistemological norms (those concerning how knowledge or truth must be achieved or justified) and ethical norms (how we ought to behave). Immanuel Kant provided an additional decisive contribution: He proposed that epistemological and ethical norms were to be found and justified as conditions of possibility. The Humean problem could thus be solved: it is not that norms about how X should be are to be derived of inferred from what X actually is, but instead from what makes possible the very existence of X. 1 In this way the normative shows itself not as an empirical fact, but as what makes possible that we apprehend empirical phenomena or ascertain reasons to act in the first place.
The issue of norms has had an enormous development in the areas of epistemology and ethics since then, but this is out the scope of the present article. What is relevant to us is the naturalist turn that some of these developments took, in contrast to those trends that argued merely in terms of abstract reasons or the very structure of logic or language use. Naturalism tries to bring philosophical problems down to the grounds of natural sciences [49]: something that can be measured. And there is an empirical manifestation of epistemic (and ethical) normativity: behavior. So, if we are to assume that normativity does not belong to a supernatural sphere, then the behavior of living systems might be a good entry point for trying to explain and model it. Of course, moving into the realm of biology does not automatically solve the theoretical challenges posed by normativity, but it does reframe the questions: When we model the mechanisms of natural behavior and they lead to a specific outcome, how can we justify the claim that the animal has failed or that it should had done something else? Where do the norms against which natural behavior is evaluated or judged come from? How do they relate to behavior-generating mechanisms? Can new norms appear in nature? How does the actual functioning, here and now, of a behavioral mechanism differ from how it ought to function? Is it possible to address these questions in a manner that is centered on the organism and not on aspects of its evolutionary history or on other agentsʼ intentions?

Intrinsic and Derived Norms in Natural and Artificial Systems
While artificial systems can be judged to operate in relation to norms, these norms have (thus far ) always been predefined by the designer of the artificial system or interpreted by an external observer or user. In other words, what is good or bad functioning for a robot, a car, or a coffee machine has been a matter of the design specifications, which are largely independent from the ongoing operation of the artifact. In a sense we can talk of derived or extrinsic normativity: The artifact does not possess or define any purpose of and by itself, but always in relation to some external agent or social context from which such a purpose or norm is derived. So, for instance, the decision of when and how to repair the artifact (i.e., what counts as broken) depends on the design specifications of the designer, the social context of its use, or simply the pragmatic intentions of the end user, but not on any intrinsic property of the system itself. The same physical object can be either broken if intended to be used for a particular purpose or perfectly fine in a different context. This is unlike biological organisms, which seem to "respond" to norms that are more closely related to their organization.
There is a possible analogy between the norms that a designer or context imposes upon an artifact and the norms that natural evolutionary "design" might impose upon organisms. And this analogy has inspired what currently stands as the mainstream approach to naturalizing norms. The most celebrated exponent of the so-called selected-function or evolutionary approach is Millikan [46]. She defends a whole philosophical program to naturalize biological, cognitive, and epistemological norms in evolutionary theory. Roughly speaking, behavior is here considered to be normative or adaptive if it has been selected by evolution. Under this view, adaptation is ultimately a result of natural selection, and it is only as a result of a process of selection that a character or process (e.g., a pattern of behavior ) can be said to be adaptive or maladaptive. What sets up the norm is the history of population dynamics. In this view an organ or a particular behavioral mechanism might be said to be broken, be maladaptive, or fail to carry out its proper function if it is currently working in a manner that was not the mode of functioning that evolution selected for. Artificial life models have contributed to clarifying and quantifying this notion of normativity and teleology [11]. 2 An alternative conception of normativity that is intrinsic, or non-derived, can be developed with criteria that are independent from history or the social context of design and use, and are instead grounded on the current organization of the system and its ongoing dynamics. This is precisely the motivation underlying the main alternative approach to normativity and adaptation. The organizational approach (as it might be called) puts the idea of autonomy (from the Greek autos = self and nomos = norm) at its center [9,25,55,60]. This approach sees the norms of a system as being determined by its present organization. This results in an inversion of the selectedfunction approach: A trait does not acquire a normative function because it has been selected, but it has been selected because it came to fill a normative function within a living systemʼs organization.

The Organismic Approach to Biological Autonomy: A Short
Historical Introduction The origins of this organism-centered or autonomous-organizational approach can be traced back to the works of Aristotle and, particularly, to Kant (Critique of Judgement) and his account of teleology as emerging from the self-organization (a term that is due precisely to him) of living systems. This trend of thought was further developed within German idealism and the so-called Naturphilosophie school, but it was not until the late nineteenth century that it started to take on a more detailed empirical taste, with the rise of cell theory and experimental physiology-for example, the work of Claude Bernard and his conception of the milieu interieur [13]. We can observe an influential integration of this organismic trend in biology in the work of Walter B. Cannon and his development of the concept of homeostasis as the "coordinated physiological processes which maintain most of the steady states in the organism" [20].
But it is perhaps the work of Canguilhem [18] that most directly addressed the issue of normativity in biology. As noted in [35], Canguilhem further develops some of the central ideas already anticipated in Kurt Goldsteinʼs masterpiece Der Aufbau des Organismus, where the issue of normativity is directly linked to the individual organism: "there is only one relevant norm; that which includes the total concrete individuality; that which takes the individual as its measure" [36, p. 269]. 3 Canguilhem rejects a statistical notion of normality and the idea that the deviation from a population mean is a valid departure point for approaching pathologies in organisms. Instead, he introduces the notion of normativity at the very core of the living individual in a sense that connects directly with the contemporary debate on norms: [L]ife is in fact a normative activity. The normative, in philosophy, includes every judgment which evaluates or qualifies a fact in relation to a norm, but this mode of judgment is essentially subordinate to that which establishes norms. The normative, in the fullest sense of the word, is that which establishes norms. And it is in that sense that we plan to talk about biological normativity. [19, pp. 126-127] A fundamental distinction introduced by Canguilhem is that between norm-establishing and normfollowing [50]. Organisms do not only operate according to norms, that is, they do not only seem to follow norms, but they also establish these norms autonomously. How exactly this occurs (and how it contrasts with the processes described by physics) was a central contribution of Hans Jonasʼ philosophical biology. Metabolism defines the organism as an individual that escapes the determination of its constituent parts by renewing them continuously, leaving the form or organization of the system as the persistent reference identity. The more complex it is, the more precarious and the more threatened by the environment a living system becomes, and yet, the more in need of its environment. To do is both the consequence and the condition of possibility of the very existence of an organism: "To be" is its intrinsic goal. Teleology comes in where the continuous identity of being is not assured by mere inertial persistence of a substance, but is continually executed by something done, and by something which has to be done in order to stay on at all: it is a matter of to-be-or-not-to-be whether what is to be done is done. [39, p. 243] In this way, Jonas shows how individuality and normativity get intertwined: norms apply to natural identities, which, unlike inanimate objects, depend on their own, continuous action of metabolic renewal and resource seeking for their ongoing existence. This recursive nature of living organization allows one to derive the norms, not directly from isolated events (as the problem was originally posed by Hume), but from the set of organized and interdependent processes that constitute an organism (i.e., its conditions of possibility).
These arguments suggest inspiring and convincing research avenues, but there remains a large gap between this philosophy and scientific modeling and experimentation. Notions of "identity," "precariousness," "preservation," or "need" require far more detailed development to be consistently assembled to deliver a clear scientific notion of normative action. It is through the relatively modern development of cybernetics, theoretical biology, and the physics and chemistry of far-from-equilibrium systems that this tradition is progressively entering, albeit partially and not without difficulties, into a proper scientific treatment of normative action.
A branch of contemporary theoretical biology, under the label of autopoietic theory or enactivism [16,26,44,60,61] and autonomous systems [8,10,22,23,42,54,55] has further elaborated some of the central tenets of the organicist school, connecting them with contemporary biology, artificial life, and cognitive science. Of particular relevance is the work by Christensen and Bickhard [21], who provide a detailed philosophical discussion of the notion of normative function within the context of contemporary philosophy of biology and cognitive science-see also [47] for a more recent version of "normative function" in biology. The central claim of these developments is that the normative nature of biological functions stems from the far-from-equilibrium and self-sustaining nature of the networked processes that constitute living organization.
However, putting these theoretical approaches to work in scientific modeling requires bridging the gap that separates conceptual and philosophical contributions from mathematical and computational modeling. The goal of this article is to bring this organicist development of the notion of normativity to a formulation that can shed light on some of its most intricate problems.

Adaptive Behavior and Viability: Contemporary Theories and
Modeling Frameworks The contemporary conception of the organizational approach contends that norms are to be found as conditions for the viability of a system: conditions of self-maintenance or closure conditions. Examples and theoretical tools are often drawn from the adaptive behavior literature, where such norms are pictured as "viability constraints" or "viable region(s)" within a "viability space" defined by different physiological variables (see Figure 1). Ashbyʼs influential work [3] contributed to the formalization of this framework. First, he translated into the concept of "essential variables" what Claude Bernard called "internal milieu" and Walter Cannon called "homeostatic variables." Within the multidimensional space defined by those variables he defined the "viable or homeostatic region" as the subspace where the system was to remain alive or viable.
This way of conceiving and formalizing adaptivity had considerable influence. In ethology, animal behavior has been modeled as responding to the maintenance of these essential variables (physiologically measured or measurable: e.g., temperature, food or hydration levels) within their viability boundaries [45]. A similar approach is taken by Beerʼs research in adaptive behavior, where he defines a systemʼs behavior as adaptive "only so long as it succeeds in maintaining its trajectory within this viability constraint, that is, only so long as it succeeds in maintaining the conditions necessary for its continued existence" [12, p. 266]. Abstractions of these conditions (or functionally specified boundaries of viability) have been used to design robot controllers [14]. And a detailed formal development of some of these issues is due to Jean-Pierre Aubinʼs viability theory-see [4] for the latest version.

Current Problems with Modeling Viability Boundaries and the Nature of Norms
In terms of modeling, the organizational or organismic account of natural norms has translated into the notion of boundaries of viability (following the work of Ashby). The fundamental norm is "do not cross the boundary"; the viability constraints have to be respected. The origin of these boundaries is given by chemical rates (energetic, kinetic, etc.) and physical constraints and their interdependences Figure 1. Standard representation of viability space: A viable region (gray area) and adaptive behavior (trajectories) are illustrated for two essential variables (food and water ). Outside the viable region the system will die. Viable trajectories are those that remain within the viability boundaries.
in metabolism and, for a more complex multicellular organism, by the physiological conditions of recursive self-maintenance of the organism, such as blood circulation and respiration. 4 There is, however, a problem with how this notion of boundaries of viability has been modeled and conceptualized so far: 1. The problem of boundaries. If the notion of viability constraint or boundary is normatively rigid (meaning the system is dead once crossed), then the norm is all-or-nothing and there is no room for reversible failure, no gradation. If the boundary is not rigid, then how can we quantify its "flexibility"? Are there alternative conceptions of the viability space that can accommodate gradation-intermediate spaces or wide "boundaries" before an irreversible state is reached? How can we quantify and qualify a richer topology of the viability space?
Ezequiel Di Paolo has identified a similar problem in autopoietic theory, and he attempted to solve it by providing what remains perhaps the most complete and explicit of current definitions of adaptivity [26]: a systemʼs capacity, in some circumstances, to regulate its states and its relation to the environment with the result that, if the states are sufficiently close to the boundary of viability, 1. tendencies are distinguished and acted upon depending on whether the states will approach or recede from the boundary and, as a consequence, 2. tendencies of the first kind are moved closer to or transformed into tendencies of the second and so future states are prevented from reaching the boundary with an outward velocity. [26, p. 436] The issue of normativity can be explicitly re-framed under this definition as the requirement to "prevent" those tendencies of essential variables that might approximate the boundaries "with an outward velocity." There remain, however, some ambiguities in Di Paoloʼs definition that prevent straightforward quantification and measurement of normative action: How much is "sufficiently close" to the boundaries of viability so that a regulation becomes adaptive? How are "velocities" and "tendencies" defined and modeled? Are there different types of boundaries within the viability space? Are cases like the trajectory that repeatedly crosses the boundary of viability in Figure 1 possible?
The crucial contribution of Di Paoloʼs definition is that it introduces a dimension of temporality and velocity into the notion of adaptivity and norms that deserves more attention than what previous conceptions of viability have paid. Essential variables have tendencies or velocities in relation to which normative action has to be defined. Normativity is not only a question of boundaries or constraints that need just to be respected, but, so to speak, a question of forces and tendencies that have to be compensated for.
In addition, most of the models that have been previously developed suffer a different type of problem (yet somewhat related to the previous one): 2. The problem of dissociation between norm-establishing and norm-following processes. Existing models of viability appear dissociated into two categories: (i) those where viability boundaries appear as given or defined from without and where the models focus on how to shape adaptive dynamics to maintain the trajectories of essential variables within those boundaries; and (ii) those where a set of viability conditions emerge out of metabolic or physiological process but do not address behavior in relation to them. As a consequence, the organismic dynamics that define viability and the dynamics that control adaptive behavior remain dissociated in existing models.
This problem is rarely acknowledged in the literature and even when recognized it is assumed as a necessary pragmatic compromise and a valid assumption. So, for instance, Beer states that: [T]his explicit separation between an animalʼs behavioural dynamics and its viability constraints is fundamentally somewhat artificial.
[…] However, if we are willing to take the existence of an animal for granted, at least provisionally, then we can assume that its viability constraint is given a priori, and focus instead on the behavioural dynamics necessary to maintain that existence. [12, p. 265] To be fair, this is a reasonable simplification when the main focus of interest is the exploration of behavioral dynamics. However, making progress in the conceptual development of the notion of normative action and adaptive agency might require one to integrate these two aspects.
In previous work [29,30] we have explored the relationship between the viability boundary determining metabolic dynamics and the dynamics that drive organismic behavior. But a further problem remained in these and related models: Although the boundaries of viability were directly linked to the modeled system, they were only defined by the system in a relatively trivial way. The boundaries of our models and similar ones by others (see, e.g., [53]) were the result of rough physical magnitudes: disappearance of the protocell due to complete lack of catalysts or bursting disintegration of the protocell marked by the upper limit of the tension of the membrane. 5 The boundaries were not intrinsically emergent from interactions between system processes in the holistic system-interdependent manner that characterizes integrity and systemic identity in real organisms.
In natural systems, the limits of viability do not map with the physical disintegration of a system, but rather with the loss of the capacity of the system to sustain itself. To lose viability is not to disappear altogether, but to cross a much more subtle boundary where self-maintenance is overwhelmed by processes of degradation-where the system is on a trajectory toward death that can only be averted by changing the environmental conditions. Often, once viability has been lost, it becomes increasingly difficult to regain-the loss sometimes reaching a point of no return and becoming irreversible. This problem connects back to the first one, the nature and rigidity of the boundaries as standardly conceived by the adaptive behavior literature needs to be revisited. Additional regions within the space defined by essential variables might be required, and the notion of boundary complemented with that of a field.
In what follows we shall introduce a minimal model of agency capable of addressing the problems just presented in this section. We first introduce a model of a minimal protocell-like metabolism whose dynamics define a precise topology of its viability space. This allows us to identify and illustrate norm-establishing as emerging from metabolism. For fixed concentrations of available resources, we can plot a bifurcation diagram of the chemodynamics that indicates the intrinsic boundaries of viability of the system. Different regions within the viability space will be identified and the adaptive norms of the system clearly defined and quantified. Next we add chemotactic capacities to the protocell, we put it in an environment with a chemical metabolic resource gradient, and we study how behavior maps into the viability space and consider how it can be interpreted as an instance of norm-following behavior. We then abstract away from our model to more general cases, and we finally come back to the problems and theoretical discussion opened above.
3 Norm-Establishing: The Emergence of Norms from Metabolic Dynamics

Model Design
The metabolic organization is one of the most fundamental properties of living systems and has been studied as such by many under the form of autocatalytic networks [41], autopoietic or autonomous systems [16,44,55,60], fluid machineries like the chemoton model [34], or systems closed to efficient causation [43,48,52]-to mention but some of the most relevant contributions to a systemic conception that takes metabolic self-organization as the core principle of living organization.
Real metabolisms are complex networks involving hundreds of metabolites and even more reactions. But we are interested in general abstract properties of metabolic networks, so we do not need to model all of the intricacies. Minimal models with equivalent or similar dynamical properties to the one we are just about to introduce have been studied before, with theoretical goals that are continuous with ours, but without explicitly addressing the issue of normativity [16,48,63].
We start with the simplest conceivable model, involving the simulation of the concentrations of two categories of chemical components: autocatalysts (A) and resources or food chemicals (F ). The abstraction here is high-level; [A] is intended to represent the concentration of all of the different chemicals involved in an autocatalytic metabolic network, conceptually equivalent to the biomass of metabolites or an order parameter globally representing metabolic dynamics. Similarly, [F] represents the concentration of all material and energetic resources necessary for the ongoing functioning of metabolism.
We consider two processes: (i) the autocatalysis of A, whereby it transforms F into more A, and (ii) the spontaneous degradation of A into a quickly dissipating, non-reactive waste that has no subsequent effect upon the system. These two processes are described by the following reaction equations: The following differential equation describes how the concentration of A changes over time: The first term on the right represents the backward reaction 3A → 2A + F, the second term represents the forward reaction 2A + F → 3A, and the third term represents the degradation of A into a nonreactive waste (A → Ø). The constants (k b = 0.45, k f = 1.0, k d = 1.0) were assigned so that, for low [A], the forward autocatalytic reaction occurs more rapidly than the backward reaction. This stoichiometry of the autocatalysis was selected because it results in a bistable system; when the concentration of A is high, the backward reaction runs more quickly than the forward reaction. Figure 2A shows

It is clear in this equation that one equilibrium occurs when [A]
= 0, and that any other equilibria occur when what is inside the parentheses equals 0. When we apply the quadratic formula, to identify the roots of the quadratic inside the parentheses of Equation 3, it is apparent that there is only a single additional root when ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi b 2 − 4ac p ¼ 0. Plugging the coefficients into this equation and then solving for [F], we find the equation that specifies the value of [F] at the bifurcation point: When we substitute the constants specified above, we find that at the bifurcation point, [F] ≈ 1.09.

Interpretation
3.2.1 The Topology of the Viability Space: Living, Dead, Precarious, and Viable Regions Such a simple model of metabolism suffices to generate a viability space where living, viable, precarious, and dead regions can be clearly identified. These are highlighted in Figure 4. The dead region corresponds to zero concentration of the required metabolites (complete disintegration of the system). The arrows in Figure 4 indicate the general tendency of metabolic dynamics for different regions of the viability space, superimposed upon the bifurcation diagram shown in Figure 3. A viable region can be precisely

defined for a range of the parameter [F] and a range of initial conditions [A]
as the subregion of the living region where, for each point, the evolution of the system will tend toward the stable living equilibrium. The unstable equilibrium at the bottom of the viable region defines a boundary of viability below which the system tends to the dead state.
For small values of [A] and [F] we can distinguish what we have termed a precarious region (medium gray area in Figure 4), where the system is still alive but will tend to die if the parameter [F] is kept constant, but can still recover if [F] is appropriately modulated. Underneath the precarious region a terminal region can also be distinguished (dark gray area in Figure 4). If [A] falls in this region, the system will be alive for some time, but will irreversibly die. The reasons for this impossibility of recovering viability are intrinsic limiting factors on the dynamics of change of [F]. So, for instance, in a protocell, the maximum increase in [F] could be limited by the maximum permeability of the membrane. Note that at this point we are still not considering any limitations coming from the structure of the environment itself or the capacities of the system to change those conditions (e.g., by moving up a gradient of [F]).
There is a significant difference between the standard representation of a viability space in Figure 1 and our representation in Figure 4. The first thing to note is that the viability space in our approach is not only defined by a set of essential variables or just a constraint, but by a relational property between internal variables ([A]) and environmental parameters or boundary conditions ([F]). In addition, a viable region is not specified by a set of ad hoc or experimentally determined boundaries: It is the result of the internal dynamics of the system in relation to the environment. The bifurcation point and the boundary between two basins of attraction (the viable and the dead one) are the result of the dynamics specified by metabolism. If we further explore the dynamics of the system within this region we can move beyond a description of normativity in terms of boundaries and regions and enrich it with the concept of field.

The Normative Field
We can now introduce the notion of a normative field, defined by the minimal constant increase of a behaviorally influenced variable, in this case [F], that is required at each point of the precarious region in order to move the state of the system into a viable region before the system reaches the   figure), the required increase of [F] is very big, since the tendency of [A] will soon push the system into the terminal region. If the concentration of F is low but there is a lot of A, the required constant increase in [F] is low, because the system has sufficient time to reach the viability boundary before the tendency to die becomes irreversible. There are still rigid boundaries of a binary nature, like the dead-living boundary at [A] = 0, but the notion of the normative field along the precarious region permits us to define and quantify norms in a graded manner.
We have provided a minimal model of norm-establishing processes and how they define different normative regions and dynamic normative fields. The goal of the next section is to see how normfollowing behavior can occur and how it maps into the normative field.

Norm-Following: Chemotaxis and Normative Behavior
We now consider the possibility of influencing or modulating [F] through behavior. There are many ways in which this could be accomplished. We consider two cases, both of which involve a motile protocell (or bacterium) performing chemotaxis on a gradient of [F]. The first mechanism is a stochastic mechanism inspired by the run-and-tumble behavior observed in bacteria such as Escherichia coli. The second is a non-stochastic, perfect gradient-climbing mechanism in which the simulated organism always moves directly up the [F] gradient. 6

Stochastic Gradient-Climbing
In this section, we extend the model described above, simulating the metabolism as being located within a bacterium-like body that can move around a 2D environment. In the environment, F is distributed according to a Gaussian gradient specified by the following formula (also, see Figure 6A): The stochastic gradient-climbing behavior is inspired by the chemotaxis mechanism of E. coli [31], which is based on the on the modulation of two types of motility: running, where the organism moves in a straight line, and tumbling, a random reorientation of the organism. Running is a linear, constantvelocity motion defined by the following differential equations: dx dt ¼ cosðaÞ; dy dt ¼ sinðaÞ; d a dt ¼ 0, where a represents the orientation of the organism. A tumble is simulated by selecting a new value for a from a flat distribution between 0 and 2k. Each tumble is always followed by a short cooldown period (0.1t ) of running.
The run-tumble behavior of the organism is modulated by change in the concentration of F. When [F] is increasing, the organism tends to run, and when it is decreasing, the organism tends to tumble. This is simulated by calculating, at every iteration, an approximation U of the recent change in the concentration of F that includes an error term: When U ≥ 0, the simulated organism runs; otherwise (as long as it is not cooling down from a recent tumble), it tumbles. The calculation of U includes the error term q, which is a random number selected from a normal distribution with a mean of 0 and a standard deviation of 0.01. This error term means that the organism does not always make a correct evaluation of the change in the concentration [F]. In most cases, if the organism starts reasonably close to the gradient, it exhibits statistical chemotactic behavior, moving up the resource gradient. Figure 6A shows typical paths of three agents, each started at a different distance from the peak resource. Figure 6B shows the evolution of essential variables as a result of the agentʼs behavior, through the viability space, the same space as in Figures 3−5, and 7. We see in Figure 6B that trajectory 2 starts in the precarious region, moves for some time in correlation with the normative field, then moves away from the peak (marked with *), and then turns into the viable region. Once there, it comes out of it (marked **) to return to the viable region where it remains. It has survived a state that, in the absence of behavioral influence on [F], would have led to its death, but it can be said to have made mistakes (marked * and **) on its way toward regaining viability.
It is also the case that sometimes, despite performing chemotaxis, the simulated agents fail to reach sufficiently high concentrations of F before they fall into the terminal irreversible region. Such is the case of trajectory 1 in Figure 6, at the bottom line, where, after having moved slightly against the gradient, it reaches a point around [A] = 2; there, despite being positively correlated with the normative field, the overall behavior has been insufficient to reach the viable region, the agent cannot behaviorally compensate for its precariousness, and it enters the terminal region.

Direct Gradient-Climbing
The mechanism just described has the overall statistical effect of moving the protocell up the resource gradient. To simplify the analysis, we abstract the stochasticity of the mechanism, replacing it with a continuous behavior that performs direct gradient-climbing. During this behavior, the organism always moves directly up the resource gradient at a rate proportional to its distance from the peak-an approximation of the overall effect of the stochastic gradient-climbing behavior. The removal of stochasticity simplifies the dynamical analysis and allows us to more easily visualize the effect of behavior upon the viability of the simulated agent.
The radial symmetry of the [F] gradient and the fact that the direct gradient-climbing only involves axial motion allow us to reduce the number of spatial dimensions in our model. Now, a single variable x, representing the distance of the organism from the peak of the resource gradient, suffices. The following equation describes the gradient, relating [F] to x: We define dx dt ¼ −k m x, formalizing our definition of direct gradient-climbing mechanism. We then derive the following differential equation in terms of [F], eliminating x from the model: In our analysis, we take k m , a constant indicating the speed of motion of the simulated organism, to be 0.1.
To explore the influence of the direct gradient-climbing behavior on the viability space of the simulated organism, we have superimposed streamlines indicating the trajectories with perfect gradientclimbing behavior over the bifurcation diagram for fixed [F] (see Figure 7). With perfect gradient climbing, the entire system has two stable equilibria (dead and living ) that are indicated with solid circles, and a saddle node that is indicated with an open circle. All of these points are on the line [F] = 2.0. Also indicated on this diagram by a bold curve is the streamline that ends at the saddle node. This path lies along the basin boundary between initial conditions that lead to the living equilibrium and those that lead to the dead equilibrium. The bold vertical arrows indicate the effect of a small perturbation from the saddle node.
The effect of the direct gradient-climbing behavior upon the viability space of the simulated organism can be seen in Figure 7. With [F] no longer fixed, but increasing as a result of behavior, the space of initial values of [F] and [A] that fall into a stable living equilibrium point is greater than that of the non-behaving model.

Normative Behavior
Since [F] can be modulated by behavior (provided that the environment displays a gradient of [F]), a sense of normative behavior can be precisely defined for every state of the system in the precarious region: the amount of increase of [F] that behavior should achieve to compensate for its precariousness, that is, the required movement in space that increases available [F] in accordance with the normative field. To remove the ambiguity of the expression "in accordance with," we can more precisely define norm-following behavior as the course of behavior whose effect on the viability space positively correlates with the normative field. Quantitative and qualitative aspects of this correlation can be distinguished. At the qualitative level, the projection of behavioral effects on the normative field can take three different forms of correlation: positive, negative, or null. Whereas norm-following behavior is that which is positively correlated with the normative field, negatively correlated behavior can be said to be against the norm, and behavior that has no correlation can just be said to be neutral (only in a sense relative to the correlation with the norm, since this "neutrality" is deadly for the system). 7 At the quantitative level one can measure how much a behavior matches the normative field. Note that the system can fail to meet the norm, (i.e., to adapt) for a variety of causes (e.g., because there is not 7 A proper sense of "neutrality" is found within the viable region where the normative field is null. enough F in the environment, or because it cannot move sufficiently fast or does not manage to move up the gradient-as in the case of the experiments illustrated in Figure 6). And yet, for some of these cases of failure, behavior can be said to be in accordance with the norm, (i.e., a norm-following behavior) if it positively correlates with the normative field. Failure might be quantitative in these cases: Despite the correlation with the normative field being positive, the agent does not move sufficiently fast, and the behavior matches the direction of the normative vectors but not their length.

The Adapted, Adaptive, and Futile Regions
We have projected the relational agent-environment coupling effect on the viability space (as shown in Figure 7) for a specific and stable environment and a deterministic and perfect behavioral capacity of the agent. The resulting extension of the viable region we have termed the adaptive region, because this is the area within the precarious region where behavior is capable of compensating for decay (see Figure 8). Whereas the viable region can be said to be the space where the system is adapted (there is nothing that the system has to do), this is the region where the system is adaptive (there is something that the system has to do and it can do). Unlike the viable region, which is defined for all possible environmental conditions, the adaptive region has to be defined in relation to both the structure of the environment (the gradient defined by Equation 8) and the behavioral capacities of the system (defined by Equation 9).
An additional region can be added if we consider the limits of the agentʼs behavioral capacities and the structure and limits of the environment (the space to the left of the gray area in Figure 7, where all trajectories lead to death). We call this the futile region: no matter what it does, the agent will die, because it lacks the appropriate behavioral capacity to modulate its environmental conditions-or simply because the environment does not allow for it. So, for instance, if there is insufficient F in the environment or if the chemotactic mechanism does not allow the protocell to move up the gradient of [F] fast enough, the agent will die. Note that we have defined normative behavior as behavior whose effects positively correlate with the normative field. As a result, within the futile region the agent will be behaving normatively but without a real chance to survive (i.e., it is out of the adaptive region). The futile region is specified by the organization of the metabolic system as well as Figure 8. Adapted, adaptive, and futile regions of the viability space. The projection of the agent-environment coupling upon the viability space defines an adaptive region where the system is capable of compensating for its precariousness for that particular coupling, and a futile region where the agent will inevitably die given its behavioral capacities and the structure of the environment. behavioral or environmental dynamics. This is unlike the terminal region, which is specified by factors that are intrinsic to the organization of the system in a way that is independent of behavioral capacities and environmental structure.

Recapitulation: The Topology of the Viability Space; Some Generic Definitions
Abstracting away from our model and bringing together the set of distinctions we have made throughout this article, we can now provide a set of generic definitions that might be applied to other models of adaptive or normative agency and behavior.
• Viability space: The space defined by the relationship between (a) the set of essential variables representing the components, processes, or relationships that determine the systemʼs organization and (b) the set of external parameters representing the environmental conditions that are necessary for the systemʼs self-maintenance.
• Living region: The region of the viability space where a system maintains its ongoing organization as a unity.
• Viable region: A region of the viability space where the evolution of essential variables, ceteris paribus (i.e., given fixed environmental conditions), will remain within the living region. That is, ∀xðtÞ ∈ V ⊂ L; xðt þ 1Þ ∈ V (where L and V stand for the living and viable regions, respectively).
• Precarious region: A subregion of the living region where the system will reach death if the environmental conditions do not change.
• Terminal region: A subregion of the living region where the system will reach death no matter how the environmental conditions change.
• Normative field: For each point of the precarious region, the required minimal change of environmental conditions necessary to bring the system back to a viable region or to a precarious region of an opposing tendency.
• Normative action or norm-following behavior: System-driven modulation of the systemʼs coupling with the environment whose effect on the viability space positively correlates with the normative field.
• Adaptive region: A projection of the agent-environment coupling dynamics into the viability space, specifying a subregion of the precarious region where from each point at least one possible trajectory leads to viability.
• Futile region: A projection of the agent-environment coupling dynamics into the viability space, specifying a subregion of the precarious region where all trajectories lead to death.
Some aspects of the above definitions remain in need of further development; we shall briefly mention some of them. The definitions of viability space and living region were provided for completeness, but a more precise and justified definition of these terms is out of the scope of this article. Similarly the term "essential variable" demands further explanation. For a protocellular system where a membrane or some enclosing property (e.g., phase separation between oil droplet and water ) distinguishes a self-sustaining network of reactions from the environment, essential variables represent those observables of a system that correspond to the capacity of the system to keep producing or maintaining itself as a system distinguishable from its environment. The expression "ongoing organization as a unity" also remains open to discussion and further clarification. For the purpose of this article it suffices to indicate that at some point the essential variables of the system cease to represent an observable of the system. This could be just because the system disappears as a result of the observables disappearing altogether ([A] = 0 in our model) or for more complex reasons. So, for instance, if beyond a certain level of [A] the protocell bursts, then [A] is no longer the amount of metabolites inside the membrane of an individual cell, but a measure of the concentration of A in the environment (e.g., the Petri dish or pond where the bacterium was found before the bursting). We have committed to a specific conception of life-inspired by [34,38,44,57]-but the definitions just provided do not strictly depend on it; other attempts to define life might be equally valid as long as different processes could define and be mapped into the viability space. One could even proceed, to some extent, without a definition of life and establish essential variables, environmental conditions, and their boundaries experimentally (e.g., heart rate = 0).
We are aware that the conceptual framework of these definitions could be applied to biological function more generally and not only to the issue of agency and normative behavior. Because our focus of interest is agency, we have decided to narrow down the framework, making some simplifications that could lead to confusion. So, for instance, we defined the viability space and its topology in relation to "environmental conditions." But what we consider to be environmental conditions could perfectly well be internal conditions, and the relationship between agent and environment might be translated into a part-whole relationship (e.g., the heart versus the conditions that the rest of the body creates for it: oxygen supply, hormonal and neuronal activity, etc.).
We have modeled normativity at a single level of description and organization, one involving a macroscopic description of metabolic dynamics and its coupling with the environment. What was defined as a viable region at this level could perfectly well correspond to a precarious region at a lower level of description and organization if observed in more detail. Most cases of natural normativity are cases where various nested hierarchies of precarious processes constitute an organism, and so it is only by fixing or assuming the normative functioning of lower levels that non-precarious viable regions might be defined.
In our case the normative field was represented as a vector field, but could equally well have been a scalar field, since there was a single viability parameter. More complex cases might require a tensor field to express different routes to viability that are equally valid when different viable regions coexist within the same viability space. It could also be the case that the viable region is very small or nonexistent, depending on the particular topology of the viability space. For such cases, and others where viable regions might still be in place, we have adapted the definition of normative field to include the condition "to bring the system to a precarious region of an opposing tendency." Although in our model it turns out to be impossible to reach an opposing tendency without directly falling into the viable region, more complex cases might permit the organism to remain indefinitely alive within the precarious region by navigating opposing tendencies of essential variables. For more complicated models where there are multiple environmental variables that influence the viability of the organism, identifying the normative field might not be trivial. We do not address the problem of finding a universal way to determine a normative field in this article. More complex models will be required to make progress in this direction.

Discussion
The model we have presented here is still limited in a number of ways that we consider important. However, we are now in a position to address them within the framework we have sketched. We have no space to deal with these topics in detail, but we shall point them out briefly.
6.1 Virtuality, Multidimensionality, and Plasticity of Norms 6.1.1 The Virtual Nature of Norms It follows from our analysis that norms have a virtual character. Norms are not mechanisms operating on the system. There is nothing in the current state of the system that specifies the norm. Rather, it is in the space of the possible dynamic evolutions of the system for different environmental conditions that the normative vector field is found, in a conditional manner: what would happen if the system were to evolve under such and such conditions, and how it would have to compensate such an evolution if it were to avoid death in the future. The organization of the system is the source of norms, but it makes no sense to claim that norms themselves have any causal or regulatory effect.

Multidimensionality of the Viability Space
The present model benefits from its low dimensionality in that it is easier to understand, but it also suffers, perhaps, from being oversimplified in that there are really only two ways that the system can vary. Real organisms are of course much more complex and would display a multidimensional normative field and viability boundaries or surfaces. Richer topologies of the viability space might be able to give rise to different phenomena. One of them is the existence of different viable regions (in the form of islands in a viability space, separated by precarious regions) for the same organism, that could define forms or regimes of viability and their associated forms of adaptation (that is, modes in which behavior can compensate for precariousness around those viable regions).

Plasticity of Norms
We have considered only fixed organizations that have no potential for change. Living systems are plastic, meaning that certain organic functions can be expanded or reduced in their speed, tolerance, rate, resistance, and so on. It is well known that animals, if progressively exposed to a poisonous environment, can significantly increase their tolerance and survive levels of exposure that would kill them quickly were those to arise suddenly. We have not considered a protocell that can change its metabolism or its relationship with the environment under different conditions. But our model could be expanded to address such cases. Examples of metabolic plasticity could include the activation of different metabolic routes under different conditions so that the viable region might change (e.g., F is metabolized faster). An example of plasticity at the system-environment relationship level could be the alteration of membrane properties: for example, if the protocell is exposed repeatedly to some chemical in the environment that could, in turn, increase (or decrease) the inflow of F, the terminal region could be expanded or reduced. Ruiz-Mirazo and Mavelli [53] provide a perfect illustration of this notion of plasticity of boundaries; their model of a protocell includes the production of peptides that change the elastic properties of the membrane, modifying the upper limit of the internal concentration of metabolites that the protocell can reach before it bursts.

Moving beyond Normative Behavior: Agency and Teleology
Up to this point we have studied norm-establishing dynamics on the one hand and behavior on the other, and we were able to define behavior as normative, or norm-following, by mapping the effects of behavior on the viability space and its normative field. Behavioral mechanisms in our model were directly sensitive to [F], meaning they were assumed to be causally correlated with some environmental quantity or value (as transmembrane receptors do in bacteria). In a sense, the mechanisms and dynamics that produced behavior in our model were still dissociated from norm-establishing processes. In our minimal system it happens to be the case that an increase in [F] is always good. As a result, mechanisms as simple as those that can transduce variation in [F] into the appropriate motor output were capable of following the norms (i.e., generating behavior that positively correlated with the normative field), independently of the dynamics of [A]. A crucial point that we had to leave out of this article is the requirement that some kind of causal-dynamical integration between norm-establishing and behaviorgenerating mechanisms might be necessary to achieve a full sense of agency and teleology. A system can follow a norm, yet do so blindly, without any causally relevant effect of norm-establishing processes upon norm-following processes. Can we really say that such systems are agents? In cases like bacterial chemotaxis it is assumed that natural selection has tuned sensor transduction mechanisms, sensorimotor chemical pathways, and flagellar rotation speed and probabilities so that behavior turns out to be adaptive. But is there any possibility to address intrinsic teleology other than the blind watchmaker creating organisms that behave normatively but are blind to their own norms?
A deeper dynamic integration between norm-establishing and norm-following processes is indeed possible. We can use the present model to explain this possibility. So, for instance, instead of just tracking the increase of [F] in the environment, it is possible to make behavior dynamically sensitive to [A] (or even some combination of [A] and [F]). In this way the system would be capable of modulating its behavior in direct causal correlation with its viability dynamics and not just by responding to external conditions. Note that this cannot happen if viability boundaries are externally defined or contingently imposed on the viability space. This is why modeling the emergence of intrinsic norms is crucial. In our model, the unstable equilibrium boundary that separates the viable from the precarious region implies radically different dynamics for metabolism. Only when norms emerge in this manner can norm-establishing and norm-following processes be connected in a dynamically integrated manner.
The goal of this article has been to clarify the notion of "a system following norms that are generated by itself." This is part of a larger body of research, where a great deal of interesting and exciting work lies ahead. We have published on and continue to study the relationship between metabolic dynamics and behavioral control in what we refer to as self-sensitivity-see [27] for a comprehensive view. This research explores some of the consequences of what we have called metabolism-based chemotaxis in bacteria [29] as opposed to the inherited concept of metabolism-independent forms of chemotaxis-a view that has recently been challenged by experimental confirmation of many species of bacteria performing various types of metabolism-dependent chemotaxis; see [1,62]. The evolutionary and adaptive potential of integrating metabolism and behavior was also explored in [28].
Also, previous work [5,6] has focused on the possible limitations of biological agency in terms of integrating norm-establishing processes with behavior-generating mechanisms, exploring instead the concepts of autonomy and viability at the level of neurodynamics and behavior as a new domain where properly behavioral or cognitive norms are to be found. We believe that exploring a deeper connection between norm-establishing and norm-following processes will provide an insightful research avenue to clarify the nature of teleology and agency.
Finally, we want to clarify that issues of normativity only relate to one of the three conditions that, according to [7], define a system as an agent: (i) individuality, (ii) interactional asymmetry, and (iii) normativity. The model presented in this article is therefore partial and is not intended to capture agency as such, but just to address some of its constitutive properties (those related to normativity). 8

Conclusions
One of the central challenges to naturalizing and synthesizing natural agents is the issue of normativity. This article started with a short summary of some of the most relevant contributions to the characterization of natural norms, focusing on the organizational, or organismic, approach. This school of thought contends that norms emerge from the precarious and self-sustaining nature of living individuals, whose existence depends on the satisfaction of some environmental or boundary conditions. But further progress is needed to articulate a scientifically fruitful theory that can be formalized and modeled.
We identified two problems of previous modeling approaches to norms via the notion of viability boundaries. The first problem has to do with the ad hoc imposition of norms on modeled systems and the resultant rigidity of viability boundaries and absence of a normative gradation. In a circumvention of this problem, we modeled the norm-establishing processes explicitly, allowing us to generate a viability space with intrinsic boundaries and dynamic regions. This made it possible for us to derive the notion of a normative field, enabling the quantification of normative behavior in a dynamical way that is richer than the crossing of a binary "life or death" boundary-as suggested by [26]. In this way the precarious region avoids the standard rigidity of boundaries, generating a middle ground between the boundary that separates viable and precarious regions and the boundary that separates precarious and terminal regions. It is in this precarious middle ground where tendencies of essential variables are explicit and can be compensated. Although our model did not address the possible plasticity or elasticity of such boundaries, we considered possible expansions of the model to accommodate these phenomena.
We then introduced behavior-generating mechanisms and addressed the second identified problem: the problem of dissociation between norm-establishing and norm-following dynamics. By mapping the effect of behavior upon metabolism, we were able to assess the normative nature of particular actions. We finally acknowledged the limitation of our current model, identifying the need to make further progress by integrating the dynamics of norm-establishing processes directly with those generating behavior, so as to avoid blindness to normativity and make progress toward a better understanding of natural agency. Many open questions lie ahead. But the minimal model presented here (and the conceptual definitions and distinctions it has made possible) allow us, now, to formulate those questions more precisely and move beyond the inspiring, but also limited, theoretical and philosophical approaches that put the autonomy of the life, and its capacity to create norms, at the center of living phenomenology.